Cylindrical Shell Buckling Under External Pressure
External-pressure and vacuum buckling of cylinders — ovalisation, the classical pressure buckling formula, length and end-cap effects, and imperfection sensitivity.
External pressure loading
A cylindrical shell under external pressure experiences a uniform inward pressure on its outer surface. The loading arises in vacuum vessels, in submarine hulls, in launch vehicle tanks under external aerodynamic pressure, and in pipes under external soil or hydrostatic pressure. The membrane stress state is circumferential compression (hoop compression) and, if the pressure acts on end caps, axial compression as well. The shell buckles when the circumferential compression reaches a critical value — the circular cross-section loses stability and deforms into a non-circular (oval or lobed) shape. This is an ovalisation buckling, distinct from the diamond-pattern axial buckling.
Ovalisation and collapse mechanism
Under external pressure, a perfect cylinder is in uniform circumferential compression. At the critical pressure, the circular shape becomes unstable and the shell preferentially deforms into a mode with n lobes around the circumference (n = 2 is simple ovalisation, n = 3 is a tri-lobed shape, and so on). The lobed shape relieves the circumferential compression by developing bending at the lobe boundaries. The collapse is typically a limit-point snap-through: the pressure reaches a maximum and then the shell snaps inward into a heavily deformed shape. The post-buckling path is unstable, and the collapse is sudden and complete — the shell can fold inwards in a progressive manner once it starts.
Classical critical pressure
For a perfect, simply supported, medium-length isotropic cylinder under uniform external pressure, the classical critical pressure for the n-lobed mode is found by minimising over the number of circumferential waves n. The result for a cylinder with end caps (so that axial compression from the pressure on the caps is included) is approximately:
p_cr = (E / (4 * (1 - nu^2))) * (t / R)^3 * (1 / (n^2 - 1 + (pi*R/(n*L))^2)^2)
* [ (n^2 - 1)^2 + ((2*pi*R/L)^2) / (n^2 - 1) ]
where:
p_cr = critical external pressure [MPa]
E = elastic modulus [MPa]
nu = Poisson ratio [-]
t = wall thickness [mm]
R = cylinder radius [mm]
L = cylinder length between end supports [mm]
n = number of circumferential lobes (integer >= 2)
The governing n is found by minimising p_cr
over n = 2, 3, 4, ...
For a long cylinder, n = 2 (ovalisation)
and the formula simplifies.Long cylinders — simple ovalisation
For a long cylinder (L large relative to R), the dominant mode is n = 2 (simple ovalisation) and the end effects are negligible. The critical pressure reduces to a form that depends on the bending stiffness per unit length and the curvature:
For a long cylinder (n = 2, L >> R): p_cr ≈ (2 * E) / (sqrt(3) * (1 - nu^2)) * (t / (2*R))^3 ≈ (E / (4 * (1 - nu^2))) * (t / R)^3 [approximate] For nu = 0.3: p_cr ≈ 0.275 * E * (t / R)^3 Note the cubic dependence on t/R — doubling the thickness increases the pressure buckling resistance by a factor of eight.
Effect of length and stiffening rings
The length of the cylinder has a strong effect on the external-pressure buckling load — unlike axial compression, where length is unimportant for medium cylinders. A longer cylinder buckles at a lower pressure because the lobe wavelength can be longer and the bending energy is lower. This is the reason pressure vessels and submarine hulls use circumferential stiffening rings: the rings subdivide the cylinder into shorter lengths, and the buckling length L is replaced by the ring spacing. Reducing the ring spacing raises the critical pressure dramatically. The ring spacing is a primary design parameter for external-pressure resistance.
Imperfection sensitivity
External-pressure buckling is imperfection-sensitive, but generally less severely than axial-compression buckling. The dominant imperfection is initial out-of-roundness (ovality) — a cylinder that is already slightly oval will ovalise further under pressure and reach the limit load at a lower pressure than a perfect cylinder. Typical design reductions account for an out-of-roundness of a few percent of the radius. Other imperfections — local dents, thickness variations, residual stresses — also reduce the collapse pressure. The design approach combines an empirical reduction factor with limits on the permitted out-of-roundness from manufacturing.
Design and analysis
Design codes for external-pressure buckling (e.g. ASME BPVC Section VIII, EN 13445, the Navy submarine hull codes) provide charts or formulas that give an allowable pressure as a fraction of the classical critical pressure, with the fraction depending on R/t, L/R and the material. The codes also specify the permitted out-of-roundness and require that stiffening rings have adequate moment of inertia to act as effective nodal lines. For non-standard geometries, a nonlinear FEA with an initial ovality imperfection is used to establish the collapse pressure. The FEA must use an arc-length or similar method to pass the limit point and capture the unstable post-buckling path.
Pressure differential, vacuum and load definition
External-pressure buckling depends on the pressure differential across the shell, not simply on the absolute external pressure. Vacuum vessels, submerged structures and evacuated ducts can therefore experience the same fundamental stability problem under very different operating environments. The analyst should establish whether internal pressure, trapped gas, hydrostatic variation or transient pressure changes alter the effective differential during the critical event. For large deformation, pressure direction and area can also change with the deformed geometry; the finite-element pressure formulation should be checked so that the load follows the intended surface behaviour. An apparently small error in load definition can be important because collapse pressure is often strongly geometry-sensitive.
Ring stiffeners, frames and compartment interaction
Ring stiffeners do more than increase local section stiffness: they subdivide a long shell into shorter effective bays and can change the governing mode from overall ovalisation to inter-ring shell buckling or to stiffener instability. Their effectiveness depends on ring bending stiffness, attachment continuity and the flexibility of the supporting frame or bulkhead. A very stiff ring can anchor the shell and raise the bay buckling pressure, while an under-designed ring may distort with the shell and provide much less restraint than assumed. Where several bays are connected, adjacent compartments can interact through the rings. The assessment should therefore check shell, ring and combined modes rather than treating the ring spacing as a purely geometric input to a closed-form equation.
Nonlinear collapse assessment under external pressure
For imperfection-sensitive cylinders, a nonlinear geometric analysis is normally the most informative way to establish collapse pressure. A practical sequence is to solve the pre-buckling pressure state, obtain representative eigenmodes, introduce measured or bounded imperfections, and then increase the differential pressure using an appropriate path-following method. The result should be checked against mesh refinement, imperfection amplitude and shape, boundary stiffness and material behaviour if yielding can occur before collapse. The reported capacity should be the governing result from a justified sensitivity envelope, not simply the highest converged pressure from one model. Deformation plots should be examined to confirm that the calculated collapse mechanism is mechanically credible and consistent with the expected shell mode.