Pressure Vessel Calculator

✓ Verified Content: All equations, formulas, and reference data in this simulation have been verified by the Simulations4All engineering team against authoritative sources including ASME Section VIII standards and peer-reviewed publications. See verification log

Quick Answer

What is hoop stress in a pressure vessel?

Hoop stress (circumferential stress) is the stress acting tangentially around a pressure vessel wall. For thin-walled cylindrical vessels (t/r < 0.1), hoop stress is calculated as σh = Pr/t, where P is internal pressure, r is internal radius, and t is wall thickness. Hoop stress is always twice the longitudinal stress in cylinders, making it the critical design stress. For thick-walled vessels, use the Lame equations which account for stress variation through the wall thickness.

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Introduction to Pressure Vessel Analysis

Here is something that keeps me up at night: a seemingly innocuous steel tank, sitting quietly in an industrial facility, containing enough potential energy to level a city block. Pressure vessels are everywhere around us, from the propane tank on your backyard grill to the massive reactor vessels in chemical plants, and they all share one terrifying reality. The internal pressure wants out. Badly.

I have spent years analyzing failed pressure vessels, and every single failure taught me the same lesson. The load has to go somewhere, and if you have not designed a proper path for that load, the vessel will create its own path in spectacular and often tragic fashion. This simulator lets you explore the stress distributions that determine whether a vessel safely contains its contents or becomes a catastrophic failure waiting to happen.

What makes pressure vessels particularly fascinating from a structural perspective is the elegant simplicity of thin-wall theory combined with the brutal complexity of what happens when those thin-wall assumptions break down. When the wall thickness exceeds about 10% of the radius, we enter thick-wall territory where the stress distribution through the wall becomes non-uniform, and suddenly those simple formulas that every engineering student memorizes become dangerously optimistic [1].

Thin-Wall vs. Thick-Wall Analysis

The first decision any pressure vessel designer must make is whether thin-wall approximations are valid. This is where structures fail when engineers blindly apply thin-wall formulas to vessels that clearly violate the underlying assumptions.

Thin-Wall Theory (t/r ≤ 0.1)

For thin-walled cylindrical vessels, the stress distribution is remarkably simple. The hoop stress (circumferential stress) is given by:

σ_h = Pr/t

where P is the internal pressure, r is the internal radius, and t is the wall thickness. The longitudinal stress is exactly half:

σ_l = Pr/2t

This 2:1 ratio between hoop and longitudinal stress explains why cylindrical pressure vessels almost always fail along longitudinal seams rather than circumferential ones. The metal is working twice as hard in the hoop direction [2].

Thick-Wall Theory (Lamé Equations)

When t/r exceeds 0.1, the stress distribution through the wall thickness becomes significant. Gabriel Lamé developed the equations that describe this behavior in 1833, and they remain the gold standard for thick-wall analysis [3]:

σ_h = (P_i × r_i² - P_o × r_o²)/(r_o² - r_i²) + (P_i - P_o) × r_i² × r_o² / (r² × (r_o² - r_i²))

The key insight from Lamé equations is that the maximum stress occurs at the inner surface of the vessel wall, not uniformly through the thickness as thin-wall theory assumes. This means a thick-walled vessel designed using thin-wall formulas will be under-designed where it matters most.

Key Parameters

Von Mises Equivalent Stress

Real materials do not fail according to simple uniaxial stress criteria. They fail according to distortion energy theory, which is why we use the Von Mises equivalent stress to assess safety [4]:

σ_vm = √[((σ_h - σ_l)² + (σ_l - σ_r)² + (σ_h - σ_r)²) / 2]

This equivalent stress represents the uniaxial tensile stress that would produce the same distortion energy as the actual multiaxial stress state. For cylindrical pressure vessels under internal pressure only, the Von Mises stress is approximately 0.866 times the hoop stress. A vessel that looks safe based on hoop stress alone might actually be operating closer to yield than you think.

