The hinge squeezed while it turns

The hinge a squeezed section has to work for put a column’s axial load on two sections and bent them. An I-section that develops 98% of its plastic moment at 1.2 times its first-yield curvature as a beam needed 6.4 times it under six-tenths of its squash load, because the compression had already yielded its web before any bending began. A rectangle’s reserve above first yield grew instead. Every curve there held the axial load fixed while the section bent.

The column that most needs a hinge does not get a fixed load. A frame swaying under an earthquake overturns: the columns on the side it leans towards are pressed harder and the ones it leans away from are relieved, by an amount that grows with the sway. The hinge at the foot of a corner column is bent and squeezed at the same time, and the squeeze grows as the hinge turns. The question that essay left was whether such a hinge follows the curve of its final load — needing the larger curvature from the start — or the curve of its initial load until its moment is taken away from under it.

Following every fibre #

A curve at a fixed axial load can be computed without any history at all. In a steel that is elastic and then perfectly plastic, a section bent at constant axial force never asks a fibre to unload: every fibre’s strain only grows in the direction it started, and the stress is a function of the strain alone. Change the axial force while the section bends and that stops being true. A fibre on the stretched face that has yielded in tension is pushed back by the extra compression. It comes back elastically, carrying a stress that is no longer the stress at its total strain but the stress at its strain less the plastic stretch it has kept.

So the section here is followed step by step. It is cut into 200 fibres through its depth. At each step its curvature and its axial force are both advanced, the axial strain at the centroid is found that makes the fibres’ forces add up to the axial force, and every fibre that reaches the yield stress has its plastic strain updated. The sections are the earlier essay’s — an I-section and a rectangle, each of 3,000 mm² and 200 mm deep, in a steel of 275 N/mm² — taken as perfectly plastic so that nothing but the load path can make the moment fall.

The hinge forms early and then gives way #

The column’s compression rises from a tenth of its squash load to six-tenths as its curvature goes from nothing to twelve times its unloaded first-yield curvature, in proportion. That is the leeward column of a frame that starts from a moderate gravity load and is pushed until the overturning has added half its squash load.

The rising path follows neither curve. At first it follows the low-load curve: the compression has hardly grown when the section first yields, and it turns its corner at about the curvature the earlier essay found for the I-section as a beam. By twice its first-yield curvature it carries 0.86 of its unloaded plastic moment, where the curve at the final load carries 0.43 and is a long way from having formed its hinge. The hinge forms early, at the load it started with.

Then the moment falls. The section is fully plastic, and the moment a fully plastic section can carry is fixed by its axial force: the plastic moment at that load, read off the section’s interaction curve. As the compression rises, the plastic moment falls, and the hinge rides down it. The dashed line is the plastic moment at the compression the path has reached at each curvature, and from about four first-yield curvatures on the rising path is on it to within a per cent. At six times first yield it carries 0.70, at twelve 0.43. The steel has not softened at all.

A rectangle reaches the curve later #

The rectangle tells the same story with a different start.

A rectangle takes several times its first-yield curvature to approach its plastic moment even as a beam — the section that yields from the outside in priced its reserve in curvature — because the material it recruits last is the material nearest its axis, which has the least lever arm. On the rising path the compression grows while it is still climbing, so it reaches its own falling plastic moment later than the I-section does. It is still climbing at 0.88 at twice first yield and is near its plastic moment from about six times first yield. From there it rides down the same way, to 0.63.

The rectangle’s interaction curve is shallower at the start — its plastic moment falls as $1 - n^2$, so the first tenths of compression cost it little — and steeper later. The I-section’s falls almost in a straight line, because its web, which carries the compression first, is the part of it that carries least moment.

The order does not matter to the moment #

If the moment on the falling branch is simply the plastic moment at the current load, it should not matter how the section got to its curvature and its load. It does not.

Squeezed first and then bent, the section follows the constant-load curve at 60% and reaches 0.43. Bent first to six times first yield at the low load, it carries 0.96. Then the compression is raised with the curvature held, and the moment falls straight down to 0.43 without the section turning at all. Bent the rest of the way it stays there. All three paths end at the same moment. Once a steel section is fully plastic, its moment is the plastic moment at whatever load it carries, however it got there.

