V. Direct Analysis of a Beam

Verify the maximum bending moment and deflection of a beam 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 1

Details

A 40' long, simply-supported beam is subject to a uniformly distributed load of 100 lbs/ft and axial compression of 100 kips. The section is a W10x60 (I_z = 341 in⁴).

Validation

The maximum moment and deflection both occur at mid-span.

Maximum moment

M_{MAX} = M_{0} [\frac{2 (sec u - 1)}{u^{2}}]

where

$M_{0}$	=	$\frac{w L^{2}}{8} = \frac{0.1 {(40)}^{2}}{8} = 20 ft·k$
$u$	=	$\sqrt[]{\frac{P L^{2}}{4 E I}} = \sqrt[]{\frac{100 \times {(40)}^{2} \times {(12)}^{2}}{4 \times 29,000 \times 341}} = 0.763$
$\frac{2 (sec u - 1)}{u^{2}}$	=	$\frac{2 [sec (0.763) - 1]}{{0.763}^{2}} = 1.318$

Therefore, the maximum moment is: $20 \times 1.318 = 26.36 ft·k$

Maximum deflection

δ_{MAX} = δ_{0} [\frac{12 (2 sec u - u^{2} - 2)}{5 u^{4}}]

where

$δ_{0}$	=	$\frac{5 w L^{4}}{384 E I} = \frac{5 (0.1) {(40)}^{4} {(12)}^{3}}{384 (29,000) (341)} = 0.582 in$
$\frac{12 (2 sec u - u^{2} - 2)}{5 u^{4}}$	=	$\frac{12 (2 sec (0.763) - {(0.763)}^{2} - 2)}{5 {(0.763)}^{4}} = 1.310$

Therefore, the maximum deflection is: $0.582 \times 1.310 = 0.763 in$

Results

Table 1. Comparison of results
Result Type	Reference	STAAD.Pro	Difference	Comments
M_MAX (ft·k)	26.36	26.308	negligible
δ_MAX (in)	0.763	0.757	negligible

Note: The maximum moment and deflection results are taken from the Postprocessing workflow Beam Results and Displacements pages, respectively.

STAAD.Pro Input

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

STAAD PLANE
START JOB INFORMATION
ENGINEER DATE 17-Apr-08
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 40 0 0;
MEMBER INCIDENCES
1 1 2;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 4.176e+06
POISSON 0.3
DENSITY 0.489024
ALPHA 6.5e-06
DAMP 0.03
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
1 TABLE ST W10X60
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT FX MX MY MZ
DEFINE DIRECT ANALYSIS
FYLD 7200 ALL
END
LOAD 1 LOADTYPE None  TITLE APPLIED LOADS
MEMBER LOAD
1 UNI GY -0.1
JOINT LOAD
1 FX 100
2 FX -100
PERFORM DIRECT ANALYSIS LRFD PDITER 15 REDUCEDEI 1
SECTION 0.5 ALL
PRINT MEMBER SECTION FORCES ALL
PRINT SECTION DISPL ALL
FINISH

STAAD.Pro Output

   MEMBER FORCES AT INTERMEDIATE SECTIONS
   --------------------------------------
   ALL UNITS ARE -- KIP  FEET
 MEMB  LOAD  SEC       AXIAL   SHEAR-Y   SHEAR-Z       MOM-X       MOM-Y       MOM-Z
     1    1  0.50     100.00     -0.00      0.00        0.00        0.00      -20.00
   ************** END OF LATEST ANALYSIS RESULT **************
    38. PRINT SECTION DISPL ALL
  SECTION  DISPL    ALL               
      STAAD PLANE                                              -- PAGE NO.    4
     MEMBER SECTION DISPLACEMENTS
     ----------------------------
     UNITS ARE - INCH
     MEMB  LOAD     GLOBAL X,Y,Z DISPL FROM START TO END JOINTS AT 1/12TH PTS
       1     1       0.0000    0.0000    0.0000   -0.0078   -0.1984    0.0000
                    -0.0156   -0.3822    0.0000   -0.0234   -0.5386    0.0000
                    -0.0312   -0.6577    0.0000   -0.0390   -0.7321    0.0000
                    -0.0468   -0.7574    0.0000   -0.0545   -0.7321    0.0000
                    -0.0623   -0.6577    0.0000   -0.0701   -0.5386    0.0000
                    -0.0779   -0.3822    0.0000   -0.0857   -0.1984    0.0000
                    -0.0935    0.0000    0.0000
     MAX LOCAL  DISP =    0.75736   AT     240.00  LOAD    1   L/DISP=    633
         ************ END OF SECT DISPL RESULTS ***********
    39. FINISH