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92T,v dx2 dy2 dxdy

Eliminating rvv between eqns (4) and (1) we obtain:

E dy2 E dy2 E dx2

or d2oxx 32<7v

dx2 dx2

E dx2 d2o dx2

do yy dy2

A similar development for cylindrical coordinates yields the stress equation of compability which in the case of axial symmetry (where stresses are independent of 9) reduces to:

From the earlier work of this chapter it should now be evident that in elastic stress analysis there are generally fifteen unknown quantities to be determined; six stresses, six strains and three displacements. These are functions of the independent variables x, y and z (in cartesian coordinates) or r, 0 and z (in cylindrical polar coordinates). A quick look at the governing equations presented earlier in the chapter will convince the reader that the equations are difficult to solve for these unknowns, except for a number of relatively simple problems.

In order to extend the range of useful solutions several techniques are available. In the first instance one may make certain assumptions about the physical problem in an effort to simplify the equations. For example, are the loading and boundary conditions such that:

(i) the plane stress assumption is adequate - as in a thin-walled pressure vessel? or,

(ii) does plane strain exist - as in the case of a pressurised thick cylinder?

If we can convince ourselves that these assumptions are valid we reduce the three-dimensional problem to the two-dimensional case.

Having simplified the governing differential equations one must then devise techniques to solve, or further reduce, their complexity. One such concept was that proposed by Sir George B. Airy.t. His approach was to assume that the stresses in the two-dimensional problem axx, oyy and r, v could be described by a single function of x and y. This function 4> is referred to as a "stress function" (later the "Airy stress function") and it appears to be the first time that such a concept was used. Airy's approach was later generalised for the three-dimensional case by Clerk Maxwell.^

Airy proposed that the stresses be derived from a particular function <p such that:

a2 19 1

dr2 r dr

8.27. The stress function concept a20 df d24>

a20 df d24>

t G.B. Airy, Brit. Assoc. Advancement of Sei. Rep. 1862; Phil. Trans. Roy. Soc. 153 (1863), 49-80 i J.C. Maxwell Edinburgh Roy. Soc. Trans., 26 (1872), 1 -40.

It should be noted that these equations satisfy the two-dimensional versions of equilibrium equations (8.38):

do-xx

+

dXXy

= 0

dx

dy

0 0

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