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., 3^rd 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.

E_steel = 30×(10)⁶ psi, E_copper = 16×(10)⁶ psi

α_steel = 70E-7 in/in/°F, α_copper = 92E-7 in/in/°F

A_X = 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
σ_Steel (psi)	19,695	19,698.3	negligible
σ_Copper (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