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- 1. Review of the Concepts of Stress, Strain, and Hookes Law by: Prof. Mark E. Tuttle Dept. Mechanical Engineering Box 352600 University of Washington Seattle, WA 98195 1
- 2. Prof. M. E. Tuttle University of WashingtonForce Vectors• A force, F , is a vector (also called a "1st-order tensor")• The description of any vector (or any tensor) depends on the coordinate system used to describe the vector 100 lbf 100 lbf 100 lbf +y +x" +x +y 45 degs +y" 60 deg +x F = 70.7 lbf (x-direction) F = 86.6 lbf (x"-direction) F = 100 lbf + + (y-direction) 70.7 lbf (y-direction) 50 lbf (y"-direction) 2
- 3. Prof. M. E. Tuttle University of WashingtonNormal and Shear Forces• A "normal" force acts perpindicular to a surface• A "shear" force acts tangent to a surface P = Normal Force V = Shear Force 3
- 4. Prof. M. E. Tuttle University of Washington Forces Inclined to a Plane• Since forces are vectors, a force inclined to a plane can always be described as a combination of normal and shear forces Inclined Force P = Normal Force V = Shear Force 4
- 5. Prof. M. E. Tuttle University of WashingtonMoments• A moment (also called a "torque" or a "couple") is a force which tends to cause rotation of a rigid body• A moment is also vectoral quantity (i.e., a 1st-order tensor)... M M 5
- 6. Prof. M. E. Tuttle University of WashingtonStatic Equilibrium• A rigid solid body is in "static equilibrium" if it is either - at rest, or - moves with a constant velocity• Static equilibrium exists if: Σ F = 0 and Σ M = 0 50 lbf 50 lbf +y (40 lbf) (40 lbf) (30 lbf) +x (30 lbf) 60 lbf (30 lbf) (BALL ACCELERATES) (30 lbf) (NO ACCELERATION) (40 lbf) (40 lbf) 50 lbf 50 lbf 6
- 7. Prof. M. E. Tuttle University of WashingtonFree Body Diagramsand Internal Forces• An imaginary "cut" is made at plane of interest• Apply Σ F = 0 and Σ M = 0 to either half to determine internal forces F F R (= F) "cut" (or) R (= F) F F 7
- 8. Prof. M. E. Tuttle University of WashingtonFree Body Diagramsand Internal Forces• The imaginary cut can be made along an arbitrary plane• Internal force R can be decomposed to determine the normal and shear forces acting on the arbitrary plane F R (= F) P "cut" V F F F 8
- 9. Prof. M. E. Tuttle University of WashingtonStress:Fundamental Definitions• Two "types" of stress: normal stress = σ = P/A shear stress = τ = V/A where P and V must be uniformly distributed over A P = Normal Force σ = P/A V = Shear Force τ = V/A 9 A = Cross-Sectional Area
- 10. Prof. M. E. Tuttle University of WashingtonDistribution of Internal Forces• Forces are distributed over the internal plane...they may or may not be uniformly distributed F M σ = F/A σ= ? "cut" "cut" F F M M 10
- 11. Prof. M. E. Tuttle University of WashingtonInfinitesimal Elements• A free-body diagram of an "infinitesimal element" is used to define "stress at a point"• Forces can be considered "uniform" over the infinitesimally small elemental surfaces +x +z +y dx dy dz 11
- 12. Prof. M. E. Tuttle University of WashingtonLabeling Stress Components• Two subscripts are used to identify a stress component, e.g., "σxx" or "τxy" (note: for convenience, we sometimes write σx = σxx, or σxy = τxy) • 1st subscript: identifies element face • 2nd subscript: identifies "direction" of stress τ xy +y σ xx +x σ xx 12
