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A circular tube of aluminum is subjected to torsion by torques T applied at the ends (see figure). The bar is 0.75 m long, and the inside and outside diameters are 28 mm and 45 mm, respectively. It is determined by measurement that the angle of twist is 4 degree when the torque is 700 N middot m. (a) Calculate the maximum shear stress tau_max in the tube, the shear modulus of elasticity G, and the maximum shear strain gamma_max (in radians). (b) If the maximum shear strain in the tube is limited to 2.2 times 10^-3 and the inside diameter is increased to 35 mm, what is the maximum permissible torque?

Two sections of steel drill pipe, joined by bolted flange plates at B, are subjectd to a concentrated torque 500 kN*m. at x = 0.5m, and a uniformly distributed torque t₀ = 250 kN *m/m is applied on pipe BC. Let G = 75 GPa and assume that pipes AB and BC have the same inner diamter, d = 200mm. Pipe AB has a thickness t_AB = 15mm., and pipe BC has thickness t_BC = 12 mm. Find the maximum shear stress and maximum twist of the pipe and their locations along the pipe. Assume G = 75 GPa.

For the thin nonprismatic steel pipe of constant thickness t and variable diameter d shown with applied torques at joints 2 and 3, determine the following.

(a) Find reaction moment R₁.

(b) Find an expression for twist rotation ∅₃ at joint 3. Assume that G is constant.

(c) Draw the torsional moment diagram (TMD: T(x), 0 ≤ x ≤ L).

The normal strain in the 45° direction on the surface of a circular tube (see figure) is 880 × 10⁻⁶ when the torque T = 85 N‧m. The tube is made of copper alloy with G = 42GPa and v = 0.35.

(a) If the outside diameter d₂ of the tube is 20 mm., what is the inside diameter d₁?

(b) If the allowable normal stress in the tube is 96 MPa, what is the maximum permissible inside diameter d₁?

The drive shaft for a truck (outer diameter 60 mm and inner diameter 40 mm) is running at 2500 rpm.

(a) If the shaft transmits 150 kW, what is the maximum shear stress in the shaft?

(b) If the allowable shear stress is 30 MPa, what is the maximum power that can be transmitted?

A solid steel bar of diameter d₁ = 25.0 mm is enclosed by a steel tube of outer diameter d₃ = 37.5 mm and inner diameter d₂ = 30.0 mm (see figure). Both bar and tube are held rigidly by a support at end A and joined securely to a rigid plate at end B. The composite bar, which has a length L = 550 mm, is twisted by a torque T = 400 N·m acting on the end plate.

(a) Determine the maximum shear stresses τ₁ and τ₂ in the bar and tube, respectively.

(b) Determine the angle of rotation ∅ (in degrees) of the end plate, assuming that the shear modulus of the steel is G = 80 GPa.

(c) Determine the torsional stiffness kT of the composite bar.

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