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Geometric nonlinearity occurs when axial load is applied to the displaced configuration of a structural element. This effect influences internal forces and further affects displacement. Of particular concern is when gravity loads act on the laterally-displaced configuration of a structure. Column bending stresses and interstory drifts are then magnified while deformation capacity is reduced.

Geometric nonlinearity is also known as P-Delta effect. There are two sources of P-Delta effect, contributions of which are shown in Figure 1, and described as follows:

P-δ, also known as Large-Displacement effect, or P-"small-delta", is associated with local deformation. Equilibrium conditions are evaluated about displaced-configuration relative to element chord. Of particular concern is when local behavior breaks from compatibility relationships. 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 Large-Displacement effect is to subdivide critical elements into multiple segments, which transfers behavior into P-Δ effect (Powell 2006).

P-Δ, also known as Gravity Load-Deformation effect, or P-"big-delta", is associated with story drift. This behavior is measured between 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 significantly influence structural response when the global system displaces laterally. P-Δ may contribute to loss of lateral resistance, ratcheting of residual deformations, and dynamic instability (Deierlein et al. 2010). In that laterally-displaced gravity loading magnifies internal forces, effective lateral stiffness decreases, reducing strength capacity in all phases of the force-deformation relationship (PEER/ATC 2010). This effect is shown in Figure 3. 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).

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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.

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.