Project 2
First-Principles Geometric Ring Tension in Cylindrical Concrete Shells: Making the Invisible Term Visible
Martin Reynolds BEng
Abstract
Project 1 restored the reference chain from Viktor Lewe, 1915, to the coefficient tables still used for circular concrete tanks. This paper writes the term those tables were carrying without naming it.
The classical hoop equation is
σ_θ = pr / t
For a closed ring an additional circumferential restoration is required. The augmented equation is
σ_θ = pr / t + σ_R
σ_R (sigma_R) is geometric ring tension. Its mechanism is discrete reassembly around the circumference — judder. Appendix A shows why that restoration is invisible if the continuum is treated as mono-stable. Appendix B shows how the spatial pattern of the term can be controlled. Appendix C holds the Rest Time residual from which the converter is later written.
Project 3 names the source of sigma_R: trapped water-stretch on Reynolds’ misfit surface. Project 4 names the converter:
Z_R = A_* t_* L_* τ² = m⁴ s².
σ_R = F / Z_R
Part 2 of this paper returns those results to practice. The tables remain as a check. They are no longer the method.
1. Introduction
A thin cylindrical tank wall is still designed, in much ordinary practice, from coefficient tables. The parent of those tables is Lewe’s 1915 analysis of cylindrical container walls. Project 1 set out that chain. What Project 1 could not do was write the missing term in the hoop equation.
This paper does that, and only that, in Part 1. Part 2 takes the converter derived in Project 4 and puts the completed equation back on the calculation sheet. The 2008 BEng question — can the loadings be calculated without coefficients? — is the reason the paper exists. The answer given here is yes as method.
2. Theoretical background
The classical membrane hoop for an internally pressured thin cylinder is
σ_θ = pr / t + σ_R
That equation assumes the wall is a free hoop. A tank wall is not free. It is a closed ring, joined to a base, and made of a material that had to open volume under constraint before it set. Love’s thin-shell theory already requires higher-order terms once curvature and bending are admitted. Lewe’s method requires that indeterminate moment and shear remain visible after the cut. Coefficient tables fold those requirements into K1, K2, K3.
This paper keeps them visible.
Two assumptions must be distinguished.
Mono-stable continuum: grip is permanent, slip is loss, the ring has no living reassembly. Only pr/t survives.
Bistable continuum: a contact can grip and slip. Rebuild of the contact occupies a duration. A closed circumference must reassemble. That reassembly is an extra circumferential action. It is sigma_R.
Appendix A is the continuum statement of the second assumption. It is not required for the algebraic form of the hoop equation. It is required to explain why the extra term vanished from later tables.
3. Derivation of the geometric ring-tension term
Consider a closed cylindrical wall of radius r and thickness t, under internal pressure p.
Membrane equilibrium still supplies pr/t.
Because the wall is a closed ring, circumferential displacement cannot run out to infinity. Any local opening of a contact must be restored around the ring. Write that restoration as a line tension T on a strip of height b. Then the additional wall stress is
σ_R = T / (t b)
and the hoop equation becomes
σ_θ = pr / t + F / Z_R
T is not imported from a table. It is the integral of shear along a diametral path that ends on the rim:
T = integral of s(x) dx from 0 to r
Judder is the name given to discrete reassembly travelling on that rim. Its speed is a celerity, m/s. It is the mechanism of σ_R, not yet its measured source.
The source is named in Project 3: water occupying Reynolds’ singular surface of misfit, later stitched by hydration. The converter is named in Project 4 and used in Part 2 below.
What this section claims is only the necessity of the extra term for a closed ring, and the engineering placeholder σ_R = T / (t b).
4. Visibility of the term
If the continuum is written as mono-stable, σ_R has nowhere to live. It is absorbed into a coefficient.
If both actions are kept after the cut — Lewe’s standard — σ_R is the circumferential action that belongs to the ring, and pr/t is the action that belongs to the stored liquid. Appendix B shows that the spatial pattern of the ring term can be indexed by zeros of Bessel functions. That is control, not proof of magnitude.
Recovery of the term has three immediate consequences.
- The designer sees the geometry the tables were hiding.
- Material can be judged from the two terms separately. An 18–20 percent reduction in thickness remains a geometric indication, not a measured saving.
- Safety factors can be pointed at. A number that cannot be pointed at is a coefficient.
5. Part 2 — return to practice
Project 3 supplied the source. Project 4 supplied the converter. The practice equation is therefore
σ_θ = pr / t + F / Z_R
Z_R = A_* t_* L_* τ2 = m4 s2
F is the strength-of-water residual, N m s2.
sigma_R = F / Z_R is wall stress, N/m2.
A_* is the opened contact area. t_* is auxetic thickening. L_* is the in-plane opening that supplies the fourth metre. τ2 is the rebuild delay.
The strip form remains the site reading of the same act:
σ_R = T / (t b)
The three design values of the 2008 thesis remain the three design values:
maximum hoop tension
height of maximum tension
base moment
They are no longer read from K1, K2, K3. They are read from pr/t, from sigma_R, and from the joint that spends part of those two as moment and shear.
When F is spent and Z_R is locked — after set — the extra term is a residual, not a living breath. That is why the finished wall is strong and brittle. When Z_R is taken as infinite, or F as zero, the classical equation returns. That is the check.
This does not replace a code. It states what the code absorbs when it issues a table and no working.
The converter is named in Project 4 as Z_R = m4 s2, with σ_R = F / Z_R. This paper does not derive it.
6. Conclusions
Project 1 restored the parent of the tables.
This paper wrote the missing term:
σ_θ = pr / t + σ_R
Project 3 named the source. Project 4 named the converter. Part 2 put both back on the calculation sheet.
The 2008 question, as method, is answered. The tables may stay as a check. They are no longer the last word.
What remains is measurement: one hoop-excess reading, one base rotation, one prestress case. Those are 2008 tasks 5 and 6, and the experiment reserved in the portfolio proposal.
Working close:
Write both actions after the cut.
Do not hide the stitch inside a coefficient.
σ_R is the stitch, named.
References
Lewe, V. (1915). Die Berechnung durchlaufender Träger und mehrstieliger Rahmen nach dem Verfahren des Zahlenrechtecks. Dresden: Noske.
Lewe, V. (1915). Einfache Formeln und Kurventafeln zur Berechnung zylindrischer Behälterwände. Handbuch für Eisenbetonbau, 2nd ed., Heft IV/V. Berlin: Wilhelm Ernst & Sohn.
Love, A. E. H. (1927). A Treatise on the Mathematical Theory of Elasticity. 4th ed. Cambridge: Cambridge University Press.
Portland Cement Association (1993). Circular Concrete Tanks without Prestressing. Skokie, IL: PCA. (First issued 1942/43.)
Reynolds, O. (1902). On an Inversion of Ideas as to the Structure of the Universe. The Rede Lecture. Cambridge: Cambridge University Press.
Reynolds, O. (1903). The Sub-Mechanics of the Universe. London: The Royal Society.
Reynolds, M. (2008). BEng thesis on Cylindrical Reinforced Concrete Water Tanks. South Bank University. Section §6.1.
Carpenter, H. (1927). A contribution to the calculation of circular tanks in reinforced concrete. Concrete and Constructional Engineering, 22(4), 237–241.
Project 1, Project 3, Project 4, and Appendices A–D of this portfolio.
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