Langford Analytic · Knowledge Base
Pressure & Containment Analysis
Pressure-containing structures are governed by more than nominal hoop stress. Geometry, discontinuities, thermal gradients, external loads, local bending, instability, cyclic pressure, joints, supports and transient fluid effects can all control structural integrity. This section covers the engineering methods used to assess pressure vessels, piping and containment systems from fundamental membrane behaviour through detailed finite element analysis, fatigue, collapse and dynamic pressure loading.
Pressure Structural Fundamentals
Pressure & Containment Analysis FundamentalsThe engineering basis for pressure and containment analysis — how pressure environments become structural loading through membrane response, bending, local stress, instability and the analysis workflow from requirements to verification.Pressure as a Structural LoadHow distributed normal traction on a surface becomes resultant force and moment, pressure direction, surface normals, curved and closed surfaces, and common modelling mistakes.Absolute, Gauge & Differential PressureAbsolute, gauge and differential pressure definitions, reference pressure, structural significance of each convention, and common modelling mistakes from mixing pressure conventions.Internal vs External PressureWhy internal and external pressure produce fundamentally different structural responses — membrane tension versus compression and instability — and the implications for analysis method selection.Pressure Resultant ForcesHow pressure integrates to resultant force and moment, F = pA for simple cases, projected area, closed ends, curved surfaces, and when p times local surface area is not the correct approach.Pressure Vessel Load PathsHow pressure loads travel through shells, heads, nozzles, flanges, bolts, supports, connected piping and foundations — reinforcing physical equilibrium from pressure to reaction.Thin-Walled Pressure Vessel FundamentalsThin-wall idealisation, membrane behaviour in cylindrical and spherical geometry, the r/t concept, and the conditions under which the thin-wall assumption is and is not valid.Hoop Stress in Thin-Walled CylindersDerivation and physical meaning of the hoop or circumferential stress in a thin-walled cylinder under internal pressure, with variable definitions and the assumptions behind the standard relationship.Longitudinal Stress in Closed-End CylindersDerivation of the axial membrane stress from pressure end load, relationship to hoop stress, and the difference between closed-ended and open-ended cylinders.Spherical Pressure Vessel StressMembrane stress in a spherical pressure vessel from symmetry, comparison with cylindrical vessel behaviour, and the structural efficiency of spherical geometry.Pressure End LoadsHow pressure on end areas creates axial thrust on closures, flanges, bolts and pipe terminations — the structural load path from pressure end load through the vessel and into supports.Thin-Wall Assumption & Its LimitationsThe radius-to-thickness concept, when the membrane approximation breaks down, radial stress, through-thickness variation, local features, and the transition to thick-wall theory.Thick-Walled Cylinder FundamentalsRadial and circumferential stress through the wall thickness, inner and outer radius, internal and external pressure, and why thin-wall theory becomes inadequate for thick-walled cylinders.Lamé Equations for Thick-Walled CylindersThe Lamé equations for radial and circumferential stress in a thick-walled cylinder under internal, external and combined pressure, with variable definitions, interpretation and limitations.Through-Thickness Stress in Pressure VesselsRadial, hoop and longitudinal stress variation through the wall, inner-wall peak, outer-wall response, stress gradients and the significance for assessment and fatigue.Combined Internal & External PressureStress distributions under simultaneous internal and external pressure, the role of pressure difference versus absolute pressure, and the structural implications for design.Thick-Wall vs Thin-Wall Pressure AnalysisComparison of assumptions, through-thickness gradients, geometry, computational effort and appropriate applications for thin-wall and thick-wall pressure analysis methods.
