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{live-template:Test Problem}

The\\
purpose
ofThis this test problem isdemonstrates toand demonstrateverifies and verifythe P-Delta effect and largeLarge displacementDisplacement effectseffect on a simplecantilevered modelcolumn ofwith a fixed cantilever column. Selected values were verified by freely available [Scilabfull fixity. [Scilab |http://www.scilab.org] numerical computation software is used forto generalverify numericalselected computationsvalues.

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h1. Model description

The model is fully-fixed cantilevercantilevered-column columnmodel withhas the following properties:

* lengthLength: L = 10m

* crossCross-section: 0.1m byx 0.1m square
cross-section
* materialConcrete concretemodulus: E = 30 GPa

* compressionAxial vertical load: atF free~v~ tip: T = 4kN (compression)
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F ~v~ is chosen as about 70% of the critical buckling load;, notewhere that {math}P_{cr} = \frac\{\pi^2 EI\}\{4 L^2\} = 6.168 kN{math})

* lateralLateral load: F ~H~ = 0.045 kN (
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F ~H~ is chosen to cause an elastic deflection of 0.06m;, notewhere that {math}\Delta = \frac\{PL^3\}\{3EI\}{math})

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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

The resultsResults obtained from SAP2000 V14.1.0[SAP2000|sap2000:home] and [Scilab |http://www.scilab.org] are summarized in the tabletables below. The analysis process integral to Scilab codesoftware utilizes utilizedthe stiffness and geometric matrices provideddescribed in Dr. Edward L. Wilson 2004. Please refer to's text _Static and Dynamic Analysis of Structures_. Additional details are included in the attached Scilab [input file |P-Delta effect for additional details. a cantilevered column^Scilab files.zip] attached.


h2. Lateral tip displacement
(jt.
4)

|| 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

TheFigure screenshot2 belowpresents is for a variationrealization of the cantilevered-column model with 100 steps saved and 6 kN axial force., 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

* TheResults results from SAP2000 and Scilab showindicate goodagreement agreementbetween ofdisplacement displacementsvalues. There is minor difference in base moments.
* Lateral displacements for large displacement analysis are smaller because P-delta has no restriction on lateral displacement, it movesA 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 displacementsDisplacement accountseffect, forhowever, theis rotationbound ofby thecolumn memberlength. andAs movescolumn inrotation aincreases, circle,the sotip thedisplaces lateralalong displacementa iscurvilinear approximatelyprofile. limitedAs bya theresult, lengthgiven ofLarge theDisplacement member.effect, Axialaxial displacementsdisplacement should be larger for large displacements due to the circular motion.

.


h1. References

* Edward L. Wilson:, _Static &and Dynamic Analysis of Structures_, 4th Edition, 2004, p. 120-121 (Definition of Geometricgeometric Stiffnessstiffness)


h1. Attachments

* [SAP2000 V14.2.4model model|P-Delta effect for a cantilevered column^SAP2000 V14.2.4 model.zip] (Zippedzipped SDB file)
* [VariationModel ofvariation the model with 100 steps saved and 6a kN6kN axial force |P-Delta effect for a cantilevered column^model B V14.2.4 - 100 steps and 6 kN axial force.zip] (Zippedzipped SDB file)
* [Hand calculations |P-Delta effect for a cantilevered column^Hand calcs.pdf] (PDF File, 0.7MBfile)
* [Zipped Scilab input and output files |P-Delta effect for a cantilevered column^Scilab files.zip] (zipped)