Pressure Vessels, Fatigue and Structural Decisions

Prerequisite knowledge: force equilibrium, SI dimensional analysis, algebra and first-year calculus.

Learning objectives: state a model and its validity range, calculate physically meaningful engineering quantities, reconcile equations with a free-body diagram or deformation compatibility, and identify at least two failure modes excluded by the idealization.

Unit context and engineering interpretation

Pressure vessels are a useful example of membrane equilibrium. A smooth thin cylinder under internal pressure develops circumferential tension because pressure pushes its curved wall outward. The hoop force balance across a diametral cut leads to pr/t, while the longitudinal balance across a closed end yields pr/(2t). The resulting factor of two is a geometric result of projected areas and membrane forces, not an arbitrary correction. A spherical vessel with the same thin-wall assumptions has a different stress relation.

An applied thin-wall formula does not confer pressure-equipment compliance. Real shells have seams, nozzles, local thinning, thermal gradients, corrosion and cyclic pressure variations. Thick cylinders require a radial stress distribution and Lame-type elasticity solutions. The wall-thickness ratio acts as an applicability warning. Students should explicitly document why the ideal membrane model may be satisfactory for an order-of-magnitude explanation but insufficient for certification.

Fatigue adds long-term history to a stress calculation. A static stress below yield does not preclude crack initiation after many load cycles. Stress amplitude, mean stress, surface quality, defects and environment all affect lifetime. Synthetic S-N curves show why exponent uncertainty can produce enormous predicted cycle variation. An engineering fatigue assessment needs measured or specified material data, representative load histories, uncertainty treatment and the relevant standard. A designer should not confuse a smooth curve in a learning app with a validated probabilistic life model.

Lesson 9.1 — Thin-walled cylindrical pressure vessels

Physical mechanism and governing idea

Pressure loads on a cylindrical shell create membrane tension in both hoop and longitudinal directions. When wall thickness is small relative to radius, sigma_hoop=pr/t and sigma_long=pr/(2t) approximate closed-end cylinder stresses. The hoop stress is twice the longitudinal membrane stress in that ideal geometry.

Governing relationship

sigma_hoop=p r/t; sigma_long=p r/(2t)

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

At p=2 MPa, inner radius 0.25 m and t=10 mm, thin-wall hoop stress estimate is 50 MPa.

What the simplified formula misses

Thin-wall formulas must not be used for thick vessels, nozzles, stress concentrations or pressure-code certification. This tool is for education only.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 9.1 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 9.2 — Stress concentrations and geometric details

Physical mechanism and governing idea

Holes, shoulders, fillets and weld toes redistribute stress. A theoretical stress concentration factor Kt relates local peak elastic stress to a chosen nominal value for a specified geometry. The local field depends on radius, load direction and boundary conditions. Fatigue sensitivity may differ from an elastic Kt.

Governing relationship

sigma_peak,elastic ≈ Kt sigma_nominal (only for a verified geometry and definition)

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

If a verified chart gives Kt=2.2 at a particular shoulder and sigma_nominal=60 MPa, the elastic peak estimate is 132 MPa.

What the simplified formula misses

Never choose a Kt by appearance. Finite element convergence, validated charts or experiments may be necessary, and plasticity limits elastic peak estimates.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 9.2 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 9.3 — Fatigue, S-N curves and mean stress

Physical mechanism and governing idea

A fluctuating load may cause failure below monotonic strength. S-N diagrams relate stress amplitude to cycles under particular specimen and load conditions. A log-log Basquin-type relationship is useful for teaching sensitivity. Real design requires stress range extraction, mean stress correction, material/environment effects and justified scatter allowances.

Governing relationship

S_a=C N^b, with b negative, in a fitted regime

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

With a fitted exponent b=-0.10, raising amplitude by about 26 percent decreases predicted cycles by a factor of roughly ten in the same model.

