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Thermal & Torsion stresses

Daniel D. Dasig, Jr.

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Thermal stresses

Temperature causes bodies to expand or contract.Change in length due to increase in temperaturecan be expressed as

L= L..t

Where, L is the length, (/o

C) is the coefficient oflinear expansion, and t(oC) is the temperaturechange.

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When the temperature deformation is

prevented, thermal stress developed due to

temperature change can be given as:

= E..t

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Example 1

A steel rod with a cross-sectional area of 0.25

in2is stretched between two fixed points. The

tensile load at 70F is 1200 lb. What will be

the stress at 0F? At what temperature will the

stress be zero? Assume = 6.5 10-6in/(inF)and E = 29 106psi.

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For the stress at 0F:

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For the temperature that causes zero stress

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Example 2

A steel rod is stretched between two rigid

walls and carries a tensile load of 5000 N at

20C. If the allowable stress is not to exceed

130 MPa at -20C, what is the minimum

diameter of the rod? Assume =

11.7 m/(mC) and E = 200 GPa.

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Solution

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TORSIONTORSIONAL SHEARING

STRESS,

For a solid or hollow circular shaft

subject to a twisting moment T, the

torsional shearing stress at a

distance from the center of the shaftis

whereJ is the polar moment of inertia

of the section and r is the outer radius.

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For solid cylindrical shaft

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For hollow cylindrical shaft

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ANGLE OF TWIST

The angle through which the bar length L will twist is

whereT= is the torque in Nmm,

L= is the length of shaft in mm,

G= is shear modulus in MPa,

J= is the polar moment of inertia in mm4,Dand d= are diameter in mm, and

r= is the radius in mm.

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POWER TRANSMITTED BY THE SHAFT

A shaft rotating with a constant angular velocity (in radians per

second) is being acted by a twisting moment T. The power

transmitted by the shaft is

Where

T= is the torque in Nm

f= is the number of revolutions per second, andP= is the power in watts.

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Example 1

A steel shaft 3 ft long that has a

diameter of 4 in is subjected to a

torque of 15 kipft. Determine themaximum shearing stress and the

angle of twist. Use G = 12 106psi.

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Example 2

A steel marine propeller shaft 14 in. in

diameter and 18 ft long is used to

transmit 5000 hp at 189 rpm. If G = 12 106psi, determine the maximum

shearing stress.

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QUIZ 2

1. A copper rod of 15 mm diameter, 800 mmlong is heated through 50C. (a) What is its

extension when free to expand? (b) Suppose

the expansion is prevented by gripping it at

both ends, find the stress, its nature and the

force applied by the grips when one grip yields

back by 0.5 mm.