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48 in pressure vessel at 120 psi: membrane stresses, Lamé check and the ASME VIII-1 UG-27 / UG-32 thickness and MAWP

Tool	Pressure Vessel
Source	Hibbeler, R. C. (2014). Mechanics of Materials, 9th ed. Pearson Prentice Hall, Upper Saddle River, NJ. Example 8.1 (the vessel geometry); American Society of Mechanical Engineers (2023). ASME Boiler and Pressure Vessel Code, Section VIII, Division 1. ASME, New York. UG-27(c), UG-32(d), UG-99(b); Lamé, G. (1852). Leçons sur la théorie mathématique de l'élasticité des corps solides. Bachelier, Paris.
Level	Undergraduate

Problem

A cylindrical vessel of 48 in inside diameter and 1/2 in wall (Hibbeler's Example 8.1 geometry) holds 120 psi. Find the thin-wall hoop and longitudinal stresses, the end-cap force and the load on 32 head bolts of 0.334 in² stress area, the exact Lamé stresses at the bore with the yield and burst pressures of SA-516 Grade 70 (S_y 38 ksi, S_u 70 ksi), and the ASME Section VIII Division 1 required thickness and MAWP of the shell and its 2:1 ellipsoidal heads with S = 20 ksi, E = 0.85 and a 1/16 in corrosion allowance.

Inputs to Enter

Cylinder, p_i 120 psi, D_i 48 in, t 0.5 in; bolted head 32 bolts, A_b 0.334 in², S_b 25 ksi
Lamé part: p_o 0, closed ends, S_y 38 ksi, S_u 70 ksi, S_a 20 ksi, Tresca criterion
ASME part: S 20 ksi, E 0.85, CA 0.0625 in, undertolerance 0, 2:1 ellipsoidal heads 0.5 in thick, test stress ratio 1.0

Open the Pressure Vessel tool (loaded with this example)

Answers

Thin wall: σ_θ = p r/t = 5.760 ksi, σ_z = 2.880 ksi, von Mises 4.988 ksi, t/r = 0.0208; F_end = p π D²/4 = 217,100 lb, 6,786 lb per bolt, 20.32 ksi (M.S. +0.23)
Lamé at the bore: σ_θ = 5.821 ksi (thin-wall 1.04 % low), σ_r = −0.120 ksi, σ_z = 2.850 ksi, τ_max = 2.970 ksi, von Mises 5.145 ksi; Tresca wall for 20 ksi 0.1453 in
First yield (Tresca) 767.6 psi, von Mises 886.3 psi; collapse S_y ln(b/a) = 783.5 psi, (2/√3) S_y ln(b/a) = 904.7 psi; Faupel burst 1,318 psi
UG-27(c)(1): t = PR/(SE − 0.6P) = 120 × 24.0625/(17,000 − 72) = 0.1706 in, + CA = 0.2331 in; MAWP = SEt/(R + 0.6t) with t = 0.4375 in → 305.8 psi (M.S. +1.55)
UG-32(d) 2:1 head: t = PD/(2SE − 0.2P) = 0.1700 in (+ CA 0.2325 in), MAWP 308.5 psi; vessel MAWP 305.8 psi (shell); UG-99(b) test 1.3 × MAWP = 397.5 psi

Notes

The tool's default inputs are this vessel; press Submit. Hibbeler's printed answer for the same geometry is the pressure at which the hoop stress reaches 20 ksi, 417 psi (833 psi for a sphere), which the thin-wall formula reproduces; the Code numbers are the UG-27 / UG-32 formulas evaluated independently in numpy (References/Pressure-Vessel/vessel_check.py), since no open worked ASME example was reachable.