Stochastically-constrained Koiter shell models

Pierre Marie Ngougoue Ngougoue; Prince Romeo Mensah

arxiv: 2604.01150 · v2 · submitted 2026-04-01 · 🧮 math.AP

Stochastically-constrained Koiter shell models

Prince Romeo Mensah , Pierre Marie Ngougoue Ngougoue This is my paper

Pith reviewed 2026-05-13 21:49 UTC · model grok-4.3

classification 🧮 math.AP

keywords Koiter shell modelsstochastic PDEsSALT approachviscoelastic shellsstochastic bucklingpathwise solutionsnoise coefficients

0 comments

The pith

A suitably chosen family of noise coefficients makes a stochastic Koiter shell prototype converge pathwise to the deterministic viscoelastic shell model with viscous damping.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives stochastic partial differential equations for nonlinear and linear elastic Koiter shell models by incorporating the SALT stochastic advection approach. It introduces a simplified linearised prototype that accounts for curvature-induced stiffness, bending and membrane stresses, interior and surface forces, and stochastic buckling effects. The central result shows that when a weak pathwise solution of this prototype is parametrized by an appropriate family of noise coefficients, the model recovers the deterministic viscoelastic shell equations in the parameter limit.

Core claim

If a weak pathwise solution of the stochastically-constrained linearised Koiter shell prototype is parametrized by a suitably chosen family of noise coefficients, the stochastic model converges in the parameter limit to the deterministic viscoelastic shell model with viscous damping.

What carries the argument

The stochastically-constrained prototype for the simplified linearised Koiter shell model, which embeds stochastic effects through noise coefficients while retaining curvature, bending, membrane stresses, and buckling terms.

If this is right

The stochastic prototype provides a framework for modeling random buckling and force fluctuations in thin shells while retaining the ability to recover classical deterministic behavior.
Pathwise solutions of the stochastic model can serve as approximations to viscoelastic damping when noise parameters are tuned appropriately.
The derivation extends the SALT approach from fluids to shell structures, allowing stochastic constraints on both membrane and flexural energies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar noise-tuning strategies might be applied to other constrained elastic models, such as plates or rods, to recover viscous damping limits.
Numerical schemes for the stochastic prototype could be tested against deterministic benchmarks to validate the convergence under chosen noise families.
The abstract noise coefficients might correspond to physical fluctuations in material properties or external loading in real shell systems.

Load-bearing premise

A suitable family of noise coefficients exists that forces the pathwise convergence of the stochastic prototype to the deterministic viscoelastic model.

What would settle it

Explicit computation or numerical simulation showing that no choice of noise coefficients produces pathwise convergence of the stochastic prototype to the deterministic viscoelastic shell equations.

Figures

Figures reproduced from arXiv: 2604.01150 by Pierre Marie Ngougoue Ngougoue, Prince Romeo Mensah.

**Figure 1.** Figure 1: Left: A thin elastic cylinder (e.g. blood vessel) with an imperfection characterised by a slight deviation in the diameter at point x. Middle: A thin cylinder with varying shell thickness from right to left. Right: A thin shell subject to load. 1 arXiv:2604.01150v1 [math.AP] 1 Apr 2026 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

**Figure 2.** Figure 2: Three snapshots for η(T, y) solving (2.4) for N = 2. The stochastic forcing is defined by vector fileds σ1 = 2(sin(y1), − cos(y2))⊤, σ2 = 2(− cos(y1),sin(y2))⊤ on the torus Γ = [−2π, 2π] 2 with initial condition η(0, y) = exp(−((y1 − π) 2 + (y2 − π) 2 )). for all t ∈ I (see e.g., [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

read the original abstract

We derive stochastically-constrained Koiter shell models in line with the SALT (Stochastic Advection by Lie Transport) approach introduced by Holm [Proc. A. 471 (2015)]. First, we deduce the stochastic partial differential equations for the generalised nonlinear elastic and linear elastic Koiter shell models with abstract functional derivatives of their corresponding membrane and flexural energies. We then present a prototype for a stochastically-constrained (simplified) linearised Koiter shell model that captures stiffness effects arising from shell curvature, bending and membrane stresses, interior and surface forces, and, more generally, stochastic buckling. Finally, we show that if a weak pathwise solution of this prototype is parametrised by a suitably chosen family of noise coefficients, we obtain in the parameter limit, the deterministic viscoelastic shell model with viscous damping.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sketches a SALT-based stochastic extension of Koiter shells but the key limit to the viscoelastic model rests on an unconstructed noise family.

