CIV5136 Chap.7 Plastic Collapse and Mechanism Method
Plastic Collapse and Mechanism Method
Plastic collapse analysis idealises sections as reaching a plastic moment and then rotating as hinges. A correct collapse load satisfies equilibrium, yield and mechanism conditions simultaneously.
Virtual work evaluates beam, sway, combined or partial mechanisms; the smallest valid mechanism multiplier is compared with a statically admissible moment field.
The kinematic calculation begins with possible hinge locations and a compatible motion. Assign a virtual displacement, derive each hinge's relative rotation and equate external work to the sum of plastic moment times rotation.
Shared hinges in combined mechanisms are counted once using their resulting rotation. Hinge signs must agree with the bending pattern required by equilibrium, not merely with convenient points on the structural drawing.
A mechanism calculation alone gives an upper-bound candidate. A moment field that satisfies equilibrium without exceeding capacity gives a lower bound.
When one load is both kinematically and statically admissible, the idealised collapse load is uniquely identified. This is why the course material keeps equilibrium, yield and mechanism conditions together rather than treating virtual work as a complete proof by itself.
Continuous structures can suffer partial collapse in one span before a complete mechanism forms across the whole system.
Frames can develop beam, sway or combined modes. List all plausible candidates, calculate positive load multipliers consistently and select the smallest. Then state the rigid-plastic limitations: service deflection, instability, local failure and limited hinge rotation capacity require separate judgement.
What this chapter covers
- 01
Rigid-plastic section idealisation
- 02
Plastic hinge formation
- 03
Equilibrium, yield and mechanism conditions
- 04
Upper and lower bound theorems
- 05
Virtual work for frame mechanisms
- 06
Complete and partial collapse
- 07
Beam, sway and combined modes
- 08
Critical load-factor selection
Worked example · free
Collapse of a fixed-fixed beam
- +1Place plastic hinges at both fixed ends and at midspan.
- +1For midpoint displacement delta, each half rotates theta = delta divided by 3.
- +1The midspan relative rotation is 2 theta.
- +2Internal work is 4Mp theta = 120 delta.
- +1External work is P delta, giving P = 120 kN.
- +1Check an equilibrium moment field with no section above 90 kN m.
Key terms
- Plastic moment
- The ideal section moment capacity maintained while a plastic hinge rotates in rigid-plastic analysis.
- Plastic hinge
- A concentrated rotation point formed when a section reaches the adopted plastic moment capacity.
- Mechanism condition
- The requirement that enough hinges form in compatible positions to permit collapse motion.
- Static admissibility
- A moment field that satisfies equilibrium and does not exceed plastic capacity anywhere.
- Partial collapse
- A mechanism confined to part of a continuous structure while other regions remain stable.
Plastic Collapse and Mechanism Method FAQ
Is a drawn mechanism enough to prove collapse?
No. It must be kinematically compatible, while the associated bending moments must also satisfy equilibrium and remain within the yield limits. The three conditions together establish the ideal collapse state. A virtual-work equality by itself proves only the selected mechanism calculation. Add the static check separately. The governing mode is the smallest admissible positive multiplier.
How many hinges form a complete mechanism?
A common count is one more than the degree of static indeterminacy, but the hinge positions must actually create a compatible motion. Counting alone cannot validate the geometry.
Why compare several mechanisms?
Different beam, sway, combined and partial modes can all be admissible. The smallest positive load multiplier identifies the first mechanism predicted as proportional loading increases.
What does the rigid-plastic model omit?
It does not provide elastic deflections below collapse and relies on sufficient rotation capacity. Instability, local failure and limited ductility require separate consideration in a design interpretation.
What is the difference between upper and lower bounds?
A kinematically admissible mechanism with virtual work supplies an upper-bound collapse estimate. A statically admissible moment field satisfying equilibrium and staying within plastic capacity supplies a lower bound. If the same load satisfies both descriptions, the ideal rigid-plastic collapse value is identified uniquely.
Why are combined mechanisms easy to miscalculate?
They superpose compatible beam and sway motions, so rotations at shared hinges must be added algebraically. Counting one physical hinge separately in both elementary mechanisms double-counts internal work. Draw the combined velocity field first and label the resulting relative rotation at each distinct hinge.
Exam move
Draw each possible mechanism with rotation arrows and label positive and negative plastic hinges. Use one virtual displacement to derive all rotations, then calculate the work equation without double-counting shared hinges. Rank every admissible load factor and select the smallest. For the governing mode, sketch a compatible bending moment distribution and explicitly check equilibrium and yield.
Keep partial span mechanisms in the candidate list for continuous structures because global collapse may not occur first. Build a comparison table with mechanism name, hinge positions, relative rotations, multiplier and rejection reason. Before accepting the minimum, trace the corresponding moment signs through the structure and confirm that no non-hinge section has been pushed past its plastic capacity.
