The panel a soft joint empties into the span

An interior joint of a frame carries almost nothing in its column-web panel under full gravity, because the two beams’ hogging moments either side of it are equal and their flange forces cancel across the column. Load one bay and not the other, or push the frame sideways, and they no longer cancel: the panel carries the difference, and under a load on one bay with a push from the wind it can carry more than the corner joint whose panel is the usual worry. Every joint in that calculation was rigid. The beam ends and the column turned together.

Real beam-to-column joints are not rigid. An end plate bolted to a column flange bends, the bolts stretch, the column web panel itself shears, and the beam end turns relative to the column by an amount that grows with the moment — a rotational stiffness, in kilonewton-metres per radian, that the joint that was chosen set out to classify. A joint flexible enough rotates under the unbalanced moment of a pattern load, the loaded beam’s end one way and the unloaded beam’s the other. The question the rigid frame left was whether that rotation sheds the pattern’s unbalanced moment and relieves the panel, or whether it lets the sway concentrate in fewer joints and loads the remaining panels harder.

The same frame with springs at the beam ends #

The frame is the one before: two bays of 8 m and two storeys of 4 m, fixed at the bases, with 457 mm deep beams and 254 mm columns whose webs are 8.6 mm thick. Each beam now meets each column through a rotational spring, a joint whose stiffness is stated as a multiple of the beam’s own EI/L. That is the measure EN 1993-1-8 classifies by: a joint in a frame that sways is rigid at 25 EI/L or more and nominally pinned at 0.5 EI/L or less, and anything between is semi-rigid and must be in the frame’s analysis. For these beams EI/L is 7.6 MN·m per radian, so EN 1993-1-8’s rigid limit is 190 MN·m/rad and its pinned limit 3.8.

The spring is modelled as a member 20 mm long whose bending stiffness gives exactly the joint’s rotational stiffness and whose own deflection is negligible; set very stiff, it reproduces the rigid frame’s panel shear to within 0.4 per cent, the difference being the 20 mm it moves the beam’s end.

What leaves the panel lands in the span #

Load the left bay with 20 kN/m and leave the right one empty, the arrangement that loads the interior panel most under gravity alone.

The panel is relieved, and in proportion to how soft the joint is. At EN 1993-1-8’s rigid limit the relief is 5 per cent, which is the error the classification accepts by calling such a joint rigid. At three times EI/L, a typical stiffness for a flush end plate on a beam of this size, the panel carries 104 kN rather than 154 — a third less. At EI/L it carries 63.

The relief is not free, and its price is on the same figure. The loaded beam’s largest sagging moment rises as the panel’s shear falls: from 71 kN·m with rigid joints to 102 at 3 EI/L and 126 at EI/L. The two curves are one quantity seen twice. The interior panel’s shear is the loaded beam’s hogging moment at the interior column, less what the unloaded beam returns, divided by the beam’s depth; a softer joint lowers that hogging moment, and the beam’s span moment rises by exactly what its supports have lost.

The beam’s diagrams show why. A uniformly loaded span carries a free moment of wL²/8 — 160 kN·m here — whatever its supports do, and the supports only decide where the line hangs from. Rigid joints hold the ends at 77 and 101 kN·m hogging and the middle sags to 71; joints at 3 EI/L let the ends relax to 53 and 59 and the middle sags to 102; joints near pinned hold almost nothing and the beam is simply supported, sagging 154. The moment that goes round the corner into the column has to come from somewhere, and in a beam it comes from the span.

So whether a semi-rigid joint is a saving depends on what was governing. A beam sized for its support moment, with a span moment to spare, gains from a softer joint: its support moment falls and so does the panel’s shear. A beam sized for its span moment loses. This is the trade the joint that was chosen found for one beam — a stiffness at which support and span moments are equal and the beam is lightest — appearing here at the interior panel too, which gains from every reduction in the support moment and never pays the span’s bill.

What the classification boundary means #

EN 1993-1-8’s rigid limit is not where the joint stops mattering; it is where the error of ignoring it is judged acceptable. At 25 EI/L the panel is 5 per cent lighter and the span 7 per cent heavier than the rigid analysis says, and those are the errors a designer accepts by modelling such a joint as rigid. For a frame that does not sway, braced against the wind, the limit is 8 EI/L, because there the joint’s flexibility matters only to the beam’s own moments and not to the frame’s drift; at 8 EI/L in this frame the panel would be 15 per cent lighter than rigid and the span 21 per cent heavier.