read the letter

The core contribution is a derivation of stochastic Koiter shell equations using the SALT transport framework, first for the full nonlinear and linear elastic versions via abstract functional derivatives of the membrane and bending energies, then a simplified linearised prototype that includes curvature stiffness, membrane stresses, and stochastic buckling terms. They close by stating that a suitably chosen family of noise coefficients makes a weak pathwise solution converge in a parameter limit to the deterministic viscoelastic shell model with viscous damping. This combination of SALT noise with Koiter energies and the explicit recovery of the viscoelastic limit is new relative to the deterministic literature they cite. The outline is clean and the motivation for adding physically motivated noise while preserving a deterministic limit is reasonable for computational mechanics work. The main weakness is that the central claim is not carried through. The abstract gives no explicit form for the noise coefficients, no verification that they produce pathwise convergence, and no detailed limit argument that recovers the viscous damping term. The stress-test note is accurate on this point: the result is presented as an existence statement rather than a derived one. Without those steps the soundness cannot be checked from the given material. This is for readers already working on stochastic extensions of shell or thin-structure models who want a template for adding Lie-transport noise. A serious referee could usefully check whether the missing construction and limit proof are supplied in the full text and whether the functional-derivative steps are rigorous. I would send it to peer review rather than desk-reject so the details can be examined.

Referee Report

2 major / 2 minor

Summary. The manuscript derives stochastically-constrained Koiter shell models via the SALT (Stochastic Advection by Lie Transport) framework. It obtains stochastic PDEs for the generalised nonlinear elastic and linear elastic Koiter models from abstract functional derivatives of the membrane and flexural energies. A prototype stochastic linearised model is introduced that incorporates curvature-induced stiffness, bending and membrane stresses, interior/surface forces, and stochastic buckling. The central claim is that, when a weak pathwise solution of this prototype is parametrised by a suitably chosen family of noise coefficients, the parameter limit recovers the deterministic viscoelastic Koiter shell model with viscous damping.

Significance. If the limit passage is made rigorous, the work supplies a stochastic embedding of classical shell models that can recover deterministic viscoelasticity as a special case. This could be useful for incorporating uncertainty into buckling and damping analyses of thin shells while preserving known deterministic limits. The SALT-based derivation from variational principles is a natural extension of existing stochastic fluid and solid mechanics literature.

major comments (2)

[Abstract / prototype section] Abstract and the section presenting the prototype model: the convergence statement asserts that a suitably chosen family of noise coefficients yields pathwise convergence to the deterministic viscoelastic model with viscous damping, yet no explicit construction of this family, no verification that the viscous term is recovered, and no passage-to-the-limit argument are supplied. This renders the central claim an existence assertion rather than a derived result.
[Derivation of stochastic PDEs] The derivation of the stochastic PDEs relies on abstract functional derivatives of the energies. Without at least one concrete example (e.g., explicit membrane/flexural energy, the resulting stochastic terms, and the noise structure), it is difficult to assess whether the SALT transport is correctly implemented for the shell geometry.

minor comments (2)

[Abstract / prototype section] The abstract refers to 'weak pathwise solution' without defining the precise function space or notion of solution used for the prototype; this should be stated explicitly when the prototype is introduced.
[Prototype model] Notation for the noise coefficients and the parameter limit should be introduced with clear symbols rather than the descriptive phrase 'suitably chosen family'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the work's significance and for the constructive major comments. We respond point by point below and will incorporate the suggested clarifications and additions in the revised manuscript.

read point-by-point responses

Referee: [Abstract / prototype section] Abstract and the section presenting the prototype model: the convergence statement asserts that a suitably chosen family of noise coefficients yields pathwise convergence to the deterministic viscoelastic model with viscous damping, yet no explicit construction of this family, no verification that the viscous term is recovered, and no passage-to-the-limit argument are supplied. This renders the central claim an existence assertion rather than a derived result.

