3.4 Solved Problems
59
Fig. 3.16 Refined analysis: a Simply supported and b cantilevered Euler–Bernoulli beam loaded
by a single force
Fig. 3.17 Finite difference discretization of the simply supported Euler–Bernoulli beam based on
seven grid nodes
Start from the fourth-order as well as the second-order differential equation and
compare the results. Derive from the results a general scheme for n grid points
(n > 7).
3.4 Solution
(a) The finite difference discretization of the simply supported beam is shown in
Fig. 3.17 where at first only the inner nodes will be considered.
It should be noted here that the fourth-order differential equation according to
Eq. (3.9) does not allow to account for external single forces (in our case: F 0 ). The
equivalent nodal force R i is obtained from distributed loads q X and should not be
confused with external single loads. If the derivations should be based on the fourthorder differential equation, then a modeling approach as shown in Fig. 3.18 can be
applied. Thus, we understand the single force F 0 as the integral value of a distributed
load q 0 , which is acting over a length of . Obviously, this is no more exactly
the same load case as shown in Fig. 3.16a. Nevertheless, it allows us to proceed the
derivations based on the fourth-order differential equation.
Evaluation of the finite difference approximation of the fourth-order differential
equation according to Eq. (3.9) at the inner nodes i = 2, . . . , 6 gives:
59
Fig. 3.16 Refined analysis: a Simply supported and b cantilevered Euler–Bernoulli beam loaded
by a single force
Fig. 3.17 Finite difference discretization of the simply supported Euler–Bernoulli beam based on
seven grid nodes
Start from the fourth-order as well as the second-order differential equation and
compare the results. Derive from the results a general scheme for n grid points
(n > 7).
3.4 Solution
(a) The finite difference discretization of the simply supported beam is shown in
Fig. 3.17 where at first only the inner nodes will be considered.
It should be noted here that the fourth-order differential equation according to
Eq. (3.9) does not allow to account for external single forces (in our case: F 0 ). The
equivalent nodal force R i is obtained from distributed loads q X and should not be
confused with external single loads. If the derivations should be based on the fourthorder differential equation, then a modeling approach as shown in Fig. 3.18 can be
applied. Thus, we understand the single force F 0 as the integral value of a distributed
load q 0 , which is acting over a length of . Obviously, this is no more exactly
the same load case as shown in Fig. 3.16a. Nevertheless, it allows us to proceed the
derivations based on the fourth-order differential equation.
Evaluation of the finite difference approximation of the fourth-order differential
equation according to Eq. (3.9) at the inner nodes i = 2, . . . , 6 gives: