Beer & Johnston — Pure Bending: Problem 4.32

4.32 It was assumed in Sec. 4.1B that the normal stresses \(\sigma_y\) in a member in pure bending are negligible. For an initially straight elastic member of rectangular cross section, (a) derive an approximate expression for \(\sigma_y\) as a function of \(y\), (b) show that \((\sigma_y)_{max} = -(c/2\rho)(\sigma_x)_{max}\) and, thus, that \(\sigma_y\) can be neglected in all practical situations. (Hint: Consider the free-body diagram of the portion of beam located below the surface of ordinate \(y\) and assume that the distribution of the stress \(\sigma_x\) is still linear.)