The panel that gravity leaves empty

The moment that goes round the corner cut the knee of a portal out as its own free body and found the largest shear force anywhere near it on no member diagram: the beam’s end moment arrives as a couple in its flanges, and what crosses the web of the column between them is that couple less half the column’s shear — 130.4 kN in a knee whose largest member shear is 80. It named the case it had not drawn: “the interior joint, where two beams arrive and the panel demand roughly doubles for the same member forces.”

The word “roughly” is doing a lot of work in that sentence, and the arithmetic says it does the wrong work. This essay puts the corner into a frame that has interior joints — two bays of 8 m, two storeys of 4 m, the same 457 mm beams and 254 mm columns with 8.6 mm webs, bases fixed — and follows the panel shear at every joint through every load case a designer would combine.

Four kinds of joint #

A two-bay, two-storey frame has four kinds of joint, and they differ only in what frames into them. The roof corner is the knee of the earlier essay: one beam, one column below. The roof interior has two beams and a column below. The floor exterior has one beam and columns above and below. The floor interior has everything: two beams, two columns, a cross.

The panel is the same in all of them: the rectangle of column web between the beam flanges. And the shear across it has the same origin. Each beam delivers its end moment as a flange couple, a pull in one flange and a push in the other, $M/d_b$ apart. Each column carries a shear away above and below. The design code writes the panel’s share as

$$V_{wp} = \frac{M_{b1} - M_{b2}}{z} - \frac{V_{c1} - V_{c2}}{2}$$

with the signs of the moments and shears as they fall. The half on the column shears is not a factor of safety. It is the statement that the beams’ axial forces, which carry the columns’ shears across the joint, arrive at the beams’ mid-depth — half above the middle of the panel and half below — so a column shear relieves only half the panel of itself. The knee’s 130.4 kN is 142.7 of flange couple less half of 24.5 of column shear.

Full gravity: the interior is empty #

Under the load that designs almost everything in a building frame — every beam loaded — the moment diagram looks as it always does: sagging in the middle of each beam, hogging at each end, and the columns bent by whatever the joints hand them. At the exterior joints the beam hogs into the joint and the columns take the moment away, and the panels carry 143 kN at the floor and 106 at the roof.

At the interior joints the picture is different in kind. The beam on the left hogs into the joint with 123.4 kN·m, and the beam on the right hogs into it with 123.4 kN·m the other way. Each beam’s top flange pulls away from the column and each bottom flange pushes towards it, symmetrically, and the column web between them is squeezed from both sides by equal couples. The interior panel carries nothing, and the interior column carries no moment. The frame is symmetric about it, and a symmetric frame under a symmetric load does not bend its own line of symmetry.

The panel reads the joint the way a balance reads two weights. When the beams hog on both sides, the panel sees the difference between their couples; when they bend the joint the same way, it sees the sum. Full gravity is the pure difference case, and the difference is zero.

That already refutes the sentence the earlier essay ended on. An interior joint does not carry twice a corner’s panel shear “for the same member forces”. Under the member forces that full gravity produces, it carries none.

What fills it #

Two things break the symmetry, and both are load cases a designer combines as a matter of routine.

The first is a pattern: live load on one bay and not the other. With the left bay loaded, the left beam still hogs into the interior joint with 101.2 kN·m, but the unloaded right beam brings only 22.2 kN·m, and the panel sees the difference: 154 kN at the floor, a little more than the 150 in the exterior joint of the same loaded bay. A pattern load turns an interior joint into an exterior one, because its unloaded neighbour hardly holds it.

The second is sway. A frame pushed sideways by wind bends every beam into a double curvature, hogging at one end and sagging at the other, so at an interior joint the beam on the left and the beam on the right both turn the joint in the same direction. Their couples add. Under 20 kN at each floor the interior panel carries 89 kN and the exterior 53 — 1.67 times, not twice, because a joint held by two beams rotates less than a joint held by one and each of its beams takes a smaller end moment: 23.1 kN·m against the exterior beam’s 28.3. That the interior column takes about twice the storey shear of an exterior one under sway is exactly the assumption the portal method builds its whole analysis on; the panel inherits the same factor, less the joint’s stiffness.

The combination that designs nothing else #

The design question is what happens when the cases are combined, and it has an answer the member design does not suggest.

With one bay loaded and the frame pushed sideways from the loaded side, the two mechanisms line up at the interior joint. The loaded beam hogs into it; the sway bends the unloaded beam the same way round; and the panel carries 242 kN. The picture shows why: at the floor interior the two beam diagrams arrive on opposite sides of their members and turn the joint together.

