Fluid-Structure Interaction of Composite Structures [1st ed.] 9783030576370, 9783030576387

Table of contents :
Front Matter ....Pages i-xiii
Introduction (Young W. Kwon)....Pages 1-9
Experimental Study of FSI with Impact Loading (Young W. Kwon)....Pages 11-31
Numerical Modeling of FSI Under Dynamic Loading (Young W. Kwon)....Pages 33-57
FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures (Young W. Kwon)....Pages 59-84
FSI Study with Cyclic Loading (Young W. Kwon)....Pages 85-103
FSI Study of Structures Containing Fluid (Young W. Kwon)....Pages 105-153
Structural Coupling by FSI (Young W. Kwon)....Pages 155-230
Composite Structures Moving in Fluids (Young W. Kwon)....Pages 231-280

Citation preview

Springer Tracts in Mechanical Engineering

Young W. Kwon

Fluid-Structure Interaction of Composite Structures

Springer Tracts in Mechanical Engineering Series Editors Seung-Bok Choi, College of Engineering, Inha University, Incheon, Korea (Republic of) Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Yili Fu, Harbin Institute of Technology, Harbin, China Carlos Guardiola, CMT-Motores Termicos, Polytechnic University of Valencia, Valencia, Spain Jian-Qiao Sun, University of California, Merced, CA, USA Young W. Kwon, Naval Postgraduate School, Monterey, CA, USA

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Young W. Kwon

Fluid-Structure Interaction of Composite Structures

123

Young W. Kwon Department of Mechanical and Aeronautical Engineering Naval Postgraduate School California, CA, USA

ISSN 2195-9862 ISSN 2195-9870 (electronic) Springer Tracts in Mechanical Engineering ISBN 978-3-030-57637-0 ISBN 978-3-030-57638-7 (eBook) https://doi.org/10.1007/978-3-030-57638-7 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to my family who has always stood by me with love and support

Preface

Composite materials and structures have been developed initially for the aerospace applications because of their high specific strength and stiffness, i.e., high strength and stiffness per mass density. However, those and the effective anti-corrosive properties have also pushed composite materials and structures for marine applications. One distinctive difference between air and water is their mass densities. Because of such a high density, water plays an important role on dynamic responses and failure of polymer composite structures. This book addressed many different kinds of problems associated with fluid– structure interaction (FSI) of composite structures. Each chapter started with a brief introduction about the topic to be presented in that chapter and ended with a summary of findings. Every topic was studied both experimentally and numerically to complement each other because each one of them has its own merits and limitations. Chapter 1 provided a brief introduction of composite materials and the basic concept of fluid–structure interaction, followed by the description of the scope of the book. From the second chapter, different topics of FSI of polymer composite structures were presented. Chapters 2 and 3 presented experimental and numerical impact study on composite structures including FSI. Those chapters demonstrated how critical was the effect of FSI on the dynamic response and failure of composite structures. Changes in the vibrational frequency, mode shape, and modal curvature were examined in Chap. 4. The FSI resulted in much significant changes in those characteristics of composite structures as compared to metallic structures. Cyclic fatigue failure was studied in Chap. 5 on composite structures with FSI, which reduced the number of cycles for failure significantly. Chapter 6 presented the results of composite structures containing fluid inside. The structures were impacted by a low-velocity object or penetrated by a high-velocity object. Dynamic coupling of separate composite structures through a fluid medium was investigated in Chap. 7. Without the fluid medium, the structures were independent, but the fluid medium coupled the independent structures. Finally, Chap. 8 discussed moving composite structures in water either at constant vii

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velocities or with constant accelerations. The water might include iceberg-like solid objects such that the composite structure could interact with the solid objects in water. The findings in the book are hoped to lead to a safer design and proper analysis of composite structures for marine applications because those structures could have premature failures if constructed without considering FSI. Monterey, CA, USA

Young W. Kwon

Acknowledgements

The content presented in this book is the collection of research conducted since more than a decade ago with the support by the Solid Mechanics Program of the Office of Naval Research (ONR). The Program Manager is Dr. Yapa Rajapakse. The author greatly appreciates the continued support to undertake the research leading to this publication. Many people contributed to the research. Without their contribution, the research would not be successful. The contributors are listed below in a random order: Angela Owen, Michael Violette, Ryan McCrillis, Ryan Conner, Eric Priest, Scott Knutton, Spyros Plessas, Stuart Blair, Scott Millhouse, Steven Arceneaus, Linda Craugh, Teo Hui Fen, Scott Bolstad, Jessica Rodriguez, Kangjie Roy Yang, Kyung-Jae Yoon, Joshua Bowling, Dariush Alaei, as well as the staff members at Naval Postgraduate School, Jarema (Jake) Didoszak, Chanman Park, and John Mobley. If anyone who contributed to the project is omitted, the author apologizes them sincerely. Finally, the author appreciates the support by the Springer Nature, especially Dr. Leontina Di Cecco for her help with preparing the book project.

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3 Numerical Modeling of FSI Under Dynamic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Finite Element Method . . . . . . . . . . . . . 3.1.1 Beam Bending . . . . . . . . . . . . . . 3.1.2 Plate Bending . . . . . . . . . . . . . . 3.1.3 Acoustic Wave Equation . . . . . . 3.2 Cellular Automata . . . . . . . . . . . . . . . . . 3.2.1 Acoustic Wave Equation . . . . . . 3.2.2 Beam Bending . . . . . . . . . . . . . . 3.2.3 Plate Bending . . . . . . . . . . . . . . 3.3 Lattice Boltzmann Method . . . . . . . . . . 3.3.1 Acoustic Wave Equation . . . . . . 3.3.2 Fluid Mechanics . . . . . . . . . . . .

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1 Introduction . . . . . . . . . . . . . . . . . . . . 1.1 Composite Materials . . . . . . . . . . 1.2 Material Properties of Composites 1.3 Micromechanics Models . . . . . . . 1.4 Fluid-Structure Interaction . . . . . . 1.5 Scope of the Book . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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2 Experimental Study of FSI Loading . . . . . . . . . . . . . . . 2.1 Experimental Set-Up . 2.2 Experimental Results 2.3 Summary . . . . . . . . . References . . . . . . . . . . . . .

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Examples of Coupled Fluid and Structure Domains . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Beam Supported by Fluid . . . . . . . . . 3.4.2 Numerical Study of Impact with FSI . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 FSI Effect on Frequencies, Mode Shapes, Curvatures . . . . . . . . . . . . . . . . . . . . . . . 4.1 Description of Experiments . . . . . . . 4.2 Experimental Results . . . . . . . . . . . 4.3 Numerical Study . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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5 FSI Study with Cyclic Loading . 5.1 Experimental Study . . . . . . 5.2 Numerical Study . . . . . . . . 5.3 Summary . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .

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6 FSI Study of Structures Containing Fluid . 6.1 Experimental Study . . . . . . . . . . . . . . 6.2 Numerical Study . . . . . . . . . . . . . . . . 6.3 Study of Hydrodynamic Ram . . . . . . 6.3.1 Numerical Modeling . . . . . . . 6.3.2 Model Validation . . . . . . . . . . 6.3.3 Water Filling Level . . . . . . . . 6.3.4 Wall Thickness . . . . . . . . . . . 6.3.5 Projectile Impact Velocity . . . 6.3.6 Projectile Mass . . . . . . . . . . . 6.3.7 Impact Angle . . . . . . . . . . . . . 6.3.8 Projectile Shape . . . . . . . . . . . 6.4 Summary . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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and Modal

7 Structural Coupling by FSI . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Description of Experiments for Plate Structures . . . . . . 7.2 Experimental Results of Two Plates of Same Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Experimental Results of Two Plates of Same Thickness at Different Spacings . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Experimental Results of Two Plates of Different Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Experimental Results of Three Plates . . . . . . . . . . . . . .

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7.6 Experimental Setup for Two Cylinders . . . 7.7 Experimental Results of Two Cylinders . . 7.8 Numerical Study of Flat Plates . . . . . . . . 7.9 Numerical Study of Cylindrical Structures 7.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Composite Structures Moving in Fluids . . . . . . . . . . . . . . 8.1 Experimental Study in Fluid Without Floating Solid Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Experimental Set-Up . . . . . . . . . . . . . . . . . . . 8.1.2 Results and Discussion . . . . . . . . . . . . . . . . . . 8.2 Numerical Study in Fluid Without Floating Solid Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Plate Model . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Basic Box Model . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Change in Magnitude of Constant Acceleration 8.2.4 Change in Steady State Velocity . . . . . . . . . . . 8.2.5 Intermittent Zero Acceleration . . . . . . . . . . . . 8.2.6 Monotonically Varying Acceleration . . . . . . . . 8.2.7 Geometric Variation . . . . . . . . . . . . . . . . . . . . 8.2.8 Variation in Material Property . . . . . . . . . . . . 8.2.9 Variation in Water Depth from Free Surface . . 8.3 Experimental Study in Fluid with Floating Solid Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Experimental Set-Up . . . . . . . . . . . . . . . . . . . 8.3.2 Results of Steady State Motion . . . . . . . . . . . . 8.3.3 Results of Accelerating Motion . . . . . . . . . . . . 8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

Composite materials have been used increasingly for engineering applications, especially for load-carrying structural members. Additionally, composite materials have been extended for marine applications where both structures and fluids interact during their dynamic motions. This chapter provides a brief introduction of composite materials as well as fluid-structure interaction. Then, the scope of the book is presented at the end.

1.1 Composite Materials Composite materials consist of multiple materials without their chemical reactions. Most of composite materials are made of two different materials. One of them is strong and stiff, and this material is the major load-carrying component. The other material is used to bind the former material. The former is called the reinforcing material while the latter is called the binding material. Both materials are called constituent materials. Composite materials are classified based on the shape of the reinforcing material. The reinforcing material may be continuous or discontinuous. The typical continuous fiber composite is shown in Fig. 1.1a. This is called the unidirectional composite. Another commonly used composite is the woven fabric composite as sketched in Fig. 1.1b. Depending on the woven architectures, woven fabrics may be called the plain weave, twill, or satin, etc. A composite structure is usually constructed by multiple layers, and each layer may have a different fiber orientation relative to one another. The main reason to make the diameter of the fiber so small is to minimize internal defects inside the fiber in order to enhance its strength. Typical fibers have diameters in the dimension of micrometers.

© Springer Nature Switzerland AG 2020 Y. W. Kwon, Fluid-Structure Interaction of Composite Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-57638-7_1

1

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1 Introduction

Fig. 1.1 Examples of composite materials

(a) Continuous fiber composite

(b) Woven fabric composite

(c) Discontinuous fiber composite

(d) Particulate composite

A discontinuous fiber composite is shown in Fig. 1.1c. Mostly, discontinuous fibers were oriented randomly. Figure 1.1d illustrated a particulate composite, whose reinforcements have almost the same length in all three directions such as spheres, cubes, tec.

1.2 Material Properties of Composites

3

1.2 Material Properties of Composites The anisotropic material has material properties which vary along the orientation of the material. The constitutive equation for the anisotropic material is expressed as below: ⎫ ⎡ ⎫ ⎧ ⎤⎧ ⎪ εx x ⎪ c11 c12 c13 c14 c15 c16 ⎪ σx x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ c c c c c c ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ ε yy 21 22 23 24 25 26 yy ⎪ ⎢ ⎪ ⎪ ⎪ ⎥ ⎪ ⎪ ⎪ ⎪ ⎬ ⎢ ⎬ ⎨ ⎥⎨ σzz ⎢ c31 c32 c33 c34 c35 c36 ⎥ εzz =⎢ (1.1) ⎥ ⎢ c41 c42 c43 c44 c45 c46 ⎥⎪ ⎪ σx y ⎪ εx y ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎥ ⎪ ⎪ ⎪ ⎪ε ⎪ ⎪ ⎣ c c c c c c ⎦⎪ ⎪ ⎪σ ⎪ ⎪ yz ⎪ 51 52 53 54 55 56 ⎪ yz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎭ ⎩ σx z c61 c62 c63 c64 c65 c66 εx z where σi j and εi j are the stress and strain component, ci j is the component of the material stiffness matrix which is symmetric such that ci j = c ji . Therefore, there are 21 material constants. If there is a material symmetry about one plane, for example the xy-plane, the constitutive equation becomes ⎫ ⎡ ⎧ ⎪ c11 σx x ⎪ ⎪ ⎪ ⎪ ⎪ ⎢c ⎪ ⎪ ⎪ ⎪ σ yy 21 ⎢ ⎪ ⎪ ⎪ ⎪ ⎬ ⎢ ⎨ σzz ⎢ c31 =⎢ ⎢ c41 ⎪ σx y ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 0 σ yz ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ σx z 0

c12 c22 c32 c42 0 0

c13 c23 c33 c43 0 0

c14 c24 c34 c44 0 0

0 0 0 0 c55 c65

⎫ ⎤⎧ ⎪ εx x ⎪ ⎪ 0 ⎪ ⎪ ⎪ε ⎪ ⎪ ⎪ 0 ⎥ yy ⎪ ⎥⎪ ⎪ ⎪ ⎨ ⎥⎪ ⎬ 0 ⎥ εzz ⎥ 0 ⎥⎪ εx y ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎦ c56 ⎪ ε yz ⎪ ⎪ ⎪ ⎩ ⎭ c66 εx z

(1.2)

In this case, the number of material constants becomes 13. If there are three material symmetric planes, the material is called orthotropic, and the constitutive equation becomes ⎧ ⎫ ⎡ ⎪ σx x ⎪ c11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎢c ⎪ ⎪ ⎪ σ yy 21 ⎪ ⎢ ⎪ ⎪ ⎪ ⎨ ⎬ ⎢ σzz ⎢ c31 =⎢ ⎢ 0 ⎪ σx y ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 0 σ yz ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ σx z 0

c12 c22 c32 0 0 0

c13 c23 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c55 0

⎫ ⎤⎧ ⎪ εx x ⎪ ⎪ 0 ⎪ ⎪ ⎪ε ⎪ ⎪ ⎪ 0 ⎥ yy ⎪ ⎪ ⎥⎪ ⎪ ⎨ ⎬ ⎥⎪ 0 ⎥ εzz ⎥ 0 ⎥⎪ εx y ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎦⎪ ε yz ⎪ ⎪ ⎪ ⎩ ⎭ εx z c66

(1.3)

The number of material constants becomes nine. Material properties of composite are resulted from both reinforcing and binding materials. Particulate composites are assumed to have isotropic material behaviors at the composite level when particles are distributed randomly. Discontinuous fiber composites are also assumed isotropic on the plane where the discontinuous fibers were laid randomly. However, the transverse normal direction of the plane has a different and much lower modulus because there is no reinforcement of fibers in

4

1 Introduction

that direction. On the other hand, unidirectional fiber composites have transversely isotropic material properties. If the x-axis is along the fiber direction, the material property on the yz-plane is assumed isotropic with a random distribution of fibers on that plane. The transversely isotropic material has the following constitutive equation: ⎫ ⎡ ⎧ ⎪ c11 σx x ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ c21 ⎪ ⎪ ⎢ ⎪ σ yy ⎪ ⎢ ⎬ ⎨ σzz ⎢ c21 =⎢ ⎪ 0 ⎪ ⎢ ⎪ σx y ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ σ 0 ⎪ ⎪ yz ⎪ ⎪ ⎭ ⎩ σx z 0

c12 c22 c32 0 0 0

c12 c23 c22 0 0 0

0 0 0 0 0 0 0 c44 0 c22 − c23 0 0

⎫ ⎤⎧ εx x ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ε ⎪ ⎪ ⎪ 0 ⎥ yy ⎪ ⎪ ⎥⎪ ⎪ ⎨ ⎬ ⎥⎪ 0 ⎥ εzz ⎥ 0 ⎥⎪ ε ⎪ ⎪ xy ⎪ ⎥⎪ ⎪ ⎪ ⎪ 0 ⎦⎪ ε yz ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ εx z c44

(1.4)

Therefore, there are five material properties. If the material is isotropic, the constitutive equation is simplified such as ⎧ ⎫ ⎡ ⎪ σx x ⎪ c11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ σ yy ⎪ ⎪ ⎢ c21 ⎪ ⎪ ⎨ ⎬ ⎢ σzz ⎢c = ⎢ 21 ⎪ ⎪ σ 0 x y ⎪ ⎪ ⎢ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ σ 0 ⎪ yz ⎪ ⎪ ⎪ ⎩ ⎭ σx z 0

c12 c11 c21 0 0 0

⎫ ⎤⎧ ⎪ εx x ⎪ c12 0 0 0 ⎪ ⎪ ⎪ε ⎪ ⎪ ⎥⎪ ⎪ c12 0 0 0 yy ⎪ ⎪ ⎥⎪ ⎪ ⎨ ⎬ ⎥⎪ c11 0 0 0 ⎥ εzz ⎥ ⎥⎪ 0 0 ε ⎪ 0 c11 − c12 ⎪ xy ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎦⎪ 0 ε yz ⎪ 0 0 c11 − c12 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ εx z 0 0 0 c11 − c12

(1.5)

In order to relate the material constants ci j to more familiar properties such as elastic modulus E and Poisson’s ratio υ, the constitutive equation is inversed. For the orthotropic material, the expression becomes ⎫ ⎡ ⎫ ⎧ ⎤⎧ ⎪ ⎪ σx x ⎪ 1/E x −ν yx /E y −νzx /E z 0 εx x ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ −ν /E ⎪ ⎪ε ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ 1/E −ν /E 0 0 0 σ yy x y x y zy z yy ⎪ ⎢ ⎪ ⎪ ⎪ ⎥ ⎪ ⎪ ⎪ ⎪ ⎬ ⎢ ⎬ ⎨ ⎥⎨ εzz 0 0 0 ⎥ σzz ⎢ −νx z /E x −ν yz /E y 1/E z =⎢ (1.6) ⎥ ⎢ ⎪ εx y ⎪ 0 ⎥⎪ σx y ⎪ 0 0 0 1/G x y 0 ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ε yz ⎪ σ yz ⎪ 0 0 0 0 1/G yz 0 ⎦⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎭ ⎩ εx z σx z 0 0 0 0 0 1/G x z Likewise, for the transversely isotropic material, E y = E z , ν yx = νzx , G x y = G x z , E and G yz = 2(1+νy zy ) .

1.3 Micromechanics Models

5

1.3 Micromechanics Models In order to predict the smeared effective composite materials, micromechanics models or unit cell models have been developed. Those models provide the effective composite material properties from the material properties of both constituent materials and their volume fractions. For a particulate composite, the simple rule of mixture can be used as below: E c = v p E p + vm E m

(1.7)

where subscripts ‘c’, ‘p’, and ‘m’ denote the composite, particle, and matrix materials, respectively, and v p and vm are the particle and matrix volume fractions. If there is no other material including voids, the sum of the particle and matrix volume fractions is equal to unity. For the continuous unidirectional fiber composite, the effective material properties are estimated using the following equations: E L = v f E f + vm E m

(1.8)

vf 1 vm = + ET Ef Em

(1.9)

Here, E L and E T are the longitudinal and transverse elastic moduli of the fibrous composite, E f and E m are the fiber and matrix moduli, and v f and vm are the volume fractions of the fiber and matrix, respectively. The effective material properties were also predicted using the unit-cell model, which provided more reliable predictions of effective material properties [1–5]. The unit-cell consists of eight subcells as shown in Fig. 1.2. The unit-cell has a unit Fig. 1.2 Unit-cell made of eight subcells

3

7

2

8

1 b

a

3

6

4 b

1

2 a c

d

6

1 Introduction

length in every direction. For the particulate composite, the unit-cell dimensions are assumed to be the following:  1/3 a = c = vp

(1.10)

On the other hand, the fibrous composite has the following dimensions:  1/2 a = vf

(1.11)

by assuming the fiber is oriented along the 1-axis. In that case, c and d do not matter. However, for convenience, it is assumed that c = d = 0.5. In order to compute the effective material properties from the constituent materials, each subcell is assumed to have proper material properties. For example, subcell ‘1’ has the properties of the reinforcing particle, or subcells ‘1’ and ‘2’ have the properties of the fiber material. The rest of subcells are assumed to have the matrix material properties. Considering the fibrous composite, the constitutive equations are given for the fiber and matrix materials, respectively, as below:     σf = Cf εf

(1.12)

{σm } = [Cm ]{εm }

(1.13)



Then, the objective is to find the constitutive equation for the effective composite material properties such as {σC } = [CC ]{εC }

(1.14)

where subscript ‘C’ denotes the composite level values. In order to relate Eqs. (1.11) and (1.12) to Eq. (1.13), the following equilibrium and compatibility equations are used. Each subcell is assumed to have uniform stresses and strains for mathematical simplicity. The equilibrium at the interface of any two facing subcells, for example subcells ‘1’ and ‘3’, has the following expression. 1 3 1 3 1 3 = σ33 , σ31 = σ31 , σ32 = σ32 σ33

(1.15)

Such equilibrium equations are applied to every interface of subcells. Here, superscript denotes the subcell number. The compatibility equations along the 1-axis states 1 2 3 4 5 6 7 8 + dε11 = cε11 + dε11 = cε11 + dε11 = cε11 + dε11 cε11

(1.16)

Similar compatibility equations are developed for other directions as well as shear deformations. Finally, the composite level stresses and strains are assumed to be the volume average of all subcells like

1.3 Micromechanics Models

7

Table 1.1 Constituent material properties E x (GPa)

Ey (GPa)

Gxy (GPa)

Gyz (GPa)

ν12

Carbon

221

13.8

13.8

5.5

0.2

Epoxy

4.4

4.4

0.34

0.34

1.64

Table 1.2 Fiber composite material properties E x (GPa)

Ey (GPa)

νxy

νyz

FEA

151

10.1

0.24

0.5

Unit-cell

156

10.2

0.24

0.54

{σC } =

8 

Vi {σ }i

(1.17)

Vi {ε}i

(1.18)

i=1

{εC } =

8  i=1

where V i is the volume fraction of the i-th subcell and {}i is the stress or strain of the i-th subcell. Mathematical operation of the above set of equations, Eqs. (1.12), (1.13), (1.15) though (1.18) results in the equivalent expression for Eq. (1.13) so that the effective constituent material properties can be determined. Table 1.1 lists the constituent material properties such as carbon fibers and the epoxy. Then, Table 1.2 shows the comparison of the predicted material properties using the unit-cell model and the micromechanical finite element model of a find mesh. The prediction of the unit-cell model agreed very well with the micromechanical finite element model.

1.4 Fluid-Structure Interaction When a solid body moves inside a fluid medium, there is an interaction between the two bodies. In order to illustrate the interaction between the solid and fluid media, let’s consider one dimensional example. The solid was modelled as a single degree of freedom using a mass and a linear spring, and the mass was in contact with a fluid medium as sketched in Fig. 1.3. One end of the spring was constrained from any motion while the other end was attached to a rigid mass which was also in contact with fluid. To simplify the problem, the fluid motion was assumed to be one-dimensional by neglecting the effect of the wall friction. Let’s say an external harmonic force is applied to the mass. If the interaction by the fluid medium is neglected, the equation of the motion is written as

8

1 Introduction

Fig. 1.3 One-dimensional model of fluid-structure interaction

k

m Tube filled with fluid

u

m

d 2u + ku = f sin ωt dt 2

(1.19)

where m and k are the mass and spring constant, u is the displacement of the mass, t is time, f is the magnitude of the applied force, and ω is the frequency of the applied force. The solution to the above equation of motion is   f f u˙ o cos ωn t + 2 sin ωn t + u o − 2 cos ωt u(t) = 2 ωn ωn − ω ωn − ω2

(1.20)

in which u o and u˙ o are the initial displacement and velocity of the mass, and the natural frequency of the mass-spring system ωn is expressed as  ωn =

k m

(1.21)

However, when the fluid medium is in contact with the mass, the equation of motion is modified as m

d 2u + ku = f sin ωt − P(t) dt 2

(1.22)

where P(t) is the force resulting from the pressure in the fluid medium. Because the pressure load is applied to the opposite direction of the assumed motion, the sign is negative. The difficult part of the pressure loading is that it is not constant and dependent on the pressure of the fluid at the interface between the mass and the fluid. The fluid pressure at the interface changes with the fluid motion which is also associated with the motion of the mass. Therefore, it is a coupled problem between the solid and fluid media and called Fluid-Structure Interaction (FSI).

1.5 Scope of the Book This book presented various topics on FSI of composite structures. The covered topics were dynamic responses and failures of composite structures subjected to impact loading; effect of FSI on vibrational frequency, mode shape, and modal curvature;

1.5 Scope of the Book

9

dynamic responses of structures containing fluid inside; coupling of two independent structures through a fluid medium; and moving structures in a fluid medium. Both experimental and numerical studies were presented.

References 1. Kwon YW (2016) Multiphysics and multiscale modeling: techniques and application. CRC Press, Boca Raton 2. Kwon YW, Kim C (1998) Micromechanical model for thermal analysis of particulate and fibrous composites. J Therm Stresses 21:21–39 3. Kwon YW, Park MS (2013) Versatile micromechanics model for multiscale analysis of composite structures. Appl Compos Mater 20(4):673–692 4. Park MS, Kwon YW (2013) Elastoplastic micromechanics model for multiscale analysis of metal matrix composite structures. Comput Struct 123(2013):28–38 5. Kwon YW, Darcy J (2018) Failure criteria for fibrous composites based on multiscale modeling. Multiscale Multidiscip Model Exp Des 1(1):3–17

Chapter 2

Experimental Study of FSI with Impact Loading

While low velocity impact is not a major concern for most metallic structures, it has received a broad attention to polymer composite structures because such impact loading may cause internal damage to composite structures, resulting in their unexpected premature failure as load-carrying members. As a result, study of low velocity impact on composite structures has been conducted widely [1–8]. Almost every impact study considered dry composite structures. However, some composite structures contain uids or are surrounded by uids. When those structures are dynamically loaded, FSI occurs [9–22]. This chapter presents experimental studies of composite structures subjected to impact loading when they are in direct contact with uids, mostly water.

2.1 Experimental Set-Up In order to study the FSI effect on composite structures subjected to impact loading, the structures were tested inside an anechoic water tank which is approximately 3 m × 3 m × 3 m. The wall of the water tank was constructed to minimize the re ection of the incoming wave. A portable impact test set-up was designed and fabricated, and it was installed on the water tank for impact testing [9–12]. To examine the FSI effect experimentally, dynamic responses of a composite structure were compared between the two cases under the same impact loading. One case is that the structure is in direct interaction with water and the other case is that the structure has no interaction with water. To this end, the impact test machine should be designed to exert the same impact loading whether the structure is in water or not. This suggests that the impactor should not go into the water even though the structure is inside water. As the impactor moves through the water, drag force and buoyance force would reduce the speed of the impactor before striking

© Springer Nature Switzerland AG 2020 Y. W. Kwon, Fluid-Structure Interaction of Composite Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-57638-7_2

11

12 Fig. 2.1 Schematic of impact test set-up made of an impactor, a transmitting rod and a structure

2 Experimental Study of FSI with Impact Loading

Impactor

Transmitting Rod

Structure

the composite structure. Then, the impact speed would be different depending on whether the structure is inside water or not. The impactor was designed to include both a drop weight and a transmitting rod as schematically sketched in Fig. 2.1 in order to provide the same impact loading condition whether the composite structure was in water or not. The impactor strikes the transmitting rod, and the transmitting rod impacts the structure. A load cell was attached at the bottom of the transmitting rod to measure the force time history during the impact. For the impact testing on a composite structure submerged in water, the top end of the transmitting rod was above the water surface so that the impactor did not go into water. In addition, to minimize the motion of the transmitting rod in water and the resulting drag force, the bottom end of the transmitting rod was located initially very close to the top surface of the composite plate. Before testing, the impactor has a speci ed mass and a predetermined height in relative to the top of the transmitting rod. During the test, the impactor falls freely from the initial position and strikes the top of the transmitting rod, which then hits the structure at the bottom end of the transmitting rod. The impact machine was fabricated using aluminum frames and plates, and the fabricated testing machine is shown in Fig. 2.2. The drop weight was guided by four rods with slider bearings. The transmitting rod was supported using the aluminum frame. The testing machine has two horizontal beam structures which were used to install the impact machine in the anechoic water tank as shown in Fig. 2.3. It was supported at the top of the water tank so that the major portion of the impact machine was submerged in the water. In other words, the composite structure and a larger portion of the transmitting rod were located inside the water. When dry impact tests were conducted, water was drained from the water tank so that the test could be conducted using the same equipment whether it was a dry and wet test. Four different test cases were investigated. The rst case was a dry structure without contact with water on all surfaces of the structure. The second case was a completely wet structure which was in contact with water on both top and bottom

2.1 Experimental Set-Up Fig. 2.2 Impact testing machine

Fig. 2.3 Impact test machine installed on anechoic water tank

13

14

2 Experimental Study of FSI with Impact Loading

Fig. 2.4 Enclosure box at the bottom side of a structure

sides of the structure. The two other cases were a half wet structure. One of these had the wet top surface of the structure while the other had the wet bottom surface. For the last two cases, an enclosure box was used for the impact testing machine. Depending on which side was wet, the enclosure box was attached to the dry side of the structure. The bottom enclosure box is shown in Fig. 2.4. The box was speci cally constructed for testing of the top wet and bottom dry case. This is called the airbacked wet surface impact from now on. The box was made of a Plexiglas and it was secured to the bottom aluminum plate, which was used to hold a composite plate to be tested, using eight C-clamps, and sealed with putty tape to prevent water intrusion into the box. The box completely covered the sample so that none of the sample was exposed to water on the enclosure side. A small size of hole was cut out from the side of the enclosure to feed the wiring from the strain gages to the data analyzer because the gages were attached to the bottom side of the composite plate. The hole was lled with putty to prevent water leakage. Four different test conditions are sketched in Fig. 2.5. Figure 2.5a shows the dry impact when the structure never interacts with water while Fig. 2.5b is the case when the structure is exposed to water on both sides. The former is called a dry structure while the latter is called a wet structure from now on. Figure 2.5c, d are the cases when the structure is exposed to water only at one side. The former, which is called an air-backed structure, has the wet surface on the top of the structure; and the latter has the wet surface at the bottom of the structure, which is called a water-backed structure. Laminated composite plates as well as sandwich composite plates and beams were tested. All the plates were clamped along the edges to mimic the built-in boundary condition. Some composite beams were also tested, and those beams were clamped at both ends, too. Figure 2.6 shows a carbon plate clamped around the edges using C-clamps.

2.1 Experimental Set-Up

Impactor

15

Impactor Water surface

Clamped plate

Clamped plate

Anechoic tank

Anechoic tank

(a)

(b) Water surface

Impactor

Impactor

Wall

Water surface

Clamped plate

Clamped plate

Empty containe r Anechoic tank (c)

Anechoic tank (d)

Fig. 2.5 Four different test cases; a both sides dry b both sides wet called fully wet c impact side wet while the opposite side dry called air-backed, and d impact side dry while the opposite side wet called water-backed

Composite plates and beams were fabricated using either the Vacuum Assisted Resin Transfer Molding technique (VARTM) as shown in Fig. 2.7 or the manual layup procedure. Figure 2.8 shows a ow of resin through composite layers during the VARTM process. The manual layup procedure used brushes and rollers to apply resin between every layer of a composite specimen. A vacuum process was also applied to the manually prepared composite plates at the end in order to remove air pockets in the resin. Woven fabric composites made of either carbon bers or E-glass bers were used for the sample. The fabric was in the form of plain weave. For the sandwich structure, foams or balsa wood were selected as the core materials.

16

2 Experimental Study of FSI with Impact Loading

Fig. 2.6 Carbon plate clamped to the impact equipment

Fig. 2.7 Vacuum assisted resin transfer technique process

Any wet impact testing was conducted as soon as the test specimen was inserted into the anechoic water tank so that the specimen did not have enough time to absorb moisture, which could affect the material properties of the composite. In addition, as soon as the wet tests were conducted, the specimens were taken out of water so that they stayed in the dry condition all the time except for a short testing period.

2.2 Experimental Results In order to study the dynamic motion and failure, strain gages were attached to each composite sample. Several bidirectional or 45-degree rosette gages were af xed to

2.2 Experimental Results

17

Fig. 2.8 Resin ow through composite layers

the composite samples and the strain-time history was measured. Each strain gage was properly sealed such that water does not in uence the gage reading. Figure 2.9 shows a carbon ber composite plate with strain gage rosette attached to the plate. Strain gages were, in general, attached to the center, corner, and mid-portion of each plate though the actual gage locations were changed for case by case depending on the objective of each test. Fig. 2.9 Strain gage rosettes af xed to a carbon ber composite plate

Gage #4 Gage #2

Gage #3

Gage #1

18

2 Experimental Study of FSI with Impact Loading

Every test case was repeated multiple times to make sure the results were consistent from test to test. If there was no damage to the composite structure, the repeated tests were undertaken for the same structure. Once damage was observed, tests were repeated for other structures which were fabricated in the same batch. Test results were quite consistent. Instead of taking the average values of the test results, the most representative test results were presented in the following discussion unless mentioned otherwise. First of all, the impact force was compared for a carbon ber composite plate among three different conditions: dry plate, fully wet plate and air-backed plate. All cases had the same initial impact conditions in terms of the impactor weight and its drop height. The impact force was measured using the load cell attached to the bottom of the transmission rod, and it was normalized with respect to the maximum force on the dry plate. Figure 2.10 shows the force-time history during the impact. The gure clearly suggests that fully wet and air-backed plates produced much greater impact forces than the dry plate. In other words, the effect of FSI resulted in much higher impact forces. This may be explained as follows. As the composite plate de ects down resulting from the impact force, FSI provides an added mass effect. In other words, the plate becomes heavier with the added mass, which reduces the acceleration and the motion of the composite plate. The reduced motion can increase the contact force between the impacting rod and the composite plate. The increased contact force is registered in the load cell. One thing to be also notable is that the fully wet and air-backed plates had the almost same peak impact force while the impact forces were different at later times. This result indicates that a design without considering the effect of FSI can introduce an unexpected premature failure of a composite structure when it is subjected 2

Dry Plate Fully Wet Plate Air-Backed Plate

Normalized Force

1.5

1

0.5

0

-0.5

0

0.005

0.01

0.015

Time (sec.)

Fig. 2.10 Comparison of impact forces among three different cases

0.02

2.2 Experimental Results

19

10 -4

4

Dry Plate Fully Wet Plate Air-Backed Plate

Strain

3

2

1

0

-1

0

0.01

0.02

0.03

0.04

0.05

0.06

Time(sec.) Fig. 2.11 Comparison of horizontal strains at gage #1 among three different cases

to an impact load. The strain gage responses support this statement. Figure 2.11 compares horizontal strains near the center of the plate. This is the strain gage #1 located in Fig. 2.9. All strains were also normalized in terms of the maximum strain of the dry plate. Because impactor struck the center of the plate, the strain gage was attached very close to the center. The greater impact force resulting from FSI induced signi cantly larger strains near the center of the plate as compared to that of the dry plate. Both fully wet and air-backed plates showed very close maximum strains near the center. However, the effect of FSI was not uniform over the entire composite plate. Figure 2.12 plots the strains at gage #2 which is located at the center along the horizontal direction and at the top quarter point along the vertical direction. The peak strains were very comparable with and without the effect of FSI. A similar observation was made for the gage #3. The strain gage #4, as shown in Fig. 2.13, gave a very different response resulting from FSI. The peak strains resulting from FSI were almost ve times greater than that of the dry plate. This ratio was the largest among all measured strains. In other words, the effect of FSI was the largest at gage #4. The fully wet and air-backed plates showed a major difference in the positive peaks following the initial negative peaks. The fully wet plate had a slightly greater initial negative strain than the air-backed plate, but the subsequent positive peak strain was much greater for the air-backed plate than the fully wet plate. The Fast Fourier Transform (FFT) was applied to the measured strain timehistory to determine the vibrational frequency of the structure. Once the vibrational frequencies were computed for both dry and wet structures, the Added Virtual Mass Incremental Factor (AVMIF) was computed using the following equation [23].

20

2 Experimental Study of FSI with Impact Loading

Normalized Strain

1.5

Dry Plate Fully Wet Plate Air-Backed Plate

1

0.5

0

-0.5

0

0.01

0.02

0.03

0.04

0.05

Time (sec.) Fig. 2.12 Comparison of horizontal strains at gage #2 among three different cases 5

Normalized Strain

Dry Plate Fully Wet Plate Air-Backed Plate

0

-5

0

0.02

0.04

0.06

0.08

0.1

Time (sec.)

Fig. 2.13 Comparison of horizontal strains at gage #4 among three different cases

ωw = √

ωd 1+β

(2.1)

where ωw and ωd are the frequencies of the wet (i.e. with FSI) and dry (i.e. without FSI) structures, respectively, and β is the AVMIF. There was a minor variation of the frequency among different strain gage data, but those were very small. The measured frequencies of the dry and fully wet plates were compared in Table 2.1 which also shows the AVMIF. The results showed that

2.2 Experimental Results

21

Table 2.1 Comparison of vibrational frequencies with and without FSI

Fully wet freq. (rad/s)

Dry freq. (rad/s)

β (AVMIF)

Horizontal strain at gage #3a

223.49

615.62

6.59

Horizontal strain at gage #2a

238.29

661.64

6.71

Vertical strain at gage #3a

226.99

633.44

6.79

Vertical strain at gage #2a

226.86

614.75

6.34

Average

228.91

631.36

6.61

a Gage

locations are shown in Fig. 2.9

the vibrational frequency was signi cantly reduced resulting from FSI. The average AVMIF was 6.6 which was a large value. The main reason is that the density of the composite material is very comparable to that of water, and the effect of the added mass from water is signi cant on the composite material. For a steel material, the AVMIF was much lower. The next set of tests were conducted for glass ber sandwich composite plates. The skin was fabricated using E-glass woven fabrics while the core was a balsa panel. These tests were undertaken to investigate the damage initiation and progression during the impact loading associated with FSI. The impact testing was performed as follows. As a ber glass sandwich composite was clamped along all edges, an initial drop height was set, which was low enough that no damage was expected in the sandwich structure. The sandwich plate was impacted twice from the initial drop height. The reason two identical impact tests were undertaken from the same drop height was to check the damage initiation or any further progression. If the measured responses of the two tests are identical and there is no change in damage during the visual check, it could be assumed that there was no change in the damage state in the structure. Then, the impact height was increased incrementally while the weight was remained constant. At each drop height, impact tests were also conducted twice. After each impact test, the damage was observed and measured if there was any. The state of damage was delamination between the skin and core layers at the bottom side of the sandwich plate. This set of tests were repeated for other sandwich plates that were fabricated using the same procedure and the same materials. Additionally, the same test procedure was applied to sandwich plates while they were dry or submerged in water to investigate the effect of FSI on damage. Because the damage mode was the delamination between the skin and core layers, the average diameter of the delamination was measured for each impact test, and the delamination size was plotted as a function of the impact drop height. A typical plot of delamination vs. drop height is shown in Fig. 2.14. The gure shows that the damage initiated at a lower drop height for the fully wet

22

2 Experimental Study of FSI with Impact Loading 6

Dry Plate Fully Wet Plate

Delamination (cm)

5 4 3 2 1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

Drop Height (m)

Fig. 2.14 Plot of delamination as a function of impact drop height

plate as compared to the dry plate. This indicates that FSI can cause pre-mature damage if it were not considered in the design and analysis. Figure 2.15 shows a typical damage occurred in the sandwich plate. The top side had a local and partial penetration resulting from the load cell while the back surface had delamination. Damage could not be measured for the air-backed sandwich plate because the part attached to the bottom side of the plate to provide air support prohibited the delamination measurement at the back surface of the plate. Otherwise, the part should be detached for delamination measurement and re-attached repeatedly. However, this would take too much time and effort, and this might alter the test condition. As a result, the damage measurement was skipped for the air-backed sandwich plate. Figure 2.16 shows the progressive impact forces as a function of the initial drop

Impact point

Underside delamination

Fig. 2.15 Impact damage on the front and back surfaces of sandwich plate

2.2 Experimental Results

23

Drop Height: 10.16cm

Drop Height: 20.32cm 2000

1500

Dry Plate

Dry Plate

Force (N)

Force (N)

Fully Wet Plate

Fully Wet Plate

500

0

-500

Air-Backed Plate

1500

Air-Backed Plate

1000

1000

500

0

0

20

40

60

-500

80

0

20

Time (msec)

40

Drop Height: 40.64cm 4000

Dry Plate

Dry Plate Air-Backed Plate

2000

Air-Backed Plate

3000

Fully Wet Plate

Fully Wet Plate

Force (N)

Force (N)

80

Drop Height: 60.96cm

3000

1000

0

-1000

60

Time (msec)

2000

1000

0

0

20

40

Time (msec)

60

80

-1000

0

20

40

60

80

Time (msec)

Fig. 2.16 Plot of progress impact forces as a function of initial impact drop height for three different cases

height. At the lower drop heights, the sandwich plates did not show damage. However, as the drop height increased, damage initiated and propagated. Because the same sandwich plate was impacted progressively for either a dry, fully wet, or air-backed plate, respectively, a prior damage condition in the plate changed the impact force because of its change in stiffness. As seen in Fig. 2.16, the impact force was greater with the FSI effect. However, earlier damage with FSI resulted in a reduction in its stiffness at the damage section. Therefore, at the later impact with a larger drop height, the impact force with FSI became smaller than that of the dry structure. The strain responses of the sandwich plate became more complex as damage occurred and grew in the plate. Figures 2.17 and 2.18 compare the strain responses at two locations of the sandwich plate, which is shown in Fig. 2.19. Both strain gages

24

2 Experimental Study of FSI with Impact Loading Drop Height: 10.16cm

1

Dry Plate

0.8

Dry Plate

Air-Backed Plate

0.6

Air-Backed Plate

1

Fully Wet Plate

Milistrain

Milistrain

Drop Height: 20.32cm

1.5

0.4 0.2 0

Fully Wet Plate

0.5

0

-0.2 -0.4

0

20

40

60

80

-0.5

100

0

20

Time (msec) Drop Height: 40.64cm

2

100

Air-Backed Plate

Milistrain

Milistrain

80

Dry Plate

2

Air-Backed Plate Fully Wet Plate

1 0.5

1.5

Fully Wet Plate

1 0.5

0 -0.5

60

Drop Height: 60.96cm

2.5

Dry Plate

1.5

40

Time (msec)

0

0

20

40

60

Time (msec)

80

100

-0.5

0

20

40

60

80

100

Time (msec)

Fig. 2.17 Plot of progress strains as a function of initial impact drop height for three different cases at gage location #A in Fig. 2.19

were along the diagonal line of a square composite plate. Gage A was closer to the center of the plate than Gage B. The two gages shown in Fig. 2.19 gave very different strain behaviors. At the location of gage A, the dry plate had a peak strain close to those with the FSI conditions. On the other hand, gage B showed much greater strains with FSI as compared to that without FSI. The subsequent tests were conducted for sandwich beam structures clamped at both ends. The sandwich beams were made of the E-glass bers. However, the Eglass bers used for the beam structures had thinner tows than that used for the sandwich plate. Otherwise, the rest of the materials and fabrication procedure were the same as before. Figure 2.20 shows a sandwich beam installed in the impact test equipment. The impact was applied to the center of the beam. The sandwich beams were tested under the dry condition as well as the fully wet condition. Different impact heights were applied to the beam structures so that some beams were undamaged while others

2.2 Experimental Results

25

Drop Height: 7.62cm

0.4

Drop Height: 35.56cm

0.8

Dry Plate

Dry Plate

0.3

Fully Wet Plate

Milistrain

0.2

Milistrain

0.6

Air-Backed Plate

0.1

-0.1

-0.2

20

60 40 Time (msec)

80

-0.4

100

Drop Height: 45.72cm

0.6

0

20

Dry Plate

100

Air-Backed Plate

0.6

Milistrain

0

80

Dry Plate

0.8

Fully Wet Plate

0.2

40 60 Time (msec)

Drop Height: 71.12cm

1

Air-Backed Plate

0.4

Milistrain

0.2 0

0

Fully Wet Platre

0.4

0

-0.2

Air-Backed Plate

Fully Wet Plate

0.4 0.2 0

-0.2 -0.4

-0.2

0

20

60 40 Time (msec)

80

100

-0.4

0

20

60 40 Time (msec)

80

100

Fig. 2.18 Plot of progress strains as a function of initial impact drop height for three different cases at gage location #B in Fig. 2.19 Fig. 2.19 Strain gage locations of sandwich composite plates (circles denote equal divisions along the diagonal of a square plate)

Gage A

Gage B

26

2 Experimental Study of FSI with Impact Loading

Fig. 2.20 Sandwich beam installed in the impact test equipment

were damaged. When there was damage in the beam structure, the force measured from the load cell showed a sudden drop in its graph. Figures 2.21 and 2.22 are the force and strain plots when the sandwich beam was fully wet and impacted. The former gure was for the impact loading which did not cause any noticeable damage to the beam structure when visually inspected. On the other hand, the latter gure was for the impact resulting in noticeable damage to the beam. When the damage initiated, there were decreases in both impact force and strains which were af xed to the center, a point very close to one clamped end called the boundary, and a quarter point along the span of the beam called the 1/4 span. One interesting thing observed during the testing was the failure locations of the sandwich composite beams when impacted. Impact on the dry beams induced failure at one of the clamped ends while fully wet beams showed failure at the center where impacted. Because there might be some variation from sandwich beam sample to sample, more than a dozen tests were conducted. Some possible and unavoidable variation was the balsa core material. The mechanical property of the balsa core was not quite homogeneous. However, the test results showed a clear preference in the failed location as tabulated in Table 2.2. Five tests showed the center failure for the fully wet beams, and ve tests also showed the clamped end failure for dry beams. On the contrary, two fully wet beams showed failure at the clamped boundary, and one dry beam showed failure at the center. The results suggested that the effect of FSI could alter the potential failure locations of the structures when subjected to dynamic loading. Figures 2.23 and 2.24 show typical failures of the wet and dry sandwich beams, respectively. The onset of damage initiation was determined for the sandwich beams using the progress impact tests with incremental drop heights. Table 2.3 compares the average drop height, initial impact velocity, and impact force between the dry and fully wet beams at the onset of damage initiation. The effect of FSI yielded failure at a lower impact height.

