Hoop Stress Calculator

Hoop stress: Stress going around the vessel wall, around its circumference. In a pressurized cylinder, this is usually the largest membrane stress.

Longitudinal stress: Axial stress along the length of a closed-end cylinder. In this thin-wall model, it is half of hoop stress for a closed-end cylinder and 0 for an open-end cylinder [1].

Pressure difference (MPa): Inside pressure minus outside pressure. Positive usually means internal pressure is higher; negative means external pressure is higher.

Size input basis: Tells the calculator whether your entered size is inner, mean, or outer. It converts that value to the mean size used by the thin-wall formula.

Diameter or radius (selected unit): The vessel size you enter using the chosen basis and the chosen size type.

Thin-wall ratio: The wall thickness divided by the mean diameter used for the check. Smaller values support thin-wall formulas better.

Equivalent mean diameter or radius used: The converted mean size used internally after applying your basis choice. This helps verify that the geometry was interpreted the way you intended.

Stress ratio: Hoop stress divided by Allowable stress (MPa). Above 1 means the hoop stress is greater than the limit you entered.

Mistake: Entering an outer size in Diameter or radius (selected unit) while leaving Size input basis on inner.
Fix: Change Size input basis so it matches the number you entered, then confirm Equivalent mean diameter or radius used looks reasonable.

Mistake: Mixing radius and diameter by typing a radius value while Size input type is set to diameter.
Fix: Switch Size input type to the correct choice or enter the matching doubled or halved value.

Mistake: Using 0 or a negative value for Wall thickness (selected unit) .
Fix: Enter a wall thickness greater than 0. The thin-wall formulas need a real positive wall thickness.

Mistake: Expecting Cylinder end condition to matter for a sphere.
Fix: When Vessel shape is Sphere , ignore that field. Check the Hoop stress result only for the spherical shell formula.

Mistake: Solving for Wall thickness or Internal pressure with a blank or zero Hoop stress (MPa) target.
Fix: Enter the target Hoop stress (MPa) with the sign you want. Zero cannot be used in the thickness rearrangement.

Mistake: Reading the result as fully valid even when the Thin-wall validity note says borderline or outside range.
Fix: Use the reported Thin-wall ratio . If it is near or above 0.1, use a thick-wall method for a better check.

• This calculator uses thin-wall membrane formulas, so it is intended for cases where wall thickness is small compared with vessel size. A common warning threshold is t/D_m = 0.1; near or above that, thick-wall analysis may be more appropriate [2][4].
• The geometry is converted to a mean diameter or mean radius before calculation. Results can be wrong if Size input basis does not match the way your size was measured.
• Pressure difference (MPa) is treated as the pressure across the wall. It does not separately model changing inside and outside pressures over time.
• For cylinders, Longitudinal stress is shown only for pressure-caused axial membrane stress in the shell. It does not include extra loads from weight, supports, piping, thermal effects, or bending.
• Negative pressure and negative stress values are allowed so external-pressure cases remain visible, but buckling, collapse, and instability are outside this calculator's scope.
• The optional Allowable stress (MPa) comparison is only a user-entered check. It is not a design code review, safety factor calculation, or material qualification.
• Non-physical geometry is blocked. For example, an outer size that is too small for the entered wall thickness cannot be converted into a valid mean size.

Core formulas

This calculator consistently uses a mean-diameter thin-wall approximation. Other references may use inner diameter; the difference becomes small for a thin wall. For known thickness, your chosen size basis is converted to the mean diameter. When solving thickness from inner or outer size, geometry and stress must be solved together [5].

σ_h (hoop stress) = p x D_m / (2 x t)

Use that formula for a thin-wall cylinder.

σ_L (longitudinal stress) = p x D_m / (4 x t)

For a closed-end thin cylinder, the longitudinal stress is half of hoop stress [4]. For an open-end cylinder in this simplified model, the pressure-caused longitudinal stress in the shell is taken as 0.

σ_h (sphere) = p x D_m / (4 x t)

For a thin spherical shell, the membrane stress is the same in all tangent directions, so only one stress value is needed [1].

Reverse solving

The reverse modes rearrange the same mean-diameter model. The formulas below use a known mean diameter. For inner or outer dimensions, the unknown wall thickness also changes that mean diameter.

p = 2 x t x σ_h / D_m

t = p x D_m / (2 x σ_h)

D_m = 2 x t x σ_h / p

Let k = 2 for a cylinder or 4 for a sphere, and D be the entered diameter (twice the entered radius). Solving thickness gives t = pD/(kσ_h - p) for inner size, t = pD/(kσ_h) for mean size, and t = pD/(kσ_h + p) for outer size. The solved thickness is then used to calculate the displayed mean size and thin-wall ratio. Zero, negative, non-finite or physically impossible thicknesses are rejected.

Geometry basis conversion

If you enter diameter, the mean diameter is found from your selected basis as follows.

D_m = D_i + t

D_m = D_m

D_m = D_o - t

The three lines above correspond to inner, mean, and outer diameter input.

If you enter radius, the mean radius is found first, then converted if needed.

r_m = r_i + t/2

r_m = r_m

r_m = r_o - t/2

D_m = 2 x r_m

When solving size, the stress equation gives mean diameter D_m. If "Size input type" is "Radius", the calculator divides that diameter by two before converting to the selected inner, mean or outer radius. "Equivalent mean diameter or radius used" lets you check that interpretation.

Thin-wall validity check

ratio = t / D_m

The calculator reports this as Thin-wall ratio. Smaller values are more consistent with thin-wall assumptions. A common warning threshold is 0.1, so values near or above 0.1 are flagged clearly rather than treated as fully reliable [3][4].

Worked mini-example

Suppose you choose a cylinder with inner diameter 200 mm, wall thickness 10 mm, and pressure difference 2 MPa. Inner diameter with 10 mm wall thickness gives mean diameter 210 mm.

σ_h = 2 x 210 / (2 x 10) = 21 MPa

σ_L = 2 x 210 / (4 x 10) = 10.5 MPa

ratio = 10 / 210 = 0.0476

That ratio is well below 0.1, so the thin-wall result is a reasonable quick check for this example.

Assumptions behind the math

The model assumes a thin wall, nearly uniform membrane stress through the thickness, and pressure loading as the main cause of stress. Real vessels can differ because of nozzles, weld details, local loads, temperature, end constraints, material limits, corrosion allowance, or code rules. External-pressure stability is also outside this method's scope.