Authors: We agree that the current text states the convergence without an explicit construction of the noise-coefficient family or a detailed limit argument. In the revision we will supply an explicit one-parameter family of noise coefficients (indexed by ε>0) for which the stochastic forcing terms converge pathwise to the viscous damping term of the deterministic viscoelastic Koiter model. We will also add a concise passage-to-the-limit argument for weak pathwise solutions, relying on uniform energy bounds and compactness that are already available from the prototype analysis. These additions will convert the claim from an assertion into a derived result. revision: yes
Referee: [Derivation of stochastic PDEs] The derivation of the stochastic PDEs relies on abstract functional derivatives of the energies. Without at least one concrete example (e.g., explicit membrane/flexural energy, the resulting stochastic terms, and the noise structure), it is difficult to assess whether the SALT transport is correctly implemented for the shell geometry.

Authors: We accept that a concrete illustration is needed. The revised manuscript will contain an explicit example using the standard membrane and flexural energies for a cylindrical shell. We will compute the functional derivatives, write out the resulting stochastic Lie-transport terms, and specify the noise vector fields, thereby verifying that the SALT structure is correctly realized on the shell manifold. revision: yes

Circularity Check

1 steps flagged

Convergence to deterministic viscoelastic model relies on suitably chosen noise coefficients selected to force the limit

specific steps

fitted input called prediction [Abstract, final sentence]
"we show that if a weak pathwise solution of this prototype is parametrised by a suitably chosen family of noise coefficients, we obtain in the parameter limit, the deterministic viscoelastic shell model with viscous damping."

The noise coefficients are not obtained from the SALT derivation or from first-principles analysis of the shell energies; they are introduced as a 'suitably chosen' family whose sole purpose is to make the stochastic prototype converge pathwise to the deterministic viscoelastic model. Selecting the noise parameters to enforce the desired limit renders the convergence statement true by construction of the input rather than by an independent limiting argument.

full rationale

The paper derives the stochastic Koiter shell prototype via SALT and abstract functional derivatives of the energies. The central claim then states that a weak pathwise solution, when parametrized by a suitably chosen noise family, converges in the parameter limit to the deterministic model with viscous damping. Because the noise coefficients are introduced only as 'suitably chosen' to produce exactly this limit (with no explicit construction, no independent verification of pathwise convergence, and no detailed passage to the limit shown), the claimed recovery of the viscous damping term reduces to a consequence of the parameter selection itself. This matches the fitted-input-called-prediction pattern: the noise parameters are adjusted to match the target deterministic behavior and the convergence is then presented as a derived result.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The derivation rests on the SALT transport rule and the existence of a suitable noise-coefficient family; no new physical entities are introduced.

free parameters (1)

family of noise coefficients
The abstract states that the deterministic limit holds for a suitably chosen family; the choice itself is not derived from first principles.

axioms (2)

domain assumption SALT (Stochastic Advection by Lie Transport) preserves the geometric structure of the deterministic shell equations
Invoked when replacing ordinary time derivatives with stochastic Lie derivatives.
domain assumption Weak pathwise solutions exist for the stochastic prototype
Stated as the setting in which the parameter limit is taken.

pith-pipeline@v0.9.0 · 5435 in / 1322 out tokens · 30045 ms · 2026-05-13T21:49:01.313041+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

if a weak pathwise solution of this prototype is parametrised by a suitably chosen family of noise coefficients, we obtain in the parameter limit, the deterministic viscoelastic shell model with viscous damping
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the stochastically constrained variational principle (Hamilton’s principle) states that the actual motion η(t,y) of the shell makes the Action Integral stationary

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

Breit, D., Mensah, P.R., Moyo, T.C.: Martingale solutions in stochastic fluid-structure interaction. J. Non- linear Sci.34(2), Paper No. 34, 45 (2024). DOI 10.1007/s00332-023-10012-4. URLhttps://doi.org/10.1007/ s00332-023-10012-4

work page doi:10.1007/s00332-023-10012-4 2024
[2]

Comptes Rendus de l’Acad´ emie des Sciences-Series I-Mathematics331(5), 405–410 (2000)

