Condition (10.1) on the axial strain Ex~ is the foundation of the so-called Euler-Bernoulli theory of elastic bending. It will be instructive to investigate the
relation of this condition to the axial displacement u. If we stipulate that u be a linear function of y at a given cross section x = Xo [i.e., u = f(x)y where f(x) is some function of x], then clearly plane sections remain plane during deformation. It immediately follows by simple differentiation that Exx = iJu/iJx = [df(x)/dx]y, which brings us back to Eq. (10.1) with lIR = -df(x)/dx. [It is for this reason that we refer to Eq. (10.1) as the "plane sections remain plane" condition.] Now we have reminded you that for the case of pure bending of an isotropic elastic beam, the plane sections remain plane condition (i.e., Exx = -ylR) yields via integration of the straindisplacement equations the axial displacement u = - xy/ R [see Eq. (9.128)]. Since this expression for u is linear in y at a fixed value of x, it is clear that in this case plane sections do in fact remain plane. However, we must point out that if we follow this converse procedure of assuming that Exx is linear in y and then integrating the strain-displacement equations to determine u, it does not necessarily follow that u is also linear in y. An example of this is seen in Appendix IV, where as noted above we prove that the flexure formula is exact when the bending moment is a linear function in x. It then follows from Hooke's law that Exx is linear in y, but it does not also follow that u is linear in y unless we set the bending moment equal to a constant. Thus plane sections do not in general remain plane even if the flexure formula is satisfied, unless we require in addition that the bending moment be independent of x.