The hinge a squeezed section has to work for

A section yields from the outside in. Bend a steel rectangle and its extreme fibres reach yield first, at a moment $Zf_y$; bend it further and the yielded zones grow inward until the whole section is at yield, carrying the plastic moment $Sf_y$, half again as much. An I-section, with most of its material already in its flanges, has little left to recruit and carries only a seventh or so more. That essay’s point was that the shape factor is a price rather than a bonus: the rectangle reaches its reserve only by bending four times further than its first yield, while the I-section turns its corner almost at once.

Every curve in it was a beam’s: bending alone. A column carries an axial force as well, and every section in a frame’s columns is bent while it is squeezed. The axial load changes both numbers that essay compared — the reserve and the curvature it costs — and it changes them in opposite directions for the two shapes.

A rectangle and an I, squeezed #

The two sections are the ones the earlier essay used: each 3,000 mm² of mild steel, 200 mm deep, one a solid rectangle and one an I-section with most of its area in its flanges. Each is loaded first by an axial compression $n$ times its squash load $Af_y$, and then bent while the compression stays.

Unloaded, the I-section’s curve is nearly a corner: straight up, first yield at 92 per cent of its plastic moment, and the full plastic moment almost immediately after. Under 30 per cent of its squash load the corner has rounded. The section first yields at 64 per cent of its unloaded plastic moment, but the most it can now carry is 75 per cent, and it needs more than twice the curvature to get there. Under 60 per cent of its squash load the corner is gone: it first yields at 37 per cent, carries at most 44, and needs 6.4 times the unloaded first-yield curvature to develop it. The I-section under axial load has lost the sharp corner that was its advantage.

The rectangle was always the section with a long rounded curve. Under axial load it keeps that shape, and two things happen to it. Its first yield comes down faster than its plastic moment, so the gap between them widens. And the curvature it needs to develop its plastic moment grows only a little, from 4.2 to 5.1 times the unloaded first-yield curvature at 60 per cent. Squeezed, the rectangle behaves much as it did; squeezed, the I-section behaves like a rectangle.

The reserve, as a function of the load #

The ratio of the plastic moment to the first-yield moment, which the earlier essay treated as a property of the shape, now depends on the axial load too.

For the rectangle the result is exact and pleasingly simple. First yield happens when the axial stress $nf_y$ plus the bending stress at the extreme fibre reaches $f_y$, so the moment that causes it is $Zf_y(1 - n)$, falling in a straight line to nothing at the squash load. The plastic moment, with the axial force carried by a central strip of the section and the bending by the rest, is $Sf_y(1 - n^2)$. Their ratio is

$$\frac{S(1-n^2)}{Z(1-n)} = 1.5\,(1 + n).$$

A rectangle’s reserve past first yield grows with the axial load, to nearly three times its first-yield moment under heavy compression. That is the reverse of what a check against first yield suggests. An elastic check of a heavily compressed solid section finds almost no moment capacity left — at 80 per cent of the squash load, a fifth of its unloaded first-yield moment — when the section can in fact carry more than half its unloaded first-yield moment once it has yielded through; the elastic check is not wrong, but it is describing the start of a curve whose end is nearly three times higher. The I-section’s grows only from 1.09 to about 1.2, because almost all its material is in flanges that sit at the extreme fibres: whatever the axial load does to first yield, it does to most of the section at once.

The reason the two moments fall at different rates is where the axial stress is charged. At first yield it is added to the extreme fibre, the fibre doing most of the bending, so every bit of it comes straight off the moment. In the fully plastic state the axial force is carried by a strip about the middle of the section — the material with the shortest lever arm, which was doing least of the bending anyway — so it costs the moment much less. The axis that moves when the section yields is the same strip seen from the other side: the plastic neutral axis moves off the centroid by however much area the axial force needs.

The plastic moments themselves tell the more practical story. At half its squash load the rectangle keeps three-quarters of its unloaded plastic moment and the I-section a little over half. The I-section’s web, which carries little of its bending, is the first material the axial force uses up, and once it is gone the force has to be taken from the flanges, which carry nearly all the bending. That is why the I-section’s plastic moment falls almost in a straight line, the shape familiar from the interaction curves design rules give for I-sections — and why a column section loses moment capacity faster than the solid sections the textbook derivations use.

