Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 671

TABLE 11.2
Overview of Stress and Stability Analyses of a Beam
Conditions Theory of First Order Theory of Second Order Theory of Third Order

Equilibrium Undeformed system Deformed system, with Deformed system

based on: only those terms with

the greatest influence
considered

Strains Linear Nonlinear (in part) Nonlinear

1
2 (w )

1
2 (w )

ε x = u ε x = u +

2

ε x = u +

2

/ρ = w /ρ = w /ρ = w /( + (w )

2 3 / 2

κ = 1 1 1 1 )

Force- Linear Nonlinear Nonlinear

displacement
diagram
Superposition Valid Not valid Not valid

Governing Linear Linearized Nonlinear

differential
equations
Failure due to: Stress or displacement Critical load Critical load

level

Bifurcation of Not possible Homogeneous problem Also can study

equilibrium yields eigenvalue post-critical behavior

state is taken from the linear theory, such as that given in Section 11.2.2 and then using
linearized equations, is the second order theory. The homogeneous equations render the
bifurcations of equilibrium and the critical loads. This approach is emphasized in this
chapter. A full-fledged nonlinear analysis is the third order theory and is based on an
incremental formulation. Some properties of these methods are summarized in Table 11.2.
The nonlinear strains in this table are obtained from Chapter 1, Eq. (1.15) in which the
square of x is neglected for the second order theory.

In the second order theory, the nonlinear effects are treated in a single increment, with
the full loading in which the forces are known from a previous state. The values of these
forces are then compared with the corresponding results of the analysis. If a desired rel-
ative accuracy is not reached, the analysis has to be repeated with the new forces as
initial values. Thus, for each iteration, the analysis is based on the whole virtual work
(

∂ u /∂

).
The desired relations arise from the global form corresponding to the differential equa-
tions of Section 11.2.2 using the forces V and H referred to the undeformed configuration.
The principle of the virtual work then appears as

W ≈ W

u + δw + δθ )

− δ W = ( H δ V M dx

w 0 δw + κδθ )

] b

− (

p x δ

+

p z δw +

− δw + δθ

u H EI dx [ V M

= 0 (11.55)

EAu ,V w ≈ Introduce the approximations of Eq. (11.12), leading to H ≈ N = = Q + N

w =

EAu w for a beam in which the shear terms in the material law for Q are neglected.

N