Thin and thick cylinders

In short

A cylindrical shell under internal fluid pressure carries that pressure as tensile stress in its wall: hoop (circumferential) stress around the circumference and longitudinal (axial) stress along the axis. When the wall is thin (t/d ≤ 1/20) the stress is assumed uniform across the wall and simple formulas apply; when the wall is thick, stress varies sharply from bore to outside and Lamé's theory is needed. This decides the plate thickness of every boiler drum, LPG cylinder, air receiver, hydraulic cylinder, pipeline and reactor vessel. It is a guaranteed-marks topic in Strength of Materials, a repeat GATE numerical, and standard interview ground for BHEL, L&T Heavy Engineering, ISRO, IOCL/ONGC, Thermax, Kirloskar and boiler/pressure-vessel design houses.

Explained

Thin cylinders — assumptions and the two stresses

• Classification: thin if wall thickness t ≤ d/20 (d/t ≥ 20); otherwise thick. Boiler shells and LPG cylinders are thin; gun barrels and high-pressure hydraulic rams are thick.
• Assumptions: stress uniform through the thickness; radial stress neglected (σr ≈ 0 against σh, σl); the shell is a membrane — no bending; material homogeneous, isotropic, elastic; far from ends and openings.
• Hoop stress σh = pd/2t: from a longitudinal diametral section — bursting force p·d·L is resisted by two wall sections 2·t·L·σh.
• Longitudinal stress σl = pd/4t: from a transverse section — end force p·(π/4)d² resisted by πdt·σl. Hence σh = 2σl, which is why a pressurised shell splits along a line parallel to its axis, and why the longitudinal (welded or riveted) joint is the critical one.
• Element state: biaxial tension (σh, σl, both principal, σr ≈ 0). Maximum in-plane shear = (σh − σl)/2 = pd/8t; the true maximum shear in three dimensions = σh/2 = pd/4t, on a plane at 45° to the wall surface containing the axial direction.
• External pressure is a different problem — a thin shell fails by elastic buckling (collapse), not by yielding, so thickness is set by stiffening rings and code charts, not by pd/2t.

Deformations, volumetric strain and joints

• Hoop strain εh = (σh − νσl)/E = pd(2 − ν)/4tE equals δd/d, the diametral strain.
• Longitudinal strain εl = (σl − νσh)/E = pd(1 − 2ν)/4tE equals δL/L. For ν = 0.5 it would be zero; for steel (ν ≈ 0.3) the shell elongates slightly.
• Volumetric strain εv = 2εh + εl = pd(5 − 4ν)/4tE — used for "extra fluid pumped in" and hydraulic-test problems (add the fluid's own compression p·V/K if asked).
• Joint efficiency η = strength of joint / strength of solid plate. Design uses σh = pd/(2tηl) for the longitudinal joint and σl = pd/(4tηc) for the circumferential joint; the longitudinal joint almost always governs.
• Riveted boiler shells gave η ≈ 60–80%; modern fully radiographed welds give J = 1.0, spot-radiographed 0.85, un-radiographed single-V about 0.70.
• Corrosion allowance (commonly 1.5–3 mm for carbon steel in Indian practice) is added to the calculated thickness, then rounded to a standard plate size.

Spherical shells and wire-wound cylinders

• A thin sphere has the same stress in every direction: σ = pd/4t — half the hoop stress of a cylinder of the same d and t, so a sphere needs about half the thickness. That is why LPG bullets/Horton spheres are used for bulk storage.
• Sphere strain ε = pd(1 − ν)/4tE; volumetric strain εv = 3ε = 3pd(1 − ν)/4tE.
• Cylinder with hemispherical ends: for no discontinuity in diametral strain the hemisphere must be thinner than the cylinder (th/tc = (1 − ν)/(2 − ν) ≈ 0.412 for ν = 0.3); otherwise a bending discontinuity stress arises at the junction.
• Wire-wound cylinders: winding wire in tension puts the cylinder wall into initial compression, so the net hoop stress under pressure is reduced — the same idea as a compound cylinder, used in old gun barrels and high-pressure moulds.

