Pressure-Containing Components and Mechanical Integrity

Introduction and context

Pressure-containing components are machine elements whose primary function is to contain a fluid or gas at a pressure different from the surroundings. They include cylinders, pipes, pressure vessels, tube sheets, flanges, gaskets, bolted closures, plugs and welded closures. Their failure can release stored energy rapidly, causing rupture, fire, toxic exposure, projectiles or environmental damage. Consequently, pressure design is not merely a strength calculation: it is an integrity discipline combining mechanics, materials, fabrication, inspection, testing, operation and regulatory control.

The designer must establish the pressure–temperature envelope, select compatible materials, determine wall dimensions, control deformation and leakage, assess fatigue and fracture risks, and specify inspection and proof testing. Codes such as ASME BPVC Section VIII, ASME B31.3, EN 13445, EN 13480, PD 5500 and API standards provide detailed rules. A code is not a substitute for engineering judgment; the applicable legal and contractual requirements must be identified before design begins.

Learning objectives

After studying this chapter, a student should be able to:

1. Distinguish thin- and thick-walled pressure cylinders and select appropriate stress models.
2. Calculate membrane stresses, radial and hoop stresses, deformation and required wall thickness.
3. Explain flange, gasket and closure behavior and identify the sources of leakage.
4. Include temperature, cyclic loading, stress concentration, corrosion and fabrication effects in design.
5. Develop a defensible inspection, proof-test and mechanical-integrity plan.
6. Interpret the role and limitations of pressure-equipment codes.

1. Pressure design as a system problem

A pressure boundary is exposed to several simultaneous actions:

• internal or external pressure;
• dead weight, piping reactions and support loads;
• thermal expansion and temperature gradients;
• vibration, pulsation and pressure transients;
• bending, torsion and nozzle loads;
• corrosion, erosion, wear and hydrogen damage;
• residual stresses from welding, forming and heat treatment.

The design pressure is normally greater than the intended operating pressure and should represent credible overpressure. Design temperature must include both maximum and minimum metal temperatures, not only fluid temperature. At elevated temperature, strength, creep resistance, gasket behavior and bolt preload may change substantially. At low temperature, ductile materials may suffer reduced fracture toughness.

A useful design sequence is:

1. Define fluid, pressure, temperature, volume, operating cycles and hazard category.
2. Select the governing code and material specifications.
3. Identify geometry, joints, openings, supports and load combinations.
4. Calculate required thickness and stresses.
5. Check local stresses, buckling, fatigue, fracture and leakage.
6. Specify fabrication, welding, heat treatment, nondestructive examination and testing.
7. Establish inspection, monitoring, maintenance and retirement criteria.

The pressure boundary should be treated as a life-cycle system. A component that passes an initial pressure test can still fail later because of fatigue cracks, corrosion thinning, creep, gasket relaxation or unauthorized modification.

2. Stress in a cylindrical pressure boundary

Consider a cylinder of internal radius r, wall thickness t and internal pressure p. For a sufficiently long cylinder with closed ends, the pressure produces circumferential (hoop) stress and longitudinal stress. If the wall is thin, stress is assumed uniform through the thickness.

The thin-wall approximation is generally appropriate when t/r is small, commonly t/r ≤ 0.1 as an engineering guide. The exact limiting value depends on the code and required accuracy.

For a closed-end cylinder:

where σ_h is hoop stress, σ_l is longitudinal stress, p is internal gauge pressure, r is the mean radius, and t is wall thickness. Pressure and dimensions must use consistent units; for example, MPa and mm give stress in MPa because 1 MPa = 1 N/mm².

The hoop stress is twice the longitudinal stress and normally governs a cylindrical shell. These formulas assume a circular, uniform, thin wall; negligible external loads; elastic behavior; and pressure acting steadily. They do not adequately describe thick walls, openings, abrupt geometry changes, local loads or plastic collapse.

For a cylinder with an allowable membrane stress S and a weld or joint efficiency E, a preliminary pressure thickness is obtained from the hoop relation:

This is only a simplified membrane result. Code equations often use inside or outside diameter, corrosion allowance, pressure coefficients, and geometry-specific denominators. The selected code equation must govern the final design.

