Page 295 - Introduction to Continuum Mechanics

The Elastic Solid 279

From the compatibility equations, Eqs. (3.16.8), (3.16.9) and (3.16.7), we have

Thus, G(XI £1) = a K\ + J3x2 + y. In the absence of body forces, the equations of equilibrium
are obviously satisfied.

Example 5.16.3

Consider the stress function <p = a xi ^ + fi xi KI

(a) Is this an allowable stress function?
(b) Determine the associated stresses for the plane strain case.
(c) Determine a and ft in order to solve the plane strain problem of a cantilever beam with
end load P (Fig. 5.16).

Fig. 5.16

(d) If the faces x$ = ±b/2 are traction-free, are the stress components given in (b) still valid
for this case if we simply remove 733 from them ?
Solution, (a) Yes, because the stress function satisfies Eq. (5.16.8) exactly.