2.3 Summary

27 Fully Wet Beam

1200 1000 800

Force (N)

600 400 200 0 -200 0

5

10

15

20

25

30

35

40

45

50

Time (msec) 20

Strain (microstrain)

15 10 5 0 Center Boundary

-5 -10

0

5

10

15

20

25

30

35

40

45

50

Time (msec)

Fig. 2.21 Force and strain plots of a fully wet sandwich beam under impact but without damage

2.3 Summary The FSI in uenced the effect of low velocity impact on polymer composite structures signi cantly because mass densities of the polymer composite materials are quite comparable to that of a uid, i.e. water in this study. The in uence was mostly negative. Because of FSI, the polymer composite structures could fail at a lower impact loading condition. FSI could also alter potential failure locations. The vibrational frequencies of polymer composites were reduced signi cantly resulting from FSI. Therefore, it is essential to consider the effect of FSI when a composite structure is

28

2 Experimental Study of FSI with Impact Loading Fully Wet Beam 1200 1000 800

Force (N)

600 400 200 0 -200

0

5

10

15

20

25

30

35

40

45

50

Time (msec) 10 Boundary 1/4th Span

Strain (microstrain)

5

Center

0

-5

-10

0

5

10

15

20

25

30

35

40

45

50

Time (msec)

Fig. 2.22 Force and strain plots of a fully wet sandwich beam under impact but without damage Table 2.2 Number of sandwich beam samples failed at center or clamped boundary

Center

Boundary

Fully wet beams

5

2

Dry beams

1

5

2.3 Summary

29

Top side

Bottom side

Fig. 2.23 Sandwich beam failed at the center with impact to the center

Top side

Bottom side

Fig. 2.24 Sandwich beam failed at the clamped end with impact to the center

30 Table 2.3 Comparison of average impact height and peak impact force at the onset of initial delamination

2 Experimental Study of FSI with Impact Loading Impact height (cm)

Impact velocity Impact force (m/s) (N)

Dry beams 22.0

2.08

1939

Fully wet beams

1.82

1436

16.9

in contact with uid and subjected to dynamic loading. Otherwise, premature failure can occur in the composite structure.

References 1. Aslan Z, Karakuzu R, Okutan B (2003) The response of laminated composite plates under low-velocity impact loading. Compos Struct 59:119–127 2. Abrate S (1994) Impact on laminated composites; recent advances. Appl Mech Rev 47(11):517– 544 3. Strait LH, Karasek ML, Amateau MF (1992) Effects of stacking sequence on the impact resistance of carbon ber reinforced thermoplastic toughened epoxy laminates. J Compos Mater 26(12):1725–1740 4. Hosur MV, Jain K, Chowdhury F, Jeelani S, Bhat MR, Murthy CRL (2007) Low velocity impact response of carbon/epoxy laminates subjected to cold-dry and cold-moist conditioning. Compos Struct 79:300–311 5. Richardson MOW, Wisheart MJ (1996) Review of low-velocity impact properties of composite materials. Compos Part A 27A:1123–1131 6. Hosur MV, Karim MR, Jeelani S (2003) Experimental investigations on the response of stitched/unstitched woven S2-glass/SC15 epoxy composites under single repeated low velocity impact loading. Compos Struct 62:89–102 7. Abrate S (1998) Impact on composite structures. Cambridge University Press, Cambridge 8. Sjoblom PO, Hartness JT, Cordell TM (1988) On low-velocity impact testing of composite materials. J Compos Mater 22(30):30–51 9. Kwon YW, Owens AC, Kwon AS, Didoszak JM (2010) Experimental study of impact on composite plates with uid-structure interaction. Int J Multiphys 4(3):259–271 10. Kwon YW, Violette MA, McCrillis RD, Didoszak JM (2012) Transient dynamic response and failure of sandwich composite structures under impact loading with uid structure interaction. Appl Compos Mater 19(6):921–940 11. Kwon KW, Violette MA (2012) Damage initiation and growth in laminated polymer composite plates with uid-structure interaction under impact loading. Int J Multiphys 6(1):29–42 12. Kwon YW, Conner RP (2012) Low velocity impact on polymer composite plate in contact with water. Int J Multiphys 6(3):179–197 13. Kwon YW, Priest EM, Gordis JH (2013) Investigation of vibrational characteristics of composite beams with uid-structure interaction. Compos Struct 105:269–278 14. Kwon YW, Teo HF, Park C (2016) Cyclic loading on composite beams with uid structure interaction. Exp Mech 56(4):645–652 15. Kwon YW (2011) Study of uid effects on dynamics of composite structures. ASME J Pressure Vessel Technol 133:031301–031306 16. Kwon YW (2014) Dynamic responses of composite structures in contact with water while subjected to harmonic loads. Appl Compos Mater 21(1):227–245 17. Kwon YW, Plessas SD (2014) Numerical modal analysis of composite structures coupled with water. Compos Struct 116:325–335

References

31

18. Kwon YW, South T, Yun KJ (2017) Low velocity impact to composite box containing water and baf es, composite structures. ASME J Pressure Vessel Technol 139(3):031304 19. Kwon YW, Bowling JD (2018) Dynamic responses of composite structures coupled through uid medium. Multiscale Multidisc Model Exp Des 1(1):69–82 20. Kwon YW, Knutton SC (2015) Computational study of effect of transient uid force on composite structures submerged in water. J Multiphys 8(4):367–395 21. Kwon YW, Millhouse SC, Arceneaux S (2015) Study of composite plate in water with transient and steady sate motions. Compos Struct 123:393–400 22. Kwon YW (2016) Multiphysics and multiscale modeling: techniques and application. CRC Press, Boca Raton 23. Kwak MK (1996) Hydroelastic vibration of rectangular plates. J Appl Mech 63:110–115

Chapter 3

Numerical Modeling of FSI Under Dynamic Loading

Numerical analyses of FSI problems subjected to dynamic loading require coupling of two different solvers; one for the fluid domain and the other for the structural domain. There are multiple techniques to solve structural problems. Among those, the Finite Element Method (FEM) [1–6] is the most popular and powerful technique. As a result, the FEM was mostly used for the structural analysis. On the other hand, the classical Computational Fluid Dynamics (CFD) [7–9] is very common for the fluid analysis. More recently, additional new techniques have been developed. Some of those are the Cellular Automata (CA) and the Lattice Boltzmann Method (LBM). The CA technique has been applied to both the wave propagation problems [10, 11] and the beam and plate bending problems [12]. The LBM was applied for the viscous flow problem [13–16]. For the present study, either FEM or CA was used for structural analysis while either LBM or CA was adopted for fluid analysis. When the fluid flow was considered, the LBM was selected. On the other hand, when fluid flow was neglected, the CA technique was utilized. In this chapter, a brief description of the FEM was presented for structural analysis. The CA technique was presented for the structural and wave propagation analyses. Finally, LBM was discussed for viscous fluid flows. More detailed presentations of these techniques can be found in Refs. [3, 17–19].

3.1 Finite Element Method This section presents the finite element formulation for the beam and plate structure. The weighted residual formulation is used for the beam bending problem while the energy principle is applied for the plate bending formulation.

© Springer Nature Switzerland AG 2020 Y. W. Kwon, Fluid-Structure Interaction of Composite Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-57638-7_3

33

34

3 Numerical Modeling of FSI under Dynamic Loading

3.1.1 Beam Bending The Weighted Residual Method (WRM) is applied to develop the finite element formulation of structural members. One of the most frequently used structural members is the beam which is one-dimensional in its geometric character and supports the external load in the transverse direction. The equation of motion for beam bending is expressed as below:   ∂ 2u ∂2 ∂ 2u m 2 + 2 EI 2 − p = 0 ∂t ∂x ∂x

(3.1)

where m is the beam mass per unit length, u is the transverse deflection of the beam, p is the transverse load per unit length, x is the axis along the beam, t is time, E is the elastic modulus of the beam, and I is the second moment of inertia of the crosssection of the beam. The product EI is called the beam rigidity which represents the stiffness of the beam. Applying the WRM to Eq. (3.1) yields the following weighted residual.    L  2 ∂ 2u ∂ u ∂2 R = w m 2 + 2 E I 2 − p dx = 0 ∂t ∂x ∂x

(3.2)

0

where w is the test function (or called weighting function) and L is the length of the beam. Applying integrations by parts twice to the second term of the above equation gives L R= 0

∂ 2u mw 2 d x + ∂t

L EI 0

∂ 2w ∂ 2u dx ∂x2 ∂x2

 L   L L   ∂ 2u ∂ 2u ∂w ∂ EI 2 E I 2 − w pd x = 0 − + w ∂x ∂x ∂x ∂x 0 0

(3.3)

0

Now, since the shear force V =

∂ ∂x



E I ∂∂ xu2 2



and the bending moment M =

∂2u , Eq. (3.3) can be rewritten as below along with the finite element discretization ∂x2

EI of the beam domain.

R=

N  e=1 Ω e

N  ∂ 2u ∂ 2w ∂ 2u mw 2 d x+ E I 2 2 dx ∂t ∂x ∂x e=1

+ [wV ]0L −



∂w M ∂x

L

Ωe

− 0

N  e=1 Ω e

w pd x = 0

(3.4)

3.1 Finite Element Method

35

where e denotes each finite element domain, and N is the number of finite elements of the beam domain. Then, we select shape functions for w and u for every element domain. If the same shape functions are used for both functions w and u, it is called the Galerkin method, which is used for the present formulation. The shape functions depend on the type of elements, i.e. the number of nodes per element. The simplest and the most commonly used beam element has two nodes, one at each end. Each node has two degrees of freedom, one for the transverse displacement u and the other for the slope θ = ddux for the Bernoulli-Euler beam bending theory. This slope condition θ = ddux is relaxed for the Timoshenko theory by introducing the strain energy resulting from shear deformation. For the present discussion, the Bernoulli-Euler beam theory is used as expressed in Eq. (3.1). The shape functions for the linear beam element for the Bernoulli-Euler beam theory are expressed as below: 3x 2 2x 3 + l2 l3 2 x3 2x + 2 H2 (x) = x − l l 3x 2 2x 3 H3 (x) = 2 − 3 l l x3 x2 H4 (x) = − + 2 l l H1 (x) = 1 −

(3.5)

where x is the coordinate starting from the left node of the element toward the right node direction, and l is the length of the element. These shape functions are used for every beam element. The shape functions have the following characteristics: H1 (0) = 1, H2 (0) = 0, H3 (0) = 0, H4 (0) = 0 H1 (0) = 0, H2 (0) = 1, H3 (0) = 0, H4 (0) = 0 H1 (l) = 0, H2 (l) = 0, H3 (l) = 1, H4 (l) = 0 H1 (l) = 0, H2 (l) = 0, H3 (l) = 0, H4 (l) = 1

(3.6)

Then, the beam deflection u for each beam element is expressed as u = H1 (x)u i + H2 (x)θi + H3 (x)u j + H4 (x)θ j

(3.7)



where u i θi u j θ j is called the nodal variable vector and it denotes the transverse displacement and slope at the nodes i and j, respectively, which are the left and right nodes of each beam element as shown in Fig. 3.1. Applying the shape functions and the nodal variables to Eq. (3.4) produces the following equation.

36

3 Numerical Modeling of FSI under Dynamic Loading

Fig. 3.1 Linear beam element

N

N N {Fe } [Me ] U¨ e + [K e ]{Ue } =

e=1

e=1

(3.8)

e=1

Here  [Me ] =

[H ]T [H ]d x

(3.9)

Ωe

  [K e ] = Ωe

∂2 H ∂x2

T 

 ∂2 H dx ∂x2

(3.10)



{Fe } =

[H ]T pd x

(3.11)

Ωe

The matrix [Me ] is the element mass matrix, [K e ] is the element stiffness matrix, {Fe } is the element force vector, and {Ue } is the vector for the element nodal degrees of freedom. The element stiffness matrix for the linear element is determined as the shape functions, Eq. (3.5) are substituted into Eq. (3.10) as below: ⎡

12 EI ⎢ 6l [K e ] = 3 ⎢ l ⎣ −12 6l

6l 4l 2 −6l 2l 2

−12 −6l 12 −6l

⎤ 6l 2l 2 ⎥ ⎥ −6l ⎦ 4l 2

(3.12)

The element mass matrix from Eq. (3.9) is given as below: ⎡

156 ml ⎢ ⎢ 22l [M] = 420 ⎣ 54 −13l

22l 4l 2 13l −3l 2

54 13l 156 −22l

⎤ −13l −3l 2 ⎥ ⎥ −22l ⎦ 4l 2

(3.13)

This mass matrix is called the consistent mass matrix because it was determined by applying the shape functions to Eq. (3.9) as was done for the element stiffness matrix. Sometime, the diagonal mass matrix is used for computational efficiency, which is expressed as

3.1 Finite Element Method

37

⎡

39 0 2 ml ⎢ ⎢0 l [Me ] = 78 ⎣ 0 0 0 0

⎤ 0 0 0 0⎥ ⎥ 39 0 ⎦ 0 l2

(3.14)

The forcing vector depends on the applied load. If the applied load is constant within each element, the force vector is expressed as {Fe } =

T po l 6 l 6 −l 12

(3.15)

where po is the uniform load intensity per unit length. In addition, {Ue } =

T u i θi u j θ j and superimposed dot denotes temporal derivative. Finally, summation in Eq. (3.8) results in the following equation. [M] U¨ + [K ]{U } = {F}

(3.16)

If we introduce the viscous damping force, which is proportional to the velocity, Eq. (3.16) is generalized like the following: [M] U¨ + [C] U˙ + [K ]{U } = {F}

(3.17)

in which [C] is the damping matrix. This is not usually computed from any governing equation. Instead, it is estimated from engineering experience or experimental data. For the proportional damping, the damping matrix is expressed as a linear combination of the mass and stiffness matrices such as [C] = α[M] + β[K ]

(3.18)

where α and β are the proportional constants to be determined by engineering analysts. Through the finite element formulation, the partial differential equation given in Eq. (3.1) is transformed to a set of ordinary differential equations in terms of time derivatives. The final Eq. (3.17) is solved numerically using a time integration technique. There are many different time integration schemes, but only one technique is presented here. Additional techniques can be read in Refs. [2, 4–6]. The central difference technique is presented here. The computational procedure for the central difference technique is summarized below: 1. Compute the acceleration from the given displacement and velocity as follows.   t t U¨ = [M]−1 {F}t − [C] U˙ − [K ][U ]t

(3.19)

38

3 Numerical Modeling of FSI under Dynamic Loading

where superscript t denotes time. To start the process at t = 0, the initial values 0 of U˙ and {U }0 are used. 2. Compute the velocity from the acceleration. t+0.5t t−0.5t t U˙ = U˙ + U¨ t

(3.20)

where t is the time step size. For the first computation, it is assumed that −0.5t 0 U˙ = U˙ . 3. Compute the displacement from the velocity. t+0.5t {U }t+t = {U }t + U˙ t

(3.21)

4. Apply the boundary conditions as necessary. 5. Compute the average velocity such as t+0.5t t+0.5t t + U¨ U˙ U˙ = 2

(3.22)

6. Repeat the process 1 through 5 for the next time increment until the final time is reached. The central difference time integration scheme is conditionally stable so that the time step size must be controlled. For a linear system, the time step size must be smaller than the critical time step size as expressed below: tcrit =

Tmin π

(3.23)

where Tmin is the smallest period of the finite element matrix system.

3.1.2 Plate Bending The finite element formulation for plate bending problem is developed from the total potential energy. The total potential energy is expressed as

=V −W

(3.24)

in which V and W are the strain energy stored in the deformable body and the work done by external load. The strain energy for plate bending has two parts. One is the strain energy resulting from bending and the other comes from the transverse shear deformation. In the classical plate bending theory, the latter was neglected because it is much smaller than the former. However, including the shear energy is beneficial for the finite element formulation of the plate bending problem because that decouples

3.1 Finite Element Method

39

the transverse deflection and slopes. This makes the same shape functions can be used for the deflection and slopes, respectively. The strain energy is expressed as below: V =

1 2

 {σb }T {εb }dΩ + Ω

κ 2

 {σs }T {εs }dΩ

(3.25)

Ω

Here subscripts ‘b’ and ‘s’ denote the bending and shear, respectively, and κ is the shear correction factor. Stresses are related to strains by the following constitutive equations. {σb } = [E b ]{εb }

(3.26)

{σs } = [E s ]{εs }

(3.27)

For an isotropic material, the constitutive matrices are ⎡ ⎤ 1ν 0 E ⎣ [E b ] = ν0 0 ⎦ 1 − ν2 0 0 1−ν 2   E 10 [E s ] = 2(1 + ν) 0 1

(3.28)

(3.29)

in which ν is Poisson’s ratio. Substitution of Eqs. (3.26) and (3.27) into Eq. (3.25) yields 1 V = 2



κ {εb } [E b ]{εb }dΩ + 2

 {εs }T [E s ]{εs }dΩ

T

Ω

(3.30)

Ω

As the plate domain is discretized into a number of finite element domains, Eq. (3.30) is expressed as 1 2 e=1 N

V =



Ωe

{εb }T [E b ]{εb }dΩ +

N  κ {εs }T [E s ]{εs }dΩ 2 e=1

(3.31)

Ωe

The typical shapes of plate elements are either triangles or quadrilaterals. The simplest triangular element has three nodes while the simplest quadrilateral element has four nodes as shown in Fig. 3.2. Every node of a plate bending element has three degrees of freedom: one for the transverse deflection, and two slopes as sketched in terms of the 3-D perspective in Fig. 3.3 where the transverse displacement was denoted by w while two slopes are represented by θx and θ y . The inplane

40

3 Numerical Modeling of FSI under Dynamic Loading

Fig. 3.2 Two-dimensional elements; triangular (left) and quadrilateral (right) shapes

Fig. 3.3 Three degrees of freedoms per node of plate bending element

z y w

x

displacements of a plate along the x- and y-axis in Fig. 3.3 are written as u(x, y) = −zθx (x, y)

(3.32)

v(x, y) = −zθ y (x, y)

(3.33)

Both the transverse displacement and the slopes are interpolated using the same shape functions as below: w=

n

Hi (x, y)wi

(3.34)

Hi (x, y)(θx )i

(3.35)

Hi (x, y)(θY )i

(3.36)

i=1

θx =

n i=1

θY =

n i=1

The shape functions for a linear triangular element are expressed as 1 [(x2 y3 − x3 y2 ) + (y2 − y3 )x + (x3 − x2 )y] 2A 1 H2 (x, y) = [(x3 y1 − x1 y3 ) + (y3 − y1 )x + (x1 − x3 )y] 2A H1 (x, y) =

3.1 Finite Element Method

H1 (x, y) =

41

1 [(x1 y2 − x2 y1 ) + (y1 − y2 )x + (x2 − x1 )y] 2A

(3.37)

where (xi , yi ) is the coordinate of the three nodal points of an element with i = 1, 2, 3, and the nodes are numbered in the counter-clockwise direction starting from any corner node. A is the area of the triangular element. For the quadrilateral element, isoparametric elements are commonly used [1–6]. The bending strains are ⎧ ⎫ ⎧ ∂u ⎨ εx ⎬ ⎪ ⎨ ∂x ∂v {εb } = ε y = ∂y ⎩ ⎭ ⎪ ⎩ ∂u + γx y ∂y

⎫ ⎪ ⎬ ∂v ∂x

⎪ ⎭

(3.38)

and the shear strains are  {εs } =

γ yz γx z



 =

∂v ∂z ∂u ∂z

+ +

∂w ∂y ∂w ∂x

 (3.39)

Substitution of Eqs. (3.32) through (3.36) into Eqs. (3.38) and (3.39) results in the following expression. {εb } = [Bb ]{Ue }

(3.40)

{εs } = [Bs ]{Ue }

(3.41)

where ⎤ 0 0 −z ∂∂Hx2 0 0 −z ∂∂Hx3 0 0 −z ∂∂Hx1 ⎢ ∂H ∂H ∂H ⎥ [Bb ] = ⎣ 0 0 −z ∂ y1 0 0 −z ∂ y2 0 0 −z ∂ y3 ⎦ ∂ H1 ∂ H1 ∂ H2 ∂ H2 ∂ H3 0 −z ∂ y −z ∂ x 0 −z ∂ y −z ∂ x 0 −z ∂ y −z ∂∂Hx3  ∂ H1 0 −H1 ∂∂Hy2 0 −H2 ∂∂Hy3 0 −H3 [Bs ] = ∂∂Hy1 −H1 0 ∂∂Hx2 −H2 0 ∂∂Hx3 −H3 0 ∂x ⎡



T {Ue } = w1 (θx )1 (θ y )1 w2 (θx )2 (θ y )2 w3 (θx )3 (θ y )3

(3.42)

(3.43) (3.44)

Applying the above expressions to Eq. (3.31) and the minimum total potential energy principle gives the following stiffness matrix  [K b ] =

 [Bb ]T [E b ][Bb ]dΩ + κ

Ωe

[Bs ]T [E s ][Bs ]dΩ Ωe

(3.45)

42

3 Numerical Modeling of FSI under Dynamic Loading

The expression in Eq. (3.45) holds for any type of plate bending element such as whether the element has a triangular or quadrilateral shape with any number of nodes per element. The mass matrix for the plate bending can be computed as  ρh[N ]T [N ]dΩ

[Me ] =

(3.46)

Ωe

where ρ and h are the mass density and thickness of the plate element, and

[N ] = H1 0 0 H2 0 0 H3 0 0

(3.47)

for the three node triangular element, This expression for the mass matrix does not include the rotatory inertia effect associated with the slope degrees of freedom. If that is included, the mass matrix will be modified. However, that mass effect is very small compared to the linear mass terms. Once the mass and stiffness matrices are computed for each element, the same time integration scheme as used for the beam bending problem can be applied.

3.1.3 Acoustic Wave Equation The linear acoustic wave equation is given as below: 1 ∂2 p = ∇2 p c2 ∂t 2

(3.48)

where c is the speed of sound, p is the acoustic pressure, and ∇ is the Laplace operator. Applying the weighted residual formulation to Eq. (3.48) results in  R= Ω



 1 ∂2 p 2 w 2 2 − ∇ p dΩ = 0 c ∂t

(3.49)

where w is the test function and  is the acoustic domain. Applying integrations by parts to the second term of the above equation yields  R= Ω

1 ∂2 p w dΩ + c2 ∂t 2



 ∇w · ∇ pdΩ −

Ω

w Γ

∂p dΓ = 0 ∂n

(3.50)

in which  is the boundary of the domain, and n is the unit normal vector to the boundary of the domain, which is assumed positive in the outward direction.

3.1 Finite Element Method

43

Discretization of Eq. (3.50) into each finite element domain and applying proper shape functions [H] to each element produces the following matrix equation. N

N

[Me ]{ p¨e } +

e=1

[K e ]{ pe } =

e=1

N

{Fe }

(3.51)

e=1

where 

T 1 [H ] [H ]dΩ c2

(3.52)

[∇ H ]T [∇ H ]dΩ

(3.53)

[Me ] = Ωe

 [K e ] =

Ωe



{Fe } =

[H ]T Γe

∂p pd x ∂n

(3.54)

Equation (3.50) is solved using any time integration technique based on the finite difference technique as discussed in previous sections. If the boundary is insulated, the boundary integral term becomes zero.

3.2 Cellular Automata The Cellular Automata (CA) uses rules that repeat themselves from cell to cell. For example, let us consider a rectangular shape of domain which is divided into cells by parallel vertical and horizontal lines as sketched in Fig. 3.4. Every cell has its own address by (i, j) where i and j are the numbered sequentially from the left bottom corner cell which is denoted by (1, 1). Each cell is assigned either 0 or 1 randomly to begin with. Then except for the cells along the boundary, every cell has neighboring cells, which are located at the east, west, north, and south sides of the cell. One simple rule that may be applied is the following. If the sum of the numbers of the four neighboring cells are odd, 0 is assigned to the cell. Otherwise, 1 is assigned Fig. 3.4 Enclosure box at the bottom side of a structure

Cell (i,j)

44

3 Numerical Modeling of FSI under Dynamic Loading

Fig. 3.5 Node representing a cell in a CA domain

Cell (i,j)

Node (i,j)

to the cell. This rule can be applied to every cell repeatedly. Of course, this rule cannot be applied to the cells located along the boundaries because they do not have complete neighbors. Boundary conditions are provided to those cells. More specific explanation will be provided later in this section. The major point of CA is to find the proper rules which can describe the physics under consideration. Because most of other numerical techniques use grid points or nodal point rather than cells, when CA is coupled with other numerical technique, it is inconvenient to match a grid point to a cell. As a result, a representative node is considered for each cell as illustrated in Fig. 3.5. From now on, grids or nodes may be used for describing the CA formulation. In this section, CA formulations are provided for the wave equations and the structural analysis of beams and plates like the previous section.

3.2.1 Acoustic Wave Equation The linear acoustic wave equation was given in Eq. (3.48). For the one-dimensional domain, the wave propagation is modelled using the following rule. = u te + u tw − u tc u t+t c

(3.55)

where subscripts c, e, and w indicate the center node which is under consideration, and its east and west neighbor nodes. Superscript t is time and t is the time step size. The time step size depends on the speed of sound and the size between the grid points. Therefore, t =

x c

(3.56)

for a one-dimensional problem. Here x is the grid spacing and c is the speed of sound. Let all nodes be numbered sequentially starting from 1 from the leftmost node. Equation (3.55) is applied to every odd node from left to right to complete a single time step increment. The next iteration is to apply Eq. (3.55) to every even node for another time step increment. These processes repeat themselves until the termination

3.2 Cellular Automata

45

time. Because the boundary nodes do not have either its east or west node depending on its location, boundary conditions are applied to address the problem. A fictitious node is introduced next to the boundary. For example, u 0 is introduced to the west side of the first node which is the leftmost point. Then, a specific value is assigned to the fictitious nodal value u 0 depending on the boundary condition. Figure 3.6 shows the assigned values for u 0 for some common boundary conditions. The same concept is applicable for both the first and the last nodes. Furthermore, this can be applicable to 2-D or 3-D problems. For the partially reflecting boundary condition as shown in Fig. 3.6e, the value of α determines what fraction is reflected. α = 0 is for no reflection while α = 1 is for full reflection. The rule for the 2-D is given below: = u t+t c

u te + u tw + u tn + u ts − 2u tc 2

(3.57)

The rule for the 2-D application used four neighbor nodes. Likewise, the rule for 3-D is given below u t+t = c

u te + u tw + u tn + u ts + u tf + u tb − 3u tc 3

(3.58)

This rule used six neighbors including front and back sides. The time step size is x t = √ c n dim

(3.59)

with an assumption of x = y = z and n dim is the number of the problem dimension. Like the one-dimensional problem, the rules for the 2-D and 3-D cases are also applied every other node. For example, the 2-D cells are denoted by black and while cells as shown in Fig. 3.7. Then, Eq. (5.57) is applied to the white cells for the entire domain, which is followed by the back cells of the whole domain. Another way to develop the rule was using the finite difference technique for the differential equation. The governing equation, Eq. (3.48) is replaced by the following set of equations: p¨i,t j = ci,2 j

!

" "  ! t t t t 2 t t 2 pi+1, j + pi−1, j − 2 pi, j /(x) + pi, j+1 + pi, j−1 − 2 pi, j /(y)

= p˙ i,t j + p¨i,t j (t) p˙ i.t+t j = pi,t j + p˙ i,t+t pi,t+t j j (t)

(3.60)

This set of equation is applied to every node sequentially. This is not every other node like the previous scheme. The boundary conditions to this set of equations are the same as illustrated in Fig. 3.6.

46

3 Numerical Modeling of FSI under Dynamic Loading

Fig. 3.6 Different boundary condition for CA technique

3.2 Cellular Automata

47

Fig. 3.7 2-D domain with black and white cells

3.2.2 Beam Bending The governing equations for bending of a straight beam are given as below: ∂ 2w ∂x2 2 ∂ w ∂2 M m 2 =− 2 +p ∂t ∂x M = EI

(3.61)

in which M is the bending moment, EI is the flexural rigidity of the beam, w is the transverse deflection, p is the applied load per unit length, and m is the mass density per unit length. The spatial variable is x, and t is the time variable. Applying the finite difference technique to the set of equations yields ! t " t + wi−1 − 2wit /(x)2 Mit = (E I )i wi+1 " 

! t t w¨ it = −Mi+1 − Mi−1 + 2Mit /(x)2 + pit /m i w˙ it+t = w˙ it + w¨ it (t) wit+t = wit + w˙ it+t (t)

(3.62)

where t is the time step size, x is the cell size or the grid spacing, and the superimposed dot indicates the derivative with respect to time. Superscript t denotes time, and subscript i indicates the i-th cell. This set of equations are applied to every node without skipping. For parallel computations, this set of operations can be conducted simultaneously because no computation requires any updated value. Like the wave equation, a fictitious cell may be required to apply boundary conditions. Considering the leftmost cell which is denoted by 1, the fictitious cell left to the cell is called the zeroth cell. If the cell is simply supported, the boundary conditions are u 1 = 0 and M1 = 0. These boundary conditions can be applied to the leftmost

48

3 Numerical Modeling of FSI under Dynamic Loading

cell directly without any fictitious cell. On the other hand, the clamped boundary 1 = 0. This requires a fictitious cell u o for the zero conditions are u 1 = 0 and du dx slope boundary, which is expressed as u 0 = u 2 .

3.2.3 Plate Bending The governing equations for plate bending are expressed as M = D∇ 2 w m

∂ 2w = −∇ 2 M + q ∂t 2

(3.63)

where M = (Mx + M y )/(1 + ν) with Mx and M y for the bending moment per unit length about the x and y axes, respectively,ν is Poisson’s ratio, and w is the Eh 3 transverses displacement. The plate rigidity is D = 12(1−ν 2 ) with elastic modulus E and the plate thickness h. In addition, m is the mass density per unit area, and q is the applied pressure to the plate. The finite difference scheme for Eq. (3.63) results in the following set of equations.

! t " t t 2 Mi,t j = Di, j wi+1, j + wi−1, j − 2wi, j /(x) ! t "  + wi, j+1 + wi,t j−1 − 2wi,t j /(y)2

! " t t t 2 w¨ i,t j = −Mi+1, j − Mi−1, j + 2Mi, j /(x) "  ! + −Mi,t j+1 − Mi,t j−1 + 2Mi,t j /(y)2 + qi, j /m i, j w˙ i.t+t = w˙ i,t j + w¨ i,t j (t) j = wi,t j + w˙ i,t+t wi,t+t j j (t)

(3.64)

Here, subscripts i and j indicates the cell located at the grid (i,j). This set of equations are applied to every cell without skipping. Applying the boundary condition to the plate bending problem is the same as that for the beam bending problem. The simply supported boundary condition does not require any fictitious cell while the clamped boundary condition requires a fictitious cell next to the problem domain.

3.3 Lattice Boltzmann Method The Lattice Boltzmann Method (LBM) has been applied to both acoustic wave problems and the Navier-Stokes equation for fluid mechanics. The LBM uses a collection

3.3 Lattice Boltzmann Method

49

of imaginary particles. The particles move along lattice points and need to satisfy the conservation law as necessary depending on the governing equation. The general form for LBM is given below " 1 ! eq − → → → → f i (→ r +− e i t, t + t) − f i (− r , t) = r , t) − f i (− r , t) f i (− τ

(3.65)

→ → r , t) is the number of particles at lattice site − r and time t, and the Here f i (− particles move along the neighboring lattices as described by a specified instruction depending on the governing equation. The local particle velocity at each lattice point eq → → r , t) is the local equilibrium distribution of particles, and is − e i . Furthermore f i (− τ is the relaxation parameter to reach the equilibrium solution. The LBM consists of two phases. One is the collision phase and the other is the distribution phase. Each of the formulation is presented below.

3.3.1 Acoustic Wave Equation Let’s consider the 2-D wave equation to present the formulation using the LBM. In particular, the lattice scheme called D2Q5 is used for 2-D applications. Here, D2 means two-dimensional and Q5 indicates five lattice points are considered in the calculation, which are the center lattice under consideration, east, west, north, and south lattice points in terms of the center lattice. For the wave equation using the twodimensional square lattice with D2Q5, the local equilibrium solution is expressed as   eq  2c2 f o = 1 − e2s ϕ if i = 0 (3.66)  c2 eq f i = 2es2 ϕ + 21 eei ·2ζ if i = 0 These equilibrium equations are used for the general LBM expression, Eq. (3.65), and the additional variables are explained below. ϕ=



fi

(3.67)

i

− → − ei fi → ξ =

(3.68)

i

and cs is the speed of wave propagation. The local velocity distribution at each lattice point is given as follows − → e =



(0, 0) i =0 (cos{(i − 1)π/2}, sin{(i − 1)π/2}) i = 1, 2, 3, 4

(3.69)

50

3 Numerical Modeling of FSI under Dynamic Loading

where i = 0 denotes the center lattice; and 1 through 4 indicate the east, north, west, and south lattice direction. As τ = 1/2 is selected, Eq. (3.65) is reduced to 

f o ( r , t + t) = 2μ n n−1 r , t) if i = 0 2 ϕ − f o ( μ ˜ r + t ei , t + t) = 2n 2 ϕ − f i ( r , t) if i = 0 f i ( 2

(3.70)

and this set of equation is equal to the acoustic wave equation as given in Eq. (3.48) for the 2-D case such as  2  ∂ 2ϕ ∂ 2ϕ 2 ∂ ϕ (3.71) = cs + 2 ∂t 2 ∂x2 ∂y In Eq (3.70), μ is called the attenuation factor, and n (≥1) is called the refraction index. While μ = 0 means perfect reflection, μ = 1 means perfect transmission. Any value μ between the two extreme cases means that the wave is partially absorbed. In addition, the tilde in Eq. (3.70) means the direction of the local velocity is reversed. The presentation above can be extended for 3-D acoustic applications.

3.3.2 Fluid Mechanics This section presents the LBM for the 2-D Navier-Stokes equation because the 3-D formulation can be easily extended from the 2-D formulation. The most popular lattice scheme is D2Q9. Because it is Q9, the scheme involves nine lattice points which is the center one under consideration and eight neighboring lattice points. Those eight points are the neighboring lattice points in the north, south, east, west, northeast, northwest, southeast, and southwest orientations, respectively. The local velocity at a given lattice point is given as below  e = c

0 1 0 −1 0 1 −1 −1 1 0 0 1 0 −1 1 1 −1 −1

 (3.72)

where the sequence of the local velocity vectors are the center, east, north, west, south, northeast, northwest, southwest, and southeast in that order. In addition, c is the lattice speed which is 1 for the normalized problem. The relaxation parameter τ is related to the kinematic viscosity ν of the fluid like the following: ν=

  1 1 τ− 6 2

(3.73)

The local equilibrium distribution of f i is expressed as below for the Navier-Stokes equation:

3.3 Lattice Boltzmann Method

51

 eq fi

= ρwi 1 +

9( u · ei )2 u · u 3 u · ei + − 2 4 c 2c 2c2

(3.74)

where ρ is the macroscopic density of the fluid, u is the fluid velocity vector, ei is the local velocity vector of the i-th component of Eq. (3.72), and wi is the weight distribution parameters among the nine lattice orientations as below ⎡ wi = ⎣

4 9 1 9 1 36

i =0 i = 2, 3, 4, 5 i = 6, 7, 8, 9

(3.75)

The density is computed as ρ=



fi

(3.76)

i

and the fluid velocity is calculated as u =

1 f i ei ρ i

(3.77)

One of the common boundary conditions in fluid mechanics is the no-slip boundary, where the flow velocity is zero at the wall. This boundary condition is applied using the “bounce-back” technique. In this technique, all values of f i are replaced by those of f i in the opposite direction. The procedure to apply the LBM is provided as below. 1. For each lattice point, compute the macroscopic properties of the fluid such as density and viscosity. 2. Apply boundary condition. eq 3. Compute the equilibrium distribution f i . 4. Relaxed to equilibriums. 5. Conduct the re-distribution of f i .

3.4 Examples of Coupled Fluid and Structure Domains Coupling of the fluid and structural domains is conducted at the interface of the two domains by applying continuity and equilibrium conditions at the interface. The velocity should be the same between the fluid and structure at the interface. In addition, the traction should be in equilibrated at the interface. If the viscous force is neglected, the pressure should be the same between the fluid and the structure at the interface. If cavitation occurs at the interface, then such continuity and equilibrium

52

3 Numerical Modeling of FSI under Dynamic Loading

do not apply. This section presents a couple of example problems. More numerical examples of coupled FSI problems are shown in other chapters.

