Case & pressure vessel sizing

In one lineSize the wall so it does not yield at proof and does not burst below design burst — then check it leaks before it breaks.

Before thisHow a solid rocket motor works

Why this matters

The case is usually the heaviest inert part of a solid motor, and every kilogram of case is a kilogram of propellant you are not carrying. Propellant mass fraction — propellant mass over total motor mass — is the single biggest hardware lever on delivered performance. Case sizing is therefore a balance between being light enough to perform and strong enough to never fail, with no leak warning and an energetic payload inside.

The intuition: why cylinders split lengthwise

Think of it like…

A sausage in a hot pan

Overcooked sausages and frozen water pipes always split along their length, never around their circumference. That is not a weakness in the skin — it is geometry. The force trying to open a lengthwise seam grows with the diameter, but the force trying to pull the end off grows with the end area and is resisted by the whole circumference. Work it through and the lengthwise seam carries exactly twice the stress.

Two ways to cut a pressurised cylinder

Same pressure, same wall. Cut lengthwise (A), only two thin strips of wall resist a force proportional to the diameter. Cut across (B), the whole circumference resists a force proportional to the end area. The lengthwise seam ends up carrying twice the stress — which is why cylinders split along their length.

For a thin-walled cylinder of inner radius $r$ and wall $t$ at internal pressure $p$:

$$\sigma_h = \frac{p\,r}{t} \qquad\qquad \sigma_a = \frac{p\,r}{2t} = \frac{\sigma_h}{2}$$

A sphere has no “long direction” — every cut is the same as cut B — so its wall stress is $p\,r/2t$ in every direction. That is why a hemispherical dome needs only half the wall of the cylinder it closes, and why the sphere is the lightest possible pressure vessel. Domes are bubbles; flat closures are plates in bending and are heavy.

The three pressures

A case is not designed to the operating pressure. It is judged against a ladder of pressures, every one referenced to the maximum expected operating pressure (MEOP):

The pressure ladder — illustrative factors only

Everything is referenced to MEOP. The proof and burst factors come from your governing requirement — not from this chart, a textbook, or a previous programme.

• MEOP already contains the hot-day grain temperature, manufacturing spread, ignition spike and erosive burning. See why pressure is amplified.
• Proof is applied to every case (or a defined sample) to show no yield and no leak. It is also your best flaw screen — any crack big enough to fail at proof fails on the test stand, not in service.
• Design burst is the minimum ultimate capability the design must have, shown by analysis and by test to failure on qualification articles.

That gives two sizing requirements, and the thicker wall wins:

$$t_{\text{proof}}:\;\; \sigma_h(p_{\text{proof}}) \le \eta\,\sigma_y \qquad\qquad t_{\text{burst}}:\;\; \sigma_h(p_{\text{burst}}) \le \eta\,\sigma_{\text{UTS}}$$

where $\eta$ is the weld (joint) efficiency. High-strength case steels have yield close to UTS, so burst usually governs; for lower yield-to-tensile ratios, proof can.

When “thin” stops being thin

The thin-wall formula assumes the stress is uniform through the wall. It isn’t — it is highest at the bore. Lamé’s thick-cylinder solution gives the exact hoop stress at the inner surface:

$$\sigma_{h,\text{bore}} = p\,\frac{r_o^2 + r_i^2}{r_o^2 - r_i^2} \qquad\Longrightarrow\qquad t = r_i\left(\sqrt{\frac{\sigma + p}{\sigma - p}} - 1\right)$$

Below about $t/r = 0.1$ the two agree within a few percent. Motor cases are typically far thinner than that, but closures, bosses and high-pressure gas generators are not. The calculator below always uses Lamé and shows the thin-wall value alongside so you can see the difference.

Leak before break

A case can fail two ways. In leak-before-break, a crack grows through the wall, vents gas and is noticed. In break-before-leak, a crack reaches critical size while still buried in the wall and the case ruptures without warning. The simplest screen (Irwin’s) asks whether a crack that has just broken through — about as long as the wall is thick on each side, half-length $c \approx t$ — would be stable:

$$c_{\text{crit}} = \frac{1}{\pi}\left(\frac{K_{IC}}{\sigma_h}\right)^2 \qquad\text{leak before break if}\qquad c_{\text{crit}} \ge t$$

Think of it like…

A balloon versus a football

Prick a balloon and it bursts — the material is highly stressed and has little toughness, so the hole runs. Prick a football and it just goes flat — the hole stays a hole. You want your case to behave like the football. Higher strength steels let you thin the wall, but their toughness usually falls as strength rises, pushing you towards the balloon.

This is a screening check only. Real fracture-control plans use surface and embedded crack solutions, measured toughness at the right orientation, temperature and thickness, and crack-growth analysis over the motor’s pressure history.

Calculator

Motor case sizing — metallic, membrane onlyCopy link to these inputs

Material

Geometry

Internal diameter (mm)Cylinder length (mm)

Design basis

MEOP

Proof factorBurst factor

Example factors. Use the values your governing requirement sets.

