Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 651

FI GURE 11.3
P vs. φ curves for the bar of Fig. 11.2.

Similar results are found if o nly small values of φ are to be considered. Then cos φ ≈ 1,

sin φ ≈ φ , so that P = ( k φ − HL )/ L φ . This can be written as

( k − PL )φ = HL (4)

which appear s to be in a “stiffness-like” form. In (4), k corresponds to the conventional
stiffness, and PL i s t he so-called differential (or geometric or stress) stiffness, t ha t occurs in
stability theory. If P in (4) approaches k / L , then small values at applied load H will make

φ theory indicates

the same tr ends as the more accurate nonlinear theory of Figs. 11.3a and b. The bifurcation
point of P

approach infinity. The plots of Figs. 11.3c and d show that the small φ

L is evident in Fig. 11.3d.
This example c on tains the characteristics of more difficult stability problems. The case of
Fig. 11. 3a, where P is a nonlinear function of

= k /

and a unique state of equilibrium exists for
each P, would be analyzed from the standpoint of stresses if this were an elastic body. This
is sometimes said to be a third order theory . It is characterized by the nonlinear function of
φ

φ

, and that equilibrium is taken at the deformed state. When H = 0 , as in Fig. 11.3b, the

bifurcation phenomenon can be identified.
For small , another stress analysis could be utilized in the case exhibited in Fig. 11.3c. If

the influence of the nonlinearities are only approximately taken into account, this is often

φ