That is the reason the rising path is easy to describe, and it is also a warning. A section that is plastic has no moment of its own to defend. It has whatever the axial force leaves it. The column bent first and squeezed afterwards — a column whose hinge forms under gravity and wind and is then loaded by a floor above being filled — drops half its moment with no rotation at all, and the frame around it has to take the difference at once.

What the order does change is the column’s shortening. The plastic strains each fibre keeps depend on when it yielded and which way, and the section that was bent before it was squeezed ends a few per cent shorter than the others. The moment forgets the path; the length does not, quite.

How fast a column’s load grows #

The path drawn adds half the squash load over twelve first-yield curvatures, and a frame decides how fast the load really rises. A frame pushed sideways carries its overturning moment as a couple between its columns: the storey shear times the height above the base, divided by the distance between the columns. Once the frame’s hinges have formed, its lateral load stops growing, and the overturning stops growing with it. The extra compression in the leeward column then grows only through what the sway itself adds — the frame’s weight moved sideways by its own drift, which is the $P$–$\Delta$ moment — and through a lateral load that redistributes as hinges form in sequence.

So the rate is set by the frame’s proportions. A slender frame, tall over a narrow base, sends more of its overturning into axial force than a squat one, and its corner columns see their compression grow fastest. A frame with many bays shares its couple among more columns and the outer ones carry most of it, much as a portal frame divides its sway between its columns by their stiffness rather than by statics alone. A pushover analysis follows exactly this: the frame is pushed and every member’s forces are tracked as its stiffness changes, and a column whose hinge’s moment depends on its axial force is the reason a pushover has to update both together at every step rather than check each hinge against a moment fixed at the start. A hinge model read off a constant-load curve, at the gravity load or at a guessed final load, gets one end of the path right and the falling branch wrong.

The curve the hinge rides #

Everything on the falling branch is the section’s plastic interaction curve between axial force and moment, and the interaction curve itself is computed exactly by walking the fully plastic neutral axis through the section. It is usually drawn as a limit — a boundary that a combination of force and moment may not cross — and checked once, at the design force. On the rising path it is something more: the curve the hinge travels along, at a speed the frame sets. The shape of that curve, which a single check never needs, decides how steeply the hinge loses moment. And between two columns, through its curvature, it decides how much of the frame’s strength the overturning takes away.

A falling branch from statics #

A hinge whose moment falls as it rotates has negative stiffness. In a frame, a member with negative stiffness hands load to its neighbours as it deforms, and a frame whose hinges all have it can lose lateral strength as it sways, with no buckling and no fracture anywhere.

The steepness of the fall is the rate the compression grows with the rotation times the slope of the interaction curve. A column whose load rises from 0.1 to 0.35 of its squash load over twelve first-yield curvatures loses 2% of its unloaded plastic moment for every first-yield curvature of further rotation. Rising to 0.6, 4%. Rising to 0.85, 7%, and by the end it carries less than a fifth of what it could as a beam.

Codes treat the softening of a hinge as a property of its steel and its slenderness: local buckling of its flanges, fracture at its welds. They are real, and they come on top of this one. The loss of moment to a rising axial force needs neither, and in a hinge also near local buckling the two add. A column hinge whose load rises with the sway has a falling branch set by the frame’s statics, before any material model is asked anything. It is why the columns of a frame designed to sway are sized to stay elastic, and why the column that does yield in a frame that has run out of other places to hinge is the one whose loss of strength is fastest: the corner column on the leeward side, carrying the most overturning.

Two columns, one squeezed and one relieved #

The overturning that presses one column harder relieves another. The frame’s resistance to sway is the sum of its column hinges, and the relieved column’s plastic moment rises as its compression falls.

For two I-sections the exchange nearly balances. The squeezed column ends at 0.61 of its gravity-load plastic moment, the relieved one at 1.35, and their average at 0.99. The I-section’s interaction curve is almost straight, so what one loses the other almost gains.

For two rectangles it does not balance. The squeezed column loses 0.27 of its gravity-load plastic moment and the relieved one gains only 0.12, and the pair ends at 0.93. A rectangle’s plastic moment goes as $1 - n^2$, a curve bending down, and the average of a curve bending down at two loads is less than its value at their mean. What a frame’s columns lose together to overturning is the curvature of their interaction curve between the loads the overturning sends them to. It is a loss at the scale of the frame that no single column’s check can show, because each column, checked at its own load, carries exactly its plastic moment.

