Engineering calculators

Hoop Stress Calculator

Updated Sep 19, 2026 By Infinity Calculator
Rate Formulas
Configuration
Vessel Shape
Thin-wall closed vessel under internal pressure.
Calculation Mode
ASME B31.8 mode adds pipeline grade, design factors, MAOP and % SMYS.
Unit System
Sets every unit menu at once — each field can still be overridden individually.
Dimensions
Pipe Sizing Method
Outer diameter and wall thickness are filled in automatically.
d = OD − 2t
r = d / 2
Editable — back-calculates t from the current inner diameter.
Outer diameter divided by wall thickness (calculated).
Used only for the change-in-volume output.
Pressure & Material
Material Properties (optional)
Dimensionless, typically 0.25–0.35 for steel.
Drives the stress utilization gauge in Standard mode.
Adds weld joint efficiency (η) to the hoop and longitudinal stress formulas.
Results
Forward calculation: σh is computed from P, d and t.
Hoop (Circumferential) Stress σh solved
Longitudinal Stress σl
—
Radial Stress σr (inner wall)
—
Max Out-of-Plane Shear τmax
—
Hoop Stress is 2× Longitudinal Stress (σh = 2 · σl) — use Hoop Stress as the governing design stress.
Geometry check: D/t = —  |  t/d = —  |  t/r = —
Stress Utilization vs. Yield Strength
σh / σy —
—
Change in Dimensions
Change in Diameter δd
—
Change in Volume δV
—
Vessel Cross-Section
    Principal Stress Summary
    ComponentSymbolValueRelative to σh
    Hoop Stress vs. Wall Thickness
    Principal Stress Comparison
    Step-by-Step Solution

    Introduction

    When gas or liquid pushes out on the inside of a pipe or tank, the wall gets stretched around its curve. That pull is called hoop stress. It is usually the biggest stress in a pressure vessel, so engineers check it first. If hoop stress gets too high, the wall can crack or burst.

    This hoop stress calculator works out hoop stress from the pressure, the diameter, and the wall thickness. You can also flip it around and solve for wall thickness, pressure, or inner diameter instead. It works for both pipes (cylinders) and round tanks (spheres), in imperial or metric units.

    You can pick a standard pipe size and schedule, and the tool fills in the outer diameter and wall thickness for you. It also shows longitudinal stress, radial stress, and max shear stress, plus how close your design is to the yield strength of the metal. Pipeline engineers can switch to ASME B31.8 mode to find MAOP and check the percent of SMYS.

    Every answer comes with a step-by-step solution, a cross-section drawing, and charts, so you can see exactly how the numbers were found.

    How to use our Hoop Stress Calculator

    Enter your pipe or vessel size, the inside pressure, and (if you want) the material facts. The calculator gives you hoop stress, longitudinal stress, radial stress, max shear stress, size changes, a stress chart, and full step-by-step math.

    Vessel Shape: Pick Cylinder for pipes and tanks, or Sphere for round vessels. The formula changes with the shape.

    Calculation Mode: Pick Standard for basic thin-wall math. Pick ASME B31.8 to check a pipeline against code limits, MAOP, and % SMYS.

    Unit System: Pick Imperial or Metric. This sets every unit menu at once. You can still change any single unit menu.

    Pipe Sizing Method: Choose NPS + Schedule Lookup to fill in pipe sizes for you, or Custom Dimensions to type your own.

    Nominal Pipe Size: Pick the NPS or DN size of your pipe. The outer diameter is filled in for you.

    Pipe Schedule: Pick the schedule, like 40 or 80. This sets the wall thickness.

    Outer Diameter (OD): Type the outside diameter of the pipe or vessel and pick a unit.

    Inner Diameter (d): Type the inside diameter. It equals OD minus two wall thicknesses, so it updates on its own.

    Inner Radius (r): Half of the inner diameter. Type it here if that is easier.

    Wall Thickness (t): Type how thick the wall is. This is the biggest driver of hoop stress.

    t/d Ratio: Type a wall-to-diameter ratio and the calculator works out the wall thickness for you.

    D/t Ratio: This box fills itself. It shows outer diameter divided by wall thickness, used to check the thin-wall rule.

    Cylinder Length (L): Type the length of the cylinder. It is only used for the change-in-volume answer.

    Internal Pressure (P): Type the pressure inside the vessel in psi, kPa, MPa, or bar.

