MECHANICS OF MATERIALS
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1 009 The McGraw-Hill Companies, Inc. All rights reserved. Fifth SI Edition CHAPTER 7 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Transformations of Stress and Strain Lecture Notes: J. Walt Oler Texas Tech Universit
2 Introduction The most general state of stress at a point ma be represented b 6 components,,, normalstresses x, z z, zx shearingstresses (Note: x, z z, zx Same state of stress is represented b 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 analsis of the transformation of the components of strain. xz ) 7-
3 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 b,, 0. x and z zx z 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 structural element or machine component, i.e., at an point of the surface not subjected to an external force. 7-3
4 7.1 Transformation of Plane Stress Consider the conditions for equilibrium of a prismatic element with faces perpendicular to the x,, and x axes. F F x x 0 A 0 x xacos cos Acos Asin sin Asin cos AxAcos sin Acos Asin cos Asin sin The equations ma be rewritten to ield x x cos x x cos x sin cos sin sin sin cos 7-4
5 Principal Stresses The previous equations are combined to ield parametric equations for a circle, x where ave ave x R R x Principal stresses occur on the principal planes of stress with zero shearing stresses. max,min tan p x x x Note: definestwoanglesseparatedb 90 o 7-5
6 Maximum Shearing Stress Maximum shearing stress occurs for x ave max R x tan s Note:definestwoanglesseparatedb 90 offset from ave p x x b 45 o o and 7-6
7 Concept Application 7.1 Fig For the state of plane stress shown, determine (a) the principal planes, (b) the principal stresses, (c) the maximum shearing stress and the corresponding normal stress. 7-7
8 Concept Application 7.1 Fig x x 50MPa 10MPa 40MPa Fig
9 Concept Application 7.1 Fig x x 50MPa 10MPa 40MPa Fig
10 Sample Problem 7.1 A single horizontal force P of 600 N magnitude is applied to end D of lever ABD. Determine (a) the normal and shearing stresses on an element at point H having sides parallel to the x and axes, (b) the principal planes and principal stresses at the point H. SOLUTION: Determine an equivalent force-couple sstem at the center of the transverse section passing through H. Evaluate the normal and shearing stresses at H. Determine the principal planes and calculate the principal stresses. 7-10
11 Sample Problem 7.1 SOLUTION: Determine an equivalent force-couple sstem at the center of the transverse section passing through H. P T M x 600N 600N 0.45m 70Nm 600N 0.5m 150Nm Evaluate the normal and shearing stresses at H. Mc I Tc J 150Nm m 70Nm 0.015m m MPa 50.9MPa x 7-11
12 Sample Problem
13 Problems 7-19, 7-
14 Mohr s Circle for Plane Stress With the phsical significance of Mohr s circle for plane stress established, it ma be applied with simple geometric considerations. Critical values are estimated graphicall or calculated. For a known state of plane stress x,, plot the points X and Y and construct the circle centered at C. x x ave R The principal stresses are obtained at A and B. R max,min tan p x ave The direction of rotation of Ox to Oa is the same as CX to CA. 7-14
15 Mohr s Circle for Plane Stress With Mohr s circle uniquel defined, the state of stress at other axes orientations ma be depicted. For the state of stress at an angle with respect to the axes, construct a new diameter X Y at an angle with respect to XY. Normal and shear stresses are obtained from the coordinates X Y. 7-15
16 Concept Application 7. Fig For the state of plane stress shown, (a) construct Mohr s circle, determine (b) the principal planes, (c) the principal stresses, (d) the maximum shearing stress and the corresponding normal stress. 7-16
17 Concept Application
18 Concept Application 7. Maximum shear stress s p 45 s max R max 50MPa ave 0MPa 7-18
19 Mohr s Circle for Plane Stress Mohr s circle for centric axial loading: x P, 0 A x P A Mohr s circle for torsional loading: Tc Tc x 0 x 0 J J 7-19
20 Sample Problem 7. For the state of stress shown, determine (a) the principal planes and the principal stresses, (b) the stress components exerted on the element obtained b rotating the given element counterclockwise through 30 degrees. SOLUTION: Construct Mohr s circle x ave 80MPa R CF FX MPa 7-0
21 Sample Problem 7. Principal planes and stresses tan p p p XF CF clockwise max OAOCCA 805 max 13MPa max OAOC BC 805 min 8 MPa 7-1
22 Sample Problem 7. Stress components after rotation b 30 o Points X and Y on Mohr s circle that correspond to stress components on the rotated element are obtained b rotating XY counterclockwise through x OK OC KC 805cos5.6 OL OCCL805cos5.6 KX 5sin5.6 x 48.4MPa 111.6MPa 41.3MPa 7-
MECHANICS OF MATERIALS
CHAPTER MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Teas Tech Universit Transformations of Stress and Strain 006 The McGraw-Hill Companies,
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