Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 657

Further assume that

w w V H and Q N (11.11)

Then the relationships between the two sets of stress resultants N and Q and V and H
become

H ≈ N

+ w N V ≈ Q (11.12)

− w H − w N Q ≈ V ≈ V

w 0 are to be taken into account, replace w by w + w 0 .

If initial imperfections

11.2.2 Fundamental Equations (Differential Form)

In a first approximation to a fully nonlinear analysis, the most important terms need to
be retained. For beam problems, an engineering-type approach is the so-called theory of
second order.

Conditions of Equilibrium
The equilibrium conditions take into account the change from undeformed to the de-
formed geometry by introducing the transformation of Eq. (11.10) with the assumptions of
Eq. (11.12). The primary difference occurs in the summation of the moments (Fig. 11.8 and

FIGURE 11.8
Undeformed and deformed states of a bar.