Structural
Plastic Hinges and Limit Analysis
How ductile steel frames yield, redistribute moment, and collapse at a predictable load
A plastic hinge is a fully-yielded cross section that rotates at an essentially constant plastic moment Mp = Fy·Z, where Fy is the yield stress and Z is the plastic section modulus. Because Mp exceeds the first-yield moment My = Fy·S by the shape factor f = Z/S (1.5 for a rectangle, about 1.12–1.18 for a rolled I-beam), an indeterminate structure does not fail when the first section yields. Instead the hinge holds Mp, moment redistributes to still-elastic regions, and more hinges form until the structure becomes a mechanism and collapses. Limit analysis predicts that collapse load directly — a fixed-ended beam under uniform load fails at w = 16·Mp/L² — using the lower-bound (static), upper-bound (kinematic), and uniqueness theorems. Plastic design sizes members to a load factor λ ≈ 1.7 (or modern partial factors like 1.2D + 1.6L) and demands compact, laterally braced sections of ductile steel such as A36 (Fy = 250 MPa) or A992 (Fy = 345 MPa).
- Plastic momentMp = Fy · Z
- Shape factorf = Z/S = Mp/My
- Rectangle f1.50
- I-beam f≈ 1.12–1.18
- Hinges for collapseredundancy + 1
- Fixed beam, UDLw꜀ = 16 Mp / L²
- Load factorλ ≈ 1.7
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Why plastic hinges matter
Elastic design draws a line at first yield: as soon as the most-stressed fibre in the most-stressed section reaches Fy, the member is declared "failed." For a determinate beam that instinct is roughly right. But most real steel structures — continuous beams, propped cantilevers, portal frames, multi-storey moment frames — are statically indeterminate, and for them first yield is nowhere near collapse. A section that has fully yielded does not snap; it forms a plastic hinge that keeps carrying its full plastic moment Mp while rotating, quietly handing surplus load to its neighbours. That single fact is worth 15 to 100 percent extra capacity, and pretending it does not exist wastes steel.
- True collapse capacity. Limit analysis gives the real failure load, not a conservative first-yield proxy — the basis of AISC plastic and inelastic design provisions.
- Lighter, cheaper frames. Continuous beams and portals designed plastically use noticeably less material than the same frame proportioned to keep every section elastic.
- Seismic energy dissipation. Ductile moment frames are deliberately detailed so plastic hinges form in the beams ("strong-column / weak-beam"), absorbing earthquake energy through hysteretic rotation instead of fracturing.
- Robustness and warning. A structure that redistributes moment sags and deflects visibly before it drops — a safety feature brittle systems lack.
- Predictable failure. The collapse load and the collapse shape (which mechanism forms) are both computable, so engineers can steer where damage concentrates.
How a plastic hinge forms — step by step
Follow one cross section of a ductile beam as the bending moment climbs:
- 1 · Elastic. Stress varies linearly across the depth, tension on one face, compression on the other, zero at the neutral axis. Moment and curvature are proportional: M = E·I·κ.
- 2 · First yield. The extreme fibre reaches Fy. This is the yield moment My = Fy·S, with S the elastic section modulus. The interior is still elastic.
- 3 · Elastic–plastic (yield penetration). As moment grows past My, yielding spreads inward from both faces. The elastic core shrinks; the moment–curvature curve bends over.
- 4 · Fully plastic. The elastic core vanishes — the entire section carries ±Fy. The moment reaches the plastic moment Mp = Fy·Z, with Z the plastic section modulus (Z = first moment of the two equal half-areas about the plastic neutral axis). Curvature can now increase with almost no rise in moment: the section has become a plastic hinge.
- 5 · Rotation at constant Mp. The idealised hinge acts like a pin that transmits a fixed moment Mp. In an indeterminate structure this locks the moment at that section and forces additional load onto the elastic remainder — moment redistribution.
- 6 · Mechanism and collapse. Hinges keep forming until there are enough to make the structure a mechanism (one degree of freedom that moves with no load increase). At that instant the collapse load is reached.
The plastic section modulus for a solid rectangle b×d is Z = b·d²/4, versus the elastic modulus S = b·d²/6, so the shape factor is f = Z/S = 1.5 — a rectangle carries 50 percent more moment after first yield before it is fully plastic. Wide-flange I-sections put most of their area in the flanges, already close to the extreme fibre, so they gain far less: f ≈ 1.12 for deep beam shapes and up to about 1.18 for stockier ones. That small reserve is exactly why I-beams are efficient and why their plastic and yield moments are close together.
Limit-analysis theorems and the collapse load
Rather than march load step-by-step and watch hinges appear (an "incremental elastic–plastic" analysis), limit analysis jumps straight to collapse using three theorems. Let W꜀ be the true collapse load:
- Lower-bound (static) theorem. If you can find any moment distribution that is in equilibrium with a load W and satisfies M ≤ Mp everywhere, then W ≤ W꜀. Statically admissible fields give safe (conservative) estimates.
