Mathematical Physics - Volume II - Numerical Methods

3.1 Approximate analytical solution

53

w

w

w

1

3

5

w

w

w

w( ) x

2

4

6

1

2

3

x

L

(1)

(2)

(1)

(2)

w

w

w

w

1

3

1

3

(1)

(2)

(1)

(2)

w

w

w

w

2

4

2

4

Globalni broj ~vora

1

2

3

4

Element 1

Element 2

L 2

L 2

1

2

1

2

l =

l =

Lokalni broj ~vora

x

x

Figure 3.2: Finite elements.

For potential (static) and kinetic energy of the element, the following is obtained:

EI ρ A (

∂ 2 ω ∂ x 2

2

1 2 Z 1 2 Z

l

1 2

⃗ W ( e ) T

Π ( e ) =

[ K ( e ) ] ⃗ W ( e )

dx =

(3.28)

0

∂ω ∂ x

l

1 2

˙ ⃗ W ( e ) T

[ M ( e ) ] ˙ ⃗ W ( e )

T ( e ) =

) 2 dx =

(3.29)

0

where ρ is the density, A is the cross-section area, and "˙" is the derivative with respect to time. By replacing (3.27) into (3.28) and (3.29), the stiffness matrix [ K ( e ) ] is obtained, along with the mass matrix [ M ( e ) ] :

   3 l 3 l

  

6 3 l

− 6 3 l 2 l 2 − 3 l l 2

2 EI l 3

[ K ( e ) ] =

(3.30)

− 6 − 3 l

6

− 3 l

l 2

− 3 l 2 l 2

   22 l

  

156 22 l

54

− 13 l − 3 l 2

4 l 2

ρ AI 420

13 l

[ M ( e ) ] =

(3.31)

54 13 l 156 − 22 l − 13 l − 3 l 2 − 22 l 4 l 2

If finite element equations are arranged according to Figure 3.2, global stiffness matrix is