V. Direct Analysis of a Column

Verify the maximum bending moment and deflection of a column using the direct analysis method.

Reference

American Institute of Steel Construction. 2005. Specifications for Structural Steel Buildings. Chicago, IL:AISC. p.16.1-435, Case 2

Details

A 25' tall cantilevered column (i.e., a flagpole) is subject to an axial load of 30 kips. The section is a W8x18 (Iz = 61.9 in4).

Validation

A notional load factor of 0.003 is used. Therefore, the horizontal notional load at the top (free end) of the column is H = 0.03 × 30 = 0.09 kips.

Maximum moment

The maximum moment occurs at the fixed base.

$M MAX = M 0 tan α α$
where
 $M 0$ = $α$ = $P ⁢ L 2 E ⁢ I = 30 × ( 25 ) 2 × ( 12 ) 2 29,000 × 61.9 = 1.226$ $tan α α$ = $tan ( 1.226 ) 1.226 = 2.273$

Therefore, the maximum moment is:

Maximum deflection

The maximum deflection occurs at the free end.

$δ MAX = δ 0 3 tan α - α α 3$
where
 $δ 0$ = $3 tan α - α α 3$ = $3 tan ( 1.226 ) - 1.226 1.226 3 = 2.537$

Therefore, the maximum deflection is:

Results

Table 1. Comparison of results
MMAX (in·k) 61.37 61.36 negligible
δMAX (in) 1.145 1.14538 negligible

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\07 Nonlinear Analysis\Direct Analysis of a Column.std is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 22-Jun-09
END JOB INFORMATION
INPUT WIDTH 79
UNIT INCHES KIP
JOINT COORDINATES
1 0 0 0; 2 0 300 0;
MEMBER INCIDENCES
1 1 2;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 29000
POISSON 0.3
DENSITY 0.000283
ALPHA 6.5e-006
DAMP 0.03
END DEFINE MATERIAL
CONSTANTS
MATERIAL STEEL ALL
MEMBER PROPERTY AMERICAN
1 TABLE ST W8X18
SUPPORTS
1 FIXED
2 FIXED BUT FX FY MX MY MZ
DEFINE DIRECT ANALYSIS
FYLD 50 LIST 1
END
2 FY -30
perf analysi
change
1 1.0
1 X 0.003
PERFORM DIRECT ANALYSIS LRFD
print joint disp list 2
print memb force list 1
FINISH


   JOINT DISPLACEMENT (INCH RADIANS)    STRUCTURE TYPE = SPACE
------------------
JOINT  LOAD   X-TRANS   Y-TRANS   Z-TRANS   X-ROTAN   Y-ROTAN   Z-ROTAN
2    2    1.14538  -0.05900   0.00000   0.00000   0.00000  -0.00586
************** END OF LATEST ANALYSIS RESULT **************
46. PRINT MEMB FORCE LIST 1
MEMB     FORCE    LIST     1
STAAD SPACE                                              -- PAGE NO.    4
MEMBER END FORCES    STRUCTURE TYPE = SPACE
-----------------
ALL UNITS ARE -- KIP  INCH     (LOCAL )
MEMBER  LOAD  JT     AXIAL   SHEAR-Y  SHEAR-Z   TORSION     MOM-Y      MOM-Z
1    2     1     30.00      0.09     0.00      0.00      0.00      61.36
2    -30.00     -0.09     0.00      0.00      0.00       0.00
************** END OF LATEST ANALYSIS RESULT **************
47. FINISH