Between the two limits the joint is semi-rigid and EN 1993-1-8 asks for it to be in the analysis — which means a stiffness has to be estimated before the joint has been designed, since the joint’s detail is what fixes it. That is the circularity the analysis that assumes the answer described for member stiffnesses, appearing here at the connection: the frame’s moments depend on the joint’s stiffness, and the joint is detailed for the frame’s moments. In practice a first stiffness is assumed from the joint type — an end plate of a given thickness, a given bolt pattern — and the analysis is repeated once the joint has been detailed. The figures above say how much that first guess matters for the interior panel. Between 3 and 25 EI/L the panel shear changes by 40 per cent; a guess wrong by a factor of two either way from 3 EI/L moves it by a fifth to a quarter — to 79 kN at half the stiffness and 124 at double.

A joint made of springs in series is how that stiffness is estimated: the end plate in bending, the bolts in tension, the column flange in bending and the column web panel in shear, each a spring, their flexibilities added. Of those, the panel is the one this essay’s numbers act on. A panel that is itself flexible makes the joint semi-rigid, and a semi-rigid joint then carries less panel shear — a feedback the linear springs here leave out, and one that runs in the direction of relief.

The rotation a soft joint has to supply #

A joint that takes less moment turns further, and the turning is not free either.

The moment a joint carries falls as it softens, but its rotation, the moment over the stiffness, rises faster. At 3 EI/L the interior joint carries 62 kN·m and turns 2.6 milliradians relative to the column; at EI/L, 33 kN·m and 4.6. Those are not large rotations for an end plate in the elastic range, but they are not zero, and a joint relied on to supply them must still be within the stiffness that was assumed when it does. For a partial-strength joint — one whose moment resistance is below the beam’s — the rotation is also what decides whether it can reach its resistance before something brittle happens, a redistribution nobody chose that has to be ductile to be safe.

Under sway, the columns take what the joints give up #

Push the frame sideways instead, 20 kN at each floor, and the joints’ stiffness decides something different.

A sway frame resists a push by bending its columns in double curvature, held square at the top by the beams. Soften the joints and the beams can hold the column tops less, so each column bends more as a cantilever from its fixed base: the largest column moment rises by a third at EI/L and doubles near pinned, and the roof’s drift goes from 7.5 mm to 19 and then 46. The panels see less of the sway — the beams’ end moments, which are what the panels carry, fall with the joints’ stiffness — but the columns see more, and the drift, which governs a sway frame long before its strength does, grows fastest of all.

That was already true of a rigid frame with flexible beams; the joints simply add their flexibility to the beams’. What is new about joints is that they need not all be the same.

Soften some joints and the sway finds the others #

A frame’s joints are usually not alike. The interior joints, where two beams meet a column, are often the ones a designer tries to lighten with a partial-strength end plate; the exterior joints, with one beam, might be fully welded or stiffened. Or the other way round: exterior joints detailed as flexible end plates and interior ones as rigid moment connections.

Softening every joint to 3 EI/L relieves every panel under the combined load: the floor interior panel, the worst, falls from 242 kN to 178. Softening only the interior joints relieves the interior panels about as much — 181 kN — but the exterior joints, still rigid, carry more than they did in the all-rigid frame: 223 kN at the floor exterior rather than 203, and 155 at the roof corner rather than 141. Soften only the exterior joints and the interior panels are barely relieved, while the exterior ones drop sharply.

The sway alone makes the pattern plainer.

A push shared among joints goes to the stiffest of them. With every joint alike, softening them all just makes the frame softer, and its columns take the difference. With some joints softened and others not, the frame’s resistance to sway is redistributed toward the stiff joints, because a stiff joint resists the column’s rotation more than a soft one and so attracts more of the sway moment: the roof interior joint’s panel carries 28 per cent more with rigid interior joints and soft exterior ones than in the all-rigid frame. Even softening every joint does not relieve every panel under sway: the roof panels carry slightly more, because the floor beams, now softer at their ends, pass more of the frame’s sway resistance up to the roof.

