Some Engineering Applications in Random Vibrations and Random Structures [1 ed.] 9781600864308, 9781563472589

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Some Engineering Applications in Random Vibrations and Random Structures

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Some Engineering Applications in Random Vibrations and Random Structures

Giora Maymon RAFAEL Haifa, Israel

Volume 178 PROGRESS IN ASTRONAUTICS AND AERONAUTICS Paul Zarchan, Editor-in-Chief Charles Stark Draper Laboratory, Inc. Cambridge, Massachusetts

Published by the American Institute of Aeronautics and Astronautics, Inc. 1801 Alexander Bell Drive, Reston, Virginia 20191-4344

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Copyright O 1998 by the American Institute of Aeronautics and Astronautics, Inc. Printed in the United States of America. All rights reserved. Reproduction or @anslationof any part of this work beyond that permitted by Sections 107 and 108 of the U.S. Copyright Law without the permission of the copyright owner is unlawful. The code following this statement indicates the copyright owner's consent that copies of articles in this volume may be made for personal or internal use, on condition that the copier pay the per-copy fee ($2.00) plus the per-page fee ($0.50) through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, Massachusetts 01923. This consent does not extend to other kinds of copying, for which permission requests should be addressed to the publisher. Users should employ the following code when reporting copying from the volume to the Coypright Clearance Center:

Data and information appearing in this book are for informational purposes only. AIAA is not responsible for any injury or damage resulting from use or reliance, nor does AIAA warrant that use or reliance will be free from privately owned rights.

ISBN 1-56347-258-9

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Progress in Astronautics and Aeronautics Editor-in-Chief Paul Zarchan Charles Stark Draper Laboratory, Inc.

Editorial Board Richard G. Bradley Lockheed Martin Fort Worth Company

Leroy S. Fletcher Texas A&M University

William Brandon MITRE Corporation

Allen E. Fuhs Camel, California

Clarence B. Cohen Redondo Beach, California

Ira D. Jacobson Embry-Riddle Aeronautical University

Luigi De Luca Politechnico di Milano, Italy

John L. Junkins Texas ABM University

Philip D. Hattis Charles Stark Draper Laboratory, lnc.

Pradip M. Sagdeo University of Michigan

Vigor Yang Pennsylvania State University

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Foreword The uncertainties that occur in the design process, in the models employed for analysis of the loads, and in the geometric parameters of a structure, have been dealt with for generations by experience and safety factors. In recent years, however, the deterministic approach has been enhanced, and in some cases even replaced, by a probabilistic approach in which the various uncertainties are treated more rigorously. These methods of design and analysis require the analyst and designer to have a better understanding of random processes, random vibrations, and random structures. In this book, the author tries to provide this knowledge and brings together a careful introduction to the fundamental theory with many different practical techniques for analysis of random vibrations in structures. Though it is assumed that the reader has a rudimentary knowledge of random vibration theory and probability theory, the basic concepts and definitions are outlined (with the relevant references) and only then employed in the development of practical analysis techniques. The presentation therefore addresses the needs of both experienced design engineers and newcomers to the field. The emphasis is on engineering applications, some of which are presented in considerable detail, including flowcharts for numerical procedures. The matching with commercially available, large finite element programs is also extensively discussed, as well as the capabilities of commercially available special programs for the analysis of probabilistic structures. Failure analyses of dynamic systems and models for crack growth, both deterministic and stochastic, are treated, accentuating the advantages of the stochastic models. The focus is on engineering applications, as stated in the title. Based on his extensive experience as a designer and researcher, Giora Maymon presents the material in a manner appealing to the design-oriented engineer, while preserving the necessary sound mathematical basis. It is hoped that the book will familiarize the engineering community with random vibrations and with stochastic methods and will encourage engineers to study and adopt them widely in design procedures. Josef Singer Professor Emeritus Technion-Israel Institute of Technology

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Table of Contents Preface

..................................

Chapter 1 Deterministic Single-Degree-of-Freedom System I. I1. 111. IV . V.

V.

...........

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential Equations of Continuous System . . . . . . . . . . . . Base Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stress Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4 Random Functions and Excitation I. I1. I11. IV .

.

Differential Equations and Normal Modes of an MDOF System Generalized Masses. Dampings. Rigidities. and Forces ...... Uncoupled Differential Equations . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3 Deterministic Continuous System I. I1. 111. IV . V.

1

SDOF System Subjected to External Harmonic Excitation . . . . ............ SDOF System Subjected to Base Excitation Response of an SDOF System to General Force . . . . . . . . . . Stresses in an SDOF System ..................... Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2 Deterministic Multiple-Degree-of-Freedom System I. I1. 111. IV .

..

...........

Basic Concepts of Random Functions . . . . . . . . . . . . . . . . Practical Characterization of Random Excitation . . . . . . . . . . Important Excitation Functions . . . . . . . . . . . . . . . . . . . . Boundary-Layer Excitation Model . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 5 Response of Linear Systems to Stationary Random Excitations

.................................

I . Response of a Linear SDOF System . . . . . . . . . . . . . . . . . I1. Response of a Linear MDOF System . . . . . . . . . . . . . . .

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I11. Response of a Linear Structure to Stationary Random Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

Chapter 6 Nonlinear Single-Degree-of-Freedom and Multiple-Degree-of-Freedom Systems

................

I. I1. I11. IV .

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear Behavior of an SDOF Oscillator . . . . . . . . . . . . . Nonlinear Coefficients of a Structure ................ Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 7 Statistical Linearization Method

............

I . Statistical Linearization Method for an MDOF System . . . . . . I1. Nonlinear Response of an SDOF System to Random Gaussian Force .................................. I11. Nonlinear Random Response of Two-DOF System to Random Gaussian Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Nonlinear Random Response of an Elastic System . . . . . . . . V . Computational Procedure . . . . . . . . . . . . . . . . . . . . . . . VI . Calculation of Stress Response . . . . . . . . . . . . . . . . . . . . VII . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 8 Nondeterministic Structures: Basic Concepts I. I1. I11. IV .

Introduction . . . . . . . . . . . . Failure Surface: Basic Case . . . Reliability Index ......... Summary . . . . . . . . . . . . . .

................... ................... ................... ...................

Chapter 9 Calculation of the Probability of Failure I. I1. I11. IV . V.

.....

.......

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lagrange Multiplier Method ..................... Demonstration of the Iterative Process ............... Numerical Programs for Probabilistic Structural Analysis . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 10 Taylor Series Expansion of the Failure Surface I. I1. I11. IV . V.

...

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Taylor Series Expansion . . . . . . . . . . . . . . . . . . . . . . . . Selection of the Evaluation Point . . . . . . . . . . . . . . . . . . . Detailed Examples of the Taylor Series Expansion Method ... Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTS

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Chapter 11 Direct Calculation of the Probability of Failure Using an Existing Finite Element Program

.................

I. I1. 111. IV .

Introduction . . . . . . . . MJPDFMethod . . . . . . Numerical Examples . . . Summary . . . . . . . . . .

.......................

.......................

157 158 163 178

.....

179

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical Behavior of a Stationary Gaussian Process . . . . . . . Spectral Moments of a Determinstic MDOF System . . . . . . . . Probability of Threshold Crossing (Failure) . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179 179 184 185 194

...........

195

Introduction .............................. .................. Stochastic Crack Growth Models ...................... Crack Length Distribution Failure Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probability of Failure Calculation . . . . . . . . . . . . . . . . . . Required Information for the Analysis . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 196 202 209 212 215 217

..................

219

..............

221

....................... .......................

Chapter 12 Probability of Failure of Dynamic Systems I. I1. 111. IV . V.

Chapter 13 Stochastic Crack Growth Models I. I1. 111. IV . V. VI . VII .

157

Chapter 14 Concluding Remarks Appendix A:

Some Important Integrals

. . . . . . . . . . . . . . . . . . . .

I . Constant Value PSD Functions I1. Filtered One-sided White Noise

Appendix B:

....................

221 222

Conversion Between Acoustic Decibels and PSD

Appendix C: Finite Element Input Files for MJDPF Method 1. I1. I11. IV . V. VI .

File File File File File File

for for for for for for

Example 1 Example 2 Example 3 Example 4 Example 5 Example 6

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

............... ............... ............... ............... ............... . . . . . . . . . . . . . . .

................................ ...................................

References Index

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Preface

The purpose of this book is to present design engineers with some of the basic tools for the solution of practical problems they may encounter when analyzing random vibration of deterministic and random structures. The content of this book is presented so that it enables them practical use of the procedures and expressions required for daily engineering work. Having more than 35 years of experience in structural dynamics research and development, the author believes that a good understanding of the basic concepts and the physical meaning of the behavior of a structure is crucial to a successful work of engineers, especially in the beginning of their careers, and the material presented in this book is directed toward this goal. Nevertheless, in many practical cases, numerical solutions are the only way to solve some of the practical cases, so flowcharts for such numerical procedures are presented for many of the discussed cases. Using these flowcharts, each user can take advantage of a favorite mathematical solver, i.e., MATLAB,'" TK Solver, and others, or write a Fortran program to build a numerical procedure suitable for the design needs. The design of any engineering system is a process of decision making, under constraints of uncertainty. The uncertainty in the design process result from the lack of deterministic knowledge of different physical parameters and the uncertainty in the models with which the design is performed. This is true of many of the disciplines involved in any design such as electronics, mechanics, aerodynamics, and structures. The main sources of uncertainties in structural analysis and design are 1) uncertainties in the determination of the physical and mathematical model used for the analysis, including uncertainties in the failure criteria (model); 2) uncertainties in the determination of the magnitudes, locations, frequency content and correlations of the external (static and dynamic) loads (load); and 3) uncertainties in various structural parameters such as geometries, dimensions, material properties, and allowables (stochastic structure). These three categories do not include other more subjective uncertainties such as human errors in the design and production. Uncertainties in the model were treated by the scientific and engineering communities by exploring in three major directions: 1) The first uses analytical models for linear and nonlinear behavior of various structural elements i.e., beams, plates, and shells, with various boundary conditions, loaded statically and dynamically. Thousands of papers, books, and reports have been published on these subjects.

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G.MAYMON

2) The second uses analytical and experimental models for various failure criteria, allowable stress envelopes, accumulated damage effects, and crack propagation processes. The introduction of fiber-reinforced composite materials enhanced these research efforts, because of the more complex failure process of these materials. 3) The third uses algorithms developed for numerical computations of structures. In the last 3Gyears tremendous research and development efforts were invested in developing large finite element programs, which reduce the need to apply approximate analytical solutions. Today, programs like NASTRAN, ANSYS, and others are basic tools in the industry. This direction is not explored in this book, although results that are required for the understanding of the content are introduced. To obtain in-depth information on this subject, the interested reader should explore the existing literature. Uncertainties in the loads has been treated by academic institutions and industry during the last 20-30 years. Parameters of spectral density of external excitations (such as acoustic noise, wind loadings, ocean waves, and earthquake loads) were defined, using accumulated realistic data. Statistical methods for analysis of structures under random excitations were introduced into most of the qualification standards and into the design codes. These methods were also introduced into the commercial finite element programs, which are now used extensively in the industry. Chapters 1-7 deal with the resDonse of deterministic structures to random excitation. Subjects covered include single-degree-of-freedom systems, multiple-degrees-of-freedom systems, continuous structures, and the treatment of nonlinear structures. The response of an elastic structure to external dynamic loads plays an important role in the total analysis of a designed structure. The loads acting on an aerospace structure are both static and dynamic. Whereas the static (or very slowly varying) loads create static deflections and stresses, the structural response to dynamic loads are vibrations-time-varying displacements and stresses. In most practical cases the loads are not deterministic and have some statistical dispersion. A complete analysis of a practical structure should involve bothstatic and dynamic behavior. In many cases, as a result of the life history of the structure, both static and dynamic loads act simultaneously, and this should be taken into account during the design process. Resonance of the structure is sometimes responsible for high values of response, even for small dynamic loads. Aerospace structures are excited by aerodynamic loads, which are usually random in nature. Flow around a structure creates pressure fluctuations, which have a wide range of frequency and amplitude content. The same phenomena are caused by acoustic noise created by rocket and jet outlet flows. Rotating elements such as engines and rotors create excitations with a better defined frequency content that are in many cases random in amplitudes. The dynamic response of a structure to any loads can be analyzed by many ways such as energy methods, differential equations methods, and integral equations solutions. The most common method for solving the

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PREFACE

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response of practical structures to external excitations is the normal mode approach, which is based on a superposition of normal modes that are assumed to be characteristic of the structure. By its nature, the normal modes superposition can be applied to linear structures, although nonlinear cases can be treated by equivalent linear systems, as shown later in this book. The normal mode method is effective and therefore popular, because the normal modes and natural frequencies of the structure can be calculated without any reference to the external excitation (thus, they characterize the structure) and each mode can be treated separately as a single-degreeof-freedom oscillator. The computation methods for the determination of the normal modes and natural frequencies are well established, and all the commercially available finite element computer codes include routines for their computation for the practical cases where an analytic solution does not exist. Uncertainties in the structural parameters, or stochastic structures, have been treated more extensively only during the last 10-20 years and are described better in the introduction to Chapter 8. Chapters 8-13 deal with various aspects of stochastic structures. Problems of the response of a deterministic structure to random excitations can now be solved by most of the commercially available finite element codes. It could be asked: Why should I bother with mathematics, equations, and approximate solutions when data can be keyed into the computer and the program can provide the required answers? It is the author's opinion, supported by more than 35 years of practical experience, that the computer must be treated as a tool to speed up the task and not as a replacement for the human brain. Knowledge of the basic assumptions, the capability to quickly solve simplified models during the preliminary design stage (and compare them to computer answers, like a process of calibration), the ability to analyze results and draw smart conclusions, and the confidence to decide on necessary changes to the design depend on the physical understanding of the phenomena involved. All these are the benefits of engineers who do not let the computer think for them. The understanding of the basic concepts is crucial to a correct method of working. Therefore, in Chapter 1 the basic behavior of a single-degreeof-freedom system is described. Basic behavior of a multiple-degrees-offreedom system is described in Chapter 2, and continuous structures are detailed in Chapter 3. In Chapter 4, the reader is introduced to some guidelines of random functions and the representation of random excitations in specifications. Also included is a model for the excitation generated by the boundary-layer flow over flat and lightly curved surfaces. The response of a single-degree-of-freedom system, a multiple-degrees-of-freedom system, and a structure to random excitations is presented in Chapter 5. In Chapter 6, behavior of a geometric nonlinear oscillator is described, and in Chapter 7 the statistical linearization method is applied to demonstrate the response of nonlinear single-degree-of-freedom systems, multiple-degrees-of-freedom systems, and continuous structures to stationary random excitations. It is shown that use of nonlinear analysis may often result in a less conservative structure.

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In Chapter 8, the basic concepts of the analysis of nondeterministic structures are outlined and demonstrated. A possible analytical method for the solution of static stochastic structures is described in Chapter 9. Chapter 10 describes a Taylor series expansion method, which can be used for the solution of more complex cases. Chapter 11 describes the modified joint probability density function method, which enables the use of an existing, commercially available finite element computer code to solve stochastic structures problems. In Chapter 12 some aspects of dynamic behavior of stochastic structures are described, and in Chapter 13, some aspects of a stochastic crack propagation model are shown. The assumptions and the most important practical conclusions are summarized at the end of each chapter. Mathematical proofs are not always included as these are often less important to the practical user who, in most cases, is interested in the final practical result. Nevertheless, the more interested reader is encouraged to explore the cited references. It is believed that design engineers who read this book and understand the included examples will be able to extend the knowledge gained by this reading to practical problems they encounter during daily engineering work. Greater understanding of the physical meaning of the solutions will undoubtedly result in a much better designer. Giora Maymon October 1997

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Chapter 1

Deterministic Single-Degree-of-Freedom System I. SDOF System Subjected to External Harmonic Excitation

A

GOOD physical understanding of the behavior of vibrating structures can be achieved by analyzing the behavior of an single-degree-offreedom (SDOF) oscillator. The SDOF system is covered by an extremely large number of textbooks, e.g., Refs. 1-6, and is the basis for every course in vibration analysis. It will be discussed briefly in the first chapters to create a baseline for the analysis of random vibration. The classical oscillator contains a point mass rn, i.e., all of the mass is concentrated in a single point, which is connected to a rigid support through two elements: a linear massless spring with stiffness k and a viscous damper c (which creates a force proportional to the velocity) or a structural damper h (which creates a force proportional to the displacement and in a 90-deg phase lag behind it). The system can be excited either by a force f acting on the mass or by a base movement x,. This type of system is described in Fig. 1.1. Note that in Fig. 1.1 two elements connect the mass to the support. This is only a schematic representation. The spring element represents the stiffness of the structure, and the internal (structural) damping is represented by the viscous damping. In the following evaluation, the classical viscous damping is assumed (although treatment of structural damping is similar and can be found in many references). The basic equation of motion is

The natural frequency of the undamped system is

The system is excited by a harmonic force of amplitude fo and frequency R

Assume a solution in the form

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Fig. 1.1 SDOF system subjected to excitation force f ( t ) .

which yields the following result for xo:

where

This expression is complex, i.e., there is a phase difference between the excitation force and the displacement. Using classical solution methods, one can obtain the absolute value of xo,

where

where 5 is the damping coefficient given by

and is used extensively in engineering applications. The phase angle between the displacement and the force is

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DETERMINISTIC SDOF SYSTEM

3

Note that in Eq. (1.6), folmwi = folk is the static deflection xStaticof the SDOF system under a static force of magnitude& and, therefore, a dynamic load factor (DLF) can be defined by

The DLF expresses the dynamic amplification of a static deflection. A plot of Eq. (1.9) is shown in Fig. 1.2. These are the well-known resonance curves. Examination of Eq. (1.8) shows that, at very low-excitation frequencies (SL/oo+ 0), x is in phase with the excitation. At resonance (fl/oo = 1) the displacement is 90 deg ahead of the excitation, and in very high-excitation frequencies (SL/oo 4 a ) there is a 180-deg lag between the displacement and the force, or antiphase. In Fig. 1.3 the vectors f,x, f, and Y are shown for R/oo -+ 0, SL/wo = 1, and fl/oo -+ a . Note that, in resonance, the velocity vector is in phase with the excitation force. This result is used in many experimental methods for the determination of resonance frequencies and modes shapes, e.g., Refs. 7 and 8.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Excitation to Resonance Frequencies Ratio Fig. 1.2 Dynamic load factor.

1.6

1.8

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Fig. 1.3 Phase between force vector and displacement, velocity, and acceleration.

11. SDOF System Subjected to Base Excitation Many engineering applications involve base excitation rather than force excitation. Such are the cases of structures subjected to earthquakes, vehicles moving on rough roads, and structural subsystems mounted on a main structure. It is convenient to express the excitation by an input acceleration xs(t) as shown in Fig. 1.4. It can be shown that, when u = x(t) - xs(t), the relative displacement between the mass and the support obeys the following differential equation: mii

+ czi + ku = -mfs(t)

(1.10)

In many cases, it is the relative displacement between the mass and the support, and not the absolute displacement of the mass, that is responsible for the stresses in a spring (or in a structure). Equation (1.10) describes a system that is equivalent to that described in Eq. (1.1), with an equivalent force equal to the mass multiplied by the base acceleration, in a direction opposite to the base excitation. Thus, when xs = xsoei"', the term (-mfSo) can replace fo in Eq. (1.6) whereas u, the relative displacement, replaces

Fig. 1.4 SDOF system subjected to base excitation.

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DETERMINISTIC SDOF SYSTEM

x; thus,

whereas the expression for the phase angle remains unchanged. When the support is moved by a harmonic displacement (rather than by acceleration) of amplitude xsoand frequency

a

the acceleration is obtained by double differentiation with respect to the time t

and, therefore,

Equation (1.13) is described in Fig. 1.5.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Excitation to Resonance Frequencies Ratio

Fig. 1.5 Amplification of base displacement.

1.6

1.8

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Fig. 1.6 Phase between excitation and relative displacement, velocity, and acceleration.

The equivalent excitation force has a negative sign and, therefore, the vector plots of x,, is, $, u, ir, and ii are as described in Fig. 1.6. For very low base excitation frequencies u + 0 (no relative displacement occurs between the moving base and the mass and they move together). For very high base excitation frequencies u tends to xso,resulting in a relative displacement of -xSo,i.e., the mass does not move relative to the external world and all of the relative movement between the base and the mass is due to the base movement. 111. Response of an SDOF System to General Force The most general method to find the response of an SDOF system to a general force input is to use the response h(t) of such a system to a unit impulse S(t). A unit impulse is defined as an infinitely large force acting during an infinitely small period, so that the total impulse is one unit. The solution for h(t) input is described in most of the available textbooks, and the result for an SDOF system initially at rest, with damping coefficient < 1 (which is usually the case in structural analysis) is

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DETERMINISTIC SDOF SYSTEM

7

The response to a general force f (t) can be represented by a series of repeated impulses of magnitude ~ ( T ) A Tapplied , at t = T. In vibration textbooks it is shown that the displacement of the SDOF system can be written by either

Equations (1.15) are known as the convolution or Duhamel's integral. As h(t - T) is identically zero for t < T (the time instants preceding the excitation of the system), Eqs. (1.15) can also be written as

where, for a system initially at rest, h ( r ) is given by Eq. (1.14).

Example 1.1 The use of Duhamel's integral (1.15b) is demonstrated for an SDOF system with resonance frequency ooand a damping ratio l,subjected to a step load, e.g., a load fo that is applied at t = 0 and remains constant. For this case

h(r)

=

1

mw,

rne-@oTsin (

w o r n T)

Substituting these expressions into Eq. (1.15b) yields

Evaluation of this integral yields the following expression for the DLF:

This expression is described in Fig. 1.7.

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8

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (sec) Fig. 1.7 DLF for the example.

The integrals of Eqs. (1.15) and (1.16) cannot always be evaluated explicitly but numerical integration can be performed to determine the response of an SDOF system to a general force. In Fig. 1.8 a flow chart that enables simple programming of a procedure for this type of integration is shown. When an analysis of random vibration of structures is performed, it is common to d o many of the calculations in the frequency domain and not in the time domain. Definition of external random inputs is usually given in the former. Therefore, it is useful to demonstrate the evaluation of the behavior of a SDOF system in the frequency domain and to demonstrate the transformations between it and the time domain. Equation (1.1) can be written as

where L2(dldt) is a second-order differential operator

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10

When f (t) is an harmonic excitation f(t) =foei"' A solution in the following form is sought:

x(t)

= X(R)eint

Substitution of Eqs. (1.18) and (1.19) into Eq. (1.17) yields

as dldt = iR. Denoting

one obtains

where H(R) (called either the complex frequency response or the receptance) is given by

which is identical to the definition given in Eq. (1.5). The relation between the impulse response h(t) and complex frequency response H(R) can be determined by the following procedure. The general force f (t) is represented by a Fourier integral f(t)

= !+-

-m

F (R) ei"' dR

(1.23)

where

F(R) is a function equivalent to the Fourier coefficients in a Fourier series.

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DETERMINISTIC SDOF SYSTEM

11

The response x ( t ) can also be represented as a Fourier integral

Substitution of Eqs. (1.23) and (1.24) into Eq. (1.17) yields

and x ( t ) is, according to Eq. (1.24),

When f ( t ) is a result of a unit impulse applied at t

=

0

From Eq. (1.26) one obtains

Thus, the impulse response h ( t ) multiplied by 2n is a Fourier transform of H ( R ) . In some textbooks, Eq. (1.23) is defined slightly differently, so that the complex transfer function is the Fourier transform of h ( t ) and not of 2nh(t).Nevertheless, Eq. (1.28)are more convenient for use in the response of a system to random excitations.

IV. Stresses in an SDOF System Formally, the term "stresses" is not appropriate for an SDOF system. Nevertheless, it is clear from the behavior of an SDOF system that an internal force exists in the spring, proportional to the change in its length. The spring is the single elastic element in the SDOF system, as a point mass is assumed. This force creates stresses in the massless spring. It should be borne in mind that changes of dimensions of elastic elements, created by internal forces in the loaded system, are the reason for the existence of stresses in a system. This subject is evaluated when vibration of continuous systems such as structural elements is discussed.

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V. Summary In this chapter, the behavior of an SDOF system was described. The assumptions that are made are as follows. 1) The SDOF system is linear. 2) The mass is concentrated at one point. 3) The spring is massless. 4) A viscous damping is demonstrated, although the treatment of structural damping is similar and can be found in many textbooks, e.g., Ref. 3. It was shown that an SDOF system subjected to an harmonic force excitation vibrates in amplitudes that are different than those resulting from the static deflection of such a system. The largest amplifications are obtained when the excitation frequency is in the vicinity of the resonance frequency. Where the excitation frequency is much higher than the resonance frequency, attenuation is obtained. The response has a phase angle with the excitation force, and its magnitude depends on the ratio between the excitation and resonance frequencies. When base excitation is applied, an equivalent excitation force can be calculated, and the problem can be solved using the force response expressions. Duhamel's integral enables the computation of the response of an SDOF system to a general excitation force. Relations between the complex frequency response and the impulse response function are also presented.

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Chapter 2

Deterministic Multiple-Degree-of-FreedomSystem I. Differential Equations and Normal Modes of an MDOF System

A

MULTIPLE-DEGREE-OF-FREEDOM (MDOF) system contains several masses, interconnected by springs and dampers and excited by several external forces and/or base excitations. The degree of the MDOF is determined by the number of masses n. Each mass mi (i = 1 , 2 , . . . , n) moves with a displacement xi. The following differential equation can be written in a matrix form:

where [m] is an (n X n) mass matrix that contains a combination of the discrete masses of the system, [c] is an (n X n) damping matrix, [k] is an (n X n) stiffness matrix, {f(t)} is an (n X 1) exciting forces vector, and {x(t)) is an (n X 1) vector of the displacements of the discrete masses. A system of two masses, three springs, and three viscous dampers excited by two external forces is shown in Fig. 2.1. The two masses ml and m2 have two DOFs xl (t) and x2(t) respectively. The springs, dampers, and excitation forces are also marked in Fig. 2.1. When ml is moved by xl and m2 is moved by x2, the forces acting on these masses are shown in Fig. 2.2. Two equilibrium equations can be written

which can be rewritten as

Equation (2.2b) can be compared to Eq. (2.1).

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I ,

I ,

XI (t)

x? (t)

Fig. 2.1 Two-DOF system.

Consider the free undamped vibration of the system

[mI@(t)}+ [ k l { x ( t ) }= 0

(2.3)

{ x(t ) } = { x } eiot

(2.4)

and assume a solution

Substitution of Eq. (2.4) into Eq. (2.3) yields

( [ k ]- w2[m]){x} eio' = {0}

(2.5)

The nontrivial solution of Eq. (2.5) exists only when the determinant

from which a polynomial equation of the order n for n values of w2 :w:, . . . ,w:, . . . , is obtained. Substituting any of these frequencies back into Eq. (2.5) yields a corresponding set of relative values for {x}.

Fig. 2.2 Forces acting on the masses.

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DETERMINISTIC MDOF SYSTEM

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This is called the ith mode shape, and it is a column vector {+i}. The n mode shapes are described by a matrix [4] in which every column corresponds to the ith mode shape of the frequency mi.The matrix [4]is in the form

where the first index is the mode number and the second index indicates the system coordinate. For example, 42,3 denotes the nodal displacement of the second mode in the third DOF (coordinate) of the structure. Assume that in the example described in Fig. 2.1, ml = 2 kg, m2 = 1 kg, kl = 0.5 X lo6 Nlm, k2 = 1 X lo6 Nlm, and k3 = 0.5 X lo6 Nlm. Using Eqs. (2.2b) and (2.6) one obtains

from which of the following equation is obtained for

02:

whose solutions are of= 0.32461 X 106 ( r a d l ~ )and ~ o$ = 1.92539 X lo6 ( r a d l ~ )Using ~. (Eq. 2.5) with x2 = 1 the following mode shapes matrix is obtained:

When more than two DOFs exist, one of the many commercially available eigenvalues and eigenvectors programs can be used to calculate the frequencies and the mode shapes. 11. Generalized Masses, Dampings, Rigidities, and Forces

The resonance frequencies and the mode shapes are characteristic of the system and not of the loading. They depend only on the masses and rigidities of the system and, therefore, are attractive for use in structural dynamics analyses. The mode shapes (sometimes called normal modes) possess an important property known as orthogonality. This means that

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G.MAYMON

where the superscript Trepresents a transposed matrix and [MI is a diagonal matrix with elements known as the generalized masses. In a similar way

and as a result of Eq. (2.5) [ K ] = [o?M]

(2.9b)

where the elements of the diagonal matrix [ K ] are called generalized stiffnesses. For a system with proportional damping (damping that is pr~portional to the mass and/or the stiffness)

where the elements of the diagonal matrix [C] are called the generalized dampings. The assumption of proportional damping that yields a diagonal generalized damping matrix is not necessarily exact. Nevertheless, this assumption greatly simplifies the calculations of structural response and, therefore, is used extensively in practical engineering applications. When nodal damping coefficients & are known (from experiments or from accumulated practical knowledge) it can be shown that

It should be noted that the generalized quantities M , K , and C are not unique. They depend on the values of [+I, which is a relative set of displacements. Selection of [+] determines the generalized quantities. In some cases, a normal mode is selected so that its maximum value is 1.This implies certain generalized quantities. In other cases (usually in the large finite element programs), [+I is selected so that all Mi = 1. It is not important which normalization is made, as long as the process is consistent throughout the whole solution. For the example solved earlier

is equivalent to another modal matrix

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DETERMINISTIC MDOF SYSTEM

17

where both terms of the first column were divided by 1.17539. For the first modal matrix,

and for the second

A very useful transformation, which is used extensively in all structural dynamics analyses, is

i.e., the displacement vector {x(t)}is expressed in terms of the normal modes [4] and a generalized coordinates vector { ~ ( t ) )As . this transformation is used extensively in future chapters, it will be demonstrated in more detail for the two-DOF system. The modal matrix [4] of this system is

Equation (2.12), therefore, can be written as

or explicitly

In Eq. (2.14b) the displacements x, are expressed as a linear combination of the generalized coordinates qi, weighted by the modal shapes. Substituting Eq. (2.14b) into Eq. (2.1) and premultiplying each term with [$] yields

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The quantity on the right-hand side of Eq. (2.15b) is called the generalized forces matrix [ F]. For the preceding example

Therefore, the generalized forces represent the work done by the external forces when the masses of the system move a modal displacement. When the normal modes are known, the generalized forces { F ) can be easily calculated for a given set of external forces {f}. 111. Uncoupled Differential Equations

Writing Eq. (2.15b) explicitly yields

Mniin + Cniln + K n ~ n= Fn Using Eqs. (2.9b) and (2.11), Eq. (2.17) can be written as

M1iil-t 2llWMli71 + 4 M l r l l

= Fl

Equations (2.18) represent a set of uncoupled differential equations for the generalized coordinates vi. Each of these equations can be solved separately for qi, using all of the techniques and procedures of an SDOF system, which are well known and documented. Then the displacements { x ) can be calculated using Eq. (2.12). This is an important feature of the normal modes. It allows the engineer to solve an MDOF system using solutions of an SDOF one, once the normal modes, resonance frequencies, generalized masses, and forces are calculated, and the modal damping

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coefficients are known or assumed. It is shown in the next chapter that this approach can also be used for continuous structures. IV. Summary Using equilibrium equations, the differential equations of an MDOF system can be written. Resonance frequencies and mode shapes can then be calculated. Using the orthogonality of the normal modes, generalized masses and other generalized quantities can be calculated using Eqs. (2.8), (2.9b), and (2.11). When the displacements are expressed as a linear combination of the normal modes and general coordinates, the transformation yields a set of uncoupled linear differential equations, and each of these equations can be solved for the generalized coordinate 7,using expressions and procedures evaluated for an SDOF system. When an MDOF system is subjected to base excitations, the equivalent excitation force can be calculated, using the expressions given in the next chapter. Once these forces are known, the procedures of an SDOF system can be used for the generalized coordinates, and the system displacements can then be expressed using Eq. (2.12).