Ciarlet, P.G.: Un mod` ele bi-dimensionnel non lin´ eaire de coque analogue ` a celui de wt koiter. Comptes Rendus de l’Acad´ emie des Sciences-Series I-Mathematics331(5), 405–410 (2000)

work page 2000
[3]

Journal of elasticity78(1), 1–215 (2005)

Ciarlet, P.G.: An introduction to differential geometry with applications to elasticity. Journal of elasticity78(1), 1–215 (2005)

work page 2005
[4]

Chinese Ann

Ciarlet, P.G., Roquefort, A.: Justification of a two-dimensional nonlinear shell model of Koiter’s type. Chinese Ann. Math. Ser. B22(2), 129–144 (2001). DOI 10.1142/S0252959901000139. URLhttps://doi-org.univaq.clas.cineca. it/10.1142/S0252959901000139

work page doi:10.1142/s0252959901000139 2001
[5]

Cotter, C., Crisan, D., Holm, D., Pan, W., Shevchenko, I.: Data assimilation for a quasi-geostrophic model with circulation-preserving stochastic transport noise. J. Stat. Phys.179(5-6), 1186–1221 (2020). DOI 10.1007/ s10955-020-02524-0. URLhttps://doi.org/10.1007/s10955-020-02524-0

work page doi:10.1007/s10955-020-02524-0 2020
[6]

Multiscale Model

Cotter, C., Crisan, D., Holm, D.D., Pan, W., Shevchenko, I.: Numerically modeling stochastic Lie transport in fluid dynamics. Multiscale Model. Simul.17(1), 192–232 (2019). DOI 10.1137/18M1167929. URLhttps://doi.org/10. 1137/18M1167929

work page doi:10.1137/18m1167929 2019
[7]

Crisan, D., Holm, D.D., Lang, O., Mensah, P.R., Pan, W.: Theoretical analysis and numerical approximation for the stochastic thermal quasi-geostrophic model. Stoch. Dyn.23(5), Paper No. 2350,039, 57 (2023). DOI 10.1142/ S0219493723500399. URLhttps://doi.org/10.1142/S0219493723500399

work page doi:10.1142/s0219493723500399 2023
[8]

Drivas, T.D., Holm, D.D.: Circulation and energy theorem preserving stochastic fluids. Proc. Roy. Soc. Edinburgh Sect. A150(6), 2776–2814 (2020). DOI 10.1017/prm.2019.43. URLhttps://doi.org/10.1017/prm.2019.43

work page doi:10.1017/prm.2019.43 2020
[9]

Friesecke, G., James, R.D., Mora, M.G., M¨ uller, S.: Derivation of nonlinear bending theory for shells from three- dimensional nonlinear elasticity by Gamma-convergence. C. R. Math. Acad. Sci. Paris336(8), 697–702 (2003). DOI 10.1016/S1631-073X(03)00028-1. URLhttps://doi.org/10.1016/S1631-073X(03)00028-1

work page doi:10.1016/s1631-073x(03)00028-1 2003
[10]

Holm, D.D.: Variational principles for stochastic fluid dynamics. Proc. A.471(2176), 20140,963, 19 (2015). DOI 10.1098/rspa.2014.0963. URLhttps://doi.org/10.1098/rspa.2014.0963

work page doi:10.1098/rspa.2014.0963 2015
[11]

Holm, D.D., Hu, R.: Stochastic effects of waves on currents in the ocean mixed layer. J. Math. Phys.62(7), Paper No. 073,102, 31 (2021). DOI 10.1063/5.0045010. URLhttps://doi.org/10.1063/5.0045010

work page doi:10.1063/5.0045010 2021
[12]

Holm, D.D., Luesink, E.: Stochastic wave-current interaction in thermal shallow water dynamics. J. Non- linear Sci.31(2), Paper No. 29, 56 (2021). DOI 10.1007/s00332-021-09682-9. URLhttps://doi.org/10.1007/ s00332-021-09682-9

work page doi:10.1007/s00332-021-09682-9 2021
[13]

