Portal Frame Collapse: 2D

What plastic collapse means

A steel section behaves elastically until the extreme fibres yield, then the yielding spreads through the depth until the whole section is plastic and carries its plastic moment Mp = fy Wpl. At that point the section acts like a hinge: it keeps carrying Mp but rotates freely. A frame collapses when enough hinges have formed to turn the structure into a mechanism. The load factor λ multiplies the service loads; the collapse factor is the value at which the mechanism forms.

Hinge-by-hinge (event-to-event) analysis

The frame is 6 beam-column elements with axial and bending stiffness (EA, EI from the chosen UB section). For the current set of hinges the program solves K Δu = Pref (a hinge is a released rotation, so its moment cannot change). The element end moments grow by ΔM per unit of load; the next hinge forms at the smallest Δλ = (±Mp − M)/ΔM over all unhinged sections. Between events the response is linear, so the load-sway graph is a polyline whose kinks are the hinge events. When the stiffness matrix becomes singular the structure is a mechanism, its null vector is the collapse motion.

Mechanisms and virtual work

The upper-bound theorem says the true collapse factor is the smallest λ from any kinematically admissible mechanism: external work equals the plastic work in the hinges. With Mpj = min(Mpc, Mpb) at the eaves joints (the weaker member hinges first), a point load V at midspan and a horizontal load H at the eaves, for fixed bases: beam λ = 2(Mpj + Mpb)/(V L/2), sway λ = 2(Mpc + Mpj)/(H h), combined λ = 2(Mpc + Mpb + Mpj)/(H h + V L/2). For pinned bases the base hinges drop out. For a uniform load the lever is V L/4. The smallest one must equal the analysis result (numerical checks 1-3).

Reading the graphs

The load-sway graph kinks at each hinge and flattens at collapse; the moment graph shows each section moving toward ±1 (a hinge); the bar chart compares the three mechanisms with the analysis. Colour on the members is the utilisation M/M_p along each member; glowing red zones are hinges.

Honest simplifications

Rigid-plastic hinges with no strain hardening and no axial-force reduction of M_p, first-order geometry (no P-delta or buckling, which would lower the collapse load for slender frames), compact sections with full lateral restraint, no shear or joint flexibility, roof point loads or an equivalent nodal UDL, no self-weight, and approximate catalogue properties for the UB sections. A teaching model, not a design check.