- 13. Prof. M. E. Tuttle University of WashingtonAdmissable ShearStress States τyx +y +y +y +x τxy τxy +x τxy τxy +x τxy τyx ΣF = 0 ΣF = 0 If: τyx = τxy ΣM = 0 ΣM = 0 ΣF = 0 (inadmissable) (inadmissable) ΣM = 0 (admissable) 13
- 14. Stress Sign Conventions• The "algebraic sign" of a cube face is positive if the outward unit normal of the face "points" in a positive coordinate direction• A stress component is positive if: • stress component acts on a positive face and "points" in a positive coordinate direction, or • stress component acts on a negative face and "points" in a negative coordinate direction. 14
- 15. Stress Sign Conventions +y +y σyy σyy τxy τxy σxx σxx σxx σxx +x +x σyy σyy σxx and σ yy Positive All Stresses Positive τxy Negative 15
- 16. Prof. M. E. Tuttle University of Washington 3-Dimensional Stress States• In the most general case, six independent components of stress exist "at a point" +x F1 +z σxx F2 +y M1 τxz τzy τxy σzz F3 σyy F4 M2 16
- 17. Prof. M. E. Tuttle University of WashingtonPlane Stress• If all non-zero stress components exist in a single plane, the (3-D) state of stress is called "plane stress" +y +y σyy σyy +x τxy +z τxy σxx σxx σxx σxx +x σyy σyy 17
- 18. Prof. M. E. Tuttle University of WashingtonUniaxial Stress• If only one normal stress exists, the (3-D) state of stress is called a "uniaxial stress" +y +y +x σyy σyy +z +x σyy σyy 16
- 19. Prof. M. E. Tuttle University of WashingtonFree Body Diagram Definesthe Coordinate System F +y +y F σyy "cut" +x +x F F F 17
- 20. Prof. M. E. Tuttle University of WashingtonFree Body Diagram Definesthe Coordinate System F y y σyy P V x τxy x "cut" F F F 18
- 21. Prof. M. E. Tuttle University of WashingtonFree Body Diagram Definesthe Coordinate System F "cut" y" y" (a plane) Py"y" σy"y" x" x" Vy"x" τy"x" Vy"z" τy"z" z" z" F F F 21
- 22. Prof. M. E. Tuttle University of WashingtonStress TransformationsWithin a Plane• Given stress components in the x-y coordinate system (σxx, σyy, τxy), what are the corresponding stress components in the x-y coordinate system? σyy +y σyy τxy τxy +y σxx ? +x σxx θ +x 20
- 23. Prof. M. E. Tuttle University of WashingtonStress Transformations• Stress components in the x-y coordinate system may be related to stresses in the x-y coordinate system using a free body diagram and enforcing Σ F = 0 Σ Fx = 0 +y σyy +y +x τxy τxy θ σxx σxx σxx +x τxy "cut" σyy 23
- 24. Prof. M. E. Tuttle University of WashingtonStress TransformationEquations• By enforcing ΣFx = 0, ΣFy = 0, it can be shown: σ xx + σ yy σ xx − σ yy σ x x = + cos 2θ + τ xy sin2θ 2 2 σ xx + σ yy σ xx − σ yy σ y y = − cos2θ − τ xy sin2θ 2 2 σ xx − σ yy τ xy = − sin 2θ + τ xy cos 2θ 2 22
- 25. Prof. M. E. Tuttle University of WashingtonUse of Mohrs Circle ofStress σyy +y τxy σxx τ (σyy,τ xy) (σxx ,−τxy ) +x 2θ 2θ (Equivalent) σ σyy τxy (σxx,−τxy) (σyy ,τ xy ) +y σxx +x θ 25
- 26. Prof. M. E. Tuttle University of WashingtonPrincipal Stresses• In the principal stress coordinate system the shear stress is zero... σyy +y τxy τ σxx (σyy,τ xy) +x 2θ (Equivalent) σ22 2θ σ11 σ σ22 +2 σ11 (σxx,−τxy) +1 θ 26
