{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter 2 Review of Elasticity

# Chapter 2 Review of Elasticity - Chapter 2 Review of...

This preview shows pages 1–5. Sign up to view the full content.

Chapter 2. Review of Elasticity 1. Stress h Plane-Strain - The plane-strain distribution is based on the assumption that 0 z u z = = (2-1) where z represents the lengthwise direction of an elastic elongated body of constant cross section subjected to uniform loading (in the case of uniaxial problems, / 0 y y u = = ). xy , , x y u v u v x y y x ε ε γ = = = + (2-2) are functions of x and y only and the strain components xz yz , , z w w u w v z x z y z ε γ γ = = + = + (2-3) vanish. The corresponding strain-stress relationship becomes ( 29 ( 29 ( 29 1 1 1 1 2 1 x x y y y x xy xy xy E E G E μ ε μ σ μσ μ ε μ σ μσ τ μ γ τ + = - - + = - - + = = (2-4) Hence, ( 29 ( 29 29,000 11,154 2 1 2 1 0.3 E G μ = = = + + (2-5) 1

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

View Full Document
The stress-strain relationship is ( 29 ( 29 ( 29 1 0 1 0 1 1 2 0 0 1 2 / 2 x x y y xy xy E σ μ μ ε σ μ μ ε μ μ τ μ γ - = - + - - (2-6) ( 29 z x y σ μ σ σ = + (2-7) P Plane-Stress - The plane stress distribution is based on the assumption that 0 z yz xz σ τ τ = = = . (2-8) The stress-strain relationship is ( 29 2 1 0 1 0 1 0 0 1 / 2 x x y y xy xy E σ μ ε σ μ ε μ τ μ γ = - - (2-9) and the strain-stress relationship is ( 29 1 0 1 1 0 0 0 2 1 x x y y xy xy E ε μ σ ε μ σ γ μ τ - = - + (2-10) If we assume a state of plane stress, we need not have a corresponding plane strain and conversely, if we assume a state of plane strain, we need not have a corresponding plane stress state. These ar e entirely different assumptions. Home Work #3 2
Using the generalized Hooke’s law, prove that the maximum value of Poisson’s ratio, μ , cannot exceed 0.5. P For an analytical determination of the distribution of static or dynamic displacements and stresses in a structure (body) under prescribed external loading and temperature, we must obtain a solution to the basic equations of the theory of elasticity, satisfying the imposed boundary conditions on forces and/or displacements. P For general 3-D structures, we have 6: strain-displacement relationships; ( 29 , , f u v w ε = 6: stress-strain relationship, generalized Hooke’s law 3 : Equations of equilibrium, linear momentum balance 15 P Thus, there are 15 equations available for 15 unknowns, namely 3: displacements: u, v, w 6: stresses 6 : strains 15 3

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

View Full Document
In the case of rigid body mechanics problems, we may use 6 equations of equilibrium including three equations of linear momentum balance and three equations of angular momentum balance. Since the three equations of angular momentum balance are used up in establishing the symmetry of shear stresses ( ij ji τ τ = ), they cannot be used again along with the symmetry of shear stresses.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 20

Chapter 2 Review of Elasticity - Chapter 2 Review of...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online