    Young's Modulus (E): Type the stiffness of your material. Steel is about 29,000,000 psi or 200 GPa. Needed for size change results.

    Poisson's Ratio (μ): Type a number with no units, often 0.3 for steel.

    Material Yield Strength (σy): Type the yield strength. This runs the stress gauge that shows how close you are to yield.

    Pipe Grade (API 5L / CSA): In ASME mode, pick your pipe grade, like X52. Choose Custom to type your own value.

    SMYS: The minimum yield strength for that grade. It fills in on its own unless you picked Custom.

    Location Class → Design Factor (F): Pick the class for where the pipeline runs. Busier areas use a lower factor.

    Longitudinal Joint Factor (E): Leave it at 1.0 for seamless pipe, or tick the box to type your own value.

    Temperature Derating Factor (T): Leave it at 1.0 for normal temperatures, or tick the box to lower it for hot service.

    Advanced Mode: Turn this on to add weld joint efficiency to the stress math.

    Weld Seam Type: Pick how the vessel is welded and tested. This sets the joint efficiency (η).

    Joint Efficiency (η): A number from 0 to 1. It fills in from the seam type, or you can type it if you chose Custom.

    Solve For: Pick what you want to find: hoop stress, wall thickness, pressure, or inner diameter. The boxes you need to fill will change.

    Hoop Stress (σh): This is the main answer. If you are solving for thickness, pressure, or diameter, type your target hoop stress here instead.

    Calculate and Reset: Press Calculate to see results, or Reset to go back to the starting sample values.

    What Is Hoop Stress?

    Hoop stress is the push that pressure makes inside the wall of a pipe, tank, or tube. When gas or liquid inside pushes out, the wall stretches around the circle, like a tight hoop on a barrel. That stretch is hoop stress, also called circumferential stress.

    It is the biggest stress in a pressure vessel. That is why a pipe that bursts splits along its length, not around it. The wall pulls apart sideways first, because hoop stress is the strongest force acting on it.

    The Hoop Stress Formula

    For a thin-wall pipe or cylinder:

    σh = P × d / (2t)

    • σh = hoop stress (psi or MPa)
    • P = pressure inside
    • d = inner diameter
    • t = wall thickness

    For a sphere, the wall shares the load in two ways, so the stress is cut in half:

    σh = P × d / (4t)

    The math shows what makes sense: more pressure or a wider pipe means more stress. A thicker wall means less stress.

    Hoop Stress vs. Longitudinal Stress

    A closed pipe also gets pulled end to end. That is longitudinal (axial) stress, and it is always half of the hoop stress:

    σl = P × d / (4t) = σh / 2

    There is a third one, radial stress, which squeezes the wall inward. In thin walls it is small, so designers mostly ignore it. Hoop stress is the one that rules the design.

    Thin Wall or Thick Wall?

    The simple formula only works when the wall is thin next to the radius. The common rule is t/r less than 0.1, or D/t greater than 20. If the wall is thicker than that, stress is not even across the wall. It is higher on the inside face. For those cases engineers use Lamé's thick-wall equations instead.

    ASME B31.8 and Pipelines

    Gas pipelines follow the ASME B31.8 code. It uses the Barlow formula with the outer diameter, which is safer because it gives a slightly higher stress:

    S = P × OD / (2t)

    The code then limits how much of the steel's strength you may use. It compares stress to SMYS (Specified Minimum Yield Strength) after cutting it down with factors:

    • F is the design factor, set by location class (0.40 to 0.80). Pipes near homes and cities get lower numbers.
    • E is the joint factor, based on how the pipe seam was welded.
    • T is the temperature factor, which drops when the pipe runs hot.

    From these you get MAOP, the highest pressure the line is allowed to carry:

    MAOP = 2 × t × SMYS × F × E × T / OD

    Weld Joint Efficiency

    A welded seam is often weaker than solid steel. Joint efficiency (η) accounts for that. Seamless pipe and fully X-rayed welds get η = 1.0. Spot-checked welds get about 0.85, and welds with no X-ray check get about 0.70. Divide by η and the stress goes up, so a weaker seam needs a thicker wall.

