P-Delta effect, also known as geometric nonlinearity, involves the equilibrium and compatibility relationships of a structural system loaded about its deflected configuration. Of particular concern is the application of gravity load on laterally displaced multi-story building structures. This condition magnifies story drift and certain mechanical behaviors while reducing deformation capacity.

P-Delta effect typically involves large external forces upon relatively small displacements. 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 illustrated 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. Small P-delta effects can affect overall structural behavior by slightly reducing the

buckling load.

Typically, P-δ only becomes significant at unreasonably large displacement values, or in especially slender columns. (Powell 2006). Small P-delta effect is important for local buckling, or for design algorithms that expect member buckling to be accounted for by analysis. This includes AISC direct-analysis method.

Small P-delta is included in frame elements, to the extent that it can be represented by a cubic curve. The frame small P-delta effect is very accurate for a single element with effective-length factor of 2 (cantilever), and moderately accurate for an effective-length factor of 1 (pinned or sway condition). When accurate small P-delta effects are important for analysis or design of a member, it is generally recommended to auto-mesh frame objects into 2 elements, especially for axial loads close to buckling. For most other purposes, small P-delta effects as they impact the overall structure are adequately considered with a single frame object between connections

P-Δ effect, or P-"big-delta", is associated with displacements relative to member ends. Large P-delta effect is important for overall structure behavior under significant axial load. 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).

Large P-delta is included in all elements – frame, shell, solid, link.

The Types of P-Delta analysis article further explains the difference between P-δ and P-Δ.

References

Powell, G. H. (2010). Modeling for Structural Analysis: Behavior and Basics (1st ed.). Berkeley, CA: Computers and Structures, Inc.
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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. Gaithersburg, MD: National Institute of Standards and Technology.

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

P-Delta effect

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References