Consider the general case of a prismatic beam subjected to

Chapter 6, Problem 6-106

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Consider the general case of a prismatic beam subjected to bending-moment components \(\mathbf{M}_{y}\) and \(\mathbf{M}_{z}\), as shown, 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 \(\sigma=a+b y+c z\). Using the equilibrium conditions \(0=\int_{A} \sigma \ d A, \ M_{y}=\int_{A} z \sigma \ d A, \ M_{z}=\int_{A}-y \sigma \ d A\), determine the constants a , b , and c , and show that the normal stress can be determined from the equation \(\sigma=\left[-\left(M_{z} I_{y}+M_{y} I_{y z}\right) y+\left(M_{y} I_{z}+M_{z} I_{y z}\right) z\right] /\left(I_{y} I_{z}-I_{y z}{ }^{2}\right)\), where the moments and products of inertia are defined in Appendix A.