**Nonlinear buckling** may be evaluated in SAP2000 using Nonlinear static analysis. This procedu=
re takes an iterative approach while implementing P-Delta and Large-Displacement effect. Structural respons=
e is shown by plotting selected joint dis=
placements against load application. A softening behavior may be observed i=
n this plot, indicating the onset of buckling, and the condition of instabi=
lity which follows.

**Symmetric structures**. When analyzing symmetrical struc= tures, a geometric or loading imperfection should be introduced to initiate= buckling.

**Subdivision**. Structural objects should be subdivided i= nto lengths small enough to capture = geometric nonlinearity. Four to eight fra= me or shell objects are typically nec= essary per span.

**Convergence**. In the = load-case definition, multiple output steps should be requested to impr= ove convergence, and to better indicate buckling response. Convergence tolerance may need to be tigh= tened, possibly to the order of 1e-6.

**Displacement control**. If a structure loses load-carryi= ng capacity, displacement control should be implemented, rather than load c= ontrol. This refers to the load-case control definition, and not how the lo= ad is actually applied.

**Extreme conditions**. When instability is severe, nonlin= ear static analysis should be converted to direct-integration time-history analysis.

**Linear (Eigenvalue) analysis**. During Linear buckling a= nalysis, perturbations are applied to the undeformed structural configurati= on. A specified set of loads are observed for which deflections could induc= e instability under P-Delta effe= ct. Linear buckling analysis produces a set of buckling factors and corresp= onding mode shapes. When loading= is multiplied by these buckling factors, the resultant scaled loading cond= itions represent those which induce buckling. Similarly, the mode shapes ar= e normalized displacement sets which indicate the configuration of the buck= led structure.

**Nonlinear analysis**. During Nonlinear-static buckling a= nalysis, the total load is applied incrementally. Stiffness and response ar= e evaluated at each increment. Between each displacement step, stiffness ma= y change due to the following effects:

**P-Delta effect**, which involves large tensile or compre= ssive stresses on transverse bending and shear behavior.**Large-Displacement effect**, in which deformed configura= tion is considered when assembling the equilibrium equations.**Nonlinear material behavior**, in which performance inco= rporates inelastic response. SAP2000 = implements material nonlinear= ity using frame hinges and nonlinear = layered-shell objects.

**Comparison**. Because Nonlinear-static buckling analysis= considers material nonlinearity while generating buckling response, result= s are often more realistic than those of Linear buckling analysis. The resu= lts of Nonlinear-static analysis are indicated by a plot of deformed config= uration against load application. This plot displays the softening behavior= which indicates the onset of buckling.

The *CSI Analy=
sis Reference Manual* is an excellent resource for information on b=
uckling. We recommended the following chapters:

- Analysis Cases > Linear Buckling Analysis, page 315

- Geometric Nonlinearity > Overview, page 365

- Geometric Nonlinearity > P-Delta Effect, page 369