Membrane Shell Stresses: Method

Membrane Shell Stresses: Method, Conventions and Limits

What the tool does

The tool gives the stresses in thin shells of revolution under pressure in two layers. The first is membrane theory: the in-plane forces that a smooth, thin wall develops to carry a smooth pressure, evaluated along the meridian for a cylinder (a vertical tank with a liquid head and an axial load), a sphere, a cone with its apex down (a conical tank), a torus, an ellipsoidal head and a torispherical head. The second is the discontinuity analysis of the junction between a cylinder and its head: where a cylinder meets a hemispherical (or shallower spherical) head or a flat head, the two parts would expand by different amounts under the pressure, so an edge shear and an edge moment arise between them and bend both walls locally. The tool finds those edge loads from the compatibility of radial displacement and rotation with the edge-load solutions compiled in NASA TR R-103 (Johns and Orange 1961), follows the bending and hoop stresses they add on the inner and outer surfaces along the cylinder and the head until they die away, forms the von Mises or Tresca stress everywhere and reports margins of safety against F_tu at ultimate load and against F_ty and F_cy at limit load, on the membrane field and on the junction stresses. Two charts show the stresses along the meridian and the decay of the discontinuity stresses either side of the junction.

Membrane theory of shells of revolution

A surface of revolution is described by its meridian. At a point of the meridian, r is the radius of the parallel circle, r₁ the radius of curvature of the meridian itself and r₂ = r / sin φ the length of the normal from the surface to the axis, φ being the angle between that normal and the axis. A thin shell loaded by a pressure p normal to its surface carries it by two in-plane forces per unit length, the meridional force N_φ and the hoop (circumferential) force N_θ, which satisfy the single equilibrium equation normal to the surface,

N_φ / r₁ + N_θ / r₂ = p,

while N_φ alone follows from the vertical equilibrium of the whole part of the shell above the parallel circle: for a dome closed at the top under a uniform pressure, N_φ · 2π r sin φ = p π r², that is N_φ = p r₂ / 2 (Timoshenko and Woinowsky-Krieger, Articles 104 to 108; NASA CR-912, Section 2.32). The stresses are σ = N / t, tension positive; pressure is positive inward-acting, that is internal, and a negative value is an external pressure (every stress then changes sign). Membrane theory is exact when the loads and the geometry vary smoothly and the edges are free to move as the membrane solution requires; where they are not, bending appears in a narrow boundary zone, which is the subject of the second part.

Cylinder (r₁ = ∞, r₂ = R): N_θ = p R, the hoop stress p R / t, and from the end load of a closed cylinder N_x = p R / 2. A vertical tank with a liquid of density ρ to the depth H_l sees p + ρ g (H_l − z) below the surface, so the hoop force grows linearly to the bottom while the axial stress is unchanged (the liquid weight goes into the bottom support, not into the wall). An axial load P adds P / (2π R t) to the axial stress, tension positive.

Sphere (r₁ = r₂ = R): N_φ = N_θ = p R / 2 at every point, half the hoop force of a cylinder of the same radius.

Cone with the semi-vertex angle α, apex down, slant coordinate s from the apex and r = s sin α (r₁ = ∞, r₂ = r / cos α): N_θ = p r / cos α and N_s = p r / (2 cos α), both largest at the base. For a conical tank filled to the depth d above the apex (Timoshenko Art. 108; CR-912 Table 2.32-4 (f)), with y = s cos α the height above the apex, N_θ = ρ g (d − y) y tan α / cos α and N_s = ρ g (d − 2y/3) y tan α / (2 cos α) below the surface; above it N_s = ρ g d³ tan²α / (6 r cos α) (the whole liquid weight hanging from the wall) plus the gas-pressure term. The liquid terms peak inside the liquid, the hoop force at y = d/2 with ρ g d² tan α / (4 cos α) and the meridional force at y = 3d/4 with 3 ρ g d² tan α / (16 cos α), not at the surface.

Torus with the tube radius a and the centre-line radius b, φ measured from the top of the tube so that r = b + a sin φ: the vertical equilibrium of the ring of shell between the top circle and the parallel gives N_φ = p a (r + b) / (2 r), and with r₁ = a and r₂ = r / sin φ the hoop force is N_θ = p a / 2 everywhere (Timoshenko Art. 106; CR-912 Table 2.32-9). The meridional force is p a at the top and bottom of the tube, p a (2b + a) / (2(b + a)) at the outer equator and p a (2b − a) / (2(b − a)) at the inner equator, which is the largest: there the wall carries the pressure on the largest annulus over the smallest circumference. On the inner half of the tube r₂ is negative (the normal meets the axis on the far side); membrane theory is exact except close to φ = 0, where r₂ → ∞ and the real shell develops a small bending zone.