What the simplified formula misses

The example S-N model is synthetic. It must not be used to predict equipment service life or to approve safety-critical components.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 9.3 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 9.4 — Verification, uncertainty and responsible use

Physical mechanism and governing idea

Engineering judgment includes model verification, validation and uncertainty management. Verification asks whether equations and software were implemented correctly; validation asks whether the equations represent the real physical system. A robust design record identifies input assumptions, load combinations, manufacturing tolerances and independent checks.

Governing relationship

Compare independent equilibrium checks, dimensions, limiting cases and measured behavior

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

A beam simulation may pass unit tests and still misrepresent a real beam with a loose support or nonprismatic section.

What the simplified formula misses

These educational models are not licensed structural calculations. Design decisions must follow local codes, qualified review and appropriate physical testing.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 9.4 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Worked numerical casebook

The following 18 worked cases are intended for dimensional fluency and model interpretation, not as a substitute for unique field measurements. Cases change materials, loads and geometry and deliberately restate their validity conditions.

When reproducing a result, document a complete free body or section cut, record the input units, and calculate at least one quantity without looking at the worked result.

Case 9.01: Thin-cylinder equilibrium in a girder

A closed cylindrical shell at gauge pressure 0.8 MPa has inner radius 100 mm and wall thickness 16 mm. The thin-wall membrane equations give circumferential stress pr/t=5.00 MPa and longitudinal stress pr/(2t)=2.50 MPa. Thickness/radius=0.160. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.01: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.02: Thin-cylinder equilibrium in a roof tie

A closed cylindrical shell at gauge pressure 4 MPa has inner radius 200 mm and wall thickness 16 mm. The thin-wall membrane equations give circumferential stress pr/t=50.00 MPa and longitudinal stress pr/(2t)=25.00 MPa. Thickness/radius=0.080. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.02: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.03: Thin-cylinder equilibrium in a machine anchor

A closed cylindrical shell at gauge pressure 3 MPa has inner radius 150 mm and wall thickness 10 mm. The thin-wall membrane equations give circumferential stress pr/t=45.00 MPa and longitudinal stress pr/(2t)=22.50 MPa. Thickness/radius=0.067. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.03: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.04: Thin-cylinder equilibrium in a laboratory frame

A closed cylindrical shell at gauge pressure 0.8 MPa has inner radius 250 mm and wall thickness 16 mm. The thin-wall membrane equations give circumferential stress pr/t=12.50 MPa and longitudinal stress pr/(2t)=6.25 MPa. Thickness/radius=0.064. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.04: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.05: Thin-cylinder equilibrium in a small footbridge

A closed cylindrical shell at gauge pressure 4 MPa has inner radius 150 mm and wall thickness 8 mm. The thin-wall membrane equations give circumferential stress pr/t=75.00 MPa and longitudinal stress pr/(2t)=37.50 MPa. Thickness/radius=0.053. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.05: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.06: Thin-cylinder equilibrium in a service gantry

A closed cylindrical shell at gauge pressure 1.2 MPa has inner radius 350 mm and wall thickness 10 mm. The thin-wall membrane equations give circumferential stress pr/t=42.00 MPa and longitudinal stress pr/(2t)=21.00 MPa. Thickness/radius=0.029. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.06: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.07: Thin-cylinder equilibrium in a construction prop

A closed cylindrical shell at gauge pressure 2.5 MPa has inner radius 200 mm and wall thickness 20 mm. The thin-wall membrane equations give circumferential stress pr/t=25.00 MPa and longitudinal stress pr/(2t)=12.50 MPa. Thickness/radius=0.100. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.07: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.08: Thin-cylinder equilibrium in a testing rig

A closed cylindrical shell at gauge pressure 4 MPa has inner radius 100 mm and wall thickness 20 mm. The thin-wall membrane equations give circumferential stress pr/t=20.00 MPa and longitudinal stress pr/(2t)=10.00 MPa. Thickness/radius=0.200. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.08: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.09: Thin-cylinder equilibrium in a industrial platform