A practical nominal thickness is:

where c is corrosion or erosion allowance and t_manufacturing accounts for negative thickness tolerance, forming thinning or machining allowance. If the purchased plate has a negative tolerance fraction q, a simple adjustment is:

provided q is expressed as a decimal and the code permits this treatment.

Worked example 1: thin-wall cylinder

A cylindrical shell has internal pressure p = 2 MPa, mean radius r = 500 mm, allowable stress S = 120 MPa, joint efficiency E = 0.85 and corrosion allowance c = 2 mm. Neglecting manufacturing tolerance, the pressure thickness is:

Therefore:

A specified nominal thickness would normally be rounded upward and checked against the governing code, fabrication tolerance, minimum practical thickness and all other load cases. The longitudinal stress at the pressure thickness is approximately 51 MPa, while the hoop stress is approximately 102 MPa; thus the hoop condition governs this simplified calculation.

3. Thick-walled cylinders

When wall thickness is not small compared with radius, stress varies significantly through the wall. The Lamé solution for a long, elastic, axisymmetric cylinder is appropriate when end effects are negligible and pressure is applied uniformly.

Let r_i and r_o be the inner and outer radii, p_i the internal pressure and p_o the external pressure. At a radial position r, the radial and hoop stresses are:

where:

Compressive pressure is represented by negative radial stress under the usual tensile-positive convention. At the inner surface, the radial stress equals −p_i; at the outer surface it equals −p_o. With internal pressure greater than external pressure, the maximum tensile hoop stress occurs at the inner surface.

For a closed-end cylinder, the uniform longitudinal stress caused by pressure is:

This end-load stress must be combined with the local Lamé stresses where appropriate. The Lamé model does not account for yielding, plastic redistribution, residual stress, openings, threads, discontinuities or end effects.

Example 2: thick-wall stress calculation

A cylinder has r_i = 50 mm, r_o = 75 mm, p_i = 30 MPa and p_o = 0. First calculate:

At the inner surface:

The radial value differs slightly from −30 MPa because the numerical value of B above has been rounded incorrectly if interpreted dimensionally? Re-evaluating, 50² × 75² × 30 = 421,875,000; division by 3,125 gives B = 135,000 MPa·mm². Therefore the correct results are:

This still conflicts with the boundary condition because A must be 30 × 2,500/3,125 = 24 MPa, not 51.43 MPa. The correct values are:

Thus:

The calculation illustrates why boundary-condition checks are essential. At the outer surface, σ_r = 0 and σ_θ = 48 MPa. The inner hoop stress is therefore the maximum tensile stress for this case.

4. Design against yielding, fracture and instability

Allowable-stress design compares calculated stress with a code-defined allowable value. The allowable value incorporates material strength, temperature, uncertainty, fabrication quality and safety factors. A design based only on nominal membrane stress may be unsafe when local peaks or defects exist.

For multiaxial ductile yielding, the von Mises equivalent stress is often used:

where σ_1, σ_2 and σ_3 are principal stresses. This criterion is useful for elastic analysis of ductile metals, but pressure-equipment codes may use different stress categories and limits. Primary membrane stress, primary bending stress, local peak stress and secondary thermal stress should not be treated as interchangeable.

External pressure introduces a different failure mode: elastic or inelastic buckling. A pipe can collapse under external pressure even when its material stress is below yield. Buckling resistance depends strongly on diameter-to-thickness ratio, ovality, unsupported length, stiffening rings, imperfections and boundary conditions. Vacuum service, steam condensation and blocked-in cooling must therefore be analyzed explicitly.

Openings, nozzles, threads, keyways, weld toes and sharp transitions create stress concentrations. The elastic concentration factor is:

where σ_max is the local elastic peak stress and σ_nom is the corresponding nominal stress. Fatigue analysis must distinguish K_t from the fatigue notch factor, which also depends on material sensitivity and notch size. Smooth generous radii, qualified weld profiles, reinforcement pads and avoidance of abrupt stiffness changes reduce concentration.