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Chapter 3

Deterministic Continuous System I. Introduction

I

N A continuous system, there are an infinitely large number of infinitely small masses. The stiffness is contributed bv internal forces interacting" between these infinitely small masses, and the damping is created by internal dissipation of energy throughout the system. Therefore, this type of a system has an infinite number of DOFs. A practical structure built of interconnected beams, plates, and shells is atypical continuous system. Usually there are no discrete masses, although in some engineering applications the distributed mass is lumped into discrete masses. It has been shown in many textbooks, e.g., Refs. 1-6, that a continuous system has an infinite number of resonance frequencies and normal modes. For many simple structures such as beams with various boundary conditions, rectangular and circular plates, and simple shells, analytical expressions can be evaluated for these frequencies and mode shapes. On the other hand, it is seldom possible to find closed-form expressions for these properties for a practical structure in which different kinds of structural elements and realistic boundary conditions exist. For these reasons resonance frequencies and normal modes of a practical engineering structures are calculated numerically. The large commercial finite element computer codes (such as NASTRAN and ANSYS) are capable of computing these structural properties. As the basic concept of finite elements is the discretization of a continuous system into a large number of interconnected elements, these computer codes actually treat an MDOF system. Instead of an infinite number of resonance frequencies and normal modes, a finite (though very large) number of resonances are computed. Experience has shown that a practical continuous system can be treated as a large MDOF system with sufficient engineering accuracy. Suppose that the mode shapes {4i} (i = 1 , 2 , . . . , m) and the resonance frequencies oi(i = 1,2, . . . , m) were calculated. Then, the deflection w(x, t ) of the structure can be exvressed in terms of the normal modes and the generalized coordinates qi(t),

It should be noted that x in Eq. (3.la) does not necessarily represent a one-dimensional structure, but a location in the structure as follows: one-

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G.MAYMON

dimensional coordinate x for a one-dimensional structure (such as a beam or a frame), two-dimensional coordinates x, y or r, 8 for a two-dimensional structure (such as flat and curved plates and shells of revolution), and threedimensional coordinate x, y, z or r, 8, 9 for a three-dimensional structure (such as solids). If the modes were solved numerically for N DOFs, Eq. (3.la) takes the form

11. Differential Equations of a Continuous System The evaluation of the differential equations of motion of a continuous system are done using the Lagrange equations. For a system in which the damping forces are derived from a dissipation function D, the rth Lagrange equation is given by

where T is the kinetic energy of the system, U is the potential energy, D is the dissipation function, and N is the work done by the external force in the generalized coordinate. For clarity, the development of the equations of motion is done for a uniform beam, without any loss of generality. For such a beam, m is the mass per unit length, EZ is the flexural stiffness, c is the damping per unit length, and w(x, t ) is the displacement normal to the structure along the x axis. The kinetic and potential energies are

and the dissipation function is

where proportional damping is assumed. From Eq. (3.lb)

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DETERMINISTIC CONTINUOUS SYSTEM

From Eq. (3.2)

Changing the order of integration and summation yields

Being normal modes,

4iand +, are orthogonal; thus,

J rn+,(x)+i(x) dx = 0

for i # r

J rn+,(x)+i(x) d x

for i

= Mr

=r

(3.10)

where Mr is the rth generalized mass. Equation (3.8) takes the form

If the notation described in Chapter 1 is applied for the damping, using a modal damping coefficient, then

c = 2[,wrm

(3.12)

Substituting Eq. (3.12) into Eq. (3.5) yields

D

= 25,w,

'/z $ mw2 dx

= 25,w,T

(3.13)

and, therefore,

Using a similar procedure for the stiffness yields

To find an expression for the work done by the external forces, assume that a distributed force per unit length p(x, t ) is applied to the beam. The virtual work done by this force in a virtual displacement Sw(x, t ) is

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G.MAYMON

From Eq. (3.1a)

and, therefore,

This should be equal to the work done by N, on a virtual displacement Sqi

Comparing Eqs. (3.18) and (3.19) yields

For n discrete forces fn(t) rather than a distributed force, acting at points x = x n ,it is possible to show by the same procedure that

Thus, Ni (or Nr) is a generalized force, defined exactly as for the MDOF system (Chapter 2) is the work done by the external forces when a modal displacement +i is applied to the system. Substituting Eqs. (3.11), (3.14), and (3.15) into Eq. (3.2) yields a set of N uncoupled differential equations for the generalized coordinates

When an infinite number of DOFs is assumed, the set (3.21) contains an infinite number of equations. Equations (3.21) are identical to Eq. (2.18). Therefore, the procedure for computation of a deterministic response of a continuous system is as follows: 1) Decide how many DOFs are to replace the infinite number of a given structure. This decision must be based on the nature of the structure and should take into account the engineer's experience in designing similar structures. 2) Calculate the resonance frequencies and mode shapes. This can be done analytically for simple structural elements or numerically using a finite element computer program.

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For one-dimensional structures, i.e., beams, with mass per unit length m(x) and load of force per unit length q(x)

where the integral is taken along the whole length of the structure. For two-dimensional structures, i.e., plates, with mass per unit area m(x, y) and load of force per unit area p(x, y)

where the integral is taken over the whole surface of the structure. For three-dimensional structures, i.e., solids, with mass per unit volume p(x) and load of force per unit volume s(x)

where the integral is taken over the whole volume of the structure. 111. Base Excitation Many practical engineering problems are characterized by an input that is a base excitation rather than an external force. Examples of such cases are a vehicle traveling on a rough road, a structure excited by ground movement (earthquake), or a substructure mounted on a vibrating main structure. In Chapter 1 it was shown for an SDOF system that the equation of motion contains an equivalent excitation force, which is related to the base motion. The formulation of these equivalent forces for a continuous structure is best demonstrated on a one-dimensional structure, with two spatial coordinates, such as a frame. Assume that the frame shown in Fig. 3.2 vibrates in a mode also described in the figure. The spatial coordinate s runs along the frame. The rth normal mode 4,(s) runs along the structure. This mode has a component 4,,(s) in the x direction and +,,(s) in the y direction. The displacement of each point on the structure is then a combination of the elastic displacement Z(S,t )

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DETERMINISTIC CONTINUOUS SYSTEM

Fig. 3.2

Frame subjected to base excitation.

and the rigid body displacements wo, and w ~ ,thus, ~;

The kinetic energy of the system is

where m ( s ) is the mass per unit length of the frame and m(s) ds is the total mass of a small element of length ds. From Eq. (3.26)

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G.MAYMON

28

and, therefore,

ailr Because of the orthogonality of the normal modes and the relation +f( s ) = + f T + +f,,,the following expression is obtained, after some algebra:

and, therefore,

This expression can be identified as part of the Lagrange equations of motion [see Eq. (3.2)] originating from the kinetic energy of the system. As the potential and damping energies depend entirely on the elastic deformation of the structure, the only addition to the Lagrange equations of motion are the first and second terms on the right-hand side of Eq. (3.31). Thus, the equation of motion of the rth mode is

The equivalent generalized force is the base acceleration multiplied by the mass weighted by the normal modes in a direction opposite to the base excitation. When, for example, base excitation in the y direction exists, the integral J m ( ~ ) + ~ , ' (ds s ) can be calculated and the equivalent excitation d s()s. ) generalized force is (-wo,, $ m ( ~ ) + ~ , ~ The external base movement can be given as displacement excitation, i.e., the case of a vehicle traveling on a rough road, or as acceleration, i.e., earthquake. If harmonic displacement is imposed, say in they direction, then

(3.33a)

w o , = A, sin flt and the equivalent generalized force is a function of

w o , = -Ayf12 sin a t

f12,

because

(3.33b)

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On the other hand, if a harmonic acceleration is imposed, then w ~= ,ay~sin flt

(3.34)

and the right-hand side of Eq. (3.32) does not include f12. Once again it is demonstrated that a vibration problem of a practical structure of an infinitely large number of DOFs can be treated as a collection of SDOF systems.

IV. Stress Response Stresses in elastic structures are uniquely defined by the deflection, using the material constitutive relations and the compatibility equations. The means by which the deflection was obtained, statically or dynamically, is irrelevant. The effect of the dynamic load factor or the amplification factor, demonstrated in Chapter 1 for an SDOF system, is introduced into the system when the deflections are calculated. For a specific type of structure, the following relation is always present:

where Liis a differential operator in the spatial coordinate, K is a constant that depends on the elastic constants and the geometry of the structure, and i is an index that indicates which stress is being calculated. For instance, it can be arbitrarily defined that ui= axis axial stress in the x direction, ai= uy is bending stress in the y direction, ui= r,, is shear stress in the structure, ui= uxy is shear stress between two plies in a laminated structure, and ai= us,is stress at the edge of a hole in a plate. Two examples for K and Liin Eq. (3.35) are as follows. 1) The bending stress in the tensed side of a deflected beam is

2) The bending stress in the x direction on the tensed side of an isotropic plate is

Because it was defined in Eq. (3.la) that

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then

and according to Eq. (3.35)

which can be rewritten as

where

Equation (3.39) resembles Eq. (3.36). Therefore, the function ?P\Ir,(')(x)was designated by the author in Ref. 9 as stress mode. By the use of the stress modes, an evaluation of the stress response of a structure can be repeated in the same way it was evaluated for the displacement response. Because the normal modes $Ax) are characteristics of the structure, it can be seen from Eq. (3.40) that the stress modes are also characteristic to the structure and not to the loading condition. Once stress modes for a given structure are computed, a complete mapping of stress distribution in the randomly vibrating structure can be performed. To obtain the stress modes, one naturally tends to use Eq. (3.40), i.e., to apply the operator L on the mode shapes. There are several disadvantages in doing so. 1) The operator L is not always known in a closed-form expression. The examples given earlier were related to very simple cases. 2) The mode shapes of the structure are not always available in a closedform expression. In many cases, mode shapes are approximated by assumed functions that satisfy the boundary conditions but are still only approximations. Whereas the use of assumed modes can yield accurate results for the frequencies and deflections of a structure, their differentiation (usually double differentiation is required to obtain stress) can introduce significantly large errors in the stress results. 3) If mode shapes are obtained by a numerical solution such as a finite element computer code (a procedure that is a routine in practical engineering solutions) numerical differentiation of these modes will introduce large errors in the stress results. For these reasons, a more practical method must be found for the determination of the stress modes. Examination of Eqs. (3.39) and (3.40) shows clearly that the stress mode is the stress distribution in the structure when

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the latter is subjected to a deflection equal to the mode shape, with the generalized coordinate normalized to one. This observation leads to another practical definition of the stress mode: The stress mode is the stress distribution in the structure, when the latter is deformed statically into a deflection equal to the mode shape +,(x). The application of this definition is very simple. Most of the commercially available finite element codes can calculate stresses for an initially imposed static deflection. Therefore, the user introduces an imposed deflection to the structure with a distribution equal to the normal mode, and the required stress mode of kind i (which may vary according to the specific problem) is calculated statically. The use of stress modes calculated statistically brings the designer to a better understanding of the behavior of the structure. Past experience has shown that a design engineer usually has a good intuitive feel for static loads and is able to identify points of weakness in a design by looking at a static analysis. This feel is less reliable where dynamic stresses are concerned. In these cases, the meaning of the normal modes may be well understood, but it is difficult to visualize a physical interpretation of a weighted combination of these modes. It is even more difficult to feel random vibration, mainly because the excitation includes many components, each with a different frequency. It is interesting to note that the structural locations that experience the largest amplitudes are not necessarily those in which maximum stresses exist. The classical example is the cantilever beam, in which the largest amplitude is obtained at the free tip, while the largest stress is located at the clamped edge, where the amplitude is zero. By inspecting the stress modes, a better understanding of the dynamic stress distribution in the structure is obtained. When calculating the stress modes, care must be taken to determine the dimensions of the quantities involved in the process. In Eq. (3.36) the normal mode was assumed to be dimensionless, and the dimension of the deflection w was introduced through the general coordinate 17(t). Examination of Eq. (3.39) reveals that to obtain stresses in the correct dimension, the stress mode must have a dimension of (stressllength). This adjustment is required when using the procedure described for the determination of the stress modes and dimensional deflection must be introduced into the numerical algorithm. Thus, the introduced deflection is not +,(x), but rather Ao+,(x), where A. is a unit displacement of magnitude 1 and dimension of length. Therefore, the result of the numerical calculation is not *\Irji)(x) but Ao!P\i)(x).Practically, the numerical values are the same, but the dimensions are divided by a dimension of length. The preceding argumentation is important when mixed dimensions of length are used, e.g., length in centimeters and meters, or inches and feet. It is a good engineering practice to perform the entire analysis throughout the solution of a given structure with one set of dimensions in length, mass or force, and time. When this is done, no problems arise during the interpretation of the results. Two simple examples of a stress mode and its dimensions are demonstrated.

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Example 3.1 A simply supported beam of height h (cm) and length L (cm) is loaded by a uniformly distributed force. Young's modulus of the material is E (kg/ cm2) and the cross section of the beam has a moment of inertia Z (cm4). The first stress mode of the longitudinal bending stress on the tensed side of the beam is required. The first normal bending mode of the beam is nx ~ $ ~ (= x )sin L and the relation between the bending moment and the deflection is

The relation between the bending moment and the bending stress is

Assume that the beam is bent into the first mode w( x ) where A,

=

= A.

nx sin L

1 cm. Then

*\bend)

=n2Eh sin ?IX [(kg/cm2)/(cm)]

2L2

L

Example 3.2 A simply supported rectangular plate of length a (cm), width b (cm), and thickness h (cm) is loaded by a uniform pressure q. Young's modulus of the material is E(kglcm2) and Poisson's coefficient is v (dimensionless). The first stress mode of bending stress in the x direction is required.

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DETERMINISTIC CONTINUOUS SYSTEM

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The first normal bending mode of a simply supported rectangular plate is

(x, y)

7ix a

?Ty b

= sin - . sin -

The closed-form relation between the bending stress in the x direction and the deflection is

where D = Eh3/12(1 - v2) was substituted. Assume that the plate is bent into a deflection

w = A0 . +1(x,y) where A.

=

1 cm, then

therefore,

9F e n d )

=

[++ $1

r2Eh 2(1 - v2) a

sin

7ix

sin

7iy

[(kg/cm2)/(cm)]

When these calculations are carried out with a finite element code, and not by a closed-form expression, the stress modes are defined by a vector of values of the stress modes calculated at the nodal points of the finite element model. A simple example of this procedure is demonstrated by a numerical analysis of a beam with linearly varied height, shown in Fig. 3.3. The beam is made of a material with E = 2.1 X lo6 kg/cm2 and density of p = 7.959 x kg s2/cm4. The first mode shape is calculated using the ANSYS finite element program. The beam is divided into 10 two-dimensional beam elements and

Fig. 3.3 Beam with linearly varied height.

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calculated with the modal analysis module of the ANSYS. The first frequency obtained is ol = 168.74 radls, and the mode shape for 11 nodal points is

Next, a static case is solved by the ANSYS program, where a deflection of the described mode shape (in centimeters) is forced. The stress mode of the bending stress on the lower outer surface of the beam is

and the dimensions of this stress mode are in kilograms per square centimeters per centimeter.

V. Summary MDOF and continuous systems for which resonance frequencies and mode shapes were calculated can be represented by a set of uncoupled differential equations that include generalized masses and generalized forces. These equations are then solved as a set of SDOF systems. The treatment of base excitation of a continuous system is similar to that of the MDOF system, where an equivalent generalized force replaces the force term in the regular vibration differential equation. The concept of stress modes is presented. Use of these modes enables an easy computation of the stress response of both MDOF and continuous systems. It is again emphasized that good knowledge and experience in solving an SDOF system enables an easy solution of MDOF and continuous systems.

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Chapter 4

Random Functions and Excitation I. Basic Concepts of ,Random Functions

M

ANY practical structures in aeronautical, mechanical, and civil engineering are excited by forces, pressures, and base movements that are random in time. In many cases the spatial distribution of these excitations is also random. The time history of these excitations are not known deterministically. Nevertheless many global parameters of random excitation can be determined, including expected values, mean values, standard deviations, probability density functions of the magnitude of the excitations and their peaks, and average zero crossing rates, as well as the relations between an expected value at a certain time and the value at another time. It is not the purpose of this chapter either to repeat the basic theories of random variables, fields, and processes, or to prove basic mathematical relationships. These are well documented in numerous books, e.g., Refs. 10 and 11, papers, and reports. Only basic definitions from the theory of random functions and some practical aspects of these theories that are important to the user are repeated here. Reference 11 and its cited references is recommended for the interested reader. Assume X(t) is a random function in time with random values at different times. The probability distribution function of X(t), Fx(x; t) [sometimes called the cumulative distribution function (CDF)] is defined as

i.e., this is the probability that X(t) is smaller or equal to a given value x. The probability density function (PDF) is defined as

Note that in Eqs. (4.1) and (4.2) the superscript of the random function is written with upper case letter X, and a specific realization of the function is written in lower case x. Further details about CDF and PDF, as well as specific examples of these functions, are described in any textbook on the theory of probability. In Fig. 4.1, an example of CDF and PDF is demonstrated. The mean value of the function X(t), m,(t), is defined as the expected

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36

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

X Fig. 4.1 Example of CDF and PDF.

value of X ( t ) , and is the first statistical moment of X ( t ) , which has a PDF fx(x; t )

Thus, m x ( t )is the location of the center of the area or center of gravity of the function described in Fig. 4.1, along the x axis. The second moment of X ( t ) is

This second statistical moment is called the mean square value of the function X ( t ) and is the moment of inertia of the area bounded by fx(x; t ) around the axis x = 0. The second statistical moment of X ( t ) around the mean value m x ( t )is

This quantity is called the variance of X ( t ) and is the moment of inertia of the area bounded by fx(x; t ) around the axis x = mx(t).The variance is a measure of the spread of X ( t ) around the mean value. The root mean square of the variance ax= v ' v a r ( ~ ( t )is) known as the standard deviation. Higher-order statistical moments can also be defined, and these definitions are well documented in the literature.

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RANDOM FUNCTIONS AND EXCITATION

37

It can be easily shown that

Thus, when the mean value of X ( t ) is zero, its variance is equal to its mean square value. The autocorrelation function of X ( t ) is defined as the expected value of the product of X at times tl and t2

where E [ - ]is defined as the expected value of [ 1. A random function X ( t ) is said to be stationary when its value in t2 = tl + r depends only on the time difference 7 = t2 - t, between the two time instants tl and t2. Then the autocorrelation function is

Note from Eq. (4.1) that when r

=

0

Thus, the value of the autocorrelation function for zero time lag is the mean square of the function. When X ( t ) and Y ( t ) are two random functions, the cross-correlation between X and Y is similarly defined as

Two stationary functions are called jointly stationary when their crosscorrelation function depends only on the time difference 7; thus,

The concept of stationary functions is important when continuous vibrations of systems and structures are considered, as in this volume. The spectral density of a stationary random function Sx(w) is defined as the Fourier transform of its autocorrelation function R x ( r )

These relations, called the Wiener-Khintchine formulas, state that R x ( r ) and Sx(w) are interrelated throughout a Fourier transformation. Some

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38

G. MAYMON

textbooks present a slightly different formulation of these relations, which may differ by a factor of 27.r. The formulation in Eq. (4.12) is convenient because it can be easily shown that

thus, the mean square of a quantity X i s the integral of its spectral density over the frequency range. It can be seen that the spectral density is a function of the frequency. In effect, for a given frequency band, the spectral density function is the contribution of the quantity X to the total mean square of that quantity. In a similar way, a cross-spectral density function can be defined as the Fourier transform of the cross-correlation function.

Random excitations (forces, pressures, base movements) and random responses may be characterized by their autocorrelation and cross-correlation functions or by their spectral and cross-spectral density functions. The autocorrelation of an excitation ql is a function that describes the expected value of the product of a load at point xl at time tl, ql(xl, tl), and the load at point x2 at time t2, ql(x2, t2). It is defined as

When two excitations ql and q2 exist, the cross correlation describes the expected value of the product of one function q, at point xl at time t,, ql(xl, tl), and the load q2 at point x2 at time t2, q2(x2,t2). It is described as

Thus, when the given excitation is one force acting in one location xl it can have an autocorrelation function, but no cross correlation. On the other hand, when two or more forces are acting on a structure, there may be cross correlation between them. An external pressure acting on a structure may be thought of as an infinite number of forces, and cross-correlation functions between pair of points may be found. There are many cases in which there is a correlation between the loads on two or more points on the structure. This correlation may also be time dependent. For instance, a turbulent flow may cause a pressure fluctuation at point XI, which will influence the pressure at another point x2. Also, as the turbulence is swept

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RANDOM FUNCTIONS AND EXCITATION

39

downstream, there will be a correlation between the pressure at point x2 at time t2 and the pressure that existed at point xl at an earlier time t,. As a consequence of the earlier argumentations, a cross-spectral density matrix can be constructed for a system that is excited by n forces, each force having a spectral density function Sii(w), i = 1, 2, . . . , n, and each pair of forces having a cross-spectral density function Sij(o), i = 1.2, . . . , n, j = l , 2 , . . . , n,

where diagonal terms are spectral density functions and off-diagonal terms are cross-spectral density functions. When a system is excited by random forces, it responds with movements at all of the points. It is quite reasonable to conclude that the response to a random excitation is also random. Even when only one force is acting at one location, response is created at all of the points. Therefore, responses always have autocorrelation and cross-correlation functions, and spectral density and cross-spectral density functions. A cross-spectral density matrix similar to Eq. (4.16), therefore, can be constructed for the responses of n points on the structure. 11. Practical Characterization of Random Excitation In most practical cases, the external spectral densities of random excitations are the parameters that are given to the designer as an input. An example of this type of input are the military standards (such as Ref. 12), which provide means to generate a spectral density input, either of sound pressure levels, or of an acceleration, as function of the frequency. The definition of the spectral density inputs to the structure or its components is part of the data of the loads definition document prepared at the beginning of any project. In general cases, standards and specifications are used. In some other cases, the experience gained by a designer in previous designs, together with experimental data obtained during development of earlier projects, forms the basis for the definition of the input for the new design. The dimensions of the spectral density function are ( q ~ a n t i t y ) ~ / frequency. Thus, a spectral density of force will have the dimensions of (ne~tons)~/(radian/second), or (kilogram)2/(radian/second), or (pound)2/ (radianlsecond). The spectral density of pressure will be (Pas~al)~/(radianI second), or (atmo~phere)~I(radian/second), or (pounds per square inch)2/ (radianlsecond), and the spectral density of displacement is ent ti meter)^/ (radianlsecond), or (in~hes)~/(radian/second). In all these alternatives, the dimension of the frequency is given in radianlsecond. It should be noted

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G.MAYMON

oc=21rfc

fc

Fig. 4.2 Wideband excitation, example.

that the basic dimension of an angle is a radian, and not a cycle, which is 27r rads. Therefore, it is important to avoid the confusion that sometimes arises between radianlsecond and cyclelsecond (hertz). In practical cases, the spectral density defined by specifications and standards is given by (q~antity)~/hertz, or (quantity)2/(cycle/second). It should be noted that irrespective of the presentation of the spectral density, the mean square of the quantity, which is the integral of the spectral density with respect to the frequency, must be identical in the two possible presentations, and must have the dimension of (quantity)2. Let Sof be the value of a wideband excitation (Fig. 4.2) of a force F, a constant in (kilogram)2/(cycle/second) over a frequency range between f = 0 cps and f = f, cps. The mean square of the excitation is the area bounded by the spectral density function. For this case it is

In the same way, if So, is given in (kil~gram)~I(radian/second) over a frequency range 0 5 o 5 o,, the mean square of the excitation force is E{F2) = So,

I*[

radls

and as a result

but wc = 2nfc and, therefore,

*

o,[radls]

= So,wc[kg2]

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RANDOM FUNCTIONS AND EXCITATION

41

This result can be obtained by substituting the relation cycle = 2a rads in the dimension. For example, assume a constant spectral density of value 5 kg2/cpsbetween the frequencies 0 and 100 cps. This will be transformed into

5 kg2 - 5 cps

kg2 2a(rad/s)

=5 -kg2

2a (radls)

between the angular frequencies 0 and 100 . 2a radls. In general, when the spectral density of the excitation is dictated by specifications or standards the data is given in (quantity)'/hertz. To avoid confusion when calculating responses, it is recommended that frequencies are always transformed into radianlsecond, and the integrals in the frequency domain are performed on o (radls), and not on f (cps or Hz). Sometimes the excitation is a random acoustic pressure, which is usually defined in the specifications by the sound pressure level (SPL) over a given range of frequencies. In Appendix B, the conversion of acoustic SPLs levels (usually given as acoustic decibel) into a spectral density of pressure excitation is described.

111. Important Excitation Functions Several power spectral density (PSD) functions are frequently used in the analysis of the response of structures to random excitation. 1) A white noise excitation is one that has a constant value Sx(o) = SO= const over the whole frequency range, - w < w < + m. 2) A one-sided white noise is one that has a constant value S,(w) = So = const over the whole positive frequency range, 0 5 o < + m. 3) A band-limited white noise is one which has a constant value Sx(o) = So = const over a limited range of frequencies, -oc 5 w 5 w, and zero elsewhere. 4) A one-sided band-limited white noise is one that has a constant value S,(w) = SO= const over a limited positive range of frequencies, 0 5 o 5 o, and zero elsewhere. These four excitations are shown in Fig. 4.3. From these four constant value PSD functions, the one-sided bandlimited white noise is the most often used, because of its practical features. It is important to note that when the excitation is identically zero over part of the frequency range, the mean square value can be obtained by integration only over the relevant part of the frequency axis. Thus, for a band-limited white noise,

and for a one-sided band-limited white noise

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G.MAYMON one-sided band-limited white noise

band-limited white noise --

-1----.

--

-

---I I

I I I

I

I I

1

I I

I

I I I I

I

I

I

I

/

-

white noise

one-sided white noise

I 1

j I

1

-Wc

0

+Oc

Frequency

Fig. 4.3 Four constant value PSD functions.

In C h a ~ t e 5r it is shown that calculation of the mean value of the remonse of a linear system to random excitation involves integration of the input PSD multiplied by expressions of the complex transfer functions and their conjugates over the whole relevant frequency range. Some analytical results for cases where the input PSD is constant are shown in Appendix A, for the whole frequency range - a~ < w < + a ~ A . very important results shown in the literature, e.g., Refs. 3 and 11, is that for a system with a small damping ratio (say,-less than 0.1), which is a frequent case in structural analysis, the result for a band-limited integration can be well approximated by the unlimited band integrations shown in Appendix A, provided that the cutoff frequency o, is sufficiently higher (at least 30%) than the resonance frequency. Another important excitation PSD function is obtained when a one-sided white noise is filtered through an SDOF system that has a resonance w, and damping 5,. This filtering results with the following PSD function:

Note that w, and 5, can be used as parameters that may be selected to approximate many given excitation spectral density functions. Analytical results for integration of Eq. (4.22) together with the relevant transfer function are also included in Appendix A, based on the results of Ref. 13. An example of the shape of Zo is shown in Fig. 4.4.

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RANDOM FUNCTIONS AND EXCITATION

0

2

4

6

8

10

12

14

43

16

18

20

Frequency (radlsec) Fig. 4.4 Filtered one-sided white noise.

IV. Boundary-Layer Excitation Model In flight vehicles, the origin of a major kind of excitation is the turbulent flow that exists around the structure. The flow causes pressure fluctuations on different surfaces of the structure. These pressure fluctuations, which are random in nature, cause a vibration response of the flat and curved panels. The formation of pressure fluctuations is characterized by a correlation between the pressure at a given location and the pressure at other locations. The pressure that is built up in one location is not discrete, and other locations are influenced. Also, the turbulence that is formed in one location is swept downstream to form a pressure fluctuation at another point. It is reasonable to assume that the pressure in the latter is influenced by the first turbulence and that this effect depends on the downstream velocity. To solve the problem of response to a correlated turbulent boundary layer, the correlation functions or the cross spectral densities of the pressure fluctuations are required. It should be remembered that there is a Fourier transform relationship [the Wiener-Khintchine equations (4.12)] between these two functions. To determine the correlations and spectral densities of a turbulent boundary layer, experimental results taken from wind-tunnel measurements are

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44

G.MAYMON

used. In the past, experiments for this purpose were perf~rrned,'~.'~ and the proper functions, which contain a number of experimental constants, were fitted to the measurements. The model of Crocker14 is used in this chapter, together with measurements performed by Mae~trello.'~ Some basic concepts of turbulent boundary layer are repeated here for completeness. Further details can be found in publications on aerodynamics, e.g., Ref. 16. A boundary layer is defined as the layer of the flow in which changes in the flow velocity exist. These changes vary between zero velocity at the surface and uo, the external flow speed, as shown in Fig. 4.5. The thickness of the boundary layer is denoted as 8. The flow velocity is uo,

where M is the flow Mach number and c is the speed of sound in the unperturbed flow.

u Fig. 4.5 Velocity in the boundary layer.

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RANDOM FUNCTIONS AND EXCITATION

45

A displacement thickness 6* of the boundary layer is defined by

The displacement thickness is a function of the shape of u ( y )in the boundary layer. Experiments show that

In different models, different values of n are determined. In one of the models n = 7 is used so that

In the range 0 5 y 5 6, turbulences of different magnitudes and different frequencies are swept downstream with a speed u,, the convection velocity. Velocity u, < uo because the friction delays the average speed of the turbulences relative to the undisturbed flow. Experimental results show that in the subsonic region

There is a difference in the convection velocities of large turbulence (usually of low frequency) and small turbulence (usually of high frequency) and Eq. (4.27a) expresses a mean convection velocity. In the high-supersonic region (M > 3), experiments show that

For the low-supersonic region the value decreases to between 0.8 and 0.6. A shear stress exists between the flowing air and the surface. This shear r, is defined by

where p is the mass density of the air and cf is a friction coefficient that depends on the smoothness of the surface. For typical aeronautical structures, experiments of Maestrello15 showed a mean value of cf = 0.0021. The experiments also showed that the cross and direct PSD of the random pressure excitation is of the following nature:

Further, it was determined that the cross spectrum depends on the distance between two points rather than on their absolute location in such a way that

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G. MAYMON

46 -

where 5 = x , - x2, r] = yl - y2, and fi(O; o ) = 1, fi(O, o ) = 1. Therefore, 77 = 0. A P ( o ) is the spectral density of the pressure fluctuations for nondimensional representation of partial test results as obtained by Ref. 15 is shown in Fig. 4.6. For the nondimensionalized test results of Ref. 15, a closed-form approximation was suggested,

z=

P(")2 .

a*O' '

=A,

exp(- K, Fw)

+ A2 exp(-

K2Fo)

+ A3 exp(-

K3Fw)

7,

(4.31) where

Fig. 4.6 Nondimensionalized P(w) (partial results from Ref. 15).

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RANDOM FUNCTIONS AND EXCITATION

47

The following flow conditions were selected to plot a dimensional P(o): M = 0.78, c = 34000 cm/s = 340 m/s = 1115.5 ftls, uo = M . c = 26,520 cm/s, u, = 0 . 8 = ~ ~21,216 cm/s, S = 3.2 cm, and S* = 0.4572 cm = 0.18 in. In Figure 4.7a, P ( o ) as a function of the angular frequency is plotted on a regular scale. In Fig. 4.7b, it is plotted on a logarithmic scale. It is interesting to present the PSD of the pressure excitation in terms of acoustic SPL, a presentation that is more commonly used in acoustic excitation specifications. These presentations are usually given as a function of frequency in hertz (cycle/second) and, therefore, P(w) is also presented in Fig. 4 . 7 ~as a function of cycle/second. It should be noted that the areas under the curves in Figs. 4.7b and 4 . 7 ~should be equal, to yield the same mean square value for the excitation pressure. Applying the procedure presented in Appendix B, Fig. 4.8 is obtained for the pressure excitation in acoustic decibels, for one-third octave frequency bands, as is usually presented in acoustic specifications. In a usual computational process, the procedure should be inverted: the decibel levels are converted into pressure2/hertz, and then to pressure2/(radians/second). To find the mean square of the pressure excitation, the following integration is performed:

For the data given for Eq. (4.31) the following is obtained:

resulting in a standard deviation of the pressure fluctuations of 0.0023825 atm. Equation (4.31) expresses the spectral density of the pressure at a specific point. In Ref. 14, a cross-spectral density between two different points (xl, y,) and (x2, y2) is expressed as

where

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G. MAYMON

Angular Frequency

Angular Frequency (radhec)

Fig. 4.7 P(o)as a function of frequency.

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RANDOM FUNCTIONS AND EXCITATION

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130 1

Frequency (Hz) Fig. 4.8 Pressure excitation in acoustic decibels.

and where [and 77 are nondimensional distances along and perpendicular to the flow, respectively, and a and b are the length and the width of the rectangular plate subjected to the flow. Equation (4.33) states that the cross-spectral density of the excitation depends on the distance between points rather than on the specific location of these points. Cross correlation between the excitations in different points can be evaluated by using the real part of the Wiener-Khintchine relation

Introducing Eq. (4.33) into Eq. (4.34) yields the following expression for the correlation function:

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G.MAYMON

50

where

Ki and A, are given in Eq. (4.31). The integral Iican be expressed analytically using the formula

- exp((E

-

KiF)w)[(G - r) sin(G - r)w + ( E - K i F ) cos(G + r)w] 2[(E - Ki F)' + (G + r)']

Using the proper integration limits yields

which can be substituted into Eq. (4.35). In Fig. 4.9, the autocorrelation functions of the excitation along the flow (7= 0) and along a line r ) = 5 tan(22.5 deg) for the numerical values given for Eq. (4.31) are shown, for different values of distances 6 and r). The value of this function for 5 = r ) = 0 and r = 0 is R,(O, 0; 0) = 5.676205 X ( k g l ~ r n ~which ) ~ , is identical to the mean square of the excitation calculated by Eq. (4.32). The maximum point of the autocorrelation functions (Fig. 4.9) correspond to a time delay r, which is equal to

The physical significance of this result is that maximum correlation is obtained by a turbulence at point xl when it is swept downstream with a velocity u, and reaches point x2 after time 70.

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RANDOM FUNCTIONS AND EXCITATION

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Time Lag (sec)

0

0.0002

0.0004

0.0006

0.0008

Time Lag (sec) Fig. 4.9 Autocorrelation functions for the turbulent flow.

The results presented in Fig. 4.9 correspond to those obtained by Maestrello.15 The model presented [especially Eq. (4.33)] can be used later as a PSD input for the calculation of the response of rectangular plates to boundarylayer excitation. An example for this type of analysis can be found in Ref. 17, using the procedures described in Chapter 5, Sec. 111.

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V. Summary Some basic concepts related to random functions and external random excitations such as mean, mean square, variance, and standard deviation are presented. The concepts of autocorrelation, cross correlation, and spectral and crossspectral densities and their interrelations, the Wiener-Khintchine equations, for stationary random functions are presented. The mean square of a random process is obtained by integration of the PSD function along the frequency axis. Some practical aspects of the description of the PSD function of an external excitation are shown. Some important practical cases of random excitations are shown, including a model for the pressure fluctuations in a turbulent boundary layer.