ASME Code Compliance

ASME Section VIII Division 1 provides the design rules that govern most pressure vessels in North America [5]. The code uses a safety factor of 3.5 on ultimate tensile strength (reduced from the historical 4.0 in 1999) to determine allowable stress:

S = S_u / 3.5

The Maximum Allowable Working Pressure (MAWP) is then calculated as:

MAWP = S × E × t / (r + 0.6t)

where E is the joint efficiency (typically 0.85-1.0 depending on weld type and inspection level). The 0.6t term in the denominator accounts for the mean radius being slightly larger than the inner radius.

End Cap Stress Factors

For cylindrical vessels, the end caps represent a critical design consideration. Different head geometries produce vastly different stress concentrations [6]:

The end cap stress is calculated as: σ_head = K × Pr/t. Hemispherical heads are mechanically optimal but expensive to manufacture, which is why the 2:1 ellipsoidal head represents the practical compromise for most applications.

Learning Objectives

• Calculate hoop, longitudinal, and radial stresses for both thin and thick-walled pressure vessels
• Determine when to use Lamé equations versus thin-wall approximations based on t/r ratio
• Compute Von Mises equivalent stress for multiaxial stress states
• Apply ASME Section VIII formulas to calculate MAWP and required wall thickness
• Compare stress factors for different end cap geometries
• Understand the relationship between burst pressure, MAWP, and hydrostatic test pressure

Exploration Activities

Activity 1: Thin-Wall vs. Thick-Wall Transition

Start with a vessel of r=500mm and t=10mm (t/r=0.02, thin-wall). Gradually increase wall thickness while keeping radius constant. Note when the analysis switches to thick-wall mode (t/r>0.1) and observe how the stress distribution changes. At what thickness does the difference between thin and thick-wall hoop stress exceed 10%?

Activity 2: Material Selection Impact

Using the Boiler Drum preset, cycle through all available materials. How does the safety factor change? Which material provides the highest safety factor? Which provides the best strength-to-weight ratio (consider that aluminum is about 1/3 the density of steel)?

Activity 3: End Cap Comparison

With a cylindrical vessel at 3 MPa internal pressure, compare the end cap stresses for all four head types. Calculate the thickness required for each head type to achieve the same stress level as the hemispherical head. How much heavier would a torispherical head be?

Activity 4: Burst Pressure Investigation

Using the Scuba Tank preset (high pressure aluminum vessel), verify the burst pressure calculation. Compare the safety margin between operating pressure and burst pressure. How does this compare to the ASME safety factor of 3.5? Why might different applications require different safety margins?

Real-World Applications

Understanding pressure vessel design is critical across numerous industries [7]:

• Oil and Gas: Separators, knockout drums, and storage tanks operating from vacuum to 10+ MPa
• Chemical Processing: Reactor vessels handling corrosive materials at elevated temperatures and pressures
• Power Generation: Boiler drums and steam accumulators operating at 15-30 MPa
• Aerospace: Propellant tanks, crew capsules, and hydraulic accumulators where weight is critical
• Medical: Hyperbaric chambers, sterilizer vessels, and gas storage cylinders
• Food and Beverage: Fermentation tanks, carbonation vessels, and pasteurization equipment

Material Properties Reference

Challenge Questions

1. A cylindrical tank with r=1000mm operates at 5 MPa. What minimum wall thickness is required to maintain a safety factor of 4.0 using A516-70 steel?
2. Why is the longitudinal stress in a cylindrical vessel exactly half the hoop stress? Derive this from equilibrium considerations.
3. A spherical vessel and cylindrical vessel have the same radius and wall thickness. Which can operate at higher pressure while maintaining the same safety factor? By what ratio?
4. Calculate the hydrostatic test pressure for a vessel with MAWP of 2.5 MPa. What stress state exists in the vessel during this test?
5. A thick-walled cylinder has r_i=100mm and r_o=150mm. At what radius through the wall does the hoop stress equal the average of inner and outer surface stresses?

Common Mistakes to Avoid

• Using thin-wall formulas for thick vessels: Always check t/r ratio before selecting analysis method. This is where structures fail when engineers take shortcuts.

• Ignoring external pressure: Vacuum conditions create buckling concerns that are completely different from internal pressure design.

• Forgetting joint efficiency: Welded joints reduce effective strength. Full radiographic examination (E=1.0) is expensive but allows thinner walls.

