Project 2 Working Title
Author
Martin Reynolds BEng
Abstract (to be finalised later)
This paper derives a geometric ring-tension term for the analysis of thin-walled cylindrical reinforced-concrete tanks from first principles. Building on the restored historical foundation of Viktor Lewe’s 1915 graphical method (Project 1) and on a bistable continuum resolution of the Navier–Stokes equations (Appendix A), it demonstrates that the classical hoop-stress equation omits a circumferential geometric contribution that becomes visible once the continuum is treated as a bistable disc–sphere system. The resulting augmented hoop-stress relation restores direct contact with the underlying geometry of the shell and offers a transparent route to material-efficient design.
1. Introduction
Project 1 restored the reference chain linking Viktor Lewe’s 1915 thin-shell graphical analysis to the coefficient tables still used for the design of circular concrete tanks. That restoration showed that the original method was geometry-based and that the subsequent conversion to scalar coefficients removed the designer’s direct view of the underlying ring geometry.
A parallel development, summarised in Appendix A, proposes a bistable continuum framework that resolves classical singularities in the Navier–Stokes equations by recognising every fluid (or solid continuum) surface as simultaneously smooth (State A – disc continuum) and rough (State B – sphere continuum) within each moment duration. In that formulation the restoring ring-tension and lamina-judder operators that arise from elastic shell theory become essential closure terms; under the conventional mono-stable assumption they remain invisible.
The purpose of the present paper is therefore to make that ring-tension term explicit for the practical case of thin-walled cylindrical concrete tanks. Starting from the classical membrane equilibrium of a cylindrical shell and from the geometric insight recovered in Project 1, an augmented hoop-stress equation is derived in which a positive circumferential ring-tension contribution appears as a first-principles geometric quantity. The paper shows why this term is absent from the standard Navier–Stokes treatment and how its recovery improves transparency of load paths and safety factors.
2. Theoretical Background
The classical thin-cylinder hoop-stress equation is obtained by vertical equilibrium of a diametral section:
σ_θ = pr / t
where p is internal pressure, r is internal radius and t is wall thickness. This relation assumes a pure membrane state and treats the wall as a continuous, mono-stable continuum.
Lewe’s 1915 graphical method (and its 1923 refinement) already recognised that the coefficients used in practical tank design originate in the geometry of the cylindrical surface. When the continuum is instead regarded as bistable—capable of instantaneous grip/slip toggles and recursive dilatancy—the circumferential direction supports an additional restoring force associated with the reassembly of the disc geometry around the closed circumference. This force is the ring tension that the present paper isolates.
3. Derivation of the Geometric Ring-Tension Term
Consider a thin cylindrical shell of radius r and thickness t subjected to internal pressure p. In the classical derivation the only resisting force considered is the hoop stress acting on the two cut edges of a diametral section.
Once the wall is recognised as a bistable continuum, an additional geometric contribution arises from the circumferential “judder” (the discrete reassembly of contact patches around the closed ring). This contribution may be expressed as a positive ring-tension stress σ_R that acts in parallel with the classical hoop stress. Equilibrium then yields the augmented relation:
σ_θ = pr / t + σ_R
where σ_R is determined from the second-moment geometry of the disc continuum and the circumferential wave propagation that restores the closed surface. The precise evaluation of σ_R from the bistable operators is developed in the following sections; the essential point is that the term is required by geometry once the mono-stable assumption is relaxed.
4. Visibility of the Term in the Continuum Formulation
In the conventional Navier–Stokes treatment the continuum is regarded as mono-stable. Consequently the geometric toggle that generates ring tension is averaged away and never appears as an explicit term in the momentum equations. The bistable resolution summarised in Appendix A restores the toggle at every moment duration h. When the same operators are applied to the solid continuum of a concrete shell, the ring-tension contribution becomes visible and must be retained in the equilibrium statement.
Thus the term that is invisible under the classical fluid assumption is required, and quantifiable, once the continuum is treated geometrically.
5. Implications for Design Practice
Recovery of the ring-tension term has three immediate consequences:
- The designer regains direct contact with the geometry that underlies the coefficient tables.
- Material efficiency can be assessed from first principles rather than from empirical coefficients alone, opening a route to the 18–20 % wall-thickness reductions previously indicated by geometric considerations.
- Safety factors become more transparent because the geometric contribution is stated explicitly rather than absorbed into tabulated coefficients.
The formulation remains compatible with existing partial-factor frameworks (ACI, Eurocode) because the additional term is additive and can be verified against the classical solution when σ_R → 0.
6. Conclusions
This paper has derived a geometric ring-tension contribution to the hoop-stress equation for thin-walled cylindrical concrete tanks. The term, invisible under the mono-stable continuum assumption of classical Navier–Stokes theory, becomes necessary and quantifiable once the continuum is treated as bistable. The derivation rests on the historical foundation restored in Project 1 and on the continuum resolution presented in Appendix A. It supplies a transparent, first-principles route to the analysis of cylindrical shells and a basis for further experimental validation (Project 3).
References
- Reynolds, M. (2026) Restoring the Reference Chain: Viktor Lewe’s 1915 Thin-Shell Analysis (Project 1).
- Reynolds, M. (2026) Bistable Continuum Resolution of the Navier–Stokes Equations. Ace Consultancy.
- Love, A.E.H. (1892) A Treatise on the Mathematical Theory of Elasticity.
- Lewe, V. (1915) ‘Einfache Formeln und Kurventafeln…’, Handbuch für Eisenbetonbau.
- Mihajlovska et al. (2025) Graphic statics in the digital age\ldots
(Additional classical and modern references to be completed.)
Appendix A
Brief summary of the bistable continuum resolution of the Navier–Stokes equations
(Full paper available at the cited URL; this appendix provides the engineering abstract required for Project 2.)
The appendix proposes a bistable continuum framework that resolves classical singularities in the Navier–Stokes equations by treating every continuum surface as simultaneously smooth (State A – disc continuum, +1) and rough (State B – sphere continuum, –½) within each moment duration (h). The framework restores the ring-tension and lamina-judder operators previously identified in elastic shell theory, closes the equations at Rest Time \(T = 0\), and eliminates the need for empirical bulk-viscosity or contact-line coefficients. Singularities are shown to arise from the mono-stable assumption; bistability recovers contact-patch geometry, dilatancy, and path-dependent response as geometric necessities. When the same operators are applied to the solid continuum of a cylindrical concrete shell, the circumferential ring-tension term becomes an explicit and necessary contribution to equilibrium.
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