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Chapter 5

Response of Linear Systems to Stationary Random Excitations I. Response of a Linear SDOF System

A

SSUME that a random force F ( t ) with a mean value m F ( t ) and an autocorrelation function RF(tl,t2) is acting on a linear SDOF system, which has an impulse response function h ( t ) , described in Chapter 1. Then

~ ( f =)

1':

-

F ( r ) h ( t - 7 )d r =

jim ~ -

m

(- rt ) h ( r ) d r

(5.1)

The definition of the autocorrelation function yields

The mean function m,(t) of the response X can be similarly found as

When F ( t ) is a stationary random function, m~ is not a function of time and, therefore,

For convenience, w is used in this chapter instead of Q used in Chapter 1. Using w = Q = 0 in Eq. (1.28b) it is found that

rn,(t)

= inr. H(0) =

const

= rnx

(5.5)

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54

For the stationary function F(t), RF(tl, t2) is a function of t2 - tl, so that RF(tl - 71, t2 - 72) in Eq. (5.2b) is a function only of t2 - t1 - 7, + 7,. Denoting t2 - tl = T,

Equations (5.5) and (5.6) imply that the response of an SDOF system to stationary random excitations is also stationary. To find the PSD function of the response X , the Wiener-Khintchine equations (4.16) are used, yielding

Denoting A

= T

- 72

+

and using Eq. (1.28b),

where [ I* denotes the complex conjugate of [ 1. Using Eq.(4.12) the autocorrelation function of the response can be found,

and the mean square of the response is

Equation (5.8) is one of the most important expressions used in the response analysis of linear SDOF to stationary random excitation. It states that the PSD of the response can be obtained by multiplying the PSD of the given input by the square of the absolute value of the complex transfer function, which is easily calculated for an SDOF system using Eqs. (1.5) and (1.7) (and replacing 0 with w)

where wo is the resonance frequency of the SDOF system, 5 is its damping coefficient, and m is its mass.

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RESPONSE OF LINEAR SYSTEMS

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It can be verified easily that when units of length, force, and time are used, the units of the PSD of the excitation force are (f~rce)~/(radian/ second) and the units of the PSD of the response are (length)2/(radian/ second). For lightly damped SDOF systems, IH(w)l is a function whose values are very small over most of the frequency axis, with large values only in the vicinity of the resonance frequency, as shown in Fig. 5.1. Thus, the PSD of the response has similar features irrespective of the shape of SF(O). Even when SF(w) is not a constant, as shown in the figure, most of the PSD of the response is concentrated in the vicinity of wo. In fact, a very good approximation is obtained when SF(w) = SF(wO)is used and, thus,

Using Eq. (5.10), the mean square of the response for a constant PSD of excitation is

For realistic structures where frequencies are always taken positive,

, Transfer Function Constant Approximate Excitation

Real Excitation

0

0.5

1

1.5

Excitation to Resonance Frequencies Ratio Fig. 5.1 Transfer function of an SDOF system and PSD of excitation.

2

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G.MAYMON

56

It can be verified that

and, thus,

Equation (5.14b) provides an exact result for the mean square value of the response of an SDOF system subjected to a stationary random excitation force with constant PSD over the positive frequency axis and a very good approximation when the PSD of the excitation is not constant. The mean square response of the system whose transfer function is described in Fig. 5.1 is almost the same for the two different excitations whose PSD function are also described in this figure. The PSD of the response of an SDOF system is narrow banded and is concentrated around the resonance frequency of the system. In Chapter 1 it was shown that SDOF system responses to harmonic excitations are high when the excitation has a frequency at or near the resonance frequency. The difference between the response to random excitation and the response to a harmonic excitation is that the former results in a narrowband random response (PSD function that is narrow and has high values around the resonance). This means that the response has frequencies that are close to the resonance frequency, but the amplitudes are random, with a mean square value given by Eq. (5.14b). Thus, an SDOF system is a narrowband filter for both random and harmonic excitations. 11. Response of a Linear MDOF System

The treatment of a linear MDOF system to stationary random excitations is based on the use of the basic equations of Chapter 4, Sec. I with the differential equations of this system presented in Chapter 2. The techniques are almost identical to those presented in the previous section for the SDOF system. It was shown that the differential equations of an MDOF system [Eq. (2.1)] can be written as a set of uncoupled differential equations (2.18), containing generalized quantities instead of masses and forces. The problem then can be treated as a set of SDOF systems, solved for the generalized coordinates q(t), and the final displacement X(t) is obtained using the transformation (2.12). In the case of stationary random excitations, the force vector on the right-hand side of Eq. (2.1) is a vector of random forces. Thus, the generalized forces on the right-hand side of Eq. (2.18) are also random. Usually these forces are defined by their PSD matrix. This matrix is similar to that

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RESPONSE OF LINEAR SYSTEMS

57

given in Eq. (4.16), where the off-diagonal terms (cross-spectral density) originate from a possible correlation between pairs of forces. The MDOF system has a typical normal mode matrix [Eq. (2.7)], which is repeated here:

[4l

=

Note that the first index refers to the mode number, and the second refers to the specific location (DOF) in the MDOF system. Denoting the ith line of the matrix [ 4 ] by L4]J (a row vector),

The physical interpretation of L+,] is the first, second, third, . . . , nth modal deflection of the jth point of the system. Unlike the SDOF system, which has only one complex transfer function, an MDOF system has a diagonal matrix of transformation functions Hl(w), one diagonal term for each resonance frequency wi,

It is more convenient to exclude M, from the transfer function (5.17) and to define

which is the analog to H,(o) defined in Eq. (1.8) for the SDOF system. A generalized cross-spectrum matrix [Sd]of the external excitation matrix [SF] can be defined,

Here, L41b,lTis a column vector with terms similar to those of Eq. (5.16). The significance of the denominator M i M j is that after multiplication of the three matrices in Eq. (5.19), each term i, j in the matrix is divided by this product.

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G. MAYMON

It is shown in the literature that the spectral density of the jth point of the MDOF system is given by

The matrices in Eq. (5.20) are defined in Eqs. (5.16), (5.18), and (5.19). Equation (5.20) is the MJOF analog to Eq. (5.8) of the SDOF system. The inner multiplication [HI [S,] [H*] is the analog to [HI [SF] [H*] and the multiplication in L+j] and its transpose originates from the transformation (2.12). To solve specific problems, the designer is mainly interested in the mean square value E[X,2] of the response at point j and in its standard deviation. The mean square of the response is obtained by integrating Sx,(w) over the frequency range; thus,

When the excitation has a zero mean (which is a frequent case in random vibration analysis of practical systems), E [ X f ] is equal to the variance of X , and the standard deviation of the response is

Although Eq. (5.21) appears complicated, in effect this is not the case. When the resonance frequencies and mode shapes are known, and when the excitation is given, i.e., by specifications, all of the matrices in this equation are known. If the PSD of the excitation does not depend on the frequency, it can be taken out of the integral, and a much simpler definite integral can be solved. Expressions for important integrals of frequent cases of practical importance can be found in Appendix A. If the PSD of the excitation is frequency dependent, a numerical integration can be used. Auto- and cross correlations can also be found, using the WienerKhintchine formulas (4.12). As these quantities are of lesser importance to the practical user, their expressions are not presented here, and the interested reader can find them in the literature, e.g., Ref. 11. More features of the solution presented in Eq. (5.20) are cleared from the following numerical example.

Example 5.1 A two-DOF system is described in Fig. 5.2. In this system, there are two masses m land m2 and two springs kl and k2 whose values are selected so that the resonance frequency of each mass on its own springs is o, for the first mass and o, for the second mass. The damping coefficient of ml on the spring kl is 5, and that of the mass m2 on the spring k2 is ls.A random force f(t) with one-sided white noise (constant value over the positive

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RESPONSE OF LINEAR SYSTEMS

(t>

X2

59

(t)

Fig. 5.2 Two-DOF system.

frequency axis) spectral density of magnitude Soacts on the first mass. This is a typical model of a primary randomly excited system, which supports a secondary system. Responses x, (t) and x2(t) are selected as the generalized coordinates q, and q2. The differential equation of the system is

The following numerical values are selected: ml = 1 kg s2/cm, m2 = 0.1 kg s2/cm, wp = w, = 125 radls, l p= 5, = 0.01, and So = 550 (kg)'/(rad/s). Solving the characteristic Eq. (2.6) yields

where the modal displacements (the normal modes) of the first mass were arbitrarily selected as a unity. The normal modes matrix is, therefore,

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60

The generalized mass matrix is [Eq. (2.8)]

The modal damping coefficients are obtained by solving

therefore,

Note that the modal damping coefficients are not identical to the physical damping coefficients of the primary and the secondary systems. The excitation spectral density matrix is

and by use of Eq. (5.19)

The spectral density of the response xl of mass ml is

s x , = L41I[ @ 4 l

[S,(o)l [H(o)l*L41 IT

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and for the mass m2

Note that in both Eqs. (5.30) and (5.31) the PSD of the response depends on three terms. The first one contains IHl(w)I2and is due to the contribution of the first resonance to the vibration of the selected point. The second term contains IH3(w)I2and represents the contribution of the second resonance. The third term contains two mixed products of the transfer functions and their conjugates and is the result of the interaction between the two resonances at the selected point. For a system with more than two DOFs, n direct terms and n ( n - 1) mixed terms exist, the latter~epresengingthe - - between pairs of modes. Using the identity HjHZ + HYH, = interaction 2 Re(H,H;), the number of the mixed terms can be reduced to n(n - 1)/2. Integrating Eq. (5.31) over the frequency range, and using the preceding identity and the integrals presented in Appendix A, yields

therefore,

The same process is repeated for Eq. (5.32) to yield

E(xg) = 0.10127

+ 0.02873 - 0.0001678 = 0.129547 cm2

Therefore, the two masses vibrate so that the root mean square of the displacement of the first and second masses are 0.1065 and 0.3603 cm, respectively.

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In the preceding solutions for the mean square value of the responses, the three terms described earlier are presented separately. It can be seen that the contribution of the first resonance is larger than that of the second one. For modal dampings that are similar (though not equal) this is always the case, because of the term wq in the denominator of Eq. (Al). The contribution of the (third) interaction term is much smaller. This happens when the resonance frequencies are well separated from each other, due to the term ( w f - ~ 7 in )Eq.~(A6). This observation can be used to find an approximate solution for an MDOF system with well-separated resonance frequencies, by neglecting the interaction terms between pairs of resonances. The procedure, which can be easily proved and is not included here, takes the following steps: 1)Write the matrix S4(w)by using Eqs. (2.8), (5.15), and (5.19). Assuming that the mode shape matrix and the mass matrix are given by numerical values, the result matrix is a function of the SF matrix functions. Do not forget to divide eachterm by MiM,. 2) Build a matrix S4(w), which contains only the diagonal terms of S4. These terms contain expression of the known SqSj(w)functions. 3) For the first diagonal term, calculate a numerical value S41,1 by inserting o = wl in the SFi,,(w)functions. For the second diagonal term, calculate a numerical value S42,2 by inserting w = w2 in the SF,,,(w)functions, etc. 4) The expected mean value of the displacement response XI is then

5) For any point j,

The procedure is shown schematically in Fig. 5.3. 111. Response of a Linear Structure to Stationary Random Excitations In Chapter 3 it was shown that a continuous structure can be treated as a collection of an infinite number of SDOF systems, provided that generalized masses, modal damping coefficients, and generalized forces replace the regular quantities. It was also assumed that resonance frequencies and modal shapes of the N first resonances of the infinite number possible can be evaluated, i.e., by a finite element program, and the structural displacement can be evaluated approximately with these first N modal shapes + i ( x ) and the generalized coordinates qi(t).Therefore, a solution

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Data Masses, frequencies, mode shapes, modal darnpings, excitation PSD matrixs SF

Calculate generalized masses matrix [MI

Write expressions for Sg using (2.8), (5.15), (5.19). Remember to &vide to the generalized masses

Calculate numerical values of the diagonal terms of Sg

Calculate mean square values using (5.32) Fig. 5.3 Procedure for an approximate response solution of an MDOF system.

of a continuous system is an analog to an MDOF system, where

Suppose that a continuous structure is excited by random forces that have the cross-spectral density function Sq(xl, x2, o), which is an analog to the [SF(o)] in Eq. (5.19) for the MDOF system. This cross-spectral density function can be evaluated in terms of the normal modes and a set of coefficients SQjQkso that

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It can be shown that

Equation (5.36a) is the continuous-structure equivalent to Eq. (5.19) of the MDOF system, where each term j, k is an analog to the compatible term in the matrix [S$(o)] of the latter system. In the numerator of Eq. (5.36a), a double integration over the whole structure is performed. Thus, in a onedimensional structure, say, a beam of length L , two integrations for 0 5 xl 5 L , 0 5 x2 5 L are included. For a two-dimensional structure, say, a rectangular plate of length a and width b, two integrals for 0 5 xl 5 a, 0 5 x2 5 a, and two for 0 5 yl 5 b, 0 5 y2 5 b are included. Thus, for a one-dimensional structure (beam, frame)

and for a rectangular plate,

These equations can be written in a nondimensional form for a beam, or 5 = xla and 7 = y l b for a plate; thus,

for a beam, and

for a plate. The generalized masses can be written as

r = xlL for

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65

for the beam and the plate, respectively, where r n is ~ the mass per unit length of a beam and r n is ~ the mass per unit area of a plate. Both may be a function of the location on the relevant structure. Sometimes the spectral density of the excitation may be separable into a term that depends on the frequency and a term that depends on the location for a beam and a plate, respectively,

In this case, So(o) can be taken out of the integrals in Eqs. (5.36d) and (5.36e). All of the terms of Eqs. (5.36) can be easily calculated for a given structure once the mode shapes and the cross-spectral function of the excitation are given. The use of a finite element program results in mode shapes and generalized masses. In most of the commercially available finite elements computer codes, the modes are normalized in such a way that all of the generalized masses are equal to unity and, thus, the matrix of the generalized masses is the unit diagonal matrix [ I ] . The cross-spectral density of the response of the structure is given by an expression analog to Eq. (5.20) of the MDOF system. For a one-dimensional structure, it is

Equation (5.39) is the continuous system equivalent of Eq. (5.20) of the MDOF system. It is the cross-spectral density function between the points and 6on the structure, due to all of the modes, i, k = 1, . . . , N. The transfer functions in Eq. (5.39a) are those given in Eq. (5.18). If & Z t2, the cross-spectral density between and t2is obtained. If t1= & = 6, the spectral density at point 5 is obtained

el

For a two-dimensional structure like a plate,

which is the cross-spectral density between two points (el, 77,) and (t2, q2). If 51 = 6 = 5 and 771 = rl2 = 7, the spectral density of the response at

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point (6, 7) is obtained

The mean square value of the response is obtained by integrating the spectral density of the response over the frequency range. It should be noted that in many references the spectral density of the input is assumed to have components in both negative and positive frequencies. This is the outcome of the Wiener-Khintchine formulas (4.12), where S, is obtained from the autocorrelation function R,, which exists for both negative and positive time delays T. In cases where the spectral density is given by specifications, standards, or experimental measurements, it is one sided and, therefore, the values for negative frequencies are identically zero. In the particular engineering cases treated here, the spectral density is always taken as a one-sided function, for positive values of w only. Therefore, using Eq. (5.39b), the mean square of the response of a one-dimensional structure, such as a beam, subjected to one-sided spectral density is

A very similar expression is obtained for a two-dimensional structure, where the only difference is that the mode shapes are functions of both 6 and 7. Sometimes it is of interest to calculate the spectral density of the time derivative of the response, for instance, the spectral density of the velocity. It can be shown that

and, therefore, the mean square of the time derivative of the response is

In Eq. (5.41) two kinds of terms exist: integrals of -

H,?(w) . Hi(w) = IHj(w)12

=k

(5.44a)

when j # k

(5.44b)

when j

and integrals of B,*(w) - Hk(o) + H,(w) - H,*(w)

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which are similar to the expressions (5.30) and (5.31) in the example of the two-DOF system. Thus, direct terms (contributed from each resonance) and cross terms (contributed from the interaction of pairs of resonances) exist in the response at any given location. Because of Eqs. (5.44), Eq. (5.41) can be rewritten as

Using the identity H,HZ the form

+ H,*Hk = 2 Re[HjHz], the last

equation takes

where the first summation term contains N terms, which are the outcome of the N resonances and the second summation term contains N(N - 1)12, which are the result of intermodal interaction. Thus, Eq. (5.45b) contains N(N + 1)/2 terms, whereas Eq. (5.41) contains I\IL terms. A similar expression for the expected value of the velocity can be obtained by multiplying the integrands in Eq. (5.45b) by w2. As in the MDOF system, a quick practical approximation for cases in which the natural frequencies are well separated and the modal dampings are small can be obtained, for the cases where the spectral density does not vary strongly in the vicinity of the resonance frequencies. In these cases, the interaction between the well-separated resonances is negligible. Also the response has high values around the resonances and negligible values below and above these frequencies. Each resonance is then treated as an SDOF system, and the contribution of the jth mode to the mean square of the response is

The total mean square is obtained by summing the contribution of N modes. When the external excitation has a one-sided constant value spectral density function So(@),which does not depend on the location on the structure, Eq. (5.46a) takes the form

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It can be seen that the contribution of the jth mode to the mean square of the response is proportional to my3; thus, for well-separated resonances the main contribution to the mean square of the response comes from the first resonance. Therefore, in the cases treated by this approximation, a reasonable engineering result can be obtained using only one DOF. This conclusion is less accurate for the derivative process (velocity response), because in this case

and the contribution is proportional to o7'. The approximation is even less accurate for the second derivative process (acceleration). Therefore, it is usual to see engineering structural systems that have relatively highfrequency acceleration responses, whereas the displacement response is concentrated in the low-frequency range. In Chapter 3, Sec. IV, the concept of stress modes *f')(t) was introduced, where j is the mode number and (i) is an index that indicates which stress is being calculated. Using Eq. (3.39), expressions analog to Eqs. (5.39a), (5.39b), (5.40a), (5.40b), (5.41), (5.43), (5.45a), (5.45b), and (5.46a-5.46~) can be obtained for the stress response and the stress time derivative, by changing 4i([) with *f')([). For instance, Eq. (5.46b) becomes

for the contribution of the jth resonance to the mean square of the stress (i). Similarly (5.46~)becomes

Thus, all of the procedures, expressions, and numerical programs used for the computation of the displacement response of the structure can be used for the calculation of the stress response, using the concept of stress modes. These stress modes can be calculated using the techniques described in Chapter 3, Sec. IV.

Numerical Example A simply supported steel beam of length L with a uniform rectangular cross section of width b and height h is subjected to a stationary random excitation force per unit length whose PSD function S, is a function of only the angular frequency w, as shown in Fig. 5.4. The root mean square value of the displacements and the bending stresses at the mid- and quarter-

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RESPONSE OF LINEAR SYSTEMS

Angular Frequency Fig. 5.4 PSD of the excitation.

length of the beam is required. For simplicity, only three normal modes are used. The following numerical values are taken: b = 2 cm, h = 1 cm, L = 30 cm, density p = 7.9592 X kg s2/cm4, Young modulus = 2.1 X lo6 = 0.015, l2= 0.02, and L3 = 0.02. kg/cm2, and modal dampings From the geometry, the cross-section area is A = bh = 2 cm2, and the area moment of inertia is I = 1/12 bh3 = 0.16667 cm4.The first three frequencies and mode shapes of a simply supported beam are given by o,=

L2

/$

=

16'26.09 radls;

@I(()

= sin(n0

The generalized masses are given by (for example, see Ref. 18) M, =

yo

Ml

p A L sin2 (jn() d t = M p A L

= M2 = M3 = 2.38776

-

kg s2/cm

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To calculate SQ,Q,, Eq. (5.36d) is used. For this purpose, the following integrals are required:

1;

sin ( n n d( = -; 'A

1:

sin (2770 d c = O;

J1sin

2 ( 3 n d~c = 3n

and, therefore,

where S,(w) is given by Fig. 5.4. Note that all terms connected to the second resonance are zero. This is because the second mode is antisymmetric while the excitation is uniform (and, therefore, symmetric) along the beam. No response in the second mode and no interaction between the second modes and the first or the third modes are expected. Equation (5.46a) is used for the computation of the mean square values of the response. To use the equation, three values from Fig. 5.4 are required,

and the values of the modal displacement at the midbeam and quarterbeam are also required

Substituting the numerical values in Eq. (5.46b) yields

Therefore, the root mean square (rms) value of the displacement at the center of the beam is 0.5216 cm and at the quarter-beam is 0.1360 cm. To calculate the mean square of the bending stresses, the stress modes for bending of a beam are required. One of these has already been calculated

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71

in Example 1 of Chapter 3, Sec. IV, and is repeated here, together with the second and third bending stress modes; q@end) 1

=

1 ' r2Eh sin (23-5) 2L2

.$bald)

=

9 . v 2Eh sin (323-5) 2L2

Substituting the numerical value yields

Substituting these values into Eq. (5.47a) yields

The rms of the bending stresses are 6088 kg/cm2 and 4304 kg/cm2, respectively. It should be noted that the rms values are not the maximum value of the response. If, for instance, the excitation has a Gaussian (normal) distribution, the response is also Gaussian and values of 3 u may occur in 99.73% of the applied time. The major contribution to the response is from the first mode. As was earlier mentioned in this section, the second mode does not participate in the response. Note that the contribution of the third mode to the stresses is higher relative to its contribution to the displacements. This is due to the higher curvatures of the third mode, which result in higher bending stresses. IV. Summary In the evaluation of the expressions in this chapter, linearity is assumed for the SDOF, MDOF, and continuous systems, and the excitation is assumed to be stationary. Therefore, one can use the formulation presented for continuous, stationary random vibration, but not for the response to transient excitations. Expressions for the computation of the mean square values of the displacements of these systems are also presented. When the excitation has zero mean, the mean square values are equal to the variance of the response.

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Approximate solutions are presented for MDOF and continuous systems that have well-separated resonance frequencies. These approximations are obtained by neglecting the effects of interaction between modes. For continuous systems, the displacement response expressions can be used for stress response, by replacing the normal modes in the relevant equations with the stress modes. Stress modes can be calculated by a commercial finite element program, using the procedure described in Chapter 3, Sec. I.

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Chapter 6

Nonlinear Single-Degree-of-Freedom and Multiple-Degree-of-Freedom Systems I. Introduction

A

NONLINEAR structure behaves differently than a linear one. Factors such as geometrical nonlinearity (hardening or softening) of a structure due to geometrical changes when large deflections are introduced, material nonlinearity, e.g., plasticity, and changes in damping coefficient as a function of the structural response may increase or decrease the deformation and, thus, the stress distribution in a structure, as compared to the linear model. In many realistic structures, a linear model may result in a conservative design, both in static and dynamic analysis. Therefore, the use of nonlinear analysis may provide more efficient, less conservative designs. There are many methods that are used for the nonlinear treatment of a structure. All of the commercially available large finite element codes enable the calculation of the static behavior of a nonlinear structure. The nonlinear effects that can be handled by these programs include geometrical nonlinearity (large deflections and stiffening effects) and material nonlinearity. Fewer computational tools are available for the nonlinear vibration analysis of realistic structures. Although many analytical methods for such analyses exist, e.g., Refs. 19-21, the implementation of these methods for computation of practical structures is not straightforward. Usually, due to design considerations, good engineering practice is to avoid the nonlinear material range in vibration. Nevertheless, elastic (geometric) nonlinearity may be included in the vibration analysis to obtain significantly less conservative designs. Therefore, this chapter is limited to geometrical nonlinearity. A classical example for an elastic nonlinear system is a plate (either supported or clamped at all edges), where the supports cannot move toward each other when the plate is deformed by a perpendicular load. As the deformation increases, the length of the mean surface of the plate increases and membrane stresses are generated, in addition to the classically computed bending stresses. Because of these membrane stresses, the lateral deflection of the plate is smaller compared to the lateral deformation when the supports can move. Significantly less lateral deflection and, thus, less bending stresses are obtained. To familiarize the reader with the major phenomena of geometrical nonlinearity of a structure, a description of an SDOF oscillator is described

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in Sec. 11. In addition, the practical treatment of an MDOF system, representing a continuous structure, is described in Sec. 111. 11. Nonlinear Behavior of an SDOF Oscillator

In a linear SDOF system, the relation between the restoring force F and the displacement q is

where a is the linear rigidity of the system. In most conventional cases, the response of the structure to loads in one direction is antisymmetric to the behavior in the opposite direction. Therefore, the nonlinear behavior of the restoring force can be described by an odd series

It can be shown that in many cases the use of a cubic equation [only two terms of Eq. (6.2a)l can accurately approximate the nonlinear behavior. An oscillator with a cubic force-displacement relationship

is called a Duffing oscillator. The behavior of a Duffing SDOF oscillator of mass m, subjected to an excitation force F(t) is

where wo is the resonance frequency of the linear oscillator, P is a coefficient of nonlinearity, and l is the damping coefficient in small amplitudes. When the oscillator is excited by an harmonic force F = Fo cos ot, the steadystate response will be harmonic, with frequency o and amplitude A. It was shown19 that the following relation exists:

For a given set of Fo, M, 5, wo, and p, the relation between A and o is described in Fig. 6.1. Equation (6.4) degenerates to the classical expression

for linear systems where /3 = 0. The nonlinear response curve (6.4) has an overhang to the right when j3 > 0 (hardening), as shown in Fig. 6.1, and to the left when /3 < 0 (softening). The turning point B lies on a line, which is called the backbone curve. For A = 0 this curve intersects the frequency axis at o = o,d =

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0.7

0.8

0.9

1

1.1

1.2

1.3

Relative Frequency Fig. 6.1 Nonlinear response of an SDOF oscillator.

wo

m.The backbone curve is given by

For a damping coefficient that is small relative to 1, Eq. (6.6a) degenerates into

It can be seen in Fig. 6.1 that the resonance frequency in higher amplitudes is higher (in the case of hardening system) than the resonance frequency of the linear (small-amplitude) response, due to the higher rigidity of such a system. It is interesting to see that there is a region of o for which three solutions

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of A exist for a given excitation frequency. The stability of these three solutions was checked.19 Experimental work showed a jump behavior between the points. Thus, when a constant amplitude force, with a frequency that changes from zero upward, is applied, the amplitude of the response follows along the curve ODAB. The response jumps to B1 and follows the curve BICl. If the excitation frequency is decreased, the response follows the curve CIBIDl and then jumps to point D and fo1low.s the curve DO. Many characteristics of the oscillator can be deduced from this type of experimental procedure, some of these were described in Refs. 22 and 23. When this type of a system is excited by a wideband random excitation, its response contains components in many frequencies, and the jump phenomena, which exist when the excitation is harmonic, disappear. In some cases of narrowband random excitation, the jump can still be observed because the basic characteristic of the response to narrowband excitation is very similar to a harmonic excitation, which is a limit case of a narrowband excitation. Sometimes a continuous system can be approximated by an SDOF system. This is the case when most of the mass of the structure is concentrated around a single location and its elasticity is contributed by a well-defined elastic part. It is advantageous to create this type of SDOF approximation in the early stages of a particular design, due to the simplicity of the solution, which can enhance the understanding of the behavior of the continuous system and point out some important parametric effects. In the following example, a continuous structure is approximated by an equivalent SDOF Duffing oscillator, and response curves for different excitation levels are calculated and compared to linear analysis results. A uniform beam is simply supported on immovable supports as shown in Fig. 6.2a. The dimensions and material properties are also shown in the figure. The beam is replaced by an SDOF system described in Fig. 6.2b. In the equivalent oscillator, an equivalent concentrated mass mll is located at the center of the beam, and the springs are considered to have no mass.

Fig. 6.2 Beam with immovable supports and its SDOF equivalent: L = 60 cm, b = 8 cm, h = 0.5 cm, I = 0.08333 cm4, E = 2.1 x lo6 klcm? p = 7.959 x (specific weight 7.8), & = 0.02, W, = total weight = 1.872 kg, and M, = total mass = 1.902 x kg &em.

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The static linear relationship between the midspan force Fl and the midspan deflection r], is

and k l l can be calculated from the beam formula^.'^ With the given data one obtains

The equivalent mass is calculated so that the linear resonance frequency of the equivalent oscillator coincides with the beam first resonance frequency, also calculated by known formulas, e.g., see Ref. 1,

Therefore,

It should be noted that the generalized mass of the beam is not necessarily equal to the equivalent mass

and the difference between M and mll is 1.4%. The static nonlinear equation for the midspan deflection is

This equation can be used to calculate the coefficient of nonlinearity of the system. Although the behavior of the displacement of the mass at the center of the beam as a function of the static force Fl can be solved analytically, a finite element nonlinear module of the ANSYS program is used. This is done to demonstrate a solution process that is applicable to more general problems. The nonlinear module of the ANSYS finite element program, where large deflection and stress stiffening effects are taken into account, is used. The beam is divided into 24 beam elements and is numerically loaded by a midspan 300-kg force. Results of deflections as well as the results of a linear analysis are described in Table 6.1. Table 6.1 Linear and nonlinear midspan deflections

F, kg

Linear 7, cm, analytic

Linear 7, cm, ANSYS

Nonlinear q, cm, ANSYS

300

7.71459

7.7146

0.8822

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The nonlinear analysis result is noted to be 8.74 times smaller than the linearly calculated deflection. This indicates a significant stiffening effect for this structure. Substituting q and F from Table 6.1 and value of kll given earlier into Eq. (6.8) yields the value of bll (kg/cm3)

Therefore,

The equation of motion of the equivalent SDOF oscillator is [see Eq. (6.3)]

125

150

175

200

225

250

275

300

Frequency (radlsec) Fig. 6.3 Nonlinear and linear response of the equivalent oscillator.

325

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Assume Fl = Fo sin ot, then solutions to Eq. (6.9) can be plotted for different values of harmonic excitation forces of amplitudes Fo. This is shown in Fig. 6.3. The difference between ooand odis neglected. It should be noted that even for deflections that are smaller than the beam height ( h = 0.5 cm), significant reductions in the maximum response are obtained. 111. Nonlinear Coefficients of a Structure

To solve a nonlinear problem, a knowledge of the nonlinear coefficients of the structure is required. When a discrete system of several masses and springs are given, the nonlinearity of the springs and the nonlinearity of damping (if such exists) is usually given as data. This is not the case for a nonlinear structure. The structural geometrical nonlinearity is an outcome of the structural configuration and its boundary conditions. Although analytical expressions that describe nonlinear behavior may exist for some simple structural elements, the practical structures encountered in engineering practice must be solved through a numerical algorithm that usually requires discretization of the structure. Also, it should be kept in mind that a nonlinear system cannot be solved by a superposition of several simpler nonlinear cases, as the superposition principle exists only for linear systems. Therefore, the method for calculating the coefficients of nonlinearity is to solve static cases using a nonlinear finite element code and to extract nonlinear coefficients from these solutions. The use of finite element programs for structural analysis is practically a discretization of a continuous system (infinite number of DOFs) into an MDOF system. The method for doing so is better demonstrated by a numerical analysis of the simply supported beam described in Fig. 6.2, this time without modifying it into an SDOF system but into an MDOF system. The designer has to decide how many generalized coordinates will participate in the analysis. This decision must be based on logical considerations that depend on the nature of the problem and the past experience of the designer. Often symmetric conditions of the structure and the loads can significantly decrease the required number of generalized coordinates. If the behavior of only a few points on the structure is important, the response of only these points should be selected as general coordinates. For the demonstrated example, three general coordinates q l , q2, and q3 at quarter span, midspan, and three-quarter span, respectively, were selected. The beam is divided into 24 beam elements, as shown in Fig. 6.4.

Fig. 6.4 Discretization of a continuous beam into three-DOF system.

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First the linear case is treated,

or more specifically,

The beam is load'ed with LF1 0 01T, and Lql q2 q3] is calculcated with a linear finite element code. Substitution of this vector of results into Eq. (6.10) yields three equations with nine kij unknowns. The beam is then loaded with 10 F2 0IT to obtain three more equations, and then loaded with LO 0 F3ITto obtain another three equations. Thus, nine equations enable a solution for the nine kij. In Fig. 6.5 the numerical values of the loads and the linear deflections are shown. The nine equations are

Solution of these equations yields

Although the demonstrated problem is symmetric and can be solved with only two general coordinates, three general coordinates are shown to dem-

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NONLINEAR SDOF AND MDOF SYSTEMS

300

A

I

0

I

1

4.3395 0.6886

5.3038 0.6295

300

0 I

I

5.3038 0.5395

3.3751 0.3293

Linear Non-linear

Load

0 I

7.7146 0.8822

0

0

1

I

3.3751 0.3293

Load

0

5.3038 0.6295

Linear Non-linear

5.3038 0.5395

300

Load

I

4.3395 0.6886

Linear Non-linear

Fig. 6.5 Linear and nonlinear deflection for three loading cases.

onstrate a more general case. Also, it is not necessary to use loading vectors that contain zero values. This simplifies the set of algebraic equations but this is not necessarily required in a general computerized procedure. Now, assume that the nonlinear behavior of the structure can be formulated by

or more explicitly,

Using the same three loadings as for the linear case, but using the nonlinear module of the ANSYS program, one obtains the deflections for the nonlinear case. These are also shown in Fig. 6.5. Substituting these deflections

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into Eq. (6.13) together with the values of the linear rigidity given in Eq. (6.12) the following is obtained:

It should be emphasized that there is no connection between the number of finite elements selected for the solution of the problem (24 in this example) and the selected number of generalized coordinates. The number of elements selected is influenced by the requirements imposed on the static finite element solution, whereas the selected number of generalized coordinates is influenced by a prior estimation of the number of modes that are expected to significantly participate in the dynamic response analysis.

