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This test problem demonstrates and verifies the P-Delta and Large Displacement effect on a cantilevered column with full fixity. Scilab numerical computation software is used to verify selected values.

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 = 30 GPa

Axial load: F _v = 4kN (compression)

F _v is chosen as 70% of the critical buckling load, where
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P_

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= \frac{\pi^2 EI}{4 L^2} = 6.168 kN

Lateral load: F _H = 0.045 kN

F _H is chosen to cause an elastic deflection of 0.06m, where
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\Delta = \frac{PL^3}{3EI}

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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 Scilab input file attached.

Lateral tip displacement

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

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

Screenshots

Figure 2 presents a realization of the cantilevered-column model, subjected to a 6kN axial force. Stiffness and response are evaluated at 100 increments.

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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 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, Dr. Edward L. Static & Dynamic analysis of Structures. 4th ed. Berkeley: Computers and Structures, Inc., 2004. Print.
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Definition of geometric stiffness (pages 120-121)

Attachments

SAP2000 model (zipped SDB file)
Model variation with 100 steps and a 6kN axial force (zipped SDB file)
Hand calculations (PDF file)
Scilab input and output files (zipped)

P-Delta effect for a cantilevered column