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
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-
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
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
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
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
Concept Application 7.1 Fig. 7.13 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
Concept Application 7.1 Fig. 7.13 x x 50MPa 10MPa 40MPa Fig. 7.14 7-8
Concept Application 7.1 Fig. 7.13 x x 50MPa 10MPa 40MPa Fig. 7.16 7-9
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
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 1 4 40.015m 70Nm 0.015m 10.015m 4 0 56.6MPa 50.9MPa x 7-11
Sample Problem 7.1 7-1
Problems 7-19, 7-
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
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
Concept Application 7. Fig. 7.13 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
Concept Application 7. 7-17
Concept Application 7. Maximum shear stress s p 45 s 71. 6 max R max 50MPa ave 0MPa 7-18
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
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 100 60 ave 80MPa R CF FX 0 48 5MPa 7-0
Sample Problem 7. Principal planes and stresses tan p p p XF CF 67.4 33.7 48 0.4 clockwise max OAOCCA 805 max 13MPa max OAOC BC 805 min 8 MPa 7-1
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 60 180 60 67.4 5.6 x OK OC KC 805cos5.6 OL OCCL805cos5.6 KX 5sin5.6 x 48.4MPa 111.6MPa 41.3MPa 7-