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Lecture Mechanics of materials (Third edition) - Chapter 7: Transformations of stress and strain

The following will be discussed in this chapter: Introduction, transformation of plane stress, principal stresses, maximum shearing stress, mohr’s circle for plane stress, general state of stress, application of mohr’s circle to the three- dimensional analysis of stress, yield criteria for ductile materials under plane stress, fracture criteria for brittle materials under plane stress, stresses in thin-walled pressure vessels. | Third Edition CHAPTER MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Transformations of Stress and Strain Lecture Notes: J. Walt Oler Texas Tech University © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Transformations of Stress and Strain Introduction Transformation of Plane Stress Principal Stresses Maximum Shearing Stress Example 7.01 Sample Problem 7.1 Mohr’s Circle for Plane Stress Example 7.02 Sample Problem 7.2 General State of Stress Application of Mohr’s Circle to the Three- Dimensional Analysis of Stress Yield Criteria for Ductile Materials Under Plane Stress Fracture Criteria for Brittle Materials Under Plane Stress Stresses in Thin-Walled Pressure Vessels © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 7-2 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Introduction • The most general state of stress at a point may be represented by 6 components, σ x ,σ y ,σ z normal stresses τ xy , τ yz , τ zx shearing stresses (Note : τ xy = τ yx , τ yz = τ zy , τ zx = τ xz ) • Same state of stress is represented by a different set of components if axes are rotated. • The first part of the chapter is concerned with how the components of stress are transformed under a rotation of the coordinate axes. The second part of the chapter is devoted to a similar analysis of the transformation of the components of strain. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 7-3 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Introduction • Plane Stress - state of stress in which two faces of the cubic element are free of stress. For the illustrated example, the state of stress is defined by σ x , σ y , τ xy and σ z = τ zx = τ zy = 0. • State of plane stress occurs in a thin plate subjected to forces acting in the midplane of the plate. • State of plane stress also occurs on the free surface of a