Manual of the Theory of Elasticity

Table of contents :
FRONT COVER
TITLE PAGE
CONTENTS
NOTATION
Chapter 1 THEORY OF STRESS
I. STATIC AND DYNAMIC EQUILIBRIUM EQUATIONS
II. SURFACE CONDITIONS
III. STATE OF STRESS AT A POINT
PROBLEMS
Chapter 2 THEORY OF STRAIN
I. STRAIN EQUATIONS IN ORTHOGONAL CO-ORDINATES
II. STATE OF STRAIN AT A POINT
III. CESARO'S FORMULAS
PROBLEMS
Chapter 3 BASIC EQUATIONS OF THE THEORY OF ELASTICITY AND THEIR SOLUTION FOR SPECIAL CASES
I. ORTHOGONAL CURVILINEAR CO-ORDINATES
II. RECTANGULAR CO-ORDINATES
III. CYLINDRICAL CO-ORDINATES
IV SPHERICAL COORDINATES
PROBLEMS
Chapter 4 GENERAL SOLUTIONS OF THE BASIC EQUATIONSOF THE THEORY OF ELASTICITY. SOLUTION OF THREE-DIMENSIONAL PROBLEMS
I. HARMONIC EQUATION (LAPLACE'S)
II. BIHARMONIC EQUATION
III. BOUNDARY VALUE PROBLEMS FOR THE HARMONIC AND BIHORMONC HARMONIC
IV. VARIOUS FORMS OF THE GENERAL SOLUTIONS OF LAME'S EQUATIONS
PROBLEMS
Chapter 5 PLANE PROBLEM IN RECTANGULAR CO-ORDINATES
I. PLANE STRESS
II. PLANE STRAIN
III. SOLUTION OF BASIC EQUATIONS
PROBLEMS
Chapter 6 PLANE PROBLEM IN POLAR COORDINATES
I. PLANE STRESS
II. PLANE STRAIN
III. SOLUTION OF BASIC EQUATIONS
PROBLEMS
Chapter 7 TORSION OF PRISMATIC AND CYLINDRICAL BARS
I. PURE TORSION OF BARS OF CONSTANT SECTION
II. PURE TORSION OF CIRCULAR BARS (SHAFTS) OF VARIABLE SECTION
PROBLEMS
CHAPTER 8 THERMAL PROILEM
I. STEADY·STATE THERMAL PROCiESS
II. TRANSIENT THERMAL PROCESS
PROBLEMS
Chapter 9 CONTACT PROBLEM
I. THE ACTION OF PUNCHES ON AN ELASTIC HALF-PLAN
II. THE ACTION OF PUNCHES ON AN ELASTIC HALF-SPACE
III. CONTACT BETWEEN TWO ELASTIC BODIES
PROBLEMS
Chapter 10 DYNAMIC PROBLEM
I. SIMPLE HARMONIC MOTION
II. PROPAGATION OF VOLUME WAVES IN AN ELASTIC ISOTROPIC MEDIUM
III. WAVE PROPAGATION OVER THE SURFACE OF AN ELASTIC ISOTROPIC BODY
IV. EXCITATION OF ELASTIC WAVES BY BODY FORCES
V. DEFORMATION OF SOLIDS UNDER CENTRIFUGAL FORCES
VI. PLANE DYNAMIC PROBLEMS
VII. THERMODYNAMIC PROBLEM
PROBLEMS
REFERENCES
AUTHOR INDEX
SUBJECT INDEX
BACK COVER

Citation preview

MANUAL OP THE THEORY OF ELASTICITY

B. r . Penan npotfi. dotcm. mexn. nayK

pyHOBOflCTBO H PELUEHWO 3AflAW nO TEOPMM

y n p y ro cT M

M O C K B A « B b I C I I J A f l IIIK O J fA ,

V. G. llekach

MANUAL o f th e THEORY of ELASTICITY Translated from the Russian by M. Konyaeva

MIR PUBLISHERS MOSCOW

l'ir s t Published 1979 Rovisod from llio 1977 Russian odition

© HanaTanbCTBO «BucmaH nmojiai, 1977 © English translation, Mir Publishers, 1979

CONTENTS Notation

.

Chapter I ........................................................................................ Theory of S t r e s s ................................................................................

9 9

I. Static and Dynamic Equilibrium Equations . . . . II. Surface C o n d i tio n s ........................................................ III. Stato of Stress a t a P o i n t ............................................ P ro b le m s............................................................................

9 12 13 15

Strain Equations in Orthogonal Co-ordinates . . . Stato of Strain a t a P o i n t ........................................ Cesaro’s F o rm u la s........................................................ P r o b le m s ............................................................................

Orthogonal Curvilinear C o -o rd in ates........................ Rectangular C o - o r d in a t e s ............................................ Cylindrical C o -o rd in ates................................................ Sphoricnl C o-o rd in ates..................................................... P ro b le m s.............................................................................

