These stresses are plotted in Fig. 8.32. The longitudinal stress consists of two parts. The first term w(L2 — x2)y/2I is that given by simple bending theory (axx = My/I). The second term may be considered as a correction term which arises because of the effect of the ayy compressive stress between the longitudinal fibres. The term is independent of x and therefore constant along the beam. It thus has a value on the ends of the beam given by x — ±L. The expression for cjxx in eqn. (8.92a) is, therefore, only an exact solution if normal forces on the end exist and are distributed in such a manner as to produce the Oxx values given by eqn. (8.92a) at x = ±L. That is as shown by the correction term in Fig. 8.32. However, conditions (iv) and (v) have guaranteed that forces and moments are in equilibrium at the ends x — ±L and thus, from Saint-Venant's principle, one could conclude that at distance larger than, say, the depth of the beam, the stress distribution given by eqn. (8.92a) is accurate even when the ends are free. Such correction stresses are, however, of small magnitude compared with the simple bending terms when the span of the beam is large in comparison with its depth.

The equation for the shear stress (8.92c) predicts a parabolic distribution of rxy on every section x. This implies that at the ends x = ±L the beam must be supported in such a way that these shear stresses are developed. The values predicted by eqn. (8.92c) coincide with the simple solution. The oyy stress decreases from its maximum on the top surface to zero at the bottom edge. This again is of small magnitude compared to er^ in a thin beam type component. However, these stresses can be of importance in a deep beam, or a slab arrangement.

Derivation of the displacements in the beam

From the strain displacement relations, the constitutive relations and the derived stresses it is possible to obtain the displacements in the beam. Although this approach is not really part of the stress function concept, it is included for interest at this point in the development. The procedure is as follows:

Substituting for oxx and oyy from eqns (8.92a,b) and integrating (8.93a,b) the following is obtained:

dv 1

dv 1

0 0

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