Stability - Computational Analysis and Design of Bridge Structures - page 446

bl 4 / 16EI t = 579.658
From Timoshenko Table 2.9, 1/ m = 0.198
Then, m = 5.061
P cr / P e = m 2 = 25.61
P cr = m 2 P e = 2988.66 kip (13,293.6 kN) > 2.12 P max = 866.2956 kip (3853.3 N)
P max = 408.63
The P cr calculated here, 2988.66 kip, is far above 1.5 P max and even greatly
exceeds 2.12 P max allowed by AASHTO (2013). It can be concluded that the
response in a linear analysis step is the linear perturbation response about
the base state. The base state is the current state of the model at the end of
the last general analysis step prior to the linear perturbation step. If the first
step of an analysis is a perturbation step, the base state is determined from
the initial conditions.
14.3 fEM approach of StaBility analySiS
A technique of seeding the finite element mesh with an initial displacement
field is employed in this study to initiate out-of-plane deformations of the
flat compression panels. In this technique, the finite element mesh is sub-
jected to a linearized buckling analysis to obtain the first buckling mode.
The displacement field associated with this lowest mode is then superim-
posed on the finite element model as a seed imperfection for use in the
incremental nonlinear analysis.
As previously discussed in Chapter 3, for stiffness analysis, K T , the total
tangential stiffness matrix is the sum of three terms: (1) K 0 , the usual, small
displacements stiffness matrix; (2) K σ , initial stress matrix or geometric
matrix; and (3) K L , the initial displacement matrix or large displacement
matrix. For short-span bridges, if the large deformation is ignored, the
total tangential stiffness will have only K 0 , the elastic, small displacement
stiffness matrix, and K σ , the initial stress stiffness matrix. For a long-span
cable-stayed bridge, as the axial forces along the pylon and the girder are in
compression, K σ will reduce K T . If the loads that cause the initial stress, usu-
ally the structural weight and cable stressing, keep increasing, a critical point
will be reached, at which the determinant of the total stiffness matrix is zero.
Such a bifurcation stability problem can be solved as an eigenvalue prob-
lem (Tang 1976; Ermopoulos 1992). In actual situations, however, it rarely
happens due to the flaws in building the structure. K L should also be con-
sidered, and the full Newton-Raphson process is required. In some typical
situations, it is easy to understand. For example, the transverse stability
due to the live load of a vertically stayed cable bridge under transverse
wind loads will be enhanced after the deck moves laterally away from the