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h1. 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 transverse 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 {link-window:href=http://orders.csiberkeley.com/SearchResults.asp?Cat=2}{_}Static &and Dynamic analysisAnalysis of Structures_ {link-window}. {hidden-content}the [P-∆ implementation|kb:P-∆ implementation DRAFT] article. {hidden-content} Values for the maximum flexural response which occurs at element midspan are shown in Figure 1:

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!Figure 1.png|align=center,border=1,width=500pxpx500pxpxpxpx!

{center-text}Figure 1 - P-δ applied to a simply-supported beam{center-text}


h1. Cantilevered column P-δ

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

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{center-text}Figure 2 - P-δ applied to a cantilevered column (single curvature){center-text}

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*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 ii.png|align=center,border=1,width=500pxpx500pxpxpxpx!

{center-text}Figure 3 - P-δ applied to a cantilevered column (double curvature){center-text}


h1. 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|kb:Nonlinear].

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!Figure 4.png|align=center,border=1,width=600pxpx600pxpxpxpx!

{center-text}Figure 4 - P-∆ applied to a cantilevered column{center-text}


h1. References

* Wilson, E. L. (2004). _Static and Dynamic Analysis of Structures_ (4th ed.). Berkeley, CA: Computers and Structures, Inc.
Available for purchase on the CSI Products > {link-window:href=http://orders.csiberkeley.com/SearchResults.asp?Cat=2}Books {link-window} page