Chapter 6.5, Problem 6.106P

Consider the general case of a prismatic beam subjected to bending-moment components M_y and M_z when the x, y, z axes pass through the centroid of the cross section. If the material is linear elastic, the normal stress in the beam is a linear function of position such that σ = a + by + cz. Using the equilibrium conditions 0 = ∫_A σ dA, M_y = ∫_A zσ dA, M_x = ∫_A –yσdA, determine the constants a, b, and c, and show that the normal stress can be determined from the equation σ= [– (M_zI_y + M_yI_yz)y + (M_yI_z + M_zI_yz)z] /(I_yI_z – I_yz²), where the moments and products of inertia are defined in Appendix A.