CELE Strength of Materials — Thin-Walled Pressure VesselsSummary

A pressure vessel qualifies as thin-walled when the ratio of wall thickness to inner radius satisfies t/r ≤ 1/10 (equivalently, r/t ≥ 10). Under this condition, stress is essentially uniform across the wall thickness, and simple membrane stress analysis (based on equilibrium of forces alone) is valid. If t/r > 1/10, the vessel is thick-walled and requires the more complex Lamé (thick-cylinder) equations. Always verify this criterion after calculating required thickness.

When a cylindrical vessel is cut longitudinally and the two halves are held in equilibrium, the internal pressure creates a stress acting in the circumferential (tangential, or 'hoop') direction. This stress acts to expand the vessel circumference and tends to split it along a longitudinal seam. The formula is σₕ = pD/(2t) = pr/t, where p is gauge pressure, D is inner diameter, r is inner radius, and t is wall thickness. This is the larger of the two principal stresses in a cylinder.

When a cylindrical vessel is cut perpendicular to its axis, the internal pressure creates a stress in the axial direction. This stress acts to push the end caps outward and is given by σₗ = pD/(4t) = pr/(2t). Critically, the longitudinal stress is exactly half the hoop stress: σₗ = σₕ/2. This stress governs the strength of the circumferential (transverse) seams.

A spherical vessel, by symmetry, experiences the same stress in all directions through any great-circle cut. The single membrane stress is σ = pD/(4t) = pr/(2t). Note that this equals the longitudinal stress of a cylinder and is half the cylinder's hoop stress. For identical pressure, radius, and thickness, a sphere is stressed only half as much as a cylinder.

In pressure-vessel analysis, always use gauge pressure (internal pressure above atmospheric), not absolute pressure. If a problem states 'pressure is 2 MPa,' this is gauge. If it says '2 MPa absolute,' subtract 0.101 MPa to get gauge (or use 1 MPa if 1 MPa absolute). The formulas assume gauge pressure, because atmospheric pressure acts on both inside and outside surfaces and cancels.

Real pressure vessels have seams created by welding or riveting. These seams are weaker than the parent plate and are quantified by a joint efficiency factor η (dimensionless, 0 < η ≤ 1). When sizing a vessel wall, the allowable stress is reduced to η·σ_allow across the seam. The design formula for a cylinder becomes t = pD/(2·η·σ_allow). Typical efficiencies: double-welded butt seam η = 0.85–1.0; single-welded seam η = 0.60–0.80; riveted seam η = 0.60–0.75.

Given allowable stress σ_allow and joint efficiency η, the minimum wall thickness of a cylindrical vessel is determined by the hoop stress (the governing criterion): t_min = pD/(2·η·σ_allow). After calculating t, verify that t/r ≤ 1/10 to confirm the thin-wall assumption remains valid. If t/r > 1/10, the problem falls outside thin-wall theory and must be solved using thick-cylinder equations.

For a spherical vessel under the same design criteria, the single membrane stress governs: t_min = pD/(4·η·σ_allow). Because the stress is half that of a cylinder, the required thickness is also half (or the allowable pressure is double). This is why spheres are preferred for high-pressure applications.

In a cylindrical vessel, the two principal in-plane stresses are σₕ (hoop) and σₗ (longitudinal). The radial stress σᵣ (perpendicular to the wall) is approximately zero for thin walls. The maximum in-plane shear is τ_max = (σₕ − σₗ)/2 = pr/(4t). The absolute maximum shear (from Mohr's circle accounting for the zero radial stress) is τ_abs = σₕ/2 = pr/(2t). For a sphere, all in-plane stresses are equal, so in-plane shear is zero; absolute maximum shear is τ_abs = σ/2 = pr/(4t).

In a cylinder, hoop stress (the larger stress) acts to tear open the longitudinal seam (running along the axis). Longitudinal stress acts to separate the circumferential (transverse) seams (perpendicular to the axis). Because hoop is the larger stress, the longitudinal seam is the critical seam and governs design. In a sphere, by symmetry, there is no preferred seam — all great-circle cuts are equivalent.