So both halves of the original question are true, of different arrangements. A semi-rigid interior joint does relieve its own panel, under a pattern load and under sway. It also loads the exterior panels harder, by up to a fifth here, and the joints it relieves are the ones whose panels were already the most heavily loaded. Whether a partial-strength interior joint is a saving or a liability is therefore a question about where the frame’s sway goes when it no longer goes there, and the answer is the stiff joints that remain — which are then designed for a sway share the all-rigid analysis did not give them.

Choosing joints for a frame #

Read together, the figures give three rules for a designer choosing where to put semi-rigid joints.

The first is that a uniform choice is the safe one. Softening every joint alike relieves the floor panels under every load drawn and costs the beams’ spans and the frame’s drift. Those costs are visible in the analysis a designer already runs: a span checked for its sagging moment, a drift checked against its limit. Nothing moves somewhere it was not looked for, except a modest share of the sway to the roof.

The second is that a mixed choice must be analysed as mixed. Softening only the interior joints, the obvious place to save on a heavy moment connection, sends sway to the exterior joints and the roof corner, whose panels then carry up to a fifth more than in the all-rigid frame. An analysis that assumed every joint rigid would have sized those panels for less than they get. An envelope of load cases built from the rigid frame is the wrong envelope for the mixed one; the load cases have to be rerun with the joints the frame actually has.

The third is about rotation. A joint chosen semi-rigid has to supply its rotation — a few milliradians at service, more as it approaches its resistance — and do so repeatedly if the load that drives it comes and goes. A joint that has to keep turning under a load that reverses is a different joint from one that turns once; a flush end plate that relieves an interior panel under wind is being turned back and forth by every gust.

By hand, the exchange at one joint #

The pattern case can be checked with the beam alone. The loaded beam’s free moment is wL²/8 = 20 × 64/8 = 160 kN·m. With rigid joints its end moments are 77 and 101 kN·m hogging, averaging 89, so the mid-span sags 160 − 89 = 71 kN·m. At 3 EI/L the ends are 53 and 59, averaging 56, and the mid-span sags 104 — the figure shows 102, because the largest sagging moment is not quite at mid-span when the two ends differ.

The panel shear is the interior joint’s net beam moment over the beam’s depth, less half the difference of the column shears below and above. With rigid joints the loaded beam arrives at the interior column with 101 kN·m and the unloaded beam leaves it with 22 kN·m the other way, a net 79 kN·m; over 0.457 m that is a flange couple of 173 kN, and the columns’ shears of 10 and 28 kN take half their sum, 19, off it: 154 kN. At 3 EI/L the two beams bring 60 and 7 kN·m, a net 54 kN·m and a couple of 117 kN, and the columns take 13 off it: 104. Most of the relief is the loaded beam’s end moment falling from 101 to 60 kN·m; the unloaded beam, which the pattern bends the other way, also lets go of most of its share.

What the springs leave out #

Linear springs. Every joint is elastic and linear. A real end plate has a stiffness that falls as it approaches its resistance, and its rotation under the moments above is larger than its initial stiffness predicts once the moment is past about two thirds of the joint’s resistance.

The panel in the joint. In the component method the column-web panel in shear is itself one of the springs that make a joint semi-rigid. Here the panel’s shear is computed from the moments but its flexibility is not part of the spring; a joint whose stiffness is mostly its panel’s would shed its own panel shear by deforming in it, a coupling these numbers leave out.

First order. The frame is solved without its own sway amplifying the gravity loads. With joints softer and drift larger, the second-order moments grow, and a frame that drifts 19 mm under a modest push is a frame whose P-Δ moments are no longer negligible.

One frame, one load level. The proportions are those of one small steel frame; a taller frame, with more storeys pushing on its lower joints, sends more sway into each, and the shift toward the stiff joints grows with it.

Still open: the joint whose stiffness falls as it is loaded #

Every joint here keeps its initial stiffness however much moment it carries. A real end-plate joint softens as its bolts begin to yield in tension and its plate begins to form its own yield lines, and an interior joint under a pattern load and a push carries different moments on its two sides — one beam’s end loaded nearly to its resistance, the other lightly. The loaded side then becomes softer than the unloaded side, and the joint’s effective stiffness depends on the load that is on it. Whether the softening of the most heavily loaded joints sends still more of the sway to the stiff ones, so that the redistribution grows as the frame approaches its strength, or whether the frame settles into a pattern that a single secant stiffness per joint can describe, is the question a joint that softens under its own load puts to every result above.