The initial P-Delta analysis option accounts for the effect of a large compressive or tensile load upon the transverse stiffness of members in the structure. Compression reduces lateral stiffness, and tension stiffens it. This is a type of geometric nonlinearity known as the P-Delta effect. Initial P-Delta analysis does not include large-strain or large-rotation effects.
This option is particularly useful for considering the effect of gravity loads upon the lateral stiffness of building structures, as required by certain design codes. Other applications are possible.
Initial P-Delta analysis in ETABS considers the P-Delta effect of a single loaded state upon the structure. That load can be specified in two ways:
As a specified combination of static load cases; this is called the P-Delta Load Combination on the P-Delta Options form. For example, this may be the sum of a dead load case plus a fraction of a live load case. This approach requires an iterative solution to determine the P-Delta effect upon the structure.
As a story-by-story load upon the structure computed automatically from the mass at each level. This approach is approximate, but does not require an iterative solution.
When an initial P-Delta analysis is requested on the P-Delta Options form, it is performed before all linear-static, modal, response-spectrum, and time-history analyses in the same analysis run. The initial P-Delta analysis essentially modifies the characteristics of the structure, affecting the results of all subsequent analyses performed.
As an important exception, initial P-Delta analysis does NOT affect nonlinear-static analysis. Nonlinear static analyses consider the P-Delta effect separately, if requested.
Because the load causing the P-Delta effect is the same for all linear analysis cases, their results may be superposed in load combinations.
Initial P-Delta analysis under a P-delta load combination is iterative in nature, and may considerably increase computation time. Including initial P-Delta analysis may make interpretation of the results more difficult. It is strongly recommended that a preliminary linear analysis be performed first to check the model for correctness before using initial P-Delta analysis.
When a P-Delta load combination is specified, the following Iteration Control parameters also may be specified on the P-Delta Options form to control the iterative solution:
Convergence Tolerance (Relative): Iteration is used to make sure that equilibrium is achieved at each step of the analysis. Use this parameter to set the relative convergence tolerance that is used to compare the magnitude of force error with the magnitude of the force acting on the structure. Using a smaller value ensures better equilibrium, although the default value is usually adequate.
If the P-delta iteration does not converge during analysis, subsequent linear load cases will not be run. This is usually an indication that the structure is unstable due to one of the following causes: (a) Inadequate support or connections; (b) Global or local buckling; (c) Significant nonlinearity in links, hinges, or layered shell; or (d) Poor conditioning, such as the use of excessively large stiffnesses. The model should be improved if possible. Alternatively, a custom nonlinear load case, which allows more control over the nonlinear iteration, may be defined and used instead of the Preset P-Delta Option.
If compressive P-Delta axial forces are present and are large enough, the structure may buckle. Local buckling of individual members or global buckling of the whole structure are both possible. The program makes no distinction between local and global buckling.
If the program detects that buckling has occurred, the analysis is terminated and no results are produced. This is because the analysis of a structure that has buckled requires consideration of large-displacement effects that are only considered during non-linear static analysis.
Initial P-Delta analysis may be used to estimate buckling loads by performing a series of analyses, each time increasing the magnitude of the P-Delta load combination, until buckling is detected. The relative contributions from each static load case to the P-Delta load combination must be kept the same, increasing all load case scale factors by the same amount between runs.
It is important to understand that there is no single buckling load for a structure. Rather, there is a different buckling load corresponding to each spatial distribution of loads. If buckling of the structure is a concern under various loading situations, the buckling load should be estimated separately for each situation, as described above, by starting with different P-Delta load combinations.
For most building structures, especially tall buildings, the P-Delta effect of most concern occurs in the columns because of gravity load, including dead and live load. The column axial forces are compressive, making the structure more flexible against lateral loads.
Building codes (ACI 1999; AISC 1994) normally recognize two types of P-Delta effects: the first caused by the overall sway of the structure and the second caused by the deformation of the member between its ends. The former effect is often significant; it can be accounted for fairly accurately by considering the total vertical load at a story level, which is caused by gravity loads and is unaffected by any lateral loads. The latter effect is significant only in very slender columns or columns bent in single curvature (not the usual case); this requires consideration of axial forces in the members because of both gravity and lateral loads.
The program can analyze both of those P-Delta effects. However, it is recommended that the former effect be accounted for in the analysis, and the latter effect be accounted for in design using the applicable building-code moment-magnification factors (White and Hajjar 1991). This is how the design checks work in ETABS.