Local Features, Joints & Discontinuities
Pressure Vessel End ClosuresFlat ends, domed ends, hemispherical ends, ellipsoidal forms, pressure end load transfer, local bending at the shell-to-head junction and load path considerations.Flat End Plates Under PressureBending-dominated response of flat end plates, plate stiffness, boundary conditions, pressure and thickness relationships, and membrane effects at larger deformation.Domed & Curved Pressure ClosuresMembrane action in curved pressure closures, curvature and pressure transfer, junction stresses, structural efficiency comparisons between hemispherical, torispherical and ellipsoidal forms.Shell-to-Head Junction StressesStiffness discontinuity, membrane mismatch, local bending, geometric transition, stress concentration and modelling considerations at the shell-to-head junction.Pressure-Loaded Corners & Geometric DiscontinuitiesAbrupt stiffness changes, local bending at re-entrant regions, stress concentration, mesh interpretation at corners and structural detailing for pressure-loaded discontinuities.Openings & Nozzles in Pressure StructuresHow openings interrupt the membrane load path, local reinforcement, nozzle stiffness, pressure thrust, connected piping loads, stress concentration and FEA considerations.Pressure Loads at Nozzles & PenetrationsInternal pressure thrust at nozzles, local pressure on the nozzle neck, shell junction loads, pipe reaction forces and moments, and structural equilibrium at penetrations.Stress Concentrations in Pressure BoundariesSources of stress concentration in pressure vessels — openings, transitions, nozzles, weld geometry and attachments — with fatigue implications.Nozzle Loads & Local StressPressure and piping loads at vessel nozzles, shell deformation, nozzle stiffness, local membrane and bending response, and FEA approaches for nozzle assessment.Openings & ReinforcementInterrupted membrane load paths at openings, reinforcement concepts and limits, stress concentration at openings, multiple openings and limitations of simple analytical methods.Geometric Discontinuities in Pressure VesselsShell-to-head transitions, thickness changes, cone-to-cylinder junctions, support attachments and the local discontinuity forces that govern local bending in pressure vessels.Flange Behaviour Under PressureFlange rotation, pressure thrust, bolt loading, gasket compression, flange bending, joint stiffness, leakage risk and structural modelling of bolted flange joints.Gasket Behaviour & Sealing MechanicsGasket seating, compression, unloading, bolt preload, pressure-induced unloading, flange rotation effects, nonlinear gasket behaviour and limitations of simplified models.Bolted Pressure JointsPreload, pressure thrust, joint stiffness, separation, bolt load increase, gasket behaviour, fatigue and leakage considerations for bolted joints in pressure systems.Welded Pressure BoundariesWeld load paths in pressure vessels, butt welds, fillet welds, weld toe stress concentration, residual stress, fabrication effects and FEA representation of welded boundaries.
External Pressure, Instability & Collapse
External Pressure & CollapseExternal-pressure collapse of vessels and shells, including instability mechanisms, geometry and boundary effects, imperfection sensitivity, stiffening, nonlinear analysis and defensible collapse margins.Buckling of Cylindrical Shells Under PressureCylindrical-shell buckling under external pressure, from classical behaviour and eigenvalue screening through imperfection-seeded nonlinear collapse analysis and stiffener design.Imperfection Sensitivity of Pressure ShellsHow ovality, weld distortion, dents, thickness variation and measured geometry affect pressure-shell stability, with practical approaches to imperfection modelling and sensitivity.Plastic Collapse of Pressure ComponentsPlastic-collapse assessment of pressure components using limit concepts and nonlinear FEA, with realistic material modelling, combined loads, deformation checks and collapse criteria.Proof Pressure AssessmentEngineering assessment of hydrostatic and pneumatic proof pressure, including structural load definition, elastic-plastic response, test configuration, acceptance and analysis support.Burst Pressure AssessmentPrediction and interpretation of pressure-boundary burst, including ductile failure mechanisms, nonlinear material response, local weaknesses, testing and uncertainty.
Pressure FEA & Stress Assessment
Finite Element Analysis of Pressure VesselsPractical pressure-vessel FEA from modelling objective and idealisation through pressure loading, restraints, mesh strategy, stress recovery and verification.Stress Linearisation Through ThicknessStress classification line placement, membrane and bending extraction, interpretation in curved pressure boundaries and common mistakes in solid-model stress linearisation.Primary, Secondary & Peak Stress ConceptsEngineering meaning of primary, secondary and peak stress in pressure structures, including load control, self-limiting response, stress classification and common interpretation errors.Singularities & Local Peaks in Pressure FEAIdentification and treatment of stress singularities in pressure FEA, distinguishing numerical divergence from finite physical peaks and selecting suitable assessment methods.Pressure Vessel FEA VerificationVerification framework for pressure-vessel finite element models covering analytical checks, equilibrium, mesh convergence, boundary sensitivity, nonlinear controls and independent review.Pressure Vessel Model IdealisationSelection of axisymmetric, shell, solid and submodel representations for pressure vessels, including connections, supports, contacts, offsets and model-boundary strategy.
Piping & Connected Systems
Pipe Stress Analysis FundamentalsPipe-stress fundamentals for pressure systems, linking pressure, weight, thermal expansion, occasional loads, flexibility, support reactions and nozzle loads into a defensible structural model.Piping Flexibility & Thermal ExpansionThermal expansion and piping flexibility, including imposed strain, restraint loads, flexibility by routing, expansion loops, springs, friction and equipment-interface effects.Pressure Thrust in Piping SystemsPressure thrust in piping, including closed-end force, changes in pressure area, elbows, reducers, bellows and unrestrained expansion joints, with correct system-level equilibrium.Piping Supports, Restraints & ReactionsStructural behaviour of piping supports, anchors, guides, line stops and springs, including gaps, friction, support stiffness, load cases and reaction transfer into structures.Nozzle-Piping InteractionCoupled behaviour between piping and vessel or equipment nozzles, including interface stiffness, local shell stress, allowable loads, flexibility and submodel transfer.Branch Connections & Local Piping StressStructural assessment of tees, fabricated branches and lateral connections, including pressure reinforcement, run-and-branch load interaction, local shell stress and fatigue.
Transient Pressure & Fluid-Structure Response
Pressure Transient FundamentalsFundamentals of pressure transients in liquid and gas systems, including wave propagation, impedance, reflections, event timescale and the link between hydraulic and structural response.Water HammerWater-hammer mechanics and engineering assessment, including Joukowsky pressure rise, wave speed, valve-closure time, reflections, cavitation, supports and structural dynamics.Dynamic Response of Piping to Pressure TransientsStructural transient analysis of piping subjected to time-varying pressure and fluid-force histories, including modal content, spatial loading, supports, damping and result recovery.Fluid-Structure Interaction in Pressure SystemsFluid-structure interaction in pressure systems, from one-way pressure mapping to coupled wave propagation, pipe-wall compliance, added mass and two-way transient response.Rapid Valve Closure & Pump TransientsTransient events caused by valve actuation, pump start, trip and check-valve behaviour, including hydraulic modelling inputs, reverse flow, pressure waves and structural consequences.Pressure Pulse LoadingStructural response to short-duration pressure pulses, including pulse shape, impulse, rise time, dynamic amplification, spatial distribution and nonlinear response.
Fatigue, Thermal Effects & Structural Integrity
Pressure Cycling & FatigueFatigue assessment of cyclic pressure systems, from pressure histories and structural stress ranges through cycle counting, local discontinuities, mean stress, cumulative damage and verification.Thermal Stress in Pressure SystemsThermal stress in vessels and piping, including free expansion, restraint, through-wall gradients, differential expansion, thermal shock, coupled thermal-structural FEA and verification.Thermal Fatigue of Pressure ComponentsThermal-fatigue mechanisms in pressure equipment, including transient gradients, mixing, stratification, local cycling, thermo-mechanical stress ranges and fatigue assessment.Ratcheting & Progressive DeformationRatcheting and progressive plastic deformation under combined pressure and cyclic secondary loading, including shakedown concepts, nonlinear cyclic FEA and acceptance.Creep in High-Temperature Pressure SystemsCreep in high-temperature pressure equipment, covering time-dependent strain, stress relaxation, rupture, weld effects, creep-fatigue interaction and nonlinear life assessment.Leak-Before-Break ConceptsLeak-before-break engineering concepts for pressure boundaries, including through-wall crack development, stable tearing, leak detection, fracture stability and system-level qualification.Defect Assessment in Pressure ComponentsEngineering assessment of cracks, wall thinning, pitting, dents and other damage in pressure components using inspection data, remaining-strength methods, fracture mechanics and fitness-for-service logic.
Verification & Engineering Workflow
Pressure & Containment Analysis VerificationA systematic verification framework for pressure and containment analysis covering requirements, analytical checks, equilibrium, mesh convergence, nonlinear response, interfaces and independent review.Pressure Analysis Assumptions & LimitationsHow to identify, justify and manage geometry, material, load, support, thermal, fatigue, fracture and numerical assumptions in pressure-system analysis.Defensible Pressure & Containment Analysis WorkflowEnd-to-end workflow for pressure and containment engineering, from requirements and load definition through analytical checks, FEA, integrity assessment, verification, margins and reporting.