MECHANICS OF MATERIALS
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1 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, Inc. All rights reserved.
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 t 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 006 The McGraw-Hill Companies, Inc. All rights reserved. 7 -
3 Introduction The most general state of stress at a point ma be represented b 6 components,,, τ, τ z z, τ z normal stresses shearing stresses (Note : τ τ, τ τ, τ τ ) 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 z z z 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-3
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,, τ and τ τ 0. 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 lforce. 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-4
5 Transformation of Plane Stress Consider the conditions for equilibrium of a prismatic element with faces perpendicular to the,, and aes. F 0 A ( Acosθ ) cosθ τ ( Acosθ ) ( Asinθ ) sinθ τ ( Asinθ ) cosθ F 0 τ A + ( Acosθ ) sinθ τ ( Acosθ ) ( Asinθ ) cosθ + τ ( Asinθ ) sinθ The equations ma be rewritten to ield τ cosθ + τ + cos θ τ sin θ + τ cosθ sin θ sin θ sinθ cosθ 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-5
6 Principal Stresses The previous equations are combined to ield parametric equations for a circle, ( ) ave + τ where + ave R + τ R Principal stresses occur on the principal planes of stress with zero shearing stresses. + ma,min ± + τ τ tan θ p Note : defines two angles separated b 90 o 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-6
7 Maimum Shearing Stress Maimum shearing stress occurs for ave τ ma R tan θ s τ + τ Note : defines two angles separated b 90 offset fromθ ave p + b 45 o o and 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-7
8 Eample 7.01 SOLUTION: Find the element orientation for the principal stresses from τ tan θ p Determine the principal stresses from + ma,min ± τ For the state of plane stress shown, + determine (a) the principal planes, Calculate the maimum shearing stress with (b) the principal stresses, (c) the maimum shearing stress and the τ ma + τ corresponding normal stress The McGraw-Hill Companies, Inc. All rights reserved. 7-8
9 Eample MPa 10MPa τ + 40MPa SOLUTION: Find the element orientation for the principal stresses from τ ( + 40) tan θ p θ p 53.1, θ p , ( ) Determine the principal stresses from + ma,min ± + τ ma ( 30 ) ( 40 ) 0 ± + 70MPa min 30MPa 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-9
10 Eample MPa Calculate the maimum shearing stress with τ ma + τ τ ma 50MPa + 50MPa τ + 40MPa θ s θ 45 θ s p ( 30 ) + ( 40 ) 18.4, The corresponding normal stress is ave 0MPa The McGraw-Hill Companies, Inc. All rights reserved. 7-10
11 Sample Problem 7.1 A single horizontal lforce 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 p planes and principal stresses at the point H. SOLUTION: Determine an equivalent force-couple 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. 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-11
12 Sample Problem 7.1 SOLUTION: Determine an equivalent force-couple sstem at the center of the transverse section passing through H. M P 150lb T ( 150lb)( 18in).7 kip ( 150lb)( 10in) 1.5kip in Evaluate the normal and shearing stresses at H. Mc ( 1.5kip in )( 0.6in ) + + I 1 4 π 0.6in Tc τ + + J in ( ) 4 (.7 kip in )( 0.6in ) 1π ( 0.6in) ksi τ ksi 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-1
13 Sample Problem 7.1 Determine the principal planes and calculate the principal stresses. tan θ p θ p θ p τ 61.0, , ( 7.96) ma,min ma min + ± + τ ± ksi 4.68ksi ( 7.96) 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-13
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 t of plane stress,, plot the points X and Y and construct the circle centered at C. + ave R + τ The principal stresses are obtained at A and B. ave ± R ma,min tan θ p τ The direction of rotation of O to Oa is the same as CX to CA. 006 The McGraw-Hill Companies, Inc. All rights reserved τ
15 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 t of stress at an angle θ with respect to the aes, construct a new diameter X Y at an angle θ with respect to XY. Normal and shear stresses are obtained from the coordinates X Y. 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-15
16 Mohr s Circle for Plane Stress Mohr s circle for centric aial loading: P, τ 0 A τ P A Mohr s circle for torsional loading: Tc Tc 0 τ τ 0 J J 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-16
17 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. SOLUTION: Construction of Mohr s circle ( 50 ) + ( 10 ) + ave CF MPa R CX 0MPa FX 40MPa ( 30) + ( 40) 50MPa 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-17
18 Eample 7.0 Principal planes and stresses ma OA OC + CA ma ma 70MPa min OB OC BC min 30 MPa 0 50 tan θ θ p p θ p FX CP The McGraw-Hill Companies, Inc. All rights reserved. 7-18
19 Eample 7.0 Maimum shear stress θ s θ p + 45 τ ma R ave θ s 71.6 τ ma 50 MPa ma 0 MPa 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-19
20 Sample Problem 7. For the state of stress shown, determine (a) the principal planes and the principal p stresses,,(b) the stress components eerted on the element obtained b rotating the given element counterclockwise through 30 degrees. SOLUTION: Construct Mohr scircle ave 60 80MPa ( CF ) + ( ) ( 0) + ( 48) 5MPa R FX 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-0
21 Sample Problem 7. Principal planes and stresses tan θ θ p p θ p XF CF clockwise ma OA OC + CA ma +13 MPa ma OA OC BC 80 5 min +8 MPa min The McGraw-Hill Companies, Inc. All rights reserved. 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 db rotating XY counterclockwise through θ 60 φ τ OK OC KC 80 5cos5.6 OL OC + CL cos5.6 KX 5sin 5.6 τ MPa MPa 41.3MPa 006 The McGraw-Hill Companies, Inc. All rights reserved. 7 -
23 General State of Stress Consider the general 3D state of stress at a point and the transformation of stress from element rotation State of stress at Q defined b:,,, τ, τ, τ z z z Consider tetrahedron ed with face perpendicular to the line QN with direction cosines: λ, λ, λ The requirement leads to, n λ + + τ F n λ + 0 z λ λ + τ λ z z z λ λ + τ Form of equation guarantees that an element orientation can be found such that a b b λ + λ + n a c These are the principal aes and principal planes and the normal stresses are the principal stresses. λ c z λ z z λ 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-3
24 Application of Mohr s Circle to the Three- Dimensional i Analsis of Stress Transformation of stress for an element The three circles represent the rotated around a principal ais ma be normal and shearing stresses for represented b Mohr s circle. rotation around each principal p ais. Points A, B, and C represent the Radius of the largest circle ields the principal stresses on the principal planes maimum shearing stress. (h (shearing stress is zero) 1 τ ma ma min 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-4
25 Application of Mohr s Circle to the Three- Dimensional i Analsis of Stress In the case of plane stress, the ais perpendicular to the plane of stress is a principal ais (shearing stress equal zero). If the points A and B (representing the principal planes) are on opposite sides of the origin, then b) the maimum shearing stress for the element is equal to the maimum in- plane shearing stress a) the corresponding principal stresses are the maimum and minimum normal stresses for the element c) planes of maimum shearing stress are at 45 o to the principal i planes. 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-5
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