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Transformations of Stress

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Talha Shah
Talha Shah
Engineering
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MECHANICS OF MATERIALS Third Edition Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University CHAPTER © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 7 Transformations of Stress and Strain
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 2 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. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 3 Introduction • The most general state of stress at a point may be represented by 6 components, ),,:(Note stressesshearing,, stressesnormal,, xzzxzyyzyxxy zxyzxy zyx ττττττ τττ σσσ === • 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. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 4 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 .0,, andxy === zyzxzyx ττστσσ • 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 any point of the surface not subjected to an external force.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 5 Transformation of Plane Stress ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) θθτθθσ θθτθθστ θθτθθσ θθτθθσσ sinsincossin coscossincos0 cossinsinsin sincoscoscos0 AA AAAF AA AAAF xyy xyxyxy xyy xyxxx ∆+∆− ∆−∆+∆==∑ ∆−∆− ∆−∆−∆==∑ ′′′ ′′ • Consider the conditions for equilibrium of a prismatic element with faces perpendicular to the x, y, and x’ axes. • The equations may be rewritten to yield
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 6 Principal Stresses • The previous equations are combined to yield parametric equations for a circle, ( ) 2 2 222 22 where xy yxyx ave yxavex R R τ σσσσ σ τσσ +      − = + = =+− ′′′ • Principal stresses occur on the principal planes of stress with zero shearing stresses. o 2 2 minmax, 90byseparatedanglestwodefines:Note 2 2tan 22 yx xy p xy yxyx σσ τ θ τ σσσσ σ − = +      − ± + =
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 7 Maximum Shearing Stress Maximum shearing stress occurs for avex σσ =′ 2 45byfromoffset and90byseparatedanglestwodefines:Note 2 2tan 2 o o 2 2 max yx ave p xy yx s xy yx R σσ σσ θ τ σσ θ τ σσ τ + ==′ − −= +      − ==
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 8 Example 7.01 For the state of plane stress shown, determine (a) the principal panes, (b) the principal stresses, (c) the maximum shearing stress and the corresponding normal stress. SOLUTION: • Find the element orientation for the principal stresses from yx xy p σσ τ θ − = 2 2tan • Determine the principal stresses from 2 2 minmax, 22 xy yxyx τ σσσσ σ +      − ± + = • Calculate the maximum shearing stress with 2 2 max 2 xy yx τ σσ τ +      − = 2 yx σσ σ + =′
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 9 Example 7.01 SOLUTION: • Find the element orientation for the principal stresses from ( ) ( ) °°= = −− + = − = 1.233,1.532 333.1 1050 4022 2tan p yx xy p θ σσ τ θ °°= 6.116,6.26pθ • Determine the principal stresses from ( ) ( )22 2 2 minmax, 403020 22 +±= +      − ± + = xy yxyx τ σσσσ σ MPa30 MPa70 min max −= = σ σ MPa10 MPa40MPa50 −= +=+= x xyx σ τσ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 10 Example 7.01 MPa10 MPa40MPa50 −= +=+= x xyx σ τσ 2 1050 2 − = + ==′ yx ave σσ σσ • The corresponding normal stress is MPa20=′σ • Calculate the maximum shearing stress with ( ) ( )22 2 2 max 4030 2 += +      − = xy yx τ σσ τ MPa50max =τ 45−= ps θθ °°−= 6.71,4.18sθ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 11 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 x and y axes, (b) the principal planes and principal stresses at the point H. SOLUTION: • Determine an equivalent force-couple system 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.