---
title: "Basic theorems of limit analysis"
subject: "Theory of Structures II"
chapter: 5
key: "semester_5/theory_of_structures_ii:limit-analysis-two-theorems"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/limit-analysis-two-theorems
site: BCT Engineering
---

# Basic theorems of limit analysis

Theory of Structures II · Chapter 5: Introduction to plastic analysis

- Asked in: 2069 Chaitra, 4 marks; 2068 Chaitra, 8 marks (asked 2 times)
- Video: [Structural Mechanics: Limit Analysis](https://www.youtube.com/watch?v=Cj9R9MKsC1Q) — Qi Dang
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/limit-analysis-two-theorems
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-5

## Question

Enunciate and explain, with its uses, the two basic theorems on methods of limit analysis in plastic analysis for bending.

## Short answer

Static (lower-bound) theorem: any equilibrium moment field with |M|≤M_p gives a load factor ≤ true collapse load. Kinematic (upper-bound) theorem: virtual work on an assumed mechanism, ΣM_pθ = λΣPδ, gives load factor ≥ true collapse load; the smallest value from all possible mechanisms is the correct one. Uniqueness theorem: if both conditions are met, it is exact.

## Answer

Limit analysis finds the collapse load of a structure made of ductile material with a plastic moment $M_p$. Three conditions govern a collapse state:
1. **Equilibrium condition:** the bending moment diagram is in equilibrium with the loads.
2. **Mechanism condition:** enough plastic hinges have formed to make a mechanism.
3. **Yield (plastic moment) condition:** the moment at no section exceeds $M_p$.

#### 1. Static theorem (lower-bound theorem)
*Statement:* the load factor $\lambda_s$ obtained from any bending moment distribution that satisfies equilibrium and does not violate the yield condition ($|M|\le M_p$ everywhere, a statically admissible state) is **not greater** than the true collapse load factor: $\lambda_s \le \lambda_c$.

*Explanation:* the structure can certainly carry a load for which a safe moment field exists, so the true collapse load is at least this value. The mechanism condition is not required.

*Uses:* gives a safe (conservative) estimate; used to check the design and to confirm a result found by the kinematic method (if no section exceeds $M_p$, the load is correct).

#### 2. Kinematic theorem (upper-bound theorem)
*Statement:* the load factor $\lambda_k$ obtained from any assumed collapse mechanism, by equating the work of the external loads to the internal work in the plastic hinges, $\sum M_p\theta = \lambda \sum P\delta$ (it satisfies the equilibrium and mechanism conditions), is **not less** than the true collapse load factor: $\lambda_k \ge \lambda_c$.

*Explanation:* an assumed mechanism may not be the real one, and the real one needs less load. The correct mechanism gives the lowest value.

*Uses:* all possible mechanisms are tried (beam, sway, joint and combined) and the smallest load factor is taken. It is simple to apply by hand.

#### 3. Uniqueness theorem
If one load factor satisfies all three conditions (equilibrium, mechanism, yield), it is the true collapse load, $\lambda_s = \lambda_k = \lambda_c$.

#### Example
Propped cantilever (A fixed, C roller), span $L$, central load $P$.
- Static: choose $M_A = -0.5M_p$, then $M_{mid} = PL/4 - 0.25M_p \le M_p$, so $P = 5M_p/L$ (lower bound).
- Kinematic: hinges at A and at a section $a = 0.4L$ from A give $P = \dfrac{2M_p(2L-a)}{La} = \dfrac{2M_p(1.6L)}{0.4L^2} = 8M_p/L$ (upper bound).
- Correct mechanism (hinges at A and mid-span): $P_c = 6M_p/L$, which lies between the two.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The static (lower-bound) theorem states that a load factor from a safe moment distribution is:
   A. greater than or equal to the collapse load factor
   B. equal to the collapse load factor
   C. not greater than the true collapse load factor
   D. unrelated to the collapse load factor
   Answer: C. A statically admissible state (equilibrium, |M| <= M_p) gives lambda_s <= lambda_c. (Key point: Static theorem)

2. The kinematic theorem's load factor from an assumed mechanism is:
   A. not less than the true collapse load, so the smallest of all mechanisms is taken
   B. not greater than the true collapse load
   C. always equal to the true collapse load
   D. found by checking that no moment exceeds M_p
   Answer: A. An assumed mechanism satisfies equilibrium and mechanism conditions, giving an upper bound; the lowest value is the correct one. (Key point: Kinematic theorem)

3. When is a load factor the true collapse load?
   A. When only the mechanism condition holds
   B. When only equilibrium holds
   C. When it satisfies equilibrium, mechanism and yield conditions together
   D. When it is the highest of all trial mechanisms
   Answer: C. Uniqueness theorem: satisfying all three conditions gives lambda_s = lambda_k = lambda_c. (Key point: Uniqueness)

4. For the propped cantilever with central load, the static estimate is 5Mp/L, the kinematic trial is 6.67Mp/L. What is the correct value?
   A. 6Mp/L, between the two
   B. 5Mp/L
   C. 6.67Mp/L
   D. 8Mp/L
   Answer: A. Correct mechanism (hinges at A and mid-span) gives 6Mp/L, between lower and upper bound. (Key point: Example)

## Related in Theory of Structures II

- [Elastic vs plastic analysis; plastic hinge](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aelastic-vs-plastic-analysis-plastic-hinge) — Theory of Structures II, chapter 5
- [ILD: bending moment at B, two-span beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-bm-b-two-span-8) — Theory of Structures II, chapter 4
- [ILD: bending moment at D, two-span beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-bm-d-ab-bc-2) — Theory of Structures II, chapter 4
- [Collapse load of two-span beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-load-two-span-beam-20kn) — Theory of Structures II, chapter 5
- [Two-hinged arch: BM at section x-x](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aarch-bm-section-xx-60-85) — Theory of Structures II, chapter 2
- [Two-hinged arch: BM, thrust, radial shear, support yield](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aarch-bm-thrust-shear-yield-support) — Theory of Structures II, chapter 2

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