{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter7TransformationsofStressandStrain

# Chapter7TransformationsofStressandStrain - Chapter 7...

This preview shows pages 1–12. Sign up to view the full content.

Chapter 7 Transformations of Stress and Strain

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Examples of Plane-Stress Condition:
Thin-walled Vessels In-plane shear stress Out-of-plane shear stress Shear stress

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Max. σ x & σ y Max. τ xy (Principal stresses)
7.2 Transformation of Plane Stress

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 = + - - + =
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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)
1. σ max and σ min occur at τ = 0 2. σ max and σ min are 90 o apart.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}