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Stresses in Beams: Strains in Pure and Nonuniform Bending (continued)

This module reviews the principles of the mechanics of deformable bodies. We first review the basic concepts of equilibrium and stresses and strains in prismatic bars under axial loading. Then we discuss the major mechanical properties of common engineering materials, particularly the diagrams for normal stress and strain leading to Hooke’s Law, and their relation to lateral strain through Poisson’s ratio. Shear stresses and their relation to shear strains are then presented. We then analyze in detail deformations and stresses in axially loaded members. This includes uniform and nonuniform loading for statically determinate and indeterminate structures. Thermal effects are then considered: expansion and contraction under temperature changes and the stresses that may develop both with and without prestresses. Stresses on inclined planes under axial loadings and the resulting maximum and minimum normal and shear stresses that result are then discussed. Torsion, the twisting of circular rods and shafts by applied torques is then analyzed. We show how to calculate the angle of twist and shear stress as functions of rod properties and shape under uniform and nonuniform torsion. Applications to power transmission by rotating shafts are presented. We then discuss how shear forces and bending moments arise in beams subject to various loading types and how to calculate them. This is then generalized to local forms of the equilibrium equations leading to rules for drawing shear force and bending moment diagrams. Finally, we compute bending stresses in beams. Strains due to bending and their relation to curvature are first discussed. This is used to compute the bending stresses and their relation to the applied bending moment and beam material and cross sectional properties. This includes a review of computation of centroids and moments of inertia of various areal shapes. We complete this module with a discussion how shear stresses arise in beams subject to nonuniform bending and how to compute them. In all cases, basic ideas and equations are presented along with sample problems that illustrate the major ideas and provide practice on expected exam questions. Time: Approximately 4 hours | Difficulty Level: Medium

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