Pressure Vessel & Pipe

Q: When thin-walled (boiler formula), when thick-walled (Lamé)?

As a rule of thumb the boiler formula applies for D/s ≥ 20 or r_a/r_i ≤ 1.1. Then the stress is nearly constant across the wall and the membrane theory is accurate enough. For thicker walls the tangential stress drops markedly from inside to outside and the radial stress is no longer negligible - here the Lamé equations give the correct peak values at the inner fibre.

Q: Why is the hoop stress twice the axial stress?

It follows from two separate equilibria on the cylinder. On the longitudinal section the pressure acts over the projected area D·L against two wall strips 2·s·L, giving σ_t = p·D/(2·s). On the cross section p acts on the circular area against the ring area π·D·s, giving σ_ax = p·D/(4·s). The hoop stress therefore governs - longitudinal welds on cylinders are more highly stressed than circumferential welds.

Q: Von Mises or Tresca - which equivalent stress to use?

Both are acceptable. The shear stress hypothesis (Tresca) is more conservative and, in the thick-walled case, lies about 15 percent above von Mises; many pressure equipment codes use it. The distortion energy hypothesis (von Mises) fits ductile steels better and utilises the material somewhat further. The calculator reports both values; the safety is formed per von Mises.

Q: What do closed and open ends mean?

A vessel with heads or covers (closed ends) transfers the pressure to the end faces and thereby creates an axial stress along the wall. A flow-through pipe without end caps (open ends) has no such axial stress. Since the axial stress is one of the three principal stresses, this choice affects the equivalent stress - the calculator accounts for it via the toggle.

Q: Does the calculator replace a design to AD 2000 or EN 13445?

No. The calculator is a tool for preliminary sizing and provides the basic stresses and a first wall thickness. A certified design to the AD 2000 code or EN 13445 additionally accounts for safety factors, the weld joint efficiency, allowances for corrosion and fabrication, the elevated-temperature strength at operating temperature as well as heads, nozzles and cyclic loading. These checks are not included here.

Calculate the tangential, axial and radial stress and the required wall thickness of pressure vessels and pipes under internal pressure. Thin-walled with the boiler (membrane) formula for cylinders and spheres, thick-walled with the Lamé equations. The calculator forms the equivalent stress per von Mises and Tresca and reports the safety against yield with a traffic light - live with every input. Preliminary sizing, not a certified design to AD 2000 or EN 13445.

Calculation

Preliminary sizing

This calculator provides basic stresses and a first wall thickness. A certified design to AD 2000 or EN 13445 (safety factors, weld joint efficiency, allowances, elevated-temperature strength, heads and nozzles) is not included.

Model and load

Calculation modelShapeInternal pressure p [N/mm²]Reminder: 1 N/mm² = 10 bar.

Closed ends (heads/covers, with axial stress)

Geometry

Inner diameter D [mm]Wall thickness s [mm]

Material and check

MaterialYield strength R_e or R_p0.2 [N/mm²]Required safety factor S_reqApplies to ductile materials against the yield strength under static internal pressure, graded by the consequences of failure and how well the load is known. Cyclic pressure needs a fatigue check, brittle materials a check against tensile strength with 4.0 to 9.0 (Läpple, table 2.2). The safety factors of a certified design to AD 2000 or EN 13445 are not covered by this series.

Model: rotationally symmetric vessel or pipe under static internal pressure, linear-elastic. Boiler (membrane) formula for thin walls, Lamé for thick walls. No check of heads, nozzles, welds, external pressure (buckling) or fatigue. Preliminary sizing for machine design, not a certified pressure equipment design to AD 2000 or EN 13445.

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Formulas and fundamentals

Boiler formula (thin-walled)

For thin-walled vessels (rule of thumb D/s ≥ 20) the wall may be treated as a membrane: the stress is practically constant across the wall and the radial stress is negligible. For the closed cylinder under internal pressure p, the force balance on the longitudinal and the cross section gives the tangential or hoop stress and the axial stress, half as large, with the inner diameter D and wall thickness s:

σ_t = p·D/(2·s) σ_ax = p·D/(4·s)

The spherical shell carries equally in both directions:

σ = p·D/(4·s)

Solved for the stress, the required wall thickness is:

s = p·D/(2·σ_allow) (cylinder) s = p·D/(4·σ_allow) (sphere)

Lamé equations (thick-walled)

