Donnell's Equations for Cylindrical Shells

• Donnell’s equations are a reduced set of relations for thin cylindrical shells, capturing axial, bending, torsional, and localized load effects under the thin-shell limit.
• The framework employs separation of variables, Fourier decomposition, and a beam-on-foundation analogy to analyze both axisymmetric and non-axisymmetric deformation modes.
• These equations serve as a canonical tool for assessing buckling loads, post-buckling behavior, and nonlinear phenomena in engineering applications such as offshore pipelines and civil structures.

Donnell’s equations provide the foundational shell-theoretic framework for the analysis of thin, isotropic, homogeneous cylindrical shells under a variety of loading scenarios, including axial compression, bending, torsion, and localized forces. The equations arise as an asymptotic reduction of three-dimensional linear elasticity in the regime of small shell thickness-to-radius ratio ($t/R \ll 1$) and moderate rotations. Modern studies rigorously justify the equations’ validity at leading order, confirming their status as the canonical tool for predicting stability, deformation, and nonlinear response in cylindrical structures relevant to civil, mechanical, and offshore engineering (Mascoloa et al., 12 Dec 2025, Grabovsky et al., 2014, Lu et al., 21 Jan 2025).

1. Formulation of Donnell Shell Equations

Donnell-type shell theory considers the mid-surface of a cylindrical shell of radius $R$ and thickness $t\ll R$, parameterized by coordinates $(x,\theta)$. The midsurface displacement field is split into axial ($u(x,\theta)$), circumferential ($v(x,\theta)$), and radial ($w(x,\theta)$) components.

Under small-strain, moderate-rotation assumptions, the equilibrium of forces and moments, together with the strain-displacement and constitutive relations, yield the following coupled system for the in-plane stress resultants $(N_x, N_\theta, N_{x\theta})$, bending moment resultants $(M_x, M_\theta, M_{x\theta})$, and Airy stress function $\Phi(x,\theta)$: $$\begin{aligned} D \nabla^4 w + \frac{1}{R}\frac{\partial^2 \Phi}{\partial\theta^2} + q(x,\theta) &= 0,\ \nabla^4\Phi + \frac{E t}{R}\frac{\partial^2 w}{\partial\theta^2} &= 0, \end{aligned}$$ where $D = \frac{E t^3}{12(1-\nu^2)}$ is the flexural rigidity, and $q(x,\theta)$ is the imposed external radial load (Mascoloa et al., 12 Dec 2025). The differential operators are defined as $$\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{1}{R^2}\frac{\partial^2}{\partial\theta^2}$$, $\nabla^4 = (\nabla^2)^2$.

This formulation serves as the starting point for linear and weakly nonlinear stability analyses, localized deformation, and the characterization of global response modes in structural applications (Grabovsky et al., 2014, Lu et al., 21 Jan 2025).

2. Reduction, Decomposition, and Analytic Methods

The Donnell equations admit further specialization via separation of variables and Fourier decomposition, beneficial in scenarios with circumferential or axial symmetry, or when loads are concentrated at isolated regions:

• Fourier Decomposition: The imposed load or displacement is expanded in circumferential Fourier modes, with $n=0$ (axisymmetric) and higher $n$ representing multi-lobed deformations. For instance, an applied pair of radial forces can be represented by,

$$p(\theta) = \frac{F}{\pi R} + \frac{2F}{\pi R} \sum_{n=2,4,6,\dots} \cos(n\theta),$$

where the $n=0$ term models a uniform component and higher $n$ terms capture ovalisation and other non-axisymmetric effects (Mascoloa et al., 12 Dec 2025).

• Beam-on-Foundation Analogy: For long-wavelength modes, each Fourier component is mapped to an effective Euler–Bernoulli beam on a Winkler elastic foundation. The $n$th mode exhibits a bending stiffness and foundation modulus—explicitly,

$$B_n = \frac{\pi E t D^3}{8(1-\nu^2)n^4}, \quad m_n = \frac{8\pi\delta(n^2-1)^2}{D^3},$$

where $\delta$ is a function of shell parameters. This analytic construct enables closed-form prediction of local shell deflections and strains under concentrated loads (Mascoloa et al., 12 Dec 2025).

3. Buckling and Stability Analysis

Classical stability questions—such as the calculation of the critical buckling load for axial compression—are addressed by linearizing the Donnell equations about the trivial state and seeking eigenmode solutions: $w(x,\theta) \propto e^{i(kx + n\theta)}.$ The critical buckling load per unit circumference is given by

$$N_{\text{cr}}(k,n) = D\, \frac{(k^2 + n^2/R^2)^2}{k^2},$$

with $k$ (axial) and $n$ (circumferential) wavenumbers. In the thin-shell limit, minimization leads to the Koiter circle condition: $$\left(\frac{nR}{m}\right)^2 + (m \frac{h}{R})^2 = 2(1-\nu),$$ yielding the canonical scaling $N_{\text{cr}} \sim E h^2 / R (3(1-\nu^2))^{-1/2}$ as $h/R\to 0$ (Grabovsky et al., 2014).