Thick cylinders — Lamé's theory

• Why thin formulas fail: when d/t < 20 the radial stress (−p at the bore) is no longer negligible and hoop stress is much larger at the inside than at the outside; pd/2t underestimates the peak badly.
• Assumptions: plane sections remain plane → longitudinal strain is constant across the wall; material elastic, isotropic; long cylinder, away from ends.
• Lamé's equations: σr = A − B/r², σθ = A + B/r², tension positive, r the radius of the point. A and B come from the boundary conditions on σr.
• Internal pressure p only (bore ri, outside ro): A = p·ri²/(ro² − ri²), B = p·ri²·ro²/(ro² − ri²).
• Distribution: σθ is maximum tensile at the bore = p(ro² + ri²)/(ro² − ri²) and falls to 2A at the outside; σr = −p at the bore and 0 outside. σθ + σr = 2A = constant at every radius — a useful check.
• Longitudinal stress for closed ends is uniform, σl = p·ri²/(ro² − ri²) = A.
• Radial displacement u = (r/E)(σθ − νσr − νσl).
• Design thickness (maximum principal stress theory): t = ri[√((σθ + p)/(σθ − p)) − 1]. Note the limit: as p approaches σθ the thickness goes to infinity — no monobloc cylinder can hold a pressure equal to or above its allowable stress, however thick it is made. Beyond that, use a compound cylinder or autofrettage.
• Clavarino's equation (closed ends, ductile material, maximum strain theory): t = ri[√((σ + (1 − 2ν)p)/(σ − (1 + ν)p)) − 1]; Birnie's equation (open ends, e.g. pump cylinders, rams, gun barrels): t = ri[√((σ + (1 − ν)p)/(σ − (1 + ν)p)) − 1]. Barlow's formula t = p·ro/σ is a quick pipeline approximation.
• External pressure only: σθ and σr are both compressive, with the largest compressive hoop stress again at the bore.

Compound cylinders, shrink fit and autofrettage

• Idea: shrink an outer cylinder (bored slightly smaller) onto an inner one. The junction pressure pc leaves the inner cylinder in hoop compression and the outer in hoop tension, so when the working pressure is applied the peak hoop stress at the bore is much lower and the wall is used more evenly.
• Procedure: compute shrinkage stresses from pc alone (outer cylinder: internal pressure pc; inner cylinder: external pressure pc), compute working stresses on the whole cylinder as monobloc, then superpose algebraically.
• Shrinkage allowance: radial interference at the junction radius rc is δ = (2pc·rc³(ro² − ri²))/(E(ro² − rc²)(rc² − ri²)) for both cylinders of the same material, so the diametral interference is 2δ; different materials require each cylinder's E and ν separately.
• Assembly: heat the outer cylinder (or cool the inner in liquid nitrogen); temperature rise needed ΔT = δ/(α·rc).
• Autofrettage: over-pressurise a monobloc cylinder so the inner layers yield; on release they are left in residual compression — the same benefit without assembly. Used in gun barrels, high-pressure tubing and waterjet intensifiers.

Indian codes, standards and practice

• IS 2825 (Code for unfired pressure vessels) — the Indian design code; thin-wall shell thickness t = pD/(2fJ − p) + corrosion allowance, with f the allowable stress and J the weld joint factor. ASME BPVC Section VIII Div 1/2 is widely used by Indian fabricators for export work.
• Boilers: the Indian Boiler Regulations, 2026 (in force from October 2026, superseding the 1950 Regulations), framed under the Boilers Act, 2025 (in force from 1 May 2025, replacing the Boilers Act, 1923), govern design, materials, welding qualification and inspection of boilers and boiler-mounted piping; approval is by the State Boiler Inspectorate / Central Boilers Board.
• Gas cylinders: the Gas Cylinders Rules, 2016 and the SMPV (Unfired) Rules, 2016, administered by PESO under the Explosives Act, 1884, cover manufacture, filling and periodic testing; product standards include IS 3196 (welded low-pressure cylinders, e.g. the 14.2 kg domestic LPG cylinder) and IS 7285 (seamless cylinders for industrial gases and CNG).
• Testing: hydrostatic test at a code-specified multiple of design pressure (commonly 1.3–1.5 ×, the exact factor depending on the code and edition), plus radiography, and periodic hydro re-testing of cylinders at fixed intervals.
• Pipelines follow ASME B31.3/B31.4/B31.8 and OISD guidelines; thickness there is essentially the hoop-stress formula with a design factor.