5. Pipes, elbows, nozzles and local loads

Straight pipe pressure thickness is governed primarily by hoop stress, but piping systems also impose bending moments, axial forces, torsion and reactions at equipment nozzles. Elbows experience ovalization and pressure thrust. Branch connections disturb the membrane stress field and require reinforcement or a code-specific local analysis.

Pressure thrust at a closed end is:

where A is the projected internal area. For a circular bore of diameter d:

For p = 2 MPa and d = 100 mm, F_p = 2 × π × 100²/4 = 15,708 N, or approximately 15.7 kN. Anchors, end closures and supports must resist this load. A pipe wall calculation alone cannot demonstrate adequacy of the complete system.

Thermal expansion produces displacement that may be accommodated by flexibility, expansion loops, sliding supports or expansion joints. Excessive restraint can generate large secondary stresses and nozzle loads. Flexibility analysis should consider sustained, occasional and displacement-controlled load cases according to the relevant piping code.

6. Flanges, gaskets and bolted closures

A flange joint must simultaneously contain pressure, maintain gasket compression and transmit external loads without excessive rotation. Internal pressure tends to separate the flanges and reduce gasket compression. The joint therefore depends on bolt preload, flange stiffness, gasket seating behavior, surface finish, alignment and operating temperature.

The pressure-separating force for a simple circular effective diameter D_e is approximately:

The required bolt load is not simply equal to F_p. It must also maintain gasket compression. A simplified service condition is:

where W_operating is the total bolt load in service and F_g is the gasket load required to maintain sealing. Actual flange design uses code-specific gasket factors, effective gasket diameter, bolt stress, flange flexibility and assembly conditions. The familiar gasket relation W = yA or W = 2bπGmP is associated with particular design methods and should not be applied outside their assumptions.

Gaskets may be soft, semi-metallic or metallic. Important properties include compressibility, recovery, creep relaxation, chemical compatibility, temperature resistance and resistance to blowout. A gasket can leak despite adequate bolt strength if the flange faces are damaged, the gasket is incorrectly installed, preload is uneven or the gasket relaxes.

Bolt preload is commonly estimated by:

where T is tightening torque, F is desired bolt preload, d is nominal bolt diameter and K is a nut-factor incorporating thread and bearing friction. Because friction dominates this relationship, torque alone is an imprecise method of controlling preload. Calibrated tensioning, hydraulic tensioners or direct elongation measurement can provide better control for critical joints.

Bolted closures must also be checked for thread stripping, bolt fatigue, flange rotation, gasket seating, thermal differential expansion and accidental loosening. A closure that is pressure-tight at ambient temperature may leak at operating temperature because the bolt and shell expand differently.

7. Temperature and cyclic service

Temperature affects pressure components in three principal ways: it changes material strength, produces thermal stress and changes gasket and bolt behavior. A restrained thermal strain is approximately:

where α is the coefficient of thermal expansion and ∆T is the temperature change. If expansion is completely restrained in a simple uniaxial elastic idealization, the stress magnitude is:

where E_m is Young’s modulus. This expression is an ideal upper-bound elastic result; real components redistribute stress through flexibility, yielding, contact and creep.

Cyclic pressure and temperature can initiate fatigue cracks at weld toes, nozzles, threads and geometric transitions. A basic stress-life approach uses alternating stress amplitude and mean stress, but pressure codes prescribe particular fatigue curves, stress ranges and cycle-counting procedures. The cumulative usage concept is often expressed as:

where n_i is the expected number of cycles of type i and N_i is the allowable number of cycles for that stress range. A common screening criterion is U ≤ 1, but the applicable code may impose additional rules for environmental effects, weld classification and ratcheting.

Pressure cycling should be evaluated using realistic transients: startup, shutdown, relief-valve operation, trips, thermal shocks, pressure pulsation and maintenance events. Fatigue design is especially important for thin shells, welded attachments and components with high stress concentration.

8. Materials, fabrication and degradation

Material selection must consider yield strength, tensile strength, fracture toughness, ductility, weldability, corrosion resistance, thermal expansion and compatibility with the contained fluid. The highest-strength alloy is not automatically the safest choice; poor toughness, weldability or environmental resistance may govern.

