Basic theorems of limit analysis

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.