# V. Deflection and Reactions in a Beam

To find the deflection and support reactions due to a trapezoidally varying load applied on part of the span of a pinned-fixed beam.

## Reference

Hand calculation using the following reference:

Roark's Formulas for Stress and Strain, Warren C. Young, 6th edition, McGraw-Hill, Table 3, Case (2c), p.103

## Problem

The beam in the following geometric, load, and section properties: a = 3 m, b =4.5 m, wa = 4 KN/m, wl = 7 KN/m, IZ=5,000 cm4 , E = 200 KN/mm2 .

## Theoretical Solution

$R A = W a 8 l 3 ( l − a ) 3 ( 3 l + a ) + W l − W a 40 l 3 ( l − a ) 3 ( 4 l + a )$
$O A = − W a 48 E I l ( l − a ) 3 ( l + 3 a ) − W l − W a 240 E I l ( l − a ) 3 ( 2 l + 3 a )$
$R B = W a − W l 2 ( l − a ) − R A$
$M B = R A l − W a 2 ( l − a ) 2 − W l − W a 6 ( l − a ) 2$

where

l = a + b

## Comparison

Table 1. Comparison of results
Rotation at A, OA (radians) 0.0020 0.0020 none
Vertical reaction at A, RA (kN) 3.2886 3.2886 none
Vertical reaction at B, RB (kN) 21.461 21.461 none
Moment at B, MB (kN·m) 25.9605 25.9605 none
Note: In the STAAD model, two load cases are used. In case 1, the load is applied using the MEMBER LOAD - TRAP option. In case 2, the load is applied using a combination of MEMBER LOAD - UNI and MEMBER LOAD - LIN options. Both cases yield identical results.

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\01 Beams\Deflection and Reactions in a Beam.STD is typically installed with the program.

STAAD PLANE REACTIONS AND DISPLACEMENTS OF A PINNED-FIXED BEAM
START JOB INFORMATION
ENGINEER DATE 18-Sep-18
END JOB INFORMATION
*
* REFERENCE : ROARK'S FORMULAS FOR STRESS &amp; STRAIN
* WARREN C. YOUNG, 6TH EDITION, MCGRAW-HILL
*
* TABLE 3, CASE (2C), LOAD ON PARTIAL SPAN
*
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 3 0 0; 3 7.5 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3;
UNIT CM KN
MEMBER PROPERTY AMERICAN
1 2 PRIS AX 50 IZ 5000
UNIT METER NEWTON
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 2e+11
POISSON 0.3
END DEFINE MATERIAL
UNIT METER KN
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 PINNED
3 FIXED
2 TRAP GY -4 -7
2 UNI GY -4
2 LIN GY 0 -3
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS
UNIT METER NEWTON
PRINT SUPPORT REACTION
FINISH


   JOINT DISPLACEMENT (CM   RADIANS)    STRUCTURE TYPE = PLANE
------------------
JOINT  LOAD   X-TRANS   Y-TRANS   Z-TRANS   X-ROTAN   Y-ROTAN   Z-ROTAN
1    1     0.0000    0.0000    0.0000    0.0000    0.0000   -0.0020
2     0.0000    0.0000    0.0000    0.0000    0.0000   -0.0020
2    1     0.0000   -0.4626    0.0000    0.0000    0.0000   -0.0006
2     0.0000   -0.4626    0.0000    0.0000    0.0000   -0.0006
3    1     0.0000    0.0000    0.0000    0.0000    0.0000    0.0000
2     0.0000    0.0000    0.0000    0.0000    0.0000    0.0000
************** END OF LATEST ANALYSIS RESULT **************
40. UNIT METER NEWTON
41. PRINT SUPPORT REACTION
SUPPORT  REACTION
REACTIONS AND DISPLACEMENTS OF A PINNED-FIXED BEAM       -- PAGE NO.    4
SUPPORT REACTIONS -UNIT NEWT METE    STRUCTURE TYPE = PLANE
-----------------
JOINT  LOAD   FORCE-X   FORCE-Y   FORCE-Z     MOM-X     MOM-Y     MOM Z
1    1      0.00   3288.60      0.00      0.00      0.00      0.00
2      0.00   3288.60      0.00      0.00      0.00      0.00
3    1      0.00  21461.40      0.00      0.00      0.00 -25960.51
2      0.00  21461.40      0.00      0.00      0.00 -25960.51