A solid steel bar of circular cross section has diameter d 5 2.5 in., L 5 60 in., and shear modulus of elasticity G 5 11.5 3 106 psi. The bar is subjected to torques T 5 300 lb-ft at the ends. Calculate the angle of twist between the ends. What is the maxi - mum shear stress and the shear stress at a distance rA 5 1.0 in. measured from the center of the bar

A) θ = 4.9 x 10^ (-3) rad

B) τ_max = 1.173 ksi

C) τ_a = 4.786 ksi

Explanation:

We are given;

diameter; d = 2.5 inches = 0.2083 ft

Length; L = 60 inches = 5 ft

Torque; T = 300 lb. ft

Shear modulus; G = 11.5 x 10^ (6) psi = 11.5 x 144 x 10^ (6) lb/ft² = 1.656 x 10^ (9) lb/ft²

A) Now, formula to determine angle of twist is given as;

T/I_p = Gθ/L

Where I_p is polar moment of inertia

θ is angle of twist.

Now I_p = πd⁴/32 = π (0.2083) ⁴/32 = 1.85 x 10^ (-4) ft⁴

Thus, making θ the subject, we have;

TL/GI_p = θ

θ = (300 x 5) / (1.656 x 10^ (9) x 1.85 x 10^ (-4))

θ = 4.9 x 10^ (-3) rad

B) Maximum shear stress is given by the formula;

τ_max = (Gθ/L) (d/2)

From earlier, (Gθ/L) = T/I_p

Thus, (Gθ/L) = 300/1.85 x 10^ (-4) = 1621621.6216

Thus,

τ_max = 1621621.6216 x (0.2083/2)

τ_max = 168891.89 lbf/ft²

Converting to ksi = 168891.89/144000 ksi = 1.173 ksi

C) Shear stress at radial distance is given as;

τ_a = (Gθ/L) •r_a

r_a is given as 5.1 inches = 0.425m

τ_a = 1621621.6216 x 0.425 = 689189.189 lbf/ft²

Converting to ksi = 689189.189/144000 ksi = 4.786 ksi