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 (I_z = 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} [\frac{tan α}{α}]

where

$M_{0}$	=	$H L = 0.09 \times 25 = 2.25 ft·k$
$α$	=	$\sqrt[]{\frac{P L^{2}}{E I}} = \sqrt[]{\frac{30 \times {(25)}^{2} \times {(12)}^{2}}{29,000 \times 61.9}} = 1.226$
$\frac{tan α}{α}$	=	$\frac{tan (1.226)}{1.226} = 2.273$

Therefore, the maximum moment is: $2.25 \times 2.273 = 5.115 ft·k = 61.37 in·k$

Maximum deflection

The maximum deflection occurs at the free end.

δ_{MAX} = δ_{0} [\frac{3 (tan α - α)}{α^{3}}]

where

$δ_{0}$	=	$\frac{H L^{3}}{3 E I} = \frac{(0.09) {(25)}^{3} {(12)}^{3}}{3 (29,000) (61.9)} = 0.451 in$
$\frac{3 (tan α - α)}{α^{3}}$	=	$\frac{3 (tan (1.226) - 1.226)}{{1.226}^{3}} = 2.537$

Therefore, the maximum deflection is: $0.451 \times 2.537 = 1.145 in$

Results

Table 1. Comparison of results
Result Type	Reference	STAAD.Pro	Difference	Comments
M_MAX (in·k)	61.37	61.36	negligible
δ_MAX (in)	1.145	1.14538	negligible

STAAD.Pro Input

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
NOTIONAL LOAD FACTOR 0.002
END
LOAD 1 LOADTYPE None  TITLE VERTICAL LOAD
JOINT LOAD
2 FY -30
perf analysi
change
LOAD 2 LOADTYPE Live REDUCIBLE TITLE VERT + NOTIONAL
REPEAT LOAD
1 1.0
NOTIONAL LOAD
1 X 0.003 
PERFORM DIRECT ANALYSIS LRFD 
load list 2
print joint disp list 2
print memb force list 1
FINISH

STAAD.Pro Output

   THE TAU-b"s HAVE CONVERGED FOR LOAD CASE      2
    44. LOAD LIST 2
    45. PRINT JOINT DISP LIST 2
  JOINT    DISP     LIST     2        
      STAAD SPACE                                              -- PAGE NO.    3
   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