The root that belongs to the column

The flange that swings is in tension treated a cantilever’s root and tip as separate conditions and found that the root decides more than any table admits. A 3 m cantilever of 457 mm universal beam, loaded at its tip, buckles laterally at 521 kN·m if its root holds the lateral slope but lets the flanges warp, and at 1,266 kN·m if the root holds both. The difference between them is whether the two flanges at the root can turn in plan in opposite directions — the warping that the section that cannot stay flat found every open section does when it twists. Measured against the cantilever’s own warping stiffness over its length, $EI_w/L$, a root had to be about fifty times stiffer to deliver nine tenths of what holding the warping is worth.

That essay stopped there, because the stiffness of a real root against warping is not a number any table gives. “Built in” is a name on a drawing. The usual root is a flush end plate bolted to a column flange, and whether it sits near the bottom of the warping curve or near its top decides whether the cantilever is good for something near 521 kN·m or something near 1,266.

How a root holds warping #

Warping at the root is the two flanges turning in plan in opposite directions. With the flanges $h$ apart, a rate of twist φ′ at the root turns each flange through $h\varphi'/2$; holding it takes an equal and opposite moment in plan in each flange, $M_f$, and the pair of them is the bimoment $B = M_f h$. So a root that resists each flange’s turning with a stiffness $S_f$ in plan resists warping with a stiffness $k_w = S_f h^2/2$, in the same units as the cantilever’s $EI_w/L$.

A flange welded across a flush end plate turns in plan by bearing on the plate at one tip and pulling the plate away from the column at the bolt beside the web on the other side. The bolt line is half the gauge from the web, the bearing tip half the flange’s width, so the lever is 50 + 95 = 145 mm for M20 bolts at a 100 mm gauge on a 190 mm flange. What resists the pull is the chain that EN 1993-1-8 already describes for a bolted joint in bending: the bolt stretching, the end plate bending between the bolt and the flange, and the column flange bending behind it, three springs in series.

Then there is the column. The two flanges’ moments in plan arrive at the column as two equal and opposite torques about its axis, 444 mm apart, and the column twists between them. Each flange turns by its own spring and by half the column’s twist between the two levels, and the root’s stiffness is the two in series.

Where the root gives #

With a 254 mm universal column running a storey above and below, the bolt, the plate and the column flange together are a quarter of the root’s flexibility. The column twisting between the two flange levels is the other three quarters. The root a drawing calls an end plate is mostly a column in torsion.

That is because an open section is very poor in torsion. A 254 mm universal column has a torsion constant of about $10^6\ \mathrm{mm^4}$ — the thin-walled arithmetic that the slit that costs a factor of six hundred found makes an open section hundreds of times weaker in torsion than a closed one of the same steel. Close the column between the two flange levels, with plates welded across its flanges on both sides, and its torsion constant there rises about two hundredfold. The column’s share falls to 13%, and the plate and the bolt become most of what is left.

The column twists most a storey-fraction away #

The way the column twists is not the way a shaft twists. Two opposite torques close together are, to a bar whose warping stiffness matters, much what two opposite forces close together are to a beam: a concentrated pair. A simply supported beam under a concentrated moment deflects most not at the moment but some way either side of it, and the column, held against twisting at the floors above and below, twists most about 1.6 m from the beam, three times as far as it twists at the flange levels. What the root feels is only the difference between the two flange levels, half a milliradian per kN·m in each flange for the open column. The column’s warping stiffness — its own flanges bending in their planes — is what keeps that difference small, by spreading the twist over the storey instead of concentrating it between the beam’s flanges.

Boxed, the column barely twists at all between the flange levels, 0.01 mrad each way. The box does not touch the column above and below the beam; it closes the one stretch of column across which the root’s two torques are applied.

A thicker plate buys almost nothing #

On an open column that runs on, a 10 mm plate gives the root 4.4 times the cantilever’s warping stiffness and a 40 mm plate 6.8: quadrupling the plate’s thickness, and multiplying its bending stiffness by sixty-four, adds half again to the root. A column that ends at the beam, as at a roof, is worth less than half as much — 2.2 to 2.6 — because it has no length above the beam to spread its twist into. Boxed, the same plates give 9.2 and 34.6, and now the plate’s thickness does matter, because the column has stopped being the weak link.

The box that changes this is small. Two plates 10 mm thick, welded between the column’s flanges on either side of its web over the depth of the beam, turn 444 mm of open column into a closed tube, and the torsion constant of that short tube is about two hundred times the open column’s. Plates of that kind are already welded into many columns, where beams frame into the column’s web from both sides or a moment connection needs the web stiffened, and where they are already there the cantilever’s root has been boxed without anybody intending it.

What makes a root stiff against warping is the column, and the plate only once the column has been stiffened. No ordinary detail reaches fifty times the cantilever’s own stiffness on this beam: the best here, a 40 mm plate on a boxed column, is at about two thirds of it.

Where real roots sit on the curve #

On the curve the earlier essay drew, the ordinary root — a 15 mm flush plate on a column that runs on — is at 5.9 times the cantilever’s warping stiffness and gives a critical moment of 945 kN·m, 57% of the way from free to built in. The same plate at a top-storey column that ends at the beam gives 791, 36% of the way. A 40 mm plate on the running column gives 970, three per cent more than the 15 mm one. Boxing the column under the 15 mm plate gives 1,124, 81% of the way.

So the answer to the question the earlier essay left is that an ordinary bolted cantilever is in the middle: neither near 521 nor near 1,266, and well short of built in. The tables, which give a bolted root either a built-in effective length or a free one, are wrong in both directions about the same detail, and a designer who wants the built-in number should stiffen the column rather than the plate.