When subjected to combined axial force and torque, the Donnell framework admits further generalization: the critical load locus is determined via eigenvalue analysis within a Galerkin method, with the interplay between torsional and compressive effects modifying buckling wavenumber selection, mode shapes, and post-buckling response (Lu et al., 21 Jan 2025).

4. Nonlinear Phenomena and the von Kármán/Brazier Effects

Beyond the linearized regime, Donnell’s equations facilitate the analysis of nonlinear deformation effects, such as ovalisation under bending (Brazier effect) and its suppression via structural stiffening (the reversed von Kármán/Brazier effect):

• The nonlinear pressure resulting from imposed shell curvature is quantified by (Mascoloa et al., 12 Dec 2025)

$q(\chi) \approx \chi^2 E t \frac{D}{2},$

producing a change in the vertical diameter. When ovalisation is artificially constrained (e.g., by discrete ring stiffeners), the system develops an equivalent pair of forces $P_{\mathrm{eq}}$ that restores the original diameter:

$$P_{\mathrm{eq}} = E R^4 \chi^2 \left(\frac{t}{R}\right)^{1/2} \left[\frac{1}{1.244} + \frac{1}{4.774}\frac{t/R}{(1-\nu^2)^{3/4}}\right].$$

This additional membrane force leads to asymmetry in compressive and tensile stresses under bending. Comparison with finite element simulations and experimental data confirms the formula’s accuracy to within 10–15% in relevant parameter ranges.

5. Key Assumptions and Validity Regimes

The Donnell equations rest on several critical hypotheses:

• Thin-shell limit $t/R\ll 1$ (formally justified via rigorous asymptotics from 3D elasticity (Grabovsky et al., 2014))
• Small strains and moderate rotations
• Planar and elliptical cross-sectional assumptions for nonlinear effects
• Superposition of uniform and Fourier-mode solutions, valid for linear and weakly nonlinear regimes
• Neglect of higher-order (“short wave” or inextensional) modes in beam-on-foundation analogies
• Boundary conditions consistent with decay at infinity or prescribed constraints at supports/stiffeners

Within these constraints, the Donnell formulation is validated both theoretically and against finite element analysis and experimental results for buckling loads, modes, and localized deformation phenomena (Mascoloa et al., 12 Dec 2025, Grabovsky et al., 2014, Lu et al., 21 Jan 2025).

6. Applications in Engineering Design

Donnell’s equations are indispensable for the analytic prediction and design of cylindrical shell components in offshore pipelines, civil infrastructure, and energy systems. Specifically:

• Pipeline Integrity and Stiffener Design: The reversed von Kármán/Brazier formula allows for the optimal placement and sizing of ring stiffeners, enabling designers to tune ovalisation and maximize compressive strength in pipelines subjected to environmental loads. Calculations based on Donnell’s equations provide close agreement with both experimental strain asymmetries and FEA predictions for real-world geometries (Mascoloa et al., 12 Dec 2025).
• Buckling Under Combined Loading: The calculation of critical loads and mode shapes in scenarios of combined axial and torsional loading is informed by Donnell-based linear stability analysis, including parametric dependence on shell dimensions, material properties, and loading ratios (Lu et al., 21 Jan 2025).

Numerical example: For a $D=608.6$ mm, $t=18.9$ mm offshore pipeline under a 1 MN four-point bending load, the predicted imposed-ovalisation strain asymmetry is 1.42 (theory) versus 1.28 (experiment); the prevented-ovalisation case yields $P_{\rm eq}=27.48$ kN and a diameter shortening of 0.662 mm, matching ANSYS simulations (Mascoloa et al., 12 Dec 2025).

7. Sensitivity, Limitations, and Research Directions

The theoretical analysis demonstrates that Donnell’s equations give an exact leading-order representation for buckling and deformation in the thin-shell regime, with all corrections entering at higher orders in $t/R$ (Grabovsky et al., 2014). However, cylindrical shells exhibit extreme imperfection sensitivity; real-world critical loads can be substantially reduced by small geometric or loading imperfections, a phenomenon rooted in the flatness of the post-buckling energy landscape (the Koiter mountain-pass geometry). This sensitivity is not due to missing terms in Donnell’s model but arises from structural modal degeneracy and interaction effects.

Ongoing research extends Donnell’s framework to multi-physics environments, localization under dynamic and cyclic loading, and further nonlinear effects, particularly for high-curvature, multi-stiffened systems, and combined load cases (Lu et al., 21 Jan 2025).

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References:

• (Mascoloa et al., 12 Dec 2025) The reversed "von Karman/Brazier effect" and a compendiary analytical solution.
• (Grabovsky et al., 2014) Rigorous derivation of the formula for the buckling load in axially compressed circular cylindrical shells.
• (Lu et al., 21 Jan 2025) Buckling and post-buckling of cylindrical shells under combined torsional and axial loads.