## 212 Structural Stability

2.12.1 Stability Analysis Methods

Several stability analysis methods have been utilized in research and practice. Figure 2.111 shows schematic representations of the load-displacement results of a sway frame obtained from each type of analysis to be considered.

### 2.12.1.1 Elastic Buckling Analysis

The elastic buckling load is calculated by linear buckling or bifurcation (or eigenvalue) analysis. The buckling loads are obtained from the solutions of idealized elastic frames subjected to loads that do not produce direct bending in the structure. The only displacements that occur before buckling occurs are those in the directions of the applied loads. When buckling (bifurcation) occurs, the displacements increase without bound, assuming linearized theory of elasticity and small displacement as shown by the horizontal straight line in Figure 2.111. The load at which these displacements occur is known as the buckling load, or is commonly referred to as bifurcation load. For structural models that actually exhibit a bifurcation from the primary load path, the elastic buckling load is the largest load that the model can sustain, at least within the vicinity of the bifurcation point, provided that the postbuckling path is in unstable equilibrium. If the secondary path is in stable equilibrium, the load can still increase beyond the critical load value.

Buckling analysis is a common tool for calculations of column effective lengths. The effective length factor of a column member can be calculated using the procedure described in Section 2.12.2.5. The buckling analysis provides useful indices of the stability behavior of structures; however, it does not predict actual behavior of all but idealized structures with gravity loads applied only at the joints.

### 2.12.1.2 Second-Order Elastic Analysis

The analysis is formulated based on the deformed configuration of the structure. When derived rigorously, a second-order analysis can include both the member curvature (PS) and the side sway

Linear elastic analysis

Load factor

Deflection, A

Linear elastic analysis

Load factor

Deflection, A

FIGURE 2.111 Categorization of stability analysis methods.

FIGURE 2.112 Behavior of frame in compression and tension.

(P-D) stability effects. The P-d effect is associated with the influence of the axial force acting through the member displacement with respect to the rotated chord, whereas the P-D effect is the influence of axial force acting through the relative side sway displacements of the member ends. A structural system will become stiffer when its members are subjected to tension. Conversely, the structure will become softer when its members are in compression. Such a behavior can be illustrated by a simple model shown in Figure 2.112. There is a clear advantage for a designer in making use of the stiffer behavior of tension structures. However, the detrimental effects associated with second-order deformations due to compression forces must be considered in designing structures subjected to predominant gravity loads.

Unlike first-order analysis in which solutions can be obtained in a rather simple and direct manner, a second-order analysis often requires an iterative procedure to obtain solutions. The load-displacement curve generated from a second-order elastic analysis will gradually approach the horizontal straight line that represents the buckling load obtained from the elastic buckling analysis, as shown in Figure 2.111. Differences in the two limit loads may arise from the fact that the elastic stability limit is calculated for equilibrium based on the deformed configuration whereas the elastic critical load is calculated as a bifurcation from equilibrium on the undeformed geometry of the frame.

The load-displacement response of many practical structures usually does not involve any bifurcation of equilibrium path. In some cases, the second-order elastic incremental response may not have yielded any limit. The reader is referred to Chen and Lui (1987) for a basic discussion of these behavioral issues.

Recent work on second-order elastic analysis have been reported by Chen and Lui (1991), Liew et al. (1991), White and Hajjar (1991), and Chen and Toma (1994), among others. Second-order analysis programs that can take into consideration connection flexibility are also available (Chen et al. 1996; Chen and Kim 1997; Faella et al. 2000).

### 2.12.1.3 Second-Order Inelastic Analysis

Second-order inelastic analysis refers to methods of analysis that can capture geometrical and material nonlinearities of the structures. The most rigorous inelastic analysis method is called spread-of-plasticity analysis. It involves discretization of a member into many line segments and the cross-section of each segment into a number of finite elements. Inelasticity is captured within the cross-sections and along the member length. The calculation of forces and deformations in the structure after yielding requires iterative trial-and-error processes because of the nonlinearity of the load-deformation response, and the change in the effective stiffness of the cross-section at inelastic regions associated with the increase in the r

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