IV. Summary The behavior of a nonlinear SDOF oscillator is described. Geometric antisymmetric nonlinearity is assumed. In particular, a Duffing oscillator is treated. As a result of this nonlinearity, the natural frequency of the oscillator is amplitude dependent, with an increased frequency for a positive coefficient of nonlinearity and a decreased frequency for a negative coefficient. A jump behavior may exist in a geometric nonlinear oscillator. A practical method for calculation of the nonlinear coefficients of a practical structure using a finite element program is described and demonstrated for a simple structure, which is represented by a three-DOF system.

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Chapter 7

Statistical Linearization Method I. Statistical Linearization Method for an MDOF System

A

WIDELY used methodz4for the treatment of nonlinear systems is the statistical linearization method. As the analysis of a realistic structure is usually performed by discretization of the continuous system, the analysis procedure is demonstrated for an MDOF system. The general equation of motion of a nonlinear MDOF system can be written as

The functions @ are nonlinear functions of the vector of the generalized coordinates { q ) and its first and second time derivatives. There are only a few cases where an analytical solution of Eq. (7.1) can be obtained knowing a. Several methods for a solution were suggested in the literat~re.'~ Some of them are based on perturbations, energy balance, and slowly varying parameters. Nevertheless, these techniques fail in many practical cases, or develop into very complicated procedures. The concept of statistical linearization permits a systematic approach, which can be mechanized into a general approximate solution. According to this method, the nonlinear equations are replaced by equivalent linear equations, which are selected in such a way as to minimize the difference between these and the original equations. When an equivalent set of linear equations is obtained, they are solved by known techniques of linear equations. The equivalent linear equations are defined as

where me, c,, and k, express additional equivalent mass, damping, and rigidity, respectively. The difference between the original Eq. (7.1) and the equivalent Eq. (7.2) gives a difference vector {E), which is

The mean square of the difference

E

is minimized

E { s T . E ) = minimum

(7.4a)

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E{E:

+ E$ + . . . + E:}

= minimum

(7.4b)

where ci are the elements of the vector {F). Equation (7.4b) can be written as

2 DT

= minimum

i=l

where D; = E ( E ~ )i, = 1, 2, . n. Denoting m;, c;, and k;, the elements of [m,],[c,],and [k,],respectively, and denoting Qi the elements of {+I, D: takes the form

Minimization of Df is obtained when

Introducing Eq. (7.5) into Eq. (7.6) yields the following set of equationsz4 after some algebraic manipulations:

where qT = Lq, q, ql and m$, c$, and &* are the ith row of the matrices [m,], [c,], and [k,], respectively. Equation (7.7) is a set of equations for andk'.(1' ce. me. 11 I] The equations for the equivalent system can be simplified when the excitation is Gaussian, which is a frequent assumption in the analysis of the response of structures to random excitation. For linear systems, the response to Gaussian excitation is also Gaussian. Therefore, an approximation of Gaussian response for a nonlinear system excited by Gaussian 9

7

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STATISTICAL LINEARIZATION METHOD

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excitation is quite logical. It was shown in the literaturez4that the following formula exists for a Gaussian vector variable r]:

E { f( d . r ] )

= E{r].

v T ) E{Vf . (7))

(7.8)

where

When (4) in Eq. (7.7) is assumed to be Gaussian, comparison of Eqs. (7.7) and (7.8) leads to

where

and similar relations are obtained for d @ J d q and dQiIaq.

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Comparison of Eq. (7.9) to Eq. (7.7) yields

In the evaluation of the former equations, it was assumedz4that the nonlinear functions are antisymmetric

Therefore, the response, which was assumed to be Gaussian, has a zero mean, which also requires the Gaussian excitation to have a zero mean. This is a reasonable assumption when analysis of the response of structures to external Gaussian excitation is performed. When the excitation does not have a zero mean, it must be separated into a zero mean excitation and a static bias. the treatment of the latter is not included in the present discussion. Equation (7.10) includes the expected values of the derivatives of the nonlinear functions @withrespect to the expected values of the generalized coordinate vector {q} and its time derivatives. These expected values are not known before the solution is determined. The expression that relates the Gaussian excitation to the expected values of the response was already presented in Chapter 5, Sec. 11. If the excitation forces {Q(t)}in Eq. (7.1) are random Gaussian and have a spectral density matrix SQ(w),the weighted generalized spectral density matrix of the generalized forces is given by [see Eq. (5.19)]

and the spectral density of the response is given by [see Eq. (5.20)]

Here H ( o ) rather than H ( w ) is used, because during the solution process the generalized process the generalized masses are changed. The variance of the generalized coordinate q, is given by the integral of the relevant term of the response spectral density over the frequency range

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STATISTICAL LINEARIZATION METHOD

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Equations (7.10), (7.12), and (7.13) provide expressions for the computation of the expected values of the response {q} of the equivalent linear system, which must be solved iteratively. First, the nonlinear effects are neglected and the variance of the response of the linear system is calculated. Then, m&,c& and k;, are calculated with Eq. (7.14) and added to the equivalent equation of motion. New values of the variance are computed, and a new set of m;, c;, and k&,are computed. The process is repeated until a convergence of the variance is obtained, according to the required accuracy. Continuous structures are usually defined by their normal modes and can be treated as MDOF systems, where the number of DOFs taken into account is the number of modes involved in the response process and where the generalized masses, rigidities, and modal damping coefficients are used. The integral in Eq. (7.14) takes the form

When S+(w) is constant, integrals such as Eq. (7.15) can be performed using the formulas of Appendix A. When S+(w) can be represented as filtered white noise, Appendix A can also be used. If S+(w) is a rational function of w, a method of solution for the integral is described in Appendix B of Ref. 24. Nevertheless, numerical integration may be required for complex practical structures. It should be borne in mind that S+(w) is not the original matrix of the spectral density of the excitation, but a matrix obtained by a suitable process, where the original matrix is weighted by the normal modes, as seen in Eq. (7.12). During the iterative process of solution of the equivalent linear equations, modes are recomputed in every iteration and, therefore, the generalized masses, modal frequencies, and modal damping coefficients are changed during the process. 11. Nonlinear Response of an SDOF System to Random

Gaussian Force The iterative process is demonstrated for the beam that was represented as an SDOF in Fig. 6.2. For this beam it was found that

Therefore,

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According to Eq. (7.10)

defining

where ull is the standard deviation of the response. The equivalent rigidity single term matrix is degenerated into [k + k,,] = kIl + k;? = 38.8873 + 1160.9172~:~

(7.20)

For an SDOF system, the normal mode is always equal to one

and the generalized mass

Assuming that the excitation force F has a one-sided white noise spectral density of value S1 = 0.065 kg2/(rad/s), the matrix S4(w) is, therefore,

and the spectral density of the response is

in Eq. (7.24) is the equivalent modal damping coefficient. The variable lleq For the original system, the damping is assumed to have no nonlinear effects. As a result, cll in Eq. (7.16) is a constant, although this is not necessarily true for the modal damping coefficient. This coefficient is defined for a linear SDOF system as

and for the present case

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Therefore, during the iterative solution process, lie, changes as ole, changes, so that

= 0.02 is the damping coefficient of the linear system and llIin where for this case (see Fig. 6.2). The mean square value (the variance) of the response is given by Eqs. (7.14) and (7.24). Denoting this (T: (a, being the standard deviation of the response) the following i s obtained:

Using Eq. (7.26) this yields

The resonance frequency is calculated by the characteristic equation

or, by introducing the numerical values,

Therefore, the iterative process is as follows: 1) Calculate allin for the linear system olli,

2) Calculate

(T:

=

d38.887319.4118 X

=

203.2668 radls

by Eq. (7.27b)

3) Substitute this value into Eq. (7.28) and calculate a new value for the frequency

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4) Calculate new

CT;

with Eq. (7.27b)

5) Repeat the process until convergence is obtained. In Fig. 7.1 the consecutive values of cri and ole,are shown. The values are also repeated in Table 7.1. Because the response is sought, convergence criteria should be applied to the mean square of the response but other parameters such as the equivalent frequency and the equivalent rigidity should also be checked.

Iteration

0

5

10

15

20

25

30

35

40

45

Iteration

Fig. 7.1 Convergence of the variance and the equivalent frequency.

50

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STATISTICAL LINEARIZATION METHOD

Table 7.1 Iterative converging process, SDOF Iteration

kfP

Weq

(4

The solution converges to

i.e., the standard deviation of the Gaussian response of the mass (or the midspan of the beam) is

with a governing frequency of 393 radls. It should be noted that the standard deviation of the linear system is a,

= 0.5858

cm

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with a governing frequency of 203 radls. Thus, the nonlinear response is about 52% of that calculated by a linear analysis, emphasizing the fact that a less conservative result is obtained using a nonlinear analysis. The shift in the frequency (hardening) is also significant. A closed-form analytical solution for an SDOF oscillator is presented in Ref. 24. For the present case the solution yields the following result for the variance of the response:

which differs from the numerical iterative result by only 1.2%. 111. Nonlinear Random Response of a Two-DOF System to Random

Gaussian Force A nonlinear two-DOF system is described in Fig. 7.2. Each spring is nonlinear, with its elastic restoring force equal to

The viscous dampers are linear. The damping is cl = 2101% and c2 = 2 l o 2 S , where lol and are the damping coefficients of each damper (and not the modal damping). Also yl and y, are relative to the fixed base. It can easily be shown that the equation of motion of this system is

co2

Fig. 7.2 Two-DOF system.

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It is more reasonable to select general coordinates {q) so that

where ql is the displacement of the first mass relative to the fixed base and qz is the displacement of the second mass relative to the first mass. It should be noted that elastic forces in the springs (and, therefore, stresses if required) can be more easily calculated with these generalized coordinates. Equation (7.30) then takes the form

These equations can take a simpler form in which the damping matrix is symmetric, if both sides are multiplied by a matrix

which yields

These are the equations for which equivalent linearization is performed. The nonlinear functions @ are

and, therefore, according to Eq. (7.10) = 3clklE(q:) =

=

where

CT:

3c2k2E(qz) = 3c2k2ffz

and a$ are the variances of ql and q2, respectively.

(7.35)

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The normal modes matrix of this system is

The generalized matrix is

The generalized damping matrix is

and the modal damping coefficients are

M1 and M2 are the where CI1 and CZ2are the diagonal terms of [q, generalized masses, and wl and w2 are the natural frequencies. Assuming that forces Fl and F2 are uncorrelated zero mean stationary Gaussian processes, with one-sided white noise spectral densities of magnitudes S1and S2,the spectral density matrix of the excitations for Eq. (7.33) is

The generalized spectral density matrix S+(w) is given by Eq. (7.12), and the spectral densities of the response at the first and second masses are

where [&I, [+2] are given by Eqs. (7.36). To demonstrate the solution process, numerical values are introduced: ml = 0.010 kg . s2/cm (weight of 9.80 kg), rn2 = 0.015 kg . s2/cm (weight

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STATISTICAL LINEARIZATION METHOD

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of 14.7 kg), k1 = 1000 kglcm, k2 = 500 kglcm, q k l = 200 kg/cm3, e2k2= 100 kg/cm3, lol= lO2 = 0.01, and S1 = S2 = 180 kg2/(rad/s). The characteristic equation of the undamped linear part of Eq. (7.33) is solved for the linear natural frequencies and the normal modes

which yields the following results: qlin = 143.06 radls;

wzlin = 403.567 radls

By Eq. (7.37)

Using Eqs. (7.38) and (7.39), the linear modal damping coefficients are obtained

Substituting the normal modes and the input spectral density (7.40) into Eq. (7.12) yields

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Using the procedure described in Chapter 5, Sec. 11, the following expressions are obtained for the mean square values (variances) of ql and 9,:

In Eq. (7.47) H(o) is equivalent to H(w) as M I and M2 were already introduced in Eq. (7.46). The values of the integrals can be calculated by the expressions given in Appendix A. The results obtained for the linear system are

It should be noted that products like leqo;,and similar are presented in the denominator of the frequency integrals. The calculation for systems with linear damping can be shortened if the term ~ l i n ~ l i , is~ ~introduced , instead of this product [see Eq. (7.28)]. Therefore, the values obtained in Eqs. (7.43) and (7.45) can be used during the iterative process, saving repeated computations of l,,. The iterative process was implemented on a personal computer, using the TK+ Mathematical Solver program. Because only two DOFs exist, the characteristic equation was solved explicitly as a quadratic equation and then the mode shapes were calculated. Multiplication of matrices was done explicitly. When more DOFs exist, an eigenvalue and eigenvector subroutines can be used, as well as matrix multiplication modules. Other mathematical solver programs can also be used. In Table 7.2, results of the first 20 iterations are presented for o l , M I , &, and 4and 02, M2, h 2 , and (T:. Here +11 and 421are defined as unity. The hardening effect can be observed clearly by comparing the final nonlinear frequencies

to the corresponding linear frequencies

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Table 7.2 Iterative converging process, 2 DOFs

Iteration

01

412

MI

d

0 2

422

M 2

d

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98

Also, the rms (standard deviations) of the responses of the nonlinear system are al =

= 1.6981 cm;

a2=

= 2.0797 cm

These are significantly lower than those of the linear system crl =

= 2.5643 cm;

a2=

= 3.6132 cm

IV. Nonlinear Random Response of an Elastic System As the procedure described in the preceding section is based on finite element solution of the coefficients of nonlinearity, and normal modes are used in this procedure, there is no loss of generality in describing a beam structure as a representative example of other continuous elastic structures. In Chapter 6, Sec. 111, the linear rigidity matrix [ki,][Eq. (6.12)] and the nonlinear rigidity matrix [bij][Eq. (6.14)] of a simply supported beam were calculated, using a finite element program. Nonlinear response of this beam is now calculated, assuming the beam is subjected to a midspan random exciting force with one-sided white noise spectral density of value S2 = 0.1 kg2/(rad/s). Linear modal damping coefficients of the three first modes are assumed as = l2= l3= 0.02. The beam is represented by a three-DOF equivalent system (shown in Fig. 7.3), where the total mass of the beam MT was lumped as shown. Other methods of lumping masses are also possible. The treatment of a beam is used to demonstrate the process of a nonlinear solution of an elastic structure. Using Eqs. (7.10), (6.12), and (6.14), the equivalent linearized rigidity matrix is

and CT:~,at ml, m2, and m3, The expected mean square values a $ l , respectively, are to be calculated. Because of beam symmetry, a;, = CT:~, only two mean square values, a:, and a:2, are calculated. The cross-spectral response between y, and y2 is also calculated.

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STATISTICAL LINEARIZATION METHOD

Fig. 7.3 Three-DOF equivalent beam.

The characteristic equation from which the equivalent natural frequencies and normal modes are calculated is

This equation is solved to yield three natural frequencies, wl, w2, and w3, and three normal modes, with the following matrices:

It is convenient to define the normal modes so that

The generalized masses are obtained by

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The spectral density matrix of the excitation is

[:z 0

0

0

[SQ(~)]=

(7.52)

The procedure, which was already applied for two DOFs is repeated for three DOFs. The generalized spectral density matrix is

The matrix B

=

[H(o)] [S4(o)][H*(w)] [see Eq. (7.13)] is

Here Hi is equivalent to Hi, because division to MiMj were already performed in Eq. (7.53). The spectral density of the response at ml is sql(u) = s411H11~(411)~ 1 + sh21H212(421>2 + s433/H312(43i)2

+ 2S412421411Re(HTH2) + 2SdI3431411 Re(H1*Hd + 2S423d'21431 Re(H2*H3)

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and the mean square of the response is given by

The integral can be calculated using the expressions in Appendix A. Similarly, the spectral density of the response at m2 is Sq2(w)= S4111H112(412)2 + S4221H212(422)2 + fSaIH312($32)2

and the mean square of the response at this point is

The cross-spectral density between points 1 and 2 can be calculated using Sq12= 1 4 1 1 [ ~ 1 1 4 2 l ~

(7.59)

where B is taken from Eq. (7.54). This yields Sq12(0) = S411412411IHI I 2 + S4224224211H212+ S433 $32431

IH3

l2

and the expected value of yly2 is

The iterative procedure of solution is used again. The first step is to compute the relevant quantities of the linear system. The first three natural frequencies'' are ollln = 206.013 radls;

ozlin = 807.132 radls;

031,n = 1713.38 radls

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102

and the matrix of the normal modes is

The variances and the covariance of the linear response are, therefore,

The iterative process is solved on a personal computer using the TK+ Mathematical Solver program. Eigenvalues (natural frequencies) were solved using an explicit solution of a third power equation. If more than three DOFs are taken into account, a different algorithm for the eigenvalues and eigenvectors must be used. Other mathematical solvers may be used. Results are described in Fig. 7.4. The hardening effect is clearly seen by computing the linear and nonlinear frequencies and responses, as described in Tables 7.3 and 7.4, respectively. It can be seen that nonlinear analysis yields a response with rms values (standard deviation) that are less than 50% of the calculated linear analysis values. This justifies the claim that the use of nonlinear analysis is less conservative.

Table 7.3 Linear and nonlinear frequencies Frequencies, radls

w, =

Mode

1

2

3

linear

206.013 446.456 2.1671

807.132 930.289 1.1526

1713.38 1790.10 1.0448

a,,= nonlinear wndwi

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Iteration

Iteration

Fig. 7.4 Convergence a) of the frequencies and b) of the variances.

A designer should be aware of the importance of the rms of the response. The response of a system subjected to a stationary Gaussian excitation with zero mean is a stationary Gaussian process, with a mean of zero. Its standard deviation is the rms value. This implies that the response values are less than the rms value in 66.23% of time (la), less than twice the rms value for 95.45% of time (2a), and less than three times the rms value for 99.73% of time (3a). Thus, for the described example, the center of the beam (point 2) can vibrate with responses that are lower or equal to 3 X 0.3214 = 0.9642 cm for 99.73% of the time. The distribution of the maximum values of the response is treated more extensively in Chapter 12.

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1 04

Table 7.4 Linear and nonlinear rms values of the responses

Mean squares, cm2

Linear response Nonlinear response Ratio nonlinearllinear

0.24568 0.05262 0.2142

a:?

a:,?

0.49324 0.10330 0.2094

0.34690 0.07263 0.2094

Root mean squares, cm ~

Y

I

0.4957 0.2294 0.4628

0.7023 0.3214 0.4576

0.5890 0.2695 0.4576

V. Computational Procedure The iterative procedure described in the preceding chapters for the solution of the nonlinear behavior of a beam is based on approximating the continuous structural system by a discrete MDOF equivalent system. In doing so, the use of the finite element program is limited only to the steps reauired for the determination of the discrete elastic linear and nonlinear matrices of the structure and the relationship between these matrices and the variances and covariances of the response. Once these relationships are formulated, the iterative computational procedure does not include further use of a finite element code. This implies that the user selects a finite number of DOFs that participate in the analysis. The fewer DOFs selected, the easier and shorter the computational procedure. An efficient estimation of the required finite number of D O F ~ must be based on the user's experience and on intelligent estimations based on the particular structure analyzed. The flow chart in Fig. 7.5 describes the basic steps for a nonlinear analysis of a continuous structure, with cubic elastic nonlinearity.

VI. Calculation of Stress Response In Chapter 3, Sec. IV, it was shown that stresses in a vibrating structure can be solved using the stress modes approach. Once stress modes for a particular required stress are determined (analytically, numerically, or experimentally), the stress response of a nonlinear vibrating structure can be calculated by the same procedures used to determine the displacement response. During the development of the basic expressions for the displacement response, multiplication of certain matrices with L4,j and L4ilTwere performed twice. First, the spectral density of the excitation was premultiplied by L4il* and postmultiplied by L&l to obtain L s ~ ~This . was done to obtain a matrix of the spectral density of the generalized forces. Second, the matrix [B] was premultiplied by L4i] and postmultiplied by L4ilT.This was done to determine the spectral density of the displacement response at point i. When calculating the stress response, only the second step is performed with the stress modes instead of the normal modes. The first step, which

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STATISTICAL LINEARIZATION METHOD

- 1.

Determine the number of degrees-of-freedom to participate in the analysis -

+

I

2. Calculate, analytically or numerically, the linear elastic terms of the discrete equivalent system

I

1

-3. Calculate, analyticallyi or numerically, the non-linear elastic terms of the discrete equivalent system 4. If non-linear damping

* damping on velocity andlor dis lacement

y 5. Formulate

k? , c y , m y as function of D

1

0;by the given equations

I

I

6. Formulate solutions for Oi , [ ] and [MI 6a. Formulate solution for the stress modes, if

I

7. Defme the external power spectral density

1 8. Formulate expressions for the matrix S*;;

I

11. Calculate O iand [ 1. First iteration- linear system 1la. Calculate the stress mode, if required

1 I

I

12. Calculate SOijby the expressions of step 8

I next iteration 1

t 113. Calculate 0;withthe equations of step 9

(

I

14. Calculate new matrices for [k] [c] and [m]

1

Fig. 7.5 Computational procedure.

no

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determines the generalized spectrum of excitation, should be done with the normal modes of the system, and not the stress modes. Therefore, when applying the iterative procedure for a nonlinear stress response, both normal modes and stress modes must be recomputed in each iteration.

VII. Summary The statistical linearization method is presented. This method is based on the solution of a set of linear differential equations obtained by minimizing the difference between the original nonlinear equations and the equivalent equations. A Gaussian random excitation with zero mean is assumed. The statistical linearization method yields an iterative procedure for the computation of the response mean square values. This iterative procedure can be automated, and was demonstrated for SDOF, MDOF, and elastic systems. The demonstrated results confirm the conclusion that nonlinear analysis may result in a less conservative design.

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Chapter 8

Nondeterministic Structures: Basic Concepts I. Introduction

T

HE design of any engineering system is a process of decision making, under constraints of uncertainty. The uncertainty in the design process is due to the lack of deterministic knowledge of different physical parameters and the uncertainty in the models with which the design is performed. This is true for all disciplines such as electronics, mechanics, aerodynamics, and structures involved in any design. The uncertainty approach to the design of subsystems and complete systems was advanced by the concept of reliability. Systems are analyzed for possible failure processes and criteria, probability of occurrence, reliability of components, redundancy, possibilities of human errors in the production, and additional uncertainties. According to this type of approach, a total required reliability of a certain design is defined with proper reliability appropriation for subsytems. This required reliability certainly influences both the design cost and the product cost. Nevertheless, in most cases the structural analyst is still required to supply a design with absolute reliability, and most structural designs are performed using deterministic solutions. To compensate for the lack of knowledge, structural designers use a safety factor (lack of knowledge factor?), thus recognizing de facto the random character of many design parameters. During the last decade, the need for application of probabilistic methods for nondeterministic structures started to gain acceptance within the structural design community. Designers started to adopt the stochastic approach and the concepts of structural reliability. It is likely that by the end of the decade, this approach will dominate structural analysis procedures and, thus, the structural design will become more integrated in the total system design. The main sources of uncertainties in structural analysis and design are the model, the loads, and the uncertainties in various structural parameters. The first two were briefly discussed in the Preface. The treatment of uncertainties in structural parameters (stochastic structures) has become extensive and practical only in the last 10-20 years, although pioneering studies were published earlier, in the late 1960s and the early 1970s, especially by scientific journals of the civil engineering community. Tremendous progress has been made in the formulation of mathematical models and the establishment of several algorithms for the determination of the behavior of a stochastic structure submitted to excitation of stochastic loads. This progress was followed by the adoption of the relatively new technology for practical application in the structural design

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process in industry. It is likely that in the next decade more and more design codes and specifications will include the use of probabilistic analysis. The introduction of these methods into practical applications will be quicker than the introduction of finite element programs, because of the large infrastructure that exists in computational structural analysis and the rapid advances in computational power. For many years nondeterministic structures were tested using the Monte Carlo simulation. A problem is solved many times, each time with a set of deterministic values of the system parameters. These values are selected randomly from the legitimate space of the basic variables, including the approximate knowledge of the probability distribution functions of these variables. In some simulations, success of the system is detected, and in other simulations the system fails. The ratio between the number of failures to the total number of simulations is used to estimate the probability of failure of the system. The main disadvantage of the Monte Carlo simulation is that large numbers of deterministic simulations are required, especially in the tails of the distributions. As very low probabilities of failures are required in structural analysis, lo4-106 simulations are required for a typical problem. If one simulation run of a complex structure (using a finite element program, for example) is in the order of several minutes to hours, the use of the method becomes prohibitive for practical industrial applications as a result of the large computation time that interferes with the schedule of any project. In the last decade, numerical algorithms that use nonsimulative methods were developed. These resulted in several computer programs that solve the probabilistic structural analysis problem within a reasonable and practical time frame and that are suitable for industrial use. Some of these programs are briefly described in Chapter 9, Sec. IV. Some of the numerical examples, which are introduced later in this book, provide the reader with the basic concepts of probabilistic structural analysis and the methods that can be applied for solution of practical structural analysis problems. As most of the theoretical methods are covered extensively in several textbooks, e.g., Refs. 11 and 25-27, many conference proceedings and scientific papers, e.g., Refs. 28 and 29, only very basic mathematical evaluations are repeated in this text. The following simple example is introduced to clarify the difference between the traditional factor of safety approach and the concept of probabilistic analysis. Assume that an elastic bar of rectangular cross section A = b X h is subjected to a tensile force F, and assume that b, h, and F are normally distributed, with the following standard deviations:

where a() is the standard deviation of ( ) and ,u( ) is its mean.

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The stress in the bar is given by

and is approximately normally distributed, with a mean value psb= 2083 kg/cm2 (arbitrarily selected) and a standard deviation as = 104 kg/cm2. The maximum tensile stress in 3 a (three standard deviations) is

and in four standard deviations is

The bar is assumed to fail if the tensile stress in it is equal to or higher than the yield stress. Assume that the yield stress Sy of the bar material is normally distributed, with the following mean and standard deviation:

Thus, the minimal values of S, for three and four standard deviations are

Three types of designers who would treat this problem differently are identified. 1) The nominal designer calculates the safety factor (FS) of the bar from the nominal mean values

If this value is equal to or higher than the required FS for the specific project, the structure is approved. This approach does not take into account the dispersion expected during the production phase in the cross section dimensions b and h, the possible dispersion in the material property S, and the uncertainty in the external force F. 2) The 3a worst-case designer calculates the safety factor by

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G.MAYMON

which is not acceptable. Two kinds of design corrections are possible: a) Increase the cross section by 26% (1.210.95 = 1.26). This increases the weight by 26% and, subsequently, the product cost. b) Decrease the allowed tolerances on b and h (the designer cannot control the dispersion in F and S,). A simple calculation shows that to obtain FS = 1 (not the required 1.2), tolerances on the cross section should be decreased from 1%to 0.04%. This increases the product cost tremendously. A combination of a and b is also possible. A 4c~ worst-case designer faces much more difficult problems. 3) Assuming that the required reliability of this specific project is 99.9%, the probabilistic designer uses the probabilistic procedures described later to show that the probability of the stress in the bar to be equal to or lower than the yield stress is Pr(Sb I S,)

=

99.943%

(8.6)

This value is higher than required, and the designer is allowed to slightly decrease the cross section or slightly increase the required tolerances. Although this example is a simplification of real-life applications, it shows that contradictory measures can be taken when using each approach. It also indicates that the probabilistic approach may be more realistic and may result in less conservative designs. This approach is valid if project requirements are formulated by probabilistic methods. Although this is not the situation at present, it is believed that future requirements will contain more and more of this approach. 11. Failure Surface: Basic Case A failure surface g(R, S) is defined as a function of all random variables of the problem, so that on one side of this surface the structure is safe and on the other side it fails. The basic case includes two random variables. R is the allowable quantity (stress, displacement, etc.) in the structure and is designated the resistance term. S is the actual quantity in the structure and is designated the load effect term. Traditionally, a negative value of g(R, S) means failure and a positive value provides a safe region. The failure surface is then

This is a linear expression and for the basic case, the failure surface is a straight line, as shown in Fig. 8.1. A safety margin can be defined as

so that if M 5 0, the structure fails and if M > 0, the structure is safe. The failure surface is then

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NONDETERMINISTIC STRUCTURES: BASIC CONCEPTS

111

Fig. 8.1 Failure surface of the basic case.

Assume that R and S are normal random variables with E ( R ) and E(S) mean values and D ( R ) and D ( S ) standard deviations. Therefore, M is also random, with E ( M ) as its mean and D ( M ) as its standard deviation. Schematically, the failure and safe regions can be described on one-dimensional axis, as shown in Fig. 8.2. E ( M ) can be described in terms of the standard deviation D ( M ) ; thus,

where 0, is called the reliability index. The subscript c is after C0rnell,3~ who first formulated the problem in this form. Higher values of 0, describe systems with higher reliability, or systems with smaller probability of failure. Assume that R and S are normally distributed. Then M will also be normally distributed, with the probability density function &M), shown in Fig. 8.3. The shaded area is the probability of failure, the probability that M will be smaller than or equal to zero. The probability of failure is defined as

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Fig. 8.2 Fail and safe regions, basic case.

which can be written as

The first term in the brackets is the transformation of M into a standard normal distribution (mean equals zero, standard deviation equals one), which is denoted u

The second term in the brackets equals -6, [Eq. (3.4)]. Therefore,

where cP is the standard normal CDF. Note that the original failure functions (8.7) and (8.8) are linear. The result (8.13) is obtained because of this linearity. Thus, the probability of failure of the system can be calculated once the reliability index is known. Assume R is normally distributed having PDF &(r) with E(R) and D(R) as mean and standard deviations, respectively, and S is normally distributed

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M Fig. 8.3 Normal PDF of M.

&(s) with E ( S ) and D(S). The joint PDF is &S(r, s). The probability of failure is the volume of this function over the failure region, as shown in Fig. 8.4. Thus,

Assume that R and S are independent variables, then

Thus,

@R(s)

where DR is the CDF of r.

. &(s) ds

(8.16)

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Fig. 8.4 Basic case, two normally distributed RVs.

The integral (8.16) can only be solved explicitly in very few cases. For the two-variables basic cases presented, it can be shown that Pf can be calculated using Eq. (8.13), where

The random variables R, and S can be transformed into standard normal variables U R , and us, respectively, using

Thus,

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115

and the failure function g(R, S) can be transformed into another function G(uR, us) to yield

This function is described in Fig. 8.5. It can be easily shown (for instance, by trigonometric relations of lines in Fig. 8.5) that the reliability index P is equal to the minimum distance between the failure surface and the origin in the transformed variables system. The point on G that has minimum distance to the origin is called the design point, or the point of most probable failure, and is denoted by asterisks, u i and u:. The term design point, which is used extensively in the relevant literature, is misleading; it is somewhat incongurous to call the point of most probable failure the design point. The preceding analysis can be extended to n random variables. The failure surface is then a function of X, variables

and the transformed failure surface is

Both functions describe a hypersurface of n dimensions. The exact probability of failure is obtained by extending Eq. (8.16) to

,,..., x, (xl, x2,... ,x,) is the joint PDF of n random variables. where The reliability index is the smallest distance from the origin of the transformed (standard normal) space to the transformed hypersurface G(ul, 242, . . . , u,) = 0. If the failure surface is linear, it forms a hyperplane of n dimensions, both in the original and in the transformed spaces. If the failure function is nonlinear, a hypersurface is obtained and P is the distance to a hyperplane tangent to this hypersurface at the design point, the point closest to the origin of the transformed space. It has already been shown that instead of performing the integral (8.22) of the joint PDF over the failure region G 5 0, the reliability index P is calculated by determining the closest distance of the transformed surface to the origin, and by using Eq. (9.13) to calculate the probability of failure. This is based on the replacement of the failure hyperspace with a tangent hyperplane. Thus, Eq. (8.13) is equivalent to performing the integration of the joint PDF over the space marked first-order reliability method (FORM) in Fig. 8.6. If the failure surface is nonlinear, the integration should be

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SAFE

FAIL

Fig. 8.5 Transformation into standard normal space, a) physical space and b) standard normal space.

done over the space EXACT in the figure. In second-order reliability methods (SORM), the hyperspace is replaced by a quadratic hypersurface, and the integration should be done over the space SORM in the figure. The difference between FORM and SORM results depends on the nonlinearity of the failure surface. In most practical cases, in structural analysis, very low probability of failure should be allowed; thus, the design point D is in the tail of the joint PDF. The main contribution to Pf,therefore, is

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NONDETERMINISTIC STRUCTURES: BASIC CONCEPTS

' 'second

Order Curve

117

Design Point

Limit Function Fig. 8.6 Exact, FORM, and SORM integration spaces.

from the area above the tangent line and, thus, FORM methods give usually good approximations of the probability of failure. Expressions relating Pf and 0 for SORM were also developed and presented in the literature. These expressions contain radii of curvature of the failure function. 111. Reliability Index

In 1969, Cornel13' suggested the following formulation for the reliability index:

where pz is the value of the failure function g(Xl, X2,. . . , X,) at the mean values of all of the random variables; thus,

and

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where the derivative of g with respect to the variables Xi are taken at the mean point pi.These definitions are extensions of the basic definitions in Eq. (8.10). A major disadvantage of the Cornell reliability index is that it is not invariant to the selection of the failure surface, when several possibilities for formulations of this function are possible for the same failure criterion. This was demonstrated in Ref. 25 for the following example. Assume a bar of cross section A on which a tensile force F is acting. Assume that the allowable stress in the bar is R. Two failure functions can then be formulated.