3.4.1 Beam Supported by Fluid This example was a FSI problem between a beam and an acoustic wave domain as sketched in Fig. 3.8. The acoustic domain was confined all three boundaries by rigid walls while the top boundary was in contact with a beam which was assumed to be clamped at both ends. The beam was 1 m long with the beam rigidity 10,000 N m2 and the linear mass density 20 kg/m. A uniform load intensity of magnitude 10 kN/m was applied to the beam in the sinusoidal form. The acoustic domain had the material properties equivalent to water with the speed of sound 1500 m/s and the mass density 1000 kg/m3 . The water domain was assumed L = 1 m with varying H of 1 m, 2 m, and 3 m, respectively. Both the beam and acoustic domains were modelled using the CA technique. The sound wave propagated from the vibrating beam and reflected from the rigid walls, which also influenced the vibration of the beam. Figure 3.9 shows the plots the deflection of the beam at its center for three different domain sizes. The results indicated the shorter length of the domain size yielded a smaller deflection of the beam because of faster return of the reflected waves. Figure 3.10 compares the pressure time-history at the location 0.125 m below from the beam center. The results showed Fig. 3.8 Two-dimensional example of beam and acoustic domain

Beam

f(t)

Rigid

Rigid Water Domain

Rigid

L

H

3.4 Examples of Coupled Fluid and Structure Domains

53

10-5

4

FDCA w/ H=1 m FDCA w/ H=2 m FDCA w/ H=3 m

2

Displacement (m)

0

-2

-4

-6

-8

0

0.01

0.02

0.03

0.04

0.05

Time (sec) Fig. 3.9 Plot of the center displacement of beam with reflected boundaries of different acoustic domain size H in Fig. 3.8 104

2.5

FDCA w/ H=1m FDCA w/ H=2m FDCA w/ H=3m

2

Pressure (Pa)

1.5

1

0.5

0

-0.5 0

0.005

0.01

0.015

0.02

Time (sec)

Fig. 3.10 Plot of acoustic pressure at 0.125 m below the beam center with reflected boundaries of different acoustic domain size H in Fig. 3.8

54

3 Numerical Modeling of FSI under Dynamic Loading

Plate

Water

Fig. 3.11 Plate supported by water

the return of the reflected pressure wave at different times. The time delay was proportional to the height H of the domain.

3.4.2 Numerical Study of Impact with FSI A numerical study was conducted for a plate with an impact load while the plate was supported by water as sketched in Fig. 3.11. The plate was modelled using FEM while the water domain was modelled using CA. The plate was 30.48 cm × 30.48 cm made of E-glass woven fabric composites. The thickness of the plate was 3.5 mm, and the plate was clamped along all edges. An impact force was applied to the center by dropping a 10.8 kg mass. This numerical study was to simulate the experiment which was described in Chap. 2 [11]. The impact force was applied to the numerical model from the measured data during the experiment. The impact force time-history is shown in Fig. 3.12. Because the experiment measured strains using strain gages, the comparison was made for the strain at the top surface of the plate. The gage was located near to the center of the plate because impact struck the plate at the center. Figure 3.13 shows the comparison of the strain between the experimental and numerical studies. Even though there was some difference between the two results, the numerical modeling was acceptable considering all unknowns. For example, some potential differences between the experimental and numerical studies are listed below. The experiment did not provide complete constrains for the clamped boundary. Non-reflected boundary

3.4 Examples of Coupled Fluid and Structure Domains

55

1000

800

Force (N)

600

400

200

0

-200

0

0.005

0.01

0.015

Time (s)

Fig. 3.12 Applied impact force to the E-glass composite plate at the center 0.5

Strain (x1000)

calculated measured

0

-0.5

-1 0

0.002

0.004

0.006

0.008

0.01

0.012

Time (s)

Fig. 3.13 Comparison of numerical and experimental strains measured near the center of the clamped plate

conditions were assumed in the model. The material property of the composite was not fully uniform for the test sample. A local delamination was assumed around the center of the E-glass composite plate. The delamination was modelled using multilayer plates. In other words, the plate was modelled using three layers including a very thin resin layer where delamination was introduced. The composite plate was subjected to a concentrated force

56

3 Numerical Modeling of FSI under Dynamic Loading 10

5

-3

4

3

Strain

wet damaged wet undamaged 2

1

0

0

0.5

1

1.5

Time (s)

2

10

2.5 -3

Fig. 3.14 Strain of clamped E-glass plate with and without delamination at center

of 1 kN at the center. The results with and without delamination were compared as shown in Fig. 3.14. The strain at the center of the composite plate was compared between the two cases. The local delamination resulted in a slight change in the strain response.

3.5 Summary This chapter presented different numerical techniques which were used to study FSI of composite structures. Those were the finite element method, cellular automata, and lattice Boltzmann method. Among those, the finite element method and cellular automata were selected to model structures. For the acoustic domain, both the cellular automata and lattice Boltzmann method were utilized. Finally, the lattice Boltzmann method was also used to solve the Navier-Stokes equation of fluid mechanics. Some example problems for fluid-structure interaction were presented, which included the impact loading to a composite plate supported by water. This impact loading model was compared to the experimental result. By considering all the potential differences between the numerical and experimental models, the numerical results agreed with the experimental data. More numerical results were presented in the following chapters when numerical studies were conducted to complement the experimental work.

References

57

References 1. Akin JE (1986) Finite element analysis for undergraduates. Academic Press, London 2. Kwon YW, Bang H (2000) The finite element method using matlab, 2nd edn. CRC Press, Boca Raton 3. Kwon YW (2016) Multiphysics and multiscale modeling: techniques and application. CRC Press, Boca Raton 4. Zienkiewicz OC, Taylor RL (1991) The finite element method, 4th edn. McGraw-Hill, London 5. Bathe K-J (1996) Finite element procedures. Prentice-Hall, Upper Saddle River 6. Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, ENglwwod Cliffs 7. Patankar SV (1980) Numerical heat transfer and fluid flow. Taylor & Francis 8. Ferziger JH, Peric M (1996) Computational methods for fluid dynamics. Springer, Berlin 9. Tu J, Yeoh G, Liu C (2013) Computational fluid dynamics: a practical approach, 2nd edn. Elsevier, Waltham 10. Kwon YW, Hosoglu S (2008) Application of Lattice Boltzmann method, finite element method, and cellular automata and their coupling to wave propagation problems. Comput Struct 86:663– 670 11. Craugh LE, Kwon YW (2013) Coupled finite element and cellular automata methods for analysis of composite structures with fluid-structure interaction. Compos Struct 102:124–137 12. Kwon YW (2017) Finite difference based cellular automaton technique for structural and fluid-structure interaction applications. ASME J Pressure Vessel Technol 139:041301 13. Blair SR, Kwon YW (2015) Modeling of fluid-structure interaction using lattice boltzmann and finite element methods. ASME J Pressure Vessel Technol 137:021302 14. Kwon YW (2008) Coupling of Lattice Boltzmann and finite element methods for fluid-structure interaction application. ASME J Pressure Vessel Technol 130(1):011302 15. Kwon YW, Jo JC (2008) 3-D modeling of fluid-structure interaction with external flow using coupled LBM and FEM. ASME J Pressure Vessel Technol 130(2):021301 16. Kwon YW, Jo JC (2009) Development of weighted residual based lattice Boltzmann techniques for fluid-structure interaction application. ASME J Pressure Vessel Technol 131(3):031304 17. Wolf-Gladrow DA (2000) Lattics-gas cellular automata and lattice Boltzmann modes: introduction. Springer, Berlin 18. Succi S (2001) The Lattice Boltzmann equation: for fluid dynamics and beyond. Clarendon Press, Oxford 19. Guo Z, Shu C (2013) Lattice Boltzmann method and its applications in engineering. World Scientific, Singapore

Chapter 4

FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures

When a structure moves dynamically inside a fluid medium, there is an added mass effect. The added mass influences the vibrational frequency of the structure. This effect is significant for polymer composite structures in water. This chapter investigates the effect of FSI on vibrational frequencies, mode shapes, and modal curvatures of composite structures. Both experimental and numerical studies were undertaken.

4.1 Description of Experiments It was shown in Table 2.1 that FSI significantly reduced the vibrational frequency of the structure. The vibrational frequencies were computed from the strain gage measurement. In order to further study the effect on the frequency, mode shape and modal curvature resulting from FSI, additional experiments were conducted. In order to evaluate the effect of FSI, the same composite structures were tested while they are in air or water, respectively [1]. Two types of experiments were conducted for this study. The first experiment was undertaken for cantilever beams using the Digital Image Correlation (DIC) technique to measure the dynamic motions of the cantilever beams. The second experiment was performed for free-free beams using the experimental modal analysis technique. Each test was conducted while the beam specimens were in air or in water, respectively, to compare their dynamic characteristics. Composite beams were made of plain weave woven fabrics made of glass fibers using the vacuum assisted resin transfer molding technique. One set of composite beams had the lamination angle 0°/90° which were repeated depending on the number of layers. The other set of composite beams had +45°/−45° layers. All the samples had symmetric layers with respect to their mid-planes. The dimensions of the test samples are provided in Table 4.1, which were the test samples for both the DIC

© Springer Nature Switzerland AG 2020 Y. W. Kwon, Fluid-Structure Interaction of Composite Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-57638-7_4

59

60

4 FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures

Table 4.1 Geometric dimensions of composite samples # layers

Orientation

length in

Width

Thickness

mm

in

mm

in

mm

222.25

1

25.4

0.048

1.219

279.4

1

25.4

0.105

2.667

DIC test composite samples 6

0°/90°

6

±45°

8.75

Modal analysis test composite samples 16

0°/90°

16

±45°

11

Fig. 4.1 Speckles on composite beams

tests and the modal analysis tests. An aluminum alloy beam was also tested, and the results were compared to those of the composite beams. To measure displacements of the cantilever beams accurately using DIC, a good speckle pattern must be applied to the surface of the test specimen. The spray paint technique was used for this study. In order to have a good contrast between the speckles and the surface of the sample, a proper color choice is necessary. To achieve this, a very thin base coat of flat white paint was applied to the surface of the composite samples. Once the coating was dried, a thin black mist was sprayed on the white surface of the composite samples to create speckle patterns. Figure 4.1 shows samples with speckles. The DIC technique provides both the whole field deformation as well as time history of the deformation fields. Displacements and strains can be also plotted at some selected locations as a function of time. From the test results of the transverse displacements, vibrational frequencies and the damping factor were determined. A force was applied to the free tip of the cantilever beam to generate an initial deflection of the beam. Then, the force was released to result in free vibration of the beam. The initial force was applied by attaching a very thin fishing line at the tip of the cantilever structure and pulling the tip using the fishing line. To make the initial tip deflection consistent, a reference rigid structure made of an I-beam was installed behind the cantilever beam, and the tip was deflection until it just came in contact with the reference I-beam. Then the tip was released for free vibration. Figure 4.2 gives a sketch of the DIC test set-up. Two cameras were utilized to measure the out-of-plane displacement which was the most important parameter of the beam deflection. The stereo angle of the cameras is based on the type of lens

4.1 Description of Experiments

61

Fig. 4.2 Sketch of DIC test arrangement

Fig. 4.3 Stereo angle of two cameras for DIC tests

being used. Because 35 mm lenses were used for this experiment, the stereo angle was determined between 20° and 60°. The test set-up used the stereo angle 25° as sketched in Fig. 4.3. In order to conduct the free vibration analysis in water, a water tank was constructed using transparent 12.7 mm thick Plexiglas so that the light can penetrate for the DIC technique. The water tank had the inner dimensions of 584.2 × 279.4 × 457.2 mm. Figure 4.4 shows the DIC set-up for the beam vibration inside a water tank. The DIC application to the FSI example posed a challenge because of the different optical properties of air, water, and Plexiglas. First of all, the water medium magnifies the image captured by the DIC system. The magnification factor M of the image can be calculated using the following equation [2].  M =m

[1 + (d + R)/D] [1 + m(d + R)/D]

 (4.1)

62

4 FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures

Fig. 4.4 DIC set-up for beam vibration inside a water tank

where R = 0.8 is the radius of curvature of the camera lens, m = 1.33 is the index of refraction for water, D is the distance in centimeters from the beam specimen to Plexiglas wall, and d is the distance in centimeters from the camera to the Plexiglas. The stereo angle was also affected by the refraction of Plexiglas and water, and it was determined using Snell’s law that gives the refraction angle from one medium to another as given below n 1 sin(θ 1 ) = n 2 sin(θ 2 )

(4.2)

in which n is the index of refraction and θ is the angle of the light in each of the two media. In order to maintain the stereo angle 25° for the image, the actual stereo angle of the two cameras should be 33.4° as explained in Fig. 4.5 resulting from the refractions of the light from water to Plexiglas and Plexiglas to air. The flexural rigidity EI of composite beams were determined from three-point bending tests by measuring the applied force and deflection and using the following equation EI =

Fig. 4.5 Refraction of light from water to air by way of Plexiglas

P L3 48δ

(4.3)

4.1 Description of Experiments Table 4.2 Beam rigidity of composite specimens

63 EI (Nm2 )

Sample 6-layer 16-layer

0°/90°

0.0738

±45°

0.0594

0°/90°

0.2666

±45°

0.1529

where P and δ are the applied force and measured deflection at the center; and L is the distance between the two supports. The measured bending rigidity of four composite beams are listed in Table 4.2. The experimental modal analysis was undertaken for free-free beams. In order to represent free-free boundary conditions, beam samples were supported by a thin flexible wire at both ends. Ten accelerometers were attached to every beam to be tested in a uniform spacing from the centerline of the beam. An impact hammer was used to strike the beam sample at the center. The schematic of the test set-up is shown in Fig. 4.6. As an impact force was applied to the specimen, the attached accelerometers collected the dynamic responses. Then, the data were analyzed to determine the natural frequencies and mode shapes. If more sensors were attached to a specimen, the analysis would be more accurate. However, too many sensors may change the mass and stiffness of the original structure being tested. Therefore, it is necessary to use an optimum number of sensors. When the test was conducted in water, the accelerometers were waterproofed before being inserted into the water tank. For waterproofing, two light coats of barrier A was applied to the accelerometers with cables attached, and let them settle for three days. This was followed by two light coats of RTV with two days of set time between the two coatings. Because the impact hammer is not waterproofed, it cannot be used inside the water tank when the beam specimens are submerged in water. As a result, a steel rod was used to be rested on the beam specimens, and the other end, which is above the

Fig. 4.6 Schematic of modal analysis test set-up

64

4 FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures

water line, was impacted using the impact hammer. After the impact, the rod was removed immediately prior to the rebound of the beam. In order to check the validity of this technique, the same beam was tested with and without the steel rod while the beam was in air. The two responses were compared. Their results were identical suggesting that the usage of the steel rod did not alter the test results.

4.2 Experimental Results Five composite beam specimens were tested for each layer orientation, resulting in ten for the total specimens tested using the DIC technique. First of all, the DIC test result of the specimen in water was compared to that in air when the composite cantilever beam had the same static displacement in air and water, respectively. Both displacements were the same. This provided the validity of the test set-up discussed in the prior section as all the refractions were considered for the correct measurement. The fringe of the transverse displacement of the cantilever beam is shown in Fig. 4.7. The vibrational frequency was measured from the tip deflection, and the damping ratio ζ was also computed using the following equation. 1 n−1

ζ = 4π 2

+

  ln



1 n−1

x1 xn

ln

 2

(4.4)

x1 xn

where x1 and xn are the amplitudes of the deflection at the tip of the beam of the first and the n-th peaks. Figure 4.8 shows a typical vibrational plot of a vibrational beam. The damped natural frequency and undamped natural frequency are related as shown below.

Fig. 4.7 Fringe of deflection of a cantilever beam

4.2 Experimental Results

65

10 First peak

Displacement (mm)

5

Fourth peak

0

-5

-10

-15

-20

0

5

10

20

15

25

30

Time(x 200ms)

Fig. 4.8 Plot of vibrational motion to compute the damping ratio

fd = fn 1 − ζ 2

(4.5)

where f d is the damped natural frequency, and f n is the undamped natural frequency. Once the measured damped natural frequency is obtained, Eq. (4.5) determines the undamped natural frequency, which was compared to the theoretical natural frequency. Table 4.3 provides the summary of the measured natural frequencies in air, which were compared to the theoretical values which were determined from the following equation. Table 4.3 Comparison of natural frequencies of cantilever beam in air Theory

Experiment

Sample

Layers

fn (Hz)

fd (Hz)

ζ

fn (Hz)

Error (%)

I

0°/90°

13.49

13.16

0.1856

13.39

0.74

±45°

11.61

12.16

0.1102

12.24

5.40

0°/90°

13.59

13.29

0.1569

13.46

0.95

±45°

11.44

11.08

0.2938

11.60

1.36

III

0°/90°

13.57

13.26

0.1640

13.44

0.98

±45°

11.62

11.26

0.2782

11.73

0.92

IV

0°/90°

13.53

13.12

0.2310

13.48

0.34

±45°

11.68

11.39

0.2431

11.74

0.56

0°/90°

13.46

13.27

0.1192

13.37

0.67

±45°

11.54

11.39

0.2134

11.66

1.05

II

V

66

4 FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures

(βn L)2 fn = 2π

EI ρ AL 4

(4.6)

where EI is the beam rigidity, ρ is the mass density, L and A are the length and the cross-sectional area of the beam, and βn L is the parameter which is determined from the values in Table 4.4. Only the first natural frequencies were compared in Table 4.3. Except for one result, all measured frequencies were within approximately 1% error when compared to the theoretical values. As a result, the measured natural frequencies were in very good agreement with the theory. The average natural frequencies of the 0°/90° layered beams and +45o /−45° layered beams were 13.43 Hz and 11.68 Hz, respectively. Table 4.5 lists the frequencies measure when the cantilever beams were submerged in water. The average frequency for the 0°/90° layered beams was 4.92 Hz while that for the +45°/−45° layered beams was 3.59 Hz. Consistently, +45°/−45° layered beams showed lower frequencies because of the lower beam rigidity. The natural frequency was significantly reduced in water because of the added mass effect. The ratio of the natural frequency in water to that in air was approximately 1/3. Comparing the damping factors between Tables 4.3 and 4.5 also shows an increase in damping factor in overall resulting from FSI in water even though a couple of specimens in water showed reduced damping ratios. The cause of this was not known but did not represent the overall behavior. Table 4.4 Values for βn L for natural frequencies of the cantilever beam Mode

βn L

1

1.875

2

4.694

3

7.855

Table 4.5 Natural frequencies of cantilever beam in water Sample #

Layer

fd (Hz)

ζ

fn (Hz)

I

0°/90°

3.85

0.6502

5.07

±45°

3.68

0.0410

3.69

II

0°/90°

4.09

0.6884

5.63

±45°

3.37

0.0399

3.38

0°/90°

3.87

0.5914

4.80

±45°

3.35

0.0475

3.35

IV

0°/90°

3.85

0.5913

4.77

±45°

3.43

0.4172

3.77

V

0°/90°

3.97

0.4020

4.33

±45°

3.57

0.2980

3.74

III

4.2 Experimental Results

67

The next study examined the free vibrational motions of the cantilever beams in air and water, respectively. In other words, the deflections of the beams were compared while they vibrated in air or water, receptively. When a beam vibrates freely in air, the vibrational motion will be described as a linear combination of its mode shapes. When the vibrational motion contains some prominent natural frequencies, the mode shapes associated with those frequencies will make a major contribution to the vibrational motion. The first three theoretical mode shapes of the cantilever beam are sketched in Fig. 4.9. The initial deflection was generated by applying a load at the free tip of the cantilever beam, which makes the initial deflection very close to the first mode shape. Therefore, it was expected that the free vibrational mode of the cantilever beam would resemble the first mode shape. This expectation was correct as plotted in Fig. 4.10, which shows the deflection of a +45°/−45° layered cantilever beam at different times. The top plot of Fig. 4.19 shows early times and its bottom plot shows late times. The overall deformed shapes resembled that of the first mode shape. However, when the cantilever beam vibrated in water, the vibrational motion was very different. Figure 4.11 shows the vibrational motion of a +45°/−45° layered cantilever beam at different times in water. At early times, the vibrational motion was close to the first mode shape, but at late times higher mode shapes showed up in the vibrational motion. Figure 4.12 shows the three-dimensional plot of a deformation of the cantilever beam at a given instant of time, which also contains high modes of mode shapes. Because the effect of the added mass, i.e. the effect of FSI, was not uniform along the beam, that caused higher modes of vibrational motions. The effect of FSI is associated with the motion of the structure. The cantilever beam has nonuniform 2

Deflection

1

0

-1 Mode #1 Mode #2

-2

Mode #3

-3 0

0.2

0.4

0.6

0.8

Normalized distance

Fig. 4.9 Plot of first three theoretical mode shapes of a cantilever beam

1

68

4 FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures

Amplitude (mm)

5

0 -5 t = 1.2s

Initial Disp.

t = 1.4s

t = 1.6s

-10

-15

0

50

100

150

200

250

300

Distance from fixed end of beam (x 0.67414mm)

Amplitude (mm)

0 -0.05 -0.1 -0.15 t = 6s

-0.2

t = 8s

t = 12s

t = 14s

-0.25 0

50

100

150

200

250

300

Distance from fixed end of beam (x 0.67414mm)

Fig. 4.10 Plot of free vibrational motion of a cantilever beam in air

motion such as acceleration and velocity along the beam. This resulted in a nonuniform FSI effect along the beam. If the FSI effect were uniform, i.e., the added mass were uniformly applied to the beams in water, the characteristics of the vibrational motions would be the same in water even though the frequency was different. This test results demonstrated how complex the effect of FSI on the composite structure. The next results are from the experimental modal analysis. The test used the freefree boundary conditions, and the theoretical natural frequencies of such beam can be also computed from Eq. (4.6) using the βn L values listed in Table 4.6. The first three modes were studied using the modal tests. Before testing the composite beams, an aluminum alloy beam with free-free boundary conditions was tested using the modal analysis. Their first three natural frequencies were measured and compared to the theoretical values as shown in Table 4.7. The measure values agree very well with the theoretical values. The first three mode shapes of the aluminum beam are plotted in Fig. 4.13. The mode shapes in air match well with theoretical mode shapes as shown in Fig. 4.14. When the aluminum beam was submerged in water, its vibrational frequencies were varied. Table 4.8 shows the comparison of the frequencies in air and water. The frequencies were as measured without correction by damping factor. The aluminum beam showed a quite uniform reduction in its first three natural frequencies resulting

4.2 Experimental Results

69

Free Vibration Response Shape in Water (Sample III +/-45) 5

Amplitude (mm)

0 -5 -10 -15 t = 1s

Initial Disp.

t = 1.2s

-20 -25

0

50

100

150

200

250

Distance from fixed end of beam (x 0.71582mm)

Amplitude (mm)

0

-0.05

-0.1

-0.15 t = 6s

-0.2

0

50

t = 8s

100

t = 10s

150

t = 12s

200

250

300

Distance from fixed end of beam (x 0.71582mm)

Fig. 4.11 Plot of free vibrational motion of a cantilever beam in water

Fig. 4.12 Three-dimensional representation of the free vibration response shape at 2.4 s after initial displacement in water for a beam made of 0°/90° layers

70

4 FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures

Table 4.6 Values for βn L for natural frequencies of the free-free beam βn L

Mode 1

4.730

2

7.853

3

11.00

Table 4.7 Comparison of natural frequencies of aluminum beam with free-free boundary conditions in air Mode

Theory (Hz)

Experiment fd (Hz)

ζ

fn (Hz)

% error

1

144.7

142

0.0083

144.7

0.00

2

398.8

389

0.0092

398.8

0.00

3

781.7

767

0.0047

781.7

0.00

First Mode 0.5 In Air In Water 0

-0.5 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

0.7

0.8

0.9

1

Second Mode 0.5 In Air In Water 0

-0.5 0.1

0.2

0.3

0.4

0.5

0.6

Third Mode 0.5 In Air In Water 0

-0.5 0.1

0.2

0.3

0.4

0.5

0.6

Fig. 4.13 Plot of mode shapes of aluminum free-free beam

4.2 Experimental Results

71 Mode shapes of the Free-free beam

1 0.8 0.6

n

Mode shape X (x)

0.4 0.2 0 -0.2 -0.4 -0.6

Mode #1 Mode #2 Mode #3

-0.8 -1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/L

Fig. 4.14 Analytical mode shapes of free-free beam

Table 4.8 Comparison of damped natural frequencies of free-free aluminum beam

Mode

Air (Hz)

Water (Hz)

Ratio

1

142

98.6

0.694

2

389

274

0.704

3

767

538

0.701

from the effect of FSI. The vibrational frequencies in water were 70% of those in air. Figure 4.13 also compares the mode shapes of the aluminum beam which was in air or water, respectively. Both mode shapes were very close each other. The next results are for the carbon fiber composite beams. Table 4.9 compares the directly measured natural frequencies of the composite beam with free-free boundary conditions when the beam was in air or in water, respectively. The added mass reduced the natural frequencies by approximately 40%. When this reduction was compared to that of the aluminum beam, the composite beam had a greater added mass effect with a larger reduction in natural frequencies. The first three mode shapes of the carbon composite beam were plotted in Fig. 4.15 Table 4.9 Comparison of damped natural frequencies of free-free carbon fiber composite beam

Mode

Air (Hz)

1

103

Water (Hz) 60

Ratio 0.582

2

272

166

0.610

3

552

329

0.596

72

4 FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures First Mode 0.5 In Air In Water

0

-0.5 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Second Mode 0.5

0

In Air In Water

-0.5 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

Third Mode 0.5 In Air In Water 0

-0.5 0.1

0.2

0.3

0.4

0.5

0.6

Fig. 4.15 Plot of mode shapes of carbon composite free-free beam

when the beam was in air and water, respectively. The overall characteristics were close regardless of air or water. However, by comparing to the aluminum beam, the composite beam showed a greater difference between the two mode shapes in air or water. This difference was magnified when the modal curvatures were plotted. The modal curvatures were the second derivatives of the mode shape, and it is directly related to the bending strain. Modal curvatures were plotted in Figs. 4.16 and 4.17 for the aluminum and composite beams. The difference in the modal curvature while the beam was in air vs. water was generally larger for the composite beam than the aluminum beam. This suggested the effect of FSI could affect the bending strain of the beam.

4.3 Numerical Study

73 First Modal Curvature

0.2

In Air

0

In Water

-0.2 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.8

0.9

1

0.8

0.9

1

Second Modal Curvature 0.2

In Air In Water

0

-0.2 0.1

0.2

0.3

0.4

0.5

0.6

0.7

Third Modal Curvature 2 In Air In Water

1

0 -1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 4.16 Plot of modal curvatures of aluminum free-free beam

4.3 Numerical Study A numerical modal analysis was also conducted to simulate the experimental modal analysis [4]. The advantage of the numerical modal analysis is that there is no limitation in the number of sensors and their locations because the numerical model can compute all the variables such as displacement, velocity and acceleration at any location of the model with proper meshing. On the other hand, the disadvantage is that something like material damping cannot be described accurately. In this modal analysis, material damping was neglected. The structure to be analyzed was modeled using the finite element method. The fluid domain was modeled as an acoustic medium by neglecting the fluid flow and any viscosity effect. As a result, the acoustic wave equation was used for the fluid and the CA technique was applied to the acoustic wave equation. The interface between the fluid and structure was treated as discussed in Chap. 3. The numerical modal analysis program was developed using the MATLAB [3] program, and it was verified against theoretical solutions. To this end, beams were modeled and analyzed using the modal analysis program. Three different boundary

74

4 FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures First Modal Curvature 0.1 In Air 0

-0.1 0.1

In Water

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

0.7

0.8

0.9

1

Second Modal Curvature 1 In Air 0.5

In Water

0

0.1

0.2

0.3

0.4

0.5

0.6

Second Modal Curvature 2 in Air 1

In Water

0

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 4.17 Plot of modal curvatures of carbon composite free-free beam

conditions were considered; fixed-free like a cantilever beam, simply supported at both ends, and clamped at both ends. Their natural frequencies and mode shapes were computed using the numerical modal analysis, which used the transient responses at the nodal points when an impact load was applied to the structure like the experimental modal analysis. Then, the natural frequencies and mode shapes were determined from the transient response data. Eigenvalue analyses were also conducted numerically to find the natural frequencies and mode shapes. Therefore, three solutions were compared: theoretical solution, eigenvalue solution, and numerical modal analysis solution. No FSI was considered in these test cases. The beam studied was 1.0 m long, 0.05 m thick, and 0.1 m wide. It had the modulus of elasticity E = 20 × 109 Pa, density 2000 kg/m3 and Poisson’s ratio 0.3. Different numbers of finite elements were used for the cantilever beam. Table 4.10, 4.11 and 4.12 compare the first five natural frequencies using 10, 20, or 40 elements, respectively. The results showed that eigenvalue analyses resulted in more accurate solutions than modal analyses as compared to the theoretical solutions. The errors decreased as the number of finite elements increased, as expected. The same statement was also true for the mode shapes. When 40 elements were used, the first three mode

4.3 Numerical Study Table 4.10 Comparison of natural frequencies of a cantilever beam using 10 elements

Table 4.11 Comparison of natural frequencies of a cantilever beam using 20 elements

Table 4.12 Comparison of natural frequencies of a cantilever beam using 40 elements

75 Theoretical solution (Hz)

Eigenvalue analysis (Hz)

Modal analysis (Hz)

25.54 (0%)

27.30 (6.49%)

Mode 1

25.54

Mode 2

160.06

160.07 (0.006%)

169.00 (6.20%)

Mode 3

448.19

448.30 (0.024%)

480.00 (7.09%)

Mode 4

878.27

879.10 (0.113%)

940.00 (7.028%)

Mode 5

1451.85

1455.51 (0.252%)

1560.00 (7.449%)

Theoretical solution (Hz)

Eigenvalue analysis (Hz)

Modal analysis (Hz)

Mode 1

25.54

25.54 (0%)

26.70 (4.54%)

Mode 2

160.06

160.06 (0%)

170.00 (6.21%)

Mode 3

448.19

448.20 (0.002%)

460.00 (2.63%)

Mode 4

878.27

878.33 (0.006%)

910.00 (3.61%)

Mode 5

1451.85

1452.10 (0.017%)

1500.00 (3.31%)

Theoretical solution (Hz)

Eigenvalue analysis (Hz)

Modal analysis (Hz)

Mode 1

25.54

25.54 (0%)

26.20 (2.58%)

Mode 2

160.06

160.06 (0%)

160.00 (0.03%)

Mode 3

448.19

448.19 (0%)

460.00 (2.63%)

Mode 4

878.27

878.28 (0.001%)

890.00 (1.33%)

Mode 5

1451.85

1451.87 (0.001%)

1480.00 (1.93%)

76

4 FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures

shapes among the three solutions were so close that all the curves were virtually on top of one another. Therefore, 40 elements were also used for other boundary conditions. Both simply supported and clamped beams were also analyzed and three different solutions were compared in Tables 4.13 and 4.14. Mode shapes also agreed well among three solutions using 40 elements. Once the numerical modal analysis was verified against the known solutions, the same technique was applied to the FSI problem. The first case was a clamped beam whose bottom side was in contact with a fluid medium, which was modelled as an acoustic domain using the CA technique. The fluid medium had the density 1000 kg/m3 and the speed of sound 1500 m/s. The dynamic characteristics of a composite beam was compared between the two cases, with and without FSI. Figures 4.18 and 4.19 compare the first two mode shapes of a clamped beam while the beam was in air or water, respectively. There was slight difference between the two mode shapes with and without FSI. However, when the modal curvatures were compared in Figs. 4.20 and 4.21, the difference resulting from the effect of FSI was magnified. Table 4.13 Comparison of natural frequencies of a simply supported beam using 40 elements

Table 4.14 Comparison of natural frequencies of a clamped beam using 40 elements

Theoretical solution (Hz)

Eigenvalue analysis (Hz)

Modal analysis (Hz)

Mode 1

71.69

71.69 (0%)

70.00 (2.35%)

Mode 2

286.78

286.78 (0%)

290.00 (1.12%)

Mode 3

645.27

645.27 (0%)

660.00 (2.28%)

Mode 4

1147.14

1147.15 (0.0008%)

1160.00 (1.12%)

Mode 5

1792.41

1792.44 (0.001%)

1820.00 (1.53%)

Theoretical solution (Hz)

Eigenvalue analysis (Hz)

Modal analysis (Hz)

Mode 1

162.52

162.52 (0%)

170.00 (4.6%)

Mode 2

448.01

448.01 (0%)

460.00 (2.67%)

Mode 3

878.29

878.29 (0%)

910.00 (3.61%)

Mode 4

1500.41

1451.87 (3.23%)

1500.00 (0.027%)

Mode 5

2168.82

2168.87 (0.002%)

2240.00 (3.28%)

4.3 Numerical Study

77

0 No FSI With FSI

Noramlized Deflection

-0.2

-0.4

-0.6

-0.8

-1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalize Beam Length

Fig. 4.18 Comparison of the first mode shape of a clamped beam with and without FSI effect

1

No FSI

Normalized deflection

With FSI 0.5

0

-0.5

-1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Noramlized Beam Length

Fig. 4.19 Comparison of the second mode shape of a clamped beam with and without FSI effect

Furthermore, when the beam vibrated freely in water, the vibrational mode shape was different from that in air. For example, a clamped beam was displaced with initial force at the center such that the initial deflection was very close to the first mode shape. Then, the force was removed immediately from the beam so that it could have free vibration. As observed from the experiment of a cantilever beam, the vibrating beam in water showed high frequency components on top of the major vibrating mode shape. That is, the effect of FSI induced higher modes to the vibrating beam as shown in Fig. 4.22. The deflection in the figure did not consider the continuity of the slope. In other words, the transverse displacement degrees of freedom were only plotted. That is why the curve does not have a smooth deflection shape. Instead, it looks like a zig-zag curve. However, this clearly shows the components of higher mode shapes in the beam deflection in water.

78

4 FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures 3

Modal Curvature

2 1 0 No FSI -1

With FSI

-2 -3 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized Beam length

Fig. 4.20 Comparison of the first modal curvature of a clamped beam with and without FSI effect 60 40

Modal Curvature

20 0 No FSI

-20

With FSI

-40 -60 -80

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized Beam Length

Fig. 4.21 Comparison of the second modal curvature of a clamped beam with and without FSI effect

The frequency spectrum was plotted in Figs. 4.23 and 4.24 for a clamped beam without and with the effect of FSI. The plots showed the beam vibrated with much more high frequency components when the structure interacted with water. The frequency plots also showed that the major vibrating frequency was 170 Hz in air and 64 Hz in water, respectively. The major frequency in water was reduced by approximately 60%. The next example was a square composite plate clamped along all the boundaries. The finite element model had 12 × 12 elements. A static force was applied to the center of the plate to generate the initial deflection of the plate as shown in Fig. 4.25. Then, the plate was also released from the initial deflection. The free vibrational shape at an instant of time is shown in Figs. 4.26 and 4.27 when the plate vibrated in air or water. When the plate vibrated in the water, higher mode shapes were also

4.3 Numerical Study 10

79

-7

1

Beam Deflection

0 -1 -2 -3 -4 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized Beam Length

Fig. 4.22 Comparison of the second modal curvature of a clamped beam with and without FSI effect Fig. 4.23 Frequency spectrum of a clamped beam without FSI effect

Fig. 4.24 Frequency spectrum of a clamped beam with FSI effect

80

4 FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures

-5

0.5

x 10

0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 1 0.8 0.6 0.4 0.2 0

0.1

0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1

0.9

Fig. 4.25 Initial condition of clamped square plate

Plate Displacement

x 10

-5

1

0.5

1 0.8

0 0

0.6 0.4

0.2 0.4 0.2

0.6 0.8 1

Norm. Plate Length

Norm. Plate Width

0

Fig. 4.26 Vibrational shape of a clamped square plate in air -7

Plate Displacement

x 10 4 3

1

2 0.8

1 0.6

0 0

0.4

0.2 0.4

0.2

0.6 Norm. Plate Length

0.8 1

Fig. 4.27 Vibrational shape of a clamped square plate in water

0

Norm. Plate Width

4.3 Numerical Study

81

excited as shown in the figure. Both deflection curves were plotted by considering the transverse displacement degrees of freedom only without using the slope degrees of freedom as before in order to amplify the higher mode shapes in the deflection. The next example examined an impact loading applied to the same clamped square plate at its center. The first vibrational mode is shown in Fig. 4.28. This mode shape is close with and without FSI. However, the next symmetric mode shape was very different when the effect of FSI was included. Figures 4.29 and 4.30 compare the second symmetric mode shapes in air and water, respectively. The major difference

Fig. 4.28 First mode of clamped square plate

Fig. 4.29 Second symmetric mode shape of clamped square plate in air

82

4 FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures

Fig. 4.30 Second symmetric mode shape of clamped square plate in air

between the two mode shapes occurred near the corners. This difference explains some experimental results. Figures 4.31 and 4.32 compared the measured strains from a square composite plate which was clamped along the boundary and subjected to an impact load at the center. Because the impact was applied to the center, strains were measured at a location very close to the center. Those strain responses were different when the

Fig. 4.31 Comparison of strains measured very near the center of clamped square plate with impact at the center

4.3 Numerical Study

83

Fig. 4.32 Comparison of strains measured near a corner of clamped square plate with impact at the center

plate was in air or water as shown in Fig. 4.31. However, their peak magnitudes were relatively close each other with out of phase resulting from the change in the vibrational frequency. On the other hand, the strain response near a corner of the plate was drastically influenced by FSI. The strain with FSI was significantly greater than the strain without FSI. This agrees well with the computed mode shape shown in Fig. 4.30. This mode shape will make the modal curvature very different around the corner, which is directly related to the bending strain of the plate. Therefore, the modal analysis provided some explanation why the strain behavior was so different near the corner because of the effect of FSI.

4.4 Summary This chapter studied the effect of FSI on the vibrational and modal characteristics of composite structures using both experimental and numerical studies. For the experimental analysis, both the DIC technique and experimental modal analysis were undertaken. On the other hand, transient analysis of a coupled fluid-structure system was conducted numerically, and a numerical modal analysis was also performed. Both studies provided confirmations of findings from each other. The magnitude of the FSI effect is related to the transient motion of the structure because both the fluid and structure should maintain their compatibility at their interface unless cavity occurs at the interface. Since the vibrational motion of a flexible structure varies from location to location of the structure, the magnitude of the FSI effect also varies over

84

4 FSI Effect on Frequencies, Mode Shapes, and Modal Curvatures

the structure. The nonuniform effect of FSI on composite structures activated high frequency mode shapes during their free vibrations even though the structures were initially exited as a very low frequency mode shape such as the first mode shape. The FSI also influenced the mode shapes of composite structures. Depending on the situation, some mode shapes were quite different with and without FSI. Especially, the modal curvatures, which are the second order derivatives of the mode shapes, were much more different resulting from FSI. This modal curvature is related to bending strain and stress. Such a variation in mode shape and curvature could lead to the change in the potential failure locations.

References 1. Kwon YW, Priest EM, Gordis JH (2013) Investigation of vibrational characteristics of composite beams with fluid-structure interaction. Compos Struct 105:269–278 2. Ross HE, Franklin SS, Weltman G, Lennie P (1970) Adaptation of divers to size distortion under water. Br J Psychol 61:365–373 3. MATLAB (2018) The mathWorks. Inc. Natick, Massachusetts 4. Kwon YW, Plessas SD (2014) Numerical modal analysis of composite structures coupled with water. Compos Struct 116:325–335

Chapter 5

FSI Study with Cyclic Loading

Fatigue failure is one of the most common failure modes for structures. This is also true for composite structures that are in contact with water. This chapter presents the experimental results of composite structures subjected to cyclic loading while they were under water [1]. Then, a numerical study was also conducted to further explain the test results [2].

5.1 Experimental Study In order to conduct cyclic tests of composite structures, composite beams were fabricated using the vacuum assisted resin transfer molding technique as sketched in Fig. 5.1. Glass fiber plain weave fabrics were used along with resins. The fabricated composite panel had 23 layers of glass fabrics resulting in 5 mm thickness. Each composite panel had the dimensions of 533 mm by 406 mm. All panels had the layer orientations of 0°/90°. Pro-Set M1002 resin was mixed with the hardener Pro-Set 237 at room temperature. Each panel was cut into beams measuring 229 mm in length and 38 mm in width as shown in Fig. 5.2. In order to minimize the error associated with the fabrication technique, the composite beams made out of the same panel were tested with and without water, respectively. Experiments were conducted using the three-point bending set-up with cyclic loading. Some beams were tested in air while others were tested in water. A hydraulic powered MTS uniaxial testing machine was used for the cyclic loading. For testing in water, a water tank was installed for the MTS machine as shown in Fig. 5.3, which compares the testing in air and water, respectively. The lower part of the three-point bending set-up was installed inside the water tank while its grip section to be held by the lower end of the MTS came out of the tank with a hole to the tank at its bottom.

© Springer Nature Switzerland AG 2020 Y. W. Kwon, Fluid-Structure Interaction of Composite Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-57638-7_5

85

86

5 FSI Study with Cyclic Loading

Fig. 5.1 Schematic of vacuum assisted resin transfer technique

38 mm

229 mm Fig. 5.2 Dimension of e-glass composite beams

Fig. 5.3 Three-point bending tests in air (left) and water (right) for cyclic loading

5.1 Experimental Study

87

Then, the gap between the hole and the grip was properly sealed so that water does not leak from the water tank. The composite beams were placed in the water tank only for the duration of the cyclic testing in order to avoid any hydrothermal effect on the composite beams. To this end, the number of cyclic loads was limited so that the composite beams were not placed in water for a long time. Additionally, the composite beams were weighed before and after cyclic tests in water to make sure there was no weight change. The same cyclic loading conditions were applied to the composite beams tested in air or water, respectively. In order to determine the strength and stiffness of the composite beams, static three-point bending tests were conducted at the loading rate 2 mm/min until the failure of the specimens. Five beams were tested in air, and their results are listed in Table 5.1. The results were relatively consistent. The maximum load was the failure load under the three-point bending test, and the maximum displacement was the deflection at the center. The Stiffness was the slope of the load-displacement curve at its linear section. The load-displacement curves were mostly linear near to the failure point as seen in Fig. 5.4, which is the curve of one of the samples. Table 5.1 Static three point-bending test results of composite beams in air Sample No.