    Why It Matters

    Hoop stress sets the wall thickness of boilers, gas lines, water mains, air tanks, scuba cylinders, and hydraulic tubing. Pick a wall that is too thin and the part yields, bulges, or bursts. Pick one far too thick and you waste steel and money. Good design keeps hoop stress well under the yield strength, usually with a safety factor of about 1.5 or more.

    Formulas used

    Hoop (circumferential) stress — thin-wall cylinder
    \sigma_h = \frac{P\,d}{2\,t\,\eta}
    Hoop stress — thin-wall sphere
    \sigma_h = \frac{P\,d}{4\,t\,\eta}
    Longitudinal (axial) stress — cylinder
    \sigma_l = \frac{P\,d}{4\,t\,\eta} = \frac{\sigma_h}{2}
    Radial stress at inner wall and maximum out-of-plane shear
    \sigma_r = -\frac{P}{2}, \qquad \tau_{max} = \frac{\sigma_h - \sigma_r}{2}
    ASME B31.8 Barlow hoop stress (OD basis) and MAOP
    S = \frac{P \cdot OD}{2t}, \qquad MAOP = \frac{2\,t\,SMYS\,F\,E\,T}{OD}
    Change in diameter (cylinder / sphere)
    \delta d_{cyl} = \frac{P\,d^{2}}{2tE}\left(1-\frac{\mu}{2}\right), \qquad \delta d_{sph} = \frac{P\,d^{2}}{4tE}\left(1-\mu\right)
    Volumetric strain and change in volume
    \frac{\delta V}{V}\Big|_{cyl} = \frac{P\,d}{2tE}\left(2.5-2\mu\right), \qquad \frac{\delta V}{V}\Big|_{sph} = \frac{3P\,d}{4tE}\left(1-\mu\right), \qquad \delta V = V \cdot \frac{\delta V}{V}
    Geometry relations and utilization ratios
    d = OD - 2t,\quad r = \frac{d}{2},\quad \frac{D}{t} = \frac{OD}{t},\quad \frac{\sigma_h}{\sigma_y}\times 100\%

    Frequently asked questions

    Why is hoop stress twice the longitudinal stress in a pipe?

    It comes from how much wall area fights each load.

    • Hoop: pressure pushes out on the whole length of the pipe, and only two thin strips of wall hold it together.
    • Longitudinal: pressure pushes on the end cap, and the full ring of wall holds it.

    The ring has twice as much metal working for it, so the end-to-end stress is half as big. That is why σh = 2 × σl for any closed cylinder, no matter the size.

    Should I use the inside or outside diameter in the hoop stress formula?

    Both are used, and each gives a slightly different number.

    • Inside diameter (d): the classic thin-wall answer, σ = Pd/2t. Gives the lowest value.
    • Outside diameter (OD): the Barlow form, S = P·OD/2t. Used by ASME B31.8 and B31.4 because it is a bit more conservative.
    • Mean diameter: sits between the two and is the most exact for thin walls.

    For a thin pipe the three answers are within a few percent. For thick walls the gap grows, so match the diameter your code asks for.

    How do I find the wall thickness a pipe needs for a given pressure?

    Flip the hoop stress formula around:

    t = P × d / (2 × σallow)

    Where σallow is the stress you allow, not the yield strength. Most designs use yield divided by a safety factor, or the code allowable.

    Example: 1,000 psi in a 12 in pipe with a 20,000 psi allowable stress needs t = (1000 × 12) / (2 × 20000) = 0.30 in. Then add a corrosion allowance and round up to the next real pipe schedule.

    How do you calculate the burst pressure of a pipe?

    Use the same Barlow formula, but swap in the ultimate tensile strength instead of yield:

    Pburst = 2 × t × UTS / OD

    Example: a 6.625 in OD pipe with a 0.280 in wall and 60,000 psi UTS bursts near 2 × 0.280 × 60,000 / 6.625 = 5,070 psi.

    This is a theoretical number for new, perfect pipe. Real pipe has weld seams, dents, and rust, so never run near it. Working pressure is usually a quarter of burst or less.

    What does 72% SMYS mean for a gas pipeline?

    It means the hoop stress in the pipe wall is allowed to reach 72% of the steel's specified minimum yield strength.

    That is the design factor F = 0.72, used for Class 1 Division 2 areas (open country with few homes). The rest of the yield strength is the safety margin.

