A 304 stainless steel (yield strength 30 ksi) cylinder has an inner diameter of 4 in and a wall thickness of 0.1 in. If it is subjected to an axial load of 500 lb and a torque of 70 lb-ft, determine if yielding occurs according to the maximum shear stress and distortional energy theories.

The value we got is 1.423 ksi which is less than 30 and so the material hasn't failed and yielding hasn't occured.

Explanation:

This is a cylindrical thin walled vessel. Thus;

Hoop stress; σ1 = pr/t

Where;

p is internal pressure

r_o is radius = 4/2 = 2 in

t is thickness of wall = 0.1 in

Thus;

σ1 = 70 x 2/0.1 = 1400 ksi

Longitudinal stress; σ2 = pr/2t

σ2 = 70 x 2 / (0.1 x 2) = 700 ksi

Now, we want to find the normal stress. The inner radius of the circle will be; r_i = r_o - t = 2 - 0.1 = 1.9 in

So, normal stress by axial force is given by;

σ_fx = F/A = F / (π (r_o² - r_i²))

We are given that F = 500 lb

σ_fx = 500 / (π (2² - 1.9²))

σ_fx = 408.1 ksi

We can now find the torsion from the formula;

τ = (T•r_o) / J

We are given that T = 70 lb. ft = 70 x 12 lb. in = 840 lb. in

J is the polar moment of inertia and has a formula; ((π/2) (r_o⁴ - r_i⁴))

So, J = ((π/2) (2⁴ - 1.9⁴)) = 4.662 in⁴

Thus,τ_xy = (840 x 2) / 4.662 = 360.4 ksi

σ1 is in the y direction and σ2 is in the x direction. Thus;

σ_x = σ1 + σ_fx = 700 + 408.1 = 1108.1 ksi

Also, σ_y = σ1 = 1400 ksi

Now distortion energy can be expressed as;

σ_Y² = σ_x - σ_x•σ_y + (σ_y) ² + 3 (τ_xy) ²

Plugging in the relevant values, we obtain;

σ_Y² = 1108.1² - (1108.1*1400) + 1400² + 3 (360.4) ²

So, σ_Y² = 2.02 x 10^ (6) psi²

σ_Y = √2.02 x 10^ (6)

σ_Y = 1423 psi = 1.423 ksi

The question says the yield strength of the material is 30 ksi.

The value we got is less than 30 and so the material hasn't failed and yielding hasn't occured.