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 12 Sample Problem 7.1 SOLUTION: • Determine an equivalent force-couple system at the center of the transverse section passing through H. ( )( ) ( )( ) inkip5.1in10lb150 inkip7.2in18lb150 lb150 ⋅== ⋅== = xM T P • Evaluate the normal and shearing stresses at H. ( )( ) ( ) ( )( ) ( )4 2 1 4 4 1 in6.0 in6.0inkip7.2 in6.0 in6.0inkip5.1 π τ π σ ⋅ +=+= ⋅ +=+= J Tc I Mc xy y ksi96.7ksi84.80 +=+== yyx τσσ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 13 Sample Problem 7.1 • Determine the principal planes and calculate the principal stresses. ( ) °°−= −= − = − = 119,0.612 8.1 84.80 96.722 2tan p yx xy p θ σσ τ θ °°−= 5.59,5.30pθ ( )2 2 2 2 minmax, 96.7 2 84.80 2 84.80 22 +      − ± + = +      − ± + = xy yxyx τ σσσσ σ ksi68.4 ksi52.13 min max −= += σ σ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 14 Mohr’s Circle for Plane Stress • With the physical significance of Mohr’s circle for plane stress established, it may be applied with simple geometric considerations. Critical values are estimated graphically or calculated. • For a known state of plane stress plot the points X and Y and construct the circle centered at C. xyyx τσσ ,, 2 2 22 xy yxyx ave R τ σσσσ σ +      − = + = • The principal stresses are obtained at A and B. yx xy p ave R σσ τ θ σσ − = ±= 2 2tan minmax, The direction of rotation of Ox to Oa is the same as CX to CA.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 15 Mohr’s Circle for Plane Stress • With Mohr’s circle uniquely defined, the state of stress at other axes orientations may be depicted. • For the state of stress at an angle θ with respect to the xy axes, construct a new diameter X’Y’ at an angle 2θ with respect to XY. • Normal and shear stresses are obtained from the coordinates X’Y’.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 16 Mohr’s Circle for Plane Stress • Mohr’s circle for centric axial loading: 0, === xyyx A P τσσ A P xyyx 2 === τσσ • Mohr’s circle for torsional loading: J Tc xyyx === τσσ 0 0=== xyyx J Tc τσσ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 17 Example 7.02 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. SOLUTION: • Construction of Mohr’s circle ( ) ( ) ( ) ( ) MPa504030 MPa40MPa302050 MPa20 2 1050 2 22 =+== ==−= = −+ = + = CXR FXCF yx ave σσ σ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 18 Example 7.02 • Principal planes and stresses 5020max +=+== CAOCOAσ MPa70max =σ 5020max −=−== BCOCOBσ MPa30max −=σ °= == 1.532 30 40 2tan p p CP FX θ θ °= 6.26pθ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 19 Example 7.02 • Maximum shear stress °+= 45ps θθ °= 6.71sθ R=maxτ MPa50max =τ aveσσ =′ MPa20=′σ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 20 Sample Problem 7.2 For the state of stress shown, determine (a) the principal planes and the principal stresses, (b) the stress components exerted on the element obtained by rotating the given element counterclockwise through 30 degrees. SOLUTION: • Construct Mohr’s circle ( ) ( ) ( ) ( ) MPa524820 MPa80 2 60100 2 2222 =+=+= = + = + = FXCFR yx ave σσ σ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 21 Sample Problem 7.2 • Principal planes and stresses °= === 4.672 4.2 20 48 2tan p p CF XF θ θ clockwise7.33 °=pθ 5280 max += +== CAOCOAσ 5280 max −= −== BCOCOAσ MPa132max +=σ MPa28min +=σ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 22 Sample Problem 7.2 °=′= °+=+== °−=−== °=°−°−°= ′′ ′ ′ 6.52sin52 6.52cos5280 6.52cos5280 6.524.6760180 XK CLOCOL KCOCOK yx y x τ σ σ φ • Stress components after rotation by 30o Points X’ and Y’ on Mohr’s circle that correspond to stress components on the rotated element are obtained by rotating XY counterclockwise through°= 602θ MPa3.41 MPa6.111 MPa4.48 = += += ′′ ′ ′ yx y x τ σ σ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 23 • State of stress at Q defined by: zxyzxyzyx τττσσσ ,,,,, General State of Stress • Consider the general 3D state of stress at a point and the transformation of stress from element rotation • Consider tetrahedron with face perpendicular to the line QN with direction cosines: zyx λλλ ,, • The requirement leads to,∑ = 0nF xzzxzyyzyxxy zzyyxxn λλτλλτλλτ λσλσλσσ 222 222 +++ ++= • Form of equation guarantees that an element orientation can be found such that 222 ccbbaan λσλσλσσ ++= These are the principal axes and principal planes and the normal stresses are the principal stresses.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 24 Application of Mohr’s Circle to the Three- Dimensional Analysis of Stress • Transformation of stress for an element rotated around a principal axis may be represented by Mohr’s circle. • The three circles represent the normal and shearing stresses for rotation around each principal axis. • Points A, B, and C represent the principal stresses on the principal planes (shearing stress is zero) minmaxmax 2 1 σστ −= • Radius of the largest circle yields the maximum shearing stress.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 25 Application of Mohr’s Circle to the Three- Dimensional Analysis of Stress • In the case of plane stress, the axis perpendicular to the plane of stress is a principal axis (shearing stress equal zero). b) the maximum shearing stress for the element is equal to the maximum “in- plane” shearing stress a) the corresponding principal stresses are the maximum and minimum normal stresses for the element • If the points A and B (representing the principal planes) are on opposite sides of the origin, then c) planes of maximum shearing stress are at 45o to the principal planes.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 26 Application of Mohr’s Circle to the Three- Dimensional Analysis of Stress • If A and B are on the same side of the origin (i.e., have the same sign), then c) planes of maximum shearing stress are at 45 degrees to the plane of stress b) maximum shearing stress for the element is equal to half of the maximum stress a) the circle defining σmax, σmin, and τmax for the element is not the circle corresponding to transformations within the plane of stress
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 27 Yield Criteria for Ductile Materials Under Plane Stress • Failure of a machine component subjected to uniaxial stress is directly predicted from an equivalent tensile test • Failure of a machine component subjected to plane stress cannot be directly predicted from the uniaxial state of stress in a tensile test specimen • It is convenient to determine the principal stresses and to base the failure criteria on the corresponding biaxial stress state • Failure criteria are based on the mechanism of failure. Allows comparison of the failure conditions for a uniaxial stress test and biaxial component loading
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 28 Yield Criteria for Ductile Materials Under Plane Stress Maximum shearing stress criteria: Structural component is safe as long as the maximum shearing stress is less than the maximum shearing stress in a tensile test specimen at yield, i.e., 2 max Y Y σ ττ =< For σa and σb with the same sign, 22 or 2 max Yba σσσ τ <= For σa and σb with opposite signs, 22 max Yba σσσ τ < − =
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 29 Yield Criteria for Ductile Materials Under Plane Stress Maximum distortion energy criteria: Structural component is safe as long as the distortion energy per unit volume is less than that occurring in a tensile test specimen at yield. ( ) ( ) 222 2222 00 6 1 6 1 Ybbaa YYbbaa Yd GG uu σσσσσ σσσσσσ <+− +×−<+− <
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 30 Fracture Criteria for Brittle Materials Under Plane Stress Maximum normal stress criteria: Structural component is safe as long as the maximum normal stress is less than the ultimate strength of a tensile test specimen. Ub Ua σσ σσ < < Brittle materials fail suddenly through rupture or fracture in a tensile test. The failure condition is characterized by the ultimate strength σU.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 31 Stresses in Thin-Walled Pressure Vessels • Cylindrical vessel with principal stresses σ1 = hoop stress σ2 = longitudinal stress ( ) ( ) t pr xrpxtFz = ∆−∆==∑ 1 1 220 σ σ • Hoop stress: ( ) ( ) 21 2 2 2 2 2 20 σσ σ ππσ = = −==∑ t pr rprtFx • Longitudinal stress:
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 32 Stresses in Thin-Walled Pressure Vessels • Points A and B correspond to hoop stress, σ1, and longitudinal stress, σ2 • Maximum in-plane shearing stress: t pr 42 1 2)planeinmax( ==− στ • Maximum out-of-plane shearing stress corresponds to a 45o rotation of the plane stress element around a longitudinal axis t pr 2 2max == στ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 33 Stresses in Thin-Walled Pressure Vessels • Spherical pressure vessel: t pr 2 21 == σσ • Mohr’s circle for in-plane transformations reduces to a point 0 constant plane)-max(in 21 = === τ σσσ • Maximum out-of-plane shearing stress t pr 4 12 1 max == στ
