What is the criterion for thin-wall vs thick-wall pressure vessels?

A vessel is classified as thin-wall when the ratio of wall thickness t to inner radius r_i is less than 0.1 (t/r_i < 0.1). In this regime the simple hoop-stress formula σ_θ = p·r/t gives acceptable accuracy. When t/r_i ≥ 0.1 the vessel is thick-wall and the Lamé equations must be used for an exact solution. This tool evaluates the ratio automatically and always applies the full Lamé solution regardless of classification.

What are the Lamé equations for a thick-wall pressure vessel?

For a thick-wall cylinder under internal pressure p_i and external pressure p_o, the Lamé equations give hoop stress σ_θ(r) = A + B/r² and radial stress σ_r(r) = A − B/r², where A = (p_i·r_i² − p_o·r_o²)/(r_o² − r_i²) and B = (p_i − p_o)·r_i²·r_o²/(r_o² − r_i²). The hoop stress reaches its maximum at the inner wall (r = r_i) and decreases toward the outer wall.

How is the ASME Section VIII minimum wall thickness calculated?

The ASME Section VIII Div. 1 formula for minimum required wall thickness is t_min = p·R / (S·E − 0.6p), where S is the allowable stress (approximately yield stress / 3), E is the weld joint efficiency (1.0 for seamless), p is the design pressure, and R is the inner radius. This tool automatically sets S from the selected material yield stress and computes t_min in real time.

How is von Mises stress computed and what does the safety factor mean?

Under plane-stress approximation the von Mises equivalent stress is σ_vM = √(σ_θ² − σ_θ·σ_r + σ_r²). The safety factor S.F. = σ_Y / σ_vM_max, where σ_Y is the material yield stress. The tool color-codes the result: green (S.F. ≥ 2.0, safe), orange (1.0 ≤ S.F. < 2.0, caution), and red (S.F. < 1.0, yielding). For three-dimensional stress states including axial stress, a full triaxial von Mises calculation should be performed.

/ Pressure Vessel Stress Calculator Tools Index

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Structural Analysis Tool

Pressure Vessel Stress Calculator
Thin-Wall, Thick-Wall & Lamé Equations

Real-time calculation of through-wall stress distributions in pressure vessels using the Lamé equations. Automatic thin/thick-wall classification, ASME minimum wall thickness evaluation, and von Mises stress distribution visualization.

$$\sigma_\theta(r) = \frac{p_i r_i^2}{r_o^2 - r_i^2}\!\left(1 + \frac{r_o^2}{r^2}\right)$$

Parameter Settings

Inner radius r_i 250 mm

Wall thickness t 25 mm

Internal pressure p_i 10.0 MPa

External pressure p_o 0.0 MPa

Material

Geometry

Cylinder Sphere

Classification criterion
t/r_i < 0.1 → Thin-wall approximation valid
t/r_i ≥ 0.1 → Thick-wall (Lamé required)

σ_θ max (hoop)

—

MPa

σ_r inner wall

—

MPa

σ_vM max

—

MPa

Safety factor S.F.

—

σ_Y / σ_vM

t/R ratio

—

Thin-wall check

ASME t_min

—

Through-Wall Stress Distribution — Lamé Equations

Cross-section (inner wall = high stress)

Theory — Lamé Equations / ASME Evaluation

Hoop stress σ_θ (Lamé)

$$\sigma_\theta(r) = \frac{p_i r_i^2}{r_o^2 - r_i^2}\!\left(1 + \frac{r_o^2}{r^2}\right)$$

Maximum at the inner wall (r = r_i).

Radial stress σ_r

$$\sigma_r(r) = \frac{p_i r_i^2}{r_o^2 - r_i^2}\!\left(1 - \frac{r_o^2}{r^2}\right)$$

Equal to −p_i at the inner wall, zero at the outer wall (when p_o = 0).

ASME Sec. VIII Minimum Wall Thickness

$$t_{\min} = \frac{p \cdot R}{S \cdot E - 0.6p}$$

S = σ_Y / 3 (allowable stress), E = 1 (seamless)

von Mises Equivalent Stress

$$\sigma_{vM} = \sqrt{\sigma_\theta^2 - \sigma_\theta\sigma_r + \sigma_r^2}$$

Plane stress approximation (axial stress treated separately if required).

Pressure Vessel Stress CalculatorThin-Wall, Thick-Wall & Lamé Equations

Hoop stress σ_θ (Lamé)

Radial stress σ_r

ASME Sec. VIII Minimum Wall Thickness

von Mises Equivalent Stress

Pressure Vessel Stress Calculator
Thin-Wall, Thick-Wall & Lamé Equations