The curvature the hinge needs #

The second number the earlier essay compared was the curvature needed to reach the plastic moment. That is what decides how much a plastic hinge has to rotate before it is fully formed — and in a frame designed to form hinges, whether it can.

Unloaded, the I-section develops its hinge at 1.2 times its first-yield curvature, a quarter of what the rectangle needs. At 60 per cent of its squash load it needs 6.4 times, more than the rectangle at the same load. Squeezed hard enough, the I-section needs more curvature to form its hinge than the solid rectangle does. The ranking the earlier essay established — the I-section turns its corner at once, the rectangle has to work for it — is reversed by the axial load.

In rotations the difference is easy to picture. The unloaded first-yield curvature of a 200 mm deep mild-steel section is $2\varepsilon_y/h$, 0.0000131 per millimetre. A plastic region roughly as long as the section is deep turns through that curvature times its length, so the beam-like I-section forms its hinge after about 0.003 radians of local rotation, and the same section under three-fifths of its squash load after about 0.017 — a fifth of a degree against a degree. In a frame the hinge rotation demanded by a storey drift is set by the frame, not the section, so a column hinge that needs five times the rotation to form is a column hinge whose moment arrives later in the sway, after the frame has already relied on it.

The mechanism is visible in what the axial load does before any moment is applied. At 60 per cent of the squash load, the uniform compression has already taken the web to yield, because the web is a thin plate whose share of the section is small. The flanges are at 60 per cent of yield. Bending then has to make one flange go from compression to tension past yield and the other further into compression, with no help from the web, and the curvature that takes is large. The rectangle had no web to lose.

The compressed part takes over #

Where the neutral axis sits in the fully plastic state makes the same point as a picture.

Unloaded, a fully plastic section is half in compression and half in tension, and the moment is the two halves’ forces times the distance between their centroids. Under axial load the compressed part grows, because it has to supply the axial force as well, and what is left in tension shrinks to a strip at one face. For the rectangle the compressed depth is exactly $(1 + n)/2$ of the whole. For the I-section it jumps: at 40 per cent of the squash load the compressed zone already covers the web and most of the tension flange, and the moment is carried by a tension zone a few millimetres deep.

That thin tension zone is why the hinge needs so much curvature. The strain at the tension face has to reach yield across a tiny lever arm from the neutral axis, while the compression face is driven far past yield — and the compressive strain at the face, not the curvature, is what decides whether the flange buckles locally before the hinge has formed. A plate that ripples does so at a strain, and a column flange squeezed to 60 per cent before it bends reaches that strain sooner than any beam flange.

The web that goes first #

The I-section’s web is 6 mm thick and 180 mm deep, 1,080 mm² of its 3,000, a little over a third of the section. That fraction decides the shape of everything above.

While the axial load is less than the web could carry on its own — up to about 36 per cent of the squash load — the fully plastic section can put the whole axial force into a band of web around the middle and keep both flanges for bending. The plastic moment then falls slowly, because the web material it gives up was doing very little bending. Past that load the web is used up and the axial force has to start taking flange; every increment of axial force now comes out of the material that was carrying the moment, and the plastic moment falls almost in proportion. The kink between the two regimes is at the web’s share of the area, and the straight line past it is the shape the interaction rules for I-sections are written as.

The curvature does the opposite. Below the web’s share, bending a fully plastic I-section mostly asks the flanges to swap, one into tension and one further into compression, which they do over a small curvature because they are far from the axis. Past it, the neutral axis has had to move into the tension flange, and the strip of steel left in tension is thin and close to the face, so the curvature needed to strain it past yield is large. The I-section’s whole advantage — material far from the axis — is an advantage only while the axis is in the middle, and the axial load is what moves it.

The member around the section #

Everything here is a section, bent at one place. In a column the section is part of a member that also bends between its ends, and a column that was never straight amplifies its own bending as its axial load approaches its buckling load. The two effects compound. A column carrying a large fraction of its squash load has a section whose plastic moment has shrunk, and a member that amplifies whatever moment it is given, so the moment it can be given from the frame shrinks twice over.