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 34 Transformation of Plane Strain • Plane strain - deformations of the material take place in parallel planes and are the same in each of those planes. • Example: Consider a long bar subjected to uniformly distributed transverse loads. State of plane stress exists in any transverse section not located too close to the ends of the bar. • Plane strain occurs in a plate subjected along its edges to a uniformly distributed load and restrained from expanding or contracting laterally by smooth, rigid and fixed supports ( )0 :strainofcomponents x === zyzxzxyy γγεγεε
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 35 Transformation of Plane Strain • State of strain at the point Q results in different strain components with respect to the xy and x’y’ reference frames. ( ) ( ) ( ) ( )yxOBxy xyyxOB xyyx εεεγ γεεεε θθγθεθεθε +−= ++=°= ++= 2 45 cossinsincos 2 1 22 θ γ θ εεγ θ γ θ εεεε ε θ γ θ εεεε ε 2cos 2 2sin 22 2sin 2 2cos 22 2sin 2 2cos 22 xyyxyx xyyxyx y xyyxyx x + − −= − − − + = + − + + = ′′ ′ ′ • Applying the trigonometric relations used for the transformation of stress,
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 36 Mohr’s Circle for Plane Strain • The equations for the transformation of plane strain are of the same form as the equations for the transformation of plane stress - Mohr’s circle techniques apply. • Abscissa for the center C and radius R , 22 222       +      − = + = xyyxyx ave R γεεεε ε • Principal axes of strain and principal strains, RR aveave yx xy p −=+= − = εεεε εε γ θ minmax 2tan ( ) 22 max 2 xyyxR γεεγ +−== • Maximum in-plane shearing strain,
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 37 Three-Dimensional Analysis of Strain • Previously demonstrated that three principal axes exist such that the perpendicular element faces are free of shearing stresses. • By Hooke’s Law, it follows that the shearing strains are zero as well and that the principal planes of stress are also the principal planes of strain. • Rotation about the principal axes may be represented by Mohr’s circles.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 38 Three-Dimensional Analysis of Strain • For the case of plane strain where the x and y axes are in the plane of strain, - the z axis is also a principal axis - the corresponding principal normal strain is represented by the point Z = 0 or the origin. • If the points A and B lie on opposite sides of the origin, the maximum shearing strain is the maximum in-plane shearing strain, D and E. • If the points A and B lie on the same side of the origin, the maximum shearing strain is out of the plane of strain and is represented by the points D’ and E’.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 39 Three-Dimensional Analysis of Strain • Consider the case of plane stress, 0=== zbyax σσσσσ • Corresponding normal strains, ( ) ( )babac ba b ba a E EE EE εε ν ν σσ ν ε σσν ε σνσ ε + − −=+−= +−= −= 1 • If B is located between A and C on the Mohr-circle diagram, the maximum shearing strain is equal to the diameter CA. • Strain perpendicular to the plane of stress is not zero.
© 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 40 Measurements of Strain: Strain Rosette • Strain gages indicate normal strain through changes in resistance. ( )yxOBxy εεεγ +−= 2 • With a 45o rosette, εx and εy are measured directly. γxy is obtained indirectly with, 333 2 3 2 3 222 2 2 2 2 111 2 1 2 1 cossinsincos cossinsincos cossinsincos θθγθεθεε θθγθεθεε θθγθεθεε xyyx xyyx xyyx ++= ++= ++= • Normal and shearing strains may be obtained from normal strains in any three directions,

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Transformations of Stress

  • 1. MECHANICS OF MATERIALS Third Edition Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University CHAPTER © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 7 Transformations of Stress and Strain
  • 2. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 2 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
  • 3. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 3 Introduction • The most general state of stress at a point may be represented by 6 components, ),,:(Note stressesshearing,, stressesnormal,, xzzxzyyzyxxy zxyzxy zyx ττττττ τττ σσσ === • 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.