£ £ £ £ *

nsic Equations of die Theory of Elasticity and Their Solution or Special C a se s................................................................................ I. II. III. IV.

§

§

I. II. III.

SS8K

Chapter 2 Theory of J

Chapter 4 General Solutions of the Basic Equations of the Theory of Elasti­ city. Solution or Three-dimensional P r o b le m s .........................

66

I. Harmonic Equation (Laplace’s ) ..................................... II. Biharmonic E q u a t i o n ...................................................... III. Boundary Value Problems for the Harmonic and Biharmonic E q u a tio n s ...................................................... IV. Various Forms of the General Solutions of Lame's Equations ........................................................................... Problems ...........................................................................

66 71

79 83

Chapter 5 Plane Problem in Rectangular C o-ordinntcs.................................

106 106

I. Plane S t r e s s .................................................................... II. Plane S t r a i n .................................................................... III. Solution of Basic E q u a tio n s ........................................ P r o b le m s .............................................................................

106 lofl 1(H) 11!)

66

72

Contents Plane Problem in Polar Co-ordinalcs.......................................... I. Plane S t r e s s ................................................................ II. Plane S tra in ................................................................ III. Solution of Basic E q u atio n s................................. Problems ................................................................... Chapter 7 .................................................................................. Torsion of Prismatic and Cylindrical B a r s .............................. I. Pure Torsion of Bars of Constant S e c tio n .......... II. Pure Torsion of Circular Bars (Shafts) of Variable Section ....................................................................... P ro b le m s....................................................................... Chapter 8 .................................................................................. Thermal P r o b le m ............................................................ I. Steady-state Thermal P ro cess..................................... II. Transient Thermal Pro cess......................................... Problems ................................................................... Chapter 9 ............................................ Contact P roblem ............................................. ,1 It® * cti.on of, Plmcl183 on an Elastic Half-piano II. The Action of Punches on an Elastic I-IaU-spaco III. Contact Between Two Elastic Bodies . . . Problems ................................................................... Chapter 1 0 ......................................... Dynamic Problem ................... I. Simple Harmonic M o tio n .............................. 11 MoSfum'0" °l Volume Wttvcs in an Elastic Isotropic l n ‘ £ £ l T I y i0n 6v" ) he SUrfaCC ° f «" Elastic' IV. Excitation of Elastic Waves b y ' Body' Forces ‘ Vi' P?Inrmn 10n 0t Snolii? Under Centrifugal Forces . . P'ano Dynamic Problems . . . . VII. Thermodynamic P r o b le m ...................... ... References Author Index Subject Index

302 308 310

NOTATION - /!„+ /?„ c o la +

!+ * - o ( - p £ )

Ck. 1. Theory of Slrett 9Br , 1 *00 , 1 + X > , Y . - Y J + Y ^ + Y*, Zv = z xl + Zym + z,n, whore I = cos (i, v), m = cos (y, v

= 1.'

(1.2)

® (a. v), P -1-

„ n h°( inrlT nl b?undary cona . -^ -X . + -£ r X i + [ - £ r + ( o - 2 ) V» |y,3= 0 Theory p. 314

Problem« and

( - £ - l-aV, ) « P , - ( ^ + ^ - ) ( q . t I ri,1,pi equations in isoslatic co-ordinates for a mroo-dimcnsional problem, soo tho monogranh 1. p. 42.

Problems 1 10 The stresses at a point of an elastic body are: X x =

= 50 N/cm*, Y u = 0, Z t = - 3 0 N/cm*, X v = 50 N/cm», y = —75 N/cm*, Zx = 80 N/cm*. Find the principal normal and shearing stresses. Use Eqs. (1.4), (1.5), and (1.8) to solve the problem.

Answer. CTj = 99-3 N/cm*, a2 = 58.8 N/cm*, o3 = = —138 N/cm*, Tmai = 118.6 N/cm*.

Chapter 2 THEORY OF STRAIN

I.

STRAIN EQUATIONS IN ORTHOGONAL CO-ORDINATES

. _

1

hh

Vgh

Uh ,

-CT

1

2a Yghfv

dVgh 3av

ehv= ] / r 7 7 a 5 r ( y = ') + V ^ t o ^ [ ~ y ¥ ) ’ where ehh = linear strains, eAv = shearing strains. The d ilatatio n is ^ = ^ e hh = \ \ ^ { V e ^ x ) - \ - ^ { V g ^ l u - 2 + (2 . 2 )

+

The components of elem entary rotation are “• = w z z

[ - s i r (1V 7iUs]- i

( v 7 i U *]] ■

= iv b r

0,31“

w

t

^_ [■^

(2.3)

(

- i k ' ^

u ' }] ■

Note th a t on the basis of formulas of the calculus of vectors (div rot u = 0) the components of rotation identi-

Strain Equations In Orthogonal Co-ordinates

25

cally satisfy the equality (Vgaga wi) +

{V gagi