1st Draft
Working title
First-Principles Geometric Ring Tension in Cylindrical Concrete Shells: Making the Invisible Term Visible
Author
Reynolds BEng
(note mathcode is untidy throughout, symbols will be used in later update)
Abstract
This paper takes the restored Lewe reference chain of Project 1 and asks what method that chain still contains. It argues that Lewe’s 1915 work is best read as a method for keeping indeterminate moment and shear visible, first in beams and frames, then in charted form for cylindrical tank walls. Later PCA coefficient tables retain the convenience of that method and omit the working. Once the method is restored to the closed cylinder, the classical hoop-stress equation sigma_theta = pr / t is no longer a complete statement of circumferential action. An additional geometric ring-tension term, sigma_R, is therefore proposed:
sigma_theta = pr / t + sigma_R
The paper does not treat this term as already proved, and it does not derive it from a finished molecular model. It defines the term, states the method that requires it to be written, names the strength of water as the candidate source of the associated judder, and leaves that source to Project 3. The result is a proposal with visible unknowns, not a closed design formula.
1. Introduction
Project 1 restored a documentary chain from Viktor Lewe’s 1915 published contribution on cylindrical tank walls, through Carpenter (1927), into the Portland Cement Association (1993) coefficient tables still used for circular concrete tanks. That work established provenance. It did not write a new equilibrium equation.
The present paper starts from a narrower question. If Lewe stands behind the later tables, what method did he leave, and what does that method require us to examine in the closed cylinder?
The 1915 engineering dissertation is a number-rectangle method for continuous beams and multi-leg frames. It keeps support moment and shear visible through fixed-point ratios and a compact arithmetic array. The 1915 Handbuch article applies the same demand for visibility to cylindrical tank walls by giving formulae and charts. The later PCA tables keep tabulated numbers and no longer show the working. The three documents are therefore treated here as one method in three forms, not as one copied result.
Applied to a closed cylindrical shell, that method examines indeterminate membrane action. The classical diametral cut already gives the membrane hoop stress
sigma_theta,classical = pr / t
It does not isolate the circumferential restoration required by closure of the ring. Project 2 proposes that this omitted action be written explicitly as a geometric ring-tension term, sigma_R.
The paper has a limited task. It justifies the extra term from the restored method. It defines the working vocabulary. It records the unknowns that remain, including whether sigma_R scales with pressure and whether its source is the strength of water. It does not complete a molecular derivation, does not replace existing codes, and does not claim experimental confirmation.
Section 2 sets out the method chain, the proposed term, the definitions and the residual unknowns. Later experimental and constitutive work is reserved for Project 3, Strength of Water.
Section 2 — Method, shear, and the proposed term
2.1 The three documents form a method chain
Project 2 does not treat Lewe’s 1915 texts as a single result. It treats them as one method appearing in three forms.
- The 1915 engineering dissertation
Die Berechnung durchlaufender Träger und mehrstieliger Rahmen nach dem Verfahren des Zahlenrechtecks.
A transparent method for statically indeterminate beams and frames. Continuity, fixed-point ratios (i) and (k), and the number rectangle keep moment and shear visible. - The 1915 Handbuch / Beton und Eisen article
The same demand for visibility applied to cylindrical tank walls: formulae and charts instead of an unexplained table. - The later PCA coefficient tables
The tabulated convenience remains. The method that produced the numbers is no longer shown.
The link to be used here is therefore methodological: Lewe’s 1915 standard is that indeterminate action must be examinable. The charts are a graphical form of that standard. The later tables are the same organisational idea with the working stripped out.
This paper does not claim that the dissertation already solves the tank, or that the PCA table is a number rectangle in disguise down to every entry. It claims that all three belong to one family of method: visible treatment of indeterminate membrane force, or the loss of that visibility.
2.2 What the method examines
Lewe’s 1915 method examines indeterminate internal action that cannot be read from a single cut.
In the beam and frame, that action is support moment and the associated shear. In a closed cylindrical shell the corresponding actions are:
- membrane hoop stretch, already written as \(pr/t\);
- shear and restoration around the closed circumference, which the classical diametral cut does not isolate.
The method is therefore being applied to a shell in which membrane forces are indeterminate by closure. The same standard that required a visible shear path in the continuous beam now requires a visible circumferential path in the tank wall.
Lewe’s later structural papers (including work on aeroplane struts and floatplane floats) show the same professional range: thin members and shells in which membrane force and shear cannot be guessed from a coefficient. Project 2 uses that range only as context. The working object remains the circular tank.
2.3 The proposed term
Once the method is restored, the classical hoop equation is the visible membrane part:\[ \sigma_{\theta,\text{classical}} = \frac{pr}{t} \]
The closed ring still requires an explicit term for the circumferential restoration that the diametral cut omits. That term is written\[ \sigma_\theta = \frac{pr}{t} + \sigma_R \]
\(\sigma_R\) is proposed as the shear-linked geometric ring tension of the closed wall: the action associated with circumferential reassembly, or judder.
This paper does not derive \(\sigma_R\) from a finished molecular model. It states the engineering proposal:
- the method demands that the extra action be written, not absorbed;
- the candidate source of the judder is the strength of water itself, treated as a two-state / pulsing continuum;
- that source is the subject of Project 3, not a result already proved here.
