When 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 340
A tube 2 mm thick has the shape shown in Fig. P-340. Find the shearing stress caused by a torque of 600 N·m.
 

Problem 339
A torque of 450 lb·ft is applied to the square section shown in Fig. P-339. Determine the smallest permissible dimension a if the shearing stress is limited to 6000 psi.
 

Problem 338
A tube 0.10 in. thick has an elliptical shape shown in Fig. P-338. What torque will cause a shearing stress of 8000 psi?
 

The torque applied to thin-walled tubes is expressed as