Page 286 - Introduction to Continuum Mechanics

270 Pure Bending of a Beam

For the pure bending problem, we seek the state of stress that corresponds to a tractionless
lateral surface and some distribution of normal surface tractions on the end faces that is
e
statically equivalent to bending couples M^ = M^&2 + ^3 3 and M£ = -M/j (note that the
MI component is absent because MI is a twisting couple ). Guided by the state of stress
associated with simple extension, we tentatively assume that TU is the only nonzero stress
component and that it is an arbitrary function of x\.

Fig. 5.13

To satisfy equilibrium, we require

i.e., TU = TU (X2,x$). The corresponding strains are

Since we have begun with an assumption on the state of stress, we must check whether these
strains are compatible. Substituting the strains into the compatibility equations [Eq. (3.16.7-
12) we obtain

which can be satisfied only if TU is at most a linear function of the form