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Mathematical Theory f!f Elastuzty

1. S. SOKOLNIKOFF

Profe88OT oj Mathematic8
University of California

L08 Angele8

SECOND EDlTlON

TATA McGttAw.tHLL PUBLISHING COMPANY LTD.
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Page 245

234 MATHEMATICAL THEORY OF ELASTICITY

where the origin has been taken at the centroid of the section. Along the

side y == a we have 2 == 0, and hence no condition is imposed on the
boundary values of the function B(y) ruonlt this side, whereas we require
that

B(y) = W. x, = W. (2a + y)~
21. 21. 3

on x = ± 2a + y. va
Therefore we tako.

B(y) == ::;: (2a + y)2,
and from (60.7) it follows that

2 W. [2(<1 - .J1) 2]
V T(x, y) = T. 3(<1 + 1) y - 3 a ,

where we have set a = 0 and are, accordingly, solving the problem of pure
flexure by a load applied at the center of flexure.

%

FIG. 46

The differential equation and boundary conditions on T(x, y) can be
readily satisfied when Poisson's ratio takes a particular value, namely,
<1 == .J1, which corresponds to incompressible materials. In this case, we
have '

V2T(x, y) = - ~ ~. a in R,

T(x, y) = 0 on t!. : aU (2a + y)!.
We try

T(x, y) 0= k[x2 - U(2a + y)2J(y - a)
and find that the stress function is given by

(60.12) T(x, y} = ::;: [X2 - ~ (2a + y)l] (y - a).

Page 246

EXTENSION, TORSION, AND FLEXURE OF BEAMS 235

Equation (60.11) now yields the position of the center of flexure.
Straightforward calculations give

I. = 3.y3 a'
2

[f T(x, y) dx dy = 3 ~I~·a' = a:~,
( x'ydy = 2 (av'3 x'(.y3x - 2a) .y3dx = ~.Y3_a·,

Jc Jo 5
and therefore y., = O. Since the cross section is symmetrical about the
y-axis, we see that the center of flexure is at the origin, and hence at the
centroid of the section. Thus, the function T(x, y), given above, fur-
nishes the solution of the flexure problem for an incompressible beam of
equilateral triangular section when the load W. is applied at the centroid.

The flexure function T(x, y) was introduced, for the case of bending by
a load along a principal axis, by Timoshenko' and was used by him to
solve the flexure problem for a number of cross sections.

It will be recalled that, when the functions R(x) and S(y) were intro-
duced, it was remarked that they might be so chosen as to yield either a
simple boundary condition or a simple differential equation for the func-
tion T(x, y). The first course led to Timoshenko's stress function
T(x, y), discussed above, which can be interpreted physically as repre-
senting the deflection of an elastic membrane stretched over an opening
of boundary C in a plane plate and subjected to a nonuniform load. We
follow now the alternative course and choose R(x) and S(y) to make
T(x, y) a harmonic function.

Let us define

(60.13) {
R(x) = -w,Ky x 2 - J1.OiX,
S(y) = p.uK.y2 - J1.Oiy.

We shall designate the function 7'(x, y) defined by Eqs. (60.3) and (60.4)
with this choice of R(x) and S(y) by lI1(x, 'y). Then these ~quations
become

(60.14) \l211f(X, y) = 0

and

ddM = [J1.(1 + rr)K.x' - J1.rrKrl/' + J1.OiV] dt!!l
8 . 8

- [J1.(1 + rr)K.y' - J1.rrK.x2 - J1.OiX) :: on C.

1 S. Timoshenko, Proceedings of the London M atherna.tical Society, aer. 2, vol. 20
(1922), p. 398.

An account of this work will be found in S. Timoshenko and J. N. Goodier, Theory
of FJastieity, Sees. 106-113.

Page 491

THE-ORY
OF ELASTICITY

IVAN S. SOKOLNIKOFF

University of California, Los Angelcr

','

A unified treatmen! of seemingly unrelated
great value to engin .;ers an{j mathematicians ccncr'~
with engin.:':"ring problems of d'?formaille bodie~
des a chapter on ane~vtical and nume~kal me
approximate solutioi l (Jf problems in e!::;stidty
tional and related technique,s.

TATA HcGItAW·HILL P~BLlSHING"COMPAN'I' ~ .
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