MECH593 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: y z 𝑡≪𝐿 𝑡≪𝑊 x

Plane Stress and Plane Strain Plane Strain - Thick Plate: 𝑡≫𝐿 𝑡≫𝑊 z y x Plane Stress: Plane Strain: Replace E by and by

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

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

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

Finite Element Formulation for Plane Elasticity Let where and

Constant-Strain Triangular (CST) Element Let

Constant-Strain Triangular (CST) Element - A mesh could be too stiff y P x P - Mesh locking y II A I x

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 6 5 B 3 4 A 1 2

Evaluation of Applied Nodal Forces

Evaluation of Applied Nodal Forces Y y 6 5 B x 3 4 A 1 2 X

Element Assembly for Plane Elasticity 5 6 B 3 4 3 4 A 1 2

Element Assembly for Plane Elasticity 1 2 3 4 6 5 A B

Imposing Boundary Conditions 1 2 3 4 6 5 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

Isoparametric Elements Example 1: Physical domain (physical element) Reference domain (master element) h h 3 4 4 3 x x y 1 1 2 2 x

Isoparametric Elements Example 2: Physical domain (physical element) Reference domain (master element) h h 3 3 1 y 1 x x 2 2 x Connection with shape functions expressed in area coordination

Isoparametric Elements Example 2: Physical domain (physical element) Reference domain (master element) h h 3 3 6 5 5 6 x 1 y x 4 2 1 4 2 x

Isoparametric Elements An element is an isoparametric element if the same shape functions are employed to approximate geometry as well as the unknown variables. Stiffness matrix and force vector calculation:

Isoparametric Rectangular Elements where

Higher Order 2-D Isoparametric Elements

Gaussian Quadrature Formula for Triangles