A ductile beam does not fail the instant its first section yields. Plastic hinges form one by one, and the structure collapses only when it becomes a mechanism. Change the support condition, load and plastic moment to see the true mechanism-method collapse load and the safety margin in real time.
Parameters
Beam support condition
Sets the required hinge count and the collapse mechanism
Load
Choose a central point load or a uniformly distributed load
Beam span L
m
Plastic moment M_p
kN·m
Resisting moment when the whole section has yielded
Applied load
Point load in kN or distributed load in kN/m; compared with the collapse load
Results
—
Collapse load
—
Required plastic hinges
—
Collapse mechanism
—
Load factor λ
—
Reserve over elastic limit
—
Safety verdict
—
Collapse mechanism animation
The beam breaks into rigid segments at the plastic hinges (○ markers) for the chosen support condition and collapses as a mechanism, looping continuously.
Collapse load vs plastic moment M_p
Load-deflection curve (elastic → hinge formation → collapse plateau)
Collapse load P_c for a central point load. The collapse load is found by equating the external virtual work of the mechanism with the internal virtual work absorbed at each plastic hinge.
Load factor λ and the basic mechanism-method identity. The external virtual work W_ext equals the internal virtual work W_int (θ_i: rotation at each hinge).
What is the Plastic Hinge Collapse Load Simulator?
🙋
For a steel beam, doesn't failure happen as soon as some point in the section reaches the yield stress?
🎓
That's exactly the interesting part. A ductile steel does not fail when its outer fibres yield. As you increase the load, the yielded zone spreads steadily inward through the section, until the whole section has yielded. The moment the section then transmits is the "full plastic moment M_p", and at that point the section becomes a "plastic hinge" — it keeps carrying M_p while rotating freely. The key idea is that failure does not happen at first yield.
🙋
If it can rotate freely, it isn't holding anything anymore, right? So isn't that collapse?
🎓
For a simply supported beam, yes — the instant one hinge forms, the beam becomes a folding mechanism and collapses. But think about a fixed-fixed beam. Both ends are clamped solidly, so even after one hinge forms at midspan, the two ends still hold the load. Collapse needs three hinges in all: both ends plus midspan. Switch the support condition on the left and watch the "required hinges" card.
🙋
I see — more hinges, harder to collapse. So how do you actually compute the collapse load itself?
🎓
Tracing the elasto-plastic loading history from start to finish is hard work, but the "mechanism method" of limit analysis gets it in one step. You assume the shape of the mechanism at the moment of collapse, then equate the virtual work done by the external loads with the virtual work absorbed by M_p at each plastic hinge. Solving that gives the collapse load directly. For a propped cantilever with a central point load it is Pc=6M_p/L. The collapse-mechanism animation below shows exactly that assumed mechanism in motion.
🙋
The load-deflection curve has a few sharp kinks along the way. What are those?
🎓
Each "kink" is the moment a plastic hinge is born. It starts as a straight elastic line. When one hinge forms, the structure becomes a little softer, so the slope flattens. With the second and third hinges it flattens further, and when the last hinge forms the slope drops to zero — a flat plateau. The load stops increasing while the deflection keeps growing: that is collapse. The more indeterminate the structure, the more kinks, and the more margin from first yield to collapse. That margin is exactly redundancy.
🙋
There's a card called "reserve over elastic limit" — is that the redundancy you mentioned?
🎓
Exactly. A simple beam collapses with just one hinge, so first yield (the elastic limit) and collapse almost coincide — the reserve is 1.0. A propped cantilever has the margin of two hinges and a fixed-fixed beam of three, giving reserves of about 1.33 and 1.5. If your design thinks "yielding means failure", you throw away all of that redundancy strength. That is why plastic design properly evaluates the collapse load.
Frequently Asked Questions
A plastic hinge is the state of a ductile steel cross-section that has reached its full plastic moment M_p: the whole section has yielded and the point can rotate freely while still transmitting M_p. Unlike a physical pin, the moment does not vanish — the hinge keeps carrying M_p as it rotates. As the load increases, plastic hinges form one by one starting at the most-stressed sections, and once enough have formed the structure turns into a mechanism and collapses.
In the mechanism method (virtual work), you assume a collapse mechanism made of plastic hinges and equate the external virtual work of the loads with the internal virtual work absorbed by M_p at each hinge to solve directly for the collapse load. For example, a central point load gives Pc=4M_p/L for a simple beam, Pc=6M_p/L for a propped cantilever and Pc=8M_p/L for a fixed-fixed beam. The strength of limit analysis is obtaining the collapse load without tracing the whole elasto-plastic loading history.
A statically determinate structure (a simple beam) becomes a mechanism and collapses the instant one plastic hinge forms. An indeterminate structure keeps carrying load with its remaining restraints after the first hinge, and collapses only once two or three hinges have formed. A propped cantilever needs 2 hinges and a fixed-fixed beam needs 3. This margin between first yield and collapse is the redundancy — the source of the safety reserve of indeterminate structures.
The load factor lambda is the collapse load divided by the applied load, showing how much margin the structure has against collapse. lambda >= 1.5 means an ample safety margin against collapse, 1.0 <= lambda < 1.5 means a small margin that needs attention, and lambda < 1.0 means the applied load already exceeds the collapse load and the structure collapses. In design you adjust M_p or span to meet a target lambda (roughly 1.5 to 2.0 for steel building frames).
Real-World Applications
Plastic design of steel buildings: The beams and columns of steel moment frames are checked not only by allowable-stress design but also by ultimate lateral strength based on plastic hinges. The classic "strong-column weak-beam" approach deliberately steers plastic hinges to the beam ends, absorbing energy during a large earthquake while controlling the collapse mechanism. The beam-mechanism collapse load this tool covers is the most basic building block of that procedure.
Reserve evaluation for seismic and impact loading: For bridges and industrial frames, the question is how far the structure holds up under earthquakes or impacts beyond the design load. Elastic analysis only answers up to first yield, but the collapse load from limit analysis gives the "true limit" including the redundancy reserve. The load factor lambda is used as an at-a-glance measure of that reserve.
Design of continuous and multi-span beams: In continuous beams, several plastic hinges appear over the supports and at midspans. The crux of design is searching with the mechanism method for which combination of hinges gives the lowest collapse load (the governing mechanism). Through three typical single-span cases, this tool lets you experience the core of that reasoning.
Pre-study and verification for non-linear FEM: Before running a detailed pushover or elasto-plastic FEM analysis, the mechanism method gives a first read on the order of magnitude of the collapse load. Conversely, if an FEM result far exceeds the mechanism-method upper bound, it serves as a sanity check pointing to errors in boundary conditions or material models. Limit analysis is also useful as a cross-check for numerical results.
Common Misconceptions and Pitfalls
The biggest misconception is assuming that "a section has yielded = the structure has failed". For a ductile steel, first yield is not the start of collapse but rather the entrance to the reserve. The yielded section becomes a plastic hinge and redistributes load to regions that have not yet yielded; only once enough hinges have formed does collapse occur. If you stop design at the "first yield" of elastic analysis, you ignore the entire redundancy strength of an indeterminate structure. The "reserve over elastic limit" in this tool is the measure of that difference.
Next, the misconception that "the mechanism method always gives a safe-side answer". The mechanism method (upper-bound theorem) returns the correct answer if the assumed mechanism is right, but a value larger than the true collapse load — an unsafe value — if it is wrong. You must investigate all possible collapse mechanisms and adopt the smallest collapse load among them. This tool deals only with typical single-span mechanisms, so it returns the correct answer; but real continuous beams and frames require an exhaustive check of beam, joint and combined mechanisms.
Finally, "meeting the collapse load is not enough to make the design complete". Limit analysis treats only the ultimate strength at collapse; the service-load deflection, buckling and low-cycle fatigue from repeated loading are separate problems. In particular, in regions where plastic hinges form, local or lateral-torsional buckling can drop the strength before M_p is reached, so checking section compactness (width-to-thickness ratio) is a prerequisite. The collapse load is only one face of a structure's limit capacity and must be evaluated together with serviceability, stability and ductility capacity.
How to Use
Enter beam span length (L) in meters using LspanNum slider (range 1–10m)
Input plastic moment capacity (Mp) in kN⋅m via MpNum field (typical range 50–500 kN⋅m for steel I-beams)
Set applied point load (P) in kN using appliedLoadNum (range 5–200 kN)
Click Simulate to generate collapse mechanism, plastic hinge locations, and load factor λ
Asymmetric loading shifts hinge formation; simulator flags unsymmetrical reserve and adjusted collapse location
For Mp values below 80 kN⋅m on spans >8m, check web buckling and lateral-torsional buckling separately—plastic hinge theory assumes local section ductility only
Design code limit states (Eurocode 2/3): accept λ≥1.5 for bending-critical members; λ<1.3 flags brittle-prone behavior