1

2

3

4

5

Max load (N)

1777

1431

1573

1536

1548

Max disp. (mm)

13.4

15.9

12.7

14.0

12.4

Stiffness (kN/m)

142.8

97.8

135.8

122.9

138.1

Fig. 5.4 Static three-point bending test of composite beam in air

88

5 FSI Study with Cyclic Loading

Table 5.2 Natural frequency of composite beams Sample

ωn (rad/s)

ωn (Hz)

1

2567.98

408.65

2

2125.19

338.19

3

2504.25

398.51

4

2382.34

379.11

5

2525.37

401.87

The cyclic loading was controlled by displacement rather than force. In order to shorten the time in submerged in water, the cyclic loading conditions were selected such that the minimum and maximum displacements of the cyclic loading would be 25 and 75% of the failure displacements. The frequency of the cyclic loading was decided after checking the natural frequency of the composite beams to avoid any resonance effect. Table 5.2 lists the first natural frequency of each composite beam. The natural frequency was approximately 400 Hz, and the applied loading cycle was decided to be 2, 5 or 10 Hz based on the capacity of the testing machine. One thing noted during the cyclic bending tests was that the beam specimen might have a slight slip during loading cycles. To prevent slippage during the experiment, the ends of the specimen are loosely secured using Velcro strips onto the three-point bending set-up. This would not affect the simply support boundary condition. Even though there might be a small effect, this was consistent for all test beams so that there was no bias introduced from test sample to sample. The data was collected at 40 Hz by the testing machine. The first set of cyclic tests were conducted at 10 Hz. The composite beams made of the same batch of the VARTM process were tested for the air and water conditions, respectively. The life cycles with 10 Hz loading in air were summarized in Table 5.3. Here, maximum and minimum forces Fmax and Fmin were listed along with the mean force Fm and the force amplitude Fa . The R-ratio was determined by the maximum force divided by the minimum force. All those forces were quite consistent for all the test beams. Figure 5.5 shows the cyclic test results for the sample A1 in Table 5.3. Because of the displacement control, the applied force continued to decrease until final failure as seen in Fig. 5.5. The force values tabulated in Table 5.3 were the average values. For example, the maximum force in the table was the average of the decreasing maximum force as plotted in Fig. 5.5. Table 5.3 Summary of 10 Hz loading in air Sample

Life cycle

R-ratio

Fm (kN)

Fa (kN)

Fmax (kN)

Fmin (kN)

A1

9500

4.11

0.69

0.421

1.11

0.27

A2

14,000

4.07

0.69

0.416

1.10

0.27

A3

11,000

4.19

0.68

0.415

1.09

0.26

5.1 Experimental Study

89

Fig. 5.5 Test result of sample A1 with 10 Hz in air

Table 5.4 Summary of 10 Hz loading in water Sample

Life cycle

R-ratio

Fm (kN)

Fa (kN)

Fmax (kN)

Fmin (kN)

W1

6200

4.15

0.67

0.41

1.08

0.26

W2

6900

4.03

0.76

0.46

1.21

0.30

W3

5100

4.00

0.78

0.47

1.24

0.31

Likewise, the samples from the same batch were also tested while the beams were submerged in water using the loading frequency 10 Hz. The wet test results are summarized in Table 5.4. When Tables 5.3 and 5.4 were compared, the life cycles are much shorter for the beams tested in water as compared to the beams tested in air. The average maximum and minimum forces were greater for the beams tested in water than those in air. This suggests that the FSI increased the applied force for the beams in water because the displacement was controlled. This resulted in the shorter life cycle. Two composite beam samples were compared, one of which was tested in air and the other in water while the average applied maximum and minimum forces were very close each other. Table 5.5 compares the two test results. Both samples experienced almost identical average forces. However, the sample in water had a 44% reduction in the failure cycles. Both samples in air and water, respectively, followed the same trend. The applied force under the constant displacement control decreased rapidly for the first 500 load cycles. During this loading period, localized delamination was observed. This seemed to cause the steep reduction of the applied force. Table 5.5 Comparison of test results with 10 Hz loading Condition

Sample

Life cycle

R-ratio

Fm (kN)

Fa (kN)

Fmax (kN)

Fmin (kN)

Air

A3

11,000

4.19

0.68

0.415

1.09

0.26

Water

W1

6200

4.15

0.67

0.410

1.08

0.26

90

5 FSI Study with Cyclic Loading

After the initial 500 load cycles, both composite beams showed a gradual, almost linear decrease in the applied force. However, the composite beams in water exhibited a steeper slope as compared to those tested in air. This steeper reduction is considered resulting from the effect of FSI. Then, the samples tested in water failed earlier than the others. However, both beams failed at the load of 0.9 kN. Similar tests were also repeated with the loading cycles of 5 Hz. The test results were summarized in Table 5.6 for the dry tests and in Table 5.7 for the wet tests. The applied forces for the beams tested showed a slight variation but not so much. Like the 10 Hz loading, there was a steep reduction in the applied force during the first 500 cycles, followed by a gradual decrease. All beams tested in air or water failed at the force range of 1.0–1.2 kN. There was no difference at the final failure loads regardless of the test conditions. Table 5.8 compares the test results in air and water, respectively, for the close applied forces. The two samples have very close maximum and minimum average forces. The failure cycle for the beam in water showed a reduction by 42% as compared to that of the dry beam. Both beams had failures at 1.05 kN. Figure 5.6 compares two typical force time histories when the beams were tested in air and water, respectively with the loading cycles of 5 Hz. This figure only plotted the maximum and minimum forces not to make the plot too much crowded. The solid line is for the beam sample tested in water while the broken line is for the beam tested in air. The top curve of each case is the applied maximum force while the bottom curve is the applied minimum force. The rate of the decrease in the applied force is greater for the wet test than the dry test. Furthermore, the differential between the Table 5.6 Summary of 5 Hz loading in air Sample

Life cycle

R-ratio

Fm (kN)

Fa (kN)

Fmax (kN)

Fmin (kN)

A4

6000

3.69

0.75

0.43

1.18

0.32

A5

8900

3.87

0.73

0.43

1.16

0.30

A6

5800

3.50

0.76

0.43

1.19

0.34

Table 5.7 Summary of 5 Hz loading in water Sample

Life cycle

R-ratio

Fm (kN)

Fa (kN)

Fmax (kN)

Fmin (kN)

W4

3500

3.84

0.75

0.44

1.19

0.31

W5

3400

3.90

0.76

0.45

1.21

0.31

W6

3700

4.26

0.71

0.44

1.15

0.27

Table 5.8 Comparison of test results with 5 Hz loading Condition

Sample

Life cycle

R-ratio

Fm (kN)

Fa (kN)

Fmax (kN)

Fmin (kN)

Air

A4

6000

3.69

0.75

0.415

1.18

0.32

Water

W4

3500

3.84

0.75

0.410

1.19

0.31

5.1 Experimental Study

91

Fig. 5.6 Comparison of cyclic tests in water and air with 5 Hz loading cycle

maximum and minimum forces at the same load cycle showed a slight decrease as the load cycle increases. This is believed to result from the reduction in the beam stiffness along with the cyclic loading. Because the applied displacements remained constant for the maximum and minimum values, the reduction in the stiffness yielded a smaller force differential. However, such a change was very slow. Table 5.9 shows the ratios of life cycles in water to air for all beam samples prepared from the same batch of fabrication. As expected, there is a variation in the test results from batch to batch. However, the general qualitative results remained the same regardless of the variation resulting from different batches. The effect of FSI clearly showed the reduction in the life cycles for composite beams. Table 5.9 Summary of the life cycle ratios of tests in air to tests in water

Batch No.

10 Hz loading

5 Hz loading

1

1.50

NA

2

1.20

NA

3

1.39

1.48

4

1.77

1.7

5

1.29

1.38

92

5 FSI Study with Cyclic Loading

5.2 Numerical Study In order to complement the experimental study, a numerical study was conducted for cyclic loading to a composite structure. The composite beam was modeled using solid-like beam elements, which have nodes at the top and bottom planes with only translational degrees of freedom [3, 4]. The conventional beam elements have displacements and slopes at the reference plane. The solid-like beam elements are easier for coupling with a surrounding fluid medium. The beam was assumed to have an orthotropic material behavior. The fluid was assumed to have a negligible fluid motion and it was modelled as an acoustic medium. The CA technique was utilized for the fluid medium. The beam elements have displacements as nodal degrees of freedom while the CA fluid domain has pressure as degrees of freedom. In order to couple the two domains, a small size of acoustic domain around the beam structure was modelled using the finite element analysis of the acoustic domain and this finite element model is directly coupled to the CA acoustic domain. Figure 5.7 provides a sketch of the analysis domain. The finite element modeling of the acoustic domain can be easily coupled with the structural motion [5]. Because it is not easy to apply the force-controlled cyclic loading at high frequency for experimental studies, the numerical study applied the force-controlled cyclic loading to complement the study. The beam was simply supported at both ends, and the cyclic loading was applied to the center of the beam using 10 Hz. Bending strains were computed for both cases. One case had a beam in the fluid acoustic domain while the other case had no fluid domain. The overall strain plots with and without the fluid domain were very close. However, a close examination showed that there is a difference in the bending strains as shown in Fig. 5.8. The next study considered a structure in contact with water while the structure supported a rotating machinery as sketched in Fig. 5.9. This was an example of an equipment installed on a ship hull structure as the outside of the hull was surrounded by water. To simplify the geometry of the structure, it was modelled as a beam or a Fig. 5.7 Sketch of analysis domain

CA Acoustic Domain

FEM Acoustic Domain

Structural Domain

5.2 Numerical Study

93

Fig. 5.8 Comparison of computed bending strains with and without FSI for cyclic loading

Rotating Equipment

Flexible Structure

Acoustic Water Medium

Semi-infinite Boundary Fig. 5.9 Flexible structure supporting rotating equipment and in contact with water

plate. Then, the rotating equipment was modelled as a single mass-spring system as sketched in Fig. 5.10. Like the previous example, the structure was modelled using solid-like beam/plate elements with displacements degrees of freedom only. The fluid domain was

94

5 FSI Study with Cyclic Loading

Fig. 5.10 Simplified computer model for cyclic loading with FSI

modelled as an acoustic domain which was divided into two subdomains. The first subdomain was just around the structure, and this was much smaller than the second subdomain. The first subdomain was modelled using FEM and the second subdomain was modelled using CA. First of all, the structure was modelled as a beam which was made of a composite, aluminum or steel. The basic material properties of those three materials are listed in Table 5.10. The beam has the geometric dimensions of 1.0 m by 0.1 m and the thickness of 0.02 m. The first natural frequency of the beam in air with both ends clamped is also listed in Table 5.10. As the mass and spring system was attached to the composite beam, the natural frequency of the beam including the mass and spring also varied. The new natural frequency was computed using the FEM. A mesh sensitivity study showed at least 20 elements were good enough to result in consistent lowest natural frequencies of the first five. The change in the natural frequency of the √beam-mass-spring system was dependent on the frequency of the mass-spring, K /M where K and M are the spring and mass of the system. The √first natural frequency of the beam-spring-mass system K /M, and the increase was approximately linear until increased as a function of √ K /M = 1000 (rad/s). The first natural frequency √ of the beam-spring-mass system was smaller than that of the beam only until K /M = 500 (rad/s). That is, the additional spring-mass reduced the first natural frequency of the beam. Table 5.10 Material properties and natural frequency with clamped boundary

Elastic modulus (GPa)

Density (kg/m3 )

Natural freq (Hz)

Composite

20

2000

65.1

Aluminum

70

2700

104.8

200

8000

102.9

Steel

5.2 Numerical Study

95

Fig. 5.11 FFT of dynamic response of a clamped composite beam in water

As stated in the previous chapter, a numerical modal analysis was undertaken to determine the vibrational frequency of the composite beam. An impulse of a short duration was applied to the beam, and its transient response was computed. Then, FFT was applied to the transient solution to determine the natural frequency. The natural frequency of the composite beam in water without the spring-mass system was found to be 16 Hz as shown in Fig. 5.11, which is about 25% of the beam only frequency without water. The next study considered a beam in contact with water while a spring-mass was attached to the center of the beam. When the spring-mass system has the value of √ K /M = 100, the lowest three natural frequencies were 25 Hz, 51 Hz, and 76 Hz, respectively as seen in Fig. 5.12. In order to select proper values for the spring and mass, the force transmissibility was studied, which is expressed as TR =

1 1 − r2

(5.1)

for a rigid base structure. Here r is the ratio of the √ frequency ω of the applied force to the frequency of the spring-mass system ωn = K /M, i.e. r = ω/ωn . To generate a large transmission to the structure, the frequency ratio r should be much smaller than

96

5 FSI Study with Cyclic Loading

Fig. 5.12 FFT of dynamic response of a clamped composite beam-spring-mass in water

1.0, which suggests the loading frequency must be much smaller than the frequency of the spring-mass system. If ωn is 100, then ω is selected to be 10 so that the ratio becomes 0.1. Therefore, the problem depicted in Fig. 5.10 had the following assigned values for the present study, unless mentioned otherwise: ωn = 100 and F = sin(10t). Using those values, the composite beam was investigated for its bending strain at the center as well as at the clamped boundary. Figures 5.13 and 5.14 compare the bending strains at those locations with and without the effect of FSI. These bending strains were computed at the bottom side of the beam. Without the effect of FSI, the bending strains were very close between at the center and at the boundary even though their signs were opposite. The maximum strain was over 20 µ in magnitude at both locations when there was no water. However, FSI increased the strain significantly. The peak magnitude of bending strains was over 30 µ at the center and over 40 µ at the boundary. The bending strain at the center was increased by around 50% and that at the clamped boundary increased by around 100%. Thus, the peak bending strain was greater at the boundary than at the center resulting from FSI. This result was opposite from that of a previous study [6]. Different kinds of loading seemed to result in different responses. To explain what was different resulting from FSI, a multiple number of parameters were compared. First of all, the spring force was compared with and without FSI

5.2 Numerical Study

97

-6

10

3

2

Bending Strain

1

0

-1

-2 In Air

-3

In Water

-4 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (sec)

Fig. 5.13 Bending strain at the center of the clamped composite beam subjected to 10 Hz vibrating force 10 -6

5

In Air

4

In Water

3

Strain

2 1 0 -1 -2 -3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (sec)

Fig. 5.14 Bending strain at the boundary of the clamped composite beam subjected to 10 Hz vibrating force

whether there was any difference in the spring force. The comparison showed that both spring forces were identical. Therefore, FSI did not affect the spring force. The next parameter was the velocity at the center of the beam. Figure 5.15 shows that the

98

5 FSI Study with Cyclic Loading 10 -3

4

In Air In Water

3

Velocity (m/s)

2

1

0

-1

-2

-3 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time

Fig. 5.15 Velocity at the center for clamped composite beam subjected to 10 Hz vibrating force

velocity of the composite beam was significantly different resulting from the effect of FSI. The velocity was much higher resulting from the interaction with water. Both aluminum and steel beams were also investigated under the same condition. The steel beam did not show much difference resulting from the effect of FSI. For example, the bending strain at the center of the steel beam was almost identical regardless of the effect of FSI as shown in Fig. 5.16. Likewise, when the aluminum 10 -7

3

In Air In Water

2

Bending Strain

1

0

-1

-2

-3 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (sec)

Fig. 5.16 Bending strain at the center of the clamped steel beam subjected to 10 Hz vibrating force

5.2 Numerical Study

99

10 -7

8

In Air

6

In Water

4

Bending Strain

2 0 -2 -4 -6 -8 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (sec)

Fig. 5.17 Bending strain at the center of the clamped aluminum beam subjected to 10 Hz vibrating force

beam was studied, the difference resulting from the FSI was slightly greater than that for the steel beam, but much smaller than the composite beam. For example, Fig. 5.17 compares the bending strains at the center of the clamped aluminum beam. Comparing the three different materials, the composite beam showed a much significant effect of FSI than other two metallic materials. This is because the lower density and modulus of the composite material as compared to the steel and aluminum. The next study examined the effect of the vibrational frequency of the equipment such that the frequency was changed to 5 from 10 Hz. Figure 5.18 compares the bending strains with and without the effect of FSI. The figure clearly shows a difference between the two strains, but the difference was much smaller than that for the 10 Hz loading. Thus, there is an effect of the frequency of the loading cycles. This makes sense that the effect of FSI disappears as the loading rate becomes quasi-static, i.e. the dynamic effect becomes negligible. The next study changed the loading frequency over a spectrum to determine its effect on the dynamic response of the beam. The maximum deflection of the beam was divided by the static deflection at the center in order to investigate the magnification of the deflection resulting from different frequency of the cyclic loading when the composite beam was in contact with water. Figure 5.19 shows the deflection ratio, which was the ratio of the maximum dynamic deflection to the static deflection, as a function of the vibrating frequency of the equipment. Throughout the whole spectrum of frequency, the deflection ratio remained greater than 1, i.e. the deflection resulting from FSI was always greater than the static deflection. Another numerical study was conducted to check the effect of boundary condition along with FSI. To this end, the boundary condition of the composite beam was changed from the clamped support to the simple support. The applied frequency was

100

5 FSI Study with Cyclic Loading 10-6

1.5

1

Bending Strain

0.5

0

-0.5

In Air

-1

In Water

-1.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (sec)

Fig. 5.18 Bending strain at the center of the clamped composite beam subjected to 5 Hz vibrating force 20 In Air In Water

Deflection Ratio

15

10

5

0

0

5

10

15

20

25

30

Frequency (Hz)

Fig. 5.19 Plot of deflection ratio as a function of the exciting vibrational frequency for a clamped composite beam

5 Hz. The bending strain at the center of the simply supported beam is plotted in Fig. 5.20. Comparison of Figs. 5.18, 5.19 and 5.20 clearly shows that the relative effect of FSI was much greater for the simply supported beam than for the clamped beam. In other words, a more flexible support generated a larger FSI effect. The

5.2 Numerical Study

101

10 -6

4 3 2

Bending Strain

1 0 -1 -2

In Air In Water

-3 -4 0

0.05

0.1

0.2

0.15

0.25

0.3

0.35

0.4

0.45

Time (sec)

Fig. 5.20 Bending strain at the center for the simply supported composite beam subjected to 5 Hz excitation

difference in strains between in water and in air was about three times greater for the simply supported beam than the clamped beam under the same loading condition. The final study considered a clamped composite plate whose dimension was 1 m × 1 m × 0.02 m. The vibration equipment was placed at the center of the plate. The same mass-spring system was used for the study. The natural frequency of a clamped plate can be determined using the following equation:  ωn = 1.655

Eh 2 (Hz) ρ L 4 (1 − v2 )

(5.2)

where E is the elastic modulus, h is the plate thickness, ρ is the mass density, L is the plate length of a square plate, and ν is Poisson’s ratio. The frequency of the composite plate itself without any attachment is 110 Hz using the equation. When the same spring-mass system was attached, the frequency of the plate changed to 16 Hz. In addition, the natural frequency of the plate including the FSI effect was 36 Hz. Considering all these, the excitation frequency of the equipment was chosen to be 10 or 20 Hz. Dynamic responses of the plate were computed at various locations over the plate as shown in Fig. 5.21. Table 5.11 compares the ratio of the maximum dynamic deflection to the static deflection at different locations and two different excitation frequency. This table shows that the largest ratio occurred at the location F which is closer to the corner of the plate while the smallest ratio was at the center of the plate. This finding is also similar to that for the clamped beam. Figure 5.22 shows the deflection at the center of the plate for the two cases, with and without FSI when the plate was subjected to 20 Hz vibrating load.

102

5 FSI Study with Cyclic Loading

A

B

C

D

E F

Fig. 5.21 Clamped square plate

Table 5.11 Ratio of maximum displacement of clamped composite plate resulting from FSI (Please see Fig. 5.21 for the location)

Location

Excitation frequency 10 Hz

20 Hz

A

1.11

1.24

B

1.14

1.29

C

1.16

1.33

D

1.14

1.31

E

1.16

1.34

F

1.20

1.41

5.3 Summary Both experimental and numerical studies were conducted to investigate the effect of FSI on cyclic loading to composite structures. Both results reinforced the findings and complement each other. The major findings were that FSI reduced the life cycles of composite structures significantly, and the reduction in the life cycles could be also influenced by the frequency of the cyclic loading as well as the boundary condition of the structure. A higher cyclic loading and flexible boundary conditions could lead to earlier failure. Furthermore, the effect of FSI was greater at or near the clamped boundary than at the center of the structure. This also suggested a potential change in the failure locations resulting from FSI.

5.3 Summary

103 10 -6

2

In Air

1.5

In WaterI

Deflection (m)

1 0.5 0 -0.5 -1 -1.5 -2 0

0.05

0.1

0.15

0.2

0.25

0.3

Time

Fig. 5.22 Deflection at the center of clamped composite plate subjected to 20 Hz vibrating load

References 1. Kwon YW, Teo HF, Park C (2016) Cyclic loading on composite beams with fluid structure interaction. Exp Mech 56(4):645–652 2. Kwon YW (2014) Dynamic responses of composite structures in contact with water while subjected to harmonic loads. Appl Compos Mater 21(1):227–245 3. Kwon YW (2013) Analysis of laminated and sandwich composite structures using solid-like shell elements. Appl Compos Mater 20(4):355–373 4. Kwon YW, Bang H (2000) The finite element method using Matlab, 2nd edn. CRC Press, Boca Raton 5. Craugh LE, Kwon YW (2013) Coupled finite element and cellular automata methods for analysis of composite structures with fluid-structure interaction. Compos Struct 102:124–137 6. Kwon YW, Violette MA, McCrillis RD, Didoszak JM (2012) Transient dynamic response and failure of sandwich composite structures under impact loading with fluid structure interaction. Appl Compos Mater 19(6):921–940

Chapter 6

FSI Study of Structures Containing Fluid

Some structures contain fluids internally. One of the examples is a fluid tank. When such structure is subjected to dynamic loading, FSI occurs for the system. Hydrodynamic ram is one of the phenomena as a projectile penetrates into a structure such as a fuel tank. This chapter investigated the dynamic responses of composite structures containing different amounts of water while the structures were impacted by mechanical loading [1]. Both experimental and numerical studies were conducted. Lastly, hydrodynamic ram effects were studied numerically [2].

6.1 Experimental Study In order to conduct experimental studies, a composite box structure was fabricated using a woven fabric composite material. It was a plain weave e-glass fiber composite. First of all, the composite box was designed as shown in Fig. 6.1. In order to fabricate the composite box, an aluminum frame was constructed, and the woven fabric was wrapped around the aluminum frame as shown in Fig. 6.2. The fabricated composite box is shown in Fig. 6.3. The structure has four sides without top and bottom sides. Because the front surface of the composite box would experience repeated impacts during testing, the impact location was reinforced in order to prevent damage. A 3.81 cm × 3.81 cm × 0.3175 cm aluminum piece was attached to the impact location, as shown in Fig. 6.3, with the same epoxy used throughout fabrication. Strain gages were attached to the front and back sides of the box as sketched in Fig. 6.4. Strain gages were also attached to the center of the side walls. The aluminum base structure was used to restrain the movement of the box during the impact loading. Figure 6.5a shows the aluminum plate with a square shape of grooves in which the composite box to be fit at its bottom side. The whole aluminum plate was constrained by angle brackets and bolted into a vibration isolation table. The top side of the composite box was fitted using a PMMA plate as shown in © Springer Nature Switzerland AG 2020 Y. W. Kwon, Fluid-Structure Interaction of Composite Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-57638-7_6

105

106

6 FSI Study of Structures Containing Fluid

Fig. 6.1 a Top-down view and b isometric view of the composite structure

Fig. 6.5b. The plate was clear so that the motion of water could be visually examined during the tests, and it also had a hole to fill water inside the composite box without detaching the top from the composite box. A pendulum type of impactor, as seen in Fig. 6.6, was used to apply the impact force. A protractor was attached to the top of the impact pendulum in order to measure the initial setting angle. A load cell was attached to the impactor to measure the impact load history. The assembled test setup is shown in Fig. 6.7. For some studies, a baffle was inserted into the composite box so as to determine the effect of the baffle to the dynamic motion of the composite box through FSI. Two different designs of baffles were designed and used. Both baffles are shown in Figs. 6.8 and 6.9. They have the same total areas of openings for fluid flow but each size and the number of openings were different. Figure 6.10 shows each baffle inserted into the composite box. Impact tests were first conducted without water inside the composite box. This was used as the control case. Then, tests were conducted as water was filled into the composite box incrementally. Table 6.1 shows different water fill levels with and without a baffle. Each case of test was conducted at least several times to check the repeatability of the test results. The results were on top of one another as the tests were repeated. First of all, the impact forces were investigated. Impact was applied from the initial 45° angle which was measured using the protractor as shown in Fig. 6.6. No baffle was inserted into the composite box for this time. Figure 6.11 compares early impact force time-histories as the composite box was filled with different amount water levels. The plot shows that the general shape of the early time curve was similar among all tests. However, the peak force in terms of its magnitude increased as the water level increased. The peak force was classified into two groups. One group included 0 and 25% water levels, which is called the lower water level; and the second group includes 75 and 100% water levels, which is called the higher water level. The 50% water level was in the middle of the separation of the two groups.

6.1 Experimental Study

107

(a)

(b) Fig. 6.2 a Aluminum frame and b composite fabric wrapped around the aluminum frame

This is because the location of the impact was at the 50% water level. In other words, when the water level was much below the impact location, the effect of the water level was small on the peak impact force. The peak impact force was very close between 0 and 25% water levels. Similarly, as the water level far exceeded the impact location, the peak impact force was close each other and almost as twice as that of the lower

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6 FSI Study of Structures Containing Fluid

Fig. 6.3 Composite box

Fig. 6.4 Strain gage locations on a the front side and b the left, right, and back

water level. The middle water level, i.e. 50% water level, was the transition from the lower to high water level. Figure 6.12 is the plot of Fig. 6.11 for a longer time period. In order not to overcrowd the figures, only three water levels were plotted in the figure, which indicates that the overall curves of the impact time-history were very different depending on the water level even though the early time curves looked similar as shown in Fig. 6.11. The duration of the contact between the impactor and the composite box varied depending on the water level. The full water case had the shortest contact time

6.1 Experimental Study

Fig. 6.5 a Aluminum bottom plate and b PMMA top plate Fig. 6.6 Pendulum impactor

109

110

6 FSI Study of Structures Containing Fluid

Fig. 6.7 Test setup for composite box under impact loading

Fig. 6.8 a Side view and b top view of baffle 1

6.1 Experimental Study

111

Fig. 6.9 a Side view and b top view of baffle 2

Fig. 6.10 a Baffle 1 and b baffle 2 installed for testing Table 6.1 Water volumes for different fill levels

Fill level (%) No baffle (mL) Baffle 1 (mL) Baffle 2 (mL) 25

3995

3834

50

7990

7644

3673 7297

75

11,985

11,453

10,921

90

14,382

13,747

13,113

95

15,181

14,505

13,829

100

15,980

15,263

14,545

112

6 FSI Study of Structures Containing Fluid

Fig. 6.11 Impact force time-history for early time with different water levels

Fig. 6.12 Impact force time-history for a longer period with different water levels

while the empty water case resulted in the longest contact time among the three cases shown in Fig. 6.12. The 50% water case had the contact time between the two other cases. The peak impact force was plotted as a function of the water level in Fig. 6.13. The peak impact force increased in general as the water level increased. However,

6.1 Experimental Study

113

Fig. 6.13 Impact force as a function of the water level

the trend shows three stages. From the no water case to the 25% water level, the increase in the peak impact force was almost none. Then, there was a steep increase in the peak impact force from the 25% water level to the 75% water level. Finally, after the 75% water level, the peak impact force was more or less the same with a slight decrease. If more refined water levels were tested, the trending curve would be more accurate with the water level. Impact loading with the initial impact angles of 25° and 45° yielded very similar shapes of graphs as shown in Fig. 6.13 even though the 45° impact resulted in a greater impact force as expected. The figure also shows a very small standard deviation of many repeated tests suggesting the test results were very repeatable. The same impact force was also compared while two different types of baffles were inserted, respectively. The baffles showed a very minor effect on the peak impact force. Figure 6.14 indicated that the baffle reduced the peak impact force slightly. This observation was the same for both types of baffles. The next study was for the strain responses on different composite box walls. Figure 6.15 shows the strain time-history of the front wall for different water levels. This strain as well as other strains discussed later were the strain component along the horizontal direction unless mentioned otherwise even though both horizontal and vertical components were measured simultaneously. The strain gage was located not at the center but at the quarter point from the bottom of the box as shown in Fig. 6.4. Therefore, the water level affected the strain response. In overall, there were two groups of strain responses. One group was when the water level was higher than the strain gage location. The other group was when the water level was at or below the strain gage location. The two groups showed a kind of opposite responses, especially at early times. For example, the higher water level

114

6 FSI Study of Structures Containing Fluid

Fig. 6.14 Impact forces with and without baffle

Fig. 6.15 Strain time-history at the front wall

group yielded initial compressive strains followed by larger tensile strains while the lower water group resulted in initial tensile strains followed by larger compressive strains. The maximum magnitude of strains occurred at compression except for the 100% water level. For the 75% water level, the maximum strain was close between tension

6.1 Experimental Study

115

Fig. 6.16 Maximum strain at the front wall as a function of water level

and compression even though the compressive strain was slightly greater than the tensile strain in terms of magnitude. A similar observation was made for the 90% water level while that graph was not included in Fig. 6.15 so as not to overcrowd the figure. As expected from the impact force, the higher water level generally resulted in the greater maximum strain in magnitude as plotted in Fig. 6.16. The plot of the maximum strain at the front wall was very similar to that of the peak impact force. The largest maximum strain occurred with 75% water for the 45° impact while the maximum happened with 90% water for the 25° impact. The maximum strain increased notably from no water to 20% water while the change in the peak impact force was negligible between the two cases. The variation of the maximum strain at the left wall was plotted in Fig. 6.17 for different water levels. The maximum strain was more of less constant until the water level was 75%. This was because the strain gage was attached to the 50% water level location. While there was a decrease after 75% water, the maximum strain started to increase at 100% water. The maximum strain was very close between 75 and 100% water. The right side wall showed somehow different strain response than the left side wall because the composite box did not have completely uniform walls around the box. However, the general characteristics were similar between the left and right wall, as expected. The strain responses of the back wall were displayed in Fig. 6.18. Again the strain gage was located at the 50% water level mark. Any water level up to 50% water was called the lower water level while any water level beyond 50% water was called higher water level. The dynamic response of the back wall was caused by two paths. One is the path though the structural wall and the other is through the water medium

116

6 FSI Study of Structures Containing Fluid

Fig. 6.17 Maximum strain at the left wall as a function of water level

if there is water. The speed of sound propagation is different between the composite and water. The stress wave propagation is faster through the composite structure than the water medium. As a result, the stress wave arrived at the back wall through the right and left walls, and it was followed by the pressure wave in the water medium. For 0 and 25% water cases, the maximum strain occurred before 0.005 s while other cases, except for the 100% water case, had the maximum strain after 0.01 s. The 100% water case showed the maximum strain around 0.005 s. The maximum strain at the back wall was plotted in Fig. 6.19. The plot shows some decreases in the maximum strain at the center of the back wall until the water level is 50%. After that, the maximum strain increased with the peak value at the 95% water level. When the water was full, there was a sizable decrease in the maximum strain. The maximum strain at the back wall was about a half of that at the front wall. Even though the effect of the baffles was very small for the impact force, the baffles gave more notable effect on the strain on the front wall, especially when the water level was at least 75%. Figure 6.20 shows the comparison of the maximum strain at the front wall with and without a baffle. The baffle reduced the maximum strain at the front wall. The reduction was approximately 15% when the water level was high. A similar but smaller reduction was noticed at the back wall resulting from a baffle. Figure 6.21 shows the change in the lowest vibrational frequency as a function of the water level. The frequency was obtained from the fast Fourier transform of the strain time-history. The graph indicates the frequency decreased along with the water level until 75% water. After that, there was no further reduction in the frequency. The change in the vibrational frequency was very significant. The frequency without water was about seven times greater than that with 75% or higher water.

6.2 Numerical Study

117

Fig. 6.18 Strain time-history of the back wall for a lower water levels and b higher water levels

6.2 Numerical Study In order to complement the experimental study, a series of numerical modeling and simulations were conducted. In the numerical model, only a half of the box was modelled in order to minimize the computational time. Therefore, the model had a half plate of front and back walls as well as one side wall. Because the composite box was so thin, a shell element was selected to model the box. There were total 10,359 shell elements to model the front, back and side walls. On top of that, 1750

118

6 FSI Study of Structures Containing Fluid

Fig. 6.19 Maximum strain at the back wall as a function of water level

Fig. 6.20 Comparison of maximum strain at the front wall with and without baffle

elements were used for the top and bottom of the box, respectively. A total 2500 solid elements were used to model the impact hammer. The inside space of the box was meshed with 12,300 brick elements. Before deciding the mesh, a numerical study was conducted to check the mesh sensitivity, based on which the final mesh was decided in terms of the balance of the accuracy and the computational efficiency.

6.2 Numerical Study

119

Fig. 6.21 Plot of the lowest vibrational frequency

Depending on the water level, some portion of the inside box was modelled to represent the water medium while the rest of the internal volume as well as some outside volume of the box were modelled as air. This was necessary to consider the deformation of the composite box as well as water sloshing inside the box. As a result, all fluid media were modelled using a multi-material formulation, which means any fluid element can be filled by water or air depending on each situation. If there were no water, the fluid medium was not modelled at all. The composite box was made of woven fabrics with equal properties in the warp and fills directions. Therefore, the composite was approximated as an isotropic material, which had the elastic modulus 25 GPa, Poisson’s ratio 0.3, and density 1900 kg/m3 . An aluminum alloy with elastic modulus 710 GPa, Poisson’s ratio 0.33, and density 2700 kg/m3 was used to model the bottom wall with full constraints. The top wall was modelled using the property of Plexiglas with the elastic modulus 3.0 GPa, Poisson’s ratio 0.35, and density 1180 kg/m3 . The impactor was not modelled for a full pendulum motion from the initial position. Instead, it was placed next to the impact site with a given initial velocity. The initial velocity was determined from the initial angle position by neglecting any loss of energy during the pendulum motion. Between the impactor and the box wall, the impact-contact condition was used without friction. The numerical analysis was conducted using LS-Dyna3d [3] using the Arbitrary Lagrangian–Eulerian (ALE) technique. Figure 6.22 shows the numerical model which included the mirror image of the half model. Figure 6.23 illustrates the numerical results of the deformed box and internal water response, if there is any, for three different cases; no water, 50% water, and 100% water.

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6 FSI Study of Structures Containing Fluid

Fig. 6.22 Numerical model

The numerical model and the experimental model were not identical. For example, the experimental composite box did not have a uniform wall thickness while the numerical model was assumed to have a uniform all thickness. Therefore, it was not expected that both numerical and experimental results would be very close. However, both were expected to be comparable at least in terms of the qualitative sense. Figures 6.24 and 6.25 compare the experimental and numerical strains. The former figure was the strain at the front wall of the empty box while the latter figure was the strain at the side wall of the box with full of water. Both numerical and experimental results agreed well regardless of such a discrepancy which was mentioned above. Something to be noted was that the empty box model showed more high frequency oscillatory motions for the experimental result while the full water model had more high frequency oscillations in the numerical result even though those oscillatory motions had small magnitudes as compared to the major motions. Figure 6.26 shows the progression of water pressure wave from the impact of the front wall when the box was full of water. The high water pressure moved from the left side, i.e. the front wall, to the right side, i.e. the back wall. Initially, the pressure distribution was hemispherical, but the shape was disturbed as the top and bottom boundary interacted with the pressure wave. The overall displacement contours of the front were plotted in Fig. 6.27 for the empty box. Because of symmetry, only a half of the wall was modelled and the right

6.2 Numerical Study Fig. 6.23 Numerical analysis with a no water, b 50% water and c 100% water

121

122

6 FSI Study of Structures Containing Fluid

Fig. 6.24 Comparison of strains at the front wall between the experimental and numerical solutions when there was no water

Fig. 6.25 Comparison of strains at the side wall between the experimental and numerical solutions when there was full water

6.2 Numerical Study

Fig. 6.26 Propagation of water pressure wave inside the 100% full water box

123

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6 FSI Study of Structures Containing Fluid

Fig. 6.27 Front wall displacement at early (left) and later (right) times of the empty box

edge of the figure was the symmetric boundary. The displacement propagated quite symmetrically from the center of the box in all directions, as expected. On the other hand, the 50% full box showed asymmetric deformation of the front wall as plotted in Fig. 6.28. The 100% water box showed similar contours like the empty box on the front wall. However, the contour shapes looked like more circular at later times as seen in Fig. 6.29 when compared to that of the empty box. The deformation of the back wall was much more complicated. Figures 6.30 and 6.31 compare the deformation contours of the back wall with 50 and 100% water cases, respectively. Their deformation contours were quite different because of different water levels. The side wall deformations were also plotted in Figs. 6.32, 6.33 and 6.34. Those plots were obtained when the impactor was about to retreat from the maximum contact. As the composite box had more water, the deformation contours were more spread out to the back wall, which is the right boundary of the figures. In other words, more internal water produced more uniform deformation of the side wall.

6.3 Study of Hydrodynamic Ram

125

Fig. 6.28 Front wall displacement at early (left) and later (right) times of the 50% full box

6.3 Study of Hydrodynamic Ram Hydrodynamic Ram (HRAM) occurs when a projectile penetrates a structure containing fluid. The whole process consists of the following sequences: projectile impact on the front wall and penetration, projectile motion through an internal fluid medium, projectile impact on the back wall and exit through its penetration as illustrated in Fig. 6.35. Depending on the kinetic energy of the projectile, strength of the structural wall, and the viscosity of the fluid medium, the whole process may not occur. Instead, only some of the sequences may occur until the projectile loses its momentum and energy completely. One of the common examples of HRAM is the penetration of a projectile into a fuel tank, and HRAM was studied intensively during mid 1970s [4–12]. They conducted a series of experiments and developed simplified theories to understand and predict the phenomenon. With the development of computational power during the past decades, numerical modeling and simulations have been conducted extensively by coupling the fluid and structures, especially using the Arbitrary Lagrangian-Eulerian (ALE) technique, which could address many of the limitations that the traditional Lagrangian and Eulerian formulations have. One of the early numerical studies on HRAM was conducted in 1980 [13] using the Lagrangian formulation. Because of the limitation of the formulation,

126

6 FSI Study of Structures Containing Fluid

Fig. 6.29 Front wall displacement at a later time of the 100% full box

the coupled Eulerian and Lagrangian formulations as well as the smoothed particle hydrodynamics technique were applied to the HRAM process [14–18].

6.3.1 Numerical Modeling The numerical analyses were conducted using LS-Dyna [3]. The model presented here was the nominal case, and it was varied for a parametric study. The whole model consisted of a box, a projectile, and a fluid medium. In order to minimize the computational time, symmetries of the structure were considered as discussed in the last section so that only a quarter of the box was studied as seen in Fig. 6.36. However, when the symmetry was not satisfied with a change in a parameter for a parametric study, either the whole or a half box was modelled as necessary. The whole box had the dimensions of 0.75 m × 0.30 m with 0.15 m depth. The thickness of the box was uniform with 0.025 m.

6.3 Study of Hydrodynamic Ram

127

Fig. 6.30 Back wall displacement at early (left) and later (right) time of the 50% full box

The box walls were modelled using eight-node brick elements with a refined mesh near the impact area. Five elements were used along the thickness direction. The projectile was a sphere and it was assumed to be rigid. Table 6.2 shows the materials and the number of elements used for the box and sphere model, and Table 6.3 lists the material properties used in this study. For the aluminum wall, the Johnson–Cook model [19] was selected as the projectile penetrates into the aluminum wall. The Johnson-Cook plasticity model is provided in Eq. (6.1).   n    m  σ y = A + B ε p (1 + C ln ε˙ ) 1 − T ∗

(6.1)

where σ y is the yield stress, ε p is the effective plastic strain, ε˙ is the strain rate, and T −Tr oom . Here, the subscript ‘room’ T ∗ is the temperature ratio defined as T ∗ = Tmelt −Tr oom denotes the room temperature and ‘melt’ indicates the melting temperature. The coefficients for Eq. (6.1) were given in Table 6.3. In this study, the effective plastic strain at failure was assumed to be 0.2. The projectile was modelled as a steel sphere with a diameter 12.5 mm and its mass 8 g. Its initial velocity was set 900 m/s. For the nominal model, the spherical projectile impacted the front wall at its center in the perpendicular direction. The fluid was modelled using the multi-material ALE formulation, which means each element might be filled by one or another fluid material or the both. In this study, water and air were used for the multi-materials. The fluid mesh is shown in

128

6 FSI Study of Structures Containing Fluid

Fig. 6.31 Back wall displacement at early (left) and later (right) time of the 100% full box

Fig. 6.32 Side wall displacement of the empty box when the impactor was about to retreat

6.3 Study of Hydrodynamic Ram

129

Fig. 6.33 Side wall displacement of the 50% full box when the impactor was about to retreat

Fig. 6.34 Side wall displacement of the 100% full box when the impactor was about to retreat

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6 FSI Study of Structures Containing Fluid

Fig. 6.35 Hydrodynamic ram process Fig. 6.36 Box model with a projectile

Table 6.2 Material and element for solid

Part name

Material name Element number

Impact_wall (entry & Exit) 6063-T5 Other_wall

19,750 1800

PMMA_window

PMMA

3750

Projectile

Rigid

1000

6.3 Study of Hydrodynamic Ram

131

Table 6.3 Material properties for the solids Material

ρ (kg/m3 )

ν

A (GPa)

B (GPa)

n

C

m

6063-T5 Aluminum

2700

71

0.33

0.2

0.144

0.62

0

1

Steel

7830

207

0.28

–

–

–

–

–

PMMA

1180

3

0.35

–

–

–

–

–

E (GPa)

Fig. 6.37. The dark region was initially filled with water while the light color region was filled with air. The Gruneisen equation of state was used for water and its constants are shown in Table 6.4. The equation of state is given below   ρo C 2 μ 1 + 1 −

γo  μ 2

− a2 μ2



p= 2 + (γ o + aμ)Φ μ2 μ3 1 − (S1 − 1)μ − S2 μ+1 − S3 (μ+1) 2

(6.2)

Fig. 6.37 Fluid mesh

Table 6.4 Material properties for water and air Material ρo (kg/m3 ) (kg/m3 ) ν (m2 /s) C S1 (m/s)

S2 S3 γ0

a C4 C5 E0 (J/m3 )

Water

1000

0.89 × 10−3

1448 1979 0

0

0.11 3 –

Air

1.22

1.77 × 10−5

–

–

–

–

–

–

–

– 0.4 0.4 2.53 × 105

132

6 FSI Study of Structures Containing Fluid

where p is the internal pressure. Φ is internal energy per initial volume; and μ = V1 − 1 with V as the relative volume. In addition, ρo is the initial density, γo is the Gruneisen gamma, and a is the volume correction to the Gruneisen gamma. The linear polynomial equation of state was used for air as below: p = (C4 + C5 μ)E

(6.3)

where all the constants were also given in Table 6.4.