That is the beam-column problem, and whether it fails by yielding or by buckling depends on its slenderness. A stocky column fails at its section, by the interaction above. A slender one fails as a member, with its section well short of the plastic state. In between, where most real columns are, the section’s reduced plastic moment and the member’s amplification together set the capacity, and the design curves for beam-columns are an interpolation between the two that no single section curve contains.

What the section curve adds to that familiar picture is the curvature. A stocky column that does reach its section’s plastic moment needs, under heavy axial load, several times the rotation to do so that the same section needs as a beam — and a strong section can still fall over while it is rotating, if that rotation is more than its flanges or the member can deliver.

Why columns are kept elastic #

Frames designed to survive earthquakes by forming plastic hinges are designed so that the hinges form in the beams and not in the columns: strong columns, weak beams. The usual reasons given are about the mechanism — a storey of column hinges is a sway mechanism with no reserve — and about the column’s axial load, which it must keep carrying however much it has rotated. The curves here add a third, which is about the section rather than the frame.

A column hinge under a high axial load is a hinge that needs much more curvature to form, in a section whose compressed flange is closer to local buckling, carrying a plastic moment much smaller than the same section’s as a beam. Everything about it is worse. That is why seismic design rules typically limit the axial load on any column expected to yield to a modest fraction of its squash load, and why, where a column must form a hinge at its base, it is the one place in the frame where the axial load is kept low and the section is chosen with a thick, stocky flange. The ductility a section offers depends on what it is measured against, and for a column the ruler is the axial load.

The rectangle, by hand #

For a rectangle $b \times h$ with $Z = bh^2/6$ and $S = bh^2/4$, first yield under an axial compression $nf_ybh$ is reached when

$$n f_y + \frac{M}{Z} = f_y \quad\Rightarrow\quad M_y = Zf_y(1 - n).$$

Fully plastic, a central strip of depth $nh$ carries the axial force at $f_y$, and the two outer parts of depth $(1-n)h/2$ each carry $f_y b(1-n)h/2$ with centroids $(1+n)h/4$ from the middle, so

$$M_p = 2 \times f_y b \frac{(1-n)h}{2} \times \frac{(1+n)h}{4} = \frac{f_y b h^2}{4}(1 - n^2).$$

At $n$ = 0.4 the first-yield moment is 0.6 of its unloaded value and the plastic moment 0.84, a ratio of 1.4 times the unloaded 1.5: 2.10. The compressed depth is the outer part above the central strip plus the strip, $(1-n)h/2 + nh = (1+n)h/2$, 0.70 of the depth at $n$ = 0.4.

An elastic-plastic steel, bent about one axis #

The calculation rests on choices that limit it.

The steel is elastic and then perfectly plastic, with a yield plateau before it hardens. Real mild steel has a plateau of a strain of ten or more times yield, then hardens, which is why the curves rise again at large curvature; a high-strength steel with no plateau rounds every corner and shrinks the difference between the sections.

There are no residual stresses. A rolled or welded I-section carries stresses locked in before any load, compression at its flange tips, which yield those tips earlier and round the unloaded curve as the axial load rounds the loaded one. Under axial load the two effects add.

The section does not buckle locally. Every curve is taken to curvatures that a thin flange would not survive; the curvatures are an upper bound on what the section can deliver, and the axial load lowers the true limit further by pushing the compressed flange’s strain up.

Bending is about one axis and the axial force is constant. In a frame the axial force in a column changes as the frame sways, rising in the columns on one side and falling on the other, so the hinge in a corner column forms under a load that is itself moving.

Still open: the hinge that forms while the load changes #

Every curve here holds the axial force fixed while the section bends. In a frame swaying under an earthquake the outer columns gain axial compression as the frame overturns, at the same time as their bases are asked to rotate, so the hinge at the foot of a corner column is bent while it is being squeezed harder. Whether a section whose axial load rises during the rotation follows the curve of its final load — needing the larger curvature from the start — or follows its initial curve and then has its moment taken away from under it as the compression grows, so that the hinge loses moment while it rotates, is a question about the order in which a column is squeezed and bent, and the answer decides whether the hinge’s moment can be counted on at all.