V. Thermal Loading on a Beam

To find the support reactions due to a temperature loads applied on a fixed-fixed beam.

References

Hand calculation using the following reference:

Matrix Analysis of Framed Structures, 3rd edition, W.Weaver Jr. & J.M.Gere, Van Nostrand Reinhold, Table B2, Appendix B, p.500

Details

The beam in the following geometric, load, and section properties: L = 7.5 m, α = 11.7(10)-6/°F, IZ = 5,000 cm4 , E =200 KN/mm2 , A = 50 cm2 .
  • Case 1: T = 40° F (uniform across section)
  • Case 2: T1 = 0° F (below beam), T2 = 50° F (above beam)
Figure 1. Fixed support beam with two different temperature load cases

Calculations

Theoretical Solution

Horizontal reactions due to case 1 loads:

RA = - RB = EAαΔT = [200(10)6]×[50(10)-4]×[11.7(10)-6]· (40) = 468 kN

Moment reactions due to case 2 loads:

Results

Table 1. Comparison of results
Result Type Theory STAAD.Pro Difference
Horizontal Reaction at Node A (kN) 468 468 none
Moment at Node A (kN·m) 19.5 19.5 none

Input

The file C:\Users\Public\Public Documents\STAAD.Pro 2026\Samples \Verification Models\01 Beams\Thermal Loading on a Beam.STD is typically installed with the program.

STAAD

STAAD PLANE TEMPERATURE LOAD ON A FIXED-FIXED BEAM
START JOB INFORMATION
ENGINEER DATE 18-Sep-18
END JOB INFORMATION
*
* REFERENCE : MATRIX ANALYSIS OF FRAMED STRUCTURES
* GERE & WEAVER, 3RD EDITION, VAN NOSTRAND REINHOLD
*
* TABLE B-2, APPENDIX B, PAGE 500
*
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 YD 30
UNIT METER KN
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 2e+08
POISSON 0.3
ALPHA 1.17e-05
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 FIXED
3 FIXED
LOAD 1
TEMPERATURE LOAD
1 2 TEMP 40
LOAD 2
TEMPERATURE LOAD
1 2 TEMP 0 50
PERFORM ANALYSIS
PRINT SUPPORT REACTION
FINISH

Output

   SUPPORT REACTIONS -UNIT KN   METE    STRUCTURE TYPE = PLANE
   -----------------
 JOINT  LOAD   FORCE-X   FORCE-Y   FORCE-Z     MOM-X     MOM-Y     MOM Z
      1    1    468.00      0.00      0.00      0.00      0.00      0.00 
           2      0.00      0.00      0.00      0.00      0.00    -19.50 
      3    1   -468.00      0.00      0.00      0.00      0.00      0.00
           2      0.00      0.00      0.00      0.00      0.00     19.50