Name:  PDelta effect for a cantilevered column 
Description:  Calculation and verification of the PDelta effects of a cantilevered column. 
Program: 
SAP2000

Version:  14.2.4 
Model ID:  109 
This test problem calculates and verifies the PDelta and LargeDisplacement effects associated with the lateral deflection of a fully fixed cantilevered column. The Scilab numericalcomputation software is used to verify selected values.
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Model description
The fully fixed cantileveredcolumn model has the following properties:
 Length: L = 10m
 Crosssection: 0.1m x 0.1m square
 Concrete modulus: E = 30GPa
Axial load: F_{v} = 4kN (compression)
F_{v} is chosen as 70% of the critical buckling load, where P_{cr} = π^{2} EI / 4L^{2} = 6.168kN
Lateral load: F_{H} = 0.045kN
F_{H} is chosen to cause an elastic deflection of 0.06m, where Δ = PL^{3} / 3EI
Figure 1  Cantileveredcolumn 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 . 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 PDelta, 1 step)  0.1677 m  0.1677 m 
NLSTATIC1(10) (nonlinear with PDelta, 10 steps)  0.1671 m  not calculated 
NLSTATIC2(1) (nonlinear with PDelta and large displacements, 1 step)  0.1676 m  not calculated 
NLSTATIC2(10) (nonlinear with PDelta and large displacements, 10 steps)  0.1675 m  not calculated 
Base moment
Load Case  SAP2000 Value  Scilab Value 

STATIC (linear)  0.45 kNm  0.45 kNm 
NLSTATIC1(1) (nonlinear with PDelta, 1 step)  1.121 kNm  1.324 kNm 
NLSTATIC1(10) (nonlinear with PDelta, 10 steps)  1.120 kNm  not calculated 
NLSTATIC2(1) (nonlinear with PDelta and large displacements, 1 step)  1.120 kNm  not calculated 
NLSTATIC2(10) (nonlinear with PDelta and large displacements, 10 steps)  1.120 kNm  not calculated 
Screenshots
Figure 2 presents a realization of the cantileveredcolumn 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 indicate agreement between displacement values. A slight discrepancy exists between the moment values returned.
 When considering LargeDisplacement effect, smaller lateral displacements result. There is no geometric limitation for the application of PDelta effect, which projects laterally from the column tip in a straight line. LargeDisplacement effect, however, is bound by column length. As column rotation increases, the tip displaces along a curvilinear profile. As a result, given LargeDisplacement effect, axial displacement should be larger.
References
 Wilson, E. L. (2004). Static and Dynamic Analysis of Structures (4th ed., pp. 120121). Berkeley, CA: Computers and Structures, Inc.
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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)