Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 676

FIGURE 11.15
Comparison of different interpolations.
Several methods are available for evaluating the element matrix for a beam element in
the framework of second order theory. An overview is given in Table 11.4. The principle
of virtual work provides an excellent basis for the exact or approximate evaluation of the
stiffness matrix for the beam element. The results are given in Section 11.2.6, Eqs. (11.50)
and (11.52) for the exact, and in Section 11.3.2, Eq. (11.65) for the approximate stiffness
matrix using trial functions. On the other hand, numerically exact transfer matrices can
be obtained by integrating the corresponding system of first order differential equations
of Eq. (11.28). They can be transformed into the corresponding numerically exact stiffness
matrix using Eq. (4.11). Alternatively, the transformation given in Eq. (4.79) can be used
to calculate a transfer matrix equivalent to the approximate linear and geometric stiffness
matrices. The transfer and stiffness matrices given in Tables 4.3 and 4.4 of Chapter 4 also
include second order effects and were obtained using exact analytical solutions, and, for
this reason, give the same information as the transfer and stiffness matrices of Eqs. (11.50)
and (11.52), although in a different notation.