- 27. Prof. M. E. Tuttle University of Washington"Stress":Summary of Key Points• Normal and shear stresses are both defined as a (force/area)• Six components of stress must be known to specify the state of stress at a point (stress is a "2nd-order tensor")• Since stress is a tensoral quantity, numerical values of individual stress components depend on the coordinate system used to describe the state of stress• Stress is defined strictly on the basis of static equilibrium; definition is independent of: material properties 27 strain
- 28. Prof. M. E. Tuttle University of WashingtonStrain:Fundamental Definitions• "Strain" is a measure of the deformation of a solid body• There are two "types" of strain; normal strain (ε) and shear strain (γ) (change in length ) ε= ; units = in / in, m / m, etc (original length) γ = (change in angle ) ; units = radians 28
- 29. Prof. M. E. Tuttle University of Washington3-Dimensional Strain States• In the most general case, six components of strain exist "at a point": εxx, εyy, εzz, γxy, γxz, γyz• Strain is a 2nd-order tensor; the numerical values of the individual strain components which define the "state of strain" depend on the coordinate system used• This presentation will primarily involve strains which exist within a single plane 29
- 30. Prof. M. E. Tuttle University of WashingtonStrain Within a Plane• We are often interested in the strains induced within a single plane; specifically, we are interested in two distinct conditions: • "Plane stress", in which all non-zero stresses lie within a plane. The plane stress condition induces four strain components: εxx, εyy, εzz, and γxy • "Plane strain", in which all non-zero strain components lies within a plane. By definition then, the plane strain condition involves three strain components (only): εxx, εyy, and γxy 30
- 31. Prof. M. E. Tuttle University of WashingtonStrain Within a Plane ∠ abc = π/2 radians ∠ abc < π/2 radians +y +y a a ly + ∆ly ly c c b b +x +x lx lx + ∆lx Original Shape Deformed Shape εxx = ∆lx/lx εyy = ∆ly/ly γxy = ∆(∠ abc) 29
- 32. Strain Sign Convention• A positive (tensile) normal strain is associated with an increase in length• A negative (compressive) normal strain is associated with a decrease in length• A shear strain is positive if the angle between two positive faces (or two negative faces) decreases +y +y +x +x All Strains Positive εxx Positive εyy and γ xy Negative 32
- 33. Prof. M. E. Tuttle University of WashingtonVisualization of Strain F +y (Loading) εyy > 0 +x εxx < 0 γxy = 0 F 33
- 34. Prof. M. E. Tuttle University of WashingtonVisualization of Strain F +y +x (Loading) εxx = εyy > 0 γxy > 0 45 deg F 34
- 35. Prof. M. E. Tuttle University of WashingtonStrain Transformations• Given strain components in the x-y coordinate system (εxx, εyy, γxy), what are the corresponding strain components in the x-y coordinate system? +y +y +x ? ly + ∆ly θ +x lx + ∆lx εxx = ? εxx = ∆lx/lx εyy = ? εyy = ∆ly/ly γxy = ? γxy = ∆(∠ abc) 35
- 36. Prof. M. E. Tuttle University of WashingtonStrain TransformationEquations • Based strictly on geometry, it can be shown: ε xx + ε yy ε xx − ε yy γ xy ε x x = + cos 2θ + sin2θ 2 2 2 ε xx + ε yy ε xx − ε yy γ xy ε y y = − cos2θ − sin2θ 2 2 2 γ xy ε xx − ε yy γ xy =− sin 2θ + cos 2θ 2 2 2 36
- 37. Prof. M. E. Tuttle University of WashingtonWell, Ill Be Darned!!!• The stress transformation equations are based strictly on the equations of static equilibrium• The strain transformation equations are based strictly on geometry• Nevertheless, the stress and strain transformation equations are nearly identical!! 37
