Stress-Linearisation Worked Example

Problem

A pressure vessel wall has a through-thickness hoop stress distribution at a nozzle junction. The stress at 5 equally-spaced points through the 20 mm wall is: 180, 140, 95, 70, 55 MPa (inner to outer surface). Calculate the membrane, bending and peak stress components.

Given

• Wall thickness t = 20 mm
• 5 points at x = 0, 5, 10, 15, 20 mm
• Stresses: 180, 140, 95, 70, 55 MPa

Step 1 — Membrane stress (average)

sigma_m = (180 + 140 + 95 + 70 + 55) / 5 = 540 / 5 = 108 MPa

Step 2 — Bending stress

The bending stress is linearly varying: sigma_b(x) = 6/t^2 * integral of sigma(x)*(x - t/2) dx. Using the trapezoidal rule with 5 points: Moment = sum of sigma_i * (x_i - 10) * delta_x = [180*(-10) + 140*(-5) + 95*0 + 70*5 + 55*10] * 5 = [-1800 - 700 + 0 + 350 + 550] * 5 = -1600 * 5 = -8000 MPa*mm^2 sigma_b = 6 * (-8000) / 20^2 = -48000/400 = -120 MPa The bending stress at the inner surface (x=0) = +120 MPa and at the outer surface (x=20) = -120 MPa.

Step 3 — Membrane + bending at each point

At inner surface (x=0): sigma_m + sigma_b = 108 + 120 = 228 MPa At outer surface (x=20): sigma_m + sigma_b = 108 - 120 = -12 MPa

Step 4 — Peak stress at each point

Peak = actual - (membrane + bending): At x=0: 180 - 228 = -48 MPa At x=5: 140 - (108 + 60) = -28 MPa At x=10: 95 - 108 = -13 MPa At x=15: 70 - (108 - 60) = 22 MPa At x=20: 55 - (-12) = 67 MPa

Result

Membrane stress = 108 MPa. Bending stress = ±120 MPa at the surfaces. Peak stress varies from -48 MPa at the inner surface to +67 MPa at the outer surface. The membrane plus bending at the inner surface is 228 MPa, which would be assessed against the local membrane plus bending allowable. The peak stress drives fatigue assessment.

Related Knowledge

• Stress Linearisation Through Thickness
• Membrane, Bending & Peak Stress