Sphere and Cone Buckling: Method

Sphere and Cone Buckling: Method, Conventions and Limits

What the tool does

The tool estimates the buckling strength of two thin unstiffened isotropic shells of revolution that the cylinder tool does not cover: a spherical shell (complete, or a cap cut off by a parallel) under uniform external pressure and, for a cap, a concentrated inward load at the apex; and a truncated circular cone under axial compression, bending, uniform hydrostatic external pressure and torsion, alone or combined, with the increase of the compression and bending strengths produced by internal pressure. It follows the two NASA Space Vehicle Design Criteria monographs that are still the basis of preliminary shell design: NASA SP-8032 (1969), Buckling of Thin-Walled Doubly Curved Shells, Section 4.2.1 for the sphere, and NASA SP-8019 (1968), Buckling of Thin-Walled Truncated Cones, Section 4.2 for the cone. Both monographs follow the same philosophy as SP-8007 for cylinders: a small-deflection (classical) buckling load multiplied by a lower-bound correlation factor γ read from the test data, a plasticity correction when the stress is high, and a straight-line interaction of the load ratios. The results page prints every equation with its monograph number and the substituted values, the design allowables, the applied stresses, the interaction sum and the margin of safety, and draws the design curves with the shell's point on them.

Why the correlation factors are so small

The axially compressed cylinder, the externally pressurized sphere and the compressed cone share a steeply falling post-buckling path: once the wall begins to buckle the load it can carry drops at once, so geometric imperfections of a fraction of the wall thickness, together with non-uniform loading and edge effects, trigger collapse far below the bifurcation load of linear theory (Koiter 1945; Budiansky and Hutchinson). The sphere is the worst case of all, because every direction on it is equally curved and every buckle shape is equally critical: tests on complete spheres and deep caps of the 1930s to 1960s gave a quarter to two thirds of the classical pressure, and their scatter was wide. Because the imperfections of a real structure are not known at the design stage, the monographs bound the test population from below with a correlation factor and ask for tests whenever the design relies on anything less conservative (SP-8032 Section 3.2, SP-8019 Section 3.2).

Spherical shells (SP-8032)

Classical pressure. Linear theory gives the complete sphere the buckling pressure p_cl = 2 E (t/R)² / √(3(1 − ν²)) (Zoelly 1915; Timoshenko and Gere Art. 11.13 print 1.21 E (t/R)² for ν = 0.3), SP-8032 eq. (2). At that pressure the membrane stress p R / (2t) equals E t / (R √(3(1 − ν²))), the classical stress of an axially compressed cylinder of the same radius: the local buckle of a sphere is the same phenomenon.

The geometry parameter λ. Shallow-shell theory (Weinitschke 1960, Budiansky 1960, Huang 1964) collapses the clamped cap's behaviour onto one parameter, λ = [12(1 − ν²)]^¼ √(R/t) · 2 sin(φ/2), eq. (3), where φ is half the included angle of the cap; λ is proportional to the base radius divided by the bending boundary-layer length √(R t), so a small λ means an edge-dominated, plate-like cap and a large λ a cap with many buckle wavelengths that behaves as a complete sphere. The tool accepts either the half-angle or λ itself, and recovers the other.

Design pressure. SP-8032 bounds the clamped-cap data (collected by Wang 1967 and others) from below with p_cr / p_cl = 0.14 + 3.2 / λ² for λ > 2, eq. (4) and the report's Figure 2, and states that although λ comes from shallow-shell analysis the curve may be applied to deep shells as well. The curve reaches 1.0 near λ = 1.9 (so shallow a cap does not snap through; the tool caps the factor at 1.0 and flags it) and tends to 0.14, the value the report assigns to complete spheres and thin oblate spheroids in eq. (8). The tool uses 0.14 for the complete sphere, which has no edge and no λ, and prints the λ of a full sphere for reference only. The report gives no separate pressure curve for caps with unrestrained edges; the tool applies eq. (4) to them with a note.

Concentrated load at the apex. A cap loaded inward at its apex snaps through only when it is deep enough (SP-8032 4.2.1.2, after Fitch 1968, Ashwell 1960, Mescall 1965, Penning and Thurston). With edges free to rotate and to expand normal to the axis, buckling does not occur below λ ≈ 3.8; for 4 ≤ λ ≤ 18 the lower bound to the collapse data is P_cr R / (E t³) = λ² / 24, eq. (5). With clamped edges theory gives no buckling below λ ≈ 8, axisymmetric snap-through for 8 < λ < 9 and asymmetric buckling beyond, the shell still carrying increasing load; the report's Figure 4 (Fitch's theoretical loads, which tests of caps loaded over a small area show to be conservative estimates of collapse) gives P_cr R / (2π D) against λ: 14 at λ = 8, a peak of 17.6 near 9.3, a plateau at 11.9 from 12.5 to 13.5 and a slow fall to 10.9 at 25. The tool digitized that curve (the CSV and the trace overlay are filed with the sources) and interpolates it; beyond λ = 25 it holds the last value. When the load is applied over a large area local buckling may occur at a lower load (report). For pressure and apex load together the report recommends the straight line P / P_cr + p / p_cr = 1, eq. (6).