Suppose that R has a mean pR = 62 and standard deviation vR = 6.2, and A has a mean pA = 2.8 and standard devation a, = 0.14. Also assume that F = 100 is a deterministic force. In Table 8.1, Eqs. (8.23-8.25) are applied to both gl and g,. It can be seen that the reliability index using these different functions is not identical. In 1974, Hasofer and Lind31 suggested a method to overcome the disadvantage presented by the Cornell reliability index. The random variables Xi are transformed into standard normal variables ui by

The failure function g(X) = 0 is transformed into G(u) = 0, and the smallest distance between this surface and the origin is found

pHL= minimum

Jz

u:;

given G(ui) = 0

Both Cornell and Hasofer-Lind methods use the first and second moments of the distribution of the random variables, but do not take into account the real distribution of these variables. As the probability of failure of a practical structure should be small, this happens in the tail of the distribution functions. Thus, this method yields good results when the PDF of the random variables are normal or close to normal at the tail. In 1978, Rackwitz and Fiessler3' extended the Hasofer-Lind concept to include, approximately, the PDF of the basic random variables. According to this method, nonnormal variables are transformed into normal variables in such a way that the PDF and the CDF of the original and transformed variables are equal at the design point. As this point is not known in advance, an iterative procedure is used to calculate the reliability index.

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119

Table 8.1 Differential reliability index for different equivalent failure functions Case 1 -

g, = R

g PZ

cz P C

Case 2

-

-

gz = RA

(FIA)

F

-

F

PRPA- F

PR- PA

=

62 . 2.8

-

100 = 73.60

100 = 62 - -= 26.286 2.8 .\/a; + ( F l d ) ' . u, = 6.452

dpic i + pi 0;= 19.41

26'286 - 4.07 --

-73.6 - 3.79

19.41

6.452

The algorithm which is used in the Rackwitz-Fiessler method is as follows: 1) Define the basic physical random variables Xi(i = 1, 2, . .. , n), the cumulative distribution F, and the PDF f;: of each variable. Each variable has a mean piand a standard deviation ai. Define the failure surface g(Xi) = 0 of the problem. 2) Transform each variable Xito a standard normal variable ui using

and define the transformed failure function

3) Find the minimum distance between the surface g, and the origin /3 = minimum d u :

+ ut + . . . + uz;

given gl = 0

(8.31)

This solution defines the first design point u?,u;, . .. ,u,*. 4) Calculate XF = uTui + pi. 5 ) Calculate the mean value and the standard deviation of an equivalent normal distribution of each basic variable that is not normal, using

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120

where @ is the standard normal CDF @(z) = 0.5 + erf(z) and

4 is the standard normal PDF +(z)

1

= -exp

6

(-i

z')

6) Define new standard normal variables

(8.35) 7) Calculate the reliability index Dl as

pl = minimum VET+ E$ + . . . + E;;

given g1(E?) = 0

(8.36)

5 8) Repeat steps 5-7 until a convergence criterion, such as is obtained. The last value /3 is the calculated reliability index. Experience shows that convergence is obtained quite quickly. An example of this type of iterative process is demonstrated in Chapter 9. A schematic description of the method is in Fig. 8.7. In 1981 Chen and Lind33 extended the Rackwitz-Fiessler method by equating the slope of the PDF and the transformed standard normal PDF at the design point. In many cases this method gives better results for the reliability index. In 1981 Hoenbichler and rack wit^^^ suggested the use of the Rosenblatt transformation in the analysis of stochastic structures. According to this method, each random variable can be transformed into a standard normal variable. This is also done to dependent variables, which are transformed into a system of independent standard normal variables. Therefore, a system that has had many random variables of arbitrary PDFs and interdependence of variables is transformed into standard normal independent variables. The transformed system is solved by determining the design point as the point closest to the origin in this independent variables space, and the basic variables are then calculated by inverse transformation. This is the method used today in most of the computational tools that exist for the solution of stochastic structures. When all of the basic variables Xiare mutually independent with marginal CDFs Fxr(x,), the transformation is defined by

E,

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121

(1) Define all physical random

variables and their CDF, PDF, means and standard deviations.

r(2) Transform each physical variable X into u-space using (8.29), and

prepare a failure function in the

? (4) Calculate physical "design point"

(6),(7) With results (4) and ( S ) ,

do another iteration (equivalent

(5) Calculate new means and standard

Fig. 8.7 Rackwitz-Fiessler iterative process.

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where @ is the standard distribution. This means that

and the inverse transformation

When the basic variables are not mutually independent, the first variable is transformed by using

The second is transformed by

where FX2(x2jx1)is the distribution function of X2conditioned upon Xl= xl,

Expressions for CDFs can be found in the literature. Several examples can be found in Ref. 25.

IV. Summary The rationale behind the probabilistic treatment of structures is described and discussed. It can be seen that different design decisions can be obtained using the traditional safety factors and the probabilistic approaches. The basic case of failure surface is formulated, defining resistance terms and load effect terms in the failure function. First-order and second-order reliability methods are described. The reliability index, which is a measure of the structural reliability, is defined. Three different definitions, as suggeted by different authors, are described and discussed.

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Chapter 9

Calculation of the Probability of Failure I. Introduction

T

HERE are several methods by which practical calculation of the probability of failure can be performed. The simplest method is an analytical solution of the problem. An analytical solution is possible only when the failure function can be described by a closed-form expression and is practical only when a relatively small number of random variables are involved. In such a case a set of algebraic andlor transcendental equations can be derived and solved. Sometimes the solution process itself requires a numerical computer program, and thus it is a semianalytical solution. When the PDF functions of the random variables are normal, the Hasofer-Lind method 31 can be used. When nonnormal distributions exist, the RackwitzFiessler iterative method32 is a better alternative. When the number of random variables is large (say, more than 5-6), the analytical or semianalytical solutions tend to be cumbersome. This is also the case when correlations between variables exist. In this case it is recommended to use one of the commercially available special-purpose probabilistic computer programs, four of which are described later in this chapter. When the failure function cannot be written explicitly, both analytical and existing computer programs fail to provide a practical solution. Unfortunately, these are the most common cases. Practically the load effects term S in the failure function (8.7) cannot be expressed explicitly for a practical structure and is usually calculated using a finite element computer code. The problem of incoporating the finite element codes and the probabilistic programs will certainly be solved in the near future by interfacing these two kinds of programs. Meanwhile, approximate explicit failure functions can be derived using, for example, the Taylor series expansion method. FORM solutions can be derived using the modified joint PDF described in Chapter 11. The recommended methods of solution are described in Fig. 9.1. Also chapter numbers, which describe each method shown in the figure, are listed. All of the commercially available probabilistic codes provide the possibility to calculate not only the probability of failure of a component, which includes several random variables, but also probabilities related to combinations of components into systems, where components in series and in parallel exist. This subject is not evaluated here, and the interested reader can find more details in related references, e.g., Refs. 25-27.

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I

Closed-form failure function

I

yes I

i t

small number of

large number

variables,

of variables,

no correlations

including correlations

distributions

distributions

Multiplier

sler Iterations

Method

solution

solution

distribution

+ I

large number of

variables, including correlations

El distribution

i Commercially

Taylor series

Joint

available

expansion to

Modified

computer

provide approx.

Probability

programs

closed-form

Density

expression

Function

SORM &

I

solution

Method

t

solution

Simulations

Fig. 9.1 Methods of solution for the probability of failure.

11. Lagrange Multiplier Method When the failure function is given by a closed-form expression, it is transformed into a function of standard normal and mutually independent variables. The minimum distance from the origin to the transformed failure surface can be calculated using the Lagrange multiplier procedure. The failure surface is

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CALCULATION OF THE PROBABILITY OF FAILURE

the distance between a point u l , u2, . . . , u, to the origin is

The following expression is constructed:

D

=

P2 - A . g ( u l , u 2,..., u,)

where A is a Lagrange multiplier. Equation (9.1) plus the following ( n ) equations

provide n + 1 equations for n values of u: plus the value of A. As g(ui) is not necessarily linear, the n + 1 algebraic equations for the n + 1 unknowns are also not necessarily linear. A general closed-form solution is not possible but many numerical tools can be used to solve this set of equations. Once the values of u* are known, the minimum value of @ is obtained by Eq. (9.2). Also the values of the basic random variables at the design point can be obtained by inverse transformation of uT. Some simple examples demonstrate the use of the Lagrange multiplier method for determination of the design point and the reliability index.

Example 9.1 A bar of random cross section A and a random yield stress S, is subjected to a random tensile force F. Mean values, standard deviations, and PDFs of these parameters are tabulated in Table 9.1. The failure surface is given by

Transforming the basic variables into standard normal variables using

Table 9.1 Data for Example 9.1 Parameter

Units

Distribution

Mean, p

Standard deviation, u

SY

kg/cm2 kg cm2

Normal Normal Normal

2500 4166 2

75 125 0.04

F A

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yields the failure function in the u space

Equation (9.3) takes the form

Equation (9.4) yields

Equations (9.9) and (9.7) are solved for u* to yield

Using Eq. (9.2) yields

The results in Eq. (9.10) indicate that the most probable point of failure is the combination when the yield stress is 2.636963 standard deviations less than the mean, the acting force is 2.273637 standard deviations more than the mean, and the cross-sectional area is 1.675018 standard deviations less than the mean. Using Eq. (9.6) for inverse transformation, the physical quantities at the design point are

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CALCULATION OF THE PROBABILITY OF FAILURE

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

127

0.08

Standard Deviation of Area Fig. 9.2 Reliability as a function of the standard deviation of the cross section.

Using first-order approximation, the probability of failure is

Thus, the reliability is

For parametrically varied standard deviation of the cross section, the reliability of the structure is calculated by repeating the process just described. The results are shown in Fig. 9.2. Thus, if a reliability of 99.9% is required, a value of 0.078 cm2 for the standard deviation can be selected, a value that is higher than the original design (Table 9.1), and the tolerances on the cross-sectional area can be released.

Example 9.2 Assume that the acting force F of example 1 has a uniform probability density between 3791 and 4541 kg. The first and the second moments are, therefore,

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The cumulative distribution function of F is

F(F) = 0;

elsewhere

Use of the transformation Eq. (8.37) yields

u2 = @-'[F(F)]

(9.17)

that can also be written as @ ( ~ 2 )=

0.5

F(F)

+ erf(u2) = 1/750[F- 37911;

3791 5 F 5 4541

(9.18)

Therefore,

F = 750[5.0546667 + erf(u2)]

(9.19)

As S, and A in this example remain normally distributed with the same data given in Table 9.1, the failure function in the u space is

and D is [from Eq. (9.3)]

The derivatives of D with respect to uiyield three equations

In the second equation of (9.22), the following relation was used

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129

A solution of Eqs. (9.20) and (9.22) yields

Pf(first order) = 0.04%

This reliability is smaller than that of example 1, due to the greater standard deviation of the acting force and the greater probability to obtain higher values of the force. Example 9.3 Assume that the yield stress of the material in example 1 has a Weibull distribution with mean and standard deviation equal to those given in Table 9.1;

The Weibull distribution has the following CDF: F(S,)

=

1 - exp(-as$)

where a and j3 are constants, related to the first and second moments by

where r is the gamma function. Using Eq. (9.27) with the given moments Eq. (9.25) yields a = 8.18211 X

j3 = 42.03862

(9.28)

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Using the transformation ~1

=

@-'[F(Sy)]

@(ul) = 1 - exp(-as$) 0.5 + erf(ul) = 1 - exp(-as$)

Therefore, the failure surface is

and

Differentiation with respect to ui yields

Solution of (9.30) and (9.32) yields

U:

=

-0.7743742

A*

=

1.969025 crn

Pf(first order) = 0.075% P, (first order) = 99.25%

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131

111. Demonstration of the Iterative Process In Chapter 8 Sec. I11 a solution process suggested by Rackwitz and F i e ~ s l e r ~ ~ is described. This process takes into account the true PDF of the nonnormal random variables and provides an iterative process that may be simpler for solution than the Lagrange multiplier method demonstrated in the preceding section. In example 4 the iterative process is demonstrated. Example 9.4 Example 3 in the preceding section is solved using the RackwitzF i e ~ s l e iterative r~~ method. The steps taken are those described in Fig. 8.7. 1 ) The yield stress has a Weibull distribution given by Eqs. (9.25), (9.26), and (9.28). The force F and the cross section A are identical to those given in Table 9.1. 2 ) For the first iteration, the three physical variables Sy, F, and A are transformed into ul, u2, and u3, respectively, as if they are all normal, using

The failure function g(ui) is, therefore,

3 ) Solving the algebraic equation obtained by using Eqs. (9.3), (9.4), and (9.35) the following is obtained:

The value of fl is identical to that of example 1, where normal distributions were assumed for all three variables. 4 ) Use of Eq. (9.34) yields

5) Using Eq. (8.32), new values are obtained for pll and ull, i.e., for the first variable S,, which is the only nonnormal variable in this example,

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Table 9.2 Iterative converging process for Example 9.4 Interaction

pl,

~1

l

s,*

F*

A*

P

6) These values are introduced into Eq. (9.34) instead of psy and gSy. 7 ) A new /3 is computed. 8) The iterative process continues until convergence is obtained. In Table 9.2, five iterations are listed. The convergence criterion is determined as 16, 5 0.001 and, therefore.

This is equal to the results obtained for example 3 for the reliability index and also for the three basic variables S,, F, and A at the design point.

IV. Numerical Programs for Probabilistic Structural Analysis The numerical examples presented in the preceding section were solved analytically for the first-order probability of failure. They were characterized by the small number of random variables and by the existence of a closed-form expression for the failure function. This solution processes may give the designer an insight into the effects of different random variables and their distribution functions. For a large number of random variables, the described method of solution is cumbersome. In many practical cases, the randomness of some parameters can be expressed only as random fields rather than random variables. The physical parameters may also be interdependent so that the analytical methods fail to provide a reasonably fast solution. In these cases more sophisticated methods of solution may be required. In many practical cases it is also important to calculate the sensitivity of the reliability index and the probability of failure to small changes in statistical moments of the random parameters. By calculating these sensitivities, the designer obtains a quantitative measure of the effect of mean values (basic design) and standard deviations (tolerances) on the reliability of the structure. This can focus the designer's attention on the important variables that should be treated in the design process. Therefore, a complete probabilistic analysis of a structure should focus not only on the reliability index, but also on sensitivities and importance factors, an analysis that may enable the designer to efficiently modify the design.

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Several methods and algorithms that were suggested for a comprehensive analysis of stochastic structures can be found in the relevant literature, e.g., Refs. 25-27. Based on these algorithms, several computer programs were developed; some were prepared in the academic institutions for local research and others by research and development institutes and companies, as a part of practical research efforts intended to develop design tools for the engineering community. Some of the developed computer codes are already commercially available, and a few will be described in brief in this section. It should be emphasized that this section is not a complete survey of available computer programs in this field. The most complete commercially available program for the analysis of probabilistic structures is PROBAN,35developed in Norway by Det Norske Veritas, in cooperation with academic institutions in Norway and Denmark. Version 3 of the program has been available since 1994.This is an interactive program, suitable for many computer and workstation environments, used for probabilistic structural analysis by means of FORM, SORM, and several simulation methods, including Monte Carlo. Systems in series and in parallel, as well as basic components, can be solved. The user must provide a failure function by using many library functions as well as an option to program a user's function and link it to the program. Thus, a compiler and linker program should be available on the users network. Templates for the user limit functions are included in the program. Sensitivities and importance factors are calculated. Output files and graphic displays can be printed and plotted on standard peripheral equipment. A large variety of distribution functions is included in a library, and user-defined distributions can also be treated. Random fields can be included after performing a proper discretization into random variables. Correlations between various variables defined by the user are also treated and dynamic problems solved. The main practical disadvantage of this program is its price, more than $30,000. More information on PROBAN can be obtained by writing Det Norske Veritas, Hovik, Norway. Another commercially available program is the developed in Canada by Martec, Ltd. The main features of this program are similar to those of the PROBAN. The data should be prepared on an input file, which takes considerable time with an inexperienced user. An interactive data input is available, but it is not very simple to use. The developer also has an easy interface with a commercial finite element program and has stated that an easy interface to any other finite element program can be provided. This program demands a specific compiler (Lahey Computer Systems, Inc.). Templates for user limited function are provided. The cost of the program is $3500-$5000, and it can be installed on a personal computer. A Windows 95 compatible version is being prepared at this writing. More information can be obtained from Martec, Ltd., Halifax, Nova Scotia, Canada. A program, which can be commercially purchased, the CALREL,37was developed in the University of California at Berkeley, California. This program is not interactive, and introduction of data is somewhat cumbersome. The user has to program the failure function using Fortran language

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and, therefore, a Microsoft Fortran compiler and linker are required. No graphic output is available in the 1993 version. The program serves as a tool in the research programs at the University of California, and in this context an interface between the program and a local finite element program FEAP was reported. This allows the reliability analysis of structures for which the load effects term has no closed-form expression and can be treated only by a finite element program. The finite element program is not one of the codes frequently used by the industry. The price of the CALREL is approximately $1000. More information can be obtained from the Earthquake Engineering Research Center, University of California, Berkeley, California. A large-scale development effort is being made in the United States in the NESSUSIPSAM program. This effort is led by NASA Lewis Research Center, in cooperation with the Southwest Research Institute in San Antonio, Texas, and several universities. The program3' is reported to include a finite element module as a part of the probabilistic program itself. This, paradoxically, may be a disadvantage because most practical users in the industry use one of the commercially available finite element codes such as NASTRAN, ANSYS, and ADINA, and these cannot be changed easily. The NESSUS program (and some derivatives that are suitable for personal computers) is now available for lease for an annual fee by Southwest Research Institute. More information can be obtained from Southwest Research Institute, San Antonio, Texas. There are some other probabilistic structural analysis programs reported in the literature, most of which were developed in the academic institutions by researchers who have a need for them in their research efforts. Main researchers in this field work in Austria, Germany, Italy, and Japan.

V. Summary Several methods of calculation of the probability of failure of structures are described. A Lagrange multiplier method that enables analytical calculation of the first-order probability of failure is demonstrated with three examples. The Rackwitz-Fiessler3* iteration method is demonstrated with one example. Several commercially available computer programs for probabilistic analysis of structures are presented and discussed.

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Chapter 10

Taylor Series Expansion of the Failure Surface I. Introduction

A

S EXPLAINED in the preceding chapters, no closed-form expression is available for the failure surface of most practical cases treated in the realistic design process. Thus, the commercially available probabilistic codes cannot be used directly. A finite element code can be interfaced with a probabilistic program but this type of interface is usually not practical for the user in the industry, where finite element commercial codes are usually rented and treated as black boxes. Although the interfaces could be developed by the finite element developers, the present market demand for such interfacing does not seem to warrant the effort. With the growing interest in probabilistic analysis of structures, it is likely that in the future such efforts will be economically justified and will, therefore, be initiated. Meanwhile, the structural analyst in the industry is faced with the lack of practical tools when this type of analysis is required. One of the methods available to solve the present problem is to use an approximate closed-form expression for the failure surface, based on a Taylor series expansion of the unknown failure f ~ n c t i o n . ~ The ~ - ~ coeffil cients of this expansion can be be determined by using the commercially available finite element programs for several deterministic cases. 11. Taylor Series Expansion The load term S(Xl, X2, . . . , XI) of the I random variables can be evaluated into a Taylor series around an evaluation point (EP) as follows:

I (+C ax; E ,

( x i - XEpr)2

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where SR are terms of higher order. If the evaluation is done in a range close to the EP, the terms (Xi - XEPi)are small and SR is of third and higher order of this small difference and can be neglected. Simplification of Eq. (10.1) yields

xx I-I

+

I

(Xi - XEP,). (Xj - XEP,)

a ( ~ - i / , ) ( i + ~ ) '+ j

i=l j=i+l

+

The total number of terms in Eq. (10.2) is k + 1, where k = Z(I 3)/2, and k + 1 = ( I + 2)(1 + 1)/2. It was found4' in several numerical solutions that mixed terms can also be neglected in many practical cases. For these cases, the number of terms in Eq. (10.2) is (21 + 1). Further simplification of Eq. (10.2) yields

where

Yn = (Xi - X E P , ) ~ ;

n = I + l , I + 2 ,..., 21

i = 1 , 2,...,I

and a. is associated with the EP itself. A total of m deterministic solutions can be found for S with a finite element program, where for each case the XEPiis increased or decreased by AXi. These solutions can be represented by the vector { S }

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where So is the deterministic solution for the EP. In a matrix form, the m solutions for {S}can be written as

where the matrix [Y] has m rows and k + 1 columns, which are associated with the quantities AXi around the EP. Thus, the first column of [Y] consists of ones, and the first row (less the first term) are all zeros, as this row refers to the EP itself. ~ x ~ l i c i t l ;[Y] , is

The optimal set of coefficients for Eq. (10.5a) (based on the minimum leastsquare error) is given by solving42

which is a set of linear equations for the coefficients {a}. 111. Selection of the Evaluation Point

In Eq. (10.1), the load effects term S was expanded into a Taylor series around the EPs {Xi} = {XEP,}, which were selected arbitrarily. Expansion around the design point (the most probable point of failure) is required to obtain a good approximation of the load effects term. The design point is not known in advance, and the solution of the required probabilistic problem is needed to find this point. In an iterative process, the probabilistic problem is solved with a set of Taylor series expansions obtained using Eq. (10.6),

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a new EP is selected, and the process is repeated. Completion of the process results in a condition where the design point is very close to the selected EP, or at least is within the selected range of the Taylor series expansion. The best selected EP is the design point, which is not known in advance. Therefore, the first selected EP should be chosen with care, applying engineering considerations, some experience, and some intuition. Usually the influence of most random variables on the load effects term can be predicted. In a structure that can be analyzed only numerically, the structure input parameters necessary for the numerical solution (such as loads, local and overall dimensions, material moduli, and other structural characteristics) form a list of variables on which the load effects term depends. The designer must decide which of the parameters in this list can be considered as random, according to the given case. Each of the random variables is associated with a PDF, a mean value, and a standard deviation. These can be taken from previous data or assumed. Generally, the designer can predict the general trend of influence of a given variable on the load effects term, for example, an increase of external loads results in an increase in stresses and an increase in thickness decreases the stresses and deflections. Thus, the basic random variables can be divided into nominator type and denominator type. In cases where the influence of a certain variable is not clear, it can be clarified by a few deterministic calculations. In the design point, nominator type variables take values above the mean; thus, the EP for these variables should be the mean plus a certain small difference. Denominator type variables take values smaller than the mean; thus, the EP for them should be the mean minus a certain small difference. In most practical cases, the standard deviation of the variable is small, and a good practical choice for the difference is the standard deviation; thus,

XEP;= pi + Ui;

nominator type variables

XEP~ = pi - ffi;

denominator type variables

(10.7)

The EP range should run from the mean value upward for nominator type variables and from the mean value downward for denominator type variables. A reasonable choice is

pi

-

+

2ai = XEpi- ui 5 XEP,5 XEPi ui = pi;

denominator type (10.8)

Thus, an evaluation range of u is obtained for each variable above or below the EP. It is convenient to select the rn deterministic solutions by which the vector {S) and the matrix [Y] are generated, using the following method:

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1) The first deterministic solution is at the EP itself. Thus, the terms of the first column of [Y] are ones, and the k terms in the first row are zeros. 2) Each random variable is changed by +AX and by -AX, keeping all other variables at the EP. In this case, many terms in [Y] are zeros. This procedure is called a two-point evaluation range. The form of the matrix [Y] obtained using this method is described in Eq. (10.5b). Once the coefficients { a } are determined with Eq. (10.6), the accuracy of the approximated S function relative to the exact function can be determined by using Eq. (10.3) and comparing the results to the vector {S} obtained by the deterministic finite element solution. It has been suggested39 that the existence of an extremum of S as a function of Xi can be checked. Thus X,, is defined by --

0

ax,

If X,, is within the evaluation range, the accuracy of the approximated S (say, S1 in Fig. 10.1) is lower than that of another approximation (say, S 2 ) whose extremum is outside the evaluation range as the effect of a random variable on the load effects is monotonous. This check can be performed before probabilistic computations are made. The EP and evaluation range can then be changed before these computations, saving analysis time. Another procedure that yields more accurate results for the approximated S function is to evaluate the {a}coefficients using a large number of deterministic solutions. Instead of evaluating each random variable in two points, +AX and -AX, an evaluation can be made for four points, - 2 M , -AX, +AX, and +2AX. Although this increases the number of

A :, True 2

EP-xi

8

EP

L Xici-

'

EP+xi

Fig. 10.1 S with and without extremum within evaluation range.

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required deterministic solutions to 41 + 1, it significantly improves the quality of the approximated S function. This procedure is called four-point evaluation method. Once an approximate closed-form expression is obtained for the load term, the failure function can be approximated by the closed-form expression

where X F is the resistance R random variable and the two last terms represent the approximated S term. It should be noted that the resistance term does not always depend on one parameter only. A more complex resistance term can be applied for certain cases, according to the selected failure criteria. An example of this type of expression is a von Mises formulation for an allowable stress envelope that connects failure stresses that may vary for different directions. It should be emphasized that neither deterministic nor probabilistic analysis can be performed if failure criteria are not formulated for the case in hand. This formulation depends significantly on both personal and organizational experience. Once the approximate expression for the failure surface is formulated, calculation of the probability of failure for each structural component can be performed analytically (for simple cases) or numerically. The reliability index can be found and the probability of failure determined using FORM or SORM methods. Figure 10.2 presents a computational flow chart of the Taylor series expansion and probabilistic solution.

IV. Detailed Examples of the Taylor Series Expansion Method Example 10.1 A cantilever beam of length L, rectangular cross section of width b, and the height h is loaded by a force per unit length q. The selected failure criterion is the beginning of the yield at the clamped end. The yield stress of the material is S,. In Table 10.1, data of the PDFs, means, and standard deviations of the random variables are shown. The nominal stress of the beam (calculated with the mean values of the parameters) is 3105 kg/cm2 and the traditional safety factor is 1.159. For this structural component, the bending stress at the clamped end (the load effects term in the failure function) is given analytically by

The resistance term is

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TAYLOR SERIES EXPANSION OF THE FAILURE SURFACE

I

Define the structural

141

)

system and components

i

I

I

1

+ +

For each component

4

Form a List of variables

Decide what are the

influencing the suucture

failure criteria I

I

F l Decide on the random

1 variables of the load term I

I

I

random variables of

densities, means and

the resis

Determine nominator and

I L-

denominator variables

4

Select evaluation point and range for the expansion

I

I II

,Prepare a table of deterministic cases Check Xcr and

Solve these cases

change EP and

with an FE

range, if required

t

L

I

Solve for hevector la)

1

-

-

Perform a second iteration,

7

I

1

I I Use avadable probabilistic I cd-;

(if required)

Results

Fig. 10.2 Flow chart of the computational procedure.

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Table 10.1 Data for Example 10.1 Physical variable

x,

4

XI

L b h

x2

x3

s~

xs

x 4

Mean,

Standard deviation,

Units

Distribution

PCL,

cl

kglcm cm cm cm kg/cm2

Normal Normal Normal Normal Normal

1.15 60 4 1 3600

0.0333 0.6 0.12 0.03 300

and the failure surface is

The problem was solved using four methods: 1) FORM solution of the closed-form expression (10.13) using the Lagrange multiplier method; 2) FORM, SORM, and Monte-Carlo solutions of the closed-form expression (10.13) using the PROBAN program; 3) expansion of the failure surface into a Taylor series and a FORM solution obtained by this approximate expression using the Lagrange multiplier method; and 4) introduction of the Taylor series approximate expression into the PROBAN and performing FORM and SORM, as well as Monte-Carlo solutions. Solution I : Transforming all of the basic random variables into standard normal variables is performed using Eq. (8.27) and, therefore,

The failure surface in the standard normal space is

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Creating the variable D according to Eq. (9.3) and using Eq. (9.4) yields five equations. In addition to Eq. (10.15), these equations are solved for the design point in the transformed space. Equation (10.13) is used to obtain the design point in the basic variables space, and Eq. (9.2) is used to calculate the reliability index. The results are

Solution 2: The following results were obtained using the PROBAN program with the failure surface (10.15)

Using 5000 Monte Carlo simulations yields

PMonte Carlo = 1.300; 1.261-1 .MI, 90% confidence interval PfMonte

Carlo

= 0.0968; 0.08992-0.1037,90%

confidence interval

(10.17b)

Solution 3: The variables q and L are nominator type; b and h are denominator type. The EP was selected as the mean plus or minus halfstandard deviation, as described in Table 10.2. The number of required deterministic solutions is 9 (2 X 4 variables + 1). These are listed in Table 10.3. The solutions were performed with the closed-form expression (10.11), which simulates for this example the finite

Table 10.2 Selection of EP, Example 10.1

EP

A x

EP - AXi

EP

+ AX,

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G.MAYMON Table 10.3 Nine deterministic solutions, Example 10.1

1 2 3 4 5 6 7 8 9

4

L

b

h

Sbend

1.16665 1.15 1.1833 1.16665 1.16665 1.16665 1.16665 1.16665 1.16665

60.3 60.3 60.3 60.0 60.6 60.3 60.3 60.3 60.3

3.94 3.94 3.94 3.94 3.94 3.88 4.00 3.94 3.94

0.985 0.985 0.985 0.985 0.985 0.985 0.985 0.970 1.000

3329.1072 3281.5954 3376.6190 3296.0642 3362.3151 3380.5883 3279.1706 3432.8654 3229.9830

element solution. The nine simulations are also presented in the last column of Table 10.3. The [qmatrix corresponding to the cases in Table 10.3 is

Solution of Eq. (10.6) yields the following values for {a): a0 =

a1 =

3332.4332

2853.561562;

as = -21635.79

a2 = 110.4182;

a6 = -36.03913

a3 = -84.51475;

a7 =

a4= -6762.7467;

as = -0.0004484

-709.367

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Table 10.4 A new selection of EP, Example 10.1

EP

Ax,

EP-AX,

EP+AX,

Introducing these values into Eq. (10.3) and solving the equations for the critical values using Eq. (10.9) yields

q,,

=

b,,

= 3.88;

1.2326;

LC,= 61.832 h,,

=

-7.5 x lo6

(10.20)

The critical value of b is on the edge of the evaluation range and, therefore, a new EP is selected for the width variable. This new EP and range are summarized in Table 10.4. The nine new deterministic cases to be solved, together with the calculated values of the vector Sbend, are shown in Table 10.5. The corresponding [Y] is now

The solution for the values of {a}are a.

=

al = 2897.69069;

3380.6048 a5 = -81.82272

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Table 10.5 Nine new deterministic solutions, Example 10.1

L

9

b

h

Sbend

and the critical values calculated using Eqs. (10.3) and (10.19) are q,,

=

18.873;

b,,

= 5.448;

LC,= -14.9369 h,,

=

1.3307

(10.23)

which are all well outside the evaluation range. Using Eq. (10.22) in Eq. (10.10), the Lagrange multiplier method, the design point, the reliability index, and the probability of failure were calculated. The results are

Solution 4: The Taylor series for S [Eq. (10.3)] with the coefficients of Eq. (10.22) were introduced into the PROBAN program. The results are

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TAYLOR SERIES EXPANSION OF THE FAILURE SURFACE

Table 10.6 Comparison of results for different solution methods, Example 10.1 Method

P

Analysis

pf

-

1) Lagrange multiplier, Eq. (10.13) 2) PROBAN, Eq. (10.13)

3) Lagrange multiplier, Taylor expansion Eq. (10.10) 4) PROBAN, Taylor expansion Eq. (10.10)

FORM FORM SORM Monte Carlo FORM FORM SORM Monte Carlo

and using 10,000 Monte Carlo simulations

PMonte PfMonte

Carlo

=

1.237; 1.210-1.265,90% confidence interval

= 0.1081; 0.1030-0.1132,90%

confidence interval (10.25b)

The results of all calculation methods are tabulated in Table 10.6.

Example 10.2 A simple structural component that does not have a closed-form failure function is treated in this example. The model, described in Fig. 10.3, is a cantilever beam built of 26 E-glass/epoxy plies. The thickness of each ply is h. They are all placed at an angle 8 to the main direction of the beam and loaded by a uniform pressure p, which is random in its value but not in its uniformity. It is assumed, for simplicity and without loss of generality, that all plies are at one angle 9 and that the randomness is only between different specimens. The length of the beam is L and its width is b.

Fig. 10.3 Cantilever beam model.

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.

The deterministic analysis of this model is solved with the ANSYS finite element code, using the STIFF99 element, which is an eight-node layered shell element. The data required to calculate the stresses of the deterministic case (the input data to ANSYS) is as follows: p = external applied pressure h = thickness of one ply n = number of plies L = length of the beam b = width of the beam Ex = Young's modulus along the plies, x direction Ey = Young's modulus perpendicular to the plies, y direction E, = Young's modulus perpendicular to the plies, z direction Gxy= shear modulus, directions x-y G,, = shear modulus, directions x-z Gy, = shear modulus, directions y-z vxy,u,,, vyz= Poisson's ratios

It is good engineering practice to assume

It is assumed that L, n, vxy,and v,, are deterministic, with the following values:

L = 18 cm;

n

= 26;

vxy= 0.26;

u,,

= 0.49

Thus, the number of variables in the load effects term of the failure surface is I = 6. The statistical characteristics of the remaining variables are described in Table 10.7. Many failure criteria for composite structures exist that are beyond the scope of this discussion. By solving the nominal deterministic case and comparing the obtained stresses and strains in tension, the compression and shear along and perpendicular to the direction of the plies were determined. It was concluded that this particular structural component fails if the tensile stress in the direction perpendicular to the fibers (tensile stress Table 10.7 Data for Example 10.2

Xi

XI= P X2 = h X3 = Ex Xq = Ey X5 = G,

x6= e

Units

Distribution

Mean, pi

Standard deviation, a,

kg/cm2 cm kg/cm2 kg/cm2 kg/cm2 deg

Norma1 Normal Normal Normal Normal Normal

0.14 0.019 0.386 X lo6 0.0827 X lo6 0.0414 X lo6 45

0.00333 0.001 0.0338 X lo6 0.00124 X lo6 0.00621 X lo6 1

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Table 10.8 Selection of EP, Example 10.2 EP

AX

EP

-

AX;

EP

+ AX,

in the matrix material) is higher than the allowable tensile stress in the matrix. It is, therefore, concluded that the resistance term of the failure function is

where S;, is the allowable (failure) tensile stress in the y direction. From available data, it is assumed that R is normally distributed with the following mean and standard deviation:

The EP and range is selected according to the parameters defined in Table 10.8. The I deterministic cases to be solved, together with the tensile stress perpendicular to the fibers S,,, are described in Table 10.9. In the last row of Table 10.9 the deterministic case, which uses the average values of all of the variables, is also described. The corresponding [Yl matrix is Eq. (10.28).

