20 Nov 2019 admin

Finite Element Method and Some Examples

Finite element method (FEM) is a numerical solution method which search a solution with an acceptable approach in various engineering problems. Basic logic in Finite Element Method is to divide complex expressions to small and understandable expressions. So, what is the solution steps? Solution steps are;

Mathematical model of physical problem is formed.
Formulations of handled problem are formed.
Solution area is divided into pieces which called finite element.

Below, there is an algorithm that can be used in the finite element method.

Stiffness matrices (k' ) are formed for each elements.

Truss Element;

EA/L		-EA/L
-EA/L		EA/L

Beam Element;

12EI/L³	6EI/L²	-12EI/L³	6EI/L²
6EI/L²	4EI/L	-6EI/L²	2EI/L
-12EI/L³	-6EI/L²	12EI/L³	-6EI/L²
6EI/L²	2EI/L	-6EI/L²	4EI/L

Frame Element;

EA/L	0	0	-EA/L	0	0
0	12EI/L³	6EI/L²	0	-12EI/L³	6EI/L²
0	6EI/L²	4EI/L	0	-6EI/L²	2EI/L
-EA/L	0	0	EA/L	0	0
0	-12EI/L³	-6EI/L²	0	12EI/L³	-6EI/L²
0	6EI/L²	2EI/L	0	-6EI/L²	4EI/L

Transformation matrices (T) are formed for each element.

Truss element;

cos		sin		0		0
0		0		cos		sin

Beam element;

0	0	0
1	0	0
0	cos	0
0	0	1

Frame element;

cos	sin	0	0	0	0
-sin	cos	0	0	0	0
0	0	1	0	0	0
0	0	0	cos	sin	0
0	0	0	-sin	cos	0
0	0	0	0	0	1

Transformed element stiffness matrices (k^e = T^Tk'T) are determined.
System stiffness matrix (K) and load vector (F) are formed.
The stiffness matrix is revised according to the boundary conditions.
Joints displacements (U = K^-1F) are obtained.
Displacements for each element (u' = TU) are formed.
The internal forces of the element (k'u' = f') are calculated.