  • 4. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 4 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 .0,, andxy === zyzxzyx ττστσσ • 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 any point of the surface not subjected to an external force.
  • 5. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 5 Transformation of Plane Stress ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) θθτθθσ θθτθθστ θθτθθσ θθτθθσσ sinsincossin coscossincos0 cossinsinsin sincoscoscos0 AA AAAF AA AAAF xyy xyxyxy xyy xyxxx ∆+∆− ∆−∆+∆==∑ ∆−∆− ∆−∆−∆==∑ ′′′ ′′ • Consider the conditions for equilibrium of a prismatic element with faces perpendicular to the x, y, and x’ axes. • The equations may be rewritten to yield
  • 6. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 6 Principal Stresses • The previous equations are combined to yield parametric equations for a circle, ( ) 2 2 222 22 where xy yxyx ave yxavex R R τ σσσσ σ τσσ +      − = + = =+− ′′′ • Principal stresses occur on the principal planes of stress with zero shearing stresses. o 2 2 minmax, 90byseparatedanglestwodefines:Note 2 2tan 22 yx xy p xy yxyx σσ τ θ τ σσσσ σ − = +      − ± + =
  • 7. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 7 Maximum Shearing Stress Maximum shearing stress occurs for avex σσ =′ 2 45byfromoffset and90byseparatedanglestwodefines:Note 2 2tan 2 o o 2 2 max yx ave p xy yx s xy yx R σσ σσ θ τ σσ θ τ σσ τ + ==′ − −= +      − ==
  • 8. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 8 Example 7.01 For the state of plane stress shown, determine (a) the principal panes, (b) the principal stresses, (c) the maximum shearing stress and the corresponding normal stress. SOLUTION: • Find the element orientation for the principal stresses from yx xy p σσ τ θ − = 2 2tan • Determine the principal stresses from 2 2 minmax, 22 xy yxyx τ σσσσ σ +      − ± + = • Calculate the maximum shearing stress with 2 2 max 2 xy yx τ σσ τ +      − = 2 yx σσ σ + =′
  • 9. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 9 Example 7.01 SOLUTION: • Find the element orientation for the principal stresses from ( ) ( ) °°= = −− + = − = 1.233,1.532 333.1 1050 4022 2tan p yx xy p θ σσ τ θ °°= 6.116,6.26pθ • Determine the principal stresses from ( ) ( )22 2 2 minmax, 403020 22 +±= +      − ± + = xy yxyx τ σσσσ σ MPa30 MPa70 min max −= = σ σ MPa10 MPa40MPa50 −= +=+= x xyx σ τσ
  • 10. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 10 Example 7.01 MPa10 MPa40MPa50 −= +=+= x xyx σ τσ 2 1050 2 − = + ==′ yx ave σσ σσ • The corresponding normal stress is MPa20=′σ • Calculate the maximum shearing stress with ( ) ( )22 2 2 max 4030 2 += +      − = xy yx τ σσ τ MPa50max =τ 45−= ps θθ °°−= 6.71,4.18sθ
  • 11. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 11 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 x and y axes, (b) the principal planes and principal stresses at the point H. SOLUTION: • Determine an equivalent force-couple system 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.