- Upper-bound (kinematic) theorem. For any assumed collapse mechanism, equating the external work of the loads to the internal plastic work (Σ Mp·θ at the hinges) gives a load W ≥ W꜀. Mechanisms give unsafe (over-) estimates, so you take the lowest one.
- Uniqueness theorem. A load that satisfies equilibrium (M ≤ Mp everywhere) and forms a valid mechanism at the same time equals W꜀ exactly. The lower and upper bounds have converged.
Worked example — fixed-ended beam, uniform load. A prismatic beam of span L, clamped at both ends, carries a uniformly distributed load w. It is twice indeterminate, so collapse needs three hinges: one at each support and one at midspan. Using the beam mechanism, rotate each support hinge by θ; midspan then displaces δ = (L/2)·θ and its hinge rotates 2θ. Equate work:
External work = Internal work
(w·L)·(δ/2) = Mp·θ + Mp·θ + Mp·(2θ)
(w·L)·(Lθ/4) = 4·Mp·θ ⟹ w꜀ = 16·Mp / L²
By contrast the elastic peak moment (at the supports) is wL²/12, so the first plastic hinge forms there at w = 12·Mp/L². The ratio 16/12 = 1.33 means moment redistribution alone buys another 33 percent of load beyond the first hinge — on top of whatever the shape factor already gave between My and Mp. The table below collects the standard single-span results (all with plastic hinges shown as the required count).
| Case | Redundancy | Hinges to collapse | Plastic collapse load | First-yield load |
|---|---|---|---|---|
| Simply supported, central point load P | 0 | 1 | P꜀ = 4 Mp / L | 4 My / L |
| Simply supported, UDL w | 0 | 1 | w꜀ = 8 Mp / L² | 8 My / L² |
| Propped cantilever, UDL w | 1 | 2 | w꜀ ≈ 11.66 Mp / L² | ≈ 8 My / L² |
| Fixed–fixed beam, central point load P | 2 | 3 | P꜀ = 8 Mp / L | 8 My / L (no redistribution reserve — all 3 sections yield together) |
| Fixed–fixed beam, UDL w | 2 | 3 | w꜀ = 16 Mp / L² | 12 My / L² |
Portal frames, mechanisms, and load factor
Real frames offer several ways to collapse, and the governing one is whichever gives the lowest load. For a rectangular portal frame under a vertical load on the beam and a horizontal load at the eaves, three canonical mechanisms compete:
- Beam mechanism. Hinges in the beam (ends and under the load) — governs when vertical load dominates.
- Sway mechanism. Hinges at the column tops and bases let the frame lean sideways — governs when horizontal load dominates.
- Combined mechanism. Superposing beam and sway, one hinge cancels, and the resulting mechanism often gives the true (lowest) collapse load.
The load factor λ ties analysis to design: λ = W꜀ / W_service. You size members so the frame reaches its mechanism at λ times the expected working load. Classic plastic design used a single λ ≈ 1.7 for gravity combinations and about 1.4 with wind. Modern limit-state codes split that lumped number into partial factors on each action — for example the ASCE 7 / AISC LRFD combination 1.2·D + 1.6·L — checked against a full-plastic or plastic-collapse limit state, with resistance factors (φ) trimming the strength side.
Common misconceptions and failure modes
- "A plastic hinge is a real pin." No — it still transmits the moment Mp. It removes rotational stiffness, not moment. Drop it to zero moment in your analysis and you get the wrong mechanism.
- "First yield equals failure." Only for determinate members. For indeterminate structures first yield is the start of redistribution, not the end of usefulness — often 30 to 60 percent of reserve remains.
- "Any assumed mechanism gives the answer." The kinematic theorem is an upper bound. A plausible-looking wrong mechanism overestimates capacity — unsafe. Always check several and take the lowest, then verify M ≤ Mp everywhere.
- "Plastic design works for any steel section." It needs ductility and stability. A slender (non-compact) flange or web will locally buckle before reaching Mp, and an unbraced beam will fail by lateral-torsional buckling — either one caps the section below Mp and voids the analysis. Codes limit plastic design to compact sections with lateral bracing near hinges.
- "Axial load doesn't matter." In columns, axial force reduces the available moment: the section reaches a reduced plastic moment Mpc < Mp along an M–P interaction curve. Ignore it and column hinges form early.
- "The hinge has infinite rotation capacity." It is large but finite. If required rotation exceeds what the detail can supply, the section fractures or buckles before the last hinge forms — which is why seismic detailing (continuity plates, compact webs, weld quality) is so tightly controlled.