Plasticity. SP-8032 gives no plasticity factor for spheres and asks for tests where plasticity effects occur. The tool applies, with a note, the factor the companion monographs use for compressed shells, η = √(E_sec E_tan) / E (SP-8019 eq. 3, SP-8007 eq. 7), with the Ramberg-Osgood moduli at the membrane stress p R / (2t), solving the fixed point σ_cr = η(σ_cr) σ_e by bisection, and caps the membrane stress at F_cy; with F_cy entered and no exponent only the cutoff is applied, and with neither the results are purely elastic.

Truncated cones (SP-8019)

Geometry. A cone of semivertex angle α has at a parallel of radius r the radius of curvature ρ = r / cos α normal to the wall; the monograph refers the allowables to ρ₁ = r₁ / cos α at the small end, where the membrane stresses are highest, and to the average radius of curvature ρ̄ = (r₁ + r₂) / (2 cos α) with the slant length L = (r₂ − r₁) / sin α for the equivalent cylinder. The axial length ℓ = (r₂ − r₁) / tan α enters the torsion equation.

Axial compression. Seide (1956) showed that the axisymmetric buckling load of a long cone is P_cr = 2π E t² cos²α / √(3(1 − ν²)), eq. (1): the classical cylinder stress at the local radius of curvature ρ₁, reached by the meridional membrane stress σ = P / (2π r₁ t cos α) at the small end (eq. 4). The tests of Weingarten, Morgan and Seide (1965) and of Hausrath and Dittoe show no effect of the cone geometry on the ratio of test to theory, so SP-8019 takes the constant lower bound γ = 0.33 for 10° < α < 75° (eq. 2); for α < 10° it allows the equivalent cylinder's factor, which the tool takes as SP-8007 eq. (5) at ρ̄/t, and above 75° it requires tests (the tool keeps 0.33 and warns).

Bending. No theory existed; SP-8019 reasons that the cone buckles when the maximum compressive stress at or near the small end reaches the critical stress of a cylinder with the same wall and the same local radius of curvature, which gives M_cr = π E t² r₁ cos²α / √(3(1 − ν²)), eq. (5), the stress being M / (π r₁² t cos α). The tests of Seide, Weingarten and Morgan (1960) confirm the parameter and give the lower bound γ = 0.41 for 10° < α < 60° (eq. 6), higher than in compression because only the most compressed part of the wall is near its critical stress, as for cylinders.

Hydrostatic pressure. Seide (1960) found that a cone buckling into several circumferential waves has p_cr = 0.92 E γ / [(L/ρ̄)(ρ̄/t)^(5/2)], eq. (7), which with γ = 1 is the moderate-length hydrostatic pressure of the equivalent cylinder of radius ρ̄, length L and the same wall (Batdorf, NACA Report 874; the SP-8007 straight line 0.855 E √γ / [(1 − ν²)^(3/4) (r/t)^(5/2) (L/r)] gives 0.918 at ν = 0.3, which the report rounds to 0.92). The theoretical γ is above 1 and depends on the ratio of the end radii; the tests of Weingarten and Seide (1965) and Singer and Eckstein (1963) scatter widely and SP-8019 takes γ = 0.75 as the lower bound (eq. 8). The hoop stress at the large end, p ρ₂ / t (eq. 10), is the one the plasticity factor is read at. The tool reports the Batdorf parameter of the equivalent cylinder and notes when it is below 100, where the moderate-length form is conservative.

Torsion. Seide (1962) gives T_cr = 52.8 γ D (t/ℓ)^(1/2) (r/t)^(5/4), eq. (11), with ℓ the axial length and r an equivalent radius r = r₂ cos α {1 + [½(1 + r₂/r₁)]^(1/2) − [½(1 + r₂/r₁)]^(−1/2)} r₁/r₂, eq. (11a), whose bracketed function the report plots against the taper ratio 1 − r₁/r₂ in its Figure 1 (1.0 for a cylinder, 0.70 at a taper of 0.5, 0.43 at 0.8). For a cylinder the equation is the Donnell moderate-length torsion line of SP-8007 eq. (26) within 3 % (52.8 against 51.3 implied by the 0.747 coefficient). The recommended factor is γ = 0.67 (eq. 12), the cylinder value; the shear stress at the small end is T / (2π r₁² t) (eq. 14).

Internal pressure. Internal pressure helps a compressed cone in two ways (SP-8019 4.2.5.1, after Seide 1962): the pressurization load π r₁² p on the small end must be overcome before the wall is compressed, and the hoop tension suppresses the imperfection-driven buckles, expressed as an increase Δγ of the buckling-stress coefficient read from the report's Figure 2 against (p/E)(r₁/(t cos α))², the curve of the equivalent cylinder. The report keeps only part of the theoretical increase and writes P_press = [γ/√(3(1 − ν²)) + Δγ] 2π E t² cos²α + π r₁² p (eq. 15) and M_press = [γ_b/√(3(1 − ν²)) + Δγ] π E r₁ (t cos α)² + p π r₁³ / 2 (eq. 16), with γ = 0.33 and 0.41. The tool digitized Figure 2 (0.023 at 0.015, 0.090 at 0.1, 0.215 at 1.0, a plateau of 0.246 near 4 to 5; the cylinder tool's digitization of the SP-8007 Figure 6 print agrees within 2 to 7 %) and interpolates it in log-log; internal pressure is never multiplied by the ultimate factor (Section 3.2).

