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Geometric nonlinearity involves the equilibrium and compatibility relationships of a structural system loaded about its deflected configuration. Of particular concern is gravity-load application on laterally-displaced multi-story-building structures. Mechanical behaviors and story drift may then be magnified while deformation capacity is reduced.

P-Delta effect concerns conditions where relatively small displacements are subject to large external forces. If deformations become sufficiently large as to break from linear compatibility relationships, then Large-Displacement and Large-Deformation analyses become necessary. The two sources of P-Delta effect are shown in Figure 1, and described as follows:

P-δ effect, or P-"small-delta", is associated with local deformation relative to the element chord between end nodes. Typically, P-δ only becomes significant at unreasonably large displacement values, or in especially slender columns. So long as a structure adheres to the slenderness requirements pertinent to earthquake engineering, it is not advisable to model P-δ, since it may significantly increase computational time without providing the benefit of useful information. An easier way to capture this behavior is to subdivide critical elements into multiple segments, transferring behavior into P-Δ effect (Powell 2006).

P-Δ effect, or P-"big-delta", is associated with displacements relative to member ends. Unlike P-δ, this type of P-Delta effect is critical to nonlinear modeling and analysis. As indicated intuitively by Figure 2, gravity loading will influence structural response under significant lateral displacement. P-Δ may contribute to loss of lateral resistance, ratcheting of residual deformations, and dynamic instability (Deierlein et al. 2010). As shown in Figure 3, effective lateral stiffness decreases, reducing strength capacity in all phases of the force-deformation relationship (PEER/ATC 2010). To consider P-Δ effect directly, gravity load should be present during nonlinear analysis. Application will cause minimal increase to computational time, and will remain accurate for drift levels up to 10% (Powell 2006).

The difference between P-δ and P-Δ is explained in the [Difference between P-Delta] article. This article also strives to visually demonstrate the logic behind P-∆ emphasis over P-δ application during multi-story-building analysis and design.

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P-Δ implementation is described in the P-Δ implementation article.

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References

Powell, G. (2006). Nonlinear Dynamic Analysis Capabilities and Limitations, Computers and Structures, Inc., Berkeley, CA

Deierlein, G. G., Reinhorn, A. M., and Willford, M. R. (2010). Nonlinear Structural Analysis For Seismic Design, NEHRP Seismic Design Technical Brief No. 4., NIST GCR 10-917-5, National Institute of Standards and Technology, Gaithersburg, MD.

PEER/ATC (2010). Modeling and acceptance criteria for seismic design and analysis of tall buildings, PEER/ATC 72-1 Report, Applied Technology Council, Redwood City, CA, October 2010.