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# P-Delta effect for a cantilevered column

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Test Problem
 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

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.

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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 . Additional details are included in the attached Scilab input file .

## 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. 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, E. L. (2004). Static and Dynamic Analysis of Structures (4th ed., pp. 120-121). Berkeley, CA: Computers and Structures, Inc.
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