- 38. Prof. M. E. Tuttle University of Washington "Strain": Summary of Key Points• ε = (∆ length)/(original length) γ = (∆ angle)• Six components of strain specify the "state of strain"• Strain is a 2nd-order tensor; numerical values of individual strain components depend on the coordinate system used• Strain is defined strictly on the basis of a change in shape; definition is independent of: material properties stress• "Suprisingly," the stress and strain transformation equations are nearly identical 38
- 39. Prof. M. E. Tuttle University of Washington"Constitutive Models"• The structural engineer is typically interested in the state of stress induced in a structure during service• The state of stress cannot be measured directly...• The state of strain can be measured directly...• Hence, we must develop a relation between stress and strain...this relationship is called a "constitutive model," and the most common is "Hookes Law" 39
- 40. Prof. M. E. Tuttle University of Washington Material Properties: The Uniaxial Tensile Test• Specimen is subjected to an axial tensile force, inducing a uniform tensile stress in the "gage" region of the specimen• Stress is increased until fracture occurs; strains are measured throughout the test• The ASTM has developed "standard" specimen styles/procedures for most materials F F "Gage Region" 40
- 41. Prof. M. E. Tuttle University of WashingtonThe TensileStress-Strain Curve σTensile StrengthFracture Stress x0.2% OffsetYield Strength ε 41 0.2% Strain to Fracture
- 42. Prof. M. E. Tuttle University of Washington Load-Unload Cycles σ σ σTensile Strength Tensile Strength Tensile Strength0.2% Offset 0.2% Offset 0.2% OffsetYield Strength Yield Strength Yield Strength ε ε ε Elastic Strain Elastic Strain Inelastic Strain Elastic Strain Inelastic Strain 42
- 43. Prof. M. E. Tuttle University of WashingtonMaterial Property:Youngs Modulus σ• At stress levels below the yield stress the response is called "linear elastic" 0.2% Offset Yield Strength• The slope of the linear In Linear Region: region is called "Youngs E σ = Eε modulus" or the "modulus of elasticity", E 1• In the linear region (only): ε σ = Eε (or) ε = σ/E
- 44. Prof. M. E. Tuttle University of Washington Material Property: Poissons Ratio• A uniaxial tensile stress causes both a tensile axial strain (εa) as well as a compressive transverse strain (εt)• Poissons ratio is defined as: ν = -(εt/εa) F F εa εt 44
- 45. Prof. M. E. Tuttle University of WashingtonMaterial Properties:The Torsion Test• Cylindrical specimen is subjected to a torque, inducing a uniform shear stress τxy in the gage region of the specimen• Shear stress increased until fracture occurs; shear strain measured throughout test +y τ xy +x T T 45
- 46. Prof. M. E. Tuttle University of WashingtonMaterial Property:Shear Modulus• At linear levels, the slope of the shear τ stress vs shear strain curve is called the "shear modulus," G• In the linear region G (only): τ = Gγ (or) γ = τ/G γ 46
- 47. Prof. M. E. Tuttle University of Washington"Isotropic" vs"Anisotropic" Materials• A material which exhibits the same properties in all directions is called "isotropic" 2 2 1 1 2 1 1 2 E1 ≅ E2 E1 > E2 Isotropic Materials Anisotropic Materials 47