6.3.2 Model Validation The numerical model was validated against the experimental data [15]. The model for this validation was the same numerical model as described above. The movement of the projectile was compared between the experimental and numerical results. Both results showed a good agreement as shown in Fig. 6.38. The velocity of the projectile inside the fluid can be estimated using the drag force by the fluid. By integration of Newton’s second law, the velocity in the fluid can be expressed as 1 v  = ρ v vo 1 + 43 C D ρ pf dop t

(6.4)

where v and vo are the present and initial velocity of the projectile, and t is time. The initial velocity is the velocity just after the initial penetration of the front wall. In addition, ρ p is the density of the projectile mass, C D is the drag coefficient of the fluid and assumed to be constant, d p is the projectile diameter, and ρ f is the density of the fluid. Integration of the velocity yielded the travelling distance x of the projectile as below 

ρ f vo 3 1  ln C D t +1 (6.5) x= ρf 3 4 ρpdp C D 4 ρpdp The displacement of the projectile is plotted in Fig. 6.39, where the analytical solution using Eq. (6.5), and experimental and numerical results were compared. The results agreed well. The following results are presented for parametric studies with one parameter at a time.

6.3 Study of Hydrodynamic Ram

133

20mm

(a)

At time 0.084 msec (Exp. 32.12mm & 19.4mm)

41mm

(b)

29mm

At time 0.140 msec (Exp. 41.9mm & 28.2mm)

Fig. 6.38 Comaprison of experimental and numerical results for cavity sizes behind the projectile

6.3.3 Water Filling Level The first parameter was the level of water in the box as the level was changed from 0 to 25, 40, 50, 60, 75, and 100%, incrementally. When the water level was below 50%, the projectile did not interact directly with water. However, when the water level was 50%, the projectile was at the initial interface between the water and air. As a result, the motion of the projectile was significantly affected by the 50% water level. When the water level was greater than 50%, the water level influenced the trajectory of the projectile because of sloshing and pressure differential between upper and bottom parts of the projectile. When the box was full of water, the projectile moved straight toward the back wall. Figure 6.40 compares the vertical movement of the projectile between asymmetric pressure between its top and bottom sides. The projectile with 50% water significantly deviated from the horizontal trajectory. Figure 6.41 shows the comparison of the velocity of the projectile at two different

134

6 FSI Study of Structures Containing Fluid

Fig. 6.39 Displacement of projectile in water. A-analytical, B-experimental and C-numerical results

Fig. 6.40 Plot of projectile displacement in the vertical direction in the box for different water filling levels

water levels, 50 and 100%. Constant speeds of the projectile suggest that the projectile did not experience the drag force by the water. In other words, the projectile was in air. Therefore, when the water level was 50%, the projectile initially experienced the drag force by water, but the asymmetric pressure pushed the projectile out of water at a later time. Then, the projectile stayed at a constant velocity. Finally, the speed of the projectile was reduced as it penetrated the back wall. On the other hand, the 100% water case continually reduced the speed of the projectile until it penetrated

6.3 Study of Hydrodynamic Ram

135

Fig. 6.41 Comparison of velocity at two different water levels

the back wall as illustrated in the figure. Figure 6.42 shows the cavity shapes of two different water levels at the time of 1 ms. The 50% water case did not develop a full cavity while the 100% water case showed a full shape of cavity. The strain at the front wall was plotted in Fig. 6.43 at the location 50 mm left-hand side from the impact site. The strains were obtained at the backside of the surface so that they were tensile. When the water level was 100%, the strain was greater than that at the 75% water level. Water pressure was compared at the location which was 150 mm left and 75 mm front from the impact site. The 100% water case resulted in the greatest pressure, and Fig. 6.42 Comparison of shapes of cavity at two different level at the time of 1 ms

(a) 50% water

(b) 100% water

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6 FSI Study of Structures Containing Fluid

10-3

5

100% 75%

4

Strain

3 2 1 0 -1 0

1

2

3

Time (sec.)

4

5

6 10-4

Fig. 6.43 Strain at the front wall at the location 50 mm left-hand side from the impact site for different water levels

60% water case yielded the least pressure among the three cases. The peak pressure of the former case was more than twice of that of the latter case. The 75% water case was about in the middle of the two cases. When the water level was at or below 50%, the pressure was very small. From Fig. 6.44, the peak pressure pmax in the water can be expressed in terms of the water filling level h using the following empirical equation pmax = −2 × 107 (h − 0.5)2 + 2 × 107 (h − 0.5) + 5 × 105

(6.6)

when h ≥ 0.6, and h = 1 denotes the 100% water case. As the water level increased, the deformation at both entry and exit walls was greater. The deformation gradient of the penetrated edge was also greater with the higher water level even though the size of the penetrated hole was very close for all cases. The residual plastic strains at the front wall were compared for 50 and 100% water in Fig. 6.45 after the penetration of the projectile. The material failure was modelled using the erosion criterion. A larger plastic strain was resulted from a higher water level. When the front and back walls were compared for their plastic residual strains at the same water level, the back wall had a greater plastic strain than the front wall for all the cases. Even though the impact velocity was greater to the front wall because of the reduction of the speed of the projectile due to the drag force, the fluid shock pressure contributed to the greater plastic strain to the back wall. Figure 6.46 shows the pressure plot in water as the projectile reached the middle

6.3 Study of Hydrodynamic Ram

10

5

137

6

60% 75% 100%

Pressure (Pa)

4

3

2

1

0 0

0.5

1

1.5

Time (sec.)

2

2.5 10

-4

Fig. 6.44 Pressure at the location 150 mm left from the impact site and 75 mm in front

(a) 50% water

(b) 100%water

Fig. 6.45 Comparison of residual plastic strain of the front wall after penetration

of the box while the water level was different. As expected, the water level affected the pressure contours significantly. Especially the water pressure was not symmetric from the centerline unless the box was full of water.

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6 FSI Study of Structures Containing Fluid

Fig. 6.46 Water pressure contours when the projectile advanced to the middle of the box with a 50% water, b 75% water and c 100% water, respectively

6.3.4 Wall Thickness In this study, only the wall thickness of the box was changed while the rest of the parameters remained the same as their nominal values. The wall thickness was varied to 1.5 and 3.5 mm from the nominal thickness of 2.5 mm. The box had 100% water during this study. The wall thickness influences HRAM in terms of two aspects. One is the stiffness of the box and the other is the reduction in the projectile velocity during its penetration process. As a result, the speed of the projectile would be reduced during the entry penetration as well as the exit penetration. Figure 6.47 shows the velocity as a function of time as the wall thickness was varied. In this case, since the impact speed was very high, the change in the velocity during the penetration process was not significant, which resulted in a small difference in their velocity comparison. A small but notable reduction occurred during both entry and exit penetration. The average loss in the projectile velocity during each penetration was 13 m/s as the wall thickness increased by 1.0 mm. Therefore, the exit speed of the projectile from 3.5 mm wall was about 50 m/s slower than that of 1.5 mm wall.

6.3.5 Projectile Impact Velocity The next parameter was the projectile speed. Its nominal value was 900 m/s, and the speed was changed to 300 m/s, 600 m/s, and 1200 m/s, respectively. The impact speed affected the penetration time and its loss in the speed after penetration. As the impact velocity increased, the time for penetration was shorter. Figure 6.48 shows

6.3 Study of Hydrodynamic Ram

139

900 1.5 mm 2.5 mm 3.5 mm

Velocity (m/s)

800 700 600 500 400 0

0.5

1

1.5

2

2.5

3

3.5 10-4

Time (sec.) Fig. 6.47 Plot of velocities for three different wall thicknesses

1200

Velocity (m/s)

1000 800 1200m/s

600

900m/s

400 600m/s

200 0 0

300m/s

0.2

0.4

0.6

Time (sec.)

0.8

1 10

-3

Fig. 6.48 Projectile velocity plot for different initial impact speeds

the change in the projectile speed as a function of time for different initial impact speeds. The entry penetration is seen as the steep change in the velocity from the start. The duration of the initial steep change was longer as the impact speed decreased. The most contrasting case was between 1200 and 300 m/s. However, the lower speed shows the smaller reduction in speed during the travel through the fluid medium because the drag force is proportional to the square of the projectile speed.

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Table 6.5 Change of velocity after each wall penetration with different speeds

Initial velocity (m/s)

After front wall penetration

At arrival of back wall (m/s)

After back wall penetration

300

252 m/s (16%)

122

85 m/s (30.3%)

600

508 m/s (16%)

260

243 m/s (6.5%)

900

773 m/s (14.1%)

414

393 m/s (5.1%)

1200

1043 m/s (13.1%)

561

539 m/s (4.0%)

The major change in speed during each penetration as well as travelling through the water medium is listed in Table 6.5. The table shows that the percentage loss of the speed during the entry penetration is greater for the slower initial speed. The same characteristics is true at the exit wall. More important notes were the damage of the back wall resulting from the water pressure. When the impact velocity was 300 m/s, the velocity of the projectile travelling in the water was not high enough to produce damage to the back wall before the projectile arrived there. On the other hand, when the initial velocity was 600 m/s or higher, the fluid pressure resulted in damage to the back wall before the arrival of the projectile. This also contributed to a small loss in the velocity during the exit penetration when the initial speed was 600 m/s or higher as compared to that of 300 m/s. The former cases had about 5% loss during the exit penetration while the later case showed 30% loss in speed. The total change in the velocity of the projectile was more or less a linear function. Since the mass of the velocity is constant, the change in velocity is proportional to the change in the linear momentum. The empirical formula for the total loss in the linear momentum for the projectile of 8 g was expressed using the following equation: (mv)loss = 0.004v + 0.504

(6.7)

in which v is the initial impact speed in the unit of m/s, and the linear momentum is expressed in the unit of kg m/s. This equation suggests that the linear momentum of the projectile has a total loss 0.508 kg m/s as the impact speed increases by 1 m/s. Likewise, the HRAM process showed a total loss in the kinetic energy as expressed by the following empirical equation: (K E)loss = 0.0063v2 + 0.2v + 50

(6.8)

where the kinetic energy has the unit of N m. The greater speed generated a higher gradient of deformation around the edge of the penetrated hole of the front and back walls, respectively. However, the deformation gradient of the front wall seemed to approach a limit value as the speed increased.

6.3 Study of Hydrodynamic Ram

(a) 600m/s

141

(b) 1200m/s

Fig. 6.49 Plots of residual plastic strains at the exit wall with two different velocities

The increment in the deformation became less as the speed increased. On the other hand, the deformation of the back wall did not show such characteristics. The back wall showed almost the same deformation gradient along with the increase in the projectile speed. The penetration of the front wall has a smooth round shape, but the back wall did not have necessarily smooth round shape. Such a difference between the front wall and back wall was resulted from the fluid pressure applied to the back wall during the propagation of the projectile in the fluid medium. Since the residual plastic strain was shown for 900 m/s in Fig. 6.45, Fig. 6.49 shows those for 600 m/s and 1200 m/s, respectively. The plots show major diagonal tearing of the exit hole when the initial velocity was 1200 m/s. At this velocity, the fringe pattern was almost the square shape. The fluid shock pressures were compared in Fig. 6.50 for different impact velocities. The case for 900 m/s was already shown in Fig. 6.46 so that the current figure only included other velocities. The plots were made when the project reached the middle of the box. Since each case had a different projectile speed, the arrival time for the projectile to the mid-site was also different. The figure shows that the higher projectile speed produced greater pressure wave, which also influence the damage of the back wall. The peak pressure at the location 150 mm left from the impact site and 75 mm in front was fitted very well using the following quadratic function: pmax = 6.7v2 − 2000v + 9x105 where pmax is the peak pressure and v is the impact velocity, respectively.

(6.9)

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6 FSI Study of Structures Containing Fluid

Fig. 6.50 Water pressure contours when the projectile advanced to the middle of the box with the impact speed of a 300 m/s, b 600 m/s and c 1200 m/s, respectively

6.3.6 Projectile Mass The next variable was the mass of the projectile. The nominal mass was 8 g, which was varied to 4 g and 16 g, respectively. A heavier mass increases the linear momentum and kinetic energy, which made the time for wall penetration shorter. This means the projectile velocity is greater for the heavier mass such that it travels faster to the exist side wall. This is shown clearly in Fig. 6.51. Empirical equations were also developed to represent the total loss in the linear momentum and kinetic energy as the projectile mass was varied. The resultant equations are provided below: 

(mv)loss

  m − mo m − mo 2 m o − 0.2 = (0.004v + 0.504) 1 + 0.4 mo mo (6.10) 

(K E)loss = 0.0063v2 + 0.2v + 50





   m − mo 2 m − mo m o − 0.26 1 + 0.66 mo mo



(6.11) where m and m o are the projectile mass and its reference mass of 8 g as used in the equations. Combining the effect of wall thickness and the projectile mass, the equations for the total loss in the linear momentum and kinetic energy are (mv)loss

6.3 Study of Hydrodynamic Ram

143

1200 4 g of mass 8 g of mass 16 g of mass

Velocity (m/s)

1000 800 600 400 200 0

1

2

3

4

time (sec.)

5 10

-4

Fig. 6.51 Velocity vs time for different projectile masses 



= (0.004v + 0.504) 1 + 0.4



  m − mo m − mo 2 h − h0 m o − 0.2 1 + 0.13 mo mo ho

(6.12)

(K E)loss

    

 h − ho m − mo m − mo 2 1 + 0.08 = 0.0063v2 + 0.2v + 50 1 + 0.66 m o − 0.26 mo mo ho

(6.13) In the equation, h is the thickness of the wall expressed in mm, and h o is the reference wall thickness of 2.5 mm. These equations were used to predict the results as shown in Tables 6.6 and 6.7. The empirical equations represented the numerical results well. Table 6.6 Comparison of numerical and predicted loss of linear momentum Initial velocity (m/s)

Projectile mass (kg)

Wall thickness (mm)

Numerical loss (kg m/s)

Predicted loss (kg m/s)

% error

300

0.008

2.5

1.72

1.70

−1.2

600

0.008

2.5

2.86

2.90

1.4

900

0.008

2.5

4.03

4.10

1.7

1200

0.008

2.5

5.29

5.30

0.2

900

0.004

2.5

3.02

3.08

2.0

900

0.016

2.5

4.90

4.92

0.4

900

0.008

3.5

4.25

4.24

−0.2

900

0.008

1.5

3.82

3.82

0.0

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6 FSI Study of Structures Containing Fluid

Table 6.7 Comparison of numerical and predicted loss of kinetic energy Initial velocity (m/s)

Projectile mass (kg)

Wall thickness (mm)

Numerical loss (N m)

Predicted loss (N m)

% error

300

0.008

2.5

662

617

−6.8

600

0.008

2.5

2410

2320

−3.7

900

0.008

2.5

5230

5150

−1.5

1200

0.008

2.5

9200

9120

0.1

900

0.004

2.5

3160

3120

−1.3

900

0.016

2.5

7310

7220

−1.2

900

0.008

3.5

5400

5400

0.0

900

0.008

1.5

5050

5060

0.2

The strains were plotted at the entry wall in Fig. 6.52 for different projectile masses. The strain was obtained at the location 50 mm left from the impact site. As expected, the higher mass resulted in larger strain. Even though the speed in water was different with a different mass, the water pressure was not much different for different masses. 10-3

6 5

4g 8g 16 g

Strain

4 3 2 1 0 -1 0

1

2

3

time (sec.)

4

5 10

-4

Fig. 6.52 Strain at the front wall at the location 50 mm left-hand side from the impact site for different projectile masses

6.3 Study of Hydrodynamic Ram

145

6.3.7 Impact Angle So far, the projectile impacted the box in the perpendicular direction. In this study, impact angle was varied from the perpendicular orientation. That is, the impact angle is oblique as sketched in Fig. 6.53, where the angle is zero for the perpendicular impact. Now, the impact angle was varied to 30° and 45°, respectively. As the impact angle was set to 30°, the change in the projectile velocity is shown in Fig. 6.54 as the water level was also varied. Because the impact site was equivalent to the 50% water level, the projectile did not interact with water just after the entry penetration for a short period for the 40% water case. That is why there is a temporary constant velocity for that case. In addition, because the contact duration with water was shorter for the 40% water case, the exit velocity was higher as shown in the figure. On the other hand, the projectile was in contact with water just after the entry penetration Fig. 6.53 Oblique impact to the box containing fluid

Oblique impact angle H H/2

1000 30deg impact-40% full water 30deg impact-60% full water 30deg impact-100% full water

Velocity (m/s)

800

600

400

200

0 0

2

4

time (sec.)

6

8 10-4

Fig. 6.54 Projectile velocity with 30° impact to partially or fully filled water tank

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6 FSI Study of Structures Containing Fluid

1000 45deg impact-40% full water 45deg impact-60% full water 45deg impact-100% full water

Velocity (m/s)

800

600

400

200

0 0

0.2

0.4

0.6

time (sec.)

0.8

1 10

-3

Fig. 6.55 Projectile velocity with 45° impact to partially or fully filled water tank

for both 60 and 100% full cases. However, the drag force was slight greater for the 100% water case, which resulted in a slightly lower velocity. The consumption of the kinetic energy was greater as the oblique angle increased. Furthermore, the higher impact angle resulted in a longer distance inside the box before meeting the opposite wall for exit. This yielded a greater reduction in the velocity. Figure 6.55 is for the 45° impact angle. Comparing Figs. 6.54 and 6.55 confirms the discussion. As the projectile progressed through the water inside the box, its motion was deviated from the projected line of path based on the impact angle. In other words, the normal distance from the projected line was plotted for each case as shown in Figs. 6.56 and 6.57 for each impact angle. The deviated distance was greater for the oblique impact when the box had less water. While the 30° impact resulted in a continuous deviation from the projected line, the 45° impact showed initial deviation which was followed by a later return toward the projected line.

6.3.8 Projectile Shape So far, a spherical shape of projectile was studied because this shape did not require titling of the projectile during its motion. However, a bullet shape like projectile was also considered. The geometric dimension of the bullet-like projectile is shown in Fig. 6.58. The projectile had the nominal dimensions, L = 48 mm, b = 12 mm, and D = 8 mm. For this projectile, only the length L was varied while all other dimensions remained constant. This changed the ratio L/D such that it was 3, 4.5, and 6, respectively. Two scenarios were studied. The first scenario was that all three

6.3 Study of Hydrodynamic Ram

147

0.07 30deg impact-40% full water 30deg impact-60% full water 30deg impact-100% full water

Deviated Distance (m)

0.06 0.05 0.04 0.03 0.02 0.01 0 -0.01 0

0.2

0.4

0.6

0.8

1 10-3

time (sec.)

Fig. 6.56 Deviated distance from the direction of projected path of the projectile for 30° impact angle

Deviated Distance (m)

0.03

0.02

0.01 45deg impact-40% full water 45deg impact-60% full water 45deg impact-100% full water

0 0

0.2

0.4

0.6

time (sec.)

0.8

1 10

-3

Fig. 6.57 Deviated distance from the direction of projected path of the projectile for 45° impact angle

projectiles had the equal mass. This was achieved by changing the density of the material so that the total mass became 8 g as before. The other scenario assumed a constant density such that each projectile had the mass of 8 g, 11 g, and 16 g depending on the ratio L/D = 3, 4.5, and 6, respectively.

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6 FSI Study of Structures Containing Fluid

Fig. 6.58 Bullet-like projectile

For the projectiles of the same mass, the bullet-like projectile resulted in a lower drag force than the spherical projectile because the diameter of the bullet-like projectile was smaller as compared to that of the spherical shape projectile. This fact provided a lower decrease in speed in water for the bullet-like projectile. If the density remained constant, the longer projectile had a heavier mass and a smaller reduction in speed. This made that the constant density case was very similar to the previous study with the mass change. However, one thing to be noted was that the bullet-like projectile did not show any notable drop in velocity during the wall penetration. Figure 6.59 shows the velocity change of the bullet-like projectile. This suggests that the bullet-like projectile is more efficient for penetration. This is because of the more gradual change of the cross-section and a smaller cross-section of the bullet-like projectile. Another interesting aspect was that the three bullet-like projectiles arrived at the back wall with a small delay in time from one another while 950 L/D=3 L/D=4.5 L/D=6

Velocity (m/s)

900

850

800

750

700 0

0.5

1

1.5

time (sec.)

2

2.5

3 10

Fig. 6.59 Projectile velocities for different projectile lengths with different masses

-4

6.3 Study of Hydrodynamic Ram

(a)

149

(b)

Fig. 6.60 Residual plastic strain at the exit wall for bullet-like projectile with a L/D = 3 and b L/D =6

the three spherical projectiles showed much more significant delays in time to reach the back wall. When the residual plastic strain was compared at the entry and exit walls, the longer projectile yielded a larger plastic strain than the shorter one if both of them had the equal mass. If the projectile was both longer and heavier, the residual plastic strains were certainly greater for both walls. Figure 6.60 plots the plastic strain at the exit wall. When the plastic deformation was compared between the spherical and bullet projectiles, the bullet-like projectile produced much smaller plastic deformation than the other for the equal mass. One of the drawbacks for the bullet-like projectile is it is prone to deviate from its intended trajectory. Figures 6.61 and 6.62 plotted the deviated trajectory of the bullet-like projectile. The former was for the equal mass of projectile while the latter was for different masses of projectiles. The results showed that the deviated trajectory was influenced by both the length and mass of the projectile. A larger deviation from the projected trajectory was caused by a shorter and lighter projectile. When the projectile was long such as L/D = 6, it has a negligible deviation. The projectile with the ratio L/D = 4.5 could be deviated in either direction of the projected path depending on the mass. When the projectile was 11 g, its deviation was in the negative direction. This was the opposite when the projectile weighed 8 g. The fluid pressure influenced the trajectory deviation. The surface area of projectile subjected to high fluid pressure was almost the same regardless of the projectile length as seen in Fig. 6.63. Under that circumstance, a longer projectile was more stable in terms of changing the propagation angle. Therefore, the projectile deviation was smaller.

150

6 FSI Study of Structures Containing Fluid

10-3

Trajectory Deviation (m)

2

L/D=3 L/D=4.5 L/D=6

1

0

-1

-2 0

1

2

3

4

Time (sec.)

5 10

-4

Fig. 6.61 Plot of trajectory deviations of the bullet shape projectiles with equal mass

10-3

Trajectory Deviation (m)

2

L/D=3 L/D=4.5 L/D=6

1.5 1 0.5 0 -0.5 -1 0

1

2

3

Time (sec.)

4

5 10

-4

Fig. 6.62 Plot of trajectory deviations of the bullet shape projectiles with different masses

6.4 Summary

151

Fig. 6.63 Pressure plots around the bullet-like projectile with a L/D = 3, b L/D = 4.5 and c L/D =6

6.4 Summary This chapter presented both experimental and numerical work for fluid tank subjected to impact loading. The impactor may or may not penetrate the fluid tank. The latter case is called HRAM. During each study, some parameters were changed to find their effects on the structural responses. Impact force was dependent on the water level even though the initial impact condition remained the same. The dynamic response of a structure was also very different depending on the fluid level inside the structure, as expected. However, the response was not a monotonic change. In other words, the higher water content did not necessarily yield the largest strain. The vibrational frequency of the structure was reduced with water until the water level reached 75%. After that, any additional water did not change the vibrational frequency. Inserting a baffle into the tank did not change the impact force but influenced the strain response of the back wall, especially when the water level was greater than 60%. This was resulted from the change in the pressure wave propagation toward the back wall, which resulted from the inserted baffle. While this impact study did not consider penetration of the impactor into the fluid tank, the HRAM study considered penetration of both spherical and bulletlike projectiles into the fluid tank. A series of parametric studies were conducted to better understand HRAM effects. The parameters included the water level, thickness of structure, initial impact velocity, mass of projectile, initial impact angle, and different shapes of the projectile. Some empirical equations were developed from

152

6 FSI Study of Structures Containing Fluid

the numerical simulation to compute the loss in the linear momentum as well as kinetic energy as the projectile went thought the whole process. The projectile shape, mass, and speed influenced the wall penetration of the structure. The bullet-like projectile was more efficient than the spherical shape projectile because the cross-sectional area was smaller and of a gradual change for the former. The deviation from the projected line of path was more likely and greater for a shorter and lighter bullet-like projectile. The exit wall had greater plastic deformation after the exit of the projectile because the exit wall was affected by the pressure wave as the projectile travelled through the fluid medium. In many cases, such fluid pressure resulted in pre-damage even before the projectile arrived at the back wall.

References 1. Kwon YW, South T, Yun KJ (2017) Low velocity impact to composite box containing water and baffles, composite structures. ASME J Pressure Vessel Technol 139(3):031304 2. Kwon Y, Yun K (2017) Numerical parametric study of hydrodynamic ram. Int J Multiphys 11(1):15–47 3. LS-DYNA Keyword User’s Manual (2015) Livermore Software Technology Corporation 4. Fuhs AE, Ball RE, Power HL (1974) FY 73 hydraulic ram studies. NPS-57Fu74021 5. Power HL (1974) FY 74 experimental hydraulic ram studies. NPS-57Ph74081 6. Ball RE (1974) Prediction of the response of the exit wall of the nwc 50 cubic feet tank to hydraulic ram. NPS Report, NPS-57BP74031 7. Ball RE (1974) Aircraft fuel tank vulnerability to hydraulic ram: modification of the north rop finite element computer code br-1 to include fluid-structure interaction—theory and user’s manual for br-1hr. NPS-57Bp74071 8. Power HL (1975) FY 75 experimental hydraulic ram studies. NPS-57Ph75061 9. Page B (1975) Entry wall strain measurements during hydraulic ram. Master’s thesis, NPS 10. Patterson JW (1975) Fuel cell pressure loading during hydraulic ram. Master’s thesis. NPS 11. Duva AN(1976) Hydraulic ram effect on composite fuel entry walls. Master’s thesis, NPS 12. Ezzard HS Jr (1976) A study of the failure of joints in composite material fuel cells due to hydraulic ram loading. Master’s thesis, NPS 13. Kimsey KD (1980) Numerical simulation of hydrodynamic ram. U.S. Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD, Report No. ARBRL-TR-02217 14. Sparks CE, Hinrichsen RL, Friedmann D (2005) Comparison and validation of smooth particle hydrodynamics (SPH) and coupled Euler Lagrange (CEL) techniques for modeling hydrodynamic ram. AIAA Paper No. 2005-2331 15. Varas D, Zaera R, Lopez-Puente J (2009) Numerical modelling of the hydrodynamic ram phenomenon. Int J Impact Eng 36(3):363–374 16. Liang C, Bifeng S, Yang P (2011) Simulation analysis of hydrodynamic ram phenomenon in composite fuel tank to fragment impact. In: 3rd international conference on measuring technology and mechatronics automation (ICMTMA), Shanghai, China, 6–7 Jan, pp 241–244

References

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17. Poehlmann-Martins F, Gabrys J, Souli M (2005) Hydrodynamic ram analysis of non-exploding projectile impacting water. ASME paper no. PVP2005-71658 18. Vignjevic R, De VT, Campbell JC, Bourne NK (2002) Modelling of impact on a fuel tank using smoothed particle hydrodynamics. In: 5th conference on dynamics and control of systems and structures in space (DCSSS), Kings College, Cambridge, UK, 18–22 July 19. Johnson GR, Cook WH (1983) A constitutive model and data for metals subjected to large strains, high strain rates, and temperatures. In: Proceedings of seventh international symposium, The Hague, The Netherlands, pp 1–7

Chapter 7

Structural Coupling by FSI

Independent structures do not interact each other unless there is a mechanism to couple their responses. One of the coupling mechanisms is a fluid medium. Dynamic response of a structure (called the primary structure) can be transferred to a neighboring structure (called the secondary structure) through a fluid medium, especially when the fluid is water. Then, the secondary structure responds to the motion of the primary structure. Conversely, the response of the secondary structure also influences the dynamic response of the primary structure. As a result, both structures influence each other through FSI. This chapter presents the coupling of independent structures via a fluid medium through FSI. One structure is excited by an external force such as impact loading. Then, the response of other structures is studied. First of all, fluid coupling of two plate structures was examined followed by coupling of multiple plate structures [1, 2]. Then, fluid coupling of two concentric cylinders was presented [3]. These studied were conducted experimentally by designing and fabricating the test set-up. Finally, numerical studies were undertaken to complement the experimental study.

7.1 Description of Experiments for Plate Structures A test setup was designed and fabricated to investigate coupling of multiple flat structures through a fluid medium. The test compartments were designed to simulate independent structures that were connected by a fluid medium. With a low velocity impact, the fluid causes a coupling of the structures with the added mass effect. The test set-up was designed such that multiple parameters could be studied using the same test set-up so that their results could be compared one another. First of all, different numbers of plates were selected for the coupling study as well as different plate thicknesses. Secondly, the volume of fluid between any two neighboring plates

© Springer Nature Switzerland AG 2020 Y. W. Kwon, Fluid-Structure Interaction of Composite Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-57638-7_7

155

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7 Structural Coupling by FSI

could be varied. Finally, the test set-up could change the distance between two neighboring plates. Such a flexible test set-up could be used to assess each parameter on the coupling effect of the structures. To be able to conduct different combinations of tests, the designed test set-up had multiple compartments so that multiple plates could be studied along with different distances between two neighboring plates. In order to meet the prescribed design criteria, the test set-up was constructed out of PMMA (Polymethyl methacrylate) and aluminum alloy. The PMMA material is used for the two walls and top of the test set-up and for holding composite sheets along their edges in their grooves. Figure 7.1 shows one of the two identical wall pieces made of the PMMA. Each wall had five grooves at equal spacing. The top panel is shown in Fig. 7.2 and has five equally spaced grooves. Locations of the grooves in the side and top panels matched one another so that composite plates to be tested could be inserted into the groves. The distance between any two neighboring grooves Fig. 7.1 PMME side panel

Fig. 7.2 PMME top panel

7.1 Description of Experiments for Plate Structures

157

Fig. 7.3 Aluminum bottom panel

was 5.94 cm. The width of the grooves was designed to fit the thickest composite plates to be tested. If composite plates were thinner than the width of the groove, shims were fabricated to hold the thin plates in the grooves. It was made sure that no fluid, i.e. water, would leak through the grooves so that the fluid could be maintained at the same level between two neighboring plates during the tests. The bottom part of the test set-up was fabricated using a solid aluminum base, which was constrained from the movement of the test setup during impact testing. The bottom plate had two major grooves such that two wall plates could be inserted in them. In addition, the bottom plate had five grooves with the same width and spacing as the side wall panels such that the bottom edge of each plate could be supported by the grooves like the side walls. Figure 7.3 shows the aluminum bottom panel. The bottom plate had also grooves around its four edges as shown in the figure. Four PMMA panels were inserted into the grooves of the bottom plate as shown in the assembled structure in Fig. 7.4. The surrounding four panels were to hold water if composite plates were broken during the test. Figure 7.4 shows all the assembled panels with two composite plates inserted at the most remote slots before putting the top panel. After the top panel was placed, the whole set-up was secured by four metal rods using bolts and nuts so that they could stay together. Then, the whole set-up was secured to a vibration isolation table to minimize its vibrations during testing as shown in Fig. 7.5. As shown in the figure, impact loading was applied to the front plate. The impact loading was applied using the pendulum motion. The pendulum impactor had four major parts. It had a supporting stand, a rotating rod and a mounting plate, a pendulum arm, and an impactor which is a hemispherical shape containing a load cell. The pendulum impactor was supported by a Lateral Excitation Stand (Model 2050A). The pendulum was attached to the collar of the lateral arm with necessary adjustment along the length of the lateral arm, which was also adjustable along the vertical axis to ensure the precise alignment of the impactor to the impact location of

158

7 Structural Coupling by FSI

Fig. 7.4 Assembled set-up

the composite plate. In this study, impact was applied to the center of the front plate. When all the system was at rest before testing, the tip of the impactor was barely in contact with the center of the composite plate. In order not to rotate the stand about the vertical axis, the stand was attached to the vibration isolation table as seen in Fig. 7.6. A mounting plate was prepared using a 6061 aluminum plate whose dimensions were 0.3048 m by 0.2032 m and 6.35 mm thick. The plate had holes drilled through as sketched in Fig. 7.7. The plate with holes was mounted to the test stand using U-bolts as shown in Fig. 7.8. An aluminum tube with the outer diameter of 2.54 cm and wall thickness of 0.635 cm was secured at both end of the tube using two pillow block roller bearings, which were attached to the mounting plate at its bottom side. The roller bearings provided the pendulum motion for the impactor arm as shown in Fig. 7.9, which shows a protractor to measure the rotational angle of the pendulum. The protractor had the 90-degree mark aligned to the vertical axis along the pendulum arm. In actual impact testing, this was referenced as the 0-degree angle because the pendulum arm was raised from the reference angle. For example, 30-degree impact meant the pendulum arm was raised 30-degrees from this reference angle. The pendulum arm as shown in Fig. 7.5 was fabricated out of an aluminum bar such that its final dimension was 0.5715 m by 50.8 mm by 12.7 mm. The aluminum bar had a 25.4 mm diameter hole so that it could fit the aluminum tube that was supported by the roller bearing. Another hole was drilled on the front side of the arm so that a hemispherical shape of impactor could be attached as a threaded connection point. This hole was located such that the impactor strikes the center of the composite plate. The impactor was made of a steel hemisphere as shown in Fig. 7.10. The outer diameter of the hemispherical shell was 38.10 mm and wall thickness 3.195 mm. Because the load cell to be used had threaded connections at both sides, a threaded nut was fitted to the steel hemisphere so that the load cell could be attached to it. The nut

7.1 Description of Experiments for Plate Structures

Fig. 7.5 Experimental set-up with pendulum type impactor

Fig. 7.6 Impact pendulum secured to table top

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Fig. 7.8 Mounting plate on pendulum support stand Fig. 7.9 Pendulum arm with protractor

7 Structural Coupling by FSI

7.1 Description of Experiments for Plate Structures

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Fig. 7.10 Steel hemisphere a isometric view and b side view

was secured to the steel hemisphere using epoxy. The completed impactor without the load cell is shown in Fig. 7.11. The impactor was attached to the pendulum arm through the load cell which is shown in Fig. 7.12. When the impactor struck the composite plate, the load cell measured the impact force time-history. Once the test set-up was completed, composite plates were prepared for testing. Nine uniaxial strain gages were attached to each composite plate to measure the strain as shown in Fig. 7.13. The strain gages had a resistance of 350  and the gage factor of 2.15. The strain gages were installed evenly spaced on the composite plate as sketched in Fig. 7.14. The surface of the composite plate with strain gages attached was installed in the test set-up such that the strain gages would not exposed to water. In other words, strain gages were located on the outer face of each composite plate. Because the impactor struck the center of the front composite plate, the strain gage could not be located at its center. As a result, the strain gage was placed a little offset from the center as shown in the right side sketch of Fig. 7.14. When each composite plate was installed into the grooves of every wall, clear silicone was placed on the grooves to make sure water tight. Thin strips made of

Fig. 7.11 Impactor connection a set-up spacer and nut b filled with epoxy

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Fig. 7.12 Load cell with a side view and b end view

Fig. 7.13 Strain gage attached to a composite plate

PMMA were also inserted into the grooves if the plate thickness was smaller than the groove width. Once all the composite plates were installed in the test set-up and secured, the impact angle was decided. The impact arm was raised to the predetermined angle measured using the attached protractor, and it was released from that position. For each impact testing, the impact force and strain readings were recorded using NI LabVIEW Signal Express 2012 software suite. Once the data were collected, they were plotted and analyzed using MATLAB. The front plate was installed at the first groove all the time. Depending on in what groove the back plate was placed, it was called differently. If the back plate was in the second groove, it was called Spacing 1. If the back plate was inserted in the third groove, the configuration was called Spacing 2. Likewise, Spacing 3 was for the back

7.1 Description of Experiments for Plate Structures

163

Fig. 7.14 Locations of strain gages attached to a composite plate

plate in the fourth groove, and Spacing 4 was for the last fifth groove. Spacing 1, 2, or 3 was considered in this study, respectively. The plates had three different thicknesses constructed out of the same woven fabrics composites. The plate thicknesses were 0.794 mm, 1.588 mm, and 3.175 mm, respectively. In order to distinguished different plate thicknesses, each thickness was given a name such as T4 for the thickest and stiffest plate, T2 for the middle thickness, T1 for the thinnest plate. The number in the notation suggests that the ratio of the thickness of each plate relative to the thinnest plate which was normalized to be unity. When two different thicknesses of the plates were studied, their spacing configuration was Spacing 2. On the other hand, three plates were tested simultaneously, the plates were placed in the first three grooves. In every test, impact was given to the front plate (or called the first plate) When two plates were tested, it was called like FT2/BT4, which indicated the front plate had the thickness of T2 while the back plate had the thickness of T4. As stated, the spacing was always Spacing 2 for this case. When three plates were placed in the test set-up, the middle plate was added to the notation such as FT2/MT1/BT4, which suggests the middle plate has the thickness T1. These notations provided a clear picture of the plates used in each test. Testing at each case was repeated ten times to make sure the test results were repeatable and reliable. The test results confirmed the repeatability. That is, the data from one test matched well with those from other tests. Figure 7.15 shows a comparison between two measured impact forces at the same condition, i.e. 75% water between two composite plates. Both graphs were almost identical. As a result, one representative results were selected for each test case and compared.

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Fig. 7.15 Two repeated impact force plots at the same condition

7.2 Experimental Results of Two Plates of Same Thickness First, two composite plates with the same thickness were tested. This configuration was called FT2/BT2 with Spacing 2. Each composite plate was a woven fabric composite made of 0 and 90°, which had elastic modulus 59.3 GPa and the density 1380 kg/m3 . The spacing between the two plates was 122 mm. The entire impactor had the mass 2.15 kg and the impact velocity was set to 1.70 m/s. This velocity was computed from the initial potential energy by neglecting any energy loss from the bearing and air. The impact force was examined as the water level between the two plates was varied. The water level was measured in terms of how much portion of the plates were in contact with water initially. In other words, if a half of the plate was in contact with the still water, it was called 50%. Of course, sloshing and/or splashing may occur during the impact on the plate, which may result in the change in the wet surface. The impact force was very dependent of the water level because the amount of water provided added mass effect as well as the coupling between the two plate structures. In order not to make the plots too crowded, the time-histories of impact forces were plotted in multiple figures. Figures 7.16, 7.17, 7.18 and 7.19 show plot the impact forces at different water levels. As seen in the figures, there is no clear correlation between the impact force and the water level. Not only the peak force but also its variation during the contact changed significantly with the water level. The maximum impact force was generally higher but occurred later when there was less water between the two plates. However, the maximum force did not gradually

7.2 Experimental Results of Two Plates of Same Thickness

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decreased as a function of the water level. Figure 7.20 plots the maximum force vs. the water level. Their correlation was complex. The largest maximum impact force occurred when there was 35% water. The smallest maximum force occurred with the 55% water level. The peak impact force showed a small variation from 75% to 100% water levels as well as 0% to 15% water levels, respectively. However, this does not necessarily mean that the effect of FSI is about the same within the range of the water level as discussed later. When the water level was lower than 60%, the impact force had one or at most two peaks. The water level at least 60% or higher showed more oscillatory shapes with multiple local peaks. One thing to be noted was that the maximum impact forces occurred earlier when the water level was high. For instance, the maximum force

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Fig. 7.19 Impact force for FT2/BT2 and Spacing 2 with 75 and 100% water levels

occurred around at 0.0025 s when the water level was 75 or 100% as seen in Fig. 7.19. The maximum force occurred around at 0.007 s if there was no water. Other cases showed the maximum forces between 0.004 s and 0.005 s. The results suggested that the rate of increase in the impact force until the first peak was greater with more water, in general. This feature in the time-history of the impact force influenced the dynamic response of the plate. Even though nine strains were attached to each plate, some gages failed during the test without collecting data. However, most of them worked through every test. Therefore, only successfully measured strains were presented here.

