Creep in Pressure Vessels
Creep in pressure vessels — membrane and local stresses, nozzles and discontinuities, thermal gradients through the wall, stress redistribution, creep rupture assessment and the determination of design life.
The pressure vessel creep problem
Pressure vessels in high-temperature service (e.g. steam drums, boiler headers, reformer vessels, reactors) operate under sustained internal pressure at temperatures where creep is significant. The vessel wall is under membrane stress (the hoop and longitudinal stresses from the pressure) and local stresses (at nozzles, attachments and discontinuities). The combination of sustained pressure stress and elevated temperature causes creep strain to accumulate over the service life. The creep may lead to excessive deformation (wall thinning, diameter increase), stress redistribution (from the elastic to the steady-state creep distribution), and eventually creep rupture. The assessment must determine whether the vessel maintains adequate strength and dimensional stability for the design life, and whether the local stress concentrations (nozzles, discontinuities) have adequate creep life.
Membrane stress and the creep rupture assessment
The membrane stress in a pressure vessel wall is the primary stress from the pressure. For a thin cylinder, the hoop membrane stress is pR/t and the longitudinal stress is pR/(2t), where p is the pressure, R is the radius and t is the wall thickness. For a thick cylinder, the Lamé equations give the through-thickness stress distribution. The membrane stress is used in the creep rupture assessment: the stress is compared to the stress-rupture curve at the design temperature and design life, with the appropriate safety factor. The design code (e.g. ASME, EN, API) specifies the allowable stress and the safety factor for creep conditions. The rupture assessment should use the steady-state creep stress distribution (which may differ from the elastic distribution) for long-term assessment, and the full time-dependent stress history for short-term or cyclic assessment.
Pressure vessel membrane stress (thin cylinder):
Hoop stress: sigma_hoop = p * R / t
Longitudinal: sigma_long = p * R / (2 * t)
Equivalent (von Mises):
sigma_eq = sqrt(3) * p * R / (2 * t)
where:
p = internal pressure [Pa]
R = mean radius [m]
t = wall thickness [m]
For thick cylinders, use the Lame equations.
Creep rupture assessment:
sigma_design <= sigma_rupture(t_design, T) / SF
where:
sigma_rupture = stress causing rupture at t_design
and T (from stress-rupture curve or LMP)
SF = safety factor (per applicable design code)
Note: for long-term assessment, use the steady-state
creep stress distribution, not the elastic distribution.
The steady-state distribution differs for Norton creep
(stress proportional to elastic stress^(1/n)).Local stresses at nozzles and discontinuities
Nozzles, attachments and geometric discontinuities introduce local stress concentrations. The elastic stress concentration factor at a nozzle may be 2–3 or higher. At elevated temperature, the local high stress creeps faster, and the stress relaxes by creep redistribution. The steady-state creep stress concentration factor is lower than the elastic factor (approximately the elastic factor raised to the power 1/n for a Norton law). However, the accumulated creep strain at the nozzle is higher (because the stress was initially high), and the local creep damage may be more advanced. The nozzle region is often the life-limiting location in a pressure vessel. The assessment must analyse the nozzle with a detailed FEA (typically a 3D model with the nozzle, the wall and the reinforcement) and assess the creep strain and the rupture life at the critical location in the nozzle.
Thermal gradients through the wall
In a pressure vessel with a hot fluid inside and a cooler exterior, the through-thickness temperature gradient produces thermal stress. The inner surface (hotter) is in compression (constrained expansion) and the outer surface (cooler) is in tension, or vice versa depending on the constraint and the geometry. The thermal stress adds to the pressure stress. At elevated temperature, the thermal stress relaxes by creep. The stress redistribution from the combined pressure and thermal stress must be captured in the analysis. A coupled thermal-structural-creep analysis is needed: the temperature field is computed from the heat transfer analysis, the thermal and pressure stresses are computed, and the creep redistribution is modelled over the service life. The through-thickness stress history determines the rupture location, which may be at the inner or outer surface depending on the combined stress and the redistribution.
Stress redistribution in the thick wall
In a thick-walled pressure vessel under internal pressure, the elastic stress distribution has the maximum hoop stress at the inner surface and a lower stress at the outer surface. At elevated temperature, the inner surface (higher stress) creeps faster, and the stress redistributes outward. The steady-state creep stress distribution is more uniform than the elastic distribution and, for a Norton law with n > 1, may have the maximum stress at the outer surface. This means that the rupture location may shift from the inner surface (elastic) to the outer surface (steady-state creep). The analysis must capture this redistribution to correctly identify the rupture location and the rupture life. A simple elastic analysis with the maximum (inner surface) stress would over-predict the stress and may incorrectly identify the rupture location.
Design life and reassessment
The design life of a high-temperature pressure vessel is determined by the creep rupture life at the critical location (which may be a nozzle, a discontinuity or the general wall), with the safety factor specified by the design code. The design life may be 100,000 hours, 200,000 hours or more, depending on the application and the code. For vessels already in service, the remaining life should be reassessed periodically, based on the operating history, the material condition (from inspection) and the forward-looking conditions. The reassessment is particularly important for vessels approaching their design life — the actual consumed life may be more or less than the design assumption, depending on the actual operating conditions. The reassessment should follow a structured remaining-life procedure.
The life-limiting location in a creep pressure vessel may not be the location of maximum elastic stress. Stress redistribution may move the critical location from the inner surface to the outer surface, or from the general wall to a nozzle. Always perform a time-dependent creep analysis to identify the actual critical location and assess the rupture life there.
Local-to-global assessment strategy
A pressure-vessel creep assessment is often most efficient when the global vessel behaviour and the local discontinuity behaviour are separated deliberately. A global model should establish membrane load, bending, thermal expansion, support reactions and the redistribution that develops over long duration. Local submodels can then resolve nozzle corners, knuckles, attachments, weld transitions and other geometrical details using boundary conditions transferred from the global solution. The hand-off must preserve the relevant deformation and load path; applying an elastic boundary field from the start of life to a local creep model can suppress the very redistribution being assessed. For long-duration cases, the transfer should therefore represent the appropriate service time or sequence. Critical results should be checked for sensitivity to the submodel boundary location, mesh density and stress extraction method. The objective is not the highest local stress value, but a stable representation of the structural mechanism that controls deformation, rupture or creep-fatigue damage.
Inspection-informed reassessment
For vessels already in service, analysis should be reconciled with inspection rather than treated as an isolated prediction. Measured wall thickness, local thinning, distortion, nozzle movement, weld condition and evidence of cracking can materially change the remaining-life assessment. Where temperatures have been monitored, the recorded distribution is generally more valuable than a nominal design temperature, provided sensor location and uncertainty are understood. Geometry from inspection can be introduced directly into the model when distortion or thinning affects the load path. Conversely, an analysis can help target inspection by identifying regions where redistributed stress, temperature and metallurgical vulnerability coincide. A defensible reassessment records which dimensions and operating histories are measured, which are inferred, and which remain conservatively bounded. This closes the loop between calculation and plant evidence and avoids assigning unjustified precision to a nominal as-designed model.