- 48. Prof. M. E. Tuttle University of WashingtonMaterial PropertiesSummary• Three material properties are commonly used to describe isotropic materials in the linear-elastic region: E, ν, and G• It can be shown that (for isotropic materials) only two of these three properties are independent. For example: E G= 2(1 + ν) 48
- 49. Prof. M. E. Tuttle University of Washington Hookes Law: 3-Dimensional Stress States• Assumptions: • Linear-elastic behavior; i.e., yielding does not occur, so the principle of superposition applies • Isotropic material behavior • Constant environmental conditions (e.g., constant temperature) 49
- 50. Strains Induced by Individual Stress Components(That is, Strain Caused During Six Different Tests) = εx xσ σxx /E xx γx y τ= τx y /G = 2(1+ν)τx y /E = -νεx x εyy Induces:= -νσx x /E xy εx x = εyy = εzz = γxz = γyz = 0 Induces: εzz = -νεx x = -νσx x /E γx y = γxz = γyz = 0 εyyσ= σyy /E yy = -νεyy εx x Induces:= -νσyy /E γxz τ= τxz /G = 2(1+ν)τxz /E xz εzz = -νεyy = -νσyy /E εx x = εyy = εzz = γx y = γyz = 0 Induces: γx y = γxz = γyz = 0 = εzzσ σzz/E zz εx x Induces:= -νσzz/E = -νεzz γyz τ= τyz /G = 2(1+ν)τyz /E yz εyy = -νεzz = -νσzz/E εx x = εyy = εzz = γx y = γxz = 0 Induces: γx y = γxz = γyz = 0
- 51. Prof. M. E. Tuttle University of Washington 3-D Form of Hookes Law: Apply Principle of Superposition• The total strain εxx caused by all stresses applied simultaneously is determined by "adding up" the strain εxx caused by each individual stress component σ xx −νσ yy −νσ zz ε xx = E + E + E + [0 ] + [0 ] + [0 ] or ε xx 1 [ = σ xx − νσ yy − νσ zz E ] 51
- 52. Prof. M. E. Tuttle University of WashingtonHookes Law:3-Dimensional Stress States• Following this procedure for all six strain components: 2(1+ ν)τ yz 1 [ ε xx = σ xx − νσ yy − νσ zz E ] γ yz = E 2(1+ ν)τ zx 1 [ ε yy = −νσ xx + σ yy − νσ zz E ] γ zx = E 2(1+ ν)τ xy 1 [ ε zz = −νσ xx − νσ yy + σ zz E ] γ xy = E 52
- 53. Prof. M. E. Tuttle University of Washington Hookes Law: 3-Dimensional Stress States ε xx 1 −ν −ν 0 0 0 σ xx σ ε yy −ν 1 −ν 0 0 0 yy ε zz 1 −ν −ν 1 0 0 0 σ zz = γ yz E 0 0 0 2(1+ ν) 0 0 τ yz γ τ zx 0 0 0 0 2(1+ ν) 0 zx γ xy 0 0 0 0 0 2(1+ ν) τ xy 53
- 54. Prof. M. E. Tuttle University of Washington Hookes Law: 3-Dimensional Stress States [ ] E Eγ yzσ xx = (1− v)ε xx + νε yy + νεzz τ yz = (1 + ν)(1− 2 ν) 2(1 + ν)σ yy = E (1+ ν)(1 − 2ν) [ vε xx + (1 − v)ε yy + νε zz ] τ zx = Eγ zx 2(1 + ν) [ ] E Eγ xyσ zz = vε xx + νε yy + (1 − v)ε zz τ xy = (1+ ν)(1− 2ν) 2(1 + ν) 54
- 55. Prof. M. E. Tuttle University of WashingtonHookes Law:3-Dimensional Stress States (1− ν) ν ν 0 0 0 σ xx ε ν (1− ν) ν 0 0 0 xx σ yy ν ε ν (1− ν) 0 0 0 yy σ ε zz 0 zz E 0 (1− 2ν) = 0 0 0 τ yz (1+ ν)(1− 2ν) 2 γ yz τ (1− 2ν) 0 0 0 0 0 γ zx zx 2 τ xy γ (1− 2ν) xy 0 0 0 0 0 2 55
- 56. Prof. M. E. Tuttle University of Washington Hookes Law: 3-Dimensional Stress States• "Shorthand" notation: {σ } = [ D]{ε } where: (1− ν) ν ν 0 0 0 ν (1− ν) ν 0 0 0 ν ν (1− ν) 0 0 0 (1− 2ν) [ D] = E 0 0 0 0 0 (1 + ν)(1− 2 ν) 2 (1− 2ν) 0 0 0 0 2 0 (1 − 2ν) 0 0 0 0 0 2 56
