Shear Stress Analysis in Elastic Beams

It is well known that the elementary theory of beams described above becomes inadequate for beams with transverse dimensions of the same order of magnitude as their length. This section deals with the theory to be applied to thick non-slender beams. This theory appears to be relevant in the context of dynamic mechanical analysis. The first fact to be considered is that when the beam is flexed it experiences a shear stress that provokes a relative sliding of the adjacent transverse sections. As a consequence, the larger the transverse section, the higher is this shear strain. The final effect is an increase in the total deflection of the beam (Fig. 17.5). 

The starting approach will be the elasticity theory (3). In the elementary theory of beams, the only component of the stress tensor differing from zero is Gxx = Ey/R, which, according to the theory developed for the elastic case, can be written as 

This expression reflects a parabohc distribution of the stresses whose maximum value Jxy,meix = Td /%I is reached at y = 0. For a rectangular section, / = M /12 and the maximum shear stress is given by 

On the other hand, Eq. (17.36) indicates that the maximum for the longitudinal stress is  

Another component of the stress, Gyy, can be obtained from the remaining equilibrium condition, 

---