7.2 Experimental Results of Two Plates of Same Thickness

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The time-history of the impact force resulted in dynamic response of each composite plate. If there was no water between the two plates, the plate at the back should have absolutely no response. However, the back plate showed a very small strain response with no water. Because the test set-up was not perfectly rigid, the test structure could have a very minor motion during the impact. This caused a minor dynamic response of the back plate. As a result, this response was neglected. While the maximum impact force was greater with less water in general, the maximum strains in magnitude was not higher with less water. In other words, the higher level water produced larger strains. This seems to be contradicting. However, the maximum strain depends on many different variables such as the magnitude of the loading, the loading rate, its duration, and the dynamic characteristics of the structure. In order to illustrate this aspect, let’s consider a single degree of freedom system made of a mass and a spring with the natural frequency ωn . Two different loading profiles as shown in Fig. 7.21 were applied to the spring-mass system. One loading was slower but a higher peak value while the other loading was faster but a smaller peak value. Each loading was applied to the spring-mass system, respectively while the natural frequency of the spring-mass system was varied incrementally. The maximum displacement was computed from each loading at every natural frequency of the spring-mass system. Then, for each natural frequency of the spring-mass system, the maximum displacement obtained from the faster rate of loading was compared to that from the slower rate of loading. To this end, the ratio of the displacement was computed by dividing the maximum displacement of the faster loading by that of the slower loading. If this ratio is greater than unity, the maximum displacement is greater for the faster loading even though its peak values is smaller than the slower loading. Figure 7.22 shows the plot of the displacement ratio. The result shows that the faster loading rate can result in a greater maximum displacement than the slower

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7 Structural Coupling by FSI Slower Loading Rate with Larger Peak Value

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loading rate depending on the dynamic characteristics of the structure even though the former had a smaller peak force than the latter. The strain gages measured the strains along the vertical direction, from bottom to top direction. The gages were attached to the outer surface of both front and back plates as sketched in Fig. 7.23. The front plate had gages on the impact side while the back plate had gages on the opposite side. As a result, if both plates deformed in the same orientation as sketched in Fig. 7.23, one set of strain gages indicated compression and the other set gave tensile strains. This may give a confusion. In order to avoid such a confusion, the measured strains on the front plate were switched

7.2 Experimental Results of Two Plates of Same Thickness

169

Fig. 7.23 Strain gages attached to front and back plates

in their signs such that both strains on the front and back plates could give the tensile values for the deformed shape in Fig. 7.23. Because of the symmetry in the geometry as well as the loading and boundary conditions, only one half side of the strain gages were discussed below. Figure 7.24 shows the maximum magnitude of strains measured at every strain gage on the front plate. The maximum strains remained more or less the same independent of the water level until the water level was 25% in all gage locations. The strain was about 300 μm/m at gage #8, and other gages showed more or less around 500 μm/m. This was because all the gages were not under the water level. All the gages were above the water level and gage #8 was at the water/air interface for the 25% water level. After the 25% water level, the maximum strains varied very differently at different strain gage locations. Since the location of the gage #8 was below the water/air interface as the water level went beyond 25%, the strain at that gage showed a very steep change just after the 25% water level until the 35% water level. The strain at gage #8 had a slow increase from 35 to 80%, and then decreased from there to 100% water somehow linearly. The largest maximum strain at gage #8 occurred at the water level 80%. Gages #4 and #5 were placed at or near the 50% water level. Gage #4 measured increasing strains from 25 to 60%. The trend in the increase was a bilinear shape with one slower slope followed by a faster slop. Then, the strain at gage #4 decreased until the water level 80%. The strain was more or less the same from 80 to 90%, and it increased to the largest maximum strain at 100%. On the other hand, the strains at gage #5 increased linearly from 25 to 45%, and they remained relatively uniform throughout the water level. The gage #2 was placed at the 75% water level, and the strains increased at the water level 75% and beyond quite linearly until the 95% water level. The strain at 100% was slightly smaller than that at 95% where the largest maximum strain occurred.

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Fig. 7.24 Maximum strains on the front plate for FT2/BT2 and Spacing 2

In general, the strain response was quite different when the gage was below the water/air interface level or above it. As the water level moved higher beyond the gage location, there was a noticeable increase in the maximum strain, which indicated the importance of FSI for the section in contact with water. The largest maximum strain occurred at different water levels at different locations. That was recorded at 95% for gage #2, 100% for gage #4, 90 and 95% for gage #5, and 80% for gage #8. The result showed that the largest maximum strain was observed when the water level was 80% or higher. Maximum strains at the back plate are shown in Fig. 7.25. When there was no or small amounts of water, the coupling effect was non-existent or negligible as shown in the figure. The strains on the back plate were zero or near zero. As observed for the front plate, the strains at the back plate were also affected by the water level whether the strain gages were below or above the water/air interface. The maximum strains increased slowly as a function of the water level until the gage location was lower than the water/air interface, and they increased sharply after the water level passed the strain gage locations. However, after the rise in the maximum strains, there were some fluctuations in the maximum strains. One of the reasons was that the dynamic modes of vibration of the plate was varied depending on the water level.

7.2 Experimental Results of Two Plates of Same Thickness

171

Fig. 7.25 Maximum strains on the back plate for FT2/BT2 and Spacing 2

The water levels where the largest maximum strains were recorded were also different at different locations. They were 95% at gage #1, 90% at gage #2, 100% at gage #4, 75% at gage #5, and 80% for gages #7 and #8. Overall, the largest maximum strains occurred at the back plate when the water level was 75% or higher. The maximum strains at the back plate was lower than those at the front plate as expected. The strain at the back plate was approximately a half of the strain at the front plate. The strain time-history plots also showed the effect of the FSI on the strains as a function of time. Figure 7.26 shows the time-histories of strains at different locations when the water level was 35%, at which level the strain gages #7 and #8 were below the water level while other gages were above the water level. As shown in Fig. 7.26, the characteristics behaviors were different between the strain responses above the water line and those below the water line. The strains above the water line had the maximum value at the first peak and decreased quickly. On the other hand, the gages below the water line showed multiple peaks of similar magnitudes until around 0.07 s and decayed. The maximum positive strain occurred at the second peak and the maximum negative strain occurred at the third peak at the gage #8. This

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Fig. 7.26 Strain history at the rear plate with 35% water level for FT2/BT2 and Spacing 2

also suggests that the FSI effect with the structural coupling resulted in different vibrational characteristics. In order to investigate the vibrational frequency of the plates, the Fast Fourier Transform (FFT) was applied to their strain time-history data. The first major vibrational frequency was the same for the front and back plates at every water level. This means both plates vibrated in coherently each other from the structural coupling resulting from FSI. With the effect of the added mass and structural coupling, the vibrational frequency decreased with the water level. In order to provide the perspective of how much reduction in the frequency, all the frequencies were normalized with respect to the frequency of the front plate without water. Figure 7.27 plotted the variation of the vibrational frequency as the water level increased from 0 to 100%. The decrease in the vibrational frequency was more or less linear until the 50% of water level. The reduction was more than 60% at the 50% water level. The frequency was constant from 50 to 60% water levels, and slightly decreased from 60 to 75% water levels followed by another constant frequency after the 75% water level. It was noticed that the frequency at the 100% water level was slightly higher than that at the 95% water level. This is believed the effect of the top plate. When the water

7.2 Experimental Results of Two Plates of Same Thickness

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level was 100%, the water was in full contact with the top wall. This did not occur until the full water level. Water sloshing was possible before water was full, but it could not occur when the water was full because there was no space for sloshing. However, because the difference is small, no further investigation was not attempted to explain or find out the cause. Even though the effect of FSI and structural coupling on the vibrational frequency was much greater until the water level reached 50%, their effects on the strain responses continued to increased even after the 50% water level. The maximum strain with full water was about three time greater than that without water on the front plate. The vibrational frequency with full water was about one third of that without water. This was an interesting observation. However, these statements cannot be generalized until further studies are completed. This may be a coincident for this particular case of experimental parameters.

7.3 Experimental Results of Two Plates of Same Thickness at Different Spacings The case for FT2/BT2 with Spacing 2 was used as the baseline in this section, which was discussed in the last section. Therefore, results from all other tests were compared to the results of FT2/BT2. First, impact forces were compared. Time-histories of the impact forces for the case of Spacing 4 were given in Fig. 7.28 for the water levels 50 and 100%. The

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Fig. 7.28 Impact force time-histories for FT2/BT2 with Spacing 4

characteristics behaviors of the plots were similar to those for Spacing 2. The timehistory plots were similar along different spacings as long as the water level was the same. However, the magnitudes of the impact forces varied depending on the spacing between the two plates. As the spacing increased between two plates, the peak impact force also increased generally. The peak impact forces were compared for different spacings as the water level changed in Fig. 7.29, where ‘Section’ indicates ‘Spacing’. Spacing 4 resulted in consistently greater impact forces than that Spacing 1. The peak impact force measured for Spacing 2 was mostly bounded by the results for Spacings 1 and 4. Both Spacings 2 and 4 had the largest impact forces at the 35% water level than any other water level. The impact forces with no water were slightly lower than those at 35% for Spacings 2 and 4. The maximum impact force among all cases occurred for Spacing 4 when the water level was 35%. On the other hand, Spacing 1 had the largest impact force with no water. When the water level was 35% for Spacing 1, the impact force was much lower than that with no water. For all spacings, the smallest impact forces occurred with the 60% water level. The minimum force out of all cases happened at 60% with Spacing 1. Furthermore, the impact force for Spacing 4 was slightly smaller than that for Spacing 2 at the water level 60%. Analysis of strain responses was conducted next. Since Fig. 7.29 showed that the largest difference in the peak impact forces occurred between Spacing 1 and Spacing

7.3 Experimental Results of Two Plates of Same Thickness at Different Spacings

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Fig. 7.29 Plot of peak impact forces versus water levels for three different spacings of FT2/BT2

4 with the 35% water level, their strain responses were examined on the front plates for their comparison. Figure 7.30 shows the strain time-histories at the front plates with the 35% water and Spacing 1 while Fig. 7.31 shows that for Spacing 4. Because the impact force was greater for the spacing 4 than Spacing 1, the former resulted in larger strains than the latter. Among all the gages, gage 8 gave the largest difference between the two spacings. This gage was located under the water/air interface. On the contrary, gage 5 located above the water line showed a little difference between the two spacings. Figure 7.32 shows the plots of the absolute maximum strains of the front plate of FT2/BT2 for all water levels and different spacings. Likewise, Fig. 7.33 shows similar plots for the back plates. Regardless of the different spacings, the maximum strains had similar trends in their variation as a function of water levels at different gages. As observed previously, the water level affected the maximum strains for all spatial distances. Because Spacing 4 yielded the largest peak impact force followed by Spacing 2 and Spacing 1 in that order, it was expected Spacing 4 would result in the largest strain values on the front plate among the three spacing cases. The results confirmed this expectation except for a few exceptional cases. Overall, the maximum strains were greatest for Spacing 4 than other spacings for all strain gage locations of the front plate except for gages 2 and 5 with 90% or higher water levels. For example, gage 2 showed that Spacing 2 had the largest maximum strains when the water level was at least 90% or higher. Even though the impact force was the smallest for Spacing 1, that did not necessarily result in the lowest strains. Depending on the water level as well as the gage location, either Spacing 2 or Spacing 1 gave the smallest strains among three different spacings. As an example, gage 8 on the front plate showed smaller maximum strains

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7 Structural Coupling by FSI Gage 2

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for Spacing 2 than Spacing 1 for almost all water levels except for 30% and 35%. On the other hand, gage 2 showed the smaller maximum strains for Spacing 1 for the most of the water levels. These results indicated that the larger impact force does not necessarily yield the larger strain because there are other factors influencing the strain magnitude. Some of the factors were the loading rate and loading history. An explanation was given previously using Fig. 7.21 to illustrate that a higher loading rate with even a lower magnitude could result in a higher strain. The strain response of the back plate was much more complex as shown in Fig. 7.33. The impact force was the major cause of the dynamic response of the front plate even though FSI effect should be also included. On the other hand, the dynamic response of the back plate was resulted only from the FSI. The FSI effect was more complicated than the impact force. This fact resulted in more unpredictable strains at the back plate.

7.3 Experimental Results of Two Plates of Same Thickness at Different Spacings

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Gages 2 and 3 of the back plate had similar responses in terms of their maximum strain plots as a function of the water level. At both gages, Spacing 1 produced the largest maximum strains while Spacing 4 yielded the smallest maximum strains when the water level was at least 90% or higher. The largest maximum strain with Spacing 1 was about 50% greater than the smallest strain with Spacing 4 at those gage locations. The maximum strain at gage 5 showed that Spacing 4 has the smallest maximum strains and the largest maximum strain was either for Spacing 1 or 2 if the water level was at least 60%. The largest maximum strains, at either Spacing 1 or 2, were as twice as greater than the smallest maximum strains at Spacing 4 at gage 5. Gage 6 gave the largest maximum strain for Spacing 2 as long as the water level was at least 50%. At that location, Spacing 1 or 2 resulted in strains relatively close to each

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Fig. 7.32 Plot of absolute maximum strains of front plates of FT2/BT2 for three different spacing distances

other. Spacing 2 produced two times larger strains than the other spacings for water fill levels greater 60%. Gage 8 showed the biggest difference in the strains among three different spacings when the water level was around 50%. At those water level, the largest maximum strain occurred for Spacing 1 and the smallest maximum strain occurred for Spacing 4. The ratio of the largest to the smallest strain was nearly 2. On the other hand, the largest maximum strain occurred for Spacing 1 or 4 while the smallest maximum strain occurred for Spacing 2 for gage 9 with the water level around 50%. The ratio of the largest to smallest strain was greater than 2 for gage 9. The strain time-histories were used to determine the vibrational frequency using FFT so as to find the effect of water level and plate spacing on the vibrational frequency of FT2/BT2. As shown in Fig. 7.34, the front and back plates of the FT2/BT2 case with full water had the same major frequency in their frequency spectra. That is, both front and back plates vibrated closely each other.

7.3 Experimental Results of Two Plates of Same Thickness at Different Spacings

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0 0

20

40

60

Fill Level [%]

80

100

0

20

40

60

Fill Level [%]

Fig. 7.33 Plot of absolute maximum strains of back plates ofFT2/BT2 for three different spacing distances

The change in the vibrational frequency is plotted in Fig. 7.35 as a function of the water level for all three spacings. The frequency plots were normalized with respect to that of the dry plate. All three spacings showed a similar trend as their frequencies were close to one another. A reduction in the frequency was the largest until 50% water followed by a small change after the water level. Spacing 2 had a lower frequency than other spacings until 20% water, but it gave a higher frequency than other spacings after 50% water. In general, Spacings 1 and 4 showed closer frequencies throughout different water levels.

180

7 Structural Coupling by FSI 10

5

8

6

10

5

4

Magnitude

Magnitude

6

2

0

4

2

0 2

1

3

2

1

4

4

3

Log(Frequency) [Hz]

Log(Frequency( [Hz]

Fig. 7.34 Frequency spectra of strain responses of FT2/BT2 with full water: a front gages and b back gage 1

Spacing 1 0.9

Spacing 2 Spacing 4

Normalized Freq

0.8 0.7 0.6 0.5 0.4 0.3 0.2

0

20

40

60

80

100

Water Level [%]

Fig. 7.35 Normalized frequency of FT2/BT2 with different spacings

7.4 Experimental Results of Two Plates of Different Thickness In this series of tests, Spacing 2 was used while different combinations of plate thicknesses were tested such as FT2/BT2, FT4/BT4, FT2/BT4 and FT4/BT2. Again, the FT2/BT2 set was used as the baseline for comparison, unless mentioned otherwise. The test results indicated that the impact force was much greater with the thicker front plate FT4 than the thinner front plate FT2. That is, a stiffer front plate resulted

7.4 Experimental Results of Two Plates of Different Thickness

181

250

FT4/BT4 FT4/BT2 FT2/BT2

200

Force [N]

FT2/BT4

150

100

50

0

20

40

60

80

100

Water Level [%]

Fig. 7.36 Comparison of peak impact forces for different combinations of structural stiffness with Spacing 2

in a higher impact force because the front plate interacts with the impactor directly. If the front plate had the same thickness, the thicker back plate BT4 also yielded a greater impact force than the thinner back plate BT2. This results stated that the stiffer back plate provided more stiffness to the front plate through the FSI coupling. Figure 7.36 compares the peak impact forces as a function of the water level for four different combinations of the plates. As stated in the last paragraph, the impact force was much greater with the front plate FT4 than FT2. In addition, there was a different behavior between FT4 and FT2. For the cease with FT2, the peak impact force was largest at 35%, and the force was smaller with water as compared to the no water case, in general. However, the FT4 cases showed the smallest impact force at 35%, and the impact force with water was generally greater than that without water, especially when the water level was beyond 60%. These results suggested that the impact force was dependent on some combination of the stiffness of the front plate and the water level. There was no generalized trend in predicting the peak impact force. Figure 7.37 shows the plots of the maximum strain at the front plate for different combinations of plate thicknesses. Likewise, Fig. 7.38 shows the maximum strain at the back for the different plate thicknesses. Even though the impact force was greater for the thicker front plate FT4, the strains were smaller on the thicker front plate because of the higher bending stiffness. When the strains on the font plates were compared between FT2/BT2 and FT2/BT4, neither one of them showed consistently greater strains than the other. At some combinations of the water level and the gage location, one of the two cases had greater strains while at other combinations the other case had larger strains. For example, Gage 5 had larger strains for FT2/BT2 for

182

7 Structural Coupling by FSI Gage 2

Gage 1 1000 FT4/BT2

400

m]

m]

FT2/BT2

Strain [

Strain [

FT2/BT4

200

500

0 0 0

20

40

60

80

0

100

20

40

60

80

100

80

100

80

100

Fill Level [%]

Fill Level [%] Gage 4

Gage 5

1000

m]

m]

400

Strain [

Strain [

500

200

0

0 0

20

40

60

80

0

100

20

Fill Level [%]

40

60

Fill Level [%] Gage 8

Gage 7 1000

m] 400

Strain [

Strain [

m]

600

200 0

500

0 0

20

40

60

Fill Level [%]

80

100

0

20

40

60

Fill Level [%]

Fig. 7.37 Plot of absolute maximum strains of front plates for different combinations of plate thickness with spacing 2

water levels greater than 20% while gage 8 showed greater strains for FT2/BT4 for all water levels. Gages 2 and 4 had larger strains for FT2/BT2 for high water levels and larger strains for FT2/BT4 for low water levels. Even though the thicker back plate BT4 had generally lower strains because of its higher bending stiffness, BT4 did not necessarily have the lowest strains as compared to BT2. Among three cases FT4/BT2, FT2/BT2 and FT2/BT4, the case FT4/BT2 generally showed greater strains on the back plate. In particular, gage 8 showed much higher strains for FT4/BT2 with 80–90% water levels as compared to other cases. On the other hand, FT2/BT2 had the largest strain at gage 6 with water levels greater than 60%. The frequency analysis was also conducted for the cases of different plate thicknesses. Their frequency spectra were different between the front and back plates as their thicknesses were different. The frequency spectra for the front and back

7.4 Experimental Results of Two Plates of Different Thickness

183 Gage 3

Gage 2 600

FT2/BT2

400

m]

FT4/BT2

400

Strain [

Strain [

m]

600

200

FT2/BT4

200

0

0 0

20

40

60

80

0

100

20

40

60

Fill Level [%]

Fill Level [%]

Gage 5

Gage 6

80

100

80

100

80

100

600

m]

m]

Strain [

200

Strain [

400 400

200

0

0 0

20

40

60

80

0

100

20

40

60

Fill Level [%]

Fill Level [%]

Gage 8

Gage 9

800 400

400

Strain [

Strain [

m]

m]

600

200

200

0

0 0

20

40

60

Fill Level [%]

80

100

0

20

40

60

Fill Level [%]

Fig. 7.38 Plot of absolute maximum strains of back plates for different combinations of plate thickness with spacing 2

plates of FT2/BT4 were shown for 7.39 and 7.40 with the 50% water level. The vibrational frequencies were different for the front and back plates. Furthermore, the major vibrational frequencies were different at different strain gages locations of each plate. Frequency spectra at Gages 5 and 6 of the back plate BT4 had two major peaks as seen in Fig. 7.40. Gages 2 and 3 had different major frequency from Gages 8 and 9. The former gages had lower a vibrational frequency than the latter gages. For the same case FT2/BT4, the water level was increased to 90%. Then, the corresponding frequency spectra were plotted in Fig. 7.41 for the back plate of FT2/BT4. When Figs. 7.40 and 7.41 were compared, the vibrational frequency of the back plate showed a strong dependency on the water level between the two plates (Fig. 7.39).

184

7 Structural Coupling by FSI

4

10

Gage 1

5

10

Gage 2

5

Magnitude

Magnitude

4

2

2

0

0 1

2

3

4

1

Log(Frequency) [Hz] 10

Gage 4

4

6

10

Magnitude

Magnitude

15

5

1

10

4

Gage 5

5

4

2

2

3

4

1

Gage 7

5

10

Magnitude

10

2

3

4

Log(Frequency) [Hz]

Log(Frequency) [Hz]

Magnitude

3

0

0

10

2

Log(Frequency) [Hz]

5

0

10

Gage 8

5

5

0 1

2

3

4

1

Log(Frequency) [Hz]

2

3

4

Log(Frequency) [Hz]

Fig. 7.39 Frequency spectra of front plate of FT2/BT4 with 50% water level

7.5 Experimental Results of Three Plates Three plates were inserted into the first three grooves of the test setup as shown in Fig. 7.42. Strain gages were attached to the front and back plates, but not the middle plate. The three plate models were also compared to the two plate cases. The peak impact forces were compared in Fig. 7.43 for different combinations of plates while the water level remained the same for all the cases which included both two plate and three plate systems. The water level was set to 50% in every water compartment regardless whether there was one or two depending on the number of plates. The experimental results showed that the middle plate did not make any sizable influence on the peak impact force on the front plate. When two different plates, T1 and T2, were inserted in the middle, respectively, between FT4 and BT2, the peak impact forces of FT4/MT1/BT2 and FT4/MT2/BT2 were very close to that

7.5 Experimental Results of Three Plates 10

Gage 2

4

10

10

Magnitude

Magnitude

15

185

5

0 1

2

3

10

5

0

4

1

Gage 5

5

10

Magnitude

Magnitude

10

1

0

10

4

Gage 6

4

5

2

3

1

4

Gage 8

4

10

Magnitude

10

5

0 1

2

3

Log(Frequency) [Hz]

2

3

4

Log(Frequency) [Hz]

Log(Frequency) [Hz]

Magnitude

3

0 1

10

2

Log(Frequency) [Hz]

Log(Frequency) [Hz]

2

Gage 3

4

4

10

Gage 9

4

5

0

1

2

3

4

Log(Frequency) [Hz]

Fig. 7.40 Frequency spectra of back plate of FT2/BT4 with 50% water level

of FT4/BT2. Similarly, three cases like FT2/BT4, FT2/MT1/BT4 and FT2/MT2/BT4 also showed very similar peak impact forces. There are two compartments for water for the three plate systems while there is one compartment for the two plate system. The three plate system has one water compartment between the front and middle plate and the other compartment between the middle and the back plate. Therefore, the three plate system could have different water levels between the two compartments. In order to study the different water levels at the two compartments, three different cases of water distribution to the two compartments were tested as the total water of the two compartments remained the same. One of the three cases had 25% water for the first compartment between the front and middle plates and 75% water for the second compartment between the middle and back plates. The second case had the switch of the water levels. That is, the first compartment had 75% water while the second compartment had 25%

186

7 Structural Coupling by FSI 10

Gage 2

4

10

10

Gage 3

4

Magnitude

Magnitude

15 10 5 0

5

0 1

2

3

4

1

10

Gage 5

5

10

3

3

4

Gage 6

5

2

Magnitude

Magnitude

2

Log(Frequency) [Hz]

Log(Frequency) [Hz]

2

1

0

1

0 1

2

3

4

1

4

Gage 8

10

2

3

4

Log(Frequency) [Hz]

Log(Frequency) [Hz] 10

Gage 9

4

15

Magnitude

Magnitude

10 10

5

0

5

0 1

2

3

4

Log(Frequency) [Hz]

1

2

Fig. 7.41 Frequency spectra of back plate of FT2/BT4 with 90% water level

Fig. 7.42 Three composite plates inserted in the test set-up

3

Log(Frequency) [Hz]

4

7.5 Experimental Results of Three Plates

187

160 140

Force [N]

120 100 80 60 40 20 0

Fig. 7.43 Comparison of peak impact force for various combinations of plates with 50% water level

water. The third case had equal water levels in both compartments like 50%. These three different combinations of water levels were studied for four different cases of plate combinations. The four cases of three plate systems were FT4/MT1/BT2, FT4/MT2/BT2, FT2/MT1/BT4 and FT2/MT2/BT4. Figure 7.44 compare the peak impact forces among the four different combinations of the three plate systems as the water level in the first compartment was varied while the total water levels remained the same as 50%. The experimental results showed that the water levels in the first and the second compartments were important but the thickness of the middle plate had a minor effect on the impact force. When the front plate was thicker like FT4, 180

160

Force [N]

140

120 FT4/MT1/BT2 FT4/MT2/BT2

100

FT2/MT1/BT4 FT2/MT2/BT4 80

60 30

40

50

60

70

80

Water Level in First Compartment [%]

Fig. 7.44 Comparison of peak impact force for three plate systems with total 50% water

188

7 Structural Coupling by FSI

the largest impact force occurred when the first compartment had 75% water, but the smallest impact force occurred with 50% water levels in both compartments. The 25% water level in the first compartment resulted in the intermediate impact force among the three water level cases. When the front plate was thinner like FT2, this results were quite different from those of the thicker front plate. The largest impact force occurred as the first compartment had 25% water level, and the force decreased as the water level in the first compartment increased. These results suggested that the effect of FSI was very complex and very dependent on the arrangement of the interacting structures. The next study examined the strain responses of the three-plate systems. As the total water volume was remained 50% for all cases, the water level of the first and the second compartments were varied as described above: i.e. 25 and 75, 50 and 50, and 75 and 25%. The maximum strains were examined for the three different plate combinations as the water level was changed. The maximum strains were plotted in Figs. 7.45 and 7.46 on the front and back plates, respectively, as a function of the water level in the first compartment. The maximum strain on the front plate was affected by the middle plate when the front compartment had the 75% water level. On the contrary, the middle plate influenced the strains on the back plate as the second compartment had the 50% water level. Figure 7.47 shows the comparison of the maximum strains on the back plate for different combinations of plates when the water volume was 50%. The comparison showed that the FT2/BT2 case had the largest maximum strain and the FT2/MT2/BT4 plate had the smallest maximum strain on the back plate. When the two three-plate systems with different middle plate like FT4/MT1/BT2 and FT4/MT2/BT2 were compared, the latter yielded a larger strain on the back plate. On the other hand, when the two three-plate systems like FT2/MT1/BT4 and FT2/MT2/BT4 were compared, the former gave the larger maximum strain on the back plate. The effect of a thin or thick middle plate on the back plate depended on the selection of combination of the front and back plates. As far as the vibrational frequency spectra were concerned, the third plate in the middle affected the results. Figure 7.48 shows the vibrational frequency spectra of the back plate for the three-plate system FT2/MT2/BT4 with the 50% water level. The three-plate system showed multiple peaks in the frequency including those at higher frequencies, especially at Gages 2, 3, 5 and 6 which were located at and above the water surface. On the other hand, Gages 8 and 9 which were under the water surface, did not show the high frequency peaks,

7.6 Experimental Setup for Two Cylinders The experimental setup for testing the cylindrical shells was different from the setup for the plate tests. The plates were arranged in parallel while the cylinders were arranged in the concentric matter. One cylinder was placed inside the other cylinder such that both cylinders shared the same axis of revolution. As a result, the final assembly as seen in Fig. 7.49 does not show the inner cylinder.

7.6 Experimental Setup for Two Cylinders

189 Gage 2

Gage 1 500

300 FT4/MT1/BT2

100

0

m]

FT2/MT2/BT4

300

Strain [

FT2/MT1/BT4

m] Strain [

400

FT4/MT2/BT2

200

200 100 0 -100

-100 30

40

50

60

70

30

80

40

50

60

70

80

60

70

80

60

70

80

Fill Level [%]

Fill Level [%]

Gage 5

Gage 4 500

600

m]

200

300

Strain [

Strain [

m]

400 400

200 100 0

0

-100 30

40

50

60

70

30

80

40

50

Fill Level [%]

Fill Level [%]

Gage 8

Gage 7 1200 600

m]

800

Strain [

m]

1000

600

Strain [

400

200

400 200

0

0 30

40

50

Fill Level [%]

60

70

80

30

40

50

Fill Level [%]

Fig. 7.45 Plot of absolute maximum strains of front plates for three plate systems with total 50% water (Fill level in the figure is for the first compartment.)

The setup had two concentric cylinders of equal length, which were in the horizontal orientation. That is, the axis of revolution of the cylinders was along the horizontal axis as shown in Fig. 7.49. Both cylinders were fabricated using a filament winding machine as shown in Fig. 7.50 with ±45° layer orientations until their thickness was 1 mm. The fabricated cylinders are shown in Fig. 7.51. The cylinders were made of T700S carbon fibers and ProSetM1002 resin. Because the cylinders were fabricated from the filament winding, the inner diameter was the control parameter. The inner diameter of the outer cylinder was 88.9 mm (3.5 in.) and that of the inner cylinder was 76.2 mm (3.0 in.), respectively. Table 7.1 lists the material properties of the composite cylinders made of the carbon fibers and the resin.

190

7 Structural Coupling by FSI Gage 3

Gage 2

100

100

Strain [

Strain [

m]

200

m]

200

0

0

-100

-100 30

40

50

60

70

30

80

40

200

100

100

m]

200

Strain [

m]

60

70

80

60

70

80

60

70

80

Gage 6

Gage 5

Strain [

50

Fill Level [%]

Fill Level [%]

0

0

-100

-100 30

40

50

60

70

30

80

40

50

Fill Level [%]

Fill Level [%]

Gage 9

Gage 8 200

400

m]

m]

300 100

Strain [

Strain [

200

100

FT4/MT1/BT2

0

FT4/MT2/BT2

0

FT2/MT1/BT4 FT2/MT2/BT4

-100

-100 30

40

50

Fill Level [%]

60

70

80

30

40

50

Fill Level [%]

Fig. 7.46 Plot of absolute maximum strains of back plates for three plate systems with total 50% water (Fill level in the figure is for the first compartment.)

Strain gages were affixed to the outside surface of the outer cylinder as well as the inside surface of the inner cylinder because water were filled between the inner and outer cylinders. Therefore, the strain gages were not exposed to water. The cylinder was divided into four equal distances along the longitudinal length as sketched in Fig. 7.52. Strain gages were placed to the locations called “Middle” and “Left” in Fig. 7.52. Because of symmetry, strains at the “Right” location were not measured. Four strain gages were attached to the outer cylinder along the circumferential direction at each longitudinal location of “Middle” and “Left”, respectively. They were the front (i.e. the impact side), back, top, and bottom side as shown in Fig. 7.52. It was not easy to attach strain gages to the inside surface of the inner cylinder because the inner

7.6 Experimental Setup for Two Cylinders

191

450 400 350

Strain

300 250 200 150 100 50 0

Fig. 7.47 Comparison of maximum strains of the back plate for various combinations of plates with 50% water

diameter of the inner cylinder was small. Therefore, only three gages were placed to the inner cylinder. All strain gages measured the hoop strains of the cylinders. In order to fill water in the annulus between the two concentric cylinders, two supporting structures were fabricated using the 3-D printing technique as shown in Fig. 7.53. Each supporting structure had two grooves so that both cylinders were inserted into the grooves of the supporting structure. Proper watertight sealing was undertaken in order for water not to leak from the space between the two concentric cylinders. The supporting structures were mounted to the aluminum base plate so that the whole setup could be secured to the vibration isolation table as shown in Fig. 7.54. The base plate had grooves to adjust the length between the two supporting structures. The impact was applied using the same pendulum setup as described previously. The impact of the spherical shape had a load cell attached to measure the impact force. The drop angle was set to zero when the impactor arm was at the vertical downward position. The drop angle was changed incrementally such as 20°, 30° and 45°. The impact was applied to the front surface of the middle section as designated in Fig. 7.52. The following notations were used in the subsequent discussion. The symbol ‘M’ indicated the mid-section of a cylinder, and ‘L’ was used for the left-section along the longitudinal axis of the cylinder. Along the circumferential direction, ‘F’ was used for the front side (i.e. impact side), ‘T’ was selected for the top side, ‘B’ was denoted for the bottom side, and finally ‘P’ was chosen for the posterior side. In addition, so as to distinguish the inner and outer cylinders, ‘IC’ was used to indicate the inner cylinder while ‘OC’ was used for the outer cylinder. Then, any location of a cylinder could be defined using the combination of three symbol: one for the cylinder, one for the longitudinal direction, and one for the hoop direction. For example, the notation OC-M-F has the first symbol for the cylinder, the second symbol for the longitudinal direction and the third one for the hoop direction. As a result, OC-M-F indicates

192

7 Structural Coupling by FSI 15

10

Gage 2

4

10

10

Gage 3

4

Magnitude

Magnitude

10

5

0

5

0 1

2

3

4

1

15

10

Gage 5

4

2

3

4

Log(Frequency) [Hz]

Log(Frequency) [Hz]

10

10

Gage 6

4

Magnitude

Magnitude

10

5

0 1

2

3

5

0

4

1

10

10

Gage 8

4

2

3

4

Log(Frequency) [Hz]

Log(Frequency) [Hz]

8

10

Gage 9

4

Magnitude

Magnitude

6

5

0

4

2

0 1

2

3

Log(Frequency) [Hz]

4

1

2

3

4

Log(Frequency) [Hz]

Fig. 7.48 Frequency spectra of back plate of FT2/MT2/BT4 with 50% water level

the outer cylinder, mid-section along the longitudinal axis, and the front side of the cylinder. There was one exception because four gages could not be attached along the hoop direction at the mid-section of the inner cylinder because of the limited accessibility. Therefore, one location had the notation like IC-M-T-P. This indicated the location between top and posterior sides. Therefore, the notation consisted for four symbols. The water level was varied in the annulus, and the amount of water was denoted such as ‘50 W’ for the 50% water, which stated the 50% space of the annulus was filled by water. If water was full, it was indicated by ‘100 W’. Similarly, the impact angle was denoted by ‘20’, ‘30’ and ‘45’, respectively.

7.7 Experimental Results of Two Cylinders

193

Fig. 7.49 Assembled experimental setup for concentric cylinders

Fig. 7.50 A filament winding machine

7.7 Experimental Results of Two Cylinders In order to check the consistency of the test results, every test case was repeated at least ten times, and their test results were compared for the repeatability. The test results were very consistent with repeatability. For example, Fig. 7.55 shows ten impact test results. The impact forces were plotted when the cylinders contained 50% water inside the annulus and was impacted at the 45° angle. The plots are almost identical

194

7 Structural Coupling by FSI

Fig. 7.51 Two sizes of composite cylinders

Table 7.1 Material properties of composites made of T700S and ProSetM1002 EL

ET

GLT

GTT

νLT

νTT

ρ

139 GPa

7.35 GPa

2.18 GPa

0.432 GPa

0.236

0.216

1485 kg/m3

one another. Other measured strain data were also very repeatable. Therefore, one most representative data set were used to compare results from different test cases instead of taking average of the repeated test results. Water was filled into the space between the two concentric cylinders incrementally. Starting from no water, the annulus space had 25%, 50%, 75% and 100% water of the space volume, respectively. Impact loading was applied to the concentric cylinders with each water content. It was made sure that there was only one impact but no secondary impact for each test. Figure 7.56 compares time-histories of the impact forces as a function of different water levels between the two cylinders when the impact was given with the initial impact angle of 45°. One thing to be noted from Fig. 7.56 was that the duration of the impact was quite constant regardless of the water levels. The impact forces lasted about 16 ms for all cases. As before, the peak impact forces were compared for different water levels as well as different impact angles. The impact angles were 20°, 30° and 45°. Figure 7.57 shows the comparison. The test results indicated that the peak impact force remained relatively constant at all water levels when the impact angle was 20°. As the impact angle was increased to 45°, the peak impact force was very dependent on the water level. As the water level changed, the additional amount of water affected the overall mass and stiffness of the entire system, and those changes influenced the reaction between the impactor and the system resulting in different impact forces. The test

7.7 Experimental Results of Two Cylinders

Fig. 7.52 Strain gage locations of outer and inner cylinder Fig. 7.53 Support structures for composite cylinders

195

196

7 Structural Coupling by FSI

Fig. 7.54 Supporting structure mounted on top of the metal base plate

Fig. 7.55 Impact force plot of ten test runs with filling level 50% and drop angle 45°

results suggested that such interaction was greater for larger impact loading. This is understandable since FSI occurs during the dynamic motion of the system. When the dynamic motion is greater with larger external loading, the effect of FSI becomes greater. The test results showed that there was a decrease in the peak impact force until the water level was 75%. Such a reduction in the peak impact force until the 75%

7.7 Experimental Results of Two Cylinders

197

Force Comparison 45 Degrees

10

0

-10

Force [N]

-20 0W 25W

-30

50W 75W

-40

100W

-50

-60 0

0.005

0.01

0.015

0.02

Time [s]

Fig. 7.56 Comparison of impact force with different water levels with drop angle 45°

Fig. 7.57 Plot of peak impact force as a function of water level

198

7 Structural Coupling by FSI

water level was greater with a larger impact angle. In other words, the magnitude in the force decrease was the smallest for the 20° impact angle and the largest for the 45° impact angle. When the cylinders were full of water between them, the peak impact force became the largest among all water levels. Even though the peak impact force was smaller for the cases with partially filled water until 75% than the no water case, the measured strains on the outer cylinder were greater with the partially filled water. This was also observed from the impact testing of two parallel flat plates containing water between them. While the peak impact force was smaller with water, the resultant strains were greater on the plate with the FSI effect. Water between the two cylinders resulted in a reduction in the vibrational frequency because of the added mass effect. Figure 7.58 shows the vibrational frequency of both outer and inner cylinders as the water level was varied from no water to 100% water. Those vibrational frequencies were computed from the strain time-history data by applying the Fast Fourier Transform. All the frequencies were normalized with respect to the vibrational frequency of the outer cylinder without water. Even the vibrational frequency of the inner cylinder was also normalized with respect to that of the dry outer cylinder. As expected, the vibrational frequency decreased along with the water level. Because the outer and inner cylinders have two different diameters, their vibrational frequencies are different as shown in Fig. 7.58. When the flat plate system was compared to the cylindrical shell system, both showed the reduction in the vibrational frequency as the water level increased, However, the former showed that the change in the vibrational frequency was very small beyond the 50% water level, but the latter showed a sharp decrease in the

Fig. 7.58 Plot of vibrational frequency as a function of water level

7.7 Experimental Results of Two Cylinders

199

frequency as the water level went beyond 50% until 100%. However, the percentage in reduction was greater for the flat plate system than the cylindrical system at the same water level. As explained in Sect. 7.2, let’s consider a single degree of freedom system consisting of a single mass and a linear spring which is subjected to a harmonic force. The magnitude of the dynamic displacement of the mass d can be expressed like the following: dst  d=   1 − ωωn 

(7.1)

where dst is the static displacement of the mass when the magnitude of the harmonic loading was applied to the spring. In addition, ω is the frequency of the exciting load, and ωn is the natural frequency of the single degree of system made of the spring and mass. The natural frequencies of the present cylinders being tested were much greater than the frequency excited by the impact loading. Furthermore, the effect of added mass decreased the vibrational frequency of the system. The reduction in the vibrational frequency resulted in a decrease in the denominator of Eq. (7.1), which eventually yielded a larger dynamic displacement d. In addition, when the water level was 75% or 100%, the rate of increase in loading was greater than other cases. That is, it took a shorter time for the impact force to reach its peak value. These factors resulted in greater strains with water. Out of three impact angles, the smaller drop angle resulted in smaller impact force, as expected. If any friction in the bearing as well as in air were neglected, the ratios of the impact velocities among different impact angles were v20 /v45 = 0.206 and v30 /v45 = 0.457, respectively Here, the subscript denotes the drop angle. Three different impact force time-histories were compared in Fig. 7.59 when the cylinders had full water. The peak impact forces gave their ratios such as P20 /P45 = 0.404 and P30 /P45 = 0.532. This suggested that the impact force and impact velocity did not have clear relationship. The impact force did not show a clear peak but instead showed a gradual plateau for the drop angle 20°. On the other hand, the drop angle 45° showed a very clear peak force with sharp increase and decrease from the peak force. The 30° drop angle showed the result between the other cases, but much closer to the 20° drop angle. The duration of the contact time between the impactor and the outer cylinder was the shortest for the 20° drop angle and the longest for the 30° drop angle. The 45° drop angle had the contact time between the other cases. However, the contact time was not significantly different among the three impact angles. Additionally, when FFT was applied to the force time-histories, the frequency of the three different loading was close one another. Next, the strain responses were examined from the experiments. Unless mentioned otherwise, strain responses were compared with the impact angle 45°. Figure 7.60 shows the strains at the front side (F) of the Outer Cylinder (OC) at its mid-section

200

7 Structural Coupling by FSI Force with 100% water 10

0

Force [N]

-10

-20

-30

20 degree

-40

30 degree 45 degree -50

-60 0

0.005

0.01

0.015

0.02

Time [s]

Fig. 7.59 Comparison of forces at different drop angles with 100% water 10 -3

1

OC-M-F-45

0

Strain [m/m]

-1

-2

No water 50% Water 100% Water

-3

-4

-5

0

0.01

0.02

0.03

0.04

0.05

Time [S]

Fig. 7.60 Strains at the front side of OC at three different water levels and 45° drop angle

7.7 Experimental Results of Two Cylinders

201

(M), i.e. OC-M-F with three different water levels: no water, 50% water and 100% water. This location was the impact side. The strain curves looked like a half sign curve regardless of the water levels, and the shape somehow resembled that of the impact force time-history. This was because the strains at the front side had the direct relationship with the impact force. When the peak strains were compared among three water levels, the 50% water level resulted in larger peak strain than the 100% water level even though the difference was small. This was not expected because the peak impact force was expected to be the greatest with the 100% water level. Of course, the strain gage was not attached exactly to the front side. Instead it was placed very close to the front side not to be destroyed by the impactor. This might cause such a discrepancy between the peak force and strain. The duration of the major strain response on the front side of the outer cylinder was a little longer than the contact duration of the impactor. Thus, the strain at that location quickly diminished as the impactor was separated from the cylinder. When the drop angle was changed, the strain response at the front side of the outer cylinder was similar one another except that the peak strain was a function of the impact angle. The measured strains at the location OC-M-P, which was the posterior side of the mid-section, had more interesting behaviors. The strains at OC-M-P were plotted in Fig. 7.61 as the water level was changed from no water, 50% water to 100%, incrementally. The behavior of the strain was significantly different for different water levels. There was no similarity among three different strain responses. The 10

10

-4

OC-M-P-45

No Water

8

50% Water 100% Water

6

4

Strain [m/m]

2

0

-2

-4

-6

-8 0

0.01

0.02

0.03

0.04

0.05

Time [S]

Fig. 7.61 Strains at the posterior side of OC at three different water levels and 45° drop angle

202

7 Structural Coupling by FSI

strain showed a gradual change with time when there was no water. The shape of the strain time-history at OC-M-P was close to that at OC-M-F without water. On the other hand, the strains showed many oscillatory peaks and valleys as the cylinders contained water between them. Even though the strain behaviors were very complicated, repeated tests showed consistent responses under the same test condition. In terms of the magnitude only, the 50% water level gave the largest maximum strain among three cases, and the 100% water level resulted in a little smaller maximum strain than the 50% water case. The 50% water level gave the largest positive strain while the 100% water level yield the largest negative strain. The frequency spectra were obtained from the strain time-histories plotted in Fig. 7.61 using the FFT, and their resultant plots are given in Fig. 7.62. The frequency plots showed that major vibrational frequencies varied as the water level changed. The strains at OC-M-P were compared for three different drop angles 40°, 45° and 50° when the cylinders were full of water. Figure 7.63 shows the comparison. The figure showed that all three drop angles resulted in close strain responses at OC-M-P. The major vibrational frequency was the same for all three impact angles. Even if the magnitude of the strain was larger with a higher drop angle among the three cases, the 50o drop angle did not necessarily result in the largest value at every strain peaks of their time-history plots. OC-M-P-45 0.14

No Water 50% Water

0.12

100% Water

Magnitude of Strain

0.1

0.08

0.06

0.04

0.02

0

-1

0

1

2

3

4

5

Log(Frequency) [Hz]

Fig. 7.62 Frequency spectrum of strains at the posterior side of OC at three different water levels and 45° drop angle

7.7 Experimental Results of Two Cylinders 10

10

203

OC-M-P-100W

-4

40 degree 8

45 degree 50 degree

6

Strain [m/m]

4

2

0

-2

-4

-6

-8 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time [S]

Fig. 7.63 Strains at the posterior side of OC with 100% water level and different drop angles

The strains were compared for the locations of OC-M-F and IC-M-F as shown in Fig. 7.64. Because the outer cylinder had strain gages on the outside surface and the inner cylinder had strain gages on the inside surface, the gages on IC and OC would have opposite sign even if both OC and IC deformed in the same direction. In order to provide the same perspective for both cylinders, the sign of the strains on IC were reversed in the all the subsequent plots including Fig. 7.64 which showed that both cylinders deformed in the opposite directions. As the outer cylinder was impacted on the front side at the mid-section, the strain at OC-M-F was in compression while the strain at IC-M-F showed a short period of small compression then followed by a longer period of large tension. Both inner and outer cylinders deformed out of phase for the majority period of time at the front side of the mid-section. The outer cylinder had a larger magnitude of strain than the inner cylinder. Figure 7.65 compares the strains at the bottom of the mid-section of the inner and outer cylinders with the 100% water level. When compared to the front side, the bottom side showed more oscillatory strains with multiple peaks and valleys. Both cylinders had strains mostly in phase except for a time period between 0.005

204

7 Structural Coupling by FSI 10

3

M-F-100W45

-3

OC 2

IC

Strain [m/m]

1

0

-1

-2

-3

-4 0

0.01

0.02

0.03

0.04

0.05

Time [S]

Fig. 7.64 Comparison of strain responses at the front of the mid-section between IC and OC with 100% water and 45° drop angle

and 0.012 s where the strains were the largest in magnitude and out of phase. The magnitude of strains was much greater for the inner cylinder than the outer cylinder. Figure 7.66 compared the strains at the posterior side of the left section of the inner and outer cylinders, respectively, with 100% water. The strain responses were very different between the two cylinders. The magnitude of strains was larger for the inner cylinder than the outer cylinder. The OC had a larger tensile strain while the IC had a larger compressive strain. The magnitude of the maximum strains was shown as a function of the water level at different locations in Figs. 7.67 and 7.68. The former was for the outer cylinder while the latter was for the inner cylinder. All locations of OC except for OC-M-F showed the largest strain at the 50% water level. The inner cylinder showed more complex behaviors. The largest strain at the inner cylinder occurred at different water levels depending on their locations of IC.