  • 12. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 12 Sample Problem 7.1 SOLUTION: • Determine an equivalent force-couple system at the center of the transverse section passing through H. ( )( ) ( )( ) inkip5.1in10lb150 inkip7.2in18lb150 lb150 ⋅== ⋅== = xM T P • Evaluate the normal and shearing stresses at H. ( )( ) ( ) ( )( ) ( )4 2 1 4 4 1 in6.0 in6.0inkip7.2 in6.0 in6.0inkip5.1 π τ π σ ⋅ +=+= ⋅ +=+= J Tc I Mc xy y ksi96.7ksi84.80 +=+== yyx τσσ
  • 13. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 13 Sample Problem 7.1 • Determine the principal planes and calculate the principal stresses. ( ) °°−= −= − = − = 119,0.612 8.1 84.80 96.722 2tan p yx xy p θ σσ τ θ °°−= 5.59,5.30pθ ( )2 2 2 2 minmax, 96.7 2 84.80 2 84.80 22 +      − ± + = +      − ± + = xy yxyx τ σσσσ σ ksi68.4 ksi52.13 min max −= += σ σ
  • 14. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 14 Mohr’s Circle for Plane Stress • With the physical significance of Mohr’s circle for plane stress established, it may be applied with simple geometric considerations. Critical values are estimated graphically or calculated. • For a known state of plane stress plot the points X and Y and construct the circle centered at C. xyyx τσσ ,, 2 2 22 xy yxyx ave R τ σσσσ σ +      − = + = • The principal stresses are obtained at A and B. yx xy p ave R σσ τ θ σσ − = ±= 2 2tan minmax, The direction of rotation of Ox to Oa is the same as CX to CA.
  • 15. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 15 Mohr’s Circle for Plane Stress • With Mohr’s circle uniquely defined, the state of stress at other axes orientations may be depicted. • For the state of stress at an angle θ with respect to the xy axes, construct a new diameter X’Y’ at an angle 2θ with respect to XY. • Normal and shear stresses are obtained from the coordinates X’Y’.
  • 16. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 16 Mohr’s Circle for Plane Stress • Mohr’s circle for centric axial loading: 0, === xyyx A P τσσ A P xyyx 2 === τσσ • Mohr’s circle for torsional loading: J Tc xyyx === τσσ 0 0=== xyyx J Tc τσσ
  • 17. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 17 Example 7.02 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. SOLUTION: • Construction of Mohr’s circle ( ) ( ) ( ) ( ) MPa504030 MPa40MPa302050 MPa20 2 1050 2 22 =+== ==−= = −+ = + = CXR FXCF yx ave σσ σ
  • 18. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 18 Example 7.02 • Principal planes and stresses 5020max +=+== CAOCOAσ MPa70max =σ 5020max −=−== BCOCOBσ MPa30max −=σ °= == 1.532 30 40 2tan p p CP FX θ θ °= 6.26pθ
  • 19. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 19 Example 7.02 • Maximum shear stress °+= 45ps θθ °= 6.71sθ R=maxτ MPa50max =τ aveσσ =′ MPa20=′σ
  • 20. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 20 Sample Problem 7.2 For the state of stress shown, determine (a) the principal planes and the principal stresses, (b) the stress components exerted on the element obtained by rotating the given element counterclockwise through 30 degrees. SOLUTION: • Construct Mohr’s circle ( ) ( ) ( ) ( ) MPa524820 MPa80 2 60100 2 2222 =+=+= = + = + = FXCFR yx ave σσ σ
  • 21. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 21 Sample Problem 7.2 • Principal planes and stresses °= === 4.672 4.2 20 48 2tan p p CF XF θ θ clockwise7.33 °=pθ 5280 max += +== CAOCOAσ 5280 max −= −== BCOCOAσ MPa132max +=σ MPa28min +=σ
  • 22. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 22 Sample Problem 7.2 °=′= °+=+== °−=−== °=°−°−°= ′′ ′ ′ 6.52sin52 6.52cos5280 6.52cos5280 6.524.6760180 XK CLOCOL KCOCOK yx y x τ σ σ φ • Stress components after rotation by 30o Points X’ and Y’ on Mohr’s circle that correspond to stress components on the rotated element are obtained by rotating XY counterclockwise through°= 602θ MPa3.41 MPa6.111 MPa4.48 = += += ′′ ′ ′ yx y x τ σ σ
  • 23. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 23 • State of stress at Q defined by: zxyzxyzyx τττσσσ ,,,,, General State of Stress • Consider the general 3D state of stress at a point and the transformation of stress from element rotation • Consider tetrahedron with face perpendicular to the line QN with direction cosines: zyx λλλ ,, • The requirement leads to,∑ = 0nF xzzxzyyzyxxy zzyyxxn λλτλλτλλτ λσλσλσσ 222 222 +++ ++= • Form of equation guarantees that an element orientation can be found such that 222 ccbbaan λσλσλσσ ++= These are the principal axes and principal planes and the normal stresses are the principal stresses.
