MEG 361 CAD Finite Element Method Dr. Mostafa S. Hbib

FEM is powerful numerical technique ….. FEM uses variational and Interpolation methods for modeling and solving BVPs such as DPS (bars, beams, plates, trusses, frames, fluid flow, heat transfer …..)

…FEM is powerful numerical technique FEM is very systematic and modular. Therefore, it is easy to implement on computers. There are several FE codes packages available (Ansys, Nastran, IDEAS, ADAMS,….)

…FEM approximates structures in two ways: Structure (Field ) Discretization (into elements called FE’) Use mathematical model if known Example …

Example: The Bar Letus first review the math model of longitudinal vibrating bar

The long. Vib. Of a bar gives a simple example of how FEM is constructed and how is used to approximate the vib of a DPS with that of LPS (FEM). Two FEModels (grids of the same beam. a) Single-element and b) Three-element model.

Intergrating (1) to yield: The static (time independent) displacement of the bar element must satisfy (for 0 ≤x ≤ l): (1) Intergrating (1) to yield: (2)

The FEM proceeds with two levels: Which model to use (i.e., which mesh and size of mesh where to put elements and nodes) The choice of polynomials to use in (1) (shape functions) At each node the value of u is allowed to be time dependent, hence we use the labels u1(t) and u2(t) as boundaries to evaluate the spatial constants in the shape function: (2) Intergrating (1) to yield: At x=0 sub. Into (2):

If u1 and u2 are known then (3) would provide Subs. C1 and c2 yields the shape function: (3) If u1 and u2 are known then (3) would provide an approximate solutiion to (1). Strain energy: Subs. With u(x,t): Now consider represented by: Where:

Using u(x,t): Subs. With u(x,t): Where: Using the variational (Lagrangian) approach: Where: I is the I th coordinate of the system which is assumed to have n DOF

Again, u(x,t): Using the variational (Lagrangian) approach: Where: I is the I th coordinate of the system which is assumed to have n DOF Subs. With u(x,t) in the lagrangian (remember that u1 = 0 in this case :

Subs. With u(x,t) in the lagrangian (remember that u1 = 0 in this case yields: Which can be solved (given IC for u2 ) yields Exact solution:

The FEM has a natural freq. We have the shape function: (3) (4) Subs. the FEM solution, we get: Example: Compare the exact solution of the clamped bar and that is derived by the FEM, i. e., (4)

Example: Compare the exact solution of the clamped bar and that is derived by the FEM, i. e., (4) NB. FEM gives only one mode (One Element

This example

Example Same Cantilever Bar 3-Element, 4-Node Mesh

To use the Lagrangian approach we need to compute:

To use the Lagrangian approach we need to compute: Subs. In the Lagrangian we get:

Is the global mass matrix and the coeffecient Is the global stiffness matrix Example: Compare the natural frequencies of the 3-element FEM with the exact DPS model. the clamped-free bar determined by substituting the global stiffness matrix the global mass matrix into the FEM. …

Solution: The natural frequencies of the 3-element FEM of the clamped-free bar are determined by substituting the global stiffness matrix and the global mass matrix into the FEM. Solve the EVP: (5) (5) The natural frequencies of the 3-element FEM of the clamped-free bar are:

The exact natural frequencies of the clamped-free bar are: %Error FE Freq. Exact