FEXLA

Elastic and elastic-plastic analysis

Two ways to judge a computed stress: sort it against limits, or let the metal yield.

Principle 9 of 10 · 5 min read ·

Elastic versus elastic-plastic analysis: More capacity, more effortSide by side. Left, the elastic route: a straight load-deflection line with the first limit marked. Right, the elastic-plastic route: a curve that flattens at collapse, with the factored load below it.LoadStressYieldL2LElastic: a straight linedouble the load, double the stressAbove yield:not a real stressCut the wall, sort the stresseach class has its own limitShell wallNozzlecut linePrimarySecondaryPeakDeflectionLoadCollapseDesign loads × load factorStill carries it:no collapseLocal strainworst spot: strain keptbelow what the metal takesRatchetingCyclesPlastic strainit must stop growingElasticElastic-plasticDeflectionLoadfirst limitDeflectionLoadcollapsesort stresses, check limitsfactor loads, no collapseMore effort, often more capacity
Figure 1. More capacity, more effort.
Contents

In an elastic analysis the metal is treated as springy at any stress, so the computed stresses must be split into categories and each compared with its own limit. In an elastic-plastic analysis the metal is allowed to yield as real metal does, and the structure is judged by whether it still carries the design loads multiplied by load factors from the governing code, with separate checks on local strain and ratcheting. Both are design-by-analysis routes, usually run with a finite element model; they differ in where the engineering judgement sits.

This article compares the two routes step by step and explains why the elastic-plastic route often shows more capacity, and what it costs.

The elastic route: stress follows load

An elastic analysis treats the metal as linear at any stress: double the load and every stress doubles. It is quick, it always gives an answer, and results for separate loads can be added together, which makes load combinations cheap.

The price is that a computed stress above yield is not a real stress. Real metal would have yielded and shed load to its surroundings. The elastic number is a measure of how hard the part is being worked, to be judged against a limit, not a prediction of the stress in the metal.

Then cut, split and classify

Because the elastic stress at a point mixes membrane, bending and peak parts from several causes, it has to be sorted before it can be judged. The stress is read along a stress classification line, a line straight through the wall, usually normal to its mid-surface, and split into a membrane part, the average, and a bending part, the linear part that varies from one face to the other. What is left over is peak.

σm = (1/t) ∫ σ dx σb = (6/t²) ∫ σ · (t/2 − x) dxx through the wall, 0 to t
Elastic versus elastic-plastic analysis: Then sort and compareA section where a nozzle wall meets the shell wall, with a cut line drawn through the shell wall at the joint. Arrows fan out from the cut line to three boxes: primary, secondary and peak.LoadStressYieldL2LElastic: a straight linedouble the load, double the stressAbove yield:not a real stressCut the wall, sort the stresseach class has its own limitShell wallNozzlecut linePrimarySecondaryPeakDeflectionLoadCollapseDesign loads × load factorStill carries it:no collapseLocal strainworst spot: strain keptbelow what the metal takesRatchetingCyclesPlastic strainit must stop growingElasticElastic-plasticDeflectionLoadfirst limitDeflectionLoadcollapsesort stresses, check limitsfactor loads, no collapseMore effort, often more capacity
Figure 2. A cut line through the shell wall at a nozzle junction. Its stress is split and each part classed as primary, secondary or peak.

Each part is then classed as primary, secondary or peak by its cause and held to its own limit, as in the article on stress categories. Sorting is where most of the judgement lies. At nozzles and junctions the same bending can be argued as primary or secondary, and the placing of the cut lines changes the answer.

The elastic-plastic route: let it yield

An elastic-plastic analysis gives the model a material curve that yields, so the stress stops rising in step with the load. The design loads are multiplied by load factors from the governing code, which carry the safety margin, and applied in steps. If the model still finds a stable, converged answer at the full factored load, the structure is judged safe against collapse under that combination.

Elastic versus elastic-plastic analysis: Elastic-plastic: let it yieldLoad against deflection for an elastic-plastic analysis. The curve rises, bends over as the metal yields, and flattens at the collapse load. A line marks the design loads times the load factor; the curve passes it, so the structure still carries it.LoadStressYieldL2LElastic: a straight linedouble the load, double the stressAbove yield:not a real stressCut the wall, sort the stresseach class has its own limitShell wallNozzlecut linePrimarySecondaryPeakDeflectionLoadCollapseDesign loads × load factorStill carries it:no collapseLocal strainworst spot: strain keptbelow what the metal takesRatchetingCyclesPlastic strainit must stop growingElasticElastic-plasticDeflectionLoadfirst limitDeflectionLoadcollapsesort stresses, check limitsfactor loads, no collapseMore effort, often more capacity
Figure 3. Load against deflection: the curve bends over as the metal yields and flattens at the collapse load. The factored design load lies below it.