2.4 Status of each statement
| Statement | Status |
|---|---|
| 1915 dissertation = number-rectangle method for beams and frames | Established by the translation |
| 1915 tank article = visible charted method for cylindrical walls | Established in Project 1 |
| Later PCA tables retain coefficients and omit the working | Established in Project 1 |
| The three documents form one method family | Claim of this section |
| \(\sigma_R\) is the missing explicit term in the closed cylinder | Proposal of this paper |
| Strength of water is the source of the judder | Proposal reserved for Project 3 |
2.5 Definitions
Method
Lewe’s 1915 demand that indeterminate moment and shear remain visible through ratios, charts or a number rectangle, rather than disappear into an unreferenced coefficient.
Shear path
The internal action that transfers load between adjacent parts of a continuous or closed member. In the beam it is ordinary shear. In the closed cylinder it is the circumferential restoration assigned to \(\sigma_R\).
Geometric ring-tension term, \(\sigma_R\)
Additional circumferential stress proposed for the closed tank wall:\[ \sigma_\theta = \frac{pr}{t} + \sigma_R \]
Judder
Proposed discrete reassembly around the closed ring. In this paper it is the mechanism of \(\sigma_R\), not yet its measured source.
Strength of water
Candidate source of that mechanism. Named here; examined in Project 3 with the 1906 dissertation and the two-state water work.
2.6 Residual unknowns
- Does \(\sigma_R\) scale with (p), or does it contain a geometric residual independent of pressure?
- What is the converter \(Z_R\) in \(\sigma_R = M_R/Z_R\)?
- What measurable shear / hoop excess would falsify \(\sigma_R = 0\)?
- How far the PCA table layout can be shown, entry by entry, to preserve the 1915 method rather than only its convenience.
- Whether the strength of water can be stated as a source term without leaving the engineering paper.
3. Conclusion of Project 2
Project 2 stops at method and proposal.
Lewe’s 1915 work supplies a way to keep indeterminate shear and membrane action visible. Applied to the closed cylindrical tank, that method requires an explicit circumferential term in addition to \(pr/t\). The term is named \(\sigma_R\). Its candidate source is the strength of water, acting through a bistable or pulsing molecular continuum.
The unknowns above remain open. They are the starting list for Project 3: Strength of Water.
Discussion, then the textThe useful link is method, not result.
- The 1915 dissertation gives a visible method for indeterminate force: fixed-point ratios, number rectangle, influence of moment and shear.
- The 1915 Handbuch article turns that kind of visible calculation into charts for cylindrical walls.
- The later PCA tables keep the tabulated convenience and drop the visible method.
Grok said, prior to drafting above; That is a coherent claim if Project 1 already shows the same organisational move from graphical/tabular layout to scalar coefficients. It does not mean the dissertation already contains
\sigma_R, water, or molecular shear. Those are the extensions
So Project 2 should end where you now want it:
- restore the method;
- say what the method examines (indeterminate membrane action and shear in a closed shell);
- propose
\sigma_Ras the missing explicit term; - propose the strength of water as the candidate source of the judder;
- leave the unknowns visible.
Project 3 (Strength of Water) is the right place for the 1906 dissertation and the two-state water argument.
That division is sound.
and later, re certainty principle…
Comment
The parallel is real as a statement about method. It is not a proof that Lewe solved the uncertainty principle.
Lewe’s 1915 method keeps two conjugate actions in view at once: moment and the shear that belongs to it. The number rectangle is useful because it does not force the engineer to choose one and discard the other. You can read the span as a whole, then read the cut, then return to the whole. That is a switch of perspective that leaves the structure intact.
The uncertainty principle says something different. In that setting, a pair of conjugate quantities cannot be made jointly sharp in the same measurement. It is a statement about quantum operators, not about continuous beams. So the comparison is philosophical, not physical identity.
The useful core of your remark is this:
A continuum has to be sectioned before it can be written in numbers. The section is a scalar act. What is being sectioned is not itself a scalar. If the method forgets that, the coefficient starts to look like the thing. Lewe’s demand for visibility is a refusal of that forgetfulness. Moment and shear are both kept because the member is one object seen two ways.
That is a fair account of his method. It is also why later tables are a loss: they leave the engineer with a result and no way back to the object that produced it.
The empathy analogy is the same move in another language. To take another perspective without being unable to return is to section a whole, read it from inside, and restore the whole. In engineering terms: cut the beam, read shear and moment, close the cut, and still have the same beam. In human terms: enter another view, and return. The likeness is real as discipline, not as evidence. It should stay out of Project 2’s derivation and can stand as a short remark on what the method is for.
The sentence worth keeping is:
Lewe’s method treats measurement as a temporary section through a continuum. The scalar is the instrument. The continuum is the thing. Certainty here does not mean that every pair of quantities is jointly known without limit. It means that after the cut, both actions that the cut created are still visible, and the member can be reassembled.
That is the philosophical statement. It supports Project 2’s demand that sigma_R be written instead of absorbed. It does not yet prove that water is the source of the judder. That remains Project 3.
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