The transverse load P will produce a shear force V in the positive y direction on a mathematical cut through the beam at point “a”. This shear force is directly responsible for the shear stress τ on the x-face of a stress element at “a” in the positive y-direction. In class, we have used the Euler-Bernoulli theory of beams to calculate this shear stress. Recall that the E-B theory of beam assumes that beam cross sections always remain perpendicular to the neutral plane of the beam.

As we have seen before, the presence of a shear stress on the x-faces says that there will also be shear stress components on the y-faces of the stress element. What if we create a beam that reduces this shear stress component on the y-faces by making the beam of thin laminae not joined together? In this case, as the beam deflects, the beam cross sections do NOT remain perpendicular to the neutral plane, as see in the animation below.

CONCLUSION: If shear forces exist at a cross section of a beam, then shear stresses must develop in the beam as it deflects in order to satisfy the requirements for the E-B theory.

A magnified copy of the laminated beam deflection is shown below to assist you in seeing the inter-laminate motion as the beam deflects.