Ellipsoidal head with the horizontal semi-axis a (the shell radius) and the vertical semi-axis b, at the parallel radius r: r₂ = √(a⁴ − (a² − b²) r²) / b and r₁ = r₂³ b² / a⁴, N_φ = p r₂ / 2 and N_θ = p r₂ (1 − r₂ / (2 r₁)) (Timoshenko Art. 105). At the crown r₁ = r₂ = a²/b and both stresses are p a² / (2 b t); at the equator r₂ = a, r₁ = b²/a, N_φ = p a / 2 (continuous with the cylinder) and N_θ = p a (1 − a² / (2 b²)). The hoop force therefore becomes compressive near the equator whenever a/b exceeds √2, changing sign at r = a² / √(2(a² − b²)); for the common 2:1 head it reaches −p a at the equator, equal in magnitude to the hoop force of the attached cylinder, while the crown stress p a / t equals the cylinder's hoop stress, which is why a 2:1 head needs about the same thickness as its shell. The tool flags the compressive region; the real head redistributes the mismatch with the cylinder by bending, and a thin head with a/b well above 2 must also be checked for buckling of the knuckle region.

Torispherical head: a spherical crown of radius L_c joined by a toroidal knuckle of radius r_k to the shell of radius R, the two tangent at sin φ_k = (R − r_k) / (L_c − r_k). The crown is a sphere, N = p L_c / 2; in the knuckle r₁ = r_k, r₂ = r / sin φ with r = (R − r_k) + r_k sin φ, N_φ = p r₂ / 2 and N_θ = p r₂ (1 − r₂ / (2 r_k)). At the tangent line r₂ = L_c from both sides, so N_φ is continuous but the membrane hoop force jumps from p L_c / 2 to p L_c (1 − L_c / (2 r_k)), a large compression for every standard head (R above 2 r_k). This is what membrane theory predicts when the curvature changes abruptly, and it is not a design stress: the head carries the mismatch by local bending over a zone of order √(L_c t), the real stresses are much smaller than the membrane figure though still governing, and the codes size such heads with an empirical factor (ASME VIII-1 UG-32 and Appendix 1-4, M = ¼(3 + √(L_c/r_k))). The tool shows the membrane field so that the knuckle is recognised as the critical region, and the Pressure Vessel tool carries the code rule.

Discontinuity stresses at a junction

Under pressure a cylinder of radius R and wall t expands radially by δ_c = (1 − ν/2) p R² / (E t) (the ν/2 term is the Poisson contraction from the axial end load), a hemisphere of the same radius and wall only by δ_s = (1 − ν) p R² / (2 E t), and a flat head not at all. Since they are joined, equal and opposite edge forces act between them: a radial shear H₀ and a bending moment M₀ per unit of circumference, which bend each wall in a narrow zone next to the junction. The axisymmetric bending of a cylinder is governed by D w'''' + (E t / R²) w = 0, the equation of a beam on an elastic foundation (Hetényi 1946; Timoshenko Chapter 15), whose decaying solutions are e^−βx cos βx and e^−βx sin βx with β = [3(1 − ν²) / (R² t²)]^1/4; the disturbance has essentially vanished after βx = π, that is within about 2.4 √(R t). The edge of a spherical shell behaves in the same way in Geckeler's approximation, with the parameter λ = [3(1 − ν²) R_s² / t_h²]^1/4 and the angle φ from the edge in place of βx; a flat head is a circular plate, whose edge rotation under the pressure, p R³ / (8 (1 + ν) D_f) for a simple support, and under an edge moment, M₀ R / ((1 + ν) D_f), are the classical results of Timoshenko Article 16.

NASA TR R-103 tabulates the edge influence coefficients of all three: for the cylinder w₀ = (β M₀ + Q₀) / (2 β³ D) and V₀ = (2 β M₀ + Q₀) / (2 β² D) (its equation 30), for the spherical segment its equation 50, for the plate its equation 56. Writing the radial displacement and the rotation of each edge as the sum of the membrane value and the edge-load terms and requiring both to be equal across the junction gives two linear equations for H₀ and M₀; the report solves them in closed form (equations 17 and 18 for the cylinder and a portion of a sphere, 20 and 21 for the cylinder and a flat head), and the tool solves the same two equations and checks the closed forms. Its sign convention is kept: H₀ is positive outward on the cylinder edge and positive toward the axis on the head, M₀ is positive when it rotates the cylinder edge outward and the upper sign of a ± pair refers to the inner surface. The stresses then follow at any distance from the junction from the report's equations 26 and 27 for the cylinder,

