# 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 (Iz = 341 in4).

## Validation

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

Maximum moment

$M MAX = M 0 2 ⁢ sec u - 1 u 2$
where
 $M 0$ = $u$ = $P ⁢ L 2 4 ⁢ E ⁢ I = 100 × ( 40 ) 2 × ( 12 ) 2 4 × 29,000 × 341 = 0.763$ $2 ⁢ sec u - 1 u 2$ = $2 ⁢ sec ( 0.763 ) - 1 0.763 2 = 1.318$

Therefore, the maximum moment is:

Maximum deflection

$δ MAX = δ 0 12 2 sec u - u 2 - 2 5 u 4$
where
 $δ 0$ = $12 2 sec u - u 2 - 2 5 u 4$ = $12 2 sec ( 0.763 ) - ( 0.763 ) 2 - 2 5 ( 0.763 ) 4 = 1.310$

Therefore, the maximum deflection is:

## Results

Table 1. Comparison of results
MMAX (ft·k) 26.36 26.308 negligible
δMAX (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.

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
1 UNI GY -0.1
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


   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