Key points

• Thin/thick rule — thin if t ≤ d/20 (d/t ≥ 20); beyond that use Lamé.
• Hoop stress — σh = pd/2t, the largest stress, acts circumferentially.
• Longitudinal stress — σl = pd/4t = σh/2, acts along the axis.
• Failure plane — a thin cylinder bursts along its length because hoop stress is twice longitudinal.
• Sphere — σ = pd/4t in all directions; strongest shape for a given pressure and volume.
• Max shear (thin) — pd/8t in plane, pd/4t absolute in three dimensions.
• Volumetric strain — εv = 2εh + εl = pd(5 − 4ν)/4tE.
• Joint efficiency — longitudinal joint carries the hoop stress and governs design.
• Lamé's equations — σr = A − B/r², σθ = A + B/r²; σθ + σr = 2A is constant.
• Peak stress location — hoop stress is always maximum at the bore, both for internal and external pressure.
• Check value — σr = −p at the bore and 0 at the free outer surface (internal pressure only).
• Closed-end axial stress (thick) — σl = p·ri²/(ro² − ri²), uniform across the wall.
• Pressure limit — a monobloc thick cylinder cannot hold p ≥ allowable σθ at any thickness.
• Compound cylinder — shrink fit pre-compresses the bore; stresses superpose linearly.
• Autofrettage — controlled over-pressure leaves residual compression at the bore.
• Codes — IS 2825 for unfired vessels, IBR 2026 for boilers, PESO rules for gas cylinders.

Formulas and examples

Example 1 — thin cylinder, stresses and change in volume. d = 1,200 mm, t = 12 mm, L = 3,000 mm, p = 2 N/mm², E = 200 GPa, ν = 0.3.

• d/t = 100 ≥ 20 → thin. σh = 2 × 1200/(2 × 12) = 100 N/mm²; σl = 50 N/mm².
• εh = (100 − 0.3 × 50)/200000 = 85/200000 = 4.25 × 10⁻⁴ → δd = 4.25 × 10⁻⁴ × 1200 = 0.51 mm.
• εl = (50 − 0.3 × 100)/200000 = 20/200000 = 1.0 × 10⁻⁴ → δL = 0.30 mm.
• εv = 2(4.25 × 10⁻⁴) + 1.0 × 10⁻⁴ = 9.5 × 10⁻⁴. V = (π/4)(1200²)(3000) = 3.393 × 10⁹ mm³ → δV = 3.22 × 10⁶ mm³ = 3.22 litres.

Example 2 — design thickness with joint efficiency. Boiler shell, d = 1,000 mm, p = 1.5 N/mm², allowable stress 90 N/mm², ηl = 75%, ηc = 60%, corrosion allowance 2 mm.

• Longitudinal joint: t = pd/(2σηl) = 1.5 × 1000/(2 × 90 × 0.75) = 1500/135 = 11.11 mm.
• Circumferential joint: t = pd/(4σηc) = 1500/(4 × 90 × 0.6) = 1500/216 = 6.94 mm → longitudinal governs.
• t = 11.11 + 2 = 13.11 mm → adopt a standard 14 mm plate.

Example 3 — thick cylinder by Lamé. ri = 100 mm, ro = 150 mm, p = 30 N/mm², closed ends.

• ro² − ri² = 22500 − 10000 = 12500. A = 30 × 10000/12500 = 24; B = 30 × 10000 × 22500/12500 = 540000.
• At r = 100: σθ = 24 + 540000/10000 = 78 N/mm², σr = 24 − 54 = −30 N/mm² ✓ (= −p).
• At r = 150: σθ = 24 + 540000/22500 = 48 N/mm², σr = 24 − 24 = 0 ✓.
• σl = A = 24 N/mm². Maximum shear at bore = (78 − (−30))/2 = 54 N/mm².
• Thin-cylinder estimate would give pd/2t = 30 × 200/(2 × 50) = 60 N/mm² — 23% below the true 78 N/mm², since d/t = 4.

Example 4 — thickness of a thick cylinder. Bore 150 mm (ri = 75 mm), p = 60 N/mm², allowable hoop stress 150 N/mm².

• t = 75[√((150 + 60)/(150 − 60)) − 1] = 75[√2.3333 − 1] = 75(1.5275 − 1) = 39.6 mm → adopt 40 mm, ro = 115 mm.
• Check: σθ = 60(115² + 75²)/(115² − 75²) = 60(13225 + 5625)/7600 = 60 × 2.480 = 148.8 N/mm² < 150 ✓.

