MECH593 Introduction to Finite Element Methods Finite Element Analysis of 2D Problems Axisymmetric Problems Plate Bending

Axi-symmetric Problems Definition: A problem in which geometry, loadings, boundary conditions and materials are symmetric about one axis. Examples:

Axi-symmetric Analysis Cylindrical coordinates: quantities depend on r and z only 3-D problem 2-D problem

Axi-symmetric Analysis

Axi-symmetric Analysis – Single-Variable Problem Weak form: where

Finite Element Model – Single-Variable Problem where Ritz method: Weak form where

Single-Variable Problem – Heat Transfer Weak form where

3-Node Axi-symmetric Element 1 2

4-Node Axi-symmetric Element h 4 3 b 1 2 x a z r

Single-Variable Problem – Example z T(r,L) = T0 Step 1: Discretization R L T(R,z) = T0 r T(r,0) = T0 Step 2: Element equation Heat generation: g = 107 w/m3

Plate Bending

Governing Equations of Classical Plates From force equilibrium --- Governing Equations for Classical Plates ----- (Distributed Transverse Loading) where Bending Stiffness (Flexural Rigidity) D = Eh3/12(1-n2)

Strain Energy of Classical Plates

Weak Form of Classical Plates Governing equation: (isotropic, steady) Weak form: Note: w is the deflection of the mid-plane and u is the weight function.

Boundary Conditions of Classical Plates Essential Boundary Conditions ----- Natural Boundary Conditions ----- Examples: Clamped : Simply connected free

4-Node Rectangular Plate Element Since the governing eq. is 4th order, at each node, there should 2 EBCs and 2 NBCs in each direction (but specify just 2 of them). For displacement-based finite element formulation, the DoFs should be on generalized displacements. In total, there are 3 DoFs per node: where

Formulation of 4-Node Rectangular Plate Element Let Pascal’s Triangle ----- (incomplete 4th order polynomial)

3-Node Triangular Plate Element Let Pascal’s Triangle ----- (incomplete 3th order polynomial)