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

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 D problem

4. Axi-symmetric Analysis

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

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

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

8. 3-Node Axi-symmetric Element1 2

9. 4-Node Axi-symmetric Elementh 4 3 b 1 2 x a z r

10. Single-Variable Problem – Examplez
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

11. Plate Bending

12. 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)

13. Strain Energy of Classical Plates

14. 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.

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

16. 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

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

18. 3-Node Triangular Plate Element
Let
Pascal’s Triangle -----
(incomplete 3th order polynomial)