σ_x = p R / (2t) ± 6 [M₀ Φ(βx) + H₀ Ω(βx) / β] / t²,   σ_θ = p R / t + [2β² (R/t) Ψ(βx) ± 6ν Φ(βx) / t²] M₀ + [2β (R/t) Θ(βx) ± 6ν Ω(βx) / (β t²)] H₀,

with Θ = e^−x cos x, Φ = e^−x (cos x + sin x), Ψ = e^−x (cos x − sin x), Ω = e^−x sin x, and from equations 42 and 43 for the spherical head and the plate formulas for the flat head. The hoop stress of the cylinder changes by E w / R, the hoop strain of the local radial displacement, plus ν times the meridional bending stress.

Cylinder with hemispherical heads of the same thickness. The two shells have the same edge stiffness, so the rotations match with no moment and the shear is simply the displacement difference p R² / (2 E t) divided by the sum of the two edge flexibilities: M₀ = 0 and H₀ = −p / (8β), the cylinder edge being pulled inward (NASA TN D-1565, equations B1 and B2). The hoop stress at the junction drops to 0.75 p R / t, the mean of the two membrane values; the bending stress peaks at βx = π/4 at 0.146 p R / t (for ν = 0.3, from the moment (H₀/β) e^−π/4 sin(π/4)), and the hoop stress overshoots the membrane value by 3 % where the decaying wave reverses, at βx ≈ 1.86. The maximum von Mises stress is 1.034 times the cylinder-membrane value (√3/2) p R / t for any R/t, which is what Morgan and Bizon measured on two toriconical vessels (TN D-1565: "approximately 3½ percent higher"). The hemisphere is the mildest closure. A thicker head is stiffer and expands less, so the shear grows and a moment appears; a head shallower than a hemisphere (crown radius R_s above R) meets the cylinder at an angle, and the horizontal component (p R_s / 2) cos α of its membrane force must be reacted at the junction, which the report's equations include but which in practice calls for a ring.

Cylinder with flat heads. A flat head is a stiff restraint: its radial displacement under H₀ is only (1 − ν) R H₀ / (E t_h) and its rotation is set by the plate bending, so the cylinder edge is nearly held and large bending stresses arise at the junction. In the limit of a rigid head (t_h → ∞) the cylinder edge is built in against displacement and rotation and H₀ = −(1 − ν/2) p / β, M₀ = (1 − ν/2) p / (2 β²), a bending stress 6 M₀ / t² = 3 (1 − ν/2) / √(3(1 − ν²)) × p R / t = 1.54 p R / t for ν = 0.3 (Timoshenko Article 116 gives 1.82 p R / t for lateral pressure alone, without the end-load term); an elastic plate relaxes this. The plate itself carries −H₀ / t_h in-plane and the bending ± 6 M_r / t_h², where M_r is the simply supported plate moment (3 + ν) p (R² − ρ²) / 16 reduced by the edge moment M₀; at the centre the stress is −H₀/t_h ± 6 M₀/t_h² ∓ 3 (3 + ν) R² p / (8 t_h²) (TR R-103 equation 53). The thinner the plate the closer the junction is to a simple support and the larger the plate centre stress; the thicker, the closer to the rigid-head limit and the larger the cylinder's bending stress. Either way a flat head stresses the junction far more than a hemisphere, which is why flat heads are used only on small or low-pressure vessels.

Equivalent stress and margins of safety

At every station, and on both surfaces of the junction cases, the tool forms the von Mises stress √(σ₁² − σ₁σ₂ + σ₂²) or the Tresca stress max(|σ₁|, |σ₂|, |σ₁ − σ₂|) of the meridional and hoop stresses, neglecting the radial stress, which is of the order of p and small against p R / t in a thin shell, and reports the largest value and where it lies.

Load level and factors (site convention). The Load Level radio says what the entered pressure, axial load and liquid head are: limit, the maximum expected operating pressure (MEOP) and the loads acting with it, or ultimate, the burst pressure MEOP × FS. The loads enter the analysis as entered, since the problem is linear, and the two factors act in the margins. Strength margins compare the stress with F_tu at ultimate load and are labelled ULT: with limit loads entered the stress is multiplied by FS (default 2.0, the burst factor of MIL-STD-1522A and AIAA S-080 for metallic vessels; high-altitude and other vehicles use more), with ultimate loads entered it is taken as entered, never multiplied again. Yield margins compare the stress with F_ty (tension) or F_cy (compression) at limit load × FS_yld and are labelled LIM: with limit loads entered the stress is multiplied by FS_yld (default 1.5, the proof-pressure factor on MEOP at which the vessel must show no yielding; 1.15 is the airframe factor for no detrimental permanent set), with ultimate loads entered it is first divided by FS to recover the limit stress and then multiplied by FS_yld. FS therefore stays active whatever the radio says, because the yield margins need it. Each margin box names the level of its allowable, not of the entered loads.