Stress is proportional to diameter (or radius): σ ∝ D. This is why large-diameter vessels at the same pressure and thickness experience higher stress. Conversely, if a design specifies allowable stress, larger diameter requires thicker walls. Many errors arise from swapping D and r or forgetting to convert diameter to radius.

Large-diameter steel or concrete penstocks carry water from reservoirs at high pressure (1–3 MPa) down to turbines. Engineers must design the wall thickness to resist hoop stress from internal pressure while accounting for external loads (water weight, topography). The penstock is a classic thin-wall cylinder application. Failure of a penstock seam can cause catastrophic water loss and loss of generating capacity. Design requires careful verification of the thin-wall criterion, selection of joint-welded or riveted construction with appropriate efficiency, and often a factor of safety of 2–3 on stress.

Cylindrical water tanks store municipal or agricultural supply water. Tank walls experience hoop stress from internal water pressure (which increases with depth, but design uses the maximum pressure at the base). For typical tanks (2–10 m diameter, pressures 0.2–0.8 MPa), the thin-wall formula governs sizing. Joint efficiency is critical if the tank is welded or riveted; poor welds can lead to failure. Corrosion allowance (adding 2–3 mm to the calculated thickness) is standard practice to account for long-term deterioration.

Industrial boilers generate steam under pressure (0.5–5 MPa) in cylindrical shells. The hoop stress is the dominant stress and critical for safety. Boiler codes (e.g., ASME B&PV Code, adopted in the Philippines via RA 9296 for pressure vessels) mandate joint efficiencies, material grades, and inspection protocols. A single weld failure can cause an explosion. Boiler design is highly regulated and requires certification by professional engineers and third-party inspectors.

Liquified petroleum gas (LPG) and compressed natural gas (CNG) storage uses spherical or cylindrical tanks at pressures of 1–10 MPa. Spherical tanks are preferred because they require half the wall thickness of cylinders and distribute stress uniformly, reducing the risk of seam failure. The cost savings in material and the safety advantage make spheres the industry standard for stationary storage, despite higher fabrication complexity. Mobile LPG cylinders are often cylindrical for practical handling.

Long-distance pipelines (water, oil, natural gas) are designed as thin-wall cylinders. Pipe sizing balances pressure loss (favoring larger diameter) against wall thickness cost (favoring smaller diameter). The hoop stress formula directly determines wall thickness; allowable stress is set by material grade (e.g., API 5L steel grades B, X42, X65) and desired reliability. Pipeline failure (e.g., seam rupture) can cause environmental contamination and loss of service. Design includes corrosion allowance, external loads (burial depth, bending), and fatigue if flow is pulsating.

High-head hydroelectric systems use concrete-lined pressure tunnels instead of steel penstocks in some designs. Concrete resists external pressure and loads but is weak in tension. The internal water pressure creates hoop tension stress in the concrete; if this stress exceeds the tensile strength of concrete, reinforcing steel (or pre-stressed concrete) is added. The thin-wall hoop stress formula applies to estimate stress; design requires a hoop-reinforcement calculation per ACI 318 provisions for cylindrical sections.

Submarine hulls are cylindrical pressure vessels designed to resist external water pressure (hydrostatic pressure increases with depth: p = ρ_water·g·h). The hull acts in compression (opposite of the internal-pressure case) and failure occurs when compressive hoop stress exceeds the critical buckling stress. Design uses the thin-wall cylinder formulas with 'negative' (compressive) stress and adds buckling stability criteria. This is an advanced application but demonstrates that the thin-wall theory extends beyond simple internal-pressure cases.

Industrial filter vessels (water treatment, oil refining) are small cylindrical or spherical tanks designed to contain filtered fluid at elevated pressure. Design is straightforward: calculate hoop stress at the maximum expected pressure, apply a safety factor (typically 2–4 for pressure equipment), and size wall thickness. Corrosion allowance and connection loads add complexity. These vessels are often manufactured with multiple small seams (inlet, outlet, drain connections), each requiring joint efficiency accounting.