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This test problem demonstrates and verifies the [P-Delta|kb:Geometric nonlinearity] and Large Displacement effect on a cantilevered column with full fixity. [Scilab |http://www.scilab.org] numerical computation software is used to verify selected values.


h1. Model description

The fully-fixed cantilevered-column model has the following properties:

* Length: L = 10m

* Cross-section: 0.1m x 0.1m square

* Concrete modulus: E = 30GPa

* Axial load: F{~}v{~}v~ = 4kN (compression)
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F{~}v{~}v~ is chosen as 70% of the critical buckling load, where P{~}cr{~}cr~ = {math}\pi{math}{^}2{^}2^ EI / 4L{^}2{^}2^ = 6.168kN

* Lateral load: F{~}H{~}H~ = 0.045kN
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F{~}H{~}H~ is chosen to cause an elastic deflection of 0.06m, where {math}\Delta{math} = PL{^}3{^}3^ / 3EI

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!P-delta_and_large_displacement_for_cantilever_column.png|align=center,border=0!

{center-text}Figure 1 - Cantilevered-column buckling parameters{center-text}


h1. Results

Results obtained from [SAP2000|sap2000:home] and [Scilab |http://www.scilab.org] are summarized in the tables below. The analysis process integral to Scilab software utilizes the stiffness and geometric matrices described in Dr. Edward L. Wilson's text _Static and Dynamic Analysis of Structures_. Additional details are included in the Scilab [input file |P-Delta effect for a cantilevered column^Scilab files.zip] attached.


h2. Lateral tip displacement

|| Load Case || SAP2000 Value \\
\[m\] \\ || Scilab Value \\
\[m\] \\ ||
| *STATIC* (linear) | 0.06m | 0.06 m |
| *NLSTATIC1(1)* (nonlinear with P-Delta, 1 step) | 0.1677 m | 0.1677 m |
| *NLSTATIC1(10)* (nonlinear with P-Delta, 10 steps) | 0.1671 m | not calculated \\ |
| *NLSTATIC2(1)* (nonlinear with P-Delta and large displacements, 1 step) | 0.1676 m | not calculated |
| *NLSTATIC2(10)* (nonlinear with P-Delta and large displacements, 10 steps) | 0.1675 m \\ | not calculated \\ |


h2. Base moment

|| Load Case || SAP2000 Value \\
\[kN-m\] \\ || Scilab Value \\
\[kN-m\] \\ ||
| *STATIC* (linear) | 0.45 kN-m | 0.45 kN-m |
| *NLSTATIC1(1)* (nonlinear with P-Delta, 1 step) \\ | 1.121 kN-m | 1.324 kN-m |
| *NLSTATIC1(10)* (nonlinear with P-Delta, 10 steps) \\ | 1.120 kN-m | not calculated \\ |
| *NLSTATIC2(1)* (nonlinear with P-Delta and large displacements, 1 step) \\ | 1.120 kN-m | not calculated |
| *NLSTATIC2(10)* (nonlinear with P-Delta and large displacements, 10 steps) \\ | 1.120 kN-m \\ | not calculated \\ |


h2. Screenshots

Figure 2 presents a realization of the cantilevered-column model, subjected to a 6kN axial force. Stiffness and response are evaluated at 100 increments.

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!Evaluate_buckling_by_nonlinear_analysis.png|align=center,border=0!

{center-text}Figure 2 - Buckling analysis for a cantilevered column{center-text}


h1. Discussion

* Results from [SAP2000|sap2000:home] and Scilab indicate agreement between displacement values. A slight discrepancy exists between the moment values returned.

* When considering Large Displacement effect, smaller lateral displacements result. There is no geometric limitation for the application of P-Delta effect, which projects laterally from the column tip in a straight line. Large Displacement effect, however, is bound by column length. As column rotation increases, the tip displaces along a curvilinear profile. As a result, given Large Displacement effect, axial displacement should be larger.


h1. References

* Wilson, Dr. Edward L. _Static & Dynamic analysis of Structures_. 4th ed. Berkeley: Computers and Structures, Inc., 2004. Print.
Available for purchase on the {new-tab-link:http://www.csiberkeley.com/}CSI{new-tab-link} Products > {link-window:href=http://orders.csiberkeley.com/SearchResults.asp?Cat=2}Books {link-window} page
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Definition of geometric stiffness (pages 120-121)


h1. Attachments

* [SAP2000 V14.2.4 model |P-Delta effect for a cantilevered column^SAP2000 V14.2.4 model.zip] (zipped .SDB file)
* [Model variation with 100 steps and a 6kN axial force |P-Delta effect for a cantilevered column^model B V14.2.4 - 100 steps and 6 kN axial force.zip] (zipped .SDB file)
* [Hand calculations |P-Delta effect for a cantilevered column^Hand calcs.pdf] (PDF file)
* [Scilab input and output files |P-Delta effect for a cantilevered column^Scilab files.zip] (zipped .OUT and .SCE files)