Thick cylinder (Lame’s theory): Maximum tangential (hoop) stress at the inner surface due to internal pressure. Given: a thick cylindrical shell with inner radius r_i, outer radius r_o, internal pressure p, and negligible external pressure. Choose the correct expression for the maximum hoop stress at r = r_i.

Given data

• Thick cylinder with inner radius r_i and outer radius r_o.
• Internal pressure = p; external pressure ≈ 0.

Concept/Approach
Use Lame’s equations for thick cylinders: σ_r = A − B/r^2 and σ_θ = A + B/r^2.

Step-by-step derivation
Boundary 1 (at r = r_i): σ_r(ri) = −p ⇒ −p = A − B/r_i^2.Boundary 2 (at r = r_o): σ_r(ro) = 0 ⇒ 0 = A − B/r_o^2 ⇒ A = B/r_o^2.Substitute A into Boundary 1: −p = (B/r_o^2) − (B/r_i^2) = B,[1/r_o^2 − 1/r_i^2].Solve for B: B = p, r_i^2 r_o^2 / (r_o^2 − r_i^2).Then A = B/r_o^2 = p, r_i^2 / (r_o^2 − r_i^2).Hoop stress at the inner surface: σ_θ(ri) = A + B/r_i^2 = p(r_o^2 + r_i^2)/(r_o^2 − r_i^2).

Common pitfalls
Mixing up radial and hoop stresses or sign conventions leads to wrong expressions.

Final Answer
σ_θ(ri) = p × (r_o^2 + r_i^2) / (r_o^2 − r_i^2)