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Chapter7TransformationsofStressandStrain

Chapter7TransformationsofStressandStrain - Chapter 7...

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Chapter 7 Transformations of Stress and Strain
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7.1 Introduction General State of Stress 3 normal stresses 3 shearing stresses -- τ xy , τ yz , and τ zx -- σ x , σ y , and σ z Goals: determine: 1. Principal Stresses 2. Principle Planes 3. Max. Shearing Stresses
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2-D State of Stress Plane Stress condition Plane Strain condition A. Plane Stress State: B. Plane Stress State: σ z = 0, τ yz = τ xz = γ yz = γ xz = 0 ε z 0, τ xy 0 ε z = 0, τ yz = τ xz = γ yz = γ xz = 0 σ z 0, τ xy 0
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Examples of Plane-Stress Condition:
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Thin-walled Vessels In-plane shear stress Out-of-plane shear stress Shear stress
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Max. σ x & σ y Max. τ xy (Principal stresses)
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7.2 Transformation of Plane Stress
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0 0 ' ' : ( cos )cos ( cos )sin ( sin )sin ( sin )cos x xy x x y xy F A A A A A σ σ θ θ τ θ θ σ θ θ τ θ θ Σ = - - - - = 0 0 ' ' ' : ( cos )sin ( cos )cos ( sin )cos ( sin )sin x xy y x y y xy F A A A A A σ σ θ θ τ θ θ σ θ θ τ θ θ Σ = + - - + =
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2 2 2 ' cos sin sin cos x y xy x σ σ θ σ θ τ θ θ = + + 2 2 ' ' ( )sin cos (cos sin ) x y xy x y τ σ σ θ θ τ θ θ = - - + - 2 2 2 2 2 sin sin cos , cos cos sin θ θ θ θ θ θ = = - After rearrangement: (7.1) (7.2) 2 2 1 2 1 2 2 2 cos cos cos , sin θ θ θ θ + - = = Knowing
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2 2 2 2 ' cos sin x y x y xy x σ σ σ σ σ θ τ θ + - = + + 2 2 2 ' ' sin cos x y xy x y σ σ τ θ τ θ - = - + 2 2 2 2 ' cos sin x y x y xy y σ σ σ σ σ θ τ θ + - = - - Eqs. (7.1) and (7.2) can be simplified as: (7.5) (7.6) ' y σ Can be obtained by replacing θ with ( θ + 90 o ) in Eq. (7.5) (7.7)
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1. σ max and σ min occur at τ = 0 2. σ max and σ min are 90 o apart.
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