Holm, D.D., Tyranowski, T.M.: Variational principles for stochastic soliton dynamics. Proc. A.472(2187), 20150,827, 24 (2016). DOI 10.1098/rspa.2015.0827. URLhttps://doi.org/10.1098/rspa.2015.0827

work page doi:10.1098/rspa.2015.0827 2016
[14]

Koiter, W.T.: On the nonlinear theory of thin elastic shells. Proc. Koninkl. Ned. Akad. van Wetenschappen, Series B 69, 1–54 (1966)

work page 1966
[15]

Le Dret, H., Raoult, A.: The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci.6(1), 59–84 (1996). DOI 10.1007/s003329900003. URLhttps://doi.org/10.1007/s003329900003

work page doi:10.1007/s003329900003 1996
[16]

International Journal of Solids and Structures46(14-15), 2800–2808 (2009)

Papadopoulos, V., Stefanou, G., Papadrakakis, M.: Buckling analysis of imperfect shells with stochastic non-gaussian material and thickness properties. International Journal of Solids and Structures46(14-15), 2800–2808 (2009)

work page 2009
[17]

Thin-Walled Structures215, 113,319 (2025)

Reuter, N., Kriegesmann, B.: Probabilistic analysis of cylindrical shells under continuously varying load combinations. Thin-Walled Structures215, 113,319 (2025)

work page 2025
[18]

Roquefort, A.: Sur quelques questions li´ ees aux mod´ eles non lin´ eaires de coques minces. Ph.D. thesis, Paris 6 (2001) STOCHASTICALLY-CONSTRAINED KOITER SHELL MODELS. 11

work page 2001
[19]

Schafer, B., Graham-Brady, L.: Stochastic post-buckling of frames using koiter’s method. International Journal of Structural Stability and Dynamics6(03), 333–358 (2006) F aculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Essen, Germany F aculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Ess...

work page 2006

[1] [1]

Breit, D., Mensah, P.R., Moyo, T.C.: Martingale solutions in stochastic fluid-structure interaction. J. Non- linear Sci.34(2), Paper No. 34, 45 (2024). DOI 10.1007/s00332-023-10012-4. URLhttps://doi.org/10.1007/ s00332-023-10012-4

work page doi:10.1007/s00332-023-10012-4 2024

[2] [2]

Comptes Rendus de l’Acad´ emie des Sciences-Series I-Mathematics331(5), 405–410 (2000)

Ciarlet, P.G.: Un mod` ele bi-dimensionnel non lin´ eaire de coque analogue ` a celui de wt koiter. Comptes Rendus de l’Acad´ emie des Sciences-Series I-Mathematics331(5), 405–410 (2000)

work page 2000

[3] [3]

Journal of elasticity78(1), 1–215 (2005)

Ciarlet, P.G.: An introduction to differential geometry with applications to elasticity. Journal of elasticity78(1), 1–215 (2005)

work page 2005

[4] [4]

Chinese Ann

Ciarlet, P.G., Roquefort, A.: Justification of a two-dimensional nonlinear shell model of Koiter’s type. Chinese Ann. Math. Ser. B22(2), 129–144 (2001). DOI 10.1142/S0252959901000139. URLhttps://doi-org.univaq.clas.cineca. it/10.1142/S0252959901000139

work page doi:10.1142/s0252959901000139 2001

[5] [5]

Cotter, C., Crisan, D., Holm, D., Pan, W., Shevchenko, I.: Data assimilation for a quasi-geostrophic model with circulation-preserving stochastic transport noise. J. Stat. Phys.179(5-6), 1186–1221 (2020). DOI 10.1007/ s10955-020-02524-0. URLhttps://doi.org/10.1007/s10955-020-02524-0

work page doi:10.1007/s10955-020-02524-0 2020

[6] [6]

Multiscale Model

Cotter, C., Crisan, D., Holm, D.D., Pan, W., Shevchenko, I.: Numerically modeling stochastic Lie transport in fluid dynamics. Multiscale Model. Simul.17(1), 192–232 (2019). DOI 10.1137/18M1167929. URLhttps://doi.org/10. 1137/18M1167929

work page doi:10.1137/18m1167929 2019

[7] [7]

