Consider the general case of a prismatic beam subjected to bending-moment components and

Chapter 6, Problem 6-113

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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 \(0=\int_{\Lambda} \sigma d A, M_y=\int_A z \sigma d A, M_z=\int_A-y \sigma d A\), 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.