A closed cylindrical shell at gauge pressure 3 MPa has inner radius 150 mm and wall thickness 8 mm. The thin-wall membrane equations give circumferential stress pr/t=56.25 MPa and longitudinal stress pr/(2t)=28.12 MPa. Thickness/radius=0.053. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.09: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.10: Thin-cylinder equilibrium in a equipment mount

A closed cylindrical shell at gauge pressure 2.5 MPa has inner radius 350 mm and wall thickness 16 mm. The thin-wall membrane equations give circumferential stress pr/t=54.69 MPa and longitudinal stress pr/(2t)=27.34 MPa. Thickness/radius=0.046. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.10: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.11: Thin-cylinder equilibrium in a maintenance bracket

A closed cylindrical shell at gauge pressure 1.2 MPa has inner radius 250 mm and wall thickness 10 mm. The thin-wall membrane equations give circumferential stress pr/t=30.00 MPa and longitudinal stress pr/(2t)=15.00 MPa. Thickness/radius=0.040. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.11: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.12: Thin-cylinder equilibrium in a temporary support

A closed cylindrical shell at gauge pressure 3 MPa has inner radius 200 mm and wall thickness 12 mm. The thin-wall membrane equations give circumferential stress pr/t=50.00 MPa and longitudinal stress pr/(2t)=25.00 MPa. Thickness/radius=0.060. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.12: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.13: Thin-cylinder equilibrium in a pipe rack

A closed cylindrical shell at gauge pressure 3 MPa has inner radius 250 mm and wall thickness 16 mm. The thin-wall membrane equations give circumferential stress pr/t=46.88 MPa and longitudinal stress pr/(2t)=23.44 MPa. Thickness/radius=0.064. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.13: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.14: Thin-cylinder equilibrium in a solar panel rail

A closed cylindrical shell at gauge pressure 2.5 MPa has inner radius 350 mm and wall thickness 20 mm. The thin-wall membrane equations give circumferential stress pr/t=43.75 MPa and longitudinal stress pr/(2t)=21.88 MPa. Thickness/radius=0.057. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.14: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.15: Thin-cylinder equilibrium in a transit shelter

A closed cylindrical shell at gauge pressure 1.2 MPa has inner radius 250 mm and wall thickness 20 mm. The thin-wall membrane equations give circumferential stress pr/t=15.00 MPa and longitudinal stress pr/(2t)=7.50 MPa. Thickness/radius=0.080. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.15: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.16: Thin-cylinder equilibrium in a lightweight crane hook

A closed cylindrical shell at gauge pressure 4 MPa has inner radius 100 mm and wall thickness 20 mm. The thin-wall membrane equations give circumferential stress pr/t=20.00 MPa and longitudinal stress pr/(2t)=10.00 MPa. Thickness/radius=0.200. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.16: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.17: Thin-cylinder equilibrium in a water-tank frame

A closed cylindrical shell at gauge pressure 3 MPa has inner radius 350 mm and wall thickness 8 mm. The thin-wall membrane equations give circumferential stress pr/t=131.25 MPa and longitudinal stress pr/(2t)=65.62 MPa. Thickness/radius=0.023. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.17: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 9.18: Thin-cylinder equilibrium in a portal-frame member

A closed cylindrical shell at gauge pressure 4 MPa has inner radius 350 mm and wall thickness 12 mm. The thin-wall membrane equations give circumferential stress pr/t=116.67 MPa and longitudinal stress pr/(2t)=58.33 MPa. Thickness/radius=0.034. The calculation assumes a thin, smooth closed cylinder and excludes nozzle stresses, weld effects and inspection requirements; it is not an approved pressure design.

Extension task 9.18: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Engineering case study: Fatigue-sensitive vessel support under cycling pressure

A small cylindrical vessel experiences repeated filling and draining. Thin-wall pressure equations explain the two membrane stresses but do not model the nozzle, weld toe or support attachment where cyclic cracking may initiate. A synthetic S-N line can demonstrate how life changes with stress amplitude, but it cannot certify the vessel. The actual load cycle includes pressure, temperature and perhaps vibration.

