100
4 Investigation of Timoshenko Beams in the Elastic Range
Fig. 4.6 Simply supported Timoshenko beam with distributed load
Fig. 4.7 Cantilevered Timoshenko beam with a distributed load and b single force
• the general expression for the vertical displacement in the middle of the beam, i.e.
X =
L
2
,
• a graphical representation of the relative error between the analytical and finite
difference solution as a function of the domain node number for a squared crosssectional area of length 0.5. Other values are: E = 10000; ν = 0.0; L = 10;
q 0 = 1.
4.4 Convergence of finite difference approximation of a cantilevered Timoshenko beam
Given is a cantilevered Timoshenko beam of length L as shown in Fig. 4.7. The beam
has constant material parameters and is either loaded in the negative Z -direction by
a constant distributed load q 0 or a single force F 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
• the general expression for the vertical displacement at the right-hand end of the
beam, i.e. X = L,
• a graphical representation of the relative error between the analytical and finite
difference solution as a function of the domain node number for a squared crosssectional area of length 0.5. Other values are: E = 10000; ν = 0.0; L = 10; q 0 =
1; F 0 = 0.1.
4 Investigation of Timoshenko Beams in the Elastic Range
Fig. 4.6 Simply supported Timoshenko beam with distributed load
Fig. 4.7 Cantilevered Timoshenko beam with a distributed load and b single force
• the general expression for the vertical displacement in the middle of the beam, i.e.
X =
L
2
,
• a graphical representation of the relative error between the analytical and finite
difference solution as a function of the domain node number for a squared crosssectional area of length 0.5. Other values are: E = 10000; ν = 0.0; L = 10;
q 0 = 1.
4.4 Convergence of finite difference approximation of a cantilevered Timoshenko beam
Given is a cantilevered Timoshenko beam of length L as shown in Fig. 4.7. The beam
has constant material parameters and is either loaded in the negative Z -direction by
a constant distributed load q 0 or a single force F 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
• the general expression for the vertical displacement at the right-hand end of the
beam, i.e. X = L,
• a graphical representation of the relative error between the analytical and finite
difference solution as a function of the domain node number for a squared crosssectional area of length 0.5. Other values are: E = 10000; ν = 0.0; L = 10; q 0 =
1; F 0 = 0.1.