Applied Calculus of Variations for Engineers, 2nd Edition

202 Applied calculus of variations for engineers

global finite element matrix is then

K =

⎡

⎢

⎢

⎣

K

1

(1, 1) K

1

(1, 2) K

1

(1, 3) 0

K

1

(2, 1) K

1

(2, 2) + K

2

(1, 1) K

1

(2, 3) + K

2

(1, 2) K

2

(1, 3)

K

1

(3, 1) K

1

(3, 2) + K

2

(2, 1) K

1

(3, 3) + K

2

(2, 2) K

2

(2, 3)

0 K

2

(3, 1) K

2

(3, 2) K

2

(3, 3)

⎤

⎥

⎥

⎦

.

The assembled global load vector is similarly obtained:

F =

2



i=1

L

ge

F

g

e,i

=

⎡

⎢

⎢

⎣

F

1

(1)

F

1

(2) + F

2

(1)

F

1

(3) + F

2

(2)

F

2

(3)

⎤

⎥

⎥

⎦

.

The notation convention is the same as in the element matrix assembly.

The global solution is then obtained from the matrix equation

Ku

= F,

where K is the global stiffness matrix and F is the global force vector. The

global solution vector is

u

= K

−1

F =

⎡

⎢

⎢

⎣

u

1

u

2

u

3

u

4

⎤

⎥

⎥

⎦

,

and ...