# Simply supported beam P-δ

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***to induce vertical displacement**

_{E}**. An additional flexural contribution comes from the relationship between this deformed configuration and axial load**

*δ***. The geometric stiffness properties**

*P*

*K***which dictate this relationship are discussed further in Dr. Edward L. Wilson's text, Static and Dynamic Analysis of Structures.**

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

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.

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.

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.