Understanding Plastic Collapse: Lower Bound vs. Upper Bound Theorems

1. Introduction

Plastic analysis provides a powerful tool for estimating the collapse load of structures by using either the lower bound (static) or upper bound (kinematic) theorems. This document summarizes both methods, their differences, and uses a fixed-end beam example to explain their convergence when the structure has only one possible collapse mechanism.

2. Basic Definitions

• Lower Bound Theorem: If a set of internal moments/shears can be found that satisfies equilibrium and nowhere exceeds the ultimate (plastic) limit—not just first yield (see notes), then the corresponding load is a safe estimate of collapse.• Upper Bound Theorem: If a collapse mechanism is assumed and internal work equals external work (via virtual work), the resulting load is a possible collapse load (upper estimate).• True Collapse Load: Occurs when both theorems give the same value — indicating an accurate collapse load prediction.

3. Fixed-End Beam Example (Point Load at Midspan)

3.1 Problem Setup

• Beam: Fixed at both ends• Span: L• Point Load: P at midspan• Plastic Moment Capacity: Mp• Required for collapse: 3 plastic hinges (2 supports + midspan)

3.2 Lower Bound Solution (Static)

• Step 1: First yield at supports → PL/8 = Mp → P1 = 8Mp/L• Step 2: Continue increasing load until midspan also reaches Mp → ΔP = 8Mp/L• Collapse load = P1 + ΔP = 16Mp/L• This moment distribution satisfies equilibrium and all moments ≤ Mp → Valid lower bound

Notes;

Yield moment is when any rebar reaches the first yield. 

In co Crete structures, plastic limit is when concrete strain reaches 0.003.

3.3 Upper Bound Solution (Kinematic)

• Assume plastic hinges at both supports and midspan → mechanism forms• Internal work = 4Mpθ, External work = P(L/2)θ → P = 16Mp/L• Since assumed mechanism is kinematically admissible → Valid upper bound

3.4 Conclusion: Bounds Match

• Lower Bound Collapse Load = Upper Bound Collapse Load = 16Mp/L• This confirms the true collapse load since both conditions are satisfied.

4. When Do Bounds Match?

• If a structure has only one possible collapse mechanism, the lower and upper bound loads will always match.• Examples include: fixed-end beam, propped cantilever, simple portal frames with known failure patterns.

5. Key Differences Between Lower and Upper Bound Theorems

• Lower Bound Theorem:  - Based on internal equilibrium and yield limits  - Always gives a safe (conservative) estimate  - May underestimate collapse load• Upper Bound Theorem:  - Based on assumed mechanism and virtual work  - May overestimate collapse load unless correct mechanism is assumed  - Accurate when real mechanism is used

6. Final Conclusion

When a structure has a unique, known collapse mechanism, both the lower and upper bound theorems will yield the same collapse load. This convergence provides confidence in the plastic design approach and simplifies collapse load computation.

From the author point of view, lower bound theorem is based on force demand while upper bound is limited by ductility or deformation demand.

Concept	Force-Based View	Displacement-Based View
Traditional RC design	Capacity checks: M_u < \phi M_n	Drift and curvature limits, plastic rotations
Pushover/Seismic PBSD	Base shear, moment demands	Target drift, plastic hinge rotation, ductility
Lower bound theorem	Solve for moment distributions
	Upper bound theorem	Assume deformations to solve collapse load