Read across the chart, the four kinds of joint have different governing cases. The exterior panels are largest under a combination with gravity everywhere, which is the combination that governs the members: the leeward exterior floor panel carries 196 kN under gravity with sway. The interior panels are largest under a combination with gravity in one bay, which governs no member — the members’ largest moment, 146.5 kN·m, comes from gravity with sway, and one bay loaded with sway gives them only 124.4. A frame checked joint by joint under the combination that designs its members finds 89 kN in the floor interior panel. The panel has to carry 242.

The error is not subtle and it is not on the safe side. The interior panel’s real demand is 2.7 times what the member-governing case finds, and more than any exterior panel’s under any case. A designer who sizes doubler plates by working from the member design, and checks the exterior joints because that is where the big moments are, has left the most heavily loaded panel in the frame unchecked. The envelope is not a structure made the general version of this point about pattern loading on continuous beams: the worst value at each section comes from a different arrangement of load, and an envelope assembled from them describes no loading at all. The panel is one more section with its own worst arrangement, and it is the arrangement least like the one everything else is designed for.

How the push moves the problem inside #

The balance between the cases depends on how hard the frame is pushed, so it is worth following the two floor panels as the storey load grows.

With every beam loaded, the exterior panel starts at 143 kN from gravity alone and the interior at nothing; the interior grows only with the sway, and at the 20 kN a floor drawn here it is still well below the exterior. That is the picture a designer working from the full-gravity case would form: the exterior joints govern, and the interior ones get the same detail for tidiness.

With one bay loaded, the interior panel starts level with the exterior — 154 kN against 150 — and grows faster with every kilonewton of sway, because at the interior joint the sway’s two beam moments add to the pattern’s while at the exterior joint only one does. By 60 kN a floor the interior carries 420 kN and the exterior 309. The more a frame is asked to resist sideways load, the more its critical panels move inside. In a building whose frame is its bracing, the interior joints are where the panel doublers belong.

The panel reads the difference between the bays #

The pattern case is not all or nothing. Live load in the right bay can be anything from none to the same as the left.

Without sway the panel shear falls in a straight line from 154 kN to nothing as the right bay’s load rises to match the left’s. The interior panel sees the difference between the two bays and nothing of their sum. Double both bays’ loads and it still carries nothing; load one bay lightly and leave the other empty and it carries a share of the full pattern value in proportion. With sway the line is lifted by the sway’s own 89 kN, and the balanced case is simply where the sway is left alone.

This is also why an interior joint is so sensitive to the classification of the load. Dead load is on both bays, always, and contributes nothing to the interior panel. Only the part of the imposed load that can be present in one bay and absent from the next reaches it — so the interior panel is designed, in effect, by the live load’s variability, which is the least well known part of the loading.

Unequal bays, and gravity alone #

The emptiness under gravity is a property of equal bays. Make one bay longer and its beam hogs into the interior joint harder than its shorter neighbour, and the difference crosses the panel under gravity alone: 67 kN with bays of 8 and 6 m, 92 with 8 and 10, 208 with 8 and 12. The exterior panel on the long bay’s side grows faster still, because the long beam’s whole end moment arrives there with nothing on the other side to offset it. So in a frame of unequal bays the interior joint is less extraordinary than in the symmetric one; the symmetric frame is the one in which the interior panel’s demand is entirely a matter of patterns and sway, and therefore entirely invisible to a gravity check.

Only the lopsided part of a load reaches it #

All of the behaviour above follows from one decomposition, and it is worth stating because it applies to every interior joint of every regular frame. Any loading on the two-bay frame can be split into a part that is symmetric about the interior column line and a part that is antisymmetric about it: the average of the two bays’ loads on both bays, plus half their difference as an up-load on one bay and a down-load on the other, plus the sway, which is antisymmetric by nature. The frame is linear, so the panel shear is the sum of what each part produces — before its absolute value is taken — and the symmetric part produces none, because a symmetric frame under a symmetric load leaves its line of symmetry unbent.

So the interior panel carries only the antisymmetric part of whatever is applied. Full gravity is entirely symmetric and gives nothing. A pattern with one bay loaded is half symmetric and half antisymmetric, and the antisymmetric half — 10 kN/m up on one bay and down on the other — is what puts 154 kN through the panel. Sway is entirely antisymmetric and puts all of itself through. An exterior joint has no such filter: it sees the symmetric part and the antisymmetric part alike, which is why full gravity is its worst case and not the interior’s.

The decomposition also says what an interior joint cannot be protected by. Making the loads larger does nothing to the ratio of the parts, and neither does making the beams or columns stiffer, since both parts are resisted by the same frame. Only a genuinely symmetric loading leaves the panel alone, and in a building the only loads that are reliably symmetric are the dead ones.

