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P-δ is a local effect associated with axial load on displacement relative to element chord extending between end nodes. Figure 1 illustrates the influence of P-δ on a simply supported beam. Here, a longitudinal distributed load ω correlates with elastic bending-stiffness properties K_E to induce vertical displacement δ. An additional flexural contribution comes from the relationship between this deformed configuration and axial load P. The geometric stiffness properties K_G which dictate this relationship are discussed further in Dr. Edward L. Wilson's text, Static and Dynamic Analysis of Structures.

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Values for the maximum flexural response which occurs at element midspan are shown in Figure 1:

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Figure 1 - P-δ applied to a simply supported beam

Cantilevered column P-δ

Now, when observing P-δ effect on a cantilevered column, response is shown in Figure 2: 

Figure 2 - P-δ applied to a cantilevered column (single curvature)

However, columns seldom displace with single curvature. More commonly, especially with multi-story-building analysis and design, columns deform according to a third-order (cubic) displacement pattern under double curvature. As shown in Figure 3, P-δ effect is much less pronounced because an inflection point intersects the element chord near midspan, previously where displacement from chord was greatest.

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Figure 3 - P-δ applied to a cantilevered column (double curvature)

Cantilevered column P-∆

However, what is often of significance, given this loading condition and double-curvature displacement pattern, is P-∆ effect. Although displacement deviates from element chord much less, the lateral displacement associated with story drift is significant. With increasing levels of drift, gravity load has a greater effect on mechanical behavior, as shown in Figure 4. P-∆ effect should be implemented during design, whether static or dynamic, linear or nonlinear.

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Figure 4 - P-∆ applied to a cantilevered column

References

• Wilson, E. L. (2004). Static and Dynamic Analysis of Structures (4th ed.). Berkeley, CA: Computers and Structures, Inc.
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