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, 3^rd 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,I_Z = 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)

Fixed support beam with two different temperature load cases

Theoretical Solution

Horizontal reactions due to case 1 loads:

R_A = - R_B = 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} = \frac{α E I Δ T}{d} = \frac{[11.7 {(10)}^{- 6}] \cdot [200 {(10)}^{- 6}] \cdot [5, 000 {(10)}^{- 8}] (50)}{0.3} = 19.5 k N \cdot 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