Pressure Vessel: Method

Pressure Vessel: Theory, Conventions and the Code Rules Used

What the tool does

The tool analyses a cylindrical or spherical vessel under internal pressure (and, in the thick-wall part, external pressure) in three independent parts chosen by the checkboxes at the top of the form. The thin-wall part gives the membrane stresses of elementary strength of materials, the end-cap force and the load on the bolts of a bolted head. The thick-wall part gives the exact elastic (Lamé) stresses through the wall, the stress measures at the bore, the pressures at first yield, at full plastic collapse and at burst, and the wall needed to hold a stress measure at an allowable. The ASME part applies the design-by-rule formulas of the ASME Boiler and Pressure Vessel Code, Section VIII Division 1, for shells and formed heads under internal pressure, with corrosion allowance, joint efficiency and undertolerance, and reports the maximum allowable working pressure and the hydrostatic test pressure. The three parts share the geometry and the internal pressure; each has its own material inputs because the quantities differ (a Code allowable stress S, a yield and an ultimate strength, a bolt allowable).

Conventions

D_i is the new (uncorroded) inside diameter, t the nominal wall thickness as built; r_i = a = D_i/2 and r_o = b = a + t. Pressures are gauge pressures; p_i acts on the bore and p_o on the outer surface. Tension is positive, so the radial stress is negative (compressive) wherever a pressure acts on a surface. Thin-wall formulas use the inside radius, as Hibbeler writes them; the version with the mean radius is printed beside it in the Lamé part. Stresses are shown in ksi or MPa, pressures in psi or MPa, lengths in inches or millimetres; every input is converted to SI base units before the calculation and the results are converted back for display.

Thin-wall membrane theory

When the wall is thin compared with the radius (t/r ≤ 0.1, that is r/t ≥ 10, Hibbeler's limit) the stresses may be taken uniform through the thickness and the small radial stress neglected. Equilibrium of a half cylinder of unit length cut along a diameter gives the circumferential (hoop) stress σ_θ = p r/t, and equilibrium of the end gives the longitudinal stress σ_z = p r/(2t); for a sphere every tangential direction carries σ = p r/(2t). These are principal stresses with σ_r ≈ 0 as the third, so the maximum shear stress acts on planes at 45° through the thickness and equals σ_θ/2 (not the in-plane value (σ_θ − σ_z)/2 = σ_θ/4), and the von Mises stress of the cylinder is (√3/2) σ_θ. The thin-wall hoop stress with the inside radius is exactly the average of the Lamé hoop stress over the wall (∫σ_θ dr = p a, the equilibrium statement), so it is low at the bore by about t/(2a) for internal pressure alone: 1 % at t/a = 0.02, 5 % at t/a = 0.1.

The pressure on a closed end or a flat cover produces the axial force F = p π D_i²/4. A bolted blind flange or cover carries that force in its bolts; the tool divides it among n_b bolts and reports the tensile stress on the bolt stress area with a margin against the bolt allowable. It is the pressure load only: the gasket seating load, the preload that keeps the joint tight and the larger reaction diameter of a gasket seated outside the bore (ASME Appendix 2) are not included, so the number is a lower bound on the bolt load.

Thick-wall Lamé solution

For a long cylinder under uniform internal and external pressure the stresses are axisymmetric and independent of z, and the equilibrium equation dσ_r/dr + (σ_r − σ_θ)/r = 0 with the compatibility of the plane strains gives Lamé's solution (1852): σ_r = A − B/r², σ_θ = A + B/r², with the constants fixed by σ_r(a) = −p_i and σ_r(b) = −p_o: A = (p_ia² − p_ob²)/(b² − a²) and B = (p_i − p_o) a²b²/(b² − a²). The sum σ_r + σ_θ = 2A is constant through the wall, which is why a long cylinder with closed ends has a uniform axial stress σ_z = A, the end load over the ring area (for open ends σ_z = 0; the plane-strain value ν (σ_r + σ_θ) is not offered). The hoop stress is largest at the bore, σ_θ(a) = [p_i(a² + b²) − 2p_ob²]/(b² − a²), and the largest shear stress and von Mises stress are also at the bore for internal pressure. For a sphere the corresponding solution is σ_r = A − B/r³, σ_θ = σ_φ = A + B/(2r³) with A = (p_ia³ − p_ob³)/(b³ − a³) and B = (p_i − p_o) a³b³/(b³ − a³) (Timoshenko and Goodier, Roark Table 13.5).

Criteria and required wall. The stress measure at the bore is compared with the allowable S_a by the criterion chosen: maximum shear (Tresca, 2τ_max = s₁ − s₃), von Mises, or the largest principal stress. The wall that brings the measure to S_a is found by bisection on the Lamé field, so external pressure and the end condition are honoured; for p_o = 0 and closed ends the closed forms are b = a√(S/(S − 2p)) (Tresca), b = a√(S/(S − √3 p)) (von Mises) and b = a√((S + p)/(S − p)) (principal stress). Each measure tends to a limit as the wall grows without bound (2p, √3 p and p respectively), so an allowable below that limit cannot be met by any thickness and the tool says so.