7.8 Numerical Study of Flat Plates The fluid domains were modeled as acoustic media with the properties of water. The first example was a one-dimensional problem where two independent springmass systems were connected by fluid in a tube as sketched in Fig. 7.69. The tube

7.8 Numerical Study of Flat Plates 10

10

205 M-B-100W45

-4

OC

8

IC 6

Strain [m/m]

4

2

0

-2

-4

-6

-8 0

0.01

0.02

0.03

0.04

0.05

Time [S]

Fig. 7.65 Comparison of strain responses at the bottom of the mid-section between IC and OC with 100% water and 45° drop angle

was frictionless and the one dimensional wave equation was used for the fluid with the density 1000 kg/m3 and the speed of sound 1490 m/s. Both masses were cubes with dimensions of 0.1 m × 0.1 m × 0.1 m, and density 7850 kg/m3 . Each mass was modeled as a rigid body with 7.85 kg. The linear spring constant for the left spring was 1000 kN/m while the second spring on the right had the spring constant 10,000 kN/m. The harmonic force f (t) = 100 sin(62.8t) was applied to the left mass. Then, the right mass responded to the motion of the left mass through the FSI. The equation of motion for the left spring-mass system is m l u¨ l + kl u l = f (t) − pl A

(7.2)

where m l is the mass, kl is the spring constant, u l is the displacement of the mass, f (t) is the externally applied force to the mass, pl is the pressure, A is the crosssectional area of the pressure, and subscript l denotes the left system. Likewise, the right spring-mass system has the following equation of motion: m r u¨ r + kr u r = pr A

(7.3)

Here, all the variables are the same as described before except subscript r indicates the right side system. The pressure is in the opposite direction for Eq. (7.3) as

206

7 Structural Coupling by FSI 10

8

L-P-100W45

-4

OC

6

IC 4

2

Strain [m/m]

0

-2

-4

-6

-8

-10

-12 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time [S]

Fig. 7.66 Comparison of strain responses at the posterior of the left-section between IC and OC with 100% water and 45° drop angle

compared to Eq. (7.2) because the pressure is applied to each mass to compress the spring. The 1-D acoustic equation is expressed as follows: ∂2 p ∂2 p = c2 2 2 ∂t ∂x

(7.4)

where p is the pressure, c is the speed of sound of the medium, x is the spatial coordinate, t is time. At the interface of the acoustic domain and the spring-mass system, the following interface conditions are applied at both sides of the acoustic domain. ρ

∂p ∂ 2u = 2 ∂t ∂x

(7.5)

Here ρ is the mass density, u is the displacement, and p is the pressure. Solutions of the coupled system yielded the following results as shown in Figs. 7.70, 7.71, 7.72, 7.73 and 7.74. The first three figures, Figs. 7.70, 7.71 and

7.8 Numerical Study of Flat Plates

207

Fig. 7.67 Maximum magnitude of strains at OC as a function of water fill level with 45° drop angle

7.72 show the displacements, velocities, and accelerations of both left and right masses. The result showed that both masses moved in unison. Figure 7.73 shows the pressure time-history at the middle and the right-end of the acoustic domain. The magnitude of the pressure was very different between the two locations. The force applied to each spring is plotted in Fig. 7.74. The force applied to the right spring was much greater than the force applied to the left spring. This was because the right spring is stiffer than the left spring while both masses moved in unison. The next case considered only the left spring-mass system without the right one. The right side end of the acoustic domain was assumed either non-reflective or fully reflective. The motion of the left spring was plotted in Figs. 7.75 and 7.76 for two different boundary conditions. The displacement of the left mass is compared in Fig. 7.75. The reflective boundary condition reduced the displacement significantly and pushed the displacement toward the negative direction because of the compressive pressure wave coming from the reflected boundary at the right side. The peak to peak displacement was smaller with the reflective boundary condition. When the single spring-mass system was compared to the previous two spring-mass systems, the latter resulted in much greater displacement with an order of magnitude 10. In addition, the vibrational frequency was drastically different between the two different cases.

208

7 Structural Coupling by FSI

Fig. 7.68 Maximum magnitude of strains at IC as a function of water fill level with 45° drop angle

k1

u2

m1

Fixed

Fixed

Fluid u1

m2

k2

Fig. 7.69 One-dimensional model for structural coupling

Figure 7.76 shows that the magnitude of the velocity was not so much different between the two different boundary conditions at the right end of the acoustic domain. However, the reflective boundary condition shows more high frequency motions than the non-reflective boundary condition. Again, the single spring-mass system resulted in much smaller velocity than the two spring-mass systems at each end of the acoustic domain. The force applied to the left spring was plotted in Fig. 7.77 for two different boundary conditions. The force to the left spring had the same characteristics of the displacement of the left mass. The reflective pressure reduced the positive force and increased the negative force. The force with one spring-mass system was also much smaller than that with two spring-mass systems. Comparing Figs. 7.74 to 7.77 shows that the former had about 100 times greater force than the latter.

7.8 Numerical Study of Flat Plates

Fig. 7.70 Displacement plots of both left and right masses

Fig. 7.71 Velocity plots of both left and right masses

209

210

Fig. 7.72 Acceleration plots of both left and right masses

Fig. 7.73 Forces applied to left and right springs

7 Structural Coupling by FSI

7.8 Numerical Study of Flat Plates

211

Fig. 7.74 Forces applied to left and right springs

Fig. 7.75 Displacement of the left mass with two different boundary conditions on the acoustic domain

212

7 Structural Coupling by FSI

Fig. 7.76 Velocity of the left mass with two different boundary conditions applied to the acoustic domain

Fig. 7.77 Forces applied to left spring with two different boundary conditions applied to the acoustic domain

7.8 Numerical Study of Flat Plates

213

The next series of numerical studies were conducted for coupled plate structures as a series of parametric studies. Both structures and fluids were modeled using the CA technique. The plate structures were modeled using the following equations: M = D∇ 2 w ρh

∂ 2w + ∇2 M = q ∂t 2 Mx + M y M= 1+ν

(7.6)

where Mx and M y are the bending moments about the axes along the x- and ydirections, respectively; w is the transverse displacement; ρ is the material density of the plate; h is the thickness of the plate; q is the applied pressure loading; D is the rigidity of the plate; and ν is Poisson’s ratio. The clamped boundary has zero deflection and zero slope at the lattice point. In order to apply the zero slope condition, a fictitious lattice point is necessary next to the clamped lattice point. The simply supported boundary has zero bending moment and zero deflection at the lattice point. The simply supported boundary does not need a fictitious lattice point. The fluid medium was also solved using the linear wave equation as below: ∂2 p = c2 ∇ 2 p ∂t 2

(7.7)

where c is the speed of sound of the fluid, and p is the pressure. The boundary conditions are applied as below. The rigid wall has full reflection. This needs an artificial lattice point. If the boundary is free at the lattice point (o, j, k) in the x-direction, the following boundary condition is used. po, j,k = 2 p1, j,k − p2, j,k

(7.8)

The boundary condition for the non-reflective lattice point (o, j, k) in the xdirection is expressed as below. po, j,k = p1, j,k , p˙ 1, j,k = 0

(7.9)

The structural and fluid domains are solved in the staggered manner. For this analysis, the front plate was solved first followed by the fluid domain, Then, the back plate was solved. This process repeats itself until the solutions converge. After that, the same cycle continues with a time increment. The numerical model was designed to resemble those used in the previous experimental studies, but some variables and boundary conditions were varied to determine their effects on the results. The type of loading was varied first of all. The exponentially decaying load was applied like an explosive loading. The pressure loading was

214

7 Structural Coupling by FSI Localized Loading

10 -3

1.5

0.01

Uniform Loading

1 0.005 0.5 0

Strain

Strain

0

-0.005

-0.5

-1 -0.01 Front Plate

Front Plate

-1.5

Rear Plate

Rear Plate

-2

-0.015 0

0.02

Time(sec)

0.04

0

0.02

0.04

Time(sec)

Fig. 7.78 Comparison of numerical strain responses between front and rear plates for different loading while the space is full of water

assumed to be p = FAo e−αt where Fo and A are the applied force and the area of the applied loading, and α and t are the decay constant and the time variable. In this study, it was assumed that Fo = 1 N and α = 1000/s. Figure 7.78 shows the comparison of the strain responses of the front and back plates at their center when the front plate was subjected to two different loads with full water between the plates. One type of loading was a uniform pressure loading over the front plate, and the other loading was a highly localized force at the center of the front plate. The area of the localized loading was one hundredth of the surface area of the front plate. The two different types of loading produced very different strain responses. When the uniform pressure loading was applied, both front and back plates moved in unison. In other words, both structures were coupled very strongly. The localized loading, on the other hand, resulted in quite different responses of the front and back plates. The magnitude of strains was greater at the center of the plates with the localized loading at the center. The localized loading gave about seven times greater strain in terms of magnitude than the uniform loading when the front plate was compared. The two different types of loading were also applied when the two plates had 50% water between them. Figure 7.79 shows the comparison of the strains at the center of both plates under two different loading. The front and back plates also showed strong coupling with 50% water. However, the two plates did not move in unison in this case because water was 50% full. The localized loading gave more independent motions of the two plates. The magnitudes of the maximum strains were greater for the case of 50% water than that of 100% water for both types of loading.

7.8 Numerical Study of Flat Plates

215

Localized Loading 0.01

10

3

Uniform Loading

-3

2

0.005

1

Strain

Strain

0

-0.005

0

-1 -0.01 -2 Front Plate

Front Plate

Rear Plate

Rear Plate

-0.015

-3 0

0.01

0.02

0.03

0.04

0.05

0

0.01

Time(sec)

0.02

0.03

0.04

0.05

Time(sec)

Fig. 7.79 Comparison of numerical strain responses between front and rear plates for different loading while the space is a half full of water

The central displacements of the front and back plates, respectively, were plotted in Fig. 7.80 when the front plate was subjected to uniform pressure loading. The water level was 100 or 50%. As discussed in the previous paragraphs, both plates 50% Water Level Front Plate

Front Plate

Rear Plate

Rear Plate

0.005

Displacement (m)

0.01

Displacement (m)

100% Water Level

0.01

0.02

0

0

-0.005

-0.01

-0.01

-0.02 0

0.02

Time(sec)

0.04

0

0.02

0.04

Time(sec)

Fig. 7.80 Center displacement plots of the same thickness plates with 50 or 100% water level subjected to uniform pressure loading (both plates are 2.54 mm thick)

216

7 Structural Coupling by FSI 7 Front Plate

6

Back Plate

Magnitude

5 4 3 2 1 0

0

2

4

6

Frequency

8

10 10

4

Fig. 7.81 Frequency spectrum of the same thickness plates with 50% water level subjected to uniform pressure loading (both plates are 2.54 mm thick)

moved together when the water level was 100% but there was some difference in their motions when the water level was 50%. The FFT was applied the displacement timehistories, and the results are plotted in Fig. 7.81 for the 50% water level. The frequency spectrum was very similar between the two plates because of strong coupling. The next parametric study varied the thickness of the plates. One case under study had a 2.54 mm thick front and a 1.27 mm thick back plate, respectively. The second case considered the opposite combination of the front and back plates, i.e. thinner front plate and thicker back plate. For this study, the plates had 100% water between them and the front plate was subjected to uniform pressure loading. Figure 7.82 shows the displacements at the centers of the front and rear plates. The results were quite interesting. The two cases of different plate thicknesses showed very opposite displacements. The thin front plate of one case behaved the same way as the thin back plate of the other case. This was the same for thick front plate of one case and the thick back plate of the other case. This suggested that the plate of the same thickness behaved in the same way regardless whether it was located in the front or back. That was also consistent that when the two plates had the same thickness, the center displacements of both front and rear plates were identical. While both thin plates moved in the positive direction during the computation period, the thick plates showed initial positive displacements followed by negative displacements like a sine curve. The positive and negative displacements of the thick plates had the same magnitude. During the first 0.01 s, both front and back plates moved forward, and they moved in the opposite directions after that until around 0.04 s. The effect of different thicknesses was further investigated when the water level was 50% with the uniform pressure loading. Strains at different locations of the front and back plates of different thicknesses were plotted in Figs. 7.83 and 7.84.

7.8 Numerical Study of Flat Plates

217 0.025

0.02

0.02

0.015

0.015

0.01

Front Plate Rear Plate

0.005

Displacement (m)

Displacement (m)

Thin Front/Thick Back 0.025

Thick Front/Thin Back

0.01 Front Plate Rear Plate

0.005

0

0

-0.005

-0.005

-0.01

-0.01 0

0.02

Time(sec)

0.04

0

0.02

0.04

Time(sec)

Fig. 7.82 Center displacement plots of different thickness combination of plates with 100% water level subjected to uniform pressure loading (thick plate is 2.54 mm thick and thin plate is 1.27 mm thick)

As discussed, strains computed at the back plate were very dependent on the water level. When the front plate was thin, the strains were larger on the thin front plate than the thick back plate, as expected. Gage 2 location in Fig. 7.14 gave the largest peak strain for the thin front plate. This location was above the free water surface. The thicker back plate had smaller strains than the thin plate. However, gage 5 location in Fig. 7.14 gave almost same magnitude of the strains for the thin front and thick back plates. When the thin and thick back plates were compared out of the two cases, the thick back plate had generally smaller strains than the thin back plate as Figs. 7.83 and 7.74 were compared. The next parametric study examined the effect of plate spacing while the front plate was subjected to localized loading. The spacing between the two plates could affect the degree of coupling between them. Figure 7.85 shows the central displacements of the front and back plates for two different spacings. One was the half of the spacing which was considered in the previous study and the other was the twice of that. The results clearly showed that when the spacing was narrow, the two plates moved more closely each other even for the localized loading.

218

7 Structural Coupling by FSI 10

2

Gage #1

-3

10

5

Gage #2

-3

1

Strain

Strain

0

-1

-2

0

Front Plate

Front Plate

Rear Plate

Rear Plate

-3

-5 0

0.01

0.02

0.03

0.04

0.05

0

0.01

0.02

Time(sec) 10

2

0.04

0.05

0.04

0.05

0.04

0.05

Time(sec)

Gage #4

-3

0.03

10

4

Gage #5

-3

Front Plate

1

Rear Plate

2

Strain

Strain

0

-1

-2

0

-2

Front Plate Rear Plate

-3

-4 0

0.01

0.02

0.03

0.04

0.05

0

0.01

0.02

Time(sec) 10

2

Gage #7

-3

10

4

Gage #8

-3

Front Plate

Front Plate

Rear Plate

1

Rear Plate

2

Strain

Strain

0.03

Time(sec)

0

-1

0

-2

-2

-4 0

0.01

0.02

0.03

0.04

0.05

0

0.01

0.02

Time(sec)

0.03

Time(sec)

Fig. 7.83 Strain plots at different gage locations for 2.54 mm thick front plate and 1.27 mm thick rear plate with 50% water level subjected to uniform pressure loading (Please see Fig. 7.14 for the gage locations)

7.9 Numerical Study of Cylindrical Structures A numerical study was also undertaken for cylindrical structures. Because fluid pressures and the movements of the cylinders were not measured during the experiments, those were computed from the numerical study. The two-way FSI was conducted using the ANSYS program [4]. The composite cylinders were solved using the FEM with four-node shell elements. Each cylinder had 5408 nodes and 5304 shell elements. Two concentric cylinders are shown in Fig. 7.86 where the nodal locations for loading are indicated. The impact force was

7.9 Numerical Study of Cylindrical Structures

219 Gage #2

Gage #1 0.01

0.01 Front Plate

Front Plate

Rear Plate

Rear Plate

0.005

Strain

Strain

0.005

0

-0.005

0

-0.005

-0.01

-0.01 0

0.01

0.02

0.03

0.04

0

0.05

0.02

0.01

10

4

Gage #4

-3

10

4

0.04

0.05

0.04

0.05

Rear Plate

Rear Plate

2

Strain

Strain

0.05

Front Plate

0

-2

0

-2

-4

-4 0

0.01

0.02

0.03

0.04

0.05

0

0.02

0.01

Time(sec) 10

4

Gage #7

-3

0.03

Time(sec) 10

4

Gage #8

-3

Front Plate

Front Plate

Rear Plate

Rear Plate

2

2

Strain

Strain

0.04

Gage #5

-3

Front Plate

2

0.03

Time(sec)

Time(sec)

0

-2

0

-2

-4

-4 0

0.01

0.02

0.03

0.04

0.05

0

0.01

Time(sec)

0.02

0.03

Time(sec)

Fig. 7.84 Strain plots at different gage locations for 1.27 mm thick front plate and 2.54 mm thick rear plate with 50% water level subjected to uniform pressure loading (Please see Fig. 7.14 for the gage locations)

applied as a triangular shape in terms of time as shown in Fig. 7.87. The applied force increased linearly until 0.05 s. and after that decreased linearly until 0.1 s. The fluid was modeled using ANSYS CFX which is the computational fluid solver, and the incompressible Navier-Stokes equation was used for the fluid modeling using CFX, whose equation is shown below: 1 ∂ u + ( u · ∇) u = − ∇ p + ν∇ 2 u ∂t ρ

(7.10)

220

7 Structural Coupling by FSI 2x Spacing

1/2 Spacing 0.04

0.03

Front Plate

Front Plate

Rear Plate

Rear Plate

0.03

0.02

0.02

Displacement (m)

Displacement (m)

0.01

0

0.01

0

-0.01 -0.01

-0.02

-0.02

-0.03

-0.03 0

0.01

0.02

0.03

Time(sec)

0.04

0.05

0

0.01

0.02

0.03

0.04

0.05

Time(sec)

Fig. 7.85 Central displacement plots with two different spacing of the same thickness plates (the left plot for spacing 29.6 mm and the right plot for spacing 118.2 mm) under localized loading

Fig. 7.86 Isometric view of the cylinders with the nodal impact location

where u is the fluid velocity vector, ρ is the fluid density, p is the pressure, and ν is the kinematic viscosity. The nominal properties of water at room temperature were considered in the study. Three different water volumes such as 0%, 50% and 100% were considered for the numerical model, respectively. The space between the two cylinders were meshed to confirm the structural mesh at their boundaries, and the space was modelled either

7.9 Numerical Study of Cylindrical Structures

221

Fig. 7.87 Profile of applied force to the numerical study

Force (N)

100

0

0.05

0.1

Time (s)

water or air depending on the water level. At the fluid-structure interfaces, equilibrium of loads and deformation compatibility were applied between the two media. Pressure in the annulus was computed and plotted in Fig. 7.88 with 100% water. The pressure contour plots were obtained at different times in order to investigate the propagation of the pressure in the annulus. Figure 7.88 shows the pressure contours at the mid-length section of the cylinder at two different times. As shown in the plots, the high pressure propagated from the impacted side. Because of the 100% water level, the pressure nearly symmetric between the upper and lower parts of the annulus. The deformed shapes of the inner cylinder are shown in Fig. 7.89. The deformation was magnified by 800 times for visual clarity. The deformation of the inner cylinder was in response to the water pressure in the annulus, as expected. The deformation of the inner cylinder was greater where the water pressure was high. Figure 7.90 compares the pressure time-histories at the four different locations: front, posterior, top and bottom. The front side had the maximum pressure followed by the posterior side. The FFT was conducted to the numerical time-histories, and the frequency spectrums are plotted in Fig. 7.91 for two different water levels. The plots indicated that the first major frequency decreased significantly from the 50% to the 100% water level by approximately a half. The previous experimental study showed a 60% reduction approximately. Considering the difference between the numerical and experimental models such as the boundary conditions and uniform wall thickness of the cylinders, the difference in the frequency reduction is reasonable. The next numerical study investigated the displacements of both cylinders. The displacements at IC-M-P is shown in Fig. 7.92 for two different water levels. The results showed that the 50% water level caused a higher magnitude and frequency in the displacement at IC than 100%. Both displacements at 50 and 100% filling levels had harmonic motions.

222

7 Structural Coupling by FSI

Fig. 7.88 Water pressure contour plots at a time 0.0039 s and b time 0.0104 s

Figure 7.93 shows the displacement at OC-M-P for three different water levels. The plots also gave the higher magnitude and oscillation for the 50% water level like the inner cylinder. The filling level 50% resulted in much longer time to damp out the displacement as compared to other cases. The displacement time-history at IC-M-F is shown in Fig. 7.94. The results showed that even though the 50% water level had a more oscillatory motion, the maximum displacement occurred with 100% water. The displacement at OC-M-F

7.9 Numerical Study of Cylindrical Structures

223

Fig. 7.89 800 times magnified deformed shape of inner cylinder at a time 0.0039 s and b time 0.0104 s

was the greatest because that was the impact site as shown in Fig. 7.95. The 50% water level, in particular, resulted in the largest displacement with more oscillatory motions as compared to 0 and 100% cases. The plots of the velocity and acceleration were very similar to the displacements. The acceleration of IC generally had the largest magnitude with the water level 100%. On the other hand, the acceleration was generally small with 50% water. These characteristics can be shown in Fig. 7.96. However, OC had a different behavior of acceleration. Both locations OC-M-F and OC-M-P showed greater oscillations at the

224

7 Structural Coupling by FSI

Fig. 7.90 Pressure time history with 100% water

water level 50% as shown in Fig. 7.97. The water levels of 100 and 0% resulted in quick damped-out motions.

7.10 Summary This chapter presented both experimental and numerical solutions of coupled structures through fluid media as one structure was impacted by external loading. Both flat structures as well as cylindrical structures were studied. For the flat plates, the spacing between the structures, their thicknesses as well as more than two plates were also considered. The results indicated that structural coupling through FSI is very important because the structural behaviors are very much different resulting from the structural coupling. The resultant strains were much greater for the impacted structures as compared to those without the coupling effect. Additionally, un-impacted structures, otherwise independent, had also comparable strains by FSI as compared to the impacted structures. Besides, the water level between the structures played an important role in the coupling and the dynamic responses of the structures. There was no one water level which resulted in the largest response of the structure. The case depended on many other parameters. As far as the frequency reduction is concerned, the flat structures had a larger reduction at the same water level than the cylindrical structures. The reduction in

7.10 Summary

8

10

225 IC-M-50W

-6

F P

7

T B

Magnitude of Strain

6

5

4

3

2

1

0 0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Log10(Frequency [Hz])

(a) 10

-4

IC-M-100W

1.8 F

1.6

P T B

Magnitude of Strain

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Log10(Frequency [Hz])

(b) Fig. 7.91 Plots of strain magnitude versus frequency of the numerical results of the inner cylinder with a 50% water and b 100% water

226

Fig. 7.92 Displacement at IC-M-P at two different filling levels

Fig. 7.93 Displacement at OC-M-P at three different filling levels

7 Structural Coupling by FSI

7.10 Summary

Fig. 7.94 Displacement at IC-M-F at two different filling levels

Fig. 7.95 Displacement at OC-M-F at three different filling levels

227

228

Fig. 7.96 Acceleration plots at a IC-M-F and b IC-M-P

7 Structural Coupling by FSI

7.10 Summary

Fig. 7.97 Acceleration plots at a OC-M-F and b OC-M-P

229

230

7 Structural Coupling by FSI

the frequency of the flat structures decreased more or less linearly until 50% water and stayed relatively constant after that. On the other hand, the cylindrical structures showed continuous reduction in the frequency until 100% water.

References 1. Kwon YW, Bowling JD (2018) Dynamic responses of composite structures coupled through fluid medium. Multiscale Multidisc Model Exp Des 1(1):69–82 2. Bowling JD, Kwon YW (2018) Coupled structural response via fluid medium. Multiscale Multidisc Model Exp Des 1(3):221–236 3. Alaei D, Kwon YW, Ramezani A (2019) Fluid-structure interaction on concentric composite cylinders containing fluids in the annulus. Multiscale Multidisc Model Exp Des 2(3):185–197 4. Kohnke P (ed) (1999) Theory reference. ANSYS Release 5.6, ANSYS Inc., Canonsburg

Chapter 8

Composite Structures Moving in Fluids

Many structures such as ships move in fluids while other structures such as offshore structures are surrounded by moving fluid. In both cases, the structures are subjected to fluid loading and can experience FSI. In addition, the fluid may contain solid objects such as floating icebergs in oceans. As a ship navigates in such an environment, there are multiple interactions among a structure, fluid, and floating solid objects. This chapter presented those topics. First, experimental studies were discussed for composite structures moving in a fluid medium [1]. Then, additional experiments were presented for the fluid medium containing solid objects [2]. Finally, numerical results were provided to complement the experimental studies [3].

8.1 Experimental Study in Fluid Without Floating Solid Objects Because marine structures like ships travelling in sea are subjected to hydrodynamic loading, many studies have been undertaken to investigate the dynamic loading on structures resulting from the hydrodynamic forces [4–7]. Most of those studies considered rigid structures. However, some structures are not so stiff, and the assumption of rigidity may not produce a reliable solution. Especially, the statement may be true for flexible and light polymer composite structures because the effect of FSI has been proved significant for those structures [8–10]. The hydrodynamic loading can be either steady state or transient. However, most of marine structures are subjected to transient loading as the structures change their speed or orientations. Another example is an offshore structure which is subjected to changing currents. Some studies on transient loading were found in Refs. [11–16]. Transient hydrodynamic loading results in FSI on the structures. The interaction is bidirectional if the structure is flexible. Some studies considered rigid bodies, and some others examined sloshing and water hammer on flexible structures [17–21]. © Springer Nature Switzerland AG 2020 Y. W. Kwon, Fluid-Structure Interaction of Composite Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-57638-7_8

231

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The present study investigated dynamic response of a polymer composite plate subjected to hydrodynamic loading as the plate travelled in water at the steady state and transient motions [1]. A series of experiments were conducted for an aluminum plate and a composite plate of the same size so as to investigate the effect of FSI on both plates. Then, numerical studies were also undertaken to complement the experimental study [3].

8.1.1 Experimental Set-Up In order to conduct the experiment, a plate submerged in water was pulled in a tow tank so as to apply the hydrodynamic force to the plate. The tow tank was 11.6 m long, 0.924 m wide, and 1.22 m tall. The carriage assembly in the tow tank runs along the length of the tank. The assembly has two bearings, each of which runs along the rail of each side of the tow tank. The carriage is pulled with a series of pulley at the center of the width of the tow tank using an electric motor. The rotational speed of the motor is controlled by an AC inverter using Voltage/Hz control techniques. The controller has the pre-determined speed setting by entering the frequency which is linearly proportional to the rotational speed of the motor. The motor has the minimum speed setting at 3 Hz while the maximum speed setting at 30 Hz. Because of the limited length of the tow tank, 9 Hz was the maximum speed utilized in this study. The frequency of the motor is proportional to the speed of the carriage. The test plate was attached to the carriage using a device as shown in Fig. 8.1 which can vary the orientation angle of the test plate in relative to the towing direction. Figure 8.1a shows the angle setting device and Fig. 8.1b shows the sketch of the device affixed to the carriage with different orientation angles of the test plate. When the test plate is toward the direction of the carriage motion, it is called the positive angle. Otherwise, the angle was defined as negative as illustrated in Fig. 8.1b. The angle setting device has a range of angle from −80° to +80°. If the angle is zero, the plate is in the vertical orientation. The change in the orientation angle was considered different shapes of ship hulls travelling in the sea. The test plate was attached to the angle setting device in two different ways. One way used a frame around the test panel as shown in Fig. 8.2. Another way was using two bolted joints in order to affix the text plate. Two different attachments were considered as shown in Fig. 8.3. A set of data acquisition system was attached to the top of the carriage so that they can move together with the test panel. The hardware for the data acquisition includes a load cell to measure the hydrodynamic force applied to the test panel, quarter inch bridge adapters to be connected to strain gages attached to the test panel, a battery pack to supply power, and the wireless data acquisition interface to transfer the collected data to a computer for date collection. The entire data acquisition set-up installed on top of the carriage is shown in Fig. 8.4. The LabVIEW program was used to collect the test data which were from the load cell and strain gages.

8.1 Experimental Study in Fluid Without Floating Solid Objects

Fig. 8.1 a Device to change angle and b various positions of angle as attached to the carriage

233

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8 Composite Structures Moving in Fluids

Fig. 8.2 Aluminum frame around the test panel

Fig. 8.3 Test panel attached using bolted joints

The load cell was secured to the carriage which was pulled by the drive pulley using the motor described above. A high-speed camera was also used to check the speed of the carriage as well as to record any phenomena occurring during each test run. In order to check the speed of the carriage, distance marks were posted on the tow tank. The passage of the carriage from one mark to another was read from the high-speed camera to compute the towing velocity and acceleration. Test panels were either an aluminum or a composite plate. The composite plate was constructed from six layers of glass woven fabrics such that the composite became the quasi-isotropic material. Biaxial strain gages were attached to the test plates. Figure 8.5 shows the strain gages attached to the text plates. The square plate had the dimension 245 mm × 245 mm inside the frame while the rectangular shape had the dimension 330 mm × 178 mm. Because the strain gages will be submerged

8.1 Experimental Study in Fluid Without Floating Solid Objects

235

Fig. 8.4 Data acquisition system attached to top of the carriage

in water, water-proof coating was applied to each strain gage. The aluminum had the same dimension as the rectangular plate. Every test panel was tested with the speed setting from 3 Hz to 9 Hz. The same test was repeated in air and water, respectively so that the test result could be properly adjusted. For example, there is a frictional force for the carriage motion. The air test measured the frictional force, which was subtracted from the measured force in water. Then, the nest force represented the hydrodynamic force applied to the test panel. The water tests were conducted with the same water level. To achieve this, the tow tank was marked at the 0.68 m measured from the bottom and water was filled to the level. This resulted in the distance from the water surface to the top of the test plate to be 0.27 m. Before every test, the water level was checked and refilled as necessary as water evaporated with time. Every test was conducted with the carriage starting from the same position of the tow tank with calm water to make sure the testing environment would be consistent so that water wave generated from the previous test could not affect the results of the subsequent test. Therefore, the carriage was brought back to the same starting position and enough time was waited before the next test.

8.1.2 Results and Discussion The speed of carriage was measured as the motor speed was set at each frequency. The speed was measured using two different ways. First, an anemometer was attached to

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Fig. 8.5 Two different composite plates with strain gages attached; a square and b rectangular shape

the carriage to measure the speed. Then, a high-speed camera was used to measure the position of the carriage as a function of time. Both measurements were very close each other even though the measured speed by the high-speed camera gave a slightly higher velocity than the anemometer as shown in Fig. 8.6. Both measurements indicated that the motor frequency and the carriage speed had the linear relationship. As expected, each motion had the phase of increase in velocity until it reached the steady state velocity. Therefore, the early part was called “Transient” while the

8.1 Experimental Study in Fluid Without Floating Solid Objects

237

Fig. 8.6 Carriage speed

later part was called “Steady state”. During the transient motion, the structure had an acceleration which was almost constant. When the position of the carriage was plotted as a function of time, the transient part was very well fitted using a quadratic function while the steady state part was almost perfectly fitted using a linear function. This suggested that the initial acceleration was nearly constant. The test results were separated for the transient and steady state parts. The first study was conducted for the steady state motion of the carriage. The drag force was measured for the aluminum and composite plate as the test panels were in the top test orientation as shown in Fig. 8.3 with zero-degree orientation 0 angle. The results were compared between the two plates as shown in Fig. 8.7 because they had the same dimension (0.330 m × 0.178 m × 0.005 m) while their material properties

Fig. 8.7 Force versus velocity plot for aluminum and composite plates

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Support Plate

77 mm

Fig. 8.8 Computer model for flow over a rigid body

were different. The comparison showed that the composite plate had a greater drag force than the aluminum plate. The difference became greater as the carriage velocity increased. Those measured drag forces were also compared to a numerical model which was assumed rigid using the Ansys program [22]. Figure 8.8 shows the computer model where the rigid plate was stationary as water flows from left to right with a given velocity. This is easier to model than the opposite case. The total drag force was computed over the rigid body. Then, the drag coefficient was computed using the following standard equation for steady state motion Fd =

1 Cd ρU 2 A 2

(8.1)

Here, Fd and Cd are the drag force and the coefficient of drag, ρ is the fluid density, U is the fluid velocity, and A is the surface area of the plate. The drag coefficients were compared between the rigid numerical model and the experimental results of the aluminum plate as shown in Table 8.1. They compared well. This suggests that the aluminum plate acted like a rigid body. However, the Table 8.1 Comparison of drag coefficients between aluminum test results and numerical results for a rigid plate

Speed (m/s)

Numerical results

Test results for aluminum

1.01

1.199

1.154

1.18

1.201

1.239

1.34

1.215

1.260

8.1 Experimental Study in Fluid Without Floating Solid Objects

239

composite plate was more flexible than the aluminum plate so that the drag force was increased for the aluminum plate. The framed composite plate as shown in Fig. 8.2 was tested at steady state velocities. The measured steady state drag forces are plotted in Fig. 8.9 for different orientation angles at various carriage speeds. As expected, the drag force increased as the carriage spend increased. The measured drag force was different for the positive and negative angles of the same magnitude. In other words, the curves in Fig. 8.9 were not symmetric about the vertical line passing through the zero angle. For example, the drag force for +15° was different from that for −15°. The asymmetric behavior became greater as the carriage speed increased. The numerical analysis showed the symmetric drag force such that the drag force at +θ° was the same as that at −θ° because the model was assumed to be rigid. In addition, the experiment showed large waves behind the carriage, which is expected to change the drag force to the composite plate. The time-averaged strains at the center of the framed composite plate are also plotted in Fig. 8.10. The measured strains were in the horizonal direction, and they were also asymmetric about the vertical axis passing through the zero angle. The strain behavior was similar to the drag force. The next experiments were conducted for the transient motions of the carriage. The transient motions were the acceleration stage. The tests were repeated several

Fig. 8.9 Drag force applied to framed composite plate under steady-state velocity with different orientation angles

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Fig. 8.10 Average strain in the horizontal direction at the center of the framed composite plate under steady-state velocity with different orientation angles

times at the same condition to make sure the measured data were consistent. The results agreed well one another and confirmed the repeatability of the test data. During the transient acceleration stages, large peak forces were measured as shown in Fig. 8.11. The first peak force included the inertia force to tow the carriage from the rest condition. This was also verified from the towing test of the carriage in air, which also showed a large peak force in the beginning. As a result, the force in air was subtracted from the force in water to determine the net drag force resulting from the transient motion in water. The peak value of the net drag force during the transient motion was much greater than the steady state drag force, as expected. Figure 8.12 plotted the peak drag force of the unframed composite plate as shown in Fig. 8.5b with the 0° angle and different towing speeds. Multiple tests were conducted and the data points in Fig. 8.12 are the results from different tests. There is some scattering in the data, but it was not significant. Comparing the transient peak drag force in Fig. 8.12 to the steady state drag force in Fig. 8.9 clearly shows that the former was much greater than the latter. The strain gage at the middle of the unframed composite plate shown in Fig. 8.5b was used to measure the bending strain in the horizontal direction. Even though the force gage showed the initial peak value as the carriage was pulled in air because of the inertia of the system, the strain was not affected during the early motion. The

8.1 Experimental Study in Fluid Without Floating Solid Objects

241

250

200

Force (N)

150

100

50

0

-50 0

1

2

3

4

5

6

7

8

Time (s)

Fig. 8.11 Measured drag force time history in water for the unframed composite plate shown in Fig. 8.5b

Fig. 8.12 Measured drag force in water for the unframed composite plate shown in Fig. 8.5b with zero angle at different towing speeds

strain remained nearly zero during that period. Therefore, the measured strain in water was resulted from the hydrodynamic drag force. As expected, the peak strain during the transient motion was larger than that of the steady state motion. Figure 8.13 shows the ratios of the peak strain of the transient

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Fig. 8.13 Plot of ratios of peak transient strains to steady state strains at different angles and different towing speeds of the unframed composite plate

motion to the strain of the steady state motion at two different towing speeds. The speed indicates the final steady state speed while the transient motion is before reaching the steady state from the rest condition. The results were for the unframed composite plate. Interestingly, the strain ratio was very different depending on the orientation angle of the composite plate. However, there were some general trends. The strain ratio was greater for the lower towing speed than the higher towing speed at the same orientation angle except for the +30° case, where the higher towing speed resulted in a slightly greater strain ratio than the lower towing speed. In addition, the positive (i.e. forward) orientation angles resulted in higher ratios than the negative (i.e. backward) angles. In particular, the +45° angle yielded the greatest ratio while the −45o angle resulted in the smallest ratio. The exception was the −30° angle at 6 Hz. At the 9 Hz towing speed, the strain ratio increased as the orientation angle changed from −45° to +45° except for the angle −30°. The values used for the ratios were the average ones for each case. In other words, the peak strain was the average of multiple transient tests, and the steady state strain was also the mean value of multiple test results. All of the test results were quite repeatable so that the standard deviation of the test data was very small for both transient and steady state tests. The strain ratio at the −30° angle was very high at the towing speed 6 Hz. All those exceptions occurred at either +30° or −30° angle. This may require more research in the flow motion and the fluid-structure interaction with those orientation angles. The variation of the mean peak strain during the transient motion was plotted in Fig. 8.14 for two different angles +45° and −45°. The peak strains became more different as the speed of the carriage increased. The negative angle showed a linear increase along with the towing speed. On the other hand, the positive angle had a

8.1 Experimental Study in Fluid Without Floating Solid Objects

243

Fig. 8.14 Plot of peak transient strains versus towing speed of the unframed composite plate at two different orientation angles

lower peak strain than the negative angle, and the magnitude of the increase in the peak strain became smaller as the towing speed increased. The experimental setting moved the plate with nearly constant acceleration during the transient phase. It was not possible with the current set-up to vary the acceleration rate. As a result, some additional numerical studies were conducted by varying the acceleration rate. Figure 8.15 compares three different velocity profiles as a function of time. One is a linear function as discussed above while the other two represent

Fig. 8.15 Applied three different inlet velocities to the plate model

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monotonically increasing and decreasing accelerations, respectively. The peak transient strain to steady state strain ratios were computed from the three different acceleration cases. The constant acceleration gave the smallest value among the three, and the monotonically decreasing acceleration case yielded the largest ratio. The actual value of the strain ratio depends on the acceleration rate. For the present study with 4 m/s2 as the average reference acceleration, the monotonically decreasing acceleration case had the strain ratio of 75% greater than the constant acceleration case while the monotonically increasing acceleration case had the strain ratio of 25% greater than the constant acceleration case.

