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, w_a = 4 KN/m, w_l = 7 KN/m, I_Z=5,000 cm⁴ , E = 200 KN/mm² .

Beam with partial, trapezoidal load

Theoretical Solution

R_{A} = \frac{W_{a}}{8 l^{3}} {(l - a)}^{3} (3 l + a) + \frac{W_{l} - W_{a}}{40 l^{3}} {(l - a)}^{3} (4 l + a)

O_{A} = \frac{{- W}_{a}}{48 E I l} {(l - a)}^{3} (l + 3 a) - \frac{W_{l} - W_{a}}{240 E I l} {(l - a)}^{3} (2 l + 3 a)

R_{B} = \frac{W_{a} - W_{l}}{2} (l - a) - R_{A}

M_{B} = R_{A} l - \frac{W_{a}}{2} {(l - a)}^{2} - \frac{W_{l} - W_{a}}{6} {(l - a)}^{2}

where

l = a + b

Comparison

Table 1. Comparison of results
Result Type	Theory	STAAD.Pro	Difference
Rotation at A, O_A (radians)	0.0020	0.0020	none
Vertical reaction at A, R_A (kN)	3.2886	3.2886	none
Vertical reaction at B, R_B (kN)	21.461	21.461	none
Moment at B, M_B (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.

STAAD Input

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
LOAD 1
MEMBER LOAD
2 TRAP GY -4 -7
LOAD 2
MEMBER LOAD
2 UNI GY -4
2 LIN GY 0 -3
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS
UNIT METER NEWTON
PRINT SUPPORT REACTION
FINISH

STAAD Output

   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