MECHANICS OF MATERIALS

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1 00 The McGraw-Hill Companies, Inc. All rights reserved. Third E CHAPTER 7 Transformations MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Teas Tech Universit of Stress and Strain

2 Transformations of Stress and Strain Introduction Transformation of Plane Stress Principal Stresses Maimum Shearing Stress Eample 7.01 Sample Problem 7.1 Mohr s Circle for Plane Stress Eample 7.0 Sample Problem 7. General State of Stress Application of Mohr s Circle to the Three- Dimensional Analsis of Stress Yield Criteria for Ductile Materials Under Plane Stress Fracture Criteria for Brittle Materials Under Plane Stress Stresses in Thin-Walled Pressure Vessels 00 The McGraw-Hill Companies, Inc. All rights reserved. 7 -

Introduction The most general state of stress at a point ma be represented b 6 components,,,, z z, z normal stresses shearing stresses (Note:, z z, z Same state of stress is represented b a different

3 Introduction The most general state of stress at a point ma be represented b 6 components,,,, z z, z normal stresses shearing stresses (Note:, z z, z Same state of stress is represented b a different set of components if aes are rotated. The first part of the chapter is concerned with how the components of stress are transformed under a rotation of the coordinate aes. The second part of the chapter is devoted to a similar analsis of the transformation of the components of strain. z ) 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-3

and z z z State of plane stress occurs in a thin plate subjected to forces acting in the midplane of the plate.

4 Introduction Plane Stress - state of stress in which two faces of the cubic element are free of stress. For the illustrated eample, the state of stress is defined b,, 0. and z z 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 eternal force. 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-4

Transformation of Plane Stress Consider the cons for equilibrium of a prismatic element with

8 Principal Stresses The previous equations are combined to ield parametric equations for a circle, where ave R ave R Principal stresses occur on the principal planes of stress with zero shearing stresses. ma,min tan p Note : defines two angles separated b 90 o 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-8

9 Maimum Shearing Stress Maimum shearing stress occurs for ave ma R tan s Note :defines two angles separated b 90 offset from ave p b 45 o o and 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-9

10 Eample 7.01 For the state of plane stress shown, determine (a) the principal panes, (b) the principal stresses, (c) the maimum shearing stress and the corresponding normal stress. SOLUTION: Find the element orientation for the principal stresses from tan p Determine the principal stresses from ma,min Calculate the maimum shearing stress with ma 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-10

11 Eample MPa 10MPa 40MPa SOLUTION: Find the element orientation for the principal stresses from 40 tan p p p 53.1, , Determine the principal stresses from ma,min ma min 70MPa 30MPa 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-11

12 Eample 7.01 Calculate the maimum shearing stress with ma MPa 10MPa 40MPa ma 50MPa s 45 s p 18.4, The corresponding normal stress is ave MPa 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-1

13 Sample Problem 7.1 A single horizontal force P of 150 lb 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 and aes, (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. 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-13

Sample Problem 7.1 SOLUTION: Determine an equivalent force-couple sstem at the center of the transverse section passing through H. T M P 150lb 150lb18in.7 kip 150lb10in 1.

14 Sample Problem 7.1 SOLUTION: Determine an equivalent force-couple sstem at the center of the transverse section passing through H. T M P 150lb 150lb18in.7 kip 150lb10in 1.5kip in Evaluate the normal and shearing stresses at H. Mc I Tc J in 1.5kip in 0.6in in 4.7 kip in 0.6in 1 0.6in ksi 7.96ksi 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-14

16 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,, plot the points X and Y and construct the circle centered at C. ave R The principal stresses are obtained at A and B. ma,min tan p ave R The direction of rotation of O to Oa is the same as CX to CA. 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-16

17 Mohr s Circle for Plane Stress With Mohr s circle uniquel defined, the state of stress at other aes orientations ma be depicted. For the state of stress at an angle with respect to the aes, construct a new diameter X Y at an angle with respect to XY. Normal and shear stresses are obtained from the coordinates X Y. 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-17

Mohr s Circle for Plane Stress Mohr s circle for

for torsional loading: Tc Tc 0 0 J J 00 The

19 Eample 7.0 For the state of plane stress shown, (a) construct Mohr s circle, determine (b) the principal planes, (c) the principal stresses, (d) the maimum shearing stress and the corresponding normal stress. 00 The McGraw-Hill Companies, Inc. All rights reserved. SOLUTION: Construction of Mohr s circle ave CF MPa R CX FX 40MPa MPa 0MPa 7-19

22 Sample Problem 7. For the state of stress shown, determine (a) the principal planes and the principal stresses, (b) the stress components eerted on the element obtained b rotating the given element counterclockwise through 30 degrees. SOLUTION: Construct Mohr s circle 100 ave R 60 80MPa CF FX MPa 00 The McGraw-Hill Companies, Inc. All rights reserved. 7 -

24 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 The McGraw-Hill Companies, Inc. All rights reserved OK OC KC 80 5cos5.6 OL OC CL 80 5cos5.6 KX 5sin MPa 111.6MPa 41.3MPa 7-4

Stresses in Thin-Walled Pressure Vessels 00 The

27 Stresses in Thin-Walled Pressure Vessels Clindrical vessel with principal stresses 1 = hoop stress = longitudinal stress Hoop stress: F z pr t 0 pr t t p r 1 Longitudinal stress: F rt p r 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-7

point 1 ma(in -plane) constant 0 Maimum out-of-plane shearing

28 Stresses in Thin-Walled Pressure Vessels Spherical pressure vessel: 1 pr t Mohr s circle for in-plane transformations reduces to a point 1 ma(in -plane) constant 0 Maimum out-of-plane shearing stress ma 1 1 pr 4t 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-8

PROBLEM Solve Probs. 7.5 and 7.9, using Mohr s circle.

PROBLEM Solve Probs. 7.5 and 7.9, using Mohr s circle. PROBLEM 7.1 Solve Probs. 7.5 and 7.9, using Mohr s circle. PROBLEM 7.5 through 7.8 For the given state of stress, determine (a) the principal planes, (b) the principal stresses. PROBLEM 7.9 through 7.12