1. MECH300H Introduction to Finite Element Methods Lecture 9 Finite Element Analysis of 2-D Problems – Axi- symmetric Problems

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

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

4. Axi-symmetric Analysis

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

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

7. Single-Variable Problem – Heat Transfer Heat Transfer: Weak form where

8. 3-Node Axi-symmetric Element 1 2 3

9. 4-Node Axi-symmetric Element 1 2 3 4 a b   r z

10. Single-Variable Problem – Example Step 1: Discretization Step 2: Element equation