# V. Thermal Loading on a Beam

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

## Reference

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

## Problem

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)

## 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:

$M A = − M B = α E I Δ T d = [ 11.7 ( 10 ) − 6 ] ⋅ [ 200 ( 10 ) − 6 ] ⋅ [ 5 , 000 ( 10 ) − 8 ] ( 50 ) 0.3 = 19.5 ⁢ k N ⋅ m$

## Comparison

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
Note: In the STAAD.Pro model, two load cases are used. In case 1, the uniform expansion is applied. In case 2, the temperature change between top and bottom flanges is applied.

## STAAD Input

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

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 &amp; 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


## STAAD 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