Tools › Structures › Plates and Shells › Membrane Shell Stresses › Example
Steel cylinder with hemispherical heads under 100 psi: membrane stresses and the junction discontinuity by NASA TR R-103
| Tool | Membrane Shell Stresses |
| Source | Johns, R. H., and Orange, T. W. (1961). Theoretical Elastic Stress Distributions Arising from Discontinuities and Edge Loads in Several Shell-Type Structures. NASA Technical Report R-103, Washington, DC. Equations (17)-(18), (26)-(30), (42)-(50) and Appendix I; Morgan, W. C., and Bizon, P. T. (1963). Experimental Evaluation of Theoretical Elastic Stress Distributions for Cylinder-to-Hemisphere and Cone-to-Sphere Junctions in Pressurized Shell Structures. NASA Technical Note D-1565, Washington, DC. Appendix B (H_0 = -p/(8β), M_0 = 0) and the summary (maximum effective stress 3½ % above the cylinder-membrane value). |
| Level | Undergraduate |
Problem
A steel pressure vessel (E = 29 Msi, ν = 0.3) has a cylindrical shell of 30 in mean radius and 0.25 in wall closed by hemispherical heads of the same thickness, under 100 lb/in² internal pressure. Find the membrane stresses of the cylinder and the heads, the free radial expansions of the two parts, the edge shear and moment at the junction from the compatibility of displacement and rotation, the stresses on both surfaces along the cylinder and the head, the largest von Mises stress and the margin against a 20 ksi allowable.
Inputs to Enter
- Shape: cylinder with hemispherical heads; R = 30 in, t = 0.25 in, t_h = 0.25 in, head radius blank (hemisphere)
- p = 100 lb/in² internal, no axial load, load factor 1.0; E = 29 Msi, ν = 0.30; allowable 20 ksi, von Mises
Answers
- Membrane: cylinder σ_θ = pR/t = 12 ksi, σ_x = pR/2t = 6 ksi; head σ = pR/2t = 6 ksi
- β = [3(1 − ν²)/(R² t²)]^¼ = 0.4694 /in, decay length π/β = 6.69 in; δ_c = (1 − ν/2) p R²/(E t) = 0.01055 in, δ_s = (1 − ν) p R²/(2 E t) = 0.004345 in, difference p R²/(2 E t)
- Junction: H_0 = −p/(8β) = −26.63 lb/in (TR R-103 eq. 17, TN D-1565 eq. B1), M_0 = 0 (eq. 18 / B2); compatibility residuals below 1e-10
- Hoop stress at the junction 0.75 pR/t = 9.00 ksi on both faces; maximum hoop 1.031 pR/t = 12.38 ksi at x = 3.96 in (βx = 1.86); maximum bending 0.146 pR/t = 1.756 ksi at βx = π/4 (x = 1.67 in)
- Largest von Mises stress 10.74 ksi at x = 3.72 in on the outer surface of the cylinder = 1.0335 × the cylinder-membrane value 10.39 ksi (TN D-1565: about 3½ % higher, for any R/t); M.S. = 20/10.74 − 1 = +0.86
Notes
The tool's default inputs are this vessel; press Submit. The junction solution is checked in test_membrane_shells.php against the closed forms of TR R-103 (eqs 17-18 for five cylinder-sphere cases, eqs 20-21 for four cylinder-flat-head cases) re-transcribed in numpy, against TN D-1565's 3½ % statement, and the membrane formulas against the equilibrium equations with numerically differentiated meridians (References/Membrane-Shells/checks/membrane_check.py). In SI the same vessel reads 82.74 / 41.37 MPa, β = 0.01848 /mm, H_0 = −4.664 kN/m.