Plasticity. SP-8019 has no cone data in the yield range and borrows the cylinder factors as a conservative estimate: η = √(E_sec E_tan) / E at the small-end membrane stress for compression and bending (eq. 3-4), η = (E_sec/E)(E_tan/E_sec)^(1/4)(¼ + ¾ E_tan/E_sec)^(1/2) at the large-end hoop stress for hydrostatic pressure (eq. 9-10), and E_sec read at twice the critical shear stress for torsion (eq. 13-14). The moduli come from the Ramberg-Osgood curve ε = σ/E + 0.002 (σ/F_cy)ⁿ with the MIL-HDBK-5J compressive yield strength and exponent (a saved material fills both); the critical stress is the fixed point σ_cr = η(σ_cr) σ_e, and every allowable is also capped at F_cy (F_cy/√3 in shear).

Combined loads. SP-8019 recommends straight-line interaction for every pair it treats: R_c + R_b = 1 (eq. 17, for unpressurized and pressurized cones, the pressurized allowables of eq. 15-16 in the ratios), R_c + R_p = 1 (eq. 19; theory deviates slightly from the line, the few tests support it), R_t + R_p = 1 (eq. 21, a lower bound to scattered data) and R_t + R_c = 1 (eq. 23; the mean test curve is parabolic, the line is the lower bound). The tool sums every ratio present, R_tot = R_c + R_b + R_p + R_t, and reports M.S. = 1/R_tot − 1, the common factor by which all applied loads may grow before buckling. The report warns that with test values of P_cr and M_cr in place of the lower bounds the straight line may no longer be conservative, and that under external pressure buckling and collapse of Mylar cones differed while metal cones would collapse sooner because of plasticity.

Conventions

Sphere: R is the radius of the middle surface, t the wall thickness, φ half the included angle of the cap (90° is a hemisphere), p a positive external pressure, P_a an inward apex load; the cap edge is clamped or unrestrained (used for the apex load). Cone: r₁ and r₂ are the middle-surface radii at the small and large ends measured normal to the axis, α the semivertex angle in degrees, t the wall; P is positive in compression (a tensile load is entered negative and not credited), M and T enter as magnitudes, the pressure is a positive magnitude whose sense (external hydrostatic or internal) is chosen in the select. Angles are plain degree numbers in both unit systems; lengths, forces, moments, stresses and pressures go through the unit dictionary (USCS in, lb, lb-in, ksi, psi; SI mm, kN, N-m, MPa, kPa) and every computed quantity is stored in SI and displayed in the chosen system. Loads entered at limit level are multiplied by the ultimate factor before the comparison; internal pressure is not.

Reading the results

Each load section prints the correlation factor with its source, the classical value, the elastic design stress, the plasticity factor when it applies and the design allowable in bold, followed by the buckling load, moment, pressure or torque. The summary table lists the design allowable, the applied ultimate value and the ratio of every load; the margin box carries the interaction margin. For a sphere, Figure 2 is the SP-8032 design curve against λ with the shell's point: a point far to the right means a sphere-like shell at the 0.14 floor, a point near λ = 2 to 5 a cap whose edge restraint raises the pressure. For a cone, Figure 2 shows how the compression and bending allowables fall with the semivertex angle (γ cos²α) for the chosen wall, with the step at 10° where the report changes from the equivalent-cylinder factor to its constants, and Figure 3 is the report's torsion radius chart.

Limits

The methods are for unstiffened isotropic walls of uniform thickness and simply supported or clamped edges; the tool refuses R/t or r₁/t below 5. The correlation factors are lower bounds to the test populations of the 1930s to 1960s; the cone factors are stated for 10° to 75° (compression) and 10° to 60° (bending), and SP-8019 requires tests beyond those angles and for cutouts, elastic edge supports or minimum-weight designs. The sphere curve of eq. (4) is stated for λ > 2 and for clamped edges; ring-supported or elastically restrained edges lower the pressure (Bushnell 1967, Wang 1966) and need the report's references or a test. Not offered: ellipsoidal and torispherical heads (the oblate spheroid takes the sphere of radius B²/A with the same 0.14; prolate spheroids need SP-8032 Figures 6a-6b with a 0.75 factor), toroidal shells and segments, orthotropic, ring-stiffened and sandwich shells, cutouts, thermal stresses, combined internal pressure with torsion or external pressure, the collapse (as against buckling) of cones under external pressure, and cylinders, for which the Cylindrical Shell Buckling tool applies the SP-8007 curves directly. The analysis-based knockdown factors of the 2020 revision of SP-8007 have no counterpart for spheres and cones; a test-substantiated factor may be entered in the knockdown select.