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V. Stresses in a Cable due to Thermal Loading

A rigid bar is suspended by two copper wires and one steel wire. Find the stresses in the wires due to a rise in temperature.

Reference

Timoshenko, S., Strength of Materials, Part  1, D. Van Nostrand Co., 3rd edition, 1956, page 30, problem 9.

Problem

Assuming the horizontal member to be very rigid, determine the stresses in the copper and steel wires if the temperature rise is 10° F. Members 1 and 3 are copper and member 2 is steel.

Esteel = 30×(10)6 psi, Ecopper = 16×(10)6 psi

αsteel = 70E-7 in/in/°F,  αcopper = 92E-7 in/in/°F

AX = 0.1 in2

w = 400 lbf/in.

L = 20 in.

d= 5 in.

Tip: When modeling, assume a large moment of inertia for the horizontal rigid member and distribute of the concentrated load as uniform.

Model of a rigid wire suspended by wires

Comparison

Table 1. Comparison of results
Result Type Theory STAAD.Pro Difference
σSteel (psi) 19,695 19,698.3 negligible
σCopper (psi) 10,152 10,150.8 negligible

STAAD Input

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

STAAD PLANE :A RIGID BAR SUSPENDED BY TWO COPPER WIRES AND A STEEL WIRE
START JOB INFORMATION
ENGINEER DATE 09-Oct-17
END JOB INFORMATION
*
*  REFERENCE: 'STRENGTH OF MATERIALS', TIMOSHENKO (PART 1),
*              PAGE 30, PROB 9.
*              THE ANSWERS ARE 19700 PSI AND 10200 PSI.
*
UNIT INCHES POUND
JOINT COORDINATES
1 0 20 0; 2 5 20 0; 3 10 20 0; 4 0 0 0; 5 5 0 0; 6 10 0 0;
MEMBER INCIDENCES
1 1 4; 2 2 5; 3 3 6; 4 4 5; 5 5 6;
MEMBER PROPERTY AMERICAN
1 TO 3 PRIS AX 0.1 IZ 0.0001
4 5 PRIS AX 1 IZ 100
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 1.6e+007
POISSON 0.230769
ALPHA 9.2e-006
ISOTROPIC MATERIAL2
E 3e+007
POISSON 0.290909
ALPHA 7e-006
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 MEMB 1 3
MATERIAL MATERIAL2 MEMB 2 4 5
SUPPORTS
1 2 3 Fixed
MEMBER RELEASE
1 TO 3 START MPZ 0.99
1 TO 3 END MZ
LOAD 1 VERT LOAD + TEMP LOAD
MEMBER LOAD
4 5 UNI Y -400
TEMPERATURE LOAD
1 TO 3 TEMP 10
PERFORM ANALYSIS
PRINT MEMBER STRESSES LIST 1 TO 3
FINISH

STAAD Output

  MEMBER STRESSES
  ---------------
  ALL UNITS ARE POUN/SQ INCH
 MEMB   LD  SECT    AXIAL    BEND-Y     BEND-Z   COMBINED  SHEAR-Y  SHEAR-Z
      1    1   .0  10150.8 T      0.0       0.0   10150.8      0.0      0.0
              1.0  10150.8 T      0.0       0.0   10150.8      0.0      0.0
      2    1   .0  19698.3 T      0.0       0.0   19698.3      0.0      0.0
              1.0  19698.3 T      0.0       0.0   19698.3      0.0      0.0
      3    1   .0  10150.8 T      0.0       0.0   10150.8      0.0      0.0
              1.0  10150.8 T      0.0       0.0   10150.8      0.0      0.0