Deliverable: Separate mean and alternating stresses, explain why stress concentration and inspection access matter, identify a governing pressure code and propose a validation plan involving design calculations and nondestructive examination.

Case-study discussion

Start by differentiating data from assumptions: measured loads and geometry should be separated from guessed restraints and material properties. Build one simple calculation whose result can be checked independently. List at least three mechanisms by which the idealization could fail, and prioritize them by whether they would change the engineering conclusion. Translate the numerical demand into a practical question about safety, serviceability, testing or maintenance. Keep a clear audit trail so a reviewer can follow the unit conversions, load path and model limitations.

Calculation verification

• Check whether t/r remains suitably small and state when the membrane idealization becomes questionable.
• Verify the hoop stress equals twice the longitudinal stress for the smooth closed thin cylinder.
• Check positivity, zero-load behavior, symmetry, and asymptotic scaling whenever applicable.
• Do not use a graph or slider response as the only evidence that an equation is correct.

Structured unit assessment

Answer the following without access to the provided worked solutions. Each question needs a diagram, assumptions and a clear interpretation.

1. Use a load or material value from Case 9.02, change one geometric dimension by 20%, and calculate the resulting response with explicit units.
2. Explain how the result of Lesson 9.1 would change if one apparently rigid support had finite stiffness.
3. Choose a nonuniform or damaged member for which the idealization in Lesson 9.2 becomes inaccurate; describe a better analytical or experimental representation.
4. Derive a useful limiting case from the relationship in Lesson 9.3 and compare with a limiting setting in any associated simulation.
5. A reviewer reports a one-thousandfold discrepancy. Show one plausible N–kN, m–mm or N·m–N·mm conversion error that could create it for a calculation in this unit.
6. For the case study in Unit 9, distinguish two possible failure modes and explain why demonstrating adequate performance against one does not guarantee the other.
7. Create an experimental plan to validate one of the numerical predictions in Unit 9; name the sensor, its calibration requirement and an expected source of uncertainty.
8. State two ways that manufacturing tolerance, support flexibility or temperature could alter a result computed from one equation in Unit 9.

Associated interactive laboratories

Laboratory 10: Thin-Wall Pressure Cylinder. A closed-end thin circular cylinder develops hoop stress pr/t and longitudinal stress pr/(2t). The thickness-to-radius indicator is a model applicability warning, not a certification threshold.

Investigation: Double the radius at fixed pressure and thickness. Compare hoop and axial stress. Then make the wall relatively thick and explain why the model should not be trusted.

Open the [working HTML model](../labs/lab-10-pressure.html), record inputs and CSV data, and compare numerical outputs with an independent calculation. Identify the limiting assumptions before discussing how a real physical experiment would differ.

Laboratory 12: Fatigue S–N Explorer. This synthetic log-log S–N model demonstrates a fitted Basquin power relationship. It is not calibrated to any certified material and must not be used as a service-life prediction.

Investigation: Increase stress amplitude in increments. Compare the log-scale predicted cycles. Explain why a percentage amplitude change can cause orders-of-magnitude life changes.

Open the [working HTML model](../labs/lab-12-fatigue.html), record inputs and CSV data, and compare numerical outputs with an independent calculation. Identify the limiting assumptions before discussing how a real physical experiment would differ.

Unit research project

Prepare a technical notebook on Pressure vessels, fatigue and structural decisions for a civil engineering structure of your choice. Include a drawing with geometric dimensions, a carefully numbered input table and a free-body diagram. Develop two load or restraint scenarios, compute the most relevant stress or deformation variables, cross-check them with an independent limiting case, and identify all omissions that would prevent formal design approval. Where an associated laboratory exists, export a CSV series and include an annotated plot. Write a short conclusion that distinguishes what your simplified model genuinely predicts from what would need a professional code-based check.

Project marking rubric