Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 672

N 0 , the fa-

Based on the assumption that N is taken from the first order analysis N ≈

miliar second order theory is obtained. The internal virtual work then becomes

(Wunderlich, 1976)

u EAu + δw N 0 w + δw EI w ) − δ W i = (δ dx

.

u
···
w

x δ

x dEAd x

0

.

u T = dx (11.56)

······ ······

.

x dN 0 d x x d 2 EId x

0 +

k D

u

w with w + w 0 .

To include the effect of imperfections, replace The external virtual work is

given by

N 0 w 0 δw + κδθ ) ] b

δ W e = ( p x δ u + p z δw − EI dx − [ V δw + M δθ (11.57)

Horizontal Vertical Imperfections Imposed Concentrated Loads

Distributed Distributed Curvature at the Boundaries

Load Load

For the conditions of equilibrium to hold,

− δ W =− δ W i − δ W e = 0 (11.58)

with the internal virtual work of Eq. (11.56) and the external virtual work from Eq. (11.57).
If shear deformation is taken into account, the internal virtual work relationship would
appear as (Chapter 2, Example 2.7).

. .

x dEAd x

u

········· ······ ········· ······ ······

x δ

x dN 0 d x

. .

x dk s GA

− δ W = [ u wθ ] w dx = 0

+ x dk s GAd x

······ ······ ······ ······ ······

x dEId x

. .

θ

k s GAd x

+ k s GA

(11.59)

where N 0 0 for tensile axial forces. This expression is applied in Example 11.12, where an

appropriate bifurcation load is found with and without consideration of shear deformation.

>

11.3.2 Beam Element Matrices (Theory of Second Order)

The principle of virtual work can also be used to generate the element stiffness matri-
ces for stability analyses. For a beam the principle of virtual work takes the form of
Eq. (11.58) with the extension term deleted. With the introduction of the shape function
w = =

N u Gv Nv , the element contribution for an element of length with compressive