Plates, Shells, and Cylinders

§1Thin cylinders

Cut the shell in imagination and balance what the pressure pushes against what the wall pulls. Two cuts, two stresses — and the hoop one is double.

hoop σ_h = p rt = p d2t longitudinal σ_l = p r2t

Hoop governs — which is why pressurised tubes split along their length, never around it, and why a longitudinal weld carries twice the duty of a circumferential one (weld efficiency enters as a divisor on the allowable stress).

Example 1 — an air receiver wall

Ø600 mm receiver at 1.0 MPa, allowable 80 MPa: t = pr/σ = 1.0 × 300/80 = 3.75 mm; adding a 1 mm corrosion allowance and rounding, t = 5 mm, running at a comfortable 60 MPa hoop. Pressure equipment is code territory — the mechanics above is the skeleton the codes clothe with rules, allowances and inspection.

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§2Thin spheres

σ = p r2t — every direction at once

A sphere carries the same pressure at half the cylinder’s wall (the receiver above would need only 1.9 mm) — the shape’s whole surface works in the efficient longitudinal mode. That efficiency is why gas storage trends spherical and why cylinder ends are dished rather than flat: a hemispherical head is a sphere doing a lid’s job.

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§3Thin or thick?

The thin formulas assume the stress is uniform through the wall. That holds while the wall is a small fraction of the diameter.

Working rule: treat the shell as thin while t < d/20. Beyond that the inner fibres carry visibly more than the outer — the thin formula under-reports the bore stress and Lamé takes over. Hydraulic cylinders, gun-drill bushes and high-pressure fittings live on the thick side almost by definition.

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§4Thick cylinders — Lamé

In a thick wall the hoop stress varies through the thickness, peaking at the bore. For internal pressure p on radii rᵢ (bore) and r₀ (outside):

σ_h,max (at bore) = p × r₀² + rᵢ²r₀² − rᵢ² solved for size: r₀ = rᵢ (σ + p)/(σ − p)

Example 2 — a hydraulic cylinder barrel

Bore Ø80 mm at 35 MPa, allowable 110 MPa: r₀ = 40√(145/75) = 55.6 mm → wall 15.6 → use 16 mm (r₀ = 56: bore stress 107.9 MPa ✓). Note the formula’s warning: as p approaches σ the required r₀ runs to infinity — past that point no amount of wall helps, and the answers are stronger material, autofrettage or compound (shrink-fitted) construction.

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§5Flat circular plates

A flat cover doesn’t stretch like a shell — it bends like a plate, and its stress grows with the square of the radius-to-thickness ratio.

Uniform pressure p, radius r, ν = 0.3 edges fixed: σ_max = 0.75 p (r/t)² (at the rim) simply supported: σ_max ≈ 1.24 p (r/t)² (at the centre)

Example 3 — a bolted flat cover

Ø300 mm opening at 0.5 MPa, edges effectively fixed by the bolted flange, allowable 90 MPa: (r/t)² = 90/(0.75 × 0.5) = 240, so r/t = 15.5 and t = 150/15.5 = 9.7 → use 10 mm. A flat lid this size needs 10 mm where a dished end would need 2 — flatness is bought with thickness, which is the whole argument for dished ends.

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§6Quick reference

The working core of the page on one card rack.

Thin cylinder

σ_h = pd/2t · σ_l = pd/4t

hoop governs, ×2

Thin sphere

σ = pd/4t

half the cylinder’s wall

Regime

thin while t < d/20

Thick (Lamé)

σ_bore = p(r₀²+rᵢ²)/(r₀²−rᵢ²)

r₀ = rᵢ√((σ+p)/(σ−p))

Flat plate

fixed: 0.75 p(r/t)²

supported: ≈1.24 p(r/t)²

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Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Plates, Shells, and Cylinders. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to carry the subject from function and assumptions through design evidence, verification and controlled release. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Apply Plates, Shells, and Cylinders by beginning with the duty, not the component or software command. Convert the key ideas—thin, plates, cylinders, spheres, lamé—into measurable requirements and interfaces. Record operating and non-operating environments, duty cycle, expected life, loads, energy sources, human interaction and reasonably foreseeable abnormal conditions. When a value is not a project requirement or verified supplier datum, identify it as an assumption or illustrative value.

Create a calculation and evidence trail that another competent person can audit. Every input should carry a source, unit, revision and uncertainty or tolerance where relevant. Every model should state its boundary conditions and limitations. Keep nominal capacity separate from design capacity, and keep verification margin separate from an arbitrary safety factor. If a code or standard governs the work, confirm the applicable edition and contractual status rather than copying a number from a secondary summary.

Design for manufacture, assembly, inspection, operation and maintenance at the same time. A technically valid geometry can still fail because it cannot be fixtured, measured, cleaned, guarded, reached or replaced. Review process capability, datum or reference strategy, tolerance accumulation, access, error-proofing and changeover. Where people interact with plant, apply the hierarchy of controls and consult those who will operate, clean, maintain and recover the equipment.

Plan verification before release. Define the characteristic, method, equipment, sample or test condition, acceptance criterion, record and responsible person. Validation then asks a different question: whether the resulting system is effective and suitable in the intended use context. A passed drawing check or analysis does not by itself validate usability, maintainability or production performance.

Step-by-step operating method

Define the duty. Capture the required function, interfaces, operating environment, life, loads and unacceptable outcomes.
Establish the model. Identify governing principles, units, material or process data, assumptions and uncertainty.
Develop alternatives. Compare feasible concepts against performance, manufacturability, safety, maintainability and cost.
Verify the design. Use analysis, test, inspection or demonstration with acceptance criteria defined before execution.
Release and learn. Baseline the design, control changes, retain evidence and feed operating results into the next revision.

Illustrative design review record

Illustrative values only. Build a one-page record with the required function, input sources, assumptions, governing load or process condition, failure consequences, selected concept, verification method and acceptance criterion. Mark every numerical input as project requirement, verified supplier data, measured value, calculation output or assumption. Review the weakest evidence first. If an assumption can change safety, compliance, interchangeability or capacity, it must be resolved before release rather than buried in a calculation note.

Evidence class	Question	Release expectation
Requirement	What must the design do and under which conditions?	Approved and traceable
Input	Where did the load, property, tolerance or process limit come from?	Source, unit and revision recorded
Analysis	Which model and assumptions connect input to result?	Checkable calculation or simulation
Verification	How will conformity be demonstrated?	Method and acceptance criterion agreed
Validation	Will the solution work for intended users and conditions?	Representative use evidence

Common failure modes and recovery actions

1. Watch for

Starting detailed design before interfaces and operating limits are agreed.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Using catalogue or typical values as though they were certified project inputs.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Checking nominal performance while ignoring tolerances, degradation and foreseeable misuse.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Confusing verification of requirements with validation of user need.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Releasing drawings or procedures without configuration, inspection and change controls.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

What function and failure consequence govern this decision?
Which inputs are measured, specified, assumed or illustrative?
How will conformity be demonstrated and recorded?
What change would invalidate the current evidence?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Plates, Shells, and Cylinders?