Common degradation mechanisms include:

• general corrosion and localized pitting;
• erosion–corrosion and cavitation;
• stress-corrosion cracking;
• hydrogen embrittlement and hydrogen-induced cracking;
• creep and creep-fatigue at high temperature;
• brittle fracture at low temperature;
• thermal fatigue;
• fretting and vibration fatigue;
• gasket aging and bolt relaxation.

Welded construction requires qualified procedures, qualified welders, appropriate filler metals, preheat and interpass control, post-weld heat treatment where required, and examination of critical welds. Forming can thin a shell and introduce residual stress; the final geometry and minimum wall must be checked. Corrosion allowance is not a substitute for a corrosion-control program, and it should not be credited against mechanisms such as cracking that do not consume thickness uniformly.

9. Inspection, proof testing and mechanical integrity

Inspection begins with material certificates, traceability and dimensional checks. During fabrication, visual examination, liquid penetrant testing, magnetic-particle testing, ultrasonic testing and radiography may be selected according to material, joint type and defect mechanism. Each method has limitations: radiography is effective for volumetric discontinuities but less sensitive to some planar flaws; ultrasonic testing can detect planar defects but depends strongly on access, calibration and operator competence.

A hydrostatic pressure test uses an essentially incompressible liquid and is generally safer than a pneumatic test because stored energy is much lower. A pneumatic test may be necessary when water contamination or excessive weight is unacceptable, but it requires rigorous exclusion zones, staged pressurization and risk assessment. Test pressure, hold time, temperature correction and acceptance criteria must be obtained from the governing code rather than assumed.

Proof testing demonstrates integrity at a specified test condition; it does not prove infinite life or absence of every defect. Testing above yield can cause permanent deformation, and repeated proof testing may consume fatigue life. Test water chemistry and drying are important for stainless steels, high-strength steels and systems susceptible to corrosion or hydrogen damage.

An operating integrity program should include:

1. identification and registration of pressure equipment;
2. relief-device inspection and set-pressure verification;
3. thickness measurements at corrosion-monitoring locations;
4. inspection intervals based on damage mechanisms and risk;
5. leak detection and alarm systems;
6. management of change and repair control;
7. documented operating limits and competent personnel;
8. fitness-for-service assessment when defects or thinning are found.

Fitness-for-service methods, including recognized API and engineering-assessment practices, determine whether equipment with flaws can continue operating, be repaired, be rerated or must be retired. Assessment should consider defect dimensions, stress, material toughness, crack growth, remaining life and consequences of failure.

10. Codes, standards and governance

Codes establish minimum requirements for design, materials, fabrication, examination, testing and certification. Examples include:

The exact edition, jurisdiction and equipment classification matter. A designer must not combine equations from different codes casually because allowable stresses, joint efficiencies, gasket rules and test requirements may differ. Regulatory compliance also requires records: drawings, calculations, material certificates, weld maps, examination reports, pressure-test certificates, relief-device data and inspection history.

Ethically, engineers must communicate uncertainty, reject falsified inspection records, report unsafe conditions and avoid reducing safety margins merely to meet cost or schedule targets. Independent review is appropriate for novel designs, hazardous fluids, high-energy systems and deviations from standard practice.

11. Illustrative case: recurring leakage at a process flange

Background

An illustrative chemical-processing line operates at 1.6 MPa and 180 °C. A flanged joint using a semi-metallic gasket leaks after several thermal cycles, although the flange rating appears adequate.

Problem

Maintenance records show that leakage occurs during cooldown. Bolts were tightened by a standard torque table, but no preload verification was performed. The line also contains a rigid support close to the flange.

Analysis

The investigation identifies four interacting causes. First, friction variation caused the applied torque to produce uneven bolt preload. Second, the rigid support imposed bending on the flange during thermal expansion. Third, gasket relaxation reduced compression at temperature. Fourth, the flange faces showed minor radial scratches that created leakage paths. A pressure calculation based only on nominal flange rating could not reveal these problems.

The team performs dimensional inspection, bolt elongation checks, flange-face examination and a piping flexibility review. The effective bolt load is found to be uneven, while the support arrangement produces a significant external moment. The joint therefore loses gasket compression locally during cooldown.