Which stresses are checked. Two sets of boxes: the membrane field, and in the junction cases the total surface stress (membrane plus discontinuity) on both surfaces along the cylinder and the head. In each set the tension boxes (F_tu at ULT, F_ty at LIM) take the equivalent stress at the point of largest equivalent stress whose larger principal stress is tensile, the equivalent stress being the measure of yield and rupture under the biaxial membrane state; the compression boxes take the largest compressive principal stress, meridional or hoop, against F_cy at both levels, F_cy standing for the compression allowable of a stable shell element. Compressive stresses arise under internal pressure in the hoop direction of an ellipsoidal head with a/b above √2 and of a torispherical knuckle, and on one surface at a flat-head junction; under external pressure every stress is compressive, and the tool says so and names the buckling tools (Cylindrical Shell Buckling, Sphere and Cone Buckling) that normally govern such a shell. A blank allowable leaves its boxes out.

Secondary stresses. The discontinuity stresses are local and self-limiting. The pressure-vessel codes treat them as secondary stresses: ASME VIII-2 Part 5 limits the primary-plus-secondary stress range to 3 S, about twice the yield strength, a check against yield (shakedown) and never against the ultimate strength, and the aerospace practice of MIL-STD-1522A makes the burst check on the membrane stress and the proof check on the total. The junction boxes compare the total surface stress with the same allowables as the membrane stress and are therefore conservative at a junction; the membrane boxes carry the codes' primary-stress check, and the yield-allowable line of the equivalent-stress chart is F_ty brought to the entered load level.

Conventions and units

R is the mid-surface radius of the cylinder (and of the head at the junction), of the sphere, of the cone base, of the ellipsoid equator or of the torispherical head at the shell; t is the shell wall and t_h the head wall. The torus is defined by its tube radius a and centre-line radius b, the ellipsoid by its depth b, the torispherical head by its crown radius L_c and knuckle radius r_k, the cone by its semi-vertex angle. Lengths, pressures, forces, stresses and the liquid density pass through the unit dictionary (in, lb/in², lb, ksi, lb/in³ or mm, kPa, kN, MPa, kg/m³); the liquid pressure uses standard gravity 9.80665 m/s²; every computed quantity is stored in SI and shown in the chosen system; β is printed in the reciprocal of the length unit. In the charts a positive coordinate lies on the cylinder measured from the junction and a negative one on the head (the arc R_s φ for a spherical head, the distance from the edge toward the centre for a flat head).

Reading the results

For the membrane shapes the page prints the governing formulas with the substituted values at the characteristic points (crown, equator, base, inner equator, tangent line, junction), then the station with the largest equivalent stress and the margin boxes, and Figure 2 draws the meridional, hoop and equivalent stresses along the meridian. For the junction cases it prints the membrane stresses away from the junction, the free expansions of the two parts, β and the decay length π/β, the edge shear and moment, the stresses on both surfaces at the junction and their extremes along the walls (with the position of each), and Figures 2 and 3 show the stresses and the equivalent stress on both surfaces from the head through the junction along the cylinder until they return to the membrane values. A hoop stress at the junction below the membrane value and a hoop overshoot a little further along are the signatures of a hemispherical head; a large meridional bending stress at the junction is the signature of a flat or thick head.

Limits

The method is for thin walls (radius-to-thickness ratio of 10 or more, below which the tool refuses and the thick-wall Lamé solution of the Pressure Vessel tool applies) of uniform thickness, with the junctions far enough apart (several π/β) not to interact, which TR R-103 assumes throughout. The sphere, torus, ellipsoid and torispherical cases are membrane theory alone: their junctions with a shell and the bending zones of the knuckles are not solved, and the torispherical knuckle values in particular are membrane figures, not design stresses. The flat-head solution neglects the offset between the plate mid-plane and the cylinder edge, the plate's shear deformation and any fillet; the spherical-head solution uses Geckeler's approximation, accurate for λ above about 5 and away from the pole. An axial load enters the membrane stress of the cylinder and cone only and is not carried into the junction analysis; the liquid head applies to the cylinder and cone alone. Buckling under external pressure or axial load, nozzles, supports, local loads, ring stiffeners and thermal gradients are outside the tool.