4.2 Approximation of the Differential Equations
99
and the relative error for the given numerical values is equal to −96.514%. Using
again 17 domain nodes changes the result in a general form to
u 9 = u
L
2
= −
512E I Y + 43k s AG L
2
L F 0
2k s G A
1024E I Y + L 2 k s AG
,
(4.34)
which gives a decreased relative error of −65.875%.
4.3 Supplementary Problems
4.2 Finite difference approximation of a beam fixed at both ends
Given is a Timoshenko beam of length L which is fixed at both ends as shown in
Fig. 4.5. The beam has constant material parameters and is loaded either by a single
force F 0 or a constant distributed load q 0 . Use first five domain nodes of equidistant
spacing ((X =
L
4
) and then 17 domain nodes of equidistant spacing ((X =
L
16
) for
the finite difference approximation to determine
• the general expression for the vertical displacement in the middle of the beam, i.e.
X =
L
2
,
• the analytical solution and
• calculate the relative error between the analytical and finite difference solution
for a squared cross-sectional area of length 0.5. Other values are: E = 10000;
ν = 0.0; L = 10; q 0 = 1; F 0 = 0.1.
4.3 Convergence of finite difference approximation of a simply supported Timoshenko beam
Given is a Timoshenko beam of length L which is simply supported as shown in
Fig. 4.6. The beam has constant material parameters and is loaded by a constant
distributed load q 0 . Use the following number of equidistant domain nodes to investigate the convergence of the solution: 5, 13, 23, 33, 53, 73 and 103. The analytical
solution can be taken form [9]. Determine
Fig. 4.5 Timoshenko beam fixed at both ends: a single force case; b distributed load case
99
and the relative error for the given numerical values is equal to −96.514%. Using
again 17 domain nodes changes the result in a general form to
u 9 = u
L
2
= −
512E I Y + 43k s AG L
2
L F 0
2k s G A
1024E I Y + L 2 k s AG
,
(4.34)
which gives a decreased relative error of −65.875%.
4.3 Supplementary Problems
4.2 Finite difference approximation of a beam fixed at both ends
Given is a Timoshenko beam of length L which is fixed at both ends as shown in
Fig. 4.5. The beam has constant material parameters and is loaded either by a single
force F 0 or a constant distributed load q 0 . Use first five domain nodes of equidistant
spacing ((X =
L
4
) and then 17 domain nodes of equidistant spacing ((X =
L
16
) for
the finite difference approximation to determine
• the general expression for the vertical displacement in the middle of the beam, i.e.
X =
L
2
,
• the analytical solution and
• calculate the relative error between the analytical and finite difference solution
for a squared cross-sectional area of length 0.5. Other values are: E = 10000;
ν = 0.0; L = 10; q 0 = 1; F 0 = 0.1.
4.3 Convergence of finite difference approximation of a simply supported Timoshenko beam
Given is a Timoshenko beam of length L which is simply supported as shown in
Fig. 4.6. The beam has constant material parameters and is loaded by a constant
distributed load q 0 . Use the following number of equidistant domain nodes to investigate the convergence of the solution: 5, 13, 23, 33, 53, 73 and 103. The analytical
solution can be taken form [9]. Determine
Fig. 4.5 Timoshenko beam fixed at both ends: a single force case; b distributed load case