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Table 10.9 Thirteen deterministic solutions, Example 10.2 P

h

4

Ex

Gv

0

s,

1 (EP) 2 3 4 5 6 7 8 9 10 11 12 13 Nominal

Solution for the coefficients {a}yields a g =

321.84037

The critical values calculated using Eq. (10.3) and (10.29) are p,,

=

1.1825;

h,,

= 0.019677;

Ex,, = 0.3533 X 106

Four of these values are within the evaluation range and, therefore, a second calculation is required. This is performed by close observation of the previous calculation. The second selected E P is shown in Table 10.10.

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Table 10.10 A new selection of EP, Example 10.2

AX

EP

EP

-

AXi

EP

+ AX,

By solving an additional 13 deterministic cases with the ANSYS program and constructing a new matrix [Yl, a second set of solutions is obtained for {a} a 0 = 320.57 al = 2247.112; a, = 900.29013

a6= 12.25;

a12= -0.01392

(10.31) In this case, the critical values of two of the variables Ex and G , are very close to the end of the evaluation range. Nevertheless, the analysis is done with these values. Introducing the coefficients Eq. (10.31) into Eq. (10.10), using XF = S$ and applying the Lagrange multiplier method already described, the following results for the design point, the reliability index, and the probability of failure are obtained: p* = 0.14087 kg/cm2 h* = 0.01775 cm E,*

= 0.380376

x lo6kg/cm2

s$]= 338.4427 kg/cm2 /3 = 1.7310766

Pf= (first order) = 0.0417191

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Table 10.11 Selection of four points evaluation method, Example 10.2

AX,

EP-2AXi

EP-AX,

EP

EP+4Xi

EP-24Xi

Table 10.12 Twenty-five deterministic solutions, Example 10.2

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It can be seen that E,* is outside the evaluation range, whereas p*, E,Y, G,y, and O* are close to the end of the range. Therefore, the {a)of Eq. (10.31) are not a good set of coefficients for the approximated failure function. It is, therefore, recommended to evaluate the failure function by using four points around the EP. In Table 10.11, EP and ranges for fourpoints evaluation are shown. In this case 41 + 1 = 25 deterministic solutions are required. These are shown in Table 10.12, together with the ANSYS solution for the tensile stress perpendicular to the fibers. The matrix [Ylthat corresponds to this evaluation has 25 rows and 13 columns and is Eq. 10.33.

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Solution of Eq. (10.6) yields the following results for the Taylor series coefficients:

By introducing these values into the failure function (10.10) and applying the Lagrange multiplier method, results for the design point, reliability index, and probability of failure can be obtained. These are

O*

= 45.38 deg

Pf(first order) = 0.033356

(10.35)

The same approximate closed-form expression was used in the PROBAN program, to solve the problem by FORM, SORM, and Monte-Carlo simulations. All of the results are summarized in Table 10.13. It can be seen from the difference between the FORM and SORM results that the solved example presents nonlinear behavior. Nevertheless, the approximate expression obtained by the Taylor series expansion enables the use of a SORM analysis using an available program in which SORM solutions are

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Table 10.13 Comparison of results for different solution methods, Example 10.2 Method Lagrange multiplier two-point evaluation Lagrange multiplier four-point evaluation PROBAN, four-point evaluation

P

pf

FORM

1.7310766

0.0417191

FORM

1.8336123

0.033356

FORM SORM Monte Carlo (l0,OOO)

1.834813 2.304510 2.323

0.0332667 0.010597 0.0101

Analysis

available. The SORM solution compares well to the Monte Carlo simulation. V. Summary A Taylor series expansion method is used to create an approximate closed-form expression for the failure function of a practical structure. The expansion requires a solution of at least (21 + 1) deterministic cases, where I is the number of random variables. These cases can be solved with any analytical or numerical tool. The coefficients of the Taylor series are obtained by a least-mean-square method. The best evaluation point around which the function should be expanded is the design point, which is not known in advance. Suggestions for the practical selection of the EP are included, and an iterative process is suggested. Two examples demonstrate the application of the suggested process.

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Chapter 11

Direct Calculation of the Probability of Failure Using an Existing Finite Element Program I. Introduction

I

N the previous chapter, a Taylor series expansion of the failure surface around an arbitray selected E P was presented. The best E P for this type of expansion is the design point, which is not known in advance. Therefore, an iterative procedure was adopted. Therefore, it is clear that advance knowledge of the design point is of great benefit to the designer and the analyst. The modified joint probability density function (MJPDF) method described in this chapter can provide this knowledge. Moreover, it is shown that first-order probability analysis of practical structures can be performed by using only a commercially available finite element code, without the need for a probabilistic program. In this chapter it is shown that the design point can be calculated by finding a maximum to an objective function, which is obtained by modifying the joint PDF of the relevant problem. The maximum is searched for in the original physical random variable space X and not in the transformed space U. In case of random fields rather than random variables, discretization of these fields to random variables is performed according to the methods of stochastic finite elements, e.g., Refs. 43 and 44. Thus, a solution can be obtained using any available numerical algorithm that can find a maximum to a given function. Preferred algorithms are those that do not require calculation of numerical derivatives. The space in which this maximum is sought is the allowable range of the physical random variables, which are easily defined by the designer, and the finite element code is used as a tool for numerical simulation. Once the design point is obtained, the reliability index can be calculated, and the first-order probability of failure is obtained directly. If second-order probability of failure is required, a Taylor series expansion around the already-known design point is used. All examples presented in this chapter were solved using ANSYS$5 which is a large commercial finite element code without any modification in the code. This code (like most commercial programs) contains a built-in optimization module$6 and this particular module is used to search for the maximum of the objective function without any modification of the finite element code. The advantage of using this approach is the possibility to use an existing finite element code known to the design engineers, without modifying it

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and without using derivatives of the load effects term with respect to the random variables. This approach can encourage engineers in industry to use probabilistic structural analysis with existing tools until probabilistic analysis modules are incorporated into commercially available finite element programs. 11. MJPDF Method

Assume a typical basic failure function

where XI is the basic load effects term, normally distributed with mean pXland standard deviation uxl.Usually, XI is a function of the basic physical random variables of the structure and the loading. In case of spatially distributed random parameters, the load effects term also contains the random variables that result from a discretization of random fields by any of the methods described in the literature. X2is the basic random resistance term, normally distributed with mean px2and standard deviation gX2. If the load effects and the resistance terms are not normally distributed (and may even be dependent), a transformation to independent normal variables is available.34This is subject to the classical conditions that the marginal CDF of the variable is continuous and that the transformed correlation matrix of correlated variables is positively definite. The last condition is satisfied in nearly all cases of practical interest. Transformation into a standard normal space, e.g., Eq. (8.27), yields the limit state function in the u space

Equation (11.2b) is described in Fig. 11.1. By geometric considerations, the following quantities are calculated from Fig. 11.1: Px,

- Px,

0 = qv; "xl + cx2

- Px,

"X,(PX,

uf

= "x,

)

"X,(c~x, - P x , )

;

+ "2,

U: =

dl+ "2,

(11.3)

Inverse transformation to the original X space yields the physical design point

x;= x; =

2

P x , "x,

2 "XI

+ PX2"2,

+ "2,

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USING AN EXISTING FINITE ELEMENT PROGRAM

159

Fig. 11.1 Failure function in a standard normal space.

The joint probability density function (JPDF) fxl,x2 of XI, X2 is

The MJPDF P x , , ~is defined as the original JPDF, where the resistance variable is replaced by the load effects variable

It is easily shown that this function has a maximum when

which is identical to Eq. (11.4). Thus, the design point is the point of maximum of the MJPDF. In realistic practical cases, the load effects term XI is calculated by the required algorithm, i.e., a finite element code. This term contains many

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basic physical random variables (such as external loads, geometry, and material properties). The JPDF contains n parts, which belong to these n variables, plus one part, which originates from the resistance term. This last part is the one for which the modification is performed: the variable X2 in this part is replaced by the variable XI, which is the outcome of the finite element computation for a specific required response, i.e., stress, displacement, or moment. A special case is obtained when the resistance term is a constant (a threshold value problem). In this case, it can be easily shown that the design point is obtained by finding the point that maximizes the JPDF of the load effects term, while fulfilling the condition that the response is equal to the threshold value. The response itself is an outcome of the finite element computation. It is important for the designer to be able to represent the JPDF for the cases in hand. Three types of practical cases are possible: 1) All of the random variables are independent. In this case, the JPDF of the random variables { X } is given by

where fXn(xn)is the probability density of the nth variable. 2) All of the random variables are normal, but there is a correlation between some of them. In this case, the covariance matrix [R] is assumed to be known, and the multivariate normal distribution function is used

where det[R] is the determinant of [R]. Note the following relation between the covariance matrix and the correlation coefficients:

3) The random variables are dependent, and at least one of them is not normally distributed. The Nataf47model for known marginal distributions is used.

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Assume two dependent variables XI and X2with a correlation coefficient pl,,. The basic variables X are transformed to standard normal variables Z using

where F( ) is the marginal cumulative distribution of the variable ( ) and @ is the standard normal cumulative function. According to the Nataf model, the JPDF is approximated by

where fx,(xi) is the marginal probability density of X i and & ( z l , z2, is the biiariate normal PDF of the variables Z with correlation coefficient p ~ , , For ~ . the case of two variables xl and x2

The coefficient p,,l,2 is obtained as a function of p l , ~using the relation

In Ref. 48, expressions for F are given for many combinations of distributions. The maximum error obtained in the JPDF using these tables is usually less than 1%, although in very few combinations an error of 4.3% was obtained. Generalization of Eq. (11.1 1 ) yields

in which & ( Z , R o ) is a multivariate distribution factor similar to Eq. (11.9) with covariance matrix Ro,whose elements are obtained by using the formulations and tables published in Ref. 48. Once the JPDF is formulated, the MJPDF is obtained according to the procedure described earlier in this chapter, and the design point is computed by the finite element code. The point of maximum of MJPDF is obtained in the X space, whereas the reliability index is the distance between the

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design point in the transformed independent standard normal space u and the origin of this space. Therefore, the reliability index cannot be calculated straightforwardly, and some manipulations of the basic variables are required. For the simple case of independent normally distributed variables, the known design point {X*}is transformed to the standard normal space, using Eq. (9.27) to obtain {u*}. The reliability index is obtained using

For independent variables that are not normally distributed, the design point is transformed into the standard normal space using Eq. (11.10), and then Eq. (11.15) is used to calculate the reliability index. When the variables { X } are dependent and normally distributed with a covariance matrix such as Eq. (11.9), they are first transformed into independent normally distributed variables {Y},so that the covariance matrix of Y is diagonal. This is performedz6 using

where [A]is an orthogonal matrix of which the columns are the eigenvectors of the matrix [R].As Eq. (11.16) is a linear transformation, the mean of any variable Y i is given by

where ai are the terms in the ith row of [AITand px,are the means of the variables {X}.The standard deviations of { Y }are given by

Once the design point { X * ) is known, { Y * ) and its two first moments can be calculated, and then {u*} is calculated using

Equation (11.15) is then used to calculate the reliability index. When the variables are dependent and nonormally distributed, the results obtained by the Nataf47 model are used. In Eq. (11.10) the variables {X} with covariance matrix [R] were transformed into normal variables {Z} with covariance matrix [Ro].To calculate the reliability index, these vari-

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ables should be transformed into independent standard normal variables {Y). The following transformation between {u) and (2)is performed:

It can be shown that [To]= [Lo]-',where [Lo]is the lower triangular part of the Choleski decomposition of [Ro].For a known [Ro],the [Lo]can be calculated using [R,] = [Lo][Lo]-'. Once [ Y ]is known, Eqs. (11.15) and (11.17-11.19) can be used to complete the analysis. The preceding discussion of the MJPDF method may seem complicated, but inasmuch as this method is not covered in the literature (except for Ref. 49) it is felt that some theoretical background is required. The numerical examples in the following section demonstrate the use of the method. 111. Numerical Examples

All of the numerical examples presented in this chapter were solved using the ANSYS finite element program without any modification. ANSYS is used for calculation of the load effects term (denoted in this section by SFEto emphasize it was computed by a finite element code). Expressions for the JPDF and MJPDF were written and introduced into the proper modules of ANSYS. The optimization module of ANSYS was used to find the maximum point of the relevant function. This module creates an objective function approximation in the form of a quadratic equation, including cross-product terms. The constrained problem is then converted into an unconstrained one, using penalty functions. The loop includes a search for a minimum of the objective function, the determination of the next trial, and the calculation of the next step. Optimization looping continues until a predetermined convergence criterion is met. The algorithm was written originally for design optimization and, therefore, a minimum is sought. To use the same subroutines in the search for maximum value of the MJPDF, the objective function introduced into the program is the inverse of the MJPDF. Although ANSYS has several mathematical functions in its library, the error function is not included. Therefore, it was necessary, in some cases, to describe the error function as an approximate polynom. This is a disadvantage that does not originate from the MJPDF method and can be overcome in the future by introducing the error function into the program library. In Appendix C, input files of the ANSYS program for some of the examples are presented. The reader should note that this input is not suitable for other finite element programs, and it is presented only to show the simplicity of the application of the MJPDF method. ANSYS 4.2 version was used for the computations. In more updated versions of ANSYS (in 1996, version 5.3 was available) the procedures for optimization are slightly

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different but this does not change the major processes used by the MJPDF method. In the following subsections, details of the numerical examples are presented.

A. All Variables Uncorrelated and Normally Distributed Example 11.1 A bar of cross section A is loaded by an axial force F. The bar has an yield stress S,. Failure occurs when the stress in the bar is equal to or higher than the yield stress. All of the variables are independent. Data on the three variables are given in Table 11.1. The closed-form failure function is

g = S y - (FIA) 5 0

(11.21)

The mechanical transformation is known in this case and is FIA. Assuming that this transformation is unknown, the load effects terms is SFE,the axial stress calculated by the finite element program, and is a function of the first variable F and the second variable A. The failure function is, thus,

The JPDF of the three variables is

The MJPDF is obtained by replacing Sy with SFE

Table 11.1 Data for Example 11.1 Variable

F A SY

Distribution

Mean

Standard deviation

Normal Normal Normal

p, = 1000 PA= 2 psy= 600

a, = 33 u~ = 0.1 us?=

30

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Table 11.2 Results for Example 11.1

ANSYS

PROBAN

Error, %

and the maximum of Eq. (11.24) is sought by running the proper ANSYS case. Input files for ANSYS are described in Appendix C, Sec. I. The variable space in which this maximum is searched for is selected as mean

= 1000 5 F r

1099 = mean

+ 3 standard deviations

mean - 3 standard deviations = 1.7 5 A

5 2 = mean

(11.25)

As this example has a closed-form failure expression, it was also possible to solve it using the PROBAN program. In Table 11.2, results of the design point, reliability index, and probability of failure are presented for both the MJPDF method and the direct solution by PROBAN. The design point in the standard normal space u* was calculated using Eq. (11.27). The reliability index was calculated using Eq. (11.25). The first-order probability of failure presented in Table 11.2 is obtained using Eq. (8.13).

Example 11.2 A cantilever beam of length L is subjected to a tip force F. The cross section of the beam is rectangular, with height H and area moment of inertia I. The yield stress of the beam material is S,. H = 2 is deterministic, whereas all of the other variables are independent and normally distributed. Data are summarized in Table 11.3. Failure is defined when the bending stress in the clamped end is equal to or higher than the yield stress. The closed-form failure function can be expressed using simple beam theory as

Table 11.3 Data for Example 11.2

Variable

Distribution Normal Normal Normal Normal

Mean

Standard deviation

p~ = 1000

a~= 33

2 p~ = 50 psv= 33,000

a, = 2

PI =

al = 0.1 as"= 1000

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If no closed-form expression is known, the failure function is expressed as

where SFE is the bending moment at the clamped end, as calculated by a finite element program. The JPDF is given by the following expression:

The MJPDF is obtained by replacing Sy with SFE

A finite element beam model was run in the ANSYS program, using the optimization module where F, L, and Zwere changed and SFEwas calculated, until a maximum was found for Eq. (11.29). The input file for ANSYS is described in Appendix C, Sec. 11. Results are summarized in Table 11.4, together with PROBAN results obtained by using Eq. (11.26). The design point in the standard normal space u* was calculated in the same way as for example 1. Table 11.4 Results for Example 11.2

ANSYS

PROBAN

Error, %

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B. Normally Distributed Correlated Variables Example 11.3 Example 1 is repeated with the same data given in Table 11.1, but with a correlation coefficient p = 0.5 between the force F and the area A. The JPDF is given by

The MJPDF is

where SFEreplaced S,. The design point was obtained using the ANSYS program to find F and A that maximize Eq. (11.31). The input file for ANSYS is described in Appendix C, Sec. 111. Results are summarized in Table 11.5, and compared to PROBAN results. The u* point was calculated as follows. Table 11.5 Results for Example 11.3 ANSYS

PROBAN

Error, %

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The covariance matrix [ R ] is

The use of a 2 S,. Therefore,

X

2 matrix is sufficient, as there is no correlation with

The eigenvalues of [ R ] are

The matrix [ A ] ,which is the eigenvector matrix of [ R ] ,and its transpose [ A ] [see Eq. (11.16)] are

Therefore,

Introducing F* and A* from Table 11.5 into Eq. (11.35) yields

Applying Eqs. (11.16) and (11.17) yields

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Finally, applying Eq. (11.19) yields the values of u*, which are given in Table 11.5.

C. Nonnormal Dependent Variables Example 11.4 A bar of cross-section A and yield stress S,, both normally distributed, is subjected to an axial force T , which has a three-parameter Weibull distribution, with a minimum value of 900. There is a correlation between T and A, with a correlation coefficient 0.4. The data are summarized in Table 11.6. The three-parameter Weibull distribution has the following CDF:

F*(X)

=

1 - exp

[ (-)

X - E

-

k -E

'1

and the following PDF:

where x 2 E , p > 1, and k > E 2 0, and E , /3, and k are given constants of the specific problem. The mean and the standard deviation of the Weibull distribution is given by

Table 11.6 Data for Example 11.4 Variablea T A

s~ "Coefficient p,,

=

Distribution

Mean

Standard deviation

Weibullb Normal Normal

p~ = 1000 PA = 2 psY= 700

LTT = 50 u~ = 0.1 usY = 33

0.4. bMinimum value of T is 900.

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Tensile Force (kg) Fig. 11.2 PDF of the force T.

Sometimes F(x) and f (x) are described as follows:

Introducing the variable T (the axial force) into (11.37d) and using the data of Table 11.6 and Eq. (11.37c), the following PDF is obtained for T:

where p = 2.1013491, S = 4.8587 X and y = 900. These values correspond to a Weibull distribution with mean p~ = 1000, standard deviation UT = 50, and minimum value y = 900, as given in Table 11.6. This function is described in Fig. 11.2. The CDF for the Weibull distribution is

To write the JPDF for this case according to Eq. (11.11), Z1 and Z2,which correspond to T and A, respectively, are to be formulated. Using Eq. (11.16) the following is obtained:

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1000

1050

1100

1150

1200

Tensile Force (kg) Fig. 11.3 Polynomial solution of Z1 compared to the exact solution.

To express Z1 as a function of T inside a numerical program, a function library with the error function is required. Because the ANSYS library does not include this function, a polynomial approximation is formed, and the first of the two equations (11.39) is replaced by

In Fig. 11.3, the polynom solution of Z , is compared with the exact solution. The matrix of the variance coefficients between T and A is

Using the tables of Ref. 48, the following is obtained:

and, therefore,

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The JDPF of the two dependent variables T and A, according to Eq. (11.11), is

where $( ) is the standard normal probability density function of ( ) and Ziis given as a function of Xiby Eqs. (11.39). The JPDF of the three-variables system is given by

and the MJPDF is given by

The maximum of this function is found using the ANSYS program. The input file for ANSYS is described in Appendix C, Sec. IV. The values of T* and A * are described in Table 11.7. Using these values in Eqs. (11.39), the values of Z; and Z; are

Table 11.7 Results for Example 11.4 ANSYS

PROBAN

Error, %

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To use Eq. (11.20) to find {u), it is necessary to calculate [To].This is performed using the matrix [Ro]= [Lo][Lo]-', [To]= [Lo]-'.It can be found that

Introducing the second equation of Eq. (11.46) and the values of {Z*)from Eq. (11.45), and taking into account that S, has an independent normal distribution, the following is obtained for the vector {u*): 1.8509317 {u*) = -2.6866055 -2.9513636

I

These values enable calculation of the reliability index and the first-order probability of failure can be computed. All of the results are summarized in Table 11.7. A high reliability index indicates that the design point is at the far tail of the distribution and, therefore, the error in the probability of failure is high relative to the error in the reliability index. There is no physical significance to this error, as can be concluded when inspection of the numerical values is performed.

D. Threshold Value Problem Example 11.5 A bar of cross section A is loaded by an axial force F. The data for A and F is identical to the values presented in Table 11.1, example 1. The bar is assumed to fail when the stress in it reaches the value of 570 (deterministic). The failure function for this case is g = 570 - (FIA)

5

0

(11.48a)

If the stress in the bar FIA is not known as a closed-form expression, and the failure function is

where is a function of A and F. This is a threshold value problem, and the design point is found by determining the values of A and F that maximize

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Table 11.8 Results for Example 11.5 ANSYS

PROBAN

Error, %

the JPDF, subject to the condition that the stress is 570. The JPDF is given by

The input file is described in Appendix C, Sec. V, and is similar to that described in Appendix C, Sec. I, with the proper modifications. As the ANSYS optimization module requires a range of treated variables, the problem was solved defining a small range between 569.6 and 570 for the threshold value. The results are summarized in Table 11.8. Random Field, Correlated, Normally Distributed Variables Example 11.6 A beam of length L = 32 is clamped on both sides and has a random flexural rigidity EI with mean p~ = 1.125 X lo6 and standard deviation UE = 225,000. On the beam, a distributed load per unit length P i s applied, with mean pp = 8 and standard deviation u p = 2.4. There is a correlation in the force (per unit length) P between any two points AX apart, with a correlation coefficient given by E.

where a, is a nondimensional correlation length and a, = 0.25 is assumed for this example. Therefore, the force P is a random field rather than a random variable. The beam is assumed to fail when the bending moment at the clamped end is equal to or higher than 1100. The beam is shown in the upper part of Fig. 11.4. This example is a special case of the example solved in Ref. 44 using the stochastic finite element method. In Ref. 44, the flexural ridigity EI is also assumed to have a correlation between any two points AX apart and is, therefore, a random field. In this example, EZ is assumed to be a random variable. Using the same technique as in the application of the stochastic finite element, the beam was divided into 32 finite elements and into 4 stochastic

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Fig. 11.4 Original and discretized beam.

elements. Therefore, the case solved is the one described in the lower part of Fig. 11.4. There are many ways to discretize the random field into several random variables. In this example, the value of the force at the center of the stochastic element was selected to represent its value along the whole element. This method is less accurate when the element is too large or when changes in the value of the force over the selected element are large. Sometimes, the spatial average of the field along the element is used, and the load is applied at the center of gravity of the loading. It has been shown in the literature that the length of the stochastic element should be between a quarter and a half of the correlation length. According to this rule, at least eight stochastic elements should be selected. For simplicity of the demonstrated example, the length of the stochastic finite element is selected to be equal to the correlation length; therefore, a difference between the results of this example and Ref. 44 is expected. Nevertheless, as the case presented in Fig. 11.4 can be solved explicitly, solutions using the MJPDF method and the PROBAN program can be compared. The covariance matrix for discrete values Piis given by

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Introducing the data just given, the covariance matrix is

1 0.367879441 0.135335283 0.049787068

[R] = 5.76 .

0.367879441 0.135335283

1

1

0.367879441

1

symmetric

The determinant of the covariance matrix is 3.72362293476, and the inverse

1.156517643 -0.42549064 1.31303285

symmetric

4.0997267 X lo-"

2.9021289 X

lo-''

-0.42549064

4.0997267 X lo-''

1.31303285

-0.42549064 1.156517643

The expression within the exponent of Eq. (11.9) is, therefore,

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Table 11.9 Results for Example 11.6

ANSYS

P

;

P2

p: p:

EZ*

P pi

12.254 13.879 12.4374 10.0062 1,123,500

u; = 0.08543655 u,* = -0.36681643

uf

=

-0.75395417

u: = 2.68164 ug = -0.0066667

2.81097 0.00246962

PROBAN

Error, %

12.29 13.879 12.4374 10.0353 1,125,000 2.8250 0.00236403

0.29 0.20 0.20 0.29 0.13 0.5 4.5

Because Eq. (11.51) is proportional to the JPDF of the four discrete variables Pi, an expression that is proportional to the total JPDF of the system is obtained by multiplying the JPDF by the PDF of the rigidity EZ. An expression proportional to the JPDF of the whole system is, therefore,

The ANSYS solution, based on 32 beam elements, was solved for the bending moment at the clamp. The input file is described in Appendix C, Sec. VI. The optimization module was run to find the values of the parameters Piand EI that maximize Eq. (11.52), subject to the condition that the moment is the given threshold value of 1100. Results are described in Table 11.9. To compute the reliability index, the matrix [A] of the eigenvectors of the matrix [R] is first calculated. The results are

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Using Eqs. (11.16-1 l.l8), the following vectors can be obtained:

from which the vector {ui*}can be obtained using Eq. (11.19). Vector components are summarized in Table 11.9. As EZ is uncorrelated, the normally distributed variable, uE can be calculated directly from the physical values appearing in the same table. IV. Summary A method for a direct calculation of the first-order probability of failure of a practical structure using an existing finite element program is described. The computation is done by evaluating an MJPDF and finding the maximum of this function. It is shown that this maximum point is the design point, the most probable point of failure. The computation is based on the existing design optimization module, which is included in the commercial finite element code for quite a different application. Once the design point is found, the calculation of the first-order probability of failure is straightforward. If second-order probability of failure is required, the known design point can be used as an evaluation point for a Taylor series expansion, for use as an approximate closed-form expression in one of the probabilistic structures computer programs. A variety of examples demonstrate the application of the method for different kinds of practical structures, with different distribution functions of the random variables and different relations between these variables. These examples are solved using the ANSYS finite element program, and results are successfully compared with those calculated using the PROBAN program. Input files for the ANSYS program are presented in Appendix C. The ability to use an existing finite element program may encourage the application of probabilistic methods in the industry and encourage the suppliers of finite element programs to include a probabilistic module in the commercially available codes.

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Chapter 12

Probability of Failure of Dynamic Systems I. Introduction

I

N Chapter 5, the response of a deterministic structure to stationary random excitation was described and demonstrated for SDOF, MDOF, and continuous systems. It was assumed that parameters of the structure are known constants, and the PSD was calculated as a function of the structure parameters and the PSD of the external excitation. The calculated response can be a displacement at a certain point or, using the stress modes approach, a certain stress in the structure. In this chapter, the solution is extended to a nondeterministic structure, for which certain structural parameters, i.e., dimensions and material properties, are also treated as random variables. If a failure criterion is defined (such as a maximum allowable deflection or stress, which also may be considered as a random variable) the probability of failure can be calculated using the methods described in Chapters 8-11. A stationary Gaussian excitation process with zero mean is a common case for many practical structures. This case, which results in a stationary Gaussian response process, can be solved by explicit expressions, so that the phenomena can be demonstrated and, therefore, clearly explained. In the following paragraphs, stationary Gaussian processes with zero means are assumed. 11. Statistical Behavior of a Stationary Gaussian Process Assume that a certain structure response S(t) (which may be a displacement, a velocity, or a stress) can be described by a stationary Gaussian process with zero mean and a PSD function G s ( o )in the positive frequency domain 0 5 o 5 m. The absolute value of the maximum of this process during a period 7 is defined by the variable S, S, = max IS(t)l

(12.1)

7

S ( t ) can be a function of { z ) ,a vector of structural random variables. Assume that s is a threshold value (which may also be random and, therefore, is a part of the vector ( 2 ) ) so that when the response is higher than s, the structure fails. In Refs. 25 and 50 the conditional probability of failure (for

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given vector {z}) is found to be

where Pfo(z)is the conditioned probability that, at time t = 0, S(t) is higher than s, given the values of {z}, and v(z) is the rate (times per unit time) of the response process crossing the value s upward. It was also shown5' that Pfo(z) = 1 - exp

(

a

--

and A,' Al [which does not appear in Eq. (12.3)], and A2 are called spectral moments of the process and are defined as A,,,=

1;

omGs(o)do;

m=0,1,2

(12.4)

Thus, Eqs. (12.2-12.4) describe the conditioned probability that the process S(t) has a value higher than a prescribed values, which is defined as a failure. It can be seen that A. is the mean square value of the process, i.e., Eq. (5.13), and A2 is the mean square value of the time derivative of the process, i.e., Eq. (5.47). Spectral moment Al does not have a direct physical meaning. The spectral moments are functions of G,(o), the PSD of the response, which is a function of the PSD of the excitation and other structural parameters. Therefore, the spectral moments are functions of all of the components of the random vector {z} of the structure, including the excitation. Substituting Eq. (12.3) into Eq. (12.2) yields

where Fs, is the CDF of the maximum values ST. It was also shown that the zero crossing rate (the rate at which the process crosses the time axis) is

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The number of zero crossings at a period

181

T is

and the number of upward zero crossings (crossing the zero axis in the upward direction) is

The rate of upward crossings of the level s is given by Eq. (12.3). In the evaluation of Eq. (12.2), some limiting assumptions were One of the assumptions is a Poisson distribution for the maximum values. Using Ref. 51 some assumptions are released and an empirical coefficient q,, obtained by comparing results to numerical simulations, was introduced. In Ref. 52, following Ref. 51, a less conservative expression for Fs, was obtained,

Pf(z)= 1 -[I x exp

- exp

[ -uOT -

(&)I

1 - exp(-* . q, . s i f i ) ] exp(-s2/2Ao) - 1

where

The PDF of ST,fs,(s), is obtained by

where fs,

= exp

a2 - a2 exp(a9) exp(ao4) - 1

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and

In Fig. 12.1 the conditioned probability of failure described by Eqs. (12.5) and (12.10) are shown for different periods for the following set of spectral moments:

Figure 12.1 should be read as follows: After, say, 1000 s, the probability that there is a maximum value higher than 0.4 is 1 using either Eq. (12.5) or Eq. (12.10). The probability of having a maximum value higher than 0.5 is 0.14 [using Eq. (12.5)] and 0.1 [using Eq. (12.10)]. It is evident that for short durations the probability of having a maximum value that is higher than a relatively low threshold is small, and this probability increases as the process continues in time. In Fig. 12.2, the CDF functions Fs, [Eq. (12.9)], the maximum values of ST, are shown for different periods. As time increases the probability that the maximum value of the process is lower than a certain threshold increases. In Figure 12.3 the PDF [Eq. (12.11)] is shown. As time increases more and more maximum values of the process are included in a higher range of threshold s.

0

0.1

0.2

0.3

0.4

Threshold s Fig. 12.1 Conditional probability of failure.

0.5

0.6

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0

0.1

0.2

0.3

0.4

0.5

0.6

183

0.7

Threshold s Fig. 12.2 CDF of the maximum values of S,.

0

0.1

0.2

0.3

0.4

Threshold s Fig. 12.3 PDF of the maximum values of ST.

0.5

0.6

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111. Spectral Moments of a Deterministic MDOF System

To demonstrate the computation process of the spectral moments, an MDOF system subjected to random excitation is considered. As shown in Chapter 5, this type of system may represent a discretized continuous system. The PSD of the response is obtained using Eq. (5.20), which is repeated here

The spectral moments of this system can be calculated using

The first expression is the mean square of the displacement of the jth DOF and the last expression is the mean square of the velocity of this DOF. As an example the two-DOF system described in Fig. 5.2 is treated. In Eqs. (5.30) and (5.31), Sxl and S,, are expressed, respectively. Therefore,

-

0.24390.2

1. 0

oRe[%(o)?ii.(o)] do}

(12.14b)

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Table 12.1 Results for some dynamic properties Mass 1 ( j = 1)

Mass 2 ( j = 2 )

Using the expressions of Appendix A for the evaluation of the integrals in the last six equations, the results described in Table 12.1 are calculated. Once the values in Table 12.1 are known, Eqs. (12.5) and/or (12.10) can be solved.

IV. Probability of Threshold Crossing (Failure) In Eqs. (12.5) and (12.10), a conditioned probability of threshold crossing is formulated. This probability is conditioned on the known values of the structural random variable vector {z). In Ref. 53 it was suggested that the nonconditioned probability can be calculated using the following limit (failure) function:

@-' is the inverse standard normal distribution function and u n +is~ a standard normal variable added to the structural random variables. In Ref. 53 it was shown that when the structural random variables { z ) are transferred into { u ) variables [using, for example, Eq. (8.27)], the design point is obtained by finding the minimum distance between the origin of the u space

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and the multidimensional surface

Thus, the tools described in Chapters 8-11 can be adopted using Eq. (12.15) as a failure function. The described method of solution is possible only if the following two conditions exist: 1 ) The boundary conditions of the structure are deterministic and do not change between various specimens of the same structural design. 2 ) The parameters of the structure are random variables and not random fields. This is true when parameters are changed between specimens and not within specimens. For instance, if a thickness is a random variable, this indicates that two specimens may have different thicknesses within a prescribed statistical distribution, but the thickness does not change within one given specimen. These two conditions mean that the mode shapes are deterministic and identical for all specimens, although the natural frequency of each specimen may be random. Randomness in mode shapes is not treated herein. When only a small number of structural random variables exist, the following simple method can be used to find the probability that a displacement (or a stress) is higher than a given random threshold: 1 ) Express the PSD of the response as a function of the structural parameters and the PSD of the external excitation. 2 ) Express the spectral moments of the response. 3) Select a set of ranges for the random variables vector {z}.This selection should be in a reasonable range of each z , according to its prescribed distribution. 4) Calculate P f ( z ) using Eq. (12.5) or Eq. (12.10). 5 ) Compute k = W 1 [ P f ( z ) ] . 6) Using Eq. (12.16), set u,+~= k. 7 ) For each selected z, find the corresponding u using Eq. (8.27). 8 ) Calculate P = g u : + u? + + u: + ~ 2 , ~ . 9) Select another set of { z } values. 10) Repeat steps 5-8. 11) Find the minimum 6 for all of the possible sets of {z). Such a calculation may be very long if there are many random variables { z } and it is, therefore, practical only for simple cases where this number is small. The computation can be automated using a simple computer program. The method is really a brute force scanning of the whole space of { z } values. A flow chart for this type of calculation is shown in Fig. 12.4. When a probabilistic computer program is available, this procedure can be avoided. It should be noted that steps 1 and 2 include the natural frequencies of the system. When a closed-form expression for the frequencies exists, the rest of the calculation is simple. When this is not the case, as often occurs in practical structures, an approximate closed-form expression can be

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+

PROBABILITY OF FAILURE OF DYNAMIC SYSTEMS

Create Approximate

1. Express PSD of the

closed-form expression

response as a function of all

for the natural

the random variables

I

2. Express spectral

moments as fhnction of all the random variables

I

3. Select a set of random

I

I

4. Calculate Pf(z) using

I

(12.5) or (12.10)

i

I

5. Calculate ~=CJ-'[P~ (z)]

t 7. From selected {z} find

8. Calculate

P and store

previous one, replace stored

C e r I Exit

Fig. 12.4 Flow chart for scanning for minimum P.

187

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formulated using a Taylor series expansion. This will be demonstrated later in the section.

Example 12.1 A simply supported beam of length L, circular cross section of diameter d, made of a material of mass density p, with Young modulus E, and modal dampings 4 is subjected to a stationary random load per unit length with a PSD function of a one-sided white noise of magnitude So. Only the first three modes are considered. The nominal data are p = ylg = 7.959 X

kg s2/cm4

4 = 0.01 for all j

(12.17)

The mode shapes of the beam are1'

and the natural frequencies are18

where the cross-sectional moment of inertia I was replaced by nd4/64 and the cross-sectional area A was replaced by nd2/4. Equation (12.19) yields the following results for the first three natural frequencies: ol= 258.6566 radls (41.17 Hz) o2= 1034.6264 rad/s (164.67 Hz) o3= 2327.9099 rad/s (370.50 Hz)

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Using Eqs. (5.36) and (5.37), the following result is obtained for SQ,Qk:

It can be seen from the zero terms in Eq. (12.21) that the second mode, which is antisymmetric, does not participate in the response, because the random loading is symmetric. Substituting Eq. (12.21) into Eq. (5.39) yields the PSD function of the response at point 6

For the center of the beam, - 1; thus,

6 = 0.5, +l(e = 0.5)

=

1, and &([

=

0.5)

=

using Eq. (12.4) and the integrals in Appendix A, the first spectral moment can be obtained,

Substituting the nominal values (12.17) into Eq. (12.24) shows that the second term of Eq. (12.24), which is the contribution of the third mode, is 7.61 X times the first term, and the third term, which is the interaction times the first between the first and the third resonances, is 1.58 X

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term. Thus, the response behavior at the center of the beam is governed by the first mode only. Therefore,

The zero-crossing rate (12.6) is

and

where

Substituting the expression for the frequency (12.19) into Eqs. (12.25) and these equations into Eq. (12.5) yields

where zl is the failure threshold, i.e., when the response of the center of the beam is higher than zlthe beam fails. In practical cases the frequency is usually not known as an explicit expression. Evaluation of an approximate closed-form expression for the frequency using a Taylor series expansion is now demonstrated.

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Table 12.2 Seventeen deterministic solutions

Case 1 2 3 4

5 6 7 8 9 10 11 12 13 14 15 16 17

E 2.10 x 1.96 X 2.03 X 2.17 X 2.24 X 2.10 X 2.10 x 2.10 x 2.10 x 2.10 x 2.10 x 2.10 x 2.10 X 2.10 X 2.10 X 2.10 X 2.10 X

lo6 lo6 lo6 lo6 lo6 lo6 lo6 lo6 lo6 lo6 lo6 lo6 lo6 lo6 lo6 lo6 lo6

The resonance frequency, which is a function of E, d, p, and L, is evaluated into the following Taylor series around the nominal point p ~p d, , pp,and pL:

The four-points evaluation method described in Chapter 11 is used; therefore, 4 X 4 + 1 = 17 deterministic solutions are required. These 17 cases are described in Table 12.2. The column w,,,, in the table describes the deterministic frequencies which can be calculated for a practical structure by running a finite element program. The matrix of differences [Y] [see Eq. (10.5)J for the cases presented in Table 12.2 is shown in Table 12.3. Using

in which { w } is the column w,,,, of Table 12.2 yields the best coefficients { a ) ,

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Table 12.3 Matrix of Differences [ Y ]

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Table 12.4 Data for the random variables

Variable ZI

E

l

Distribution

Mean

Standard deviation

Range, from -30 to +3u

Normal Normal Normal

0.4 2.1 x lo6 0.01

0.0133 70000 0.000333

0.36-0.44 1.89 X lo6-2.31 X lo6 0.009-0.011

Substituting these coefficients into Eq. (12.29) yields the approximate closed-form expression for the frequency. The values obtained using this expression are shown in the last column of Table 12.2. Very good comparison can be seen between the calculated and approximated values of the frequency. To demonstrate the calculation of the probability of failure, it was assumed, without any loss of generality, that only three parameters of the problem are random, the threshold value z l , Young modulus E, and the modal damping coefficient & . The distributions of these variables and their ranges are shown in Table 12.4. The other structural parameters of the problem, So, d, L, and p, were considered deterministic and the nominal values given in Eq. (12.17) were assumed. The calculations are done for a period of t = 1 s, using the approximate closed-form expression (12.29) for the resonance frequency, with the coefficients (12.31). The scanning method described in Fig. 12.4 as well as the CALREL computer program were used. Results are shown in Table 12.5. The results show that for the given random variables, the most probable point of failure occurs when the threshold value is 0.3945 (less than the mean), Young modulus is 2.078 X lo6 (less than the mean), and the modal damping coefficient is 0.00993 (less than the mean). With this combination, the first-order probability of failure (defined as an amplitude higher than the threshold) is 1.71%, and the second-order probability of failure is 1.75%.

Table 12.5 Results for the numerical example

Scanning method Threshold value Young modulus Damping coefficient Reliability index (FORM) Probability of failure (FORM) Reliability index (SORM) Probability of failure (SORM)

CALREL

program

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V. Summary A calculation process for the probability of failure of a vibrating structure is described. Failure is defined as the case where a certain response, i.e., displacement or stress, is higher than a certain given threshold. Expressions for this probability of failure are shown for a stationary Gaussian response process with zero mean, which is a common and useful case in structural dynamics. The analysis is limited to cases for which the mode shapes are deterministic. Nevertheless, the natural frequencies may be random. The spectral moments are defined, and their calculation is demonstrated. Inasmuch as these moments are functions of the natural frequencies, a Taylor series expansion method for creating an explicit expression for these frequencies is also described. By adding an extra standard normal variable, a failure function is built. The probability of having a response higher than a prescribed threshold can then be calculated using techniques described in preceding chapters, or using one of the probabilistic structural analysis computer programs. A simple procedure suitable for application in cases where only a small number of variables participate is described and demonstrated.

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Chapter 13

Stochastic Crack Growth Models I. Introduction

E

XPERIENCE has shown that structures that are subjected to periodical loads fail after some period (or load cycles) at a stress that is significantly lower than their static failure load. The phenomenon is known as fatigue. It is common practice for designers to work with S-N curves, which are empirical data relating the failure stress levels to the number of loading cycles. The S-N curves are obtained by performing tests in which the specimens are loaded by periodic (mostly harmonic) loads with levels that are changed between specimens and observing the number of cycles to failure. Many specimens are required for the generation of a good S-N curve, and such curves can be found in many design manuals. The S-N curves are, in fact, a collection of data points with a relatively high dispersion that present a certain trend. When plotted on logarithmic scale axes, these points usually present a trend of a decreasing straight line. Practical designers can use S-N curves only as guidelines. The reason is that usually the designed structure does not have geometrical and loading conditions similar to the test specimens. Usually practical loadings are not harmonic and certainly do not have constant amplitudes. A practical structural element experiences wide spectrum loads, with a wide range of frequencies and frequency-dependent PSDs. The oscillating load does not always have a zero mean, and the ratio between maximum and minimum amplitudes does not always coincide with fatigue tested specimens with data contained in the manuals. During the past 30 years, theories for damage accumulation were developed and applied. Most of these applications were based on experimental observations, and during the early periods of the design-to-fatigue most procedures were not explained theoretically. Also during this period, models that try to explain the fatigue phenomena by fracture mechanics were developed, e.g., Refs. 53-57. According to these models, a very small crack that exists in the structure propagates during each loading cycle, due to the high stresses that exist at its tip, which are much higher than the average stress in the surrounding material. The crack grows until its length is such that the complete structure fails. Thus, the crack growth rate becomes higher toward failure, and at failure this rate is infinite. Models for crack growth have been the subject of thousands of papers published over the past 30 years. These range from simplified models (such

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196

as in Ref. 58) to more advanced models (such as in Refs. 59 and 60). Some of the models are based on the microstructure, and others are based on the analysis of experimental data. Still others are based on the theory of elasticity and use available experimental data for verification. Naturally, it is beyond the scope of this book to quote the vast number of available studies. In Ref. 61, an excellent list of 217 references is provided. The experimental work described in many papers shows that the size of a crack developed under repeated loads is of a random nature, although a repeated trend is observed. This random nature is apparent even for the experiments carried out under a strictly controlled laboratory setting using deterministic external loads. Four famous series of experimental test results are repeatedly quoted in the literature: Noronha et a1.,62 in which two sets of experiments are presented; Virkler et and Ghonem and D ~ r eFrom . ~ ~the nature of the results quoted in these references and many others, it can be seen that a good prediction of the behavior of the crack growth process must be given by a stochastic rather than a deterministic model. In engineering practice, crack growth is usually treated by the deterministic models. Expressions for the stress intensity factor required for such analyses can be found in manuals, handbooks (such as Ref. 65), and textbooks (such as Ref. 66). Stochastic crack growth models have not been yet adopted by the industry, although they describe the phenomenon better. It is the purpose of this chapter to introduce the readers to some of the concepts and procedures of the stochastic models, to show how these models can be incorporated in the calculation of the probability of structural failure due to crack propagation, and to encourage the engineering community to study and adopt such methods in practical design.

Stochastic Crack Growth Models There are three major reasons for the indeterministic nature of crack growth process: 1) The macroproperties of different structures and specimens, that is, the geometry, dimensions, and material properties (such as elastic moduli and allowable stresses), may differ slightly between specimens. Thus, the whole structure is indeterministic. 2) The external loading in practical engineering cases is usually random. 3) The microproperties of a structure or a specimen are random, which means that the microstructure is not homogenous, even for strictly controlled material production conditions. This chapter focuses mainly on the third issue, although some aspects of the first one are considered. The randomness of the external loads are not treated here but are demonstrated in the last section of this chapter. The measurement of crack size as a function of loading cycles (or time as an interchangeable parameter) has the general form shown in Fig. 13.1. The curves are characterized by two features: 1) a nonlinear increase in crack size with an increased number of loading cycles (or time) and 11.

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I

Time or Load Cycles Fig. 13.1 Generic curves for crack size as a function of time or cycles.

2) intermingling of the curves for several identical specimens. Curves similar to those of Fig. 13.1 can be seen in Refs. 62-64. Stochastic models for crack growth were suggested by many investigators over the past 15 years, (e.g., in Refs. 67-73). These include evolutionary probabilistic models (Markov chain, Markov diffusion models), cumulative jump models (Poisson process, birth process, jump-correlated models), and differential equation models. A comprehensive summary of the state of the art is given in Ref. 61. The models that can be used relatively easily in practical engineering applications are the differential equation (DE) models. Two major groups of models are used to treat the DE for the behavior of a stochastic structure due to crack growth and to evaluate the probability of failure of the structure: 1) Use the deterministic DE for the crack growth rate, while assuming that the different parameters in this equation are random variables (RV). Examples for this family are presented in Refs. 25, 74, and 75. 2) Use the modified DE for the crack growth rate, where a stochastic nature of this rate is expressed by a random process (RP). Examples for this family are presented in Refs. 66-70, 72, and 75-78. For RV methods, the basic DE for the crack growth rate is

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G.MAYMON where: a

= crack

length

t = time

N = cycles of load p = crack growth parameters AKI = range of stress intensity factor b = geometry parameters q = parameters of the external load The classic basic model of Ref. 58 is a special case of Eq. (13.la)

where Y ( a ) is a geometry function that depends on the mutual geometry of the crack and a plate structure. D and n are considered as material parameters. This model was used in Ref. 25 for the experimental results of Ref. 63, in Ref. 67 for the results of Refs. 62 and 63, and in Ref. 61 for the results of Refs. 63 and 64. When test results (crack size as a function of time or cycles) are processed to yield crack growth rate as a function of the crack length and plotted on logarithmic axes, experimental results are similar to those shown in Fig. 13.2.

Time or Load Cycles (log scale) Fig. 13.2 Generic curve for crack growth rate as a function of time or cycles, Eq. (13.2).

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Time or Load Cycles (log scale) Fig. 13.3 Generic curve for crack growth rate as a function of time or cycles, Eq. (13.3).

Another example of a D E for the crack growth process is given in Ref. 71 for the experimental results of Ref. 63,

i = 1, 2, 3,4, are constants determined by the experimental data where Ci, shown generically in Fig. 13.3. The DE (13.3) extends the range of Eq. (13.2) for values of a beyond the straight line region shown in Fig. 13.2. There are many other examples of crack growth rate DE, e g , in Refs. 61 and 71, but they are not repeated here. The solution for a stochastic crack growth rate using the RV methods is performed by assuming a certain probability distribution for n and D in Eq. (13.2). In Ref. 25 this was done for a center cracked plate, where n was assumed to have normal distribution, D was assumed to have lognormal distribution, and n and D were assumed to be correlated. In Ref. 79, a similar randomization was done for a notched beam loaded by a bending moment. The disadvantage of this type of randomization is that it does not simulate intermingling of the a-N curves, such as those shown in Fig. 13.1. Another disadvantage of the RV methods is that there is a physical inconsistency when the constants n and D in the crack growth rate equation (13.2) are randomized. The units of the stress intensity factor AKI are of stress times square root of length, i.e., psi. in."2 or MPa. mU2.The crack growth rate has units of length per unit time or per cycle. Thus, the units of D in Eq. (13.2) are [length(1+3n'2)l(s~ forcen)] or [length('+3n'2)l(cycle.forcen)].

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G.MAYMON

If n is assumed random between specimens, the units of D become random. This result has no physical meaning. The use of Eq. (13.3) presents similar problems because sinh( ) and log( ) have a physical meaning only if their argument is dimensionless; thus, randomization of the constants Cihas no physical meaning. This difficulty can be overcome if, for instance, Eq. (13.2) is written as a nondimensional equation, e.g.,

where K is a normalizing factor (which can be random) with dimensions of the stress intensity factor, e.g., Young modulus multiplied by the square root of a typical length. In this case, reported experimental results should refer to this normalization factor and its probability density, when it is not deterministic. At present, reported results do not include such information and, therefore, the use of RV methods in engineering applications can yield suspicious results. An interesting almost nondimensional crack growth model is presented in Ref. 80, the user manual of the NASAIFLAGRO computer program. In Ref. 80, curve fitting for a very large number of materials is presented, as well as crack models for many geometries. All of the data is included in the database of the FLAGRO program, which enables an interactive calculation of crack growth process, as well as boundary element analysis of various geometries. In the RP approach, the randomness of the crack growth is described by adding a nonnegative RP X(t) to Eq. (13.la). Thus, Eq. (13.la) is written as

where KI is the stress intensity factor, AKz the stress intensity factor range, S the stress amplitude, R the stress ratio, and a the crack length. Equation (13.2a) is written as

where Q includes Y(a) and AS, whereas b is related to n. It is then assumed that Q and b are constants, determined from the results as described in Fig. 13.2 by linear regression, and the dispersion around the straight line (in logarithmic scales) is contributed only by the RP X(t), e.g., Ref: 81. To achieve mathematical simplicity, X(t) is modeled as a positive lognormal

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RP, which takes values around unity. Taking the logarithm of both sides of Eq. (13.6) the following is obtained: =b

. log(a) + log(Q) + log(X(t))

(13.7a)

where

Y = log (daldt) U = log(a(t)) s =lode) z = log(X(t)) Except for the Z term in Eq. (13.7b), this equation describes a straight line with slope b and a vertical axis intersection q, as shown in Fig. 13.2. The Z = log X term describes the dispersion of data points around the straight line. Z = log X is a normal RP with zero mean and standard deviation a,. The constants b and Q can be estimated by linear regression of plots similar to those in Fig. 13.2. The regression procedure also yields a value for uz = qog(x(r)). Once qog(,(,)) is known, the mean and the standard deviation of X(t) can then be calculated using the normal-to-lognormal conversion formulas

In Ref. 67, test results that appear in Ref. 62 for aluminum fastener hole specimens were analyzed and the following results were obtained:

It is interesting to reproduce artificially crack growth rate results by using the results given in Eq. (13.9). To do this, the following procedure is performed: 1) RPs Z(a) with a normal distribution with ulogx = 0.087635 are created

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a (inches) Fig. 13.4 Four normally distributed RPs with a,,,, = 0.087635.

using a random generator computer program. For clarity, only four of these types of processes are described in Fig. 13.4. 2) Using the equation (daldt) = Q X ab X l o Z [see Eqs. (13.6) and (13.7)], values of the crack growth rate are computed. Plots of the logarithms of these values as a function of a for four processes are described in Fig. 13.5. 3) A simple integration scheme in the a direction, beginning at an initial crack length of a. = 0.004 in. is performed. Curves of a as a function of t for eight cases are plotted in Fig. 13.6. In Fig. 13.7, an insert of Fig. 13.6 is plotted, which better shows the intermingling of the curves. Using expressions of Ref. 81, it is also possible to prepare curves of statistical limits to the possible a-t curves that have the statistical properties as described in Eq. (13.9). In Fig. 13.8 such limits are described for the problem in question. The curves dictate that the probability that a specimen has a crack size growing faster than, for example, the line denoted lo%, is 10% and less. 111. Crack Length Distribution

To calculate probabilities of failure of a cracked structural element, the distribution of the crack length is required. A very convenient model for crack length distribution was presented in Ref. 67. The crack growth is approximated by a Markov chain. Whereas this approximation introduces an error associated with the crack growth rate, the error is negligible except for very short fatigue life, which is of less interest for practical good designs. These errors can be overcome if the model described in Ref. 70 is used. Nevertheless, the simplicity of application of the model presented in Ref. 67 has many advantages when practical engineering cases are solved.

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1

203

...................................................

-6.4 -6.6

-3

I

1

I

-2.5

-2

-1.5

-1

log(a) Fig. 13.5 Crack growth rate as a function of crack length.

0

5000

loo00

15000

Time (Flight Hours) Fig. 13.6 Eight simulated crack growth curves.

20000

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204

3500

4000

4500

5000

5500

6000

6500

Time (Flight Hours) Fig. 13.7 Insert from Fig. 13.6.

0

I 0

I

2500

5000

--.-

7500

----.+--10000

12500

15000

17500

Time (Flight Hours) Fig. 13.8 Limits to crack growth curves with statistical data of Eq. (13.9).

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There are two extreme cases of the RP X(t) to be studied. When this process is assumed to be completely independent at two moments in time, it was shown that the statistical variability of the crack size is the smallest possible. As a result, unconservative results that may not be used in engineering applications are obtained. On the other hand, if the process is assumed to be totally correlated at any two moments in time, this process becomes an RV, and the statistical dispersion is the largest possible. The reality is somewhere between these extreme cases, and some correlation should be assumed. It is reasonable to assume that crack growth process is local and, therefore, is correlated only within a certain distance near the crack tip in the structure. In Ref. 67 it was assumed that the random process X(t) can be described by

where p is the mean of X(t) and Y(t) is an RP with zero mean. It was assumed that the correlation coefficient of Y(t) has a triangular shape so that there is no correlation beyond a time lag A. Thus, the coefficient of correlation C, is described by

= 0;

otherwise

The numerical value of A is selected in Ref. 67 so as to best fit given experimental results. In other works, e g , Refs. 82 and 83, it is assumed that the correlation coefficient is inversely proportional to a typical value of the crack growth rate (thus, when the rate is large the correlation time is small) and the correlation function has an exponential decay

cC= exp

(- 6)

iC =

($)

aoE

where Cc is the coefficient of correlation, E(da1dt) is a typical value of daldt, and a. is a proportional constant. It was suggested to use a o = 0.32 mm = 0.012 in. for high tensile strength steel. In Fig. 13.9, Eq. (13.11) is described for A = 8000, and Eq. (13.12) is shown for three values of E(da1dt). Here log E(daldt) is selected as (-5.2), (-5.9, and (-5.8). These are typical values (as can be seen from Fig. 13.5), which correspond to T, = 2000, 4000, and 7950 flight hours, respectively.

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206

0

2000

4000

6000

8000

10000

12000

14000

Correlation Time (Flight Hours) Fig. 13.9 Correlation functions for several cases.

In Ref. 67, a DE for the PDF of a crack length a at time t is evaluated and solved given that the crack length was a. at time to. This yields

X

exp

2aZ A(t

-

to)

'3

If the function a ( a ) is known and p, A, and a: are given, Eq. (13.13) can be solved analytically or numerically. According to the assumptions made in Ref. 67 for the evaluation of the solution (13.13),values of a(t)smaller than a. can be obtained, an impossible physical result, which means that the crack length may decrease. Limiting the solutions to values a > a. yields a CDF that is larger than one. The error is large for smaller values of t - to. If long life periods, which are of

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more interest in practical engineering work, are examined, the error is small. Therefore, in Ref. 67 it is suggested to normalize Eq. (13.13) so that

where

When g = Q . a b [see Eq. (13.6)],the integral in Eq. (13.14) can be solved analytically. If the crack length at to is ao, the probability distribution of the crack length q, at time t is given by

-

qa(a,tlao, to) =

[(I

-

( a l a o ) - ( b - l ) l c ~ a-~ -' t ( 20: A(t - to)

-

forb

+1

(13.15a)

D(t - to) Q a b d 2 n a ; A(t - to)

where c = b - 1. Q and b are the coefficients appearing in Eq. (13.6), A is the correlation time of the process, and 0, is the standard deviation of the process X(t) [see Eq. (13.8)]. In Fig. 13.10, the PDF of the crack length is shown for different times, with A = 8000 flight hours, a value selected in Ref. 67 to best fit experimental results of Ref. 62. The CDF of the crack length can be determined using

Numerical integration of Eq. (13.16) was used to obtain the CDF shown in Fig. 13.11. When a crack length is selected, the probability that the crack

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Crack Length (inches) Fig. 13.10 PDFs of crack length.

Crack Length (inches) Fig. 13.11 CDF of crack length, for different flight hours.

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Crack Length (inches) Fig. 13.12 Probability of a crack being higher than a given value, for different flight hours.

length is lower or equal to this value for a given time can be determined from Fig. 13.11. The probability that a crack is larger than a given value is

These probabilities are shown in Fig. 13.12. In the described model, a relatively simple expression (13.15) is presented for a crack that follows a crack growth rate equation such as Eq. (13.6). This expression can be used in many practical applications and is demonstrated later in this chapter. More advanced crack models can be found in the literature, e.g., Refs. 82 and 83. These models assume randomness of the initial crack size and a threshold value of initial crack size below which no crack growth takes place. Also, interaction between the RP of the crack propagation and the random loading process may be taken into account. Although these models provide a more complete treatment of the crack length PDF, they are much more difficult to apply when an analysis of the probability of failure of a practical design is evaluated. The simplicity of Eq. (13.6) enables its application in the probabilistic analysis programs and, thus, a practical analysis can be performed. The reader who is interested in more advanced models is refered to the list of references, especially to Ref. 61.

IV. Failure Criterion The selection of a failure criterion for a certain analyzed structure is one of the most important decisions in engineering practice. This decision

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210

G. MAYMON

influences the nature of the design, the amount of marginal safety, and the prediction of the structure's behavior. It is this decision that requires the talent, expertise, intuition, and experience of the design engineer. In many designs it is common to require that the maximum stress in the structure will never exceed the material's yield stress (with a given margin of safety or with a predefined probability of failure). As crack growth is a process in which local plastic zones exist and local yielding of the structural element may be permitted, this is not a well-suited criterion for the phenomenon. A very simple criterion can be expressed if the demand is

where Klc is the critical stress intensity factor and KI is the stress intensity factor. The critical stress intensity factor is the stress intensity factor calculated when the crack size is such that an infinite rate of crack growth exists and the structure is ruptured. If the stress intensity factor, for a given structural element geometry, is known as a function of the crack length, the critical stress intensity factor can be calculated. The failure function g is then

where failure takes place when g 5 0. KI is a function of the structure characteristics and the loading, e.g., Eq. (13.2b). Both KIc and KI can be random. This failure criteria is now required by aerospace regulations. A more advanced criterion combines the brittle fracture, ductile fracture, or mixed failure of metals, which was presented in Ref. 84, the BurdekinStone criterion. This provides the following relation for the safety and failure regions:

where a, is the yield stress, u,is the ultimate tensile stress, and KIc is the critical stress intensity factor, for which expressions are provided in the following equations. KL corresponds to linear elastic failure and SP corresponds to plastic failure. Equation (13.20) is described in Fig. 13.13.

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21 1

STOCHASTIC CRACK GROWTH MODELS

SAFE

-

-

-

0

I

I

I

I I

I I

1

I

0.1

0.2

0.3

0.4

0.5

I

,

I

0.6

0.7

I

I

I

0.8

0.9

I

1

P Fig. 13.13 Safety and failure regions (Burdekin-Stone criterion).

In Ref. 85, expressions for KIc are given for a centered cracked plate (initial crack length 2ao, plate width b) under uniform tensile stress a

where

[ 42J (:

ub,= a, 1 - - - a,,

I

(zone 1)

(zone 2)

,

=

K

&

1 -

)

(zone 3)

KC1is the fracture toughness, which is a material property and can be found in material manuals. For demonstration, Eq. (13.22) is described in Fig. 13.14 for typical values: a, = 75,400 psi, Kc* = 100,700 lb/in.1.5, and b = 12 in. To demonstrate the calculation of the probability of failure of a structural element, the failure criterion (13.20) is used in Sec. V. Use of Eq. (13.19) yields results that are more conservative.

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212

80000

,

0

1

2

3

4

5

6

Initial Crack Length (inches) Fig. 13.14 Stress ua,as a function of initial crack length.

V. Probability of Failure Calculation The quantities and distributions defined and calculated in Sec. IV and the procedures described in the preceding chapters are used here to calculate the probability of failure of a centered cracked plate. A plate of width b is subjected to a tensile stress crand has a centered crack of length 2ao. It is assumed that the crack is in zone 1. Using Eq. (13.20) a failure function g can be written as

Failure occurs when g 5 0. Using the classical expression for a centered cracked plate, with a very small crack compared to the width b, the geometry function Y ( a ) can be assumed to be a unity; thus,

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STOCHASTIC CRACK GROWTH MODELS

213

Introducing Eqs. (13.20-13.22) and (13.24) into Eq. (13.23) yields

Denoting

the failure function can be written as

The calculation is made with the crack length PDF described in Fig. 13.10, for 4000 flight hours. It should be noted that this curve was obtained by processing the experimental results of an aluminum fastener hole62and not of a centered cracked plane and, therefore, the curve is used here for demonstration purposes only. The relevant distribution curve of Fig. 13.10 is not included in the distribution library of the probabilistic structural analysis programs and, therefore, to use these libraries, the curve was approximated by a lognormal PDF. In Fig. 13.15 the approximation is shown together with the original curve. The initial crack size is assumed as deterministic, a. = 0.004 in. Table 13.1 provides data for the distribution and their first two statistical moments of Table 13.1 Data for stochastic crack growth example

XI

x2 x 3

x 4

x5 X6

Units

Distribution

Mean

Standard deviation

psi psi psi Iblin.' in. in.

Normal Weibull Weibull Weibull Deterministic Lognormal

Varies 75,400 97,157 100,700 0.004 0.00805

Varies 3016 3886 10,070 n.a. 0.002

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G. MAYMON

0

0.005

0.01

0.0 15

0.02

Crack Length (inches)

Fig. 13.15 Lognormal approximation (+ + + +) to PDF (-)

of Fig. 13.10.

Mean Value of External Load (psi) Fig. 13.16 Probability of failure after 4000 flight hours; cov = 0.05: lower line FORM, upper line SORM, (+) linear simulation, and (0)nonlinear simulation.

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STOCHASTIC CRACK GROWTH MODELS

215

COV of Loading (%) Fig. 13.17 Probability of failure after 4000 flight hours, XI = 35,000 psi: lower line FORM, and upper line SORM.

the random variables. Equation (13.27) with the data of Table 13.1 is introduced into the CALREL program,37 and the probabilities of failure after 4000 flight hours are calculated. First, the standard deviation of XI is assumed to be 5% of the mean (cov = 0.05). The probability of failure after 4000 flight hours is shown in Fig. 13.16, as calculated by FORM, SORM, and linear and nonlinear directional simulation (20,000 simulations). Then, a mean values of 35,000 psi is assumed for X I , and different values are assumed for the coefficient of variation. Results for this calculation are shown in Fig. 13.17. The small difference between the FORM and SORM calculations means that despite the nonlinearity of the failure function (13.27), the calculated results are not influenced by this nonlinearity. Also, good agreement is obtained between simulation and approximate methods.

VI. Required Information for the Analysis In the preceeding paragraphs, a stochastic analysis of a cracked structural element was demonstrated, and the probability of failure was calculated. This process used some important items of information that are required to apply the described procedure. These items are as follows. 1) An expression, or rather a method of calculation, of the stress intensity factor, e.g., (Eq. 13.2b), is necessary. For many geometries such expressions can be found in Ref. 65. Also, many finite element computer codes can compute this factor numerically. Then an approximate closed-form expres-

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216

Define or select the DE for crack growth rate, Sec. I1 1

process X(t), Sec. I1

Built a PDF for the crack length for different times, Sec. I11

I

Select an appropriate failure criterion, Sec. IV

I

Build a failure function, Secs. IV and V I

I

*I

Calculate the probability of failure, Sec. V

I

Fig. 13.18 Process for probabilistic calculation of cracked structural element.

sion can be determined using a Taylor expansion or any other curve fitting method. 2) A crack growth DE is necessary. Equations (13.1) are of general form. Equation (13.2b) is only a simple example, used for demonstration purposes in this presentation. 3) An estimation of the correlation time of the stochastic process X ( t ) is necessary to define, for instance, the parameter A of Eq. (13.13). In Ref. 67, a triangle function was assumed, leading to Eq. (13.13), and the proper A was selected so as to best fit experimental results. In Ref. 82 it was assumed that the correlation time is inversely proportional to a typical value of the crack propagation rate, and the correlation function is an exponential decay. Thus, an estimation of r, in Eq. (13.12) is required for a solution. These required parameters must be estimated from experimental data of the particular case under consideration, or a similar known case. 4) It is important to establish the failure criterion. An example was presented in Sec. IV, but many others can be thought of. As was already

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STOCHASTIC CRACK GROWTH MODELS

mentioned, the determination of the failure criterion is of crucial importance. 5 ) Statistical data on the yield and tensile strength, fracture toughness, and initial cracks must be made available. Some constants, such as Q and b in Eq. (13.6) and the value of qogx(r) were determined using experimental data. In cases where experimental data is not available, these values can be estimated based on reported results of similar materials. At present it would be realistic to limit the demonstrated procedure to metal and metallike materials. This type of analysis is much more complicated if a composite structure is to be analyzed. The mechanism of crack growth cannot be described by a simple DE, and the calculation of the stress intensity factor is much more complex. Little experimental data is available in the literature. Also in the present analysis, effects of spectral load amplitudes on crack growth response, such as acceleration and crack delay, are not included. Thus, the method presented is applicable to load spectra that are stationary, where time histories of the load do not include sudden jumps. In Fig. 13.18 a general flow chart that demonstrates the required analysis steps, as well as the related paragraphs in this chapter, is shown. The flow chart is applicable to the general case.

VII. Summary A procedure for the calculation of the probability of failure of a cracked structural element is described and demonstrated. Although the principles of the solution process are general, they were demonstrated only for metal and metallike materials. Random process methods are recommended for the stochastic analysis of crack growth. . A relatively simple model for the calculation of the PDF of crack length is described. There are many other models that may describe the phenomena more rigorously, but they are too complex for daily routine applications. The demonstrated process is limited to stationary loadings. Some important required information is listed. It is important to bear in mind that many unknown factors still exist in this field, and results should be treated carefully. This type of analysis is suited especially for comparison between several design options. It is very important to form a data bank for the statistical parameters required in this type of analysis.

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Chapter 14

Concluding Remarks

A

N effort was made in this book to introduce readers to the basic concepts of the analysis of random vibration and random structures. The author tried to direct readers to practical use of the fundamental concepts for the daily engineering work. The book is written for design engineers who have a basic understanding of random vibration theory, and it demonstrates the practical application of the material discussed in each section. Readers are introduced to a variety of solution techniques and are directed in how to formulate their own problems, how to solve them, and what is the methodology of the solution. One of the major purposes of the presented material is to introduce readers to the physical meaning of the theoretical models of random vibration and random structures. Many (but not all) of the subjects described in the preceding chapters can be solved today by commercially available computer programs; some of these are mentioned in the text. These programs must be considered as part of the tools used by the design engineer in the process of the design. As a complete numerical analysis of every alternative and every possible option of a practical design is prohibitive, the understanding of the meaning of the structure's behavior and the basic assumptions of the theories (on which the numerical analysis is also based) may help the engineer to formulate simple models. With such models the effects of various design parameters and the sensitivity of the design to changes in these parameters can be performed very quickly, especially in the preliminary design stage. Heavy numerical analysis should be kept for verification of the close-to-final design, where detailed refinements (which are more difficult to be formulated with simple expressions) can be finally checked. The author hopes that the material presented will encourage young engineers to explore the interesting fields of random vibration, random structures, and structural probabilistic analysis. These fields are of major importance in the design of any structural system, and it is believed that their importance will grow in the near future. The understanding of the basic concepts and their physical meaning will undoubtedly result in better designs and, therefore, with better designers.

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Appendix A

Some Important Integrals

T

HE integrals summarized in this appendix are sometimes required in the solution of the response of MDOF and continuous systems to white noise, one-sided white noise, band-limited white noise, one-sided bandlimited white noise, and filtered white noise, described in Chapter 4, Sec. 111, and for the calculation of spectral moments described in Chapter 12, Sec. 11. These integrals are based on Der Kiureghian.13

I. Constant Value PSD Functions For the direct terms

n

For mixed (interaction) terms

=

j,"o ~ e [ g ~ ( w ) E ? ( odo])

1 - (2127) arctan (&I-)

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G. MAYMON In Eqs. (A4-A6),

Equations (A4-A6) degenerate to Eqs. (Al-A3), respectively, when i 11. Filtered One-sided White Noise

For this section, see Eq. (4.22). For the direct terms

where, for r

= milog.

= j.

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APPENDIX A

For mixed (interaction) terms

+ C1[2alblA3 (a?- b?)A4]+ C2[2alblA4+ (a: - b:)A3] -

blD2

In these equations

A,

=

[(a,- al)' - (bg - b:)][(ao+ a2)' - (bi - bz)]- 4bz(a0- al)(ao+ a2)

A2 = 2bo(ao+ a2)[(a0- a,)' - (b; - b?)]+ 2bo(ao - al)[(ao+ a2)'

-

(bi - b;)]

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224

A3 = [(al + ao)'

-

(b: - b$)][(al+ a2)' - (b: - b;)] - 4b?(a0 + al)(al + a2)

A4 = 2bl(al + a2)[(al+ ao)' - (b: - b;)] + 2bl(al + ao)[(al + a2)' - (b? - b;)] A,

= [(a,

+ ao)'

- (b; - bT)][(ao- a2)' - (b$ - b?)] - 4b$(a0+ al)(ao - a2)

A6 = 2bo(ao- a2)[(ao+ al)' - (b; - b:)] + 2bo(ao+ al)[(ao- a2)' A7 = [(al - aol2- (b: - b;)][(al+ ao)' - (by - b:)]

-

-

(b; - b;)]

4b:(a0 + al)(al - ao)

As = 2bl(al + ao)[(ao- al)' - (b? - b;)] + 2bl(al - ao)[(ao+ all2 - (b: - b$)] A, A,,

= [(a2- al)' -

(b; - b;)][(a2+ ao)' - (b; - b;)] - 4bg(a2 - ao)(a2+ ao)

+ a2)[(a2- aO)' - (b?

= 2b2(ao

-

b;)] + 2b2(a2- ao)[(a2+ ao)' - (b; - b;)]

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APPENDIX A

These expressions can be carefully programmed into a very simple computer program.

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Appendix B

Conversion Between Acoustic Decibels and PSDS6

T

RADITIONALLY, SPL is described by the root mean square levels of the pressure fluctuation as a function of the frequency. The units of these levels are acoustic decibels. The definition of acoustic decibel is PI,, acoustic decibel = 20 log Pref

where PI,, is the root mean square of the pressure fluctuations, and Pref= 2 x dyne/cm2 = 2 X Pa = 2.0408 X kg/m2 = 2.0408 X lo-'' kg/cm2. Acoustic SPLs are measured by special instruments in which the frequency is scanned with a certain bandwidth. Two kinds of bandwidths are usually used, octave and 113 octave. An octave is a bandwidth in which the upper frequency is twice the lower one. A 113 octave is one-third of the bandwidth of an octave, where the range between the upper and lower frequencies are divided into three bands, equally spaced on a logarithmic scale. Because PSD is squared pressure per hertz and the measurements are done for a finite bandwidth Af, which is different from 1 Hz, a correction term must be introduced when converting from acoustic decibel to pressure2/hertz and vice versa. The following equations apply:s6 LPS = SPL - AL

(B2)

where SPL is given (measured) and A L is a correction term

where Af is the bandwidth (octave or 113 octave). By the definition (Bl) PI,, LPS = 20 log Pref

from which P,,, can be obtained. This PI,, , when squared, yields the PSD of the excitation PSD

=

(Pr,,)2/(1 Hz)

(B5)

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228

I

External data

I New data

Graph of sound pressure level as a function of frequency in Hz, 1/3octave

fi,Afi,SPLi

+ fmd A Li using (B.3)

Divide the frequency range into 113 octave bandwidth strips, A f

i

find LPSi using (B.2)

For each bandwidth, find the center frequency f

For each f find the corresponding SPLi from the external data

I +

f i d Prm% using (B.4)

I

a PSDi(f)=(Prmsi)* can be converted to PSD,(o)

Fig. B1 Conversion scheme (acoustic decibels to PSD).

The PSD is obtained in pressure2/hertz as a function of the frequency f (in hertz). This should be converted to pressure2/(radian/second) by dividing the values by 227, and changed to angular frequency o by multiplying the frequency (hertz) by 227. Figure B1 schematically described this procedure. In cases where PSD is to be converted into acoustic decibel, the same equations can be used.

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Appendix C

Finite Element Input Files for MJPDF Method

I

N THIS appendix, the input files used in the examples described in Chapter 12 are listed. The examples were computed on the ANSYS 4.2 finite element program. Inasmuch as the present version of ANSYS may be different (in 1996, Version 5.3 was already available) the following files must be properly modified according to the user's available version. When another finite element program is used, the following file may be used only as examples and guidelines for the preparation of compatible files. In several lines comments are added to improve understanding of the file.

I. File for Example 1 BEGIN-INP ml=1000 sl=33 m2=2 s2=0.1 mz=600 sz=30 deml = (6.283185307)*((~1)*(~2)) dem2=sqrt(6.283185307) dem3 =(dem2)*(sz) *set,x1,1049.5 *set,x2,1.85 /PREP7 ax1 =(-0.5)*(((~1-ml)*(x1-ml))/(sl*sl)) ax2= (-0.5)*(((~2-m2)*(~2-m2))/(s2*s2)) a=(axl) + (ax2) y = (exp(a))/(deml) et,l,l ex,1,2.le6 r,l,x2 n,1,0,0 n,2,10,0 e,L2 d,l ,all,O f,2,fx,xl afwrite finish

data from Table 11.1

initial value for F initial value for A exponent of normal dist. of F exponent of normal dist. of A

finite element modeling

1

constraint at bar end axial force on the other end

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G. MAYMON

/INPUT,27 finish /POST1 stress,axst,l,3 set,l *get,z,axst,l az= (-OS)*(((z-mz)*(z-mz))/(sz*sz)) zz= (exp(az))l(dem3) t=(y)*(zz> tt= (l)/(t)

solution phase postprocessor axst is axial stress z is axial stress at node 1 first part of Eq. (12.24) MJPDF inverse of MJPDF-the objective function

finish /OPT

optimization module dv variable, range 1000-1099 dv variable, range 1.7-2 sv variable, range 540-600

opvar,xl,dv,1000,1099,0.2 opvar,x2,dv,1.7,2,0.001 opvar,z,sv,540,600,1 opvar,tt,obj,,,0.001 oplist OPCoPY opeqn,,,J oprunP0 oplist,all finish

11. File for Example 2 BEGIN-INP m l = 1000 sl=33 m2=2 s2=0.1 m3=50 s3=2 my =33000 sy= 1000 p1 =I574960995 p2=2.506628275 *set,x1,104Y.5 *set,x2,1.85 *set,x3,53 /PREP7 ax1=(-OS)*(((xl-ml)*(xl -ml))/(sl*sl)) ax2= (-0.5)*(((~2-rn2)*(~2-m2))/(~2*~2)) ax3= (-0S)*(((x3-m3)*(~3-m3))/(~3*~3)) deml =(pl*sl)*(s2*s3) dem2= (p2)*(sy) q =((axl) +(ax2)) + (ax3) Y =(exp(q)/(deml)

1

J ]

data, Table 11.3

(~qrt(257))~ sqrt(2n) initial values for load effects term exponent of normal dist. of F exponent of normal dist. of I exponent of normal dist. of L

second part of Eq. (11.29)

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APPENDIX C

et,1,3 ex,1,2.le6 a=3*x2 r,l,a,x2,2 n,1,0,0 n,ll,x3,0 fill,l,ll e,1,2 egen,lO,l,l d,l,all,O f,ll,fy,xl afwrite finish /INPUT97 finish /POST1 stress,bend,3,10 set *get,z,bend,l ax=(-OS)*(((z-my)*(z-my))/(sy*sy)) zz= (exp(az))/(dem2) t=(y>*(zz> tt =(l)/(t)

solution phase postprocessor bend is bending stress

z is bending stress at node 1 first part of Eq. (11.29) MJPDF inverse of MJPDF-the objective function

finish /OPT opvar,xl,dv,1000,1099,1

optimization module dv variable, range 1000-1099 dv variable, range 1.7-2 dv variable, range 50-56 sv variable, range 30,000-36,000

opvar,x2,dv,1.7,2.0,0.001 opvar,x3,dv,50,56,0.1 opvar,z,sv,30000,36000,100 opvar,tt,obj oplist OPCOPY opeqn,,,,l oprun,40 oplist,all finish

111. File for Example 3 BEGIN-INP ml=1000 sl=33 m2=2 s2=0.1 my=600 sy =20 r=0.5 r2=(r)*(r) rd = (2) * (r) p2= 6.283185307

A

correlation coefficient

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G.MAYMON s=sqrt((l)-(r2)) p3 =2.506628275 ci=((l)-(r2))*(-2) c=(l)l(ci) *set,x1,1033 *set,x2,1.85

/PREP7 a1 =((xl -ml)*(xl-ml))/(sl*sl) a3 = ((x2 -m2)*(x2 -m2))/(s2*s2) a21 =((XI-ml)*(x2-m2)) a22=(sl)*(s2) a2 = ((rd)*(a2l))/(a22) p l = ((al) - (a2)) + (a3) p=(c)*(pl) deml = ((p2)*(sl)*((s2)*(s)) Y = (exp(p))/(deml) et,l,l ex,1,2.le6 r,l,x2 n,1,0,0 n,2,10,0 e,1,2 d,l,all,O f,2,fx,xl afwrite finish

second part of Eq. (11.31)

1

finite element modeling

/INPUT,27 finish

solution phase

/POST1 stress,axst,l,3 set *get,z,axst,l az= (-0.5)*(((z-my)*(z-my))/((sy)*(sy))) dem2= (p3)*(sy) zz= (exp(az))/(dem2) t=(y)*(zz) tt = (l)/(t)

postprocessor axst is axial stress

finish /OPT opvar,xl,dv,1000,1040,1 opvar,x2,dv,1.7,.2.0,0.001 opvar,z,sv,540,600,1 opvar,tt,obj oplist OPCOPY opeqn,,,,l oprun,40 oplist,all finish

z is axial stress at node 1 first part of Eq. (11.31) MJPDF inverse of MJPDF-the objective function optimization module dv variable, range 1000-1040 dv variable, range 1.7-2 sv variable, range 540-600

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APPENDIX C

IV. File for Example 4 BEGIN-INP p l = - 118.213806 p2 = 0.287027632 p3 = -0.00023850733 p4=6.978248213e-8 p5 =2.506628275 del=4.8587e-5 beta=2.1013491 gama=900 ml = 1000 sl =50 m2=2 s2=0.1 my=700 sy=33 r0=0.408828 r2=0.817656 deml =OX32859666 dem2=5.73410688 *set,x1,1100 *set,x2,1.85

/PREP7 z2= ((x2) -(m2))/(s2) zll=(pl)+((p2)*(xl)) zl2=(p3)*((xl)*(xl)) zl3=((p4)*(xl))*((xl)*(xl)) zl=((zll)+(zl2))+(z13) a1 =(zl)*(zl) a2= (z2)*(z2) a12=((-l)*(r2))*((zl)*(z2)) aa=((al)+(a12))+(a2) aaa=(-OS)*((aa)i(deml)) fzz= (exp(aaa))/(dem2) ax2= (-0.5)*((~2-rn2)*(~2-m2))/(~2*(~2) fx2= exp(ax2))/((~5)*(~2)) bl =(xl)-(gamma) b2=(bl)**(beta) b3=(-del)*(b2) fl=exp(b3) b4=(bl)**((beta)-(1)) fxl = ((del)*(beta))*((b4)*(fl)) fzl = (exp((-0.5*(al)))/(p5) fz2= (exp((-OS)*(a2)))/(p5)

1

A

coefficients in Eq. (11.39b) sqrt(2n) coefficients in Eq. (11.38) lower limit of Weibull dist.

Eq. (11.39b)

terms in Eq. (11.41)

fT,* in Eq. (11.43)

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G. MAYMON et,l,l ex,1,2.le6 r,l,x2 n,l,0,0 n,2,10,0 eS,2 d,l,all,O f,2,fx,xl afwrite finish /INPUT,27 finish /POST1 stress,axst,l,3 set *get,z,axst,l ax3= (-0.5)*(((z-my)*(z-my))l((sy*sy))) fx3=(exp((ax3)>/((ps>*(sy)) jj = (fx3)*(j) tt=(l)/(jj) finish /OPT opvar,xl,dv,950,1200,0.1 opvar,x2,dv,1.7,2.0,005 opvar,z,sv,600,700,0.1 opvar,tt,obj oplist OPCOPY opeqn,,,,l oprun,40 oplist,all finish

4

solution phase postprocessor

first part of Eq. (11.43) MJPDF inverse of MJPDF-the objective function optimization module

V. File for Example 5 BEGIN-INP m l = 1000 sl=33 m2=2 s2=0.1 m3 =570 deml =(6.283l85307)*((sl)*(s2)) *set,x1,1049.5 *set,x2,1.85 IPREP7 axl=(-0.5)*(((xl -ml)*(xl -ml))/(sl*sl))

ax2=(-0.5)*(((x2-m2)*(x2-m2))l(s2*s2)) a = (axl)+ (ax2) y =(exp(a))/(deml) et,l,l ex,1,2.le6 r,l,x2

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APPENDIX

n,1,0,0 n,2,10,0 eJ,2 d,l ,a11,0 f,2,fx,xl afwrite finish /INPUT97 finish /POST1 stress,axst,l,3 set *get,z,axst,l *set,yy,y tt=(l)/(yy) finish /OPT

C

solution phase postprocessor

optimization module

opvar,xl,dv,1000,1099,0.2 opvar,x2,1.7,2,0.001

opvar,z,sv,569.6,570,0.1 opvar,tt,obj,,,le-6 oplist OPCOPY opeqn,,,,l oprun,40 oplist,all finish

VI. File for Example 6 BEGIN-INP p1 =l.l56517643 p2= 1.313035285 p3=0.850918128 p4=8.199453e-11 p5=5.8042578e-10 p6=2.50662875 p7=76.18028355

coefficients in Eq. (11.51)

sqrt(2~) ( s q r t ( 2 ~ ) ) ~ x d e t [ REq. ]~.~ (11.50)

four parts of mean and standard deviations of P

mean of El standard deviation of EI

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G. MAYMON

creation of Eq. (11.51)

first part of Eq. (11.52)

second part of Eq. (11.52) MJPDF inverse of MJPDF-the objective function et,1,3 ex,l,ee r,1,1.333333,1.3 n,1,0,0 n,33,32,0 fi11,1,33 e,12 egen,32,1,I d,l,all,O d,33,a11,0 p,1,2,x1,,8 ~,9,10,~2,,16 p,17,18,x3,,24 p,25,26,x4,,32 afwrite finish

nNPUT,27 finish

'

finite element modeling

solution phase postprocessor mom is the bending moment

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APPENDIX

set *get,ss,mom,l finish /OPT opvar,xl,dv,8,15.2,0.001 opvar,x2,dv,8,15.2,0.001 opvar,x3,dv,8,15.2,0.001 opvar,x4,dv,8,15.2,0.001 opvar,ss,sv,1096,1100,1 opvar,tt,obj oplist OPCoPY opeqn,,,S oprun,40 oplist,all finish

C

237

ss is the bending moment at node 1 optimization module

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References

' Timoshenko, S., and Young, D. H., Vibration Problems in Engineering, 3rd ed., D. Van Norstad, Princeton, NJ, 1955. Den Hartog, J. P., Mechanical Vibrations, McGraw-Hill, New York, 1956. Hurty, W. C., and Rubinstein, M. F., Dynamics of Structures, PrenticeHall, Englewood Cliffs, NJ, 1964. Biggs, J. M., Introduction to Structural Dynamics, McGraw-Hill, New York, 1964. Thomson, W. T., Theory of Vibration and Applications, Prentice-Hall, Englewood Cliffs, NJ, 1975. Vierck, R. K., Vibration Analysis, Harper and Row, Addison, IL, 1979. De Vries, G., "Emploi de la methode vectorielle d'analyse dans les essais de vibration," La Recherche Aeronautique, Vol. 74,1960, pp. 41-47. Beatrix, C., "Les Procedes Exprimentaux de I'essai Global de Vibration d'une structure," La Recherche Aeronautique, Vol. 109, 1965, pp. 57-64. Maymon, G., "Response of Aeronautical Structures to Random Acoustic Excitation-a Stress Mode Approach," Random Vibration-Status and Recent Developments, edited by I. Elishakoff and R. H. Lyon, The Stephen Harry Crandall Festschrift, Studies in Applied Mechanics, Vol. 14, Elsevier, New York, 1986, pp. 267-277. lo Prazen, E., Modern Probability Theory and its Application, Wiley, New York, 1960. " Elishakoff, I., Probabilistic Methods in the Theory of Structures, Wiley, New York, 1983. l2 "Environmental Test Methods and Engineering Guidelines," MILSTD-810D, U.S. Department of Defense, 1983. l3 Der Kiureghian, A., "Structural Response to Stationary Excitation," Journal of Engineering Mechanics, Vol. 106, No. 6, 1980, pp. 1195-1213. l4 Crocker, M. J., "The Response of Supersonic Transport Fuselage to Boundary Layer and to Reverberant Noise," Journal of Sound and Vibration, Vol. 9, No. 1, 1969, pp. 6-20. l5 Maestrello, L., "Measurement and Analysis of the Response Field of Turbulent Boundary Layer Excited Panels," Journal of Sound and Vibration, Vol. 2, No. 3, 1965, pp. 270-292. l6 Schlichting, H., Boundary Layer Theory, McGraw-Hill, New York, 1955. l7 Maymon, G., "Response of Plate-Like Structures to Correlated Random Pressure Fluctuations," Proceedings of the 33rd AIAA/ASME/ASCE/

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53 Broek, D., Elementary Engineering Fracture Mechanics, 4th ed., Martinus, Nijhoff, The Hague, Netherlands, 1985. 54 Broek, D., The Practical Use of Fracture Mechanics, Kluwer Academic, Norwell, MA, 1989. 55 Barsom. J. M.. and Rolfe. S. T.. Fracture and Fatime Control in Structures: ~ ~ ~ l i c a t i o n s~'roafc t u r e ~ e c h a n i c s , Englewood Cliffs, NJ, 1987. 56 Anderson, T. L., Fracture Mechanics: Fundamentals and Applications, CRC Press, Boca Raton, FL, 1991. 57 Parton, V. Z., and Morozov, E. M., Mechanics of Elastic-Plastic Fracture, Hemisphere, New York, 1988. 58 Paris, P. C., and Erdogan, F., "A Critical Analysis of Crack Propagation Laws," Journal of Basic Engineering, Vol. 85, No. 4, 1963, pp. 528-534. 59 Bogdanoff, J., "A New Cumulative Damage Model," Journal of Applied Mechanics, Pts. 1-3, Vol. 46, 1979, pp. 245-257, 733-739. 60 Bogdanoff, J., and Kozin, A., "A New Cumulative Damage Model," Journal of Applied Mechanics, Pt. 4, Vol. 47, 1980, pp. 40-44. 61 Sobczyk, K., and Spencer, B. F., Random Fatigue, From Data to Theory, Academic, New York, 1992. 62 Noronha, P. J., et al., "Fastener Hole Quality, I and 11, U.S. Air Force Flight Dynamics Lab. Tech. Rep. AFFDL TR-78-206, Wright-Patterson AFB, OH, 1979. 63 Virkler, D. A., et al., "The Statistical Modelling Nature of Fatigue Crack Propagation," Journal of Engineering Materials and Technology, Vol. 10, NO. 4, 1979, pp. 148-153. Ghonem, H., and Dore, S., "Experimental Study of the Constant Probability Crack Growth Under Constant Amplitude Loading," Engineering Fracture Mechanics, Vol. 27, No. 1, 1987, pp. 1-25. 65 Murukami, Y., et al. (eds.), Stress Intensity Factors Handbook, 3rd ed., Vols. 1 and 2, Pergamon, Oxford, England, UK, 1990. 66 Cherepanov, G. P., Mechanics of Brittle Fracture, McGraw-Hill, New York, 1979. 67 Lin, Y. K., and Yang, J. N., "A Stochastic Theory of Fatigue Crack Propagation," A I A A Journal, Vol. 23, No. 1, 1984, pp. 117-124. Lin, Y. K., Wu, W. F., and Yang, J. N., "Stochastic Modeling of Fatigue Crack Propagation," Proceedings of the IUTAM Symposium on Probabilistic Methods in Mechanics of Solids and Structures, Springer-Verlag, Berlin, 1984, pp. 103-110. 69 Sobcyzk, K., "Stochastic Modeling of Fatigue Crack Growth," Proceedings of the IUTAM Symposium on Probabilistic Methods in Mechanics of Solids and Structures, Springer-Verlag, Berlin, 1984, pp. 111-119. O' Spencer, B. F., and Yang, J., "Markov Model for Fatigue Crack Growth," Journal of Engineering Mechanics, Vol. 114, No. 12, 1988, pp. 2134-2157. Yang, J. N., Salivar, G. C., and Annis, C. G., "Statistical Modeling of Fatigue Crack Growth in Nickel Based Superalloy," Engineering Fracture Mechanics, Vol. 18, No. 2, 1983, pp. 257-270.

renti ice-H&

Purchased from American Institute of Aeronautics and Astronautics

REFERENCES

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72 Sobczyk, K., "Modeling of a Random Fatigue Crack Growth," Engineering Fracture Mechanics, Vol. 24, No. 4, 1986, pp. 609-623. 73 Zhu, W. Q., Lin, Y. K., and Lei, Y., "On Fatigue Crack Growth Under Random Loading," Engineering Fracture Mechanics, Vol. 43, No. 1, 1992, pp. 1-12. 74 Libiard, A. B., "Probabilistic Fracture Mechanics," Fracture Mechanics, Current State, Future Prospects, Pergamon, Oxford, England, UK;1979. 75 Varanasi, S. R., and Wittaker, I. C., "Structural Reliability Prediction Method Considering Crack Growth and Residual Strength," Fatigue Crack Growth Under Spectrum Loads, STP 595, ASTM, Philadelphia, 1976, pp. 292-305. 76 Ortiz, K., "Stochastic Modeling of Fatigue Crack Growth," Ph.D. Dissertation, Dept. of Civil Engineering, Stanford Univ., Stanford CA, 1984. 77 Ortiz, K., and Kiremidjian, A. S., "Time Series Analysis of Fatigue Crack Growth Rate Data," Engineering Fracture Mechanics, Vol. 24, No. 5, 1986, pp. 317-334. 78 Ortiz, K., and Kiremidjian, A. S., "Stochastic Modeling of Fatigue Crack Growth," Engineering Fracture Mechanics, Vol. 29, No. 3, 1986, pp. 317-334. 79 Lawrence, M., Liu, W. K., Besterfield, G., and Belytschko, T., "Fatigue Crack Growth Reliability," Journal of Engineering Mechanics, Vol. 116, NO. 3,1990, pp. 698-708. 80 "Fatigue Crack Growth Computer Program NASAIFLAGRO," Version 2, NASA Publ. JSC-22267A, Houston, TX, May 1994. Yang, J. N., and Donath, R. C., "Statistical Crack Propagation in Fastener Holes Under Spectrum Loading," Journal of Aircraft, Vol. 20, NO. 12,1983, pp. 1028-1032. 82 Tanaka, H., and Tsurui, A., "Reliability Degradation of Structural Components in the Process of Fatigue Crack Propagation Under Stationary Random Loading," Engineering Fracture Mechanics, Vol. 27, No. 5, 1987, pp. 501-516. 83 Tsurui, A., Nienstedt, J., Schueller, G. I., and Tanaka, H., "Time Variant Structural Reliability Analysis Using Diffusive Crack Growth Models," Engineering Fracture Mechanics, Vol. 34, No. 1, 1989, pp. 153-167. 84 Larsson, H., and Bernard, J., "Fracture of Longitudinally Cracked Ductile Tubes," International Journal of Pressure Vessel Piping, Vol. 6, 1978, pp. 223-243. 85 Feddersen, C. E., "Evaluation and Prediction of the Residual Strength of Center Cracked Tension Panels," STP 486, ASTM, Philadelphia, 1970, pp. 50-78. Acoustic Fatigue Sub-series Vol. 1, Data Sheets 66013, 66016, 66017, 66018, ESDU International, London, 1976, 1978.

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ANSYS, 77, 81, 148, 151, 153, 157, 163, 165-167, 171-172, 174, 177-178 Base excitation, 4-6, 2 6 2 9 Beam, simply supported, 32,68-71.98, 188 Boundary-layer excitation model, 43-51 Burdekin-Stone criterion, 210-211 CALREL, 133-134, 193,215 Cantilever beam, 31, 140, 147, 165 Chen-Lind approach, 120 Choleski decomposition, 163 Classical oscillator, 1 COMPASS, 133 Composite structures, 148 Cornell reliability index, 117-1 18 Crack growth, 195-197,210,213,216 Crack growth models, 195-196 Crack growth rate, 197-199,201-203,205,216 Crack length, 198,202-209,212-214,217 Crack size, 196-198, 209 Crocker model, 44 Cumulative distribution function (CDF), 35-36, 112-113, 118-122, 128, 158, 170, 182-183 crack length, 206-208 Design point, 157, 159, 162, 165, 173, 178 Deterministic continuous system base excitation, 26-29 differential equations, 22-26 stress response, 29-34 Deterministic loads, 25 Differential equation models, 197, 199 Dufting oscillator, 74 Duhamel integral, 7, 9 Dynamic load factor (DLF), 3, 8, 29 Dynamic response, 82 Elastic bar, 108-109 Elastic system, 98-103 Excitation functions, 41-43 Failure criterion, 209-21 1,216217 Failure surface, 110-1 17, 124, 130, 135, 142, I48 Fatigue, 195 Finite element method, 21,25, 31, 33, 72-73, 77,79,82,98, 104, 123, 133-135, 148, 157, 159-161, 163-164, 166, 174-175, 177-178,215

First-order reliability method (FORM), 115-117, 123, 133, 140, 142, 154, 214215 FLAGRO, 200 Fracture mechanics, 195 Gaussian excitations, 84-86, 103 General force input, 6 1 1 Geometric nonlinearity, 82 Harmonic excitations, external, 1-4 Hasofer-Lind method, 118, 123 Hoenbichler-Rackwitz approach, 120 Importance factors, 132- 133 Iterative procedure, 104, 131-132 Lagrange multiplier method, 124-130, 142, 147, 151, 154-155 Linear structures,62-68 Linear systems, 5 3 4 2 Load cycles, 196197 Maestrello's experiments, 44-45 Markov-chain crack growth model, 202 Microstructures, 196 Modal damping coefficients, 60,88-89.95 Modified joint probability density function (MJPDF), 157-163 numerical examples, 163-178 nonnormal dependent variables, 169-173 normally distributed, correlated variables, 167-168 normally distributed, random field, correlated variables, 174-178 normally distributed, uncorrelated variables, 164-166 threshold value problem, 173-174 Monte Carlo simulation, 108, 133, 142-143, 154155 Multiple-degree-of-freedom(MDOF) systems, 21, 34, 6365.67, 79 differential equations 13-15 equilibrium equations, 13 generalized masses, dampings, rigidities, and forces, 15-18 nonlinear random response, 98-100 normal modes, 13-16 random excitations, 56-62, 184 spectral moments, 1 8 4 185 uncoupled differential equations, 18-19

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246 Nataf model, 160, 162 NESSUS, 134 Nondeterministic structures, 107-108, 179 Nonlinear coefficients, 79-82 Nonlinear response elastic systems, 98-103 SDOF systems, 87-92 two-DOF systems, 92-98 Nonlinear structures, 73 Normal modes, 13-16.21, 24, 30, 33,94,98, 102 Numerical programs, 132-134 Panel vibration, 43 Periodic loads, 195 Power spectral density (PSD) functions, 4142,45,47, 51, 54-56.58, 61,68, 179-180, 184, 186, 189 Pressure excitations, 47, 49 Pressure fluctuations, 43 Probabilistic analysis, 107-108, 110, 132-135, 157-158 Probability density function (PDF), 35-36, 112-113, 115, 118-121, 123, 138, 140, 142, 170, 182-183 crack length, 207-209, 21 3-2 14.2 17 Probability of failure, 123-124, 151, 165, 178, 182, 193, 197.211-215.217 Probability of threshold crossing, 185-193 PROBAN, 133, 142-143, 146-147, 154-155, 165-167, 175, 178 Rackwitz-Fiessler method, 118-121, 123, 131 Random excitations, 35, 38-39,84 linear structures, 6 2 4 8 practical characterization, 3 9 4 1 SDOF systems, 76 stationary, 54-56, 6 2 4 8 Random functions, 53 basic concepts, 35-39 Random Gaussian force, 87-98 Random process (RP) method, 197,2W202, 205, 217 Random structures, 219 Random variable (RV) methods, 197, 199-200,205 Random vibrations, 1, 8, 219 Rectangular plates, 32-34,49 Reliability, structural, 107, 127 Reliability index, 117-122, 140, 143, 151, 162, 165, 173, 177 Reliability methods, 115-1 17 Resonance frequencies, 19.21, 24, 56, 89 Rosenblatt transformation, 120

INDEX Safety factor, 107-109 Second-order reliability method (SORM), 116-117, 133, 140, 142, 154-155.214-215 Sensitivity, 132-133 Single-degree-of-freedom (SDOF) systems, 34.42, 62,67 base excitation, 4-6 external harmonic excitation, 1-4, 56 general force, 6-1 1 nonlinear behavior, 74-79, 87-92 random excitations, 53-56 random Gaussian force, 87-92 stresses, 11 transfer function, 55 S-N curves, 195 Sound pressure level (SPL), 41,47 Stationary Gaussian process, 179-183 Statistical linearization method MDOF systems, 83-87 Stochastic crack growth models, 196202 differential equation models, 197, 199 Stresses, 11 Stress intensity factor, 215 Stress modes, 30-34, 70-71, 104 Stress response calculation, 104-106 Structural analysis, 107 Structural design, 107, 210 Taylor series expansion method, 123, 135-137, 157, 188, 190-191 evaluation point selection, 137-140, 143, 145, 149-153 examples, 140-155 Tensile stress, 109, 153 Threshold value problem, 173-174 Two-degree-of-freedom(two-DOF) systems, 5 8 4 2 , 6 7 , 184 random Gaussian force, 92-98 Turbulent boundary layer, 43 Turbulent flow, 38,43 autocorrelation functions, 5 1 Uncertainty approach, 107 Vibrating structures, structural vibration, 1, 8, 194 Viscous damping, 1,92 von Mises formulation, 140 Weibull distribution, 129, 131, 169 White noise, 4 1 4 2 Wideband excitation, 40 Wiener-Khintchine formulas, 37.43, 49, 54, 58,66 Wind tunnel measurements, 43 Yield stress, 129, 131