Approach and solution

The corrective design uses a qualified gasket compatible with the fluid and temperature, controlled bolt tensioning, improved flange-face finish, revised support flexibility and a documented tightening sequence. The piping system is rechecked for sustained and thermal loads. A controlled leak test is performed after assembly, and the joint is added to the inspection plan.

Outcome

No leakage occurs during subsequent operating cycles. The principal improvement is not a thicker flange but better control of preload, external loads, surface condition and assembly quality.

Learning points

Pressure-boundary integrity is multidisciplinary. A joint may satisfy a static pressure rating yet fail because of thermal movement, gasket behavior or assembly variability. Root-cause analysis should examine the entire load path and the complete manufacturing and maintenance history.

12. Contemporary developments and best practice

Modern pressure design increasingly uses finite-element analysis, digital inspection records, automated ultrasonic scanning, acoustic-emission monitoring and risk-based inspection. These tools improve evidence and prioritization, but they do not remove the need for correct boundary conditions, validated material data and competent interpretation. Artificial intelligence can assist with anomaly detection, corrosion-rate estimation and document review, but recommendations require qualified human approval because training data may be incomplete and rare failure modes may be underrepresented.

Best practice includes conservative definition of design conditions, code-consistent calculations, independent checking, robust relief protection, good fabrication records, controlled bolting, damage-mechanism-based inspection and disciplined management of change. Design calculations should state assumptions, units, material condition, code edition, load cases, allowable stresses and acceptance criteria. Every critical result should be checked against physical boundary conditions and plausible failure modes.

Chapter Summary

Pressure-containing components must be designed as systems rather than as isolated wall-thickness problems. Thin cylindrical shells use membrane hoop and longitudinal stress formulas when the wall is sufficiently thin and geometry is regular. Thick cylinders require the Lamé stress distribution, with radial and hoop stresses varying through the wall. External pressure introduces buckling risk, while openings, welds, threads and abrupt transitions create local stress concentrations.

Flanges and closures require adequate bolt preload, gasket compression, stiffness and control of external loads. Temperature changes alter material strength, produce thermal stress and affect gasket and bolt behavior. Cyclic pressure and temperature require fatigue assessment, often using code-specific fatigue curves and cumulative usage concepts. Material degradation, fabrication quality, nondestructive examination, proof testing and in-service inspection are integral to mechanical integrity.

Codes provide minimum technical and regulatory requirements, but safe engineering also requires judgment, traceability, ethical conduct and life-cycle management. The most reliable designs combine analytical calculations with sound materials engineering, qualified fabrication, controlled assembly, effective inspection and feedback from operating experience.

Key Terms

Pressure boundary: The wall or assembly that contains pressurized fluid.

Design pressure: Pressure selected for design, including credible overpressure conditions.

Design temperature: Temperature used to establish material properties and design limits.

Hoop stress: Circumferential stress produced by pressure in a cylinder.

Longitudinal stress: Axial stress produced by pressure acting on closed ends.

Thin-walled cylinder: Cylinder for which stress variation through thickness is small enough for membrane theory.

Thick-walled cylinder: Cylinder requiring a stress distribution through its wall.

Lamé solution: Elastic solution for radial and hoop stresses in a thick cylinder.

Allowable stress: Code-permitted stress incorporating material properties and safety factors.

Joint efficiency: Factor representing the strength and quality of a welded or joined region.

Corrosion allowance: Additional thickness reserved for anticipated material loss.

Stress concentration: Local amplification of stress caused by geometry or discontinuity.

Gasket: Sealing element compressed between joint faces.

Bolt preload: Initial tensile force applied to a bolt during assembly.

Pressure thrust: Force generated by pressure acting on a projected area.

Fatigue: Progressive damage caused by repeated or fluctuating loading.

Proof test: Controlled test demonstrating integrity at a specified pressure or load.

Nondestructive examination: Inspection that does not destroy the component.

Fitness for service: Engineering assessment of continued operation with known degradation.

Mechanical integrity: Ability of equipment to remain safe and functional throughout its life.