Crisan, D., Holm, D.D., Lang, O., Mensah, P.R., Pan, W.: Theoretical analysis and numerical approximation for the stochastic thermal quasi-geostrophic model. Stoch. Dyn.23(5), Paper No. 2350,039, 57 (2023). DOI 10.1142/ S0219493723500399. URLhttps://doi.org/10.1142/S0219493723500399

work page doi:10.1142/s0219493723500399 2023

[8] [8]

Drivas, T.D., Holm, D.D.: Circulation and energy theorem preserving stochastic fluids. Proc. Roy. Soc. Edinburgh Sect. A150(6), 2776–2814 (2020). DOI 10.1017/prm.2019.43. URLhttps://doi.org/10.1017/prm.2019.43

work page doi:10.1017/prm.2019.43 2020

[9] [9]

Friesecke, G., James, R.D., Mora, M.G., M¨ uller, S.: Derivation of nonlinear bending theory for shells from three- dimensional nonlinear elasticity by Gamma-convergence. C. R. Math. Acad. Sci. Paris336(8), 697–702 (2003). DOI 10.1016/S1631-073X(03)00028-1. URLhttps://doi.org/10.1016/S1631-073X(03)00028-1

work page doi:10.1016/s1631-073x(03)00028-1 2003

[10] [10]

Holm, D.D.: Variational principles for stochastic fluid dynamics. Proc. A.471(2176), 20140,963, 19 (2015). DOI 10.1098/rspa.2014.0963. URLhttps://doi.org/10.1098/rspa.2014.0963

work page doi:10.1098/rspa.2014.0963 2015

[11] [11]

Holm, D.D., Hu, R.: Stochastic effects of waves on currents in the ocean mixed layer. J. Math. Phys.62(7), Paper No. 073,102, 31 (2021). DOI 10.1063/5.0045010. URLhttps://doi.org/10.1063/5.0045010

work page doi:10.1063/5.0045010 2021

[12] [12]

Holm, D.D., Luesink, E.: Stochastic wave-current interaction in thermal shallow water dynamics. J. Non- linear Sci.31(2), Paper No. 29, 56 (2021). DOI 10.1007/s00332-021-09682-9. URLhttps://doi.org/10.1007/ s00332-021-09682-9

work page doi:10.1007/s00332-021-09682-9 2021

[13] [13]

Holm, D.D., Tyranowski, T.M.: Variational principles for stochastic soliton dynamics. Proc. A.472(2187), 20150,827, 24 (2016). DOI 10.1098/rspa.2015.0827. URLhttps://doi.org/10.1098/rspa.2015.0827

work page doi:10.1098/rspa.2015.0827 2016

[14] [14]

Koiter, W.T.: On the nonlinear theory of thin elastic shells. Proc. Koninkl. Ned. Akad. van Wetenschappen, Series B 69, 1–54 (1966)

work page 1966

[15] [15]

Le Dret, H., Raoult, A.: The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci.6(1), 59–84 (1996). DOI 10.1007/s003329900003. URLhttps://doi.org/10.1007/s003329900003

work page doi:10.1007/s003329900003 1996

[16] [16]

International Journal of Solids and Structures46(14-15), 2800–2808 (2009)

Papadopoulos, V., Stefanou, G., Papadrakakis, M.: Buckling analysis of imperfect shells with stochastic non-gaussian material and thickness properties. International Journal of Solids and Structures46(14-15), 2800–2808 (2009)

work page 2009

[17] [17]

Thin-Walled Structures215, 113,319 (2025)

Reuter, N., Kriegesmann, B.: Probabilistic analysis of cylindrical shells under continuously varying load combinations. Thin-Walled Structures215, 113,319 (2025)

work page 2025

[18] [18]

Roquefort, A.: Sur quelques questions li´ ees aux mod´ eles non lin´ eaires de coques minces. Ph.D. thesis, Paris 6 (2001) STOCHASTICALLY-CONSTRAINED KOITER SHELL MODELS. 11

work page 2001

[19] [19]

Schafer, B., Graham-Brady, L.: Stochastic post-buckling of frames using koiter’s method. International Journal of Structural Stability and Dynamics6(03), 333–358 (2006) F aculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Essen, Germany F aculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Ess...

work page 2006