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{live-template:Test Problem}


{{Test Problem
| name = P-Delta effect for fixed cantilever column |
| description = Calculation of P-Delta effect for cantilever column |
| keyword = P-Delta; cantilever |
| program = SAP2000 |
| version = V14.1.0 |
| status = finalize |
| type = test problem |
| id = ok/p-delta and large displacements for cantilever column\\
 }} |

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 Case || Value Type || SAP2000 Value || Scilab Value ||
| *STATIC* (linear) | Lateral tip displacement (jt. 4) | 0.06 m | 0.06 m |
| | base moment | 0.45 kN-m | 0.45 kN-m |
| *NLSTATIC1(1)* (nonlinear with P-Delta, 1 step) | Lateral tip displacement (jt. 4) | 0.1677 m | 0.1677 m |
| | base moment | 1.121 kN-m | 1.324 kN-m |
| *NLSTATIC1(10)* (nonlinear with P-Delta, 10 steps) | Lateral tip displacement (jt. 4) | 0.1013 m | 0.1016 m |
| | base moment | 0.707 kN-m | 0.787 kN-m |
| *NLSTATIC2(1)* (nonlinear with P-Delta and large displacements, 1 step; results for 10 steps very similar) | Lateral tip displacement (jt. 4) | 0.1286 m | not calculated |
| | base moment | 0.965 kN-m | not calculated |

h1. Discussion

* The results from SAP2000 and Scilab show good agreement of displacements, but there is some difference in base moments.
* Sufficient number of steps is required to correctly capture the 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 geometry.

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]