Representative values
| Quantity | Symbol | Typical value / range | Note |
|---|---|---|---|
| Yield stress, A36 steel | Fy | 250 MPa (36 ksi) | Common structural carbon steel |
| Yield stress, A992 W-shapes | Fy | 345 MPa (50 ksi) | Standard US wide-flange grade |
| Yield strain | εy = Fy/E | ≈ 0.12–0.17 % | E ≈ 200 GPa |
| Strain at fracture (elongation) | εu | ≈ 20–25 % | Huge ductility reserve for hinge rotation |
| Shape factor, rectangle | f | 1.50 | Z = bd²/4, S = bd²/6 |
| Shape factor, solid circle | f | ≈ 1.70 | Z = d³/6, S = πd³/32 |
| Shape factor, wide-flange I | f | ≈ 1.12–1.18 | Most area near extreme fibre |
| Classic gravity load factor | λ | ≈ 1.7 | ≈ 1.4 with wind |
Frequently asked questions
What is a plastic hinge?
A plastic hinge is a short region of a ductile member where the whole cross section has yielded — both extreme fibres and everything between them have reached the yield stress. Once fully plastic, the section can rotate almost freely while carrying an essentially constant plastic moment Mp = Fy·Z, where Fy is the yield stress and Z is the plastic section modulus. It behaves like a rusty pin: it resists a fixed moment Mp but offers no extra stiffness, so it lets the two parts of the member kink relative to each other.
What is the difference between the plastic moment Mp and the yield moment My?
My = Fy·S is the moment that first brings the extreme fibre to yield, where S is the elastic section modulus. Mp = Fy·Z is the moment when the entire section has yielded, where Z is the plastic section modulus. Their ratio is the shape factor f = Z/S = Mp/My. For a solid rectangle f = 1.5, for a solid circle f ≈ 1.70, and for a typical rolled wide-flange I-beam f ≈ 1.12–1.18 because most of the material is already near the extreme fibre. So an I-beam has only about 12–18 percent of reserve between first yield and full plasticity, whereas a rectangle has 50 percent.
How many plastic hinges are needed for collapse?
A structure collapses when enough plastic hinges turn it into a mechanism — a system that can move without adding load. The number required equals the degree of static indeterminacy plus one. A simply supported beam is determinate (redundancy 0), so one hinge collapses it. A propped cantilever has redundancy 1 and needs two hinges. A fixed-ended (clamped) beam has redundancy 2 and needs three hinges. A single-storey portal frame with fixed bases is three times redundant and typically needs three to four hinges depending on the failure mode (beam, sway, or combined).
What are the upper and lower bound theorems of limit analysis?
The lower-bound (static) theorem says any load in equilibrium with a moment field that nowhere exceeds Mp is less than or equal to the true collapse load — a safe, conservative estimate. The upper-bound (kinematic) theorem says the load computed from any assumed collapse mechanism, by equating external and internal work, is greater than or equal to the true collapse load — an unsafe overestimate. The uniqueness theorem states that when a load satisfies both equilibrium (with M ≤ Mp everywhere) and a valid mechanism simultaneously, it is exactly the collapse load. In practice engineers test several mechanisms and take the lowest upper-bound load, then verify no section exceeds Mp.
What is moment redistribution in plastic design?
In an indeterminate structure, elastic analysis concentrates the peak moment at one section (often over a support). When that section reaches Mp it becomes a plastic hinge and stops taking extra moment — but it keeps carrying Mp and keeps rotating. Any further load is shed to the still-elastic parts of the structure, which pick up more moment until they too reach Mp. This shifting of load, called moment redistribution, lets the structure carry substantially more than the load that first caused yielding. It is why the plastic collapse load of a fixed beam under uniform load is 16·Mp/L² while the first plastic hinge forms at only 12·Mp/L².
Why does plastic design require ductile, compact steel sections?
The whole method assumes each hinge can rotate through a large plastic angle while holding Mp so that later hinges have time to form. That demands ductility (structural steel yields at about 0.1–0.2 percent strain but fails near 20–25 percent, giving a rotation reserve of order 100 times) and it demands that the section not fail first by local buckling of a slender flange or web. Codes such as AISC 360 therefore restrict plastic design to compact sections (flange and web width-to-thickness ratios below λp), require lateral bracing near hinges to prevent lateral-torsional buckling, and specify steels with a guaranteed yield-to-tensile ratio and minimum elongation.
What is the load factor in plastic design?
The load factor λ is the ratio of the collapse load to the working (service) load: λ = W_collapse / W_service. Instead of keeping stresses below an allowable value, plastic design sizes members so the collapse load is λ times the expected service load, giving a direct margin against total failure. Classic plastic design used a single factor of about 1.7 for gravity load combinations and roughly 1.4 when wind was included. Modern limit-state codes replace one lumped factor with separate partial factors on each load (for example 1.2·D + 1.6·L in ASCE 7 / AISC LRFD) applied to a plastic-collapse or full-plastic-strength limit state.