    Busier places get stricter limits:

    • Class 1: F = 0.72 (0.80 in Div 1)
    • Class 2: F = 0.60
    • Class 3: F = 0.50
    • Class 4: F = 0.40

    What is the hoop stress formula for a thick-walled cylinder?

    When the wall is thick (t/r above 0.1), stress is not even. It peaks on the inside face. Use Lamé's equation:

    σh = P(ro² + ri²) / (ro² − ri²) at the inner wall

    Here ri is the inner radius and ro the outer radius. Stress drops as you move outward, so adding more metal to a very thick pipe helps less and less. The thin-wall formula under-reads this peak, which is why the t/r check matters.

    Does a longer pipe have higher hoop stress?

    No. Length does not appear in the formula at all. Hoop stress depends only on pressure, diameter, and wall thickness.

    A 10 ft pipe and a 10 mile pipe of the same size and pressure have the same hoop stress. Length only matters for volume, weight, expansion, and pressure drop along the line.

    Diameter is what changes things: double the diameter and you double the hoop stress.

    Does Schedule 80 pipe hold twice the pressure of Schedule 40?

    No. The number is not a pressure rating. Pressure follows wall thickness, and Schedule 80 walls are roughly 1.4 to 1.6 times thicker than Schedule 40, not double.

    Example, NPS 6 steel pipe:

    • Sch 40: wall 0.280 in
    • Sch 80: wall 0.432 in

    That is about 54% more wall, so about 54% more pressure capacity. Schedule also shifts with pipe size. Sch 40 on a 2 in pipe is a very different thickness than Sch 40 on a 24 in pipe.

    What is the difference between MAOP and MAWP?

    They are the same idea used in two different worlds.

    • MAOP (Maximum Allowable Operating Pressure) is for pipelines, set by ASME B31.8 or B31.4. It uses SMYS with the F, E, and T factors.
    • MAWP (Maximum Allowable Working Pressure) is for tanks and pressure vessels, set by ASME Section VIII. It uses a code allowable stress and joint efficiency.

    Both are the top pressure the equipment may see in normal service, and both are stamped or recorded so operators never exceed them.

    How does a corrosion allowance change the required wall thickness?

    You add it on top of the thickness the pressure math asks for.

    ttotal = tpressure + corrosion allowance + mill tolerance

    A common allowance is 1/16 in (0.0625 in) or 1.5 mm for carbon steel in wet or sour service. Steel pipe also comes with a mill tolerance of about 12.5% under nominal, so many designers divide by 0.875 as well.

    Example: 0.30 in needed for pressure, plus 0.0625 in for rust, gives 0.3625 in, so you pick the next schedule up.

    How much does temperature lower a pipe's pressure rating?

    Steel loses strength as it heats, so codes cut the allowable stress with a temperature derating factor T.

    ASME B31.8 values for steel:

    • Up to 250°F: T = 1.000
    • 300°F: T = 0.967
    • 350°F: T = 0.933
    • 400°F: T = 0.900
    • 450°F: T = 0.867

    At 450°F the line may only carry about 87% of its cold pressure. Below 250°F there is no cut at all.

    How much does a pipe stretch under internal pressure?

    Very little, but it is measurable. Hoop strain is:

    εh = (σh − μσl) / E

    Example: a 6 in steel pipe at 13,000 psi hoop stress, with E = 29,000,000 psi and μ = 0.3, gives a strain near 0.00038. The diameter grows about 0.002 in.

    That tiny growth still matters for pressure testing, for volume change in closed systems, and for tight-fitting seals and flanges.

    What is the von Mises stress in a pressure vessel?

    Von Mises blends the hoop, longitudinal, and radial stresses into one number you can compare to yield strength.

    For a thin cylinder where σl = σh/2 and radial stress is ignored:

    σv ≈ 0.866 × σh

    So the combined stress is a bit lower than the hoop stress alone. Checking hoop stress by itself is the safer, simpler route, which is why codes do it that way.

    Does hoop stress apply when pressure is on the outside of a pipe?

    The formula still gives a number, but the failure mode flips. With outside pressure or a vacuum inside, the wall goes into compression, not tension.

    Thin pipes do not crush evenly. They buckle and collapse into an oval long before the metal reaches yield. Collapse pressure drops sharply as D/t rises, roughly with the cube of the wall-to-diameter ratio.

    For vacuum service, subsea pipe, or jacketed lines, size the wall with external pressure buckling rules, not the hoop stress formula.

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