Manual of the Theory of Elasticity

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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