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MATHalinoEngineering Math ReviewWhen close-coiled helical spring, composed of a wire of round rod of diameter d wound into a helix of mean radius R with n number of turns, is subjected to an axial load P produces the following stresses and elongation:
The maximum shearing stress is the sum of the direct shearing stress τ1 = P/A and the torsional shearing stress τ2 = Tr/J, with T = PR.
Problem 341
Derive the torsion formula τ = Tρ / J for a solid circular section by assuming the section is composed of a series of concentric thin circular tubes. Assume that the shearing stress at any point is proportional to its radial distance.
Problem 332
In a rivet group subjected to a twisting couple T, show that the torsion formula τ = Tρ/J can be used to find the shearing stress τ at the center of any rivet. Let J = ΣAρ2, where A is the area of a rivet at the radial distance ρ from the centroid of the rivet group.
Problem 310
Show that the hollow circular shaft whose inner diameter is half the outer diameter has a torsional strength equal to 15/16 of that of a solid shaft of the same outside diameter.
Problem 305
What is the minimum diameter of a solid steel shaft that will not twist through more than 3° in a 6-m length when subjected to a torque of 12 kN·m? What maximum shearing stress is developed? Use G = 83 GPa.
TORSION
Consider a bar to be rigidly attached at one end and twisted at the other end by a torque or twisting moment T equivalent to F × d, which is applied perpendicular to the axis of the bar, as shown in the figure. Such a bar is said to be in torsion.