  • 24. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 24 Application of Mohr’s Circle to the Three- Dimensional Analysis of Stress • Transformation of stress for an element rotated around a principal axis may be represented by Mohr’s circle. • The three circles represent the normal and shearing stresses for rotation around each principal axis. • Points A, B, and C represent the principal stresses on the principal planes (shearing stress is zero) minmaxmax 2 1 σστ −= • Radius of the largest circle yields the maximum shearing stress.
  • 25. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 25 Application of Mohr’s Circle to the Three- Dimensional Analysis of Stress • In the case of plane stress, the axis perpendicular to the plane of stress is a principal axis (shearing stress equal zero). b) the maximum shearing stress for the element is equal to the maximum “in- plane” shearing stress a) the corresponding principal stresses are the maximum and minimum normal stresses for the element • If the points A and B (representing the principal planes) are on opposite sides of the origin, then c) planes of maximum shearing stress are at 45o to the principal planes.
  • 26. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 26 Application of Mohr’s Circle to the Three- Dimensional Analysis of Stress • If A and B are on the same side of the origin (i.e., have the same sign), then c) planes of maximum shearing stress are at 45 degrees to the plane of stress b) maximum shearing stress for the element is equal to half of the maximum stress a) the circle defining σmax, σmin, and τmax for the element is not the circle corresponding to transformations within the plane of stress
  • 27. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 27 Yield Criteria for Ductile Materials Under Plane Stress • Failure of a machine component subjected to uniaxial stress is directly predicted from an equivalent tensile test • Failure of a machine component subjected to plane stress cannot be directly predicted from the uniaxial state of stress in a tensile test specimen • It is convenient to determine the principal stresses and to base the failure criteria on the corresponding biaxial stress state • Failure criteria are based on the mechanism of failure. Allows comparison of the failure conditions for a uniaxial stress test and biaxial component loading
  • 28. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 28 Yield Criteria for Ductile Materials Under Plane Stress Maximum shearing stress criteria: Structural component is safe as long as the maximum shearing stress is less than the maximum shearing stress in a tensile test specimen at yield, i.e., 2 max Y Y σ ττ =< For σa and σb with the same sign, 22 or 2 max Yba σσσ τ <= For σa and σb with opposite signs, 22 max Yba σσσ τ < − =
  • 29. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 29 Yield Criteria for Ductile Materials Under Plane Stress Maximum distortion energy criteria: Structural component is safe as long as the distortion energy per unit volume is less than that occurring in a tensile test specimen at yield. ( ) ( ) 222 2222 00 6 1 6 1 Ybbaa YYbbaa Yd GG uu σσσσσ σσσσσσ <+− +×−<+− <
  • 30. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 30 Fracture Criteria for Brittle Materials Under Plane Stress Maximum normal stress criteria: Structural component is safe as long as the maximum normal stress is less than the ultimate strength of a tensile test specimen. Ub Ua σσ σσ < < Brittle materials fail suddenly through rupture or fracture in a tensile test. The failure condition is characterized by the ultimate strength σU.
