How do I calculate the exstension in a steel cable that is 8 metres long and has a 500kg weight on it. The Young's modulus of the steel is 210 GPa and the diameter is 7.2mm?

Answer 1

In order to calculate the extension of the cable we must first calculate the strain. Strain is related to Young's modulus using the following relation:

E = σA / εA

Where:
E = Young's Modulus
σA = Axial Stress
εA = Axial Strain

Therefore to calculate the strain, we must first calculate the axial stress in the cable:

σA = Force / Area

The cross sectional area (CSA) of the cable is equal to:

CSA = (pi(d2))/4 = pi x r2

CSA = pi(0.0036)2
CSA = 4.0715x10-5 m2

Axial Stress = (500 x 9.81) / 4.0715x10-5

σA = 120471449.98 Pa
Axial Stress = 120.47144998 MPa

Strain of cable (εA) = Axial Stress / Young's Modulus

εA = 120.471x106 / 210x109
εA = 5.736735713x10-4

To calculate the extension of the cable:

εA = Extension / Original Length

Therefore:

Extension = εA x Original Length

= 0.0045893885704 m

≈ 4.6 mm of extension.

Answer 2

Deformation due to axial load

DEFORMATION = (FORCE X LENGTH) / (CROSS SECTION AREA X YOUNGS MODULUS)

Assuming g=9.807m/s^2

Assuming Youngs Modulus = 210GPa

Assuming Simple Loading (straight hanging weight)

Approximating cable as homogeneous steel rod

Deformation in said cable is 6.66mm

Because of the structure of cable extension will occur due to contraction of the strands into a tighter form the actual extension of the cable will be greater than the approximation made.