Check an as-designed wall (mm, optional)

Minimum wall

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Governed by

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Thin-wall estimate

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t / r

—

Stress at MEOP

Hoop σ_h

—

Axial σ_a

—

Dome wall

—

Pressures and margins

Proof pressure

—

Design burst

—

Predicted burst

—

MS burst

—

MS yield at proof

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Fracture — Irwin leak-before-break screen

Critical crack ½-length at MEOP

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…at proof

—

Leak before break?

—

Mass and efficiency

Case mass

—

Internal volume

—

p_bV/W

—

Screening tool for early sizing. Membrane stress only — it does not size dome–cylinder junctions, closures, bosses, skirts or joints, where real cases usually fail. K_IC values are indicative. Verify against your governing standard and get stress sign-off before anything is built.

Worked example

A welded D6AC case: internal diameter 300 mm, cylinder length 1000 mm, MEOP 10 MPa (100 bar), proof factor 1.10, burst factor 1.25, full-penetration welds fully radiographed ($\eta = 0.90$). Material values from the database: UTS 1520 MPa, yield 1450 MPa, $K_{IC} \approx 70$ MPa√m, density 7.85 g/cm³.

1. Design pressures. $p_{\text{proof}} = 1.10 \times 10 = 11.0$ MPa; $p_{\text{burst}} = 1.25 \times 10 = 12.5$ MPa.
2. Allowables with weld efficiency. $0.9 \times 1520 = 1368$ MPa ultimate; $0.9 \times 1450 = 1305$ MPa yield.
3. Thin-wall thicknesses. Burst: $t = 12.5 \times 150 / 1368 = 1.371$ mm. Proof: $t = 11.0 \times 150 / 1305 = 1.264$ mm. Burst governs.
4. Lamé check. $t = 150\left(\sqrt{1380.5/1355.5} - 1\right) = 1.377$ mm — within 0.5% of thin-wall, as expected for $t/r \approx 0.009$.
5. Stress at MEOP. $\sigma_h \approx 1094$ MPa hoop, about 547 MPa axial.
6. Leak before break. $c_{\text{crit}} = \frac{1}{\pi}(70/1094)^2 = 1.30$ mm, which is less than the 1.38 mm wall. A through-crack would not be stable: this case would be expected to burst without leaking first.
7. Mass. Cylinder $\approx 10.2$ kg of steel holding about 85 litres (including hemispherical domes) at 10 MPa.

What the example teaches

The membrane sizing is easy and the answer is startlingly thin: relative to its size, this wall is thinner than an eggshell. The hard part is step 6. At this strength level the case is not leak-before-break, so its safety rests on NDT that reliably finds small flaws and on the proof test screening anything larger. Load the calculator with these inputs and try 15CDV6 or maraging 18Ni-250 instead to see how the trade moves.

Where this tool stops

Membrane stress is the easy part

The calculator sizes the cylinder away from its ends. Real cases fail at the discontinuities it does not model.

• Dome-to-cylinder junctions. A cylinder and a dome under the same pressure want to grow by different amounts. Forcing them to match creates local bending that can raise stress well above membrane values. A 2:1 ellipsoidal head also carries compressive hoop stress in its knuckle, which can buckle thin heads.
• Closures, bosses and joints. Polar bosses, bolted or pinned closure joints, clevis joints and skirt attachments are where load paths concentrate. These need their own analysis (and usually FEA).
• External and flight loads. Axial thrust, bending from flight loads, and external pressure or handling loads add to pressure stresses. Unpressurised thin cases can be governed by buckling, not burst.
• Temperature. Allowables must be at the case’s temperature during and after the burn, including heat soak-back from the insulation after burnout.
• Composite cases. Filament-wound cases are sized by netting analysis on fibre strength and wind angle, not by isotropic membrane theory. They are covered on a separate (planned) page.

Defining it on the drawing

• Say more than the alloy. Alloy, condition/temper, product form, governing specification and melt practice: “D6AC bar per AMS 6431, VAR, heat treated to 1520 MPa min UTS, double temper” is a specification; “D6AC” is a wish.
• Special processes get their own notes. Heat treatment (with pyrometry to AMS 2750), welding procedure, NDT method and acceptance class, coatings and cleanliness are each separate callouts with their own specification.
• Fracture-critical parts change the rules. Once classified fracture-critical, the drawing carries the flaw-size and NDE requirements, and material comes from a controlled quality class.
• Proof test is a drawing or acceptance requirement, with pressure, hold time, medium, temperature and acceptance criteria (no leak, permanent set limits) stated.

References and further reading

• NASA SP-8025, Solid Rocket Motor Metal Cases (1970) — open NASA design monograph covering case design criteria.
• G. P. Sutton & O. Biblarz, Rocket Propulsion Elements, 9th ed., Wiley (2017).
• W. C. Young, R. G. Budynas & A. M. Sadegh, Roark’s Formulas for Stress and Strain, 8th ed., McGraw-Hill (2011) — thick cylinders, shells and heads.
• T. L. Anderson, Fracture Mechanics: Fundamentals and Applications, 4th ed., CRC Press (2017) — leak-before-break.
• AIAA S-080 (metallic pressure vessels and pressurised structures) and AIAA S-081 (composite overwrapped pressure vessels); ASME BPVC Section VIII; BS EN 13445 — for formal design rules where they apply.

Note

Screening values for early design. Real case design is governed by the requirements of your programme and sign-off by the responsible stress and safety authorities.