6.2 Answers for Problems from Chap. 2
119
⎡
⎢
⎢
⎢
⎢
⎣
−2 1
0
0
0
1 −2
1
0
0
0
3A I +A II
4
−(A I + A II )
A I +3A II
4
0
0
0
1
−2 1
0
0
0
1 −2
⎤
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
u 5
u 6
⎤
⎥
⎥
⎥
⎥
⎦
= −
X F 0
E
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
0
1
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(6.20)
The solution of this linear system of equations gives the unknown nodal values as:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
u 5
u 6
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
F 0 L
E A
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
9
2
9
1
3
2
9
1
9
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(6.21)
(a) The evaluation of the finite difference approximation according to Eq. (2.56) at
the inner nodes i = 2, . . . , 6 gives:
node 2:
E A I
X
(u 3 − 2u 2 + u 1 ) = −p 0 X ,
(6.22)
node 3:
E A I
X
(u 4 − 2u 3 + u 2 ) = −p 0 X ,
(6.23)
node 4:
E
X
A II − A I
4
(u 5 − u 3 ) +
A I + A II
2
(u 5 − 2u 4 + u 3 )
=
− p 0 X ,
(6.24)
node 5:
E A II
X
(u 6 − 2u 5 + u 4 ) = −p 0 X ,
(6.25)
node 6:
E A II
X
(u 7 − 2u 6 + u 5 ) = −p 0 X ,
(6.26)
or in matrix notation under consideration of the boundary conditions:
⎡
⎢
⎢
⎢
⎢
⎣
−2 1
0
0
0
1 −2
1
0
0
0
3A I +A II
4
−(A I + A II )
A I +3A II
4
0
0
0
1
−2 1
0
0
0
1 −2
⎤
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
u 5
u 6
⎤
⎥
⎥
⎥
⎥
⎦
= −
X
2 p 0
E
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1/A I
1/A I
1
1/A II
1/A II
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
. (6.27)
The solution of this linear system of equations gives the unknown nodal values as: