46
3 Investigation of Euler–Bernoulli Beams in the Elastic Range
Fig. 3.5 a Simply supported and b cantilevered Euler–Bernoulli beam loaded by a single force
Fig. 3.6 Finite difference discretization of the simply supported Euler–Bernoulli beam
Fig. 3.7 Modeling approach
to consider a single force in
the fourth-order differential
equation
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
fourth-order differential equation, then a modeling approach as shown in Fig. 3.7
can be applied. Thus, we understand the single force F 0 as the integral value of a
distributed load q 0 . Obviously, this is no more exactly the same load case as shown
in Fig. 3.5a. Nevertheless, it allows us to proceed the derivations based on the fourthorder 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, . . . , 4 gives: