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{live-template:Test Problem}

The purpose of this test problem is to demonstrate and verify P-Delta effect and large displacement effects on a simple model of a fixed cantilever column. Selected values were verified by freely available [Scilab|http://www.scilab.org] software for general numerical computations.

h1. Model description

The model is fixed cantilever column with the following properties:

* length: 10m
* cross-section: 0.1m by 0.1m square cross-section
* material concrete: E = 30 GPa
* compression vertical load at free tip: T = 4kN (chosen as about 70% of critical buckling load; note that {math}P_{cr}=\frac\{\pi^2 EI\}\{4 L^2\}{math})
* lateral load: F = 0.045 kN (chosen to cause elastic deflection of 0.06m; note that {math}\Delta=\frac\{PL^3\}\{3EI\}{math})

!P-delta_and_large_displacement_for_cantilever_column.png!

h1. Results

The results obtained from SAP2000 V14.1.0 and Scilab are summarized in the table below. The Scilab code utilized stiffness and geometric matrices provided in Wilson 2004. Please refer to the attached Scilab input file for additional details.

|| Load Caseh2. Lateral tip displacement (jt. 4)

|| ValueLoad TypeCase || SAP2000 Value\\
 \[m\]\\ || Scilab Value\\
 \[m\]\\ ||
| *STATIC* (linear) | 0.06m | Lateral tip displacement (jt. 40.06 m |
| *NLSTATIC1(1)* (nonlinear with P-Delta, 1 step) | 0.061677 m | 0.061677 m |
| | base moment *NLSTATIC1(10)* (nonlinear with P-Delta, 10 steps) | 0.451671 kN-m | 0.45 kN-mnot calculated\\ |
| *NLSTATIC1NLSTATIC2(1)* (nonlinear with P-Delta and large displacements, 1 step) | 0.1676 m | Lateral tip displacement (jt. 4not calculated |
| *NLSTATIC2(10)* (nonlinear with P-Delta and large displacements, 10 steps) | 0.16771675 m\\ | 0.1677 m |
| | base moment | 1.121not calculated \\ |

h2. Base moment

|| Load Case || SAP2000 Value\\
 \[kN-m\]\\ || Scilab Value\\
 \[kN-m\]\\ ||
| *STATIC* (linear) | 0.45 kN-m | 10.32445 kN-m |
| *NLSTATIC1(101)* (nonlinear with P-Delta, 101 stepsstep) \\ | Lateral tip displacement (jt. 4)1.121 kN-m | 01.1013324 kN-m |
0.1016 m |
| | base moment | 0.707| *NLSTATIC1(10)* (nonlinear with P-Delta, 10 steps) \\ | 1.120 kN-m | 0.787 kN-mnot calculated\\ |
| *NLSTATIC2(1)* (nonlinear with P-Delta and large displacements, 1 step; results for 10 steps very similar)) \\ | 1.120 kN-m | not calculated |
Lateral tip displacement (jt. 4) | 0.1286 m | not calculated |
| | base moment | 0.965 kN-m| *NLSTATIC2(10)* (nonlinear with P-Delta and large displacements, 10 steps) \\ | 1.120 kN-m\\ | not calculated \\ |

h1. Discussion

* The results from SAP2000 and Scilab show good agreement of displacements,. but thereThere is someminor difference in base moments.
* SufficientLateral numberdisplacements offor stepslarge isdisplacement requiredanalysis toare correctlysmaller capture thebecause P-Delta effects.
* When large displacements are considered, both lateral displacement and base moment increase, which is expected, since the equilibrium equations are formulated on the deformed geometrydelta has no restriction on lateral displacement, it moves in a straight line. Large displacements accounts for the rotation of the member and moves in a circle, so the lateral displacement is approximately limited by the length of the member. Axial displacements should be larger for large displacements due to the circular motion.

h1. References

* Edward L. Wilson: Static & Dynamic Analysis of Structures, 4th Edition, 2004, p. 120-121 (Definition of Geometric Stiffness)

h1. Attachments

* [SAP2000 V14.1.0 model|P-Delta effect for fixed cantilever column^SAP2000 V14.1.0 model.zip] (Zipped SDB file)
* [Hand calculations|P-Delta effect for fixed cantilever column^Hand calcs.pdf] (PDF File, 0.7MB)
* [Zipped Scilab input and output files|P-Delta effect for fixed cantilever column^Scilab files.zip]