- 57. Prof. M. E. Tuttle University of Washington Hookes Law: Plane Stress• For thin components (e.g., web or flange of an I-beam, automobile door panel, airplane "skin", etc) the "in-plane" stresses are much higher than "out-of-plane" stresses: (σxx,σyy,τxy) >> (σzz,τxz,τyz)• It is convenient to assume the out-of-plane stresses equal zero...this is called a state of "plane stress" +y +x +z 57
- 58. Prof. M. E. Tuttle University of WashingtonHookes Law:Plane Stress 1 ε xx = (σ xx − νσ yy ) E 1 σ yy ε yy = (σ yy − νσ xx ) τ xy E −ν σ xx ε zz = (σ xx + σ yy ) E τ xy 2(1+ ν)τ xy γ xy = = G E γ xz = γ yz = 0 58
- 59. Prof. M. E. Tuttle University of WashingtonHookes Law:Plane Stress E σ xx = 2 (ε xx + νε yy ) (1- ν ) σ yy E τ xy σ yy = 2 (ε yy + νε xx ) (1- ν ) σ xx E τ xy = Gγ xy = γ xy 2(1+ ν) σ zz = τ xz = τ yz = 0 59
- 60. Prof. M. E. Tuttle University of WashingtonHookes Law:Plane Stress σ xx 1 ν 0 ε xx E σ yy = 2 ν 1 0 ε yy (1− ν ) 1 − ν τ xy 0 0 γ xy 2 60
- 61. Prof. M. E. Tuttle University of WashingtonHookes Law:Plane Stress• "Shorthand" notation: {σ } = [ D]{ε } where 1 ν 0 E [ D] = 2 ν 1 0 (1 − ν ) 1− ν 0 0 2 61
- 62. Prof. M. E. Tuttle University of Washington Hookes Law: Plane Strain• For thick or very long components (e.g., thick-walled pressure vessels, buried pipe, etc) the "in-plane" strains are much higher than "out-of-plane" strains: (εxx,εyy,γxy) >> (εzz,γxz,γyz)• It is convenient to assume the out-of-plane strains equal zero...this is called a state of "plane strain" +y +z +x 62
- 63. Prof. M. E. Tuttle University of WashingtonHookes Law:Plane Strain σ xx = E (1 + ν)(1− 2ν) [ (1− ν)ε xx + νε yy ] σ yy τ xy σ yy = E (1+ ν)(1− 2ν) [ νε xx + (1− ν)ε yy ] νE σ xx σ zz = (1+ ν)(1− 2ν) [ ] [ ε xx + ε yy = ν σ xx + σ yy ] Eγ xy τ xy = Gγ xy = 2(1+ ν) τ xz = τ yz = 0 63
- 64. Prof. M. E. Tuttle University of WashingtonHookes Law:Plane Strain σ xx (1− ν) ν 0 ε xx E ν ε σ yy = (1− ν) 0 yy (1+ ν)(1− 2ν) (1 − 2v) γ τ xy 0 0 xy 2 64
- 65. Prof. M. E. Tuttle University of WashingtonHookes Law:Plane Strain• "Shorthand" notation: {σ } = [ D]{ε } where (1− ν) ν 0 [ D] = E ν (1− ν) 0 (1 + ν)(1− 2ν) (1− 2v) 0 0 2 65
- 66. Hookes LawPlane Strain 1 − ν2 ν ε xx = σ xx − σ yy E 1− ν σ yy 1− ν 2 ν τ xy ε yy = σ yy − σ xx E 1− ν 2(1+ ν) σ xx γ xy = τ xy E ε zz = γ xz = γ yz = 0 [ ....and: σ zz = ν σ xx + σ yy ] 66
- 67. Prof. M. E. Tuttle University of WashingtonHookes Law:Uniaxial Stress• Truss members are designed to carry axial loads only (i.e., uniaxial stress)• In this case we are interested in the axial strain only (even though transverse strains are also induced) 67
- 68. Prof. M. E. Tuttle University of Washington Hookes Law: Uniaxial Stress• For: σyy = σzz = τyz = τxz = τzy = 0, "Hookes Law" becomes: σ xx = Eε xx• "Shorthand" notation: {σ } = [ D]{ε } where [ D] = [ E ] 68
- 69. Prof. M. E. Tuttle University of WashingtonHookes LawSummary of Key Points• Hookes Law is valid under linear-elastic conditions only• The mathematical form of Hookes Law depends on whether the problem involves: • Structures for which 3-D stresses are induced • Structures for which (due to loading and/or geometry) plane stress conditions can be assumed • Structures for which (due to loading and/or geometry) plane strain conditions can be assumed • Structures for which (due to loading and/or geometry) uniaxial stress conditions can be assumed 69

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