Math 2.0

Mechanics of deformable solids Definition

Limit analysis

Builds on PlasticityYield surfaceAssociated flow ruleCauchy stress tensor

Definition

Let a body consist of a rigid–perfectly plastic or elastic–perfectly plastic material with a convex yield surface and the associated flow rule. Let it carry loads λP\lambda P, where PP is a fixed load pattern and λ≥0\lambda \ge 0 is the load factor. The collapse load factor λc\lambda_c is the value of λ\lambda at which the body can undergo unrestricted plastic flow under constant loads. Limit analysis is the determination of λc\lambda_c, or of bounds on it, through the following two notions.

  1. A statically admissible stress field is a stress field that is in equilibrium with λP\lambda P and lies inside or on the yield surface everywhere. The lower bound theorem says that then λ≤λc\lambda \le \lambda_c.
  2. A kinematically admissible mechanism is a velocity field that satisfies the kinematic boundary conditions and on which PP does positive work. Equating the rate of work of λP\lambda P with the rate of plastic dissipation gives a value λ\lambda. The upper bound theorem says that then λ≥λc\lambda \ge \lambda_c.

Remarks

Example: a simply supported beam of span LL, with plastic moment MpM_p, carries a central point load λP\lambda P. The bending moment λPL/4\lambda P L / 4 at midspan reaches MpM_p when λP=4Mp/L\lambda P = 4 M_p / L, which gives a lower bound. The mechanism with a hinge at midspan gives the same value as an upper bound, so the collapse load is 4Mp/L4M_p/L. Non-example: a body of linear elastic material has no collapse load, because its displacement stays proportional to the load for every load, so it never flows without bound. The two theorems rely on the convexity of the yield surface and on the normality of the flow rule; without normality, as for many soils, the bounds can fail.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Associated flow rule, concept, no evidence levelAssociated flow ruleconceptCauchy stress tensor, concept, no evidence levelCauchy stress tensorconceptPlasticity, concept, no evidence levelPlasticityconceptYield surface, concept, no evidence levelYield surfaceconceptLimit analysis, definition, no evidence levelLimit analysisdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Associated flow rule concept
  2. Cauchy stress tensor concept
  3. Plasticity concept
  4. Yield surface concept
  5. Limit analysis definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Limit_analysis
About this page
Tag5912J8V4JW
URIm2://5912J8V4JW
Kinddefinition
WritingW0 stub
MSC74C05, 74G45
DefinesLimit analysis
UsesPlasticity, Yield surface, Associated flow rule, Cauchy stress tensor
Changed3 November 2026

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