6.2 Answers for Problems from Chap. 2
121
The evaluation of the finite difference approximation according to Eq. (2.9) at the
nodes i = 2, . . . , 7 gives:
node 2:
E A
X
(u 3 − 2u 2 + u 1 ) = 0 ,
(6.32)
node 3:
E A
X
(u 4 − 2u 3 + u 2 ) = −
p 0 X
2
,
(6.33)
node 4:
E A
X
(u 5 − 2u 4 + u 3 ) = −p 0 X ,
(6.34)
node 5:
E A
X
(u 6 − 2u 5 + u 4 ) = −
p 0 X
2
,
(6.35)
node 6:
E A
X
(u 7 − 2u 6 + u 5 ) = 0 ,
(6.36)
node 7:
E A
X
(u 8 − 2u 7 + u 6 ) = 0 ,
(6.37)
or under consideration of the conditions as the boundaries, i.e. u 1 = 0 and u 8 = u 6 ,
node 2: − 2u 2 + u 3 = 0 ,
(6.38)
node 3: u 2 − 2u 3 + u 4 = −
p 0 X
2
2E A
,
(6.39)
node 4: u 3 − 2u 4 + u 5 = −
p 0 X
2
E A
,
(6.40)
node 5: u 4 − 2u 5 + u 6 = −
p 0 X
2
2E A
,
(6.41)
node 6: u 5 − 2u 6 + u 7 = 0 ,
(6.42)
node 7: 2u 6 − 2u 7 = 0 ,
(6.43)
or in matrix form:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
−2 1 0 0 0 0
1 −2 1 0 0 0
0 1 −2 1 0 0
0 0 1 −2 1 0
0 0 0 1 −2 1
0 0 0 0 2 −2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
u 5
u 6
u 7
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
= −
p 0 X
2
E A
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
1
2
1
1
2
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(6.44)
The solution of this linear system of equations gives the unknown nodal values as: