CIV5136 Chap.8 Hinge by Hinge Frame Analysis
Hinge by Hinge Frame Analysis
Hinge-by-hinge analysis follows progressive yielding through a sequence of elastic analyses. The smallest positive capacity ratio identifies the next hinge, the stiffness model is released at that section, and accumulated actions are carried into the next load increment.
Repetition continues until a mechanism forms, with axial-force interaction iterated when it reduces plastic moment capacity.
First analyse the intact structure under a convenient proportional reference load. At every candidate section, compare the current action and incremental action rate with both positive and negative plastic limits. The smallest positive increment reaches a capacity first.
Record the load factor and full moment state at that event, because sections that have not yielded already carry part of their eventual collapse action.
Insert a rotational release at the new hinge while maintaining its plastic moment in the action state. Reanalyse only the incremental response of the modified structure, then add that increment to the accumulated actions and displacement.
The load-displacement curve becomes piecewise linear, with a reduced slope after stiffness is lost. An unchanged slope is evidence that the release was not implemented at the intended coordinate.
Portal-frame work adds moment and axial-force interaction. Recover column axial forces, update reduced plastic capacities and repeat the incremental analysis until the predicted collapse load stabilises.
Compare the resulting hinge order and response curve with upper-bound, nonlinear finite-element and observed behaviour. Differences require discussion of boundary stiffness, material properties, imperfections, connections and failure criteria, not just a percentage table.
What this chapter covers
- 01
Elastic reference-load response
- 02
First-yield scaling
- 03
Smallest positive increment selection
- 04
Accumulated action state
- 05
Stiffness release at a plastic hinge
- 06
Piecewise load-displacement response
- 07
Axial force and moment capacity interaction
- 08
Collapse convergence and project comparison
Worked example · free
Identify the first plastic hinge
- +1A reaches capacity at factor 90 divided by 45 = 2.00.
- +1B reaches capacity at factor 75 divided by 30 = 2.50.
- +1C reaches capacity at factor 80 divided by 20 = 4.00.
- +1The smallest positive factor is 2.00, so A yields first.
- +1At that state, B is 60 and C is 40 kN m.
- +1Release A and recompute incremental action rates before choosing the next hinge.
Key terms
- Load increment
- The additional load factor applied to advance the current structural state to the next yield event.
- Capacity ratio
- The remaining action capacity divided by the incremental action rate for a candidate section.
- Stiffness release
- The modification that allows rotation at a newly formed plastic hinge while retaining its plastic moment.
- Accumulated state
- The total load, actions and deformations carried from all completed incremental stages.
- Collapse convergence
- Stabilisation of predicted collapse load after repeated updates such as axial-force-dependent capacity changes.
Hinge by Hinge Frame Analysis FAQ
Why is the next hinge based on the smallest positive ratio?
That section reaches its relevant capacity first as the load increment rises. A larger ratio would skip an earlier physical event, while a negative ratio is not approached in the current increment direction. Reanalyse the released structure before selecting the following event. Preserve the accumulated action state. Every subsequent capacity comparison uses that updated state.
What changes after a hinge forms?
The section can rotate at its plastic moment, so the structural stiffness and incremental moment distribution change. Previously accumulated actions remain part of the current state.
Why can axial force require iteration?
Axial compression can reduce moment capacity. The revised capacity changes the hinge sequence and collapse load, so analysis and capacity update are repeated until the prediction stabilises.
How should analytical and experimental collapse loads be compared?
Discuss boundary stiffness, material properties, imperfections, connection behaviour and the chosen failure criterion. A percentage difference alone does not explain why one idealisation predicts a higher or lower value.
What should a hinge event table contain?
Record the cumulative load factor, current section actions, incremental action rates, remaining positive and negative capacities, selected hinge and equilibrium check. The table preserves history between releases and makes it possible to reconstruct why one section yielded before another after the stiffness distribution changed.
How is collapse recognised in the sequence?
Collapse occurs when the accumulated plastic hinges create a compatible mechanism and the incremental stiffness can no longer resist the loading pattern. Confirm the physical movement on a sketch rather than stopping only because a numerical matrix becomes singular; a modelling error can also produce singularity.
Exam move
Build a stage table with accumulated load factor, candidate action rates, remaining capacities, selected hinge and equilibrium check. Calculate both positive and negative capacity intersections and discard increments that are not positive in the current loading direction. After inserting each hinge, analyse the released model and add new actions to the existing state.
For project discussion, compare hinge sequence and load-displacement shape as well as final collapse load, then connect discrepancies to explicit modelling assumptions. Plot the accumulated points as the table grows and inspect each change in slope. When capacity depends on axial force, keep the capacity-update cycle in a second table so the convergence tolerance and the reason for stopping are visible.
