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