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{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 software for general numerical computations.
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 P_{cr}=\frac{\pi^2 EI}{4 L^2})
- lateral load: F = 0.045 kN (chosen to cause elastic deflection of 0.06m; note that \Delta=\frac{PL^3}{3EI})
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.
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Load Case
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Value Type
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SAP2000 Value
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Scilab Value
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STATIC (linear)
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Lateral tip displacement (jt. 4)
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0.06 m
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0.06 m
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base moment
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0.45 kN-m
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0.45 kN-m
This test problem calculates and verifies the P-Delta and Large-Displacement effects associated with the lateral deflection of a fully fixed cantilevered column. The Scilab numerical-computation software is used to verify selected values.
On this page:
| Table of Contents |
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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: Fv = 4kN (compression)
Fv is chosen as 70% of the critical buckling load, where Pcr = π2 EI / 4L2 = 6.168kN
Lateral load: FH = 0.045kN
FH is chosen to cause an elastic deflection of 0.06m, where Δ = PL3 / 3EI
Figure 1 - Cantilevered-column buckling parameters
Results
Results obtained from SAP2000 and Scilab 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 attached Scilab input file .
Lateral tip displacement
Load Case | SAP2000 Value | Scilab Value |
|---|---|---|
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 |
Base moment
Load Case | SAP2000 Value | Scilab Value |
|---|---|---|
STATIC (linear) | 0. |
45 kN-m | 0. |
base moment
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) |
Lateral tip displacement (jt. 4)
0.1013 m
0.1016 m
1.120 kN-m |
not calculated |
NLSTATIC2(1) (nonlinear with P-Delta and large displacements, 1 step |
) |
Lateral tip displacement (jt. 4)
1.120 kN-m | not calculated |
base moment
NLSTATIC2(10) (nonlinear with P-Delta and large displacements, 10 steps) | 1.120 kN-m | not calculated |
Screenshots
Discussion
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Figure 2 presents a realization of the cantilevered-column model, subjected to a 6kN axial force. Stiffness and response are evaluated at 100 increments.
Figure 2 - Buckling analysis for a cantilevered column
Discussion
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.
References
- Edward L. Wilson: Static & Dynamic Analysis of Structures, 4th Edition, 2004, p. 120-121 (Definition of Geometric Stiffness)
Attachments
File | filename = P-Delta effect for cantilever column SAP2000 V14.1.0 model.zip | title = SAP2000 V14.1.0 SDB file(Zipped SDB file)File | filename = P-Delta effect for cantilever column hand calculations.pdf | title = hand calculations(PDF File, 0.7MB)File | filename = P-Delta effect for cantilever column Scilab files.zip | title = Zipped Scilab input and output filesindicate 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.
References
Wilson, E. L. (2004). Static and Dynamic Analysis of Structures (4th ed., pp. 120-121). Berkeley, CA: Computers and Structures, Inc.
See Also
Attachments
SAP2000 V14.2.4 model (zipped SDB file)
Model variation with 100 steps and a 6kN axial force (zipped SDB file)
Hand calculations (PDF)
Scilab input and output files (zipped OUT and SCE files)
Metadata
Name: P-Delta effect for a cantilevered column
Description: Calculation and verification of the P-Delta effects of a cantilevered column.
Program: SAP2000
Version: 14.2.4
Model ID: 109