8.2 Numerical Study in Fluid Without Floating Solid Objects While the previous section presented the experimental study of a moving composite structures in water, the present section discusses a series of numerical studies of moving structures in water. The numerical study was to complement the experimental study rather than its validation because more diverse numerical models could be studied without restriction. The numerical models are presented below. The numerical analysis was conducted using Ansys [22]. Both Lagrangian and Eulerian models were used. The former model was applied to the structure while the latter was used for the fluid. The structure was either a plate or a box, which was described below.

8.2.1 Plate Model The plate model was first considered so that the numerical results could be compared to the experimental data. During the experiments, the plate structure was towed in the tow tank. However, the numerical study considered the opposite because of the convenience of the modelling. The structure was stationary while fluid flowed toward the structure. The relative motion is the same between the experimental and numerical studies. The plate model inside the tow tank is shown in Fig. 8.8. The plate was 328 mm long, 177.5 mm high and 3 mm thick. The composite plate was quasi-isotropic made of 10 layers of E-glass woven composites, which had the material properties of mass density 2000 kg/m3 and elastic modulus 20 GPa. The plate was attached to an aluminum bar with the cross-section of 26 mm × 26 mm. The bottom 77 mm of the aluminum bar was secured to the composite plate as sketched in Fig. 8.8. The composite plate was modelled using the mesh with 2844 nodes and 14,759 brick-shape solid elements. There were several elements through the thickness so as to represent the bending behavior properly. The water medium had the size of 0.914 m × 0.914 m × 2 m long, which was the same as the water volume in the tow tank. The fluid mesh consisted of 83,893 nodes

8.2 Numerical Study in Fluid Without Floating Solid Objects Table 8.2 Comparison of hydrodynamic forces between numerical and experimental results of rectangular composite plate

245

Max. transient force (N)

Steady state force (N)

Numerical result

28.0

5.4

Experimental result

34.2

4.9

Error (%)

−22

10

and 45,220 brick-shape elements. Before deciding those meshes, different meshes were studied. Then, the optimal mesh was selected by considering both the accuracy and computational time. The supporting condition of the plate was described above, and the fluid domain had a uniform inlet velocity which varied from zero to 0.5 m/s during the time period of 0.05 s with a constant acceleration. Then, the velocity remained the same after that. The top surface was assumed to have zero pressure as a free surface while two side walls had no-slip boundary condition. The water had the density 1000 kg/m3 and the kinematic viscosity 1.0 × 10−6 m2 /s. Finally, a uniform pressure was assumed at the outlet side. Then, the coupling technique was applied to the interface of the structure and fluid domains. The experiments were conducted in a towing tank whose dimensions were described in the previous section. The top edge of the unframed composite plate of a rectangular shape was placed approximately 0.25 m below from the free water surface. The composite plate was pulled through the carriage starting from rest to a steady state velocity of 0.5 m/s, after which the velocity remained constant. The transition period was measured using a high speed camera and was 0.05 s. The details of the test procedure were provided in the previous section. The numerical results were compared to the experimental data in terms of the hydrodynamic drag force. Table 8.2 shows the comparison between the two results for both transient and steady state periods. There was some difference between the two results. The difference for the transient peak force was about 20% while the steady state force was 10%. The transient force showed a larger difference. Even though the numerical model was attempted to represent the experimental model, there was a discrepancy between the two models. The physical test showed fluctuation of the free surface like a wave after the passage of the plate along the water. On the other hand, the numerical model did not consider such a wave motion at the top free surface. This requires a multi-material fluid model. In addition, the transient acceleration was not exactly constant as measured using the high speed camera. Considering such discrepancies between the experimental and numerical models, the difference in Table 8.2 was reasonable. Once the numerical model was confirmed, a box shape structure was studied in more details as below.

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8.2.2 Basic Box Model The next model was a box shape structure which was a cube of 1 m × 1 m × 1 m. Quadrilateral shell elements were used for the model with 3458 nodes and 3456 elements. The front face was assumed to be made of the same composite material as described above while the rest of five walls were assumed rigid and constrained from movement. Because the box was enclosed completely by six faces, the inside is empty while the outside was in contact with fluid (i.e. water). The fluid domain was extended by two meters from the front face which was the composite plate and one meter from the rest of the sides. As a result, the fluid has the domain size 3 m × 3 m × 4 m with 110,937 nodes and 100,744 brick shape elements. Fluid-structure coupling was applied to the front face of the cubic structure and no slip conditions to the other five faces. A prescribed velocity was applied to the flow inlet side while a constant pressure was applied to the flow outlet side. The rest of fluid sides were assumed to have full-slip boundary conditions by assuming there was no boundary layer effect to the cubic structure. The water properties were provided previously. The inlet velocity to the fluid domain was sketched in Fig. 8.16 where the velocity was prescribed as a function of time. The velocity increased linearly from zero to 2 m/s and it remained constant. For this study, the front composite plate was also replaced by a rigid material so that fluid flow could be examined without the effect of FSI. The no slip boundary condition was applied to the front face for this case. Then, this flow fields could be compared later with those with the effect of FSI. Figure 8.17 shows the fluid velocity contours around the completely rigid box. The contours were plotted on the fluid plane which passes through the box in the middle. There was no deformation on the box. Figure 8.18 shows the similar fluid

Fig. 8.16 Applied inlet velocity to the basic box model

8.2 Numerical Study in Fluid Without Floating Solid Objects

247

Fig. 8.17 Fluid velocity contours around the rigid box at a 0.09 s, b 0.58 s, and c 2.0 s

velocity plots on the same plane except that the front face was a flexible composite. Therefore, the deflection contours of the composite plate are shown in the figure, too. The velocity contour plots in front of the front side of the box were different between the rigid and flexible plate. This difference was resulted from the FSI of the flexible plate. The center of the composite plate showed the maximum displacement

248 Fig. 8.18 Fluid velocity contours around the box with composite front face at a 0.09 s, b 0.58 s, and c 2.0 s

8 Composite Structures Moving in Fluids

8.2 Numerical Study in Fluid Without Floating Solid Objects

249

of 45 mm toward the inward (i.e. flow) direction and the maximum displacement of 15 mm toward the outward direction. The maximum inward deflection occurred at 0.09 s while the maximum outward deflection occurred at 0.58 s. On the other hand, the steady state deflection was 5 mm in the inward direction at 2.0 s. Those deflections influenced the fluid velocity in front of the plate. When the composite plate had the maximum outward deflection at 0.58 s, the deformed composite plate affected the flow streamline and resulted in a slightly faster average velocity around the cube structure. The average velocity around the cube was 2.32 m/s for the composite plate while it was 2.22 m/s for the rigid plate. However, when the velocity stayed at a steady state for a while such as at 2.0 s, there was virtually no notable change in the velocity contours between the rigid and composite plates. The stress and strain in the composite plate was examined, too. The numerical results showed that the maximum stress or strain occurred at the edge of the composite plate which was like a clamped boundary due to the rigid side plates. The equivalent strain contours were shown in Fig. 8.19 on the composite plate. The plot clearly showed that the four edges had the largest equivalent strains. The equivalent strain has a similar expression like the von Misses stress. The equivalent strain was compared between the center and the edge of the composite plate of the base model. The strain was more than 50% greater at the edge compared to the center. During the steady state period, the strains remained constant as shown in Fig. 8.20. The peak transient equivalent strain was much greater than that at the steady state condition. The transient strain was also much larger at the boundary than at the center. Figure 8.21 shows the deflection at the center of the composite plate of the base Fig. 8.19 Plot of equivalent strain contours for the base model

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Fig. 8.20 Comparison of the equivalent strain at the center and the edge of the base model

Fig. 8.21 Central deflection as a function of time of the base model

model. As stated previously, the composite plate had the maximum inward and outward deflection as the specified time. During the acceleration period, the deflection remained in the inward direction. Just after the inlet flow velocity reached the steady state value at 0.5 s, the deflection turned into the outward direction. This was followed by an oscillatory deflection for a while until the deflection remained constant in the inward direction. The average fluid pressure on the composite plate was plotted in Fig. 8.22. The time history of the pressure profile was very similar to that of the profile of the plate deflection at the center except one was upside down of the other. This was because pressure was plotted as a positive value while the inward displacement was plotted in the negative value. The results suggested that the deflection at the center of the plate was very dependent on the applied average pressure. After the numerical study of the basic box model, a series of parametric studies were conducted by changing one parameter at a time. The cubic box model described

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Fig. 8.22 Average pressure time history on the composite plate of the base model

above was the basic model from which a variation was derived, one at a time. The parameters for change were the size of the structure, its shape, material properties, fluid boundary condition, etc. The varied models and their results were presented subsequently.

8.2.3 Change in Magnitude of Constant Acceleration This parametric study investigated the effect of the initial acceleration, which was ether increased or decreased as compared to the previous base case. The base model had the initial acceleration 4 m/s2 , and the parametric study considered the initial acceleration 2 or 8 m/s2 . The steady state velocity remained the same for all cases, which was 2 m/s. Figure 8.23 shows the sketch of the velocity profile to the flow inlet. The average pressure on the front composite plate was plotted in Fig. 8.23 for three different acceleration cases. As expected, the higher acceleration resulted in the higher average pressure on the composite plate. The peak pressure in Fig. 8.24 was linearly proportional to the acceleration rate. As the acceleration increased by two, the peak pressure became twice larger. It also took a longer time for the higher acceleration case to settle down to the steady state pressure. However, the peak pressure occurred at the same time regardless of the different acceleration rates. The equivalent stress or strain was similar to the pressure plot. The maximum stress or strain was linearly proportional to the acceleration rate. The ratio of the transient peak stress or strain to the steady state stress or strain also increased with the acceleration ratio linearly. Figure 8.25 shows the plot of the strain ratios as a function of the acceleration rate. The data were very linear. The maximum stress or

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Fig. 8.23 A parametric study of different rates of initial acceleration

Fig. 8.24 Average pressure over the composite plate for three different rates of initial acceleration

strain occurred during the acceleration stage. Therefore, structures must be designed with consideration of their acceleration rates.

8.2.4 Change in Steady State Velocity After examined the different acceleration rates, the second parametric study was to examine how the terminal steady state inlet velocity affected the response of the composite structure. The initial acceleration rate was the same while the steady state velocity was varied. One case was larger than the base case while the other case was

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Fig. 8.25 Plot of transient peak to steady state equivalent strain for three different acceleration rates

smaller than the base case. Those velocities were 2.5 and 1.5 m/s as compared to the base model velocity 2.0 m/s. As a result, the time to reach the steady state velocity became different. The steady state velocity of 2.5 m/s occurred at 0.63 s while the 1.5 m/s was reached at 0.38 s. Figure 8.26 shows three different inlet velocity profiles. As shown previously, the peak pressure and deflection occurred at 0.09 s. However, the steady-state response was dependent on the steady state velocity. As a result, the ratio of the transient peak stress or strain to the steady state stress or strain was dependent on the steady state velocity. Since the steady state stress or strain increased

Fig. 8.26 Three different steady-state inlet velocities

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with the steady state velocity, the ratio of the transient peak value to the steady state value decreased with the steady state velocity as seen in Fig. 8.27. Figure 8.28 shows the average pressure applied to the composite plate for three different terminal steady state velocities. Even though the positive peak pressure was the same for all three cases, the negative peak pressure was different. The base model gave the largest negative peak pressure even though the difference was not

Fig. 8.27 Plot of transient peak to steady state equivalent strain for three different terminal steady state velocities

Fig. 8.28 Average pressure on the composite plate with three different terminal steady state velocities

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significant. The deflection at the center of the composite plate also reflected the same behavior. After the initial maximum inward deflection, the maximum outward deflection was the greatest for the base model. In other words, increase or decrease in the terminal steady state velocity did not increase the maximum outward displacement as compared to the base case.

8.2.5 Intermittent Zero Acceleration This case considered intermittent zero accelerations. In other words, the flow profile had a linearly increasing velocity (i.e. a constant acceleration) followed by a constant velocity (i.e. no acceleration). This process continued multiple times until the final velocity reached the same steady state velocity 2.0 m/s. Figure 8.29 sketched the prescribed velocity profiles under study. The figure illustrated the transient period, but the computation continued for a much longer steady state period. One case had four steps until the velocity reached the steady state, which was called 4-step. The steps occurred at the velocities 0.5, 1.0, 1.5 and 2.0 m/s. The other case was the 2-step with velocities held at 1.0 and 2.0 m/s. The 4-step velocity profile had a greater acceleration during the acceleration period, which was followed by the 2-step velocity profile. Such a different acceleration resulted in quite a different deflection of the composite plate at its center. Figure 8.30 shows the plot of the central deflection of the composite plate for different intermittent zero accelerations. The 4-step velocity profile resulted in the largest deflection in both inward and outward directions as compared to the base case and the 2-step case. Among the latter two cases, the 2-step case resulted in the greater inward deflection while the base case yielded the larger outward deflection.

Fig. 8.29 Inlet velocity with intermittent zero acceleration

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Fig. 8.30 Central deflection of the composite plate for different inlet velocity profiles with intermittent zero acceleration

The most interesting point was the time of the maximum deflections. The previous cases consistently showed that the maximum inward deflection occurred at the same time regardless of the change in acceleration. This parametric study showed that the intermittent zero acceleration produced the largest deflection of the plate at very different time. The largest inward deflection occurred at 0.09 s for the base case, 0.24 s for the 4-step case and 0.39 s for the 2-step case. On the other hand, the most outward deflection occurred at around 0.6 s for the base model, 0.18 s for the 4-step model, and 0.62 s for the 2-step model. The 4-step model showed that the maximum outward deflection occurred followed by the maximum inward deflection. As a result, both inward and outward maximum deflections occurred at a close time interval. The base model had the longest time interval between the maximum inward and outward deflections. The 2-step case was the between the two other cases. However, the maximum inward deflection occurred at a much later time for the 2-step case as compared to other cases. The results suggested that intermittent zero accelerations yielded a very serious effect on the structural response. The maximum stress was increased by 25% for the 2-step case and 72% for the 4-step case as compared to the base case. The 4step model had the constant acceleration 6.25 m/s2 between the zero accelerations while the 2-step model had the constant acceleration 4.5 m/s2 . The base case had the acceleration 4 m/s2 . If there were no intermittent zero acceleration, the acceleration of 6.26 m/s2 would have yielded an increase in the maximum stress by 56%. Likewise, the acceleration of 4.5 m/s2 would have yielded an increase in the maximum stress by 13% without interruptions. Therefore, the interruption in the acceleration resulted greater stresses as compared to uninterrupted situations. The maximum stress occurred when the plate had the maximum deflection. The plate oscillated with different vibrational frequencies. The 4-step model showed the

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Fig. 8.31 Plot of ratios of maximum to steady state stress/strain against no. of interruptions in acceleration

highest frequency and the base model showed the lowest frequency among the three cases. Because the transition time from a constant acceleration to zero acceleration or the vice versa was a major change in acceleration, those instants influenced the dynamic motion of the plate. For example, the 2-step model started the zero acceleration at 0.22 s. and 0.5 s. The plate had a sharp increase in displacement at these times. On the other hand, fluid pressure was reduced sharply at those times, too. The ratio of the transient peak stress to steady state stress was plotted in Fig. 8.31 as a function of the acceleration interruptions. The base model had one interruption when it reached the steady state velocity. The 2-step had two interruptions and the 4-step model had four interruptions. Because the steady state value was the same for all cases, the stress ratios are larger with more interruptions, which resulted in greater stress.

8.2.6 Monotonically Varying Acceleration The previous cases considered either constant or piecewise constant accelerations. This parametric study examined non-constant accelerations. One case considered a monotonically increasing acceleration while the other case considered a monotonically decreasing acceleration as sketched in Fig. 8.15. The former case has an increasing velocity, and the latter has a decreasing velocity until both reaches the same steady-state velocity. The early portion of the monotonically decreasing acceleration was close to linear until 0.25 s like the base model, Therefore, the monotonically decreasing case was

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Fig. 8.32 Center deflection of front plate under monotonically increasing or decreasing acceleration

similar to a constant acceleration case for the early time. Figure 8.32 shows the center deflection of the front plate. The monotonically decreasing acceleration case resulted in the maximum deflection during the early time, which was very close to the base case in terms of shapes even though the magnitude was much lower for the base case. In addition, the gradually decreasing acceleration yielded less oscillatory motion as the plate reached its steady state deflection. The gradually increasing acceleration delayed the peak displacement to a later time. The peak deflection occurred when the inlet velocity reached the steady state value because the transition from the large acceleration to zero acceleration yielded more abrupt change in the fluid pressure over the plate. This also yielded a higher magnitude of oscillation before settling down to the steady static deflection. Because the gradually deceasing acceleration produced the largest strain during the early time, it also resulted in the largest ratio of peak to steady state strain. The monotonically increasing acceleration gave the second largest ratio while the base case showed the lowest ratio among the three cases. Another case was that the acceleration increased gradually in the beginning and decreased gradually later such as combining the two cases studied here. This velocity profile had an inflection point in the middle with smaller acceleration at both ends of the transient period. This type of velocity has an advantage to reduce initial and later large deflection or stress during the transient period. This kind of motion is beneficial to minimize the stress on the structure during its transition period to reach the steady state velocity.

8.2.7 Geometric Variation The geometry of the structure was varied in this study. The first case reduced the cube dimension by half such that each dimension was 0.5 m instead of 1.0 m. The second

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Fig. 8.33 Center deflection of four different geometries

case was a cylinder with either flat end or a hemispherical end. The diameter of the cylinder was 1.0 m like the base cube. The inlet velocity was the same as specified for the base model. The half size cubic model was called the cube 0.5, the flat face cylinder was called flat cylinder, and the hemispherical cylinder was called the dome cylinder. First, the location of the peak stress was determined for those models. As expected, the cube 0.5 and flat cylinder models had the peak stress near the boundary. However, the dome cylinder case showed the peak stress near the center of the dome. The maximum center deflection was larger for the base and flat cylinder models than the other two models as seen in Fig. 8.33. The change from a cube to a cylinder made a minor difference in their center deflections. The peak deflection occurred at the same time for the base and flat cylinder models. The dome cylinder showed a very small deflection. The half cube also had a small deflection because the plate was one quarter of that of the base model. The fluid pressure at the center of each plate was compared in Fig. 8.34. The cube 0.5 model had a much lower pressure at the center of the front plate. While the pressure was high for the dome cylinder, its geometric stiffness reduced the deflection at the center. The curved surface provided structural integrity. The ratio of the transient peak to steady-state stress is listed in Table 8.3, which showed the smallest ratio for the dome cylinder. However, the ratio was still nearly four.

8.2.8 Variation in Material Property This case considered different material properties of the composite plate of the base model. Both the mass density and elastic modulus were varied one by one. The inlet

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Fig. 8.34 Fluid pressure at the center of the front face of four different geometries

Table 8.3 Ratios of transient maximum to steady state stress for different geometric models Ratio

Dome cylinder

Flat cylinder

Base cube

Cube 0.5

3.9

9.8

8.1

5.5

flow speed was the same as the base case. Different combinations of the material properties are listed in Table 8.4 for this study. Figure 8.35 shows the fluid pressure at the center of the composite plate with different combinations of material properties. The change in the density and elastic modulus did not significantly influence the pressure time history. However, the response of the plate was different depending on its material properties. The equivalent strain was plotted in Fig. 8.36 for different combinations of the material properties. Even though the pressure profile was similar, the strain response was very dependent on the material properties. The strain was inversely proportional to the elastic modulus. The lower modulus resulted in the larger strain. When the modulus was the same, the lower density produced slightly larger strains. The stress plot in Fig. 8.37 was quite different from the strain in Fig. 8.36. Even though the strain was larger for the lower modulus, the stress was not the case because the stress is also related to the elastic modulus. The largest stress occurred for the highest elastic modulus among the four cases while the lowest stress occurred for the lowest modulus. When the density was higher with the same modulus, the stress was lower. The 2000 kg/m3 density case of the same modulus had approximately 15% Table 8.4 Different material property Density

(kg/m3 )

Young’s modulus (GPa)

Base

2000/50

2000/100

3000/50

2000

2000

2000

3000

20

50

100

50

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Fig. 8.35 Fluid pressure at the center of the front face of different material properties

Fig. 8.36 Equivalent strain at the center of the front face of different material properties

higher stress than the 3000 kg/m3 density case. This result indicates that a structure made of a lighter material can have higher stresses under the same hydrodynamic loading. Because material properties affect both the transient and steady state, the ratio of the transient peak value to the steady state value was more or less the same.

8.2.9 Variation in Water Depth from Free Surface The last case examined the effect of different water depth from the free surface to the structure because the free surface can influence the flow around the structure. The

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Fig. 8.37 Effective stress at the center of the front face of different material properties

base model did not have the free surface boundary condition. The free water surface was assumed to have zero pressure. The inlet velocity profile was the same as that of the base case. The depth was measured from the free surface to the top of the structure, and the first case had 1 m. Then, the depth was increased to 2 m and 3 m, respectively. In the following discussion, the n-meter model suggests n-meter depth. The free surface affected fluid pressure in front of the plate. Figure 8.38 shows the average pressure on the front plate with different water depths. As expected, the lower water depth resulted in much lower pressure. However, the qualitative shape of the pressure time history was close one another. In addition, the pressure distribution was not symmetric for the free surface model. Instead, they varied through the depth

Fig. 8.38 Average pressure on the plate with different depths

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Fig. 8.39 Center deflection of the plate with different depths

of the water. When the water depth becomes infinite and the hydrostatic pressure is neglected, the case is equal to the base case. The deflection of the plate at the center was plotted in Fig. 8.39. The time history of the deflection followed that of the average pressure as before. The stress and strain were also similar to the fluid pressure at different water depths. The ratio of the transient maximum stress to the steady state stress was computed and plotted in Fig. 8.40. The ratio increased gradually as the water depth increased until it reached the plateau value which was obtained from the base model. The base model showed the four times greater ratio than the 1 m depth model. When the water depth was about 9 m, the free surface effect was negligible.

Fig. 8.40 Plot of ratios of transient maximum to steady state stress for water depth from free surface

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8.3 Experimental Study in Fluid with Floating Solid Objects This section presented a study of a moving structure in water with floating solid objects [2]. The floating solid objects were representation of icebergs. An experimental work was performed using a tow tank with floating objects to represent icebergs, called Iceberg Equivalent Objects (IEO). The reason IEO was selected instead of actual ices was that there was no temperature control in the tow tank so that ices would be melt during tests. In that case, repeatable test results would be difficult to obtain from the experiments. On top of the experimental test, some computational modeling and simulation was performed to complement the experiment study so that the interaction between a flexible structure and the IEO in the water could be further investigated. As a result, this is a study of interaction among three things, a flexible structure, iceberg-like solids, and water.

8.3.1 Experimental Set-Up The experiment was conducted in the tow tank, which was described in the previous section, which provided the dimension of the tow tank, control of the towing speed, and the data acquisition system. The towed composite structure was also set at a designated orientation angle, as shown in Fig. 8.1, using the angle selector. During each towing experiment, a load cell measured the towing force, and strain gages were used to measure the deformation of the composite plate being towed with the sampling rate of 100 samples per second. As described in the previous section, each towing test has three stages starting from the beginning to the end of the test. The first phase is an acceleration stage while the last stage is the deceleration stage. The middle stage is the steady state stage. The composite panel was constructed using the glass-fiber woven fabric composite, which is shown in Fig. 8.41. The left side of the figure is the computer model for the numerical simulation while the right side is the composite plate. The composite plate was a square shape of 0.3048 m × 0.3048 m with ten quasi-isotropic layers of the total thickness of 5.0 mm. The composite plate had the effective modulus of 20 GPa with the density of 1475 kg/m3 . The other composite panel with a frame along the edge as shown in Fig. 8.5a was also used in this study. The composite plate was only constrained by a strip of aluminum rig to be attached to the sliding carriage. All the boundaries of the composite plate were free without any constraint. Two strain gages were affixed to the plate. One was at the center of the plate, and the other was near the left bottom corner, which was located one quarter distance from that corner to the right and the top directions, respectively. This was called the quarter point while the former was called the center point in the subsequent discussion. The strain gage at the center location was oriented in he vertical direction, and the strain gage at the quarter location had both horizontal and

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Fig. 8.41 Composite test plate used for interaction with floating ice equivalent objects

vertical direction. The number of strain gage data was limited because the available slots in the data acquisition system. To properly represent icebergs in the Arctic environment, the solid object was selected to match the mass density of the iceberg. Seawater icebergs contain pockets of brines which is a mixture of a high concentration of salts and water. Because of this nature, the density of icebergs varied within a large range such as between 720 and 960 kg/m3 [23]. The majority of the densities was close to the upper limit of the range. The seawater also has a varying density dependent on its temperature and salinity [23]. As a result, the arctic seawater has different density depending on the time of the season and the water depth. In this study, the spring and summer seasons were considered since these are the most accessible to the Arctic area. In addition, the surface layer of the seawater was also selected. Under those conditions, the salinity was between 29 and 35 PSU (Practical Salinity Unit) and the temperature was between −2 and 6 °C. The highest density resulted from the low temperature, high salinity water. As result, the density of the Arctic seawater was between 1022.8 and 1028.2 kg/m3 . Finally, those data gave the ratio of the seawater ice density to the seawater density in the Arctic area from 0.70 to 0.94. The fresh water in the tow tank has a density of 990 kg/m3 . To have the equivalent ice to water density in the Arctic condition, the IEO should have a density between 693 and 931 kg/m3 . The upper limit would be a better choice because it was more common. The selected material was Low Density Polyethylene (LDPE) of a cylindrical shape. The LDPE has the density 924 kg/m3 which resulted in the density ratio 0.933 of ice to water. For the tests, different sizes of LPDE were used. One has the diameter of 101.6 mm while the other had the diameter of 50.8 mm. The former was twice larger than the latter in terms of the diameter. One test case used two larger size LPDE pieces

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floating in the water of the tow tank while the other case used a cluster of smaller LDPE pieces which were randomly distributed in the water. The larger LDPE was to represent larger icebergs impact to the moving composite panel. The cylindrical object may float in two different configurations. One is the cylindrical axis is normal to the water surface while the other has the axis parallel to the water surface. The former is called the vertical floating condition. For this study, the vertical floating configuration was selected. In order to do that, the smaller LDPE cylinders were cut to the length 38.1 mm, which was the longest length of the cylindrical object to stay at the vertical floating condition in stable. For the same reason, the larger LDPE cylinders were cut to the length 76.2 mm. The next phase in the preparation for the experiment was the arrangement of the LDPE pieces in the tow tank. The first scenario considered a cluster of small pieces of IEO collision to a moving composite panel. Because tests were repeated multiple times at the same condition, it was important to make sure the distribution was consistent. To this end, a specific water zone in the tow tank was selected, and the surface coverage density of LDPE pieces to the water surface of the zone was computed such that the number of LDPE pieces could be determined for the water surface zone. The covered surface area of the LDPE pieces was computed from the circular cross-sectional area of one LDPE multiplied by the number of pieces. Then, the surface coverage ratio was determined by dividing the surface coverage area by LDPE pieces by the water surface area of the selected zone in the tow tank. The water zone in the tow tank was selected to be 0.94 m. Three different surface coverage ratios were considered in the study. They were called the low, medium and high ice surface coverage ratios, respectively. The low coverage ratio had 54 LDPE pieces of the smaller diameter, which resulted in the surface coverage ratio 0.125. When the LDPE pieces were doubled, the coverage ratio was 0.250. When they were tripled, the ratio became 0.375. All the pieces were placed randomly within the predefined water zone. Figure 8.42 shows an example of the distribution of LDPE pieces. The circles denote the LDPE pieces and the two vertical lines defined the water zone where the LDPE pieces were placed. Figure 8.43 is a picture of an actual distribution of the smaller LDPE pieces.

Fig. 8.42 Illustration of distributed LDPE pieces in a pre-defined water zone

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Fig. 8.43 LDPE pieces of a small dimeter in the tow tank

The other scenario used the LDPE cylinders of the larger diameter. Figure 8.44 shows different arrangements of the LDPE pieces of the larger diameter. Either single piece or two pieces were considered in this scenario. To control the larger LDPE pieces in a desired location, holes were drilled on the side of every large LDPE piece. A short soldering wire secured to the drilled hole of each LDPE piece. Then, two large LDPE pieces were connected by a very thin flexible thread to maintain the specified distance between the two by attaching the thread to the soldering wires. Each test was recorded using a video camera. If the review of a test revealed the misplaced LDPE pieces, the test result was neglected and re-tested. First, the composite plate to be tested was towed in the tow tank without water as before in order to establish the base line data without interaction with water. For example, the towing force in this test was related to the inertia force of the carriage system, and this force was subtracted from that was obtained with water such that the net force resulting from the water could be obtained. The test in water was conducted

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Fig. 8.44 Different arrangements of larger diameter LDPE pieces

without the LDPE pieces, and later those pieces were placed for the next test. By doing so, the effect of the interaction of the composite plate with water and IEO could be understood one by one. Three orientation angles were considered during each test, and those angles were −30°, 0° and +30°, which were called the negative, vertical, and positive angle, respectively in the subsequent discussion. The carriage system was pulled from one end of the tow tank toward the other end. Two locations were chosen for the IEO inside the tow tank. One location was the place where the carriage system travelled in a constant velocity of the steady state phase. The other location was where the carriage system had an increasing velocity of the acceleration phase. The latter location was the section the plate would have the accelerating transient motion. The carriage was pulled in different steady state velocities. All trials for the various plate orientations were completed first for the lower ice coverage ratio. Time was allowed between successive trials for the tow tank water and LDPE rods to reach a state of rest. Upon the completion of the low ice coverage ratio testing, the additional LDPE segments required for a higher ice density ratio were added to the tow tank. Finally, after all test data were collected, a waterproof video camera was secured to the bottom side of the test plate so as to capture the underwater footage of the interaction among the composite plate, water wave, and floating solid objects. In order to secure the video camera to the composite panel, the framed composite plate as shown in Fig. 8.5a was utilized.

8.3.2 Results of Steady State Motion Each test was repeated several times under the same condition to determine the average value and the standard deviation of the test results. A typical test set of data is plotted in Fig. 8.45 for the average value and the standard deviation. Even though

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Fig. 8.45 Average and standard deviation of test results at the speed of 8 Hz of the towing motor

the data was scattered as expected, the range was not so large. In other words, the test results were very repeatable. This observation was true with or without IEO. All the subsequent plots used the average values unless otherwise mentioned in order not to overcrowd the figures. The composite plates were submerged partially such that their top 12.7 cm was above the waterline when the water was calm. This was the same in all three different orientation angles. This top portion of the composite plate is called “freeboard” in this study. The test results in Fig. 8.45 suggested that the drag force was significantly less for the negative orientation angle regardless of with or without IEO. On the other hand, the positive and vertical angles produced very comparable drag forces which were much greater than that for the negative angle. The reduction in the drag force for the negative angle was resulted from the flow characteristics around the plate. The negatively angled plate was a more streamlined configuration than the others. In addition, the negatively angled plate pushed down the water while the positively angled plate pushed up the water as the composite plate travelled in the tow tank. Therefore, the positive angle generated larger waves in front of the plate, which was like the bow wave of a ship. This made the freeboard shorter as the composite plate moved faster. All these effects increased the drag force on the positively angled plate. The vertical plate has a larger projected normal surface area than other angles to increase the drag force. However, the effect of freeboard, i.e. drastic decrease in the length of the freeboard was less for the vertical plate than the positively angled plate, Therefore, there are two competing factors between the vertical and positive angle.

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Figure 8.46 shows both experimental and numerical results for the positively angled plate. In the experiment, the plate travelled from left to the right direction while in the numerical study, the water flowed from left to right. As a result, the orientation of the plate looks different in the figure, but both of them are the positive angle. Figure 8.47 shows the plot of drag force as the steady state towing speed was varied with different orientation angles. In addition, the results were also obtained with and without IEO. The speeds were 1.35, 1.51, and 1.68 m/s. As expected, the drag force increased as the towing speed increased. The vertical plate without IEO showed a very small increase from the speed 1.35 to 1.51 m/s. On the other hand, the vertical plate had the largest increase in the drag force from 1.50 to 1.67 m/s. The theoretical drag formula suggests the drag force is proportional to the square of the speed. The experimental results showed a nonlinear increase with the square of the velocity because generation of water surface waves as discussed above influenced the drag force. Figure 8.48 shows the strain in the vertical direction at the center of the unframed plate as the steady state towing speed was varied. The strains were measured without IEO in the tow tank. The vertical strain was the largest for the positive orientation and the smallest for the negative orientation. The trend in the vertical strain at the center followed that of the drag force. The positive orientation produced much larger vertical strains at the center than the vertical orientation while the drag force was not that much larger for the positive angle. This is because not only the total force but also its distribution resulting from the water is very different depending on the plate orientation. Therefore, the negative orientation had almost the same vertical strain at the center regardless of the towing speed.

Fig. 8.46 Flow over the positively angle plate: a experiment with the plate moving to the right, b numerical model with the plate moving to the left

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Fig. 8.47 Plot of drag force as a function of steady state towing speed for different orientation angles

Fig. 8.48 Vertical strains at the center of the plate at steady-state speeds for different plate orientation angles without equivalent objects

The next study was with the IEO with the 37.5% ice coverage. The floating solid objects increased the drag force applied to the composite plate. This occurred independent of the plate orientation angle as shown in Fig. 8.47. Because the LDPE pieces were scattered randomly in the designated zone of the tow tank, the test results

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with IEO showed more scattering than the data without IEO, but the scattering was not so significant as shown in Fig. 8.45. Figures 8.49, 8.50 and 8.51 show the plates of different orientations approaching the floating smaller LDPE pieces. The top picture of each figure was just before their contact. Then, the following pictures show progressive motions of each plate through the LDPE pieces. The arrows in each middle picture indicate how the floating LDPE pieces move in relative to the approaching plate. The LDPE pieces met the vertical plate in the normal direction. The LDPE prices moved downward along the plate of negative orientation while they moved upward along the plate of the positive orientation. Such motions make sense in terms of the orientation of the plate. A numerical study was also conducted to provide more information on the relative motion of IEO to the various moving plates. Table 8.5 compares the peak pressures in water for positively and negatively oriented plates, respectively, at different depths from the free water surface. The peak pressure was the value above the hydrostatic pressure at the given location. The fluid pressure was greater for the positively oriented plate than the negatively oriented plate at all depths. However, the two orientations gave different pressure distribution along the water depth. The negative orientation showed the decrease with the increase of the depth from the water surface. This explains why the IEO moved downward for the negatively oriented plate. On the other hand, the positive orientation showed the peak pressure at 14 cm, which was greater than the pressure near the water surface. This also resulted in push-up of IEO moving up the positively oriented plate. Contact between the IEO and plate resulted in a change in the strain on the plate. Figure 8.52 compares the strain at the corner location with and without IEO for three different plate orientations. Both vertically and negatively oriented plates had an increase in the strain resulting from IEO while the positively oriented plate showed a minor decrease. However, the strain at the center did not show any noticeable change. The next study used two large pieces of LDPE in water. Table 8.6 shows the change in the peak drag force and strains at the towing speed of 1.17 m/s for three different plate angles. The plate of the negative angle showed the largest increase in the drag force. However, the reason for such a large increase for the negatively oriented plate was that the drag force was much smaller without IEO than other plates. However, the effect on strains was very different without any clear trend. It was because the deformed shape was very dependent on the interaction of the IEO and the plate. Therefore, the strain was very dependent on the location of the plate. Some locations would have greater strains and others would have lower strains resulting from the deformed shape.

8.3.3 Results of Accelerating Motion The tests were conducted repeatedly, at least several times to check the consistency of the test results. The motion of the plate was recorded using a high-speed camera for the accelerating phase. Then, the velocity and acceleration were determined from the

8.3 Experimental Study in Fluid with Floating Solid Objects Fig. 8.49 Plate of vertical orientation passing over smaller LDPE pieces: time progression from top to bottom figures

273

274 Fig. 8.50 Plate of negative orientation passing over smaller LDPE pieces: time progression from top to bottom figures

8 Composite Structures Moving in Fluids

8.3 Experimental Study in Fluid with Floating Solid Objects Fig. 8.51 Plate of positive orientation passing over smaller LDPE pieces: time progression from top to bottom figures

275

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Table 8.5 Peak fluid pressures at different depths preceding the plates

Depth from free surface Plate of 30° (Pa) Plate of −30° (Pa) (cm) 0

519

455

7

519

408

14

560

279

21

164

109

Fig. 8.52 Strain at the horizontal corner gage at the speed 1.5 m/s

Table 8.6 Percent change for peak force and peak strain values at 1.17 m/s due to impact with large IEO Plate orientation

Vertical strain at quarter location (%)

Horizontal strain at quarter location (%)

Vertical strain at center (%)

7

−25

140

−12

−30°

43

−42

86

−17

30°

15

16

100

−9

0°

Force (%)

measured displacement. The plate moved almost quadratically as a function of time before it moved at a constant steady state motion. This suggested the acceleration phase had a constant acceleration whose average value was about 3.2 m/s2 until the speed became the steady state value 1.50 m/s. Therefore, the acceleration data were presented and compared to that at the steady state speed 1.50 m/s. Figure 8.53 shows the comparison of drag forces among three plate orientations with and without IEO. The drag forces in the acceleration phase were qualitatively similar to those in the steady state phase. That is, the negatively oriented plate still

8.3 Experimental Study in Fluid with Floating Solid Objects

277

Fig. 8.53 Drag force during accelerating motion for different plate orientations in water

showed the smallest force, and the positively orientated plate yielded the largest drag force. The acceleration phase resulted in a larger drag force than the steady state phase. For example, the vertically oriented plate showed a 30% increase, and the negatively oriented plate gave a 50% increase. The positively oriented plate resulted in a 15% increase. Therefore, the largest increase from the steady state to acceleration phase was the negative orientation. Figure 8.54 compares the vertical strains at the center of each plate with different

Fig. 8.54 Vertical strains at the plate center during acceleration phase for different plate orientations

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8 Composite Structures Moving in Fluids

orientations. The figure showed that a larger strain was resulted from a larger drag force, but the strain was not proportional to the magnitude of the drag force. This was because the deformed shapes of three plates were different because of different pressure loading over the plates. Finally, the effect of IEO was examined during the acceleration phase. The same distributions of IEO were used for both the steady state and accelerating motions. The effect of IEO on the drag force is shown in Fig. 8.53. The existence of IEO increased the drag force during the acceleration phase, as expected. In terms of the magnitude of the increase in the force, the vertical plate had the largest increase followed by the positive orientation. The negative orientation had the smallest increase in terms of magnitude. However, the latter gave the largest increase in terms of the percentage because the base force was much smaller than others. When the increase in force was compared between the steady state and accelerating motions, the latter yielded a larger effect of IEC on the drag force than the former. The contact with IEO of the plate also resulted in a change in the deformation of the plate during the acceleration phase. Therefore, the strain at the center of the plate was not directly correlated to the drag force. For the deformation is dependent on the pressure distribution over the plate rather than the resultant force. The negative orientation of the plate showed a small decrease in the vertical strain at the center because of IEO. Figure 8.55 compares the horizontal strain measurement at the quarter location of the plate for three different plate orientations as the plates moved in the steady state or accelerating phases. The figure showed that percentage change in the strain was generally greater for the steady state motion than the accelerating motion. However, this is only for the strain at the given location, and this statement cannot be generalized

Fig. 8.55 Comparison of horizontal strains at the plate quarter location at different plate orientations

8.3 Experimental Study in Fluid with Floating Solid Objects

279

for all other locations of the plates because the deformed shapes influence the local strains.

8.4 Summary This chapter presented the moving composite plates in water under various steady state velocities or accelerating conditions. The drag force was measured along with strains on some selected locations of the plate. The plate orientation relative to the towing direction was also varied. In addition, IEO was added to the tow tank in order to simulate floating icebergs. Both experimental and numerical studies were conducted to complement each other. In general, the plate orientation was very important for the drag force as well as interaction with the floating IEO. The negatively oriented plate yielded the smaller drag forces with and without IEO, in general. The accelerating phase also yielded larger drag forces than steady state conditions.

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