For thick-walled pipes (D/s < 20) the stress is no longer constant across the wall but decreases from inside to outside. For internal pressure p, inner radius r_i and outer radius r_a the Lamé equations give, at the governing inner fibre, the maximum tangential stress and the radial stress:

σ_t = p·(r_a²+r_i²)/(r_a²−r_i²) σ_r = −p (compression)

Towards the outer fibre the radial stress drops to zero and the tangential stress to σ_t = 2·p·r_i²/(r_a²−r_i²). With closed ends the constant axial stress is added:

σ_ax = p·r_i²/(r_a²−r_i²)

Equivalent stress and safety

The equivalent stress is formed from the three principal stresses: per the distortion energy hypothesis (von Mises) and per the shear stress hypothesis (Tresca):

σ_v(von Mises) = √(½·[(σ_t−σ_ax)²+(σ_ax−σ_r)²+(σ_r−σ_t)²]) σ_v(Tresca) = σ_max − σ_min

At the inner fibre of the thick-walled pipe:

σ_v(Tresca) = σ_t − σ_r = 2·p·r_a²/(r_a²−r_i²)

The check against yield compares the available safety S = R_e/σ_v with the required safety S_req; the traffic light shows green from S_req, amber from 1 and red below.

Worked example

A cylindrical vessel with inner diameter D = 500 mm and wall thickness s = 5 mm is under an internal pressure of p = 1 N/mm² (10 bar). With D/s = 100 it is clearly thin-walled and the boiler formula applies. The hoop stress is σ_t = p·D/(2·s) = 1·500/(2·5) = 50 N/mm², the axial stress σ_ax = 25 N/mm².

With σ_r ≈ 0 the von Mises equivalent stress is σ_v = √(50²−50·25+25²) = 43.3 N/mm². For a boiler steel with R_e = 235 N/mm² the safety is S = 235/43.3 = 5.4 - ample reserve at S_req = 1.5. The purely computed required wall thickness would be s = p·D/(2·σ_allow) = 500/(2·157) = 1.6 mm; the chosen 5 mm additionally cover allowances and fabrication.

A thick-walled hydraulic pipe with r_i = 50 mm and r_a = 70 mm under p = 10 N/mm² (100 bar) shows the Lamé case instead: σ_t,max at the inner fibre = 10·(70²+50²)/(70²−50²) = 30.8 N/mm², σ_r = −10 N/mm². The Tresca equivalent stress is σ_v = σ_t − σ_r = 40.8 N/mm² - noticeably larger than the tangential peak alone, because the internal pressure acts as the third principal stress.

Frequently asked questions

When thin-walled (boiler formula), when thick-walled (Lamé)?

As a rule of thumb the boiler formula applies for D/s ≥ 20 or r_a/r_i ≤ 1.1. Then the stress is nearly constant across the wall and the membrane theory is accurate enough. For thicker walls the tangential stress drops markedly from inside to outside and the radial stress is no longer negligible - here the Lamé equations give the correct peak values at the inner fibre.

Why is the hoop stress twice the axial stress?

It follows from two separate equilibria on the cylinder. On the longitudinal section the pressure acts over the projected area D·L against two wall strips 2·s·L, giving σ_t = p·D/(2·s). On the cross section p acts on the circular area against the ring area π·D·s, giving σ_ax = p·D/(4·s). The hoop stress therefore governs - longitudinal welds on cylinders are more highly stressed than circumferential welds.

Von Mises or Tresca - which equivalent stress to use?

Both are acceptable. The shear stress hypothesis (Tresca) is more conservative and, in the thick-walled case, lies about 15 percent above von Mises; many pressure equipment codes use it. The distortion energy hypothesis (von Mises) fits ductile steels better and utilises the material somewhat further. The calculator reports both values; the safety is formed per von Mises.

What do closed and open ends mean?

A vessel with heads or covers (closed ends) transfers the pressure to the end faces and thereby creates an axial stress along the wall. A flow-through pipe without end caps (open ends) has no such axial stress. Since the axial stress is one of the three principal stresses, this choice affects the equivalent stress - the calculator accounts for it via the toggle.

Does the calculator replace a design to AD 2000 or EN 13445?

No. The calculator is a tool for preliminary sizing and provides the basic stresses and a first wall thickness. A certified design to the AD 2000 code or EN 13445 additionally accounts for safety factors, the weld joint efficiency, allowances for corrosion and fabrication, the elevated-temperature strength at operating temperature as well as heads, nozzles and cyclic loading. These checks are not included here.

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