Unknown macro: {live-template}

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
Unknown macro: {math}
P_

Unknown macro: {cr}

=\frac{\pi^2 EI}{4 L^2} = 6.168 kN
)
lateral load: F = 0.045 kN (chosen to cause elastic deflection of 0.06m; note that
Unknown macro: {math}
\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.

Lateral tip displacement (jt. 4)

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

Discussion

The results from SAP2000 and Scilab show good agreement of displacements. There is minor difference in base moments.
Lateral displacements for large displacement analysis are smaller because P-delta has no restriction on lateral displacement, it moves in a straight line. Large displacements accounts for the rotation of the member and moves in a circle, so the lateral displacement is approximately limited by the length of the member. Axial displacements should be larger for large displacements due to the circular motion.

References

Edward L. Wilson: Static & Dynamic Analysis of Structures, 4th Edition, 2004, p. 120-121 (Definition of Geometric Stiffness)

Attachments

[SAP2000 V14.2.4 model|^SAP2000 V14.2.4 model.zip] (Zipped SDB file)
[Variation of the model with 100 steps saved and 6 kN axial force|^model B V14.2.4 - 100 steps and 6 kN axial force.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]

P-Delta effect for fixed cantilever column