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Unformatted text preview: 3.5 AIRY’S STRESS FUNCTION The preceding sections have demonstrated that the solution of twodimensional
problems in elasticity requires integration of the differential equations of equilibrium [Eqs. (3.6)], together with the compatibility equation [Eq. (3.9) or (3.12)] and
the boundary conditions [Eqs. (3.7)]. In the event that the body forces Fx and Fy
are negligible, these equations reduce to
0 xy
0x
+
= 0,
0x
0y ¢ 0 02
02
+ 2 ≤1
0x2
0y 0 y + 0y
x + xy 0x
y2 =0 =0 (a)
(b) together with the boundary conditions (3.7). The equations of equilibrium are identically satisfied by the stress function, £ 1x, y2, introduced by G. B. Airy, related to
the stresses as follows:
x = 02 £
,
0y2 y = 02 £
,
0x2 xy = 02 £
0x 0y (3.13) Substitution of (3.13) into the compatibility equation, Eq. (b), yields
04 £
04 £
04 £
+2 2 2+
= §4 £ = 0
0x4
0x 0y
0y4 (3.14) What has been accomplished is the formulation of a twodimensional problem in
which body forces are absent, in such a way as to require the solution of a single
biharmonic equation, which must of course satisfy the boundary conditions.
It should be noted that in the case of plane stress we have z = xz = yz = 0
and x, y, and xy independent of z. As a consequence, xz = yz = 0, and x, y,
z, and xy are independent of z. In accordance with the foregoing, from Eq. (2.9), it
is seen that in addition to Eq. (3.14), the following compatibility equations also
hold:
02 z
= 0,
0x2 02 z
= 0,
0y2 02 z
=0
0x0y (c) Clearly, these additional conditions will not be satisfied in a case of plane stress by
a solution of Eq. (3.14) alone. Therefore, such a solution of a plane stress problem
has an approximate character. However, it can be shown that for thin plates the
error introduced is negligibly small.
102 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 103 It is also important to note that, if the ends of the cylinder shown in Fig. 3.1 are
free to expand, we may assume the longitudinal strain z to be a constant. Such a
state may be called that of generalized plane strain. Therefore, we now have
= 1E 2 x = 1E 2 y xy = xy G and
z ¢ x y  1 x x =1 1 y ¢  + y2 +E ≤≤z z (3.15)
z (3.16) Introducing Eqs. (3.15) into Eq. (3.8) and simplifying, we again obtain Eq. (3.14) as
the governing differential equation. Having determined x and y, the constant
value of z can be found from the condition that the resultant force in the z direction acting on the ends of the cylinder is zero. That is,
4
where
3.6 z z dx dy = 0 (d) is given by Eq. (3.16). SOLUTION OF ELASTICITY PROBLEMS Unfortunately, solving directly the equations of elasticity derived may be a formidable task, and it is often advisable to attempt a solution by the inverse or semiinverse method. The inverse method requires examination of the assumed solutions
with a view toward finding one that will satisfy the governing equations and boundary conditions. The semiinverse method requires the assumption of a partial solution formed by expressing stress, strain, displacement, or stress function in terms of
known or undetermined coefficients. The governing equations are thus rendered
more manageable.
It is important to note that the preceding assumptions, based on the mechanics
of a particular problem, are subject to later verification. This is in contrast with the
mechanics of materials approach, in which analytical verification does not occur.
The applications of inverse, semiinverse, and direct methods are found in examples
to follow and in Chapters 5, 6, and 8.
A number of problems may be solved by using a linear combination of
polynomials in x and y and undetermined coefficients of the stress function £ .
Clearly, an assumed polynomial form must satisfy the biharmonic equation and
must be of second degree or higher in order to yield a nonzero stress solution of
Eq. (3.13), as described in the following paragraphs. In general, finding the desirable polynomial form is laborious and requires a systematic approach [Refs. 3.2 and
3.3]. The Fourier series, indispensible in the analytical treatment of many problems
in the field of applied mechanics, is also often employed (Secs. 10.10 and 13.6). 3.6 Solution of Elasticity Problems 103 ch03.qxd 12/20/02 7:20 AM Page 104 Another way to overcome the difficulty involved in the solution of Eq. (3.14) is
to use the method of finite differences. Here the governing equation is replaced by
series of finite difference equations (Sec. 7.3), which relate the stress function at
stations that are removed from one another by finite distances. These equations, although not exact, frequently lead to solutions that are close to the exact solution.
The results obtained are, however, applicable only to specific numerical problems.
Polynomial Solutions
An elementary approach to obtaining solutions of the biharmonic equation uses
polynomial functions of various degree with their coefficients adjusted so that
§4 £ = 0 is satisfied. A brief discussion of this procedure follows.
A polynomial of the second degree,
£2 = a2 2
c2 2
x + b2xy +
y
2
2 (3.17) satisfies Eq. (3.14). The associated stres...
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This note was uploaded on 01/09/2011 for the course ME 201 taught by Professor Yok during the Spring '08 term at Boğaziçi University.
 Spring '08
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