Where it is used

• Boilers and power plant: BHEL, Thermax, Cethar — drum and header thickness, IBR-certified welding and inspection.
• LPG and industrial gas cylinders: IOCL/BPCL/HPCL 14.2 kg domestic cylinders (IS 3196), seamless CNG cascades and oxygen cylinders (IS 7285), PESO-licensed filling plants.
• Process and refinery vessels: L&T Heavy Engineering (Hazira), Godrej Process Equipment, ISGEC — reactors, columns and heat exchanger shells to IS 2825 / ASME VIII.
• Hydraulics and ordnance: hydraulic cylinders and accumulators, gun barrels and pressure tubing at OFB-successor units (Munitions India, Advanced Weapons & Equipment India) using compound/autofrettage design.
• Pipelines and storage: GAIL and ONGC cross-country pipelines, Horton spheres and LPG bullets at bottling plants.
• Aerospace and automotive: ISRO propellant tanks and solid motor casings, CNG tanks and air brake receivers on commercial vehicles.

Interview questions

• Q: Why is hoop stress twice the longitudinal stress? A: The bursting area on a longitudinal section (d × L) is resisted by 2tL, while the end load on a transverse section is resisted by the full circumference πdt — the ratio works out to 2:1.
• Q: Why do pressurised pipes split lengthwise rather than around? A: Because hoop stress, which acts across a longitudinal plane, is the larger of the two principal stresses.
• Q: When must you switch from thin to thick cylinder formulas? A: When t > d/20; below d/t = 20 the radial stress and the variation of hoop stress across the wall are no longer negligible.
• Q: Where is hoop stress maximum in a thick cylinder? A: At the inner surface (bore), falling towards the outside — which is why cracks start at the bore.
• Q: What is the quickest check on a Lamé solution? A: σθ + σr = 2A at every radius, and σr must equal −p at the bore and zero at a free outer surface.
• Q: Why does increasing thickness alone fail beyond a certain pressure? A: The Lamé thickness formula requires p < allowable σθ; as p approaches it, the required thickness tends to infinity, so a compound cylinder or autofrettage is needed.
• Q: What does a shrink fit achieve? A: It pre-stresses the bore in compression so that the tensile hoop stress under working pressure is reduced and the wall material is used more uniformly.
• Q: What is autofrettage? A: Deliberately over-pressurising a cylinder so the inner layers yield; on release they are held in residual compression by the elastic outer layers.
• Q: Why is a sphere preferred for bulk gas storage? A: Its stress is pd/4t in every direction — half a cylinder's hoop stress — so it needs about half the thickness and has the least surface area per unit volume.
• Q: Why should hemispherical ends be thinner than the cylindrical shell they join? A: To match diametral strain at the junction (th/tc ≈ 0.412 for ν = 0.3) and avoid discontinuity bending stresses.
• Q: Which Indian code governs a boiler shell, and which governs an LPG cylinder? A: The Indian Boiler Regulations, 2026 under the Boilers Act, 2025 for boilers; the Gas Cylinders Rules, 2016 under PESO with IS 3196 for welded LPG cylinders.
• Q: Is the thin-cylinder formula safe for external pressure? A: No — a shell under external pressure fails by elastic buckling well before yielding, so code collapse charts and stiffening rings govern.

Common mistakes

• Using outside diameter instead of inside diameter in pd/2t (and mixing radius with diameter — pd/2t = pr/t).
• Applying thin-cylinder formulas when d/t < 20, which underestimates the bore hoop stress significantly.
• Forgetting that joint efficiency divides the strength: using pd/2t instead of pd/(2tη) for a riveted or un-radiographed welded shell.
• Taking maximum shear stress as σh/2 and (σh − σl)/2 interchangeably without stating the plane; for thick cylinders forgetting that σr = −p makes the shear at the bore (σθ + p)/2.
• Sign errors in Lamé: writing σr = A + B/r², or treating the compressive radial stress as positive.
• Confusing radial with diametral shrinkage interference in compound-cylinder problems — they differ by a factor of 2.
• Mixing units — p in bar with σ in N/mm² (1 bar = 0.1 N/mm²), or dimensions in metres with stress in N/mm².
• Omitting the corrosion allowance and standard plate rounding in a design answer, and ignoring the longitudinal stress when computing strains.

Related topics

Strength of materials: stress and strain · Principal stresses and theories of failure · Strain energy and impact loading · Bending, torsion and columns · Welding and joining processes · Design of machine elements (pressure vessels).