6.3 Answers for Problems from Chap. 3
137
Fig. 6.7 Finite difference discretization of the fixed-ended Euler–Bernoulli beam (distributed load)
based on a five and b nine grid nodes
relative error =
1
256
−
1
384
1
384
× 100 = 50.0% .
(6.140)
(b) Evaluation of the finite difference approximation of the fourth-order differential
equation according to Eq. (3.9) at the inner nodes i = 2, . . . , 8 gives:
node 2:
E I Y
X 3 (u 4 − 4u 3 + 6u 2 − 4u 1 + u 0 ) = −q 0 ,
(6.141)
node 3:
E I Y
X 3 (u 5 − 4u 4 + 6u 3 − 4u 2 + u 1 ) = −q 0 ,
(6.142)
node 4:
E I Y
X 3 (u 6 − 4u 5 + 6u 4 − 4u 3 + u 2 ) = −q 0 ,
(6.143)
137
Fig. 6.7 Finite difference discretization of the fixed-ended Euler–Bernoulli beam (distributed load)
based on a five and b nine grid nodes
relative error =
1
256
−
1
384
1
384
× 100 = 50.0% .
(6.140)
(b) Evaluation of the finite difference approximation of the fourth-order differential
equation according to Eq. (3.9) at the inner nodes i = 2, . . . , 8 gives:
node 2:
E I Y
X 3 (u 4 − 4u 3 + 6u 2 − 4u 1 + u 0 ) = −q 0 ,
(6.141)
node 3:
E I Y
X 3 (u 5 − 4u 4 + 6u 3 − 4u 2 + u 1 ) = −q 0 ,
(6.142)
node 4:
E I Y
X 3 (u 6 − 4u 5 + 6u 4 − 4u 3 + u 2 ) = −q 0 ,
(6.143)