In practice the analysis is usually run straight to the factored load, and failure to converge is the warning sign, though non-convergence can also come from the model itself, so it needs checking. A limit-load variant uses an elastic-perfectly-plastic material and small displacements; a full elastic-plastic collapse check uses a hardening curve and large displacements. The article on material models covers the difference.

Why it often shows more capacity

When the most stressed spot yields in a real structure, it stops taking extra load and passes it to the stiffer metal around it. Yielding spreads gradually, and the structure collapses only when enough of a section, or enough sections, have yielded to form a mechanism. An elastic analysis stops at the first spot; an elastic-plastic analysis follows the load as it moves.

Yielding spreading from a hole in a plate under tensionA plate with a central hole, pulled from left and right. Yielding starts in small zones at the top and bottom of the hole, spreads into larger zones as the load rises, and near collapse forms bands that cross the plate from the hole to its edges.first yieldspreadingnear collapse
Figure 4. Yielding round a hole in a plate under tension: it starts at the top and bottom of the hole, spreads, and only near collapse forms bands across the plate.

The elastic route covers this reserve only roughly, through the 1.5 factor on bending and the local membrane allowance. For shapes far from a simple bent beam, such as a heavily loaded nozzle in a thin shell, the real reserve can be larger, which is why the elastic-plastic route often passes cases the elastic route narrowly fails.

Separate checks on local strain and ratcheting

Not collapsing is not enough. A local strain check limits the plastic strain at the worst spot, compared with what the metal can take under that combination of stresses; the allowable strain falls when the stress is strongly triaxial, as at the root of a notch. A ratcheting check runs the operating loads on and off for several cycles and confirms that the plastic strain stops growing.

Elastic versus elastic-plastic analysis: Separate checks on strain and ratchetingTwo small drawings. Left, a plate with a notch under tension, with the worst spot at the notch root highlighted. Right, plastic strain against cycles: a few small steps, then flat.LoadStressYieldL2LElastic: a straight linedouble the load, double the stressAbove yield:not a real stressCut the wall, sort the stresseach class has its own limitShell wallNozzlecut linePrimarySecondaryPeakDeflectionLoadCollapseDesign loads × load factorStill carries it:no collapseLocal strainworst spot: strain keptbelow what the metal takesRatchetingCyclesPlastic strainit must stop growingElasticElastic-plasticDeflectionLoadfirst limitDeflectionLoadcollapsesort stresses, check limitsfactor loads, no collapseMore effort, often more capacity
Figure 5. Left, a notched plate with the worst spot at the notch root. Right, plastic strain against cycles: a few small steps, then flat.

Fatigue is still checked on its own in both routes, using elastic or elastic-plastic strain ranges as the governing code allows.

More capacity, more effort

The elastic route and the elastic-plastic routeTwo columns of steps. Elastic route: linear material at the design loads, stress at each point, cut lines split into membrane, bending and peak, each part classed and compared with its limit. Elastic-plastic route: a yielding material curve, loads multiplied by code load factors, a converged answer meaning no collapse, then local strain and ratcheting checks. Fatigue is checked on its own in both.Elastic routeElastic-plastic routeLinear material,design loadsYielding material curveStress at each pointLoads × codeload factorsCut lines: membrane,bending, peakConverged?Then no collapseClass each part,compare with its limitThen local strain andratcheting checksFatigue: checked on its own in both routes
Figure 6. The two routes side by side. Classification, the hardest judgement of the elastic route, does not appear in the elastic-plastic route.

The elastic-plastic route needs no stress classification, which removes the hardest judgement of the elastic route, and it often shows more capacity. The price is more effort: a material curve, loads applied in steps, convergence to watch, each load combination run separately because results cannot be added, and more results to review. A common practice is to start elastic and move to elastic-plastic where classification is unclear or the elastic margin is small.

Key points

  • An elastic analysis is quick and additive, but stresses above yield are only numbers to be judged.
  • Its stresses are cut along lines through the wall, split into membrane, bending and peak, then classified.
  • An elastic-plastic analysis lets the metal yield and checks that factored loads are still carried.
  • Local strain and ratcheting are checked separately; fatigue is checked in both routes.
  • The elastic-plastic route avoids classification and often shows more capacity, at the cost of more effort.

Where this matters in practice

  • Nozzles and openings in thin shells where the elastic local stresses are hard to classify.
  • Junctions of heads, cones and shells with stresses close to the elastic limits.
  • Supports, lugs and brackets of unusual shape with no design-by-rule formula.
  • Large openings and heavily loaded attachments where the elastic route fails by a small margin.

Corix includes a limit-load analysis on a shell model: elastic-perfectly-plastic, small displacement, each combination passing if equilibrium holds at its factored load. It does not yet run local strain or ratcheting checks. In early access, by invitation.

See the specialty module

Related principles

Next principle: Nonlinear material models

Educational article. It explains principles and is not design guidance; follow your governing code and edition.