Review Questions

Short Answer Questions

1. State the assumptions behind the thin-wall hoop-stress equation.
2. Why is hoop stress twice the longitudinal stress in a closed-end thin cylinder?
3. What is the physical meaning of corrosion allowance?
4. Identify two reasons why a flange can leak even when its pressure rating is adequate.
5. Why is hydrostatic testing generally safer than pneumatic testing?

Long Answer Questions

1. Explain the Lamé solution for a thick-walled cylinder and describe its boundary conditions.
2. Discuss how temperature affects pressure-boundary materials, bolts, gaskets and fatigue life.
3. Explain the design checks required for a bolted flange and gasketed closure.
4. Compare allowable-stress design, stress-concentration assessment and fatigue assessment.
5. Describe a complete mechanical-integrity program from fabrication through in-service inspection.

Analytical / Application Questions

1. A cylinder has p = 3 MPa, mean radius r = 400 mm, allowable stress S = 100 MPa and joint efficiency E = 0.8. Calculate the preliminary thin-wall pressure thickness, then add a 3 mm corrosion allowance. State why the result must still be checked against a pressure-equipment code.
2. A 100 mm internal-diameter pipe carries fluid at 2 MPa. Calculate the pressure thrust on a closed end and explain how this force affects anchors and connected equipment.
3. A flange leaks only during cooldown. Develop a fault tree involving bolt preload, gasket relaxation, flange rotation, piping loads, surface condition and thermal expansion, and propose an inspection or test for each suspected cause.

Multiple Choice Questions with Answers

Question 1

For a closed-end thin-walled cylinder, the hoop stress is:

A. pr/t

B. pr/(2t)

C. 2pt/r

D. pt/r²

Correct Answer: A. pr/t

Question 2

Which condition most strongly indicates the need for a thick-wall analysis?

A. Very small t/r ratio

B. Large wall thickness relative to radius

C. A perfectly uniform temperature

D. Low fluid density

Correct Answer: B. Large wall thickness relative to radius

Question 3

In a cylinder subjected only to internal pressure, the maximum tensile Lamé hoop stress generally occurs at the:

A. Mid-thickness

B. Outer surface

C. Inner surface

D. Closed end only

Correct Answer: C. Inner surface

Question 4

For p = 2 MPa and a circular projected diameter of 100 mm, the pressure thrust is approximately:

A. 1.57 kN

B. 15.7 kN

C. 157 kN

D. 1.57 MN

Correct Answer: B. 15.7 kN

Question 5

The main purpose of a gasket in a bolted flange is to:

A. Increase fluid pressure

B. Provide a seal between joint faces

C. Eliminate all bolt stress

D. Prevent thermal expansion

Correct Answer: B. Provide a seal between joint faces

Question 6

A pneumatic pressure test is generally more hazardous than a hydrostatic test because:

A. Gas has greater stored compressible energy

B. Water cannot transmit pressure

C. Gas always corrodes steel

D. Hydrostatic tests require no controls

Correct Answer: A. Gas has greater stored compressible energy

Question 7

The quantity K in the simplified relation T = KFd primarily represents:

A. Material yield strength

B. Friction and nut-factor effects

C. Pressure-vessel volume

D. Gasket thickness

Correct Answer: B. Friction and nut-factor effects

Question 8

External pressure can cause a thin pipe to fail by:

A. Buckling or collapse

B. Only longitudinal yielding

C. Gasket extrusion alone

D. Reduced fluid density

Correct Answer: A. Buckling or collapse

Question 9

In the cumulative fatigue usage expression U = ∑(n_i/N_i), n_i denotes:

A. Allowable stress

B. Actual cycles of a specified type

C. Wall thickness

D. Joint efficiency

Correct Answer: B. Actual cycles of a specified type

Question 10

Which action is most appropriate when an in-service pressure vessel has detected wall thinning?

A. Ignore it if the vessel passed its original test

B. Increase pressure to stabilize the wall

C. Perform a documented fitness-for-service or repair assessment

D. Remove all inspection records

Correct Answer: C. Perform a documented fitness-for-service or repair assessment