Yield and burst. First yield at the bore under internal pressure alone: Tresca p_y = S_y(b² − a²)/(2b²); von Mises with closed ends p_y = S_y(b² − a²)/(√3 b²), with open ends S_y(b² − a²)/√(3b⁴ + a⁴); principal stress p_y = S_y(b² − a²)/(a² + b²). With an external pressure the first-yield internal pressure is found numerically. The pressure at which a non-hardening cylinder has yielded through the whole wall (Hill 1950) is p_c = S_y ln(b/a) for Tresca and (2/√3) S_y ln(b/a) for von Mises; this is the collapse pressure of an elastic-perfectly-plastic material. Real steels harden, and Faupel's empirical burst formula p_b = (2/√3) S_y(2 − S_y/S_u) ln(b/a), fitted to burst tests of heavy-wall cylinders (Faupel 1956, as quoted by Harvey), interpolates between the von Mises collapse pressure (S_u = S_y) and twice it (S_u ≫ S_y). For a sphere first yield is at p_y = (2/3) S_y(1 − a³/b³) and the collapse pressure 2 S_y ln(b/a). Autofrettage (pre-yielding the bore to leave compressive residual stresses) is not evaluated.

ASME Section VIII Division 1 rules

The Code's design-by-rule formulas for internal pressure are written in terms of the corroded inside dimensions and the thickness available for pressure. With P the design pressure, R and D the inside radius and diameter after corrosion (R_c = D_i/2 + CA, D_c = D_i + 2CA), S the maximum allowable stress of Section II Part D at the design temperature, and E the joint efficiency of UW-12 (1.00 with full radiography, 0.85 with spot radiography, 0.70 with none, for a Type 1 double-welded butt joint):

UG-27(c)(1), cylindrical shell, circumferential stress: t = P R/(S E − 0.6P), valid for t ≤ R/2 and P ≤ 0.385 S E; (c)(2), longitudinal stress: t = P R/(2 S E + 0.4P), valid for t ≤ R/2 and P ≤ 1.25 S E; the larger t governs (the circumferential one for a single E). UG-27(d), spherical shell: t = P R/(2 S E − 0.2P), valid for t ≤ 0.356R and P ≤ 0.665 S E. UG-32(d), 2:1 ellipsoidal head: t = P D/(2 S E − 0.2P); Appendix 1-4(c) generalises it with K = (1/6)[2 + (D/2h)²] for 1.0 ≤ D/2h ≤ 3.0 (K = 1 at the 2:1 head, 1.83 at D/2h = 3, 0.50 for a hemisphere). UG-32(e), torispherical head with knuckle radius r = 6 % of the inside crown radius L: t = 0.885 P L/(S E − 0.1P); Appendix 1-4(d) generalises it with M = (1/4)[3 + √(L/r)] and t = P L M/(2 S E − 0.2P), subject to r ≥ 0.06L, r ≥ 3t and L ≤ D (M = 1.77 at L/r = 16.67, 1.00 for a hemisphere). UG-32(f), hemispherical head: t = P L/(2 S E − 0.2P). The maximum allowable working pressure of each part is the same formula inverted with the thickness actually available (nominal thickness less the mill undertolerance of UG-16(c) and the corrosion allowance of UG-25): P = S E t/(R + 0.6t) and 2 S E t/(R − 0.4t) for the cylinder, 2 S E t/(R + 0.2t) for the sphere and the hemispherical head, 2 S E t/(K D + 0.2t) for the ellipsoidal head and 2 S E t/(L M + 0.2t) for the torispherical head. The vessel MAWP is the least of the parts. UG-99(b) sets the hydrostatic test pressure at 1.3 times the MAWP times the lowest ratio of the allowable stress at test temperature to that at design temperature; the tool prints it on the MAWP and, for a vessel stamped at its design pressure, on the design pressure.

The joint efficiency entered is applied to the heads as well as the shell. A seamless head attached by a spot-examined Category B weld may take E = 1.0 for its own formula (UW-12(d)); run the head case again with that E if it applies. The required thickness printed includes the corrosion allowance (and the undertolerance is divided out when one is entered) so that it can be compared directly with a plate thickness to order. The margins compare the MAWP of each part with the design pressure.

Not covered: external pressure and vacuum (UG-28 and the Section II-D charts), nozzles and their reinforcement (UG-37), flat heads and covers (UG-34), conical sections, the Appendix 1-2 formulas for thick shells outside the UG-27 limits, the weld joint categories beyond the shell's E, the static head of the contents, wind, seismic, support and thermal loads, fatigue, and the Section II-D allowable stress tables themselves (enter S for the material and temperature).

Reading the results

The Geometry section fixes the radii. In the thin-wall part the hoop stress is the design quantity for the cylinder and t/r_i tells whether membrane theory is adequate; the von Mises stress is compared with the Lamé allowable S_a when that part runs, otherwise with the ASME S. In the Lamé part the table and Figure 1 show the stresses from the bore to the outer surface: σ_r runs from −p_i to −p_o, σ_θ decreases outward, their sum is constant, and the dotted line is the thin-wall value, so the figure shows at a glance how far the vessel is from thin. The required wall is the thickness at which the chosen bore measure equals S_a, with no corrosion allowance. The yield and burst pressures are information on the ductile reserve, not design values: a Code vessel operates far below them (the Division 1 allowable is the lesser of S_u/3.5 and 2S_y/3). In the ASME part the required thickness including corrosion allowance is compared with the thickness as built, and the MAWP is the pressure the vessel as built may be stamped for; the hydrostatic test pressure is the UG-99(b) minimum.