Shell Buckling Under Combined Loads
Interaction of axial compression, external pressure, bending, shear and torsion on cylindrical shells — interaction equations, mode switching and conservative bounding.
Combined loading on shells
Real cylindrical shells rarely carry a single load in isolation. A launch vehicle tank or aircraft fuselage carries axial compression (from vehicle weight and inertia), bending (from manoeuvre or gust loads), external pressure (aerodynamic or vacuum), and shear (from transverse loads) simultaneously. Each load component contributes to the destabilising membrane stress state, and the components interact — the combined loading buckles at a lower load than any single component alone. The interaction must be assessed with an interaction equation that accounts for the combined effect, because checking each load independently against its individual buckling load is unconservative.
Axial compression and external pressure
The most common combined loading for pressure vessels and launch vehicles is axial compression plus external pressure. Axial compression produces the diamond-pattern mode; external pressure produces the ovalisation mode. The two modes are different, and the interaction is approximately linear to mildly nonlinear. A representative interaction relationship is:
N / N_cr + (p / p_cr)^a <= 1.0
where:
N = applied axial compressive load per
unit circumference [N/mm]
N_cr = axial buckling load per unit
circumference (alone) [N/mm]
p = applied external pressure [MPa]
p_cr = external pressure buckling
pressure (alone) [MPa]
a = interaction exponent, typically 1.0
to 2.0 depending on the geometry and
the governing modes
A linear interaction (a = 1) is conservative;
a = 2 is less conservative when the two
modes are very different.Bending and axial compression
A cylinder under bending has a compression side and a tension side. The compression side experiences a membrane compressive stress that is analogous to axial compression, and the shell can buckle locally on the compression side. The bending buckling moment is related to the axial buckling load, but the stress distribution is non-uniform (linear around the circumference) rather than uniform. The bending buckling moment is typically 20–40% higher than the moment computed from the uniform axial buckling stress times the section modulus, because only part of the circumference is at the peak compressive stress and the rest is less highly stressed. The interaction between axial compression and bending is approximately linear:
Shear and torsion
A cylinder under transverse shear or torsion develops membrane shear stress. Shear buckling of a cylinder produces a diagonal wave pattern, similar to shear buckling of a flat plate but wrapped around the curvature. The torsional buckling of a cylinder is a classic problem with a known solution for the critical shear stress. Combined shear with axial compression or pressure reduces the buckling resistance — the shear stress adds to the compressive principal stress and promotes earlier buckling. The interaction is typically nonlinear (the shear term is squared), reflecting the different buckling modes.
Interaction equation summary
A general interaction relationship for combined axial compression, pressure, bending and shear takes the form of a sum of load ratios, with the shear and pressure terms typically squared:
N/N_cr + (M/M_cr)^b + (p/p_cr)^a + (T/T_cr)^c <= 1.0 where: M = applied bending moment M_cr = bending buckling moment (alone) T = applied shear or torque T_cr = shear/torsional buckling load (alone) b, a, c = interaction exponents The exponents (typically 1 to 2) depend on the mode shapes and are calibrated against test data or nonlinear FEA. Conservative practice uses linear interaction (all exponents = 1) where data is lacking.
Mode switching
Under combined loading, the governing buckling mode can switch from one type to another as the load ratio changes. For example, a cylinder under dominant axial compression buckles in a diamond pattern; adding external pressure can switch the governing mode to an ovalisation-dominated pattern. The interaction equation with fixed exponents may not capture this switch accurately. For critical applications, a nonlinear FEA with the combined load vector and an appropriate imperfection is the most reliable way to establish the collapse load across the full range of load ratios.
Conservative bounding
When detailed interaction data or nonlinear analysis is not available, conservative bounding is used. The simplest bound is a linear interaction (all exponents = 1), which is conservative for most combined-load cases. A further conservative bound is to require that each individual load ratio is below a reduced allowable (e.g. 0.8 of the individual buckling load), ensuring margin in each direction. The conservative bounding is appropriate for preliminary design; the final design should be verified with a nonlinear analysis that captures the interaction for the specific geometry and load combination.
Checking each load component against its individual buckling allowable and taking the minimum is NOT a valid interaction check — it ignores the destabilising effect of the combined stress state. A combined-load interaction equation or a nonlinear analysis is required.
Interaction surfaces rather than independent utilisation ratios
When several destabilising loads act together, the true stability boundary is an interaction surface in a multidimensional load space. Independent checks of axial compression, pressure, shear or bending can miss a combination that is safe in each single-load calculation but unstable in combination. Simple interaction equations are useful engineering approximations, but their exponent and shape depend on shell geometry, mode family and the assumptions behind the source method. For unusual geometries or high utilisation, the interaction is better obtained directly by nonlinear analysis: hold the required load proportions, scale the combined load vector and determine the collapse point. Repeating this for several proportions maps the local interaction surface and reveals whether a simple linear combination is conservative.
Load sequence and non-proportional combined loading
Combined shell loads are not always applied proportionally. A vessel may first be pressurised, then experience axial load and bending; a launch structure may carry preload before lateral acceleration; an evacuated shell may see external pressure before thermal distortion develops. The initial load can stiffen or soften the shell, change contact or boundary conditions, and create residual deformation before the later load is applied. Linear eigenvalue analyses based on one reference load vector cannot represent all such sequence effects. A nonlinear assessment should reproduce the physical loading sequence where it matters and should distinguish between a proportional collapse envelope and a path-dependent operating scenario. This is particularly important when one load introduces membrane tension that stabilises another compressive mode.
Mode switching across the interaction envelope
The controlling instability mode can change as the load ratio changes. Predominantly axial compression may produce a short-wave diamond pattern, pressure may favour ovalisation, torsion may create diagonal waves and bending may localise compression on one side of the shell. Near a mode-switching region, several eigenvalues can be close and the nonlinear collapse load can be unusually sensitive to imperfections. Verification should therefore compare deformation modes at multiple points along the interaction envelope, not just the numerical collapse factors. A smooth interaction curve accompanied by an abrupt mode change is physically possible, but it should be understood and documented because it often marks a region where small manufacturing or boundary-condition variations can alter the governing failure mechanism.