  • 31. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 31 Stresses in Thin-Walled Pressure Vessels • Cylindrical vessel with principal stresses σ1 = hoop stress σ2 = longitudinal stress ( ) ( ) t pr xrpxtFz = ∆−∆==∑ 1 1 220 σ σ • Hoop stress: ( ) ( ) 21 2 2 2 2 2 20 σσ σ ππσ = = −==∑ t pr rprtFx • Longitudinal stress:
  • 32. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 32 Stresses in Thin-Walled Pressure Vessels • Points A and B correspond to hoop stress, σ1, and longitudinal stress, σ2 • Maximum in-plane shearing stress: t pr 42 1 2)planeinmax( ==− στ • Maximum out-of-plane shearing stress corresponds to a 45o rotation of the plane stress element around a longitudinal axis t pr 2 2max == στ
  • 33. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 33 Stresses in Thin-Walled Pressure Vessels • Spherical pressure vessel: t pr 2 21 == σσ • Mohr’s circle for in-plane transformations reduces to a point 0 constant plane)-max(in 21 = === τ σσσ • Maximum out-of-plane shearing stress t pr 4 12 1 max == στ
  • 34. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 34 Transformation of Plane Strain • Plane strain - deformations of the material take place in parallel planes and are the same in each of those planes. • Example: Consider a long bar subjected to uniformly distributed transverse loads. State of plane stress exists in any transverse section not located too close to the ends of the bar. • Plane strain occurs in a plate subjected along its edges to a uniformly distributed load and restrained from expanding or contracting laterally by smooth, rigid and fixed supports ( )0 :strainofcomponents x === zyzxzxyy γγεγεε
  • 35. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 35 Transformation of Plane Strain • State of strain at the point Q results in different strain components with respect to the xy and x’y’ reference frames. ( ) ( ) ( ) ( )yxOBxy xyyxOB xyyx εεεγ γεεεε θθγθεθεθε +−= ++=°= ++= 2 45 cossinsincos 2 1 22 θ γ θ εεγ θ γ θ εεεε ε θ γ θ εεεε ε 2cos 2 2sin 22 2sin 2 2cos 22 2sin 2 2cos 22 xyyxyx xyyxyx y xyyxyx x + − −= − − − + = + − + + = ′′ ′ ′ • Applying the trigonometric relations used for the transformation of stress,
  • 36. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 36 Mohr’s Circle for Plane Strain • The equations for the transformation of plane strain are of the same form as the equations for the transformation of plane stress - Mohr’s circle techniques apply. • Abscissa for the center C and radius R , 22 222       +      − = + = xyyxyx ave R γεεεε ε • Principal axes of strain and principal strains, RR aveave yx xy p −=+= − = εεεε εε γ θ minmax 2tan ( ) 22 max 2 xyyxR γεεγ +−== • Maximum in-plane shearing strain,
  • 37. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 37 Three-Dimensional Analysis of Strain • Previously demonstrated that three principal axes exist such that the perpendicular element faces are free of shearing stresses. • By Hooke’s Law, it follows that the shearing strains are zero as well and that the principal planes of stress are also the principal planes of strain. • Rotation about the principal axes may be represented by Mohr’s circles.
  • 38. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 38 Three-Dimensional Analysis of Strain • For the case of plane strain where the x and y axes are in the plane of strain, - the z axis is also a principal axis - the corresponding principal normal strain is represented by the point Z = 0 or the origin. • If the points A and B lie on opposite sides of the origin, the maximum shearing strain is the maximum in-plane shearing strain, D and E. • If the points A and B lie on the same side of the origin, the maximum shearing strain is out of the plane of strain and is represented by the points D’ and E’.
  • 39. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 39 Three-Dimensional Analysis of Strain • Consider the case of plane stress, 0=== zbyax σσσσσ • Corresponding normal strains, ( ) ( )babac ba b ba a E EE EE εε ν ν σσ ν ε σσν ε σνσ ε + − −=+−= +−= −= 1 • If B is located between A and C on the Mohr-circle diagram, the maximum shearing strain is equal to the diameter CA. • Strain perpendicular to the plane of stress is not zero.
  • 40. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThird Edition Beer • Johnston • DeWolf 7 - 40 Measurements of Strain: Strain Rosette • Strain gages indicate normal strain through changes in resistance. ( )yxOBxy εεεγ +−= 2 • With a 45o rosette, εx and εy are measured directly. γxy is obtained indirectly with, 333 2 3 2 3 222 2 2 2 2 111 2 1 2 1 cossinsincos cossinsincos cossinsincos θθγθεθεε θθγθεθεε θθγθεθεε xyyx xyyx xyyx ++= ++= ++= • Normal and shearing strains may be obtained from normal strains in any three directions,