• Using yield instead of tensile for MAWP: ASME bases allowable stress on ultimate tensile strength divided by 3.5, not yield strength.

• Neglecting end cap stresses: The cylindrical shell might be adequate, but a poorly designed head can still fail.

• Confusing MAWP with design pressure: Design pressure is what you specify; MAWP is what the vessel can actually handle based on actual dimensions.

  Frequently Asked Questions

  Why is hoop stress twice the longitudinal stress in cylinders?

  Consider a free body diagram cutting the cylinder perpendicular to its axis versus parallel to its axis. The perpendicular cut exposes an area of 2*pi*r*t resisting pressure on an area of pi*r^2, while the parallel cut exposes pi*r^2 resisting pressure on 2*r*L. The geometry naturally produces a 2:1 stress ratio [2].

  What determines whether a vessel is "thick-walled"?

  The t/r greater than 0.1 criterion comes from error analysis. At this ratio, thin-wall formulas underestimate maximum stress by approximately 5%, which compounds with other uncertainties to become unacceptable [3].

  Why use Von Mises stress instead of principal stresses?

  Von Mises stress correlates with yielding for ductile materials regardless of the stress state. A vessel might have all stresses below yield individually but still yield due to the combined distortion energy effect [4].

  What is hydrostatic testing and why 1.3x MAWP?

  Hydrostatic testing uses water (incompressible) rather than gas to test vessels. The 1.3 factor ensures the vessel has adequate margin above operating pressure while keeping test stress below 90% of yield to avoid permanent deformation [5].

  How accurate is the burst pressure estimate?

  The Barlow formula provides a first approximation but real burst pressures depend on material ductility, strain hardening, and defects. Actual burst tests typically show 10-20% deviation from calculated values [8].

  References

  1. Shigley, J.E. and Mischke, C.R., Mechanical Engineering Design, McGraw-Hill, 10th Edition, 2014. Chapter 3: Load and Stress Analysis.
  2. Roark, R.J. and Young, W.C., Roark's Formulas for Stress and Strain, McGraw-Hill, 8th Edition, 2012. Tables for thick and thin cylinders.
  3. Lame, G., Lecons sur la theorie mathematique de l'elasticite des corps solides, Paris, 1852. Original derivation of thick-wall equations.
  4. von Mises, R., "Mechanik der festen Korper im plastisch-deformablen Zustand," Nachrichten von der Gesellschaft der Wissenschaften zu Gottingen, 1913.
  5. ASME Boiler and Pressure Vessel Code, Section VIII Division 1, American Society of Mechanical Engineers, 2023 Edition.
  6. Moss, D.R., Pressure Vessel Design Manual, Gulf Professional Publishing, 4th Edition, 2012. Chapter on head design.
  7. Megyesy, E.F., Pressure Vessel Handbook, 14th Edition, Pressure Vessel Publishing, 2008.
  8. Harvey, J.F., Theory and Design of Pressure Vessels, Van Nostrand Reinhold, 2nd Edition, 1991.

  About the Data

  Material property data in this simulation comes from ASME Section II Part D (Materials Properties) and manufacturer specifications. The MAWP calculations follow ASME Section VIII Division 1 formulas exactly. All equations have been verified against published solutions in standard mechanical engineering textbooks [1,2].

  Citation Guide

  To cite this simulation in academic work:

  Simulations4All. (2025). Pressure Vessel Calculator [Interactive Simulation]. Retrieved from https://simulations4all.com/simulations/pressure-vessel-calculator

  Verification Log

  Item	Source	Verified	Notes
  Thin-wall equations	Shigley [1], Roark [2]	2025-01-18	Standard formulas
  Lame equations	Roark [2], Lame [3]	2025-01-18	Thick-wall theory
  Von Mises criterion	von Mises [4]	2025-01-18	Distortion energy
  ASME MAWP formula	ASME VIII [5]	2025-01-18	2023 Edition
  End cap K factors	Moss [6], ASME [5]	2025-01-18	Head stress factors
  Material properties	ASME II Part D	2025-01-18	Allowable stresses