ME300H Introduction to Finite Element Methods Finite Element Analysis of Plane Elasticity

Review of Linear Elasticity Linear Elasticity: A theory to predict mechanical response of an elastic body under a general loading condition. Stress: measurement of force intensity with 2-D

Review of Linear Elasticity Traction (surface force) : Equilibrium – Newton’s Law

Review of Linear Elasticity Strain: measurement of intensity of deformation Generalized Hooke’s Law

Plane Stress and Plane Strain Plane Stress - Thin Plate:

Plane Stress and Plane Strain Plane Strain - Thick Plate: Plane Stress: Plane Strain: Replace E by and by

Equations of Plane Elasticity Governing Equations (Static Equilibrium) Constitutive Relation (Linear Elasticity) Strain-Deformation (Small Deformation)

Specification of Boundary Conditions EBC: Specify u(x,y) and/or v(x,y) on  NBC: Specify t x and/or t y on  where is the traction on the boundary  at the segment ds.

Weak Formulation for Plane Elasticity where are components of traction on the boundary 

Finite Element Formulation for Plane Elasticity Let where and

Constant-Strain Triangular (CST) Element for Plane Stress Analysis Let

Constant-Strain Triangular (CST) Element for Plane Stress Analysis

4-Node Rectangular Element for Plane Stress Analysis Let

4-Node Rectangular Element for Plane Stress Analysis For Plane Strain Analysis: and

Loading Conditions for Plane Stress Analysis

Evaluation of Applied Nodal Forces

Element Assembly for Plane Elasticity A B

A B

Comparison of Applied Nodal Forces

Discussion on Boundary Conditions Must have sufficient EBCs to suppress rigid body translation and rotation For higher order elements, the mid side nodes cannot be skipped while applying EBCs/NBCs

Plane Stress – Example 2

Plane Stress – Example 3

Evaluation of Strains

Evaluation of Stresses Plane Stress Analysis Plane Strain Analysis