Where the doubler plates go #

The consequence for the steel is a matter of where the reinforcement goes. The 8.6 mm web of the 254 mm column resists 347 kN of panel shear. At 20 kN a floor every panel in the frame is inside that, the interior one at 70 per cent. Push harder and the order of failure is not the one the member design suggests: at 50 kN a floor the floor interior panel needs 376 kN and has passed its resistance, while the worst exterior panel, at 283, still has a fifth in hand; at 60 kN a floor the interior needs 420 and the exterior 309. The interior joints need doubler plates first: the floor interior panel reaches its resistance at 43.5 kN a floor and the worst exterior one not until 74.2, so across that whole range the interior joints are the only ones in the frame that need them.

A doubler for 420 kN needs another $73/158.8 = 0.46$ thousand square millimetres of web, under 2 mm of plate over the 254 mm depth — a small plate, welded to a column web in the one place where two beams, two continuity stiffeners and a doubler all compete for the same few hundred millimetres of steel. Omitting it at the joints where the load combination that sized everything else put nothing, and fitting it at the exterior joints where it was not needed, is the characteristic mistake this arithmetic predicts.

A joint by hand #

The interior numbers can be recovered from the member end moments without the panel formula’s signs, which is the check worth having on a drawing.

Under the left-bay pattern the left beam brings 101.2 kN·m and the right beam 22.2 kN·m, turning the joint opposite ways, so the net flange couple is $(101.2 - 22.2)/0.457 = 173.0$ kN. The net moment, 79 kN·m, leaves through the columns, and in doing so they push on the joint sideways: 28.2 kN from the column above and 10.5 kN the other way from the column below, the upper storey bent harder because its roof beam is patterned too. The panel keeps half of their combined relief, $(28.2 + 10.5)/2 = 19.4$ kN, and $173.0 - 19.4 = 153.7$ kN is the frame solution’s number. Under sway the two beams bring 23.1 kN·m each, turning the joint the same way: $46.3/0.457 = 101.3$ kN of couple. The columns push with 9.6 kN above and 15.4 below, half their sum is 12.5, and the panel carries 88.8 kN. The panel’s resistance is the web area times the shear yield stress: $0.254 \times 0.0086 \times 275/\sqrt{3} = 347$ kN, so the governing 242 kN uses 70 per cent of it where the gravity-with-sway check would report 26 per cent.

Where the model stops #

The frame is elastic and first-order. Sway amplifies itself through the load that makes itself worse, and the amplification applies to the sway part of each panel’s shear, which is a larger share of the interior panel’s demand than of the exterior’s. A frame near its sway limit therefore moves its critical panels inside even faster than the sweep shows.

The panels are rigid. A real panel shears, and its distortion adds to the storey drift and changes how the joint shares its moment between the columns. An interior panel that yields under a pattern-and-sway combination is also one whose distortion was not in the analysis that computed its demand — which is why panel flexibility is an explicit spring in the design codes for sway frames.

The load cases are the five drawn. A real combination set has more patterns — alternate bays, adjacent pairs, one storey loaded and not the next — and each moves a different panel to its worst. The two-bay frame has only one way to be patterned, and in taller or wider frames the number of arrangements grows fast, which is the argument of the envelope essay again.

What the pictures cannot show #

That the panel is not only a shear problem. The flange couple arriving from each beam is also a concentrated force on the column flange and web — the tension flange trying to pull the column flange out of plane and the compression flange trying to crush or buckle the web beside it — and at an interior joint under a pattern those forces are different on the two sides, so the stiffeners that resist them are loaded unevenly. And that the panel’s thin web, if the column is a slender section, is a plate in shear, which ripples before it yields. None of that is in a panel-shear number, and all of it is at the joint that the member-governing check found empty.

Nor can they show the connection. A panel carries what the connection delivers, and a moment crossing a gap through bolts and plates is delivered with its own flexibility; the frame here assumes the beams are welded straight to the columns.

Still open: the frame whose joints are not rigid #

Every joint here is rigid, so the beams’ end moments are whatever the frame’s stiffness makes them. A semi-rigid joint — an end plate, a flush plate — rotates under its moment, and at an interior joint under a pattern load the loaded beam’s joint rotates one way and the unloaded beam’s the other. Whether a semi-rigid interior joint sheds the pattern’s unbalanced moment into its rotation and so relieves its own panel, or whether its rotation lets the sway concentrate in fewer joints and loads the remaining panels harder, is a question about the joint that was chosen rather than the one that was assumed, and it is the one that decides whether a partial-strength interior joint is a saving or a liability.