The rectangle stands for any section whose interaction curve bends: a solid steel bar, and very nearly a reinforced concrete column, whose curve bends strongly above its balance point. A concrete frame whose leeward columns are pushed above balance by the overturning loses moment in them faster than its windward columns gain it, and the frame’s base shear falls as it sways.

The I-section, by hand #

The I-section’s plastic moment under compression falls nearly in a straight line, from 1.00 of its unloaded value with no compression to 0.96 at a tenth of its squash load, 0.71 at 0.35, 0.44 at 0.6 and about 0.17 at 0.85. On the rising path to 0.6 the compression grows by $(0.6 - 0.1)/12 = 0.042$ of the squash load for every first-yield curvature, and the plastic moment falls by about 1.03 for each unit of compression between those loads, so the hinge loses about $0.042 \times 1.03 = 0.043$ of its unloaded plastic moment per first-yield curvature. The second half of the path, from six to twelve, loses $0.70 - 0.43 = 0.27$ over six curvatures, 0.045 each, a little more because the interaction curve steepens as the compression grows.

For the pair of rectangles, $1 - n^2$ at 0.6 and 0.1 is 0.64 and 0.99, an average of 0.815, against $1 - 0.35^2 = 0.8775$ at the mean load: a ratio of 0.93. For the I-sections the same arithmetic on their nearly straight curve gives 0.99.

A perfect steel, one section, a load path chosen #

The steel is perfectly plastic. Real steel hardens past yield, so a hinge rotated far enough carries more than its plastic moment, and hardening works against the fall. At the curvatures here, twelve times first yield, a mild steel’s strain hardening has only just begun at the outer fibres, and it raises the moment by a few per cent.

One section stands for the hinge. A hinge is a length of member, not a section, and along that length the axial force is the same while the moment varies. The section at the foot carries the largest moment and is the one followed here. The rotation of the hinge is its curvature integrated over a hinge length that is itself uncertain, which is why the curves here are in curvature.

The load rises in proportion to the curvature. In a frame the column’s extra compression is set by the overturning moment, which grows with the frame’s lateral load and, once the frame is yielding, with its sway. Proportion is the simplest version of that. A frame whose lateral load has peaked while its sway continues adds compression more slowly, and its column hinges fall more slowly.

Nothing buckles. A column at six-tenths of its squash load with a yielded web and flanges strained many times past yield is a candidate for local buckling well before twelve first-yield curvatures, and its effective section shrinks again when it does. The fall here is the floor of what such a hinge loses, not the whole of it.

What a plastic hinge is allowed to be #

Plastic analysis depends on hinges that hold their moment while they rotate, so that a structure can redistribute load from the hinges that form first to the members that are still elastic, and its collapse load can be found by counting mechanisms. The bound theorems guarantee the result only for a structure whose hinges behave that way. A hinge whose moment falls as it turns breaks the assumption, and the theorems no longer say what they said. The collapse load of a frame with such hinges is not the load at which a mechanism forms; it is a peak on a falling path, reached before the mechanism is complete.

The earlier essay asked how much curvature a squeezed section needs before it can be called a hinge. This one finds that the answer depends on the order of events, and that a hinge squeezed while it turns has formed long before its final load says it should, and is losing strength from then on. The displacement an earthquake asks of a frame is a demand on hinges like these. A frame whose columns’ hinges fall can meet that demand only if the beams’ hinges, whose axial load does not change, do the rotating. That is the plainest argument there is for the rule that frames are designed to hinge in their beams, and for putting the hinge where it is wanted rather than where it falls, as a hinge put in on purpose is put into an arch.

Still open: the hinge whose load swings both ways #

The overturning here grows in one direction. An earthquake reverses it every half cycle, and each column is squeezed and relieved in turn while its hinge is bent one way and then the other. A section bent and unbent while its axial load cycles accumulates plastic strain in the fibres that yield in compression each time, and the column shortens a little on every cycle whether or not its moment changes. Whether a column hinge under a cycling overturning load shortens without limit — ratcheting down storey by storey until the floor above it has sagged by more than its own drift — or settles into a fixed cycle once its fibres have found their range, is the question this one leaves for a column that is pushed both ways.