 # V. Column Pushover Displacement

Compare theoretical solution to a pushover analysis to the STAAD.Pro solution.

## Reference

Hand calculation.

## Problem ### Cantilever member model

A transverse load P is applied to a cantilever member and increased until the member fails.

Member length = 24 inch

Section = Wide flange W16X77

Material = Steel

Expected yield strength = 36 ksi ### Plastic Rotation (in radians) vs. Moment (in in·kips) plot for moment hinge

The analysis results are saved at an interval of 0.1 inch deflection of the cantilever tip. The following results are displayed in the Postprocessing workflow by selecting the Layouts > Pushover-Graphs tool in the Dynamics group on the Results ribbon tab. ### Capacity curve (Displacement at Control Joint, inches vs. Base Shear, kips) as calculated by STAAD.Pro

Table 1. Tabular form of the different points in Capacity curve as reported by STAAD.Pro
Load Step Displacement in Base Shear kip
1 0 0.153
2 0.001 1.153
3 0.078 57.837
4 0.136 61.953
5 0.243 63.979
6 0.349 65.979
7 0.455 67.979
8 0.562 69.979
9 0.668 71.979
10 0.774 73.979
11 0.880 75.979
12 0.986 77.979
13 1.065 79.479
14 1.065 21.510

## Results from hand calculation

Equation of elastic deflection at cantilever tip:
$δ z = P L 3 3 E I + P L G A v$
where
 P = 57.837 kip L = 24 in. E = 29,000 ksi I = 138.0 in.4 G = 11,154 ksi Av = 10.4323 in.2
Thus elastic deformation:
$δ z = 57.837 × 24 3 3 × 29 , 000 × 138 + 57.837 × 24 11 , 154 × 10.4323 = 0.066 + 0.012 = 0.078 in.$

Equation of elastic deflection at cantilever tip
$δ z = P L 3 3 E I + P L G A v$
where
 P = 79.479 kip
(The other values are as listed for Load Step 3)
Thus elastic deformation at cantilever tip:
$δ z = 79.479 × 24 3 3 × 29 , 000 × 138 + 79.479 × 24 11 , 154 × 10.4323 = 0.092 + 0.016 = 0.108 in.$

Since STAAD.Pro assumes small displacements (sinθ = θ), plastic deformation at cantilever tip

 δz plastic = L × θ = 24 × 0.04 = 0.96 inch

Total deflection = δz elastic + δz plastic = 0.108 + 0.96 = 1.068 inch

## Comparison

Table 2. Comparison of results
Load Step Force P (free end) kips Deflection (free end) in Percent Difference
3 57.837 0.078 0.078 none
13 79.479 1.065 1.068 negligible
Note: The deflection results from STAAD.Pro shown in the above table were obtained in the Postprocessing workflow.

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\07 Nonlinear Analysis\Column Pushover Displacement.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 20-Jul-07
END JOB INFORMATION
INPUT WIDTH 79
UNIT INCHES KIP
JOINT COORDINATES
1 0 0 0; 2 0 24 0;
MEMBER INCIDENCES
1 1 2;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 29000
POISSON 0.3
DENSITY 0.000283
ALPHA 6.5e-06
DAMP 0.03
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
1 TABLE ST W16X77
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
DEFINE PUSHOVER DATA
FRAME 2
FYE 36.000000 ALL
HINGE PROPERTY MOMENT
TYPE 1
A 0 0 B 1 1 C 11 1.3 D 11 0.32 E 15 0.32 YM 1480 YR 0.004
IO 2 LS 7 CP 10.5
HINGE TYPE 1 ALL
GNONL 1
DISP Z 2 JOINT 2
LDSTEP 1
SPECTRUM PARAMETERS
DAMPING 2.0000
SC 4
SS 1.1
S1 1.6
END PUSHOVER DATA
SELFWEIGHT Z 1
2 FZ 1
PERFORM PUSHOVER ANALYSIS
FINISH


            P R O B L E M   S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS          2  NUMBER OF MEMBERS       1
NUMBER OF PLATES          0  NUMBER OF SOLIDS        0
NUMBER OF SURFACES        0  NUMBER OF SUPPORTS      1
Using 64-bit analysis engine.
SOLVER USED IS THE IN-CORE ADVANCED MATH SOLVER
TOTAL      PRIMARY LOAD CASES =     2, TOTAL DEGREES OF FREEDOM =       6
TOTAL LOAD COMBINATION  CASES =     0  SO FAR.
MORE MODES WERE REQUESTED THAN THERE ARE FREE MASSES.
NUMBER OF MODES REQUESTED              =     6
NUMBER OF EXISTING MASSES IN THE MODEL =     3
NUMBER OF MODES THAT WILL BE USED      =     3
***  EIGENSOLUTION : ADVANCED METHOD ***
STAAD SPACE                                              -- PAGE NO.    3
CALCULATED FREQUENCIES FOR LOAD CASE       3
MODE            FREQUENCY(CYCLES/SEC)         PERIOD(SEC)
1                     306.367                  0.00326
2                     544.518                  0.00184
3                    1865.405                  0.00054
MODAL WEIGHT (MODAL MASS TIMES g) IN KIP          GENERALIZED
MODE           X             Y             Z              WEIGHT
1       0.000000E+00  0.000000E+00  7.674960E-02    7.674960E-02
2       7.674960E-02  0.000000E+00  0.000000E+00    7.674960E-02
3       0.000000E+00  7.674960E-02  0.000000E+00    7.674960E-02
MASS PARTICIPATION FACTORS
MASS  PARTICIPATION FACTORS IN PERCENT
--------------------------------------
MODE    X     Y     Z     SUMM-X   SUMM-Y   SUMM-Z
1     0.00   0.00 100.00    0.000    0.000  100.000
2   100.00   0.00   0.00  100.000    0.000  100.000
3     0.00 100.00   0.00  100.000  100.000  100.000
***WARNING: MEMBER #       1 HAS FAILED IN " DEFORMATION-CONTROLLED ACTION.
*** WARNING : STRUCTURE HAS REACHED AN UNSTABLE STATE WHERE
A VERY SMALL INCREASE IN PUSH LOAD IS CAUSING LARGE DISPLACEMENT.