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Ebook Mechanics of materials (6th edition): Part 2 - James M.Gere

(BQ) Part 2 book "Mechanics of materials" has contents: Analysis of stress and strain; applications of plane stress (pressure vessels, beams, and combined loadings); deflections of beams; statically indeterminate beams; columns; review of centroids and moments of inertia. | 7 Analysis of Stress and Strain 7.1 INTRODUCTION Normal and shear stresses in beams, shafts, and bars can be calculated from the basic formulas discussed in the preceding chapters. For instance, the stresses in a beam are given by the flexure and shear formulas (s ϭ My/I and t ϭ VQ/Ib), and the stresses in a shaft are given by the torsion formula (t ϭ Tr/IP). The stresses calculated from these formulas act on cross sections of the members, but larger stresses may occur on inclined sections. Therefore, we will begin our analysis of stresses and strains by discussing methods for finding the normal and shear stresses acting on inclined sections cut through a member. We have already derived expressions for the normal and shear stresses acting on inclined sections in both uniaxial stress and pure shear (see Sections 2.6 and 3.5, respectively). In the case of uniaxial stress, we found that the maximum shear stresses occur on planes inclined at 45° to the axis, whereas the maximum normal stresses occur on the cross sections. In the case of pure shear, we found that the maximum tensile and compressive stresses occur on 45° planes. In an analogous manner, the stresses on inclined sections cut through a beam may be larger than the stresses acting on a cross section. To calculate such stresses, we need to determine the stresses acting on inclined planes under a more general stress state known as plane stress (Section 7.2). In our discussions of plane stress we will use stress elements to represent the state of stress at a point in a body. Stress elements were discussed previously in a specialized context (see Sections 2.6 and 3.5), but now we will use them in a more formalized manner. We will begin our analysis by considering an element on which the stresses are known, and then we will derive the transformation equations that give the stresses acting on the sides of an element oriented in a different direction. 464 Copyright 2004 Thomson Learning, Inc. All Rights .