In design resistance the middle is worth more #

The critical moment is not what the cantilever is designed for. Its resistance comes from the buckling curve, which reads the critical moment through the slenderness $\sqrt{M_{pl}/M_{cr}}$ and flattens as the critical moment climbs past the plastic moment of 522 kN·m. Free to warp, the cantilever’s design resistance is 333 kN·m; built in, 449. The ordinary root gives 417 — 72% of the way, against 57% in critical moment — and the boxed column 437, 89%.

The last stretch of warping restraint lifts a critical moment that is already more than twice the plastic moment, and the buckling curve gives little for that. A partial root is worth more in design than its critical moment suggests, and the ordinary flush plate, at 6 times the cantilever’s warping stiffness, already has most of what holding the warping can give to a beam this stocky.

For this cantilever that turns the choice of detail into a smaller question than the critical moments made it look. Boxing the column lifts the critical moment by 19%, from 945 kN·m to 1,124, and the design resistance by 5%, from 417 to 437. A designer who needs those 20 kN·m can buy them with two plates welded into the column; one who does not can take the ordinary root at its real stiffness and lose almost nothing against a built-in assumption — except that the built-in assumption would have overstated the resistance by 8%, which is the error worth avoiding.

What the plate alone would have said #

A designer who did think to compute the root’s warping stiffness would most naturally compute the plate’s: the bolts, the plate and the column flange, the components a connection designer already has, put on the lever of the flange’s turning. That gives 22 times the cantilever’s warping stiffness, a critical moment of 1,139 kN·m and a design resistance of 438 kN·m — nearly the boxed column’s numbers, for a column that has not been boxed. The calculation is careful, uses the right standard, and overstates the critical moment by a fifth, because it stops at the column flange’s face.

The column’s twist is easy to miss for the same reason the shape of the diagram, and not its peak found the moment gradient easy to miss: the quantity that matters is a property of the whole, and each part examined alone looks fine. The plate is stiff, the bolts are stiff, the column is strong in bending about both its axes; it is only as a thin-walled bar under a pair of opposite torques that the column is weak, and nothing in a connection’s design asks the column to be that.

Why a drawing cannot tell these roots apart #

On a general arrangement the four roots are the same symbol: a beam meeting a column with a filled triangle at the joint. The 15 mm plate on a column in the middle of a frame and the same plate on a column that stops at the roof are the same detail on the connection drawing, and the difference between them — 945 kN·m against 791 — is in the column schedule, two sheets away. The boxed column is two extra plates welded between the column’s flanges, a detail that appears on the column’s own fabrication drawing and nowhere near the cantilever.

That is the same shape as neither pinned nor rigid, which found that a joint’s classification is a statement about the joint and the member together, never the joint alone: the same end plate is rigid on a short stiff beam and semi-rigid on a long slender one. The warping root is the same rule one level down. The root’s stiffness is a property of the plate, the bolts and the column; whether it counts as built in depends on the cantilever’s own $EI_w/L$; and a drawing shows neither the column’s torsion nor the cantilever’s warping stiffness at the point where the root is drawn.

A longer cantilever, a better root #

The root’s stiffness is a property of the plate, the bolts and the column. What it is measured against, $EI_w/L$, is the cantilever’s, and it falls as the cantilever gets longer. So the same detail is a better root for a longer cantilever: the ordinary plate delivers 47% of the built-in benefit to a 1.5 m cantilever and 66% to a 6 m one, and the boxed column 73% and 87%. A short, stocky cantilever is the case in which “built in” is least true and most claimed, because the root’s warping stiffness has to be fifty times a large number.

The warping root, by hand #

The bolt, M20 with a tensile area of 245 mm² over a 50 mm grip, is EN 1993-1-8’s $k_{10} = 1.6 \times 245/50 = 7.8$ mm. The plate, 15 mm thick with the bolt 40 mm from the flange’s weld and 100 mm of plate effective, is $k_5 = 0.9 \times 100 \times 15^3/40^3 = 4.7$ mm; the column flange, 17.3 mm thick with the bolt 30 mm from its root, $k_4 = 0.9 \times 100 \times 17.3^3/30^3 = 17.3$ mm. In series, $1/(1/7.8 + 1/4.7 + 1/17.3) = 2.5$ mm, and on a 145 mm lever each flange’s stiffness in plan is $210{,}000 \times 2.5 \times 145^2 = 1.1 \times 10^{10}$ N·mm per radian.

The column, with a torsion constant of $1.02 \times 10^6\ \mathrm{mm^4}$ and a warping constant of $7.17 \times 10^{11}\ \mathrm{mm^6}$, has a decay length $\sqrt{EI_w/GJ} = 1{,}350$ mm, three times the distance between the beam’s flanges. On an infinitely long column the pair of torques twists each flange’s level by $[h - a(1 - e^{-h/a})]/2GJ = 4.0 \times 10^{-10}$ radians per N·mm, four times the plate’s $0.9 \times 10^{-10}$; the floors 4 m away hold it to $2.5 \times 10^{-10}$. Together, $3.4 \times 10^{-10}$, a stiffness in plan of $2.9 \times 10^9$, and a root of $2.9 \times 10^9 \times 444^2/2 = 2.9 \times 10^{14}\ \mathrm{N\,mm^3}$ — 5.9 times the cantilever’s $EI_w/L = 210{,}000 \times 7.05 \times 10^{11}/3{,}000 = 4.9 \times 10^{13}$.

What the root model leaves out #

The root’s lateral slope is held. Only warping is in question. The same column that twists under the flanges’ opposite torques twists more easily still under a single torque — the beam’s whole end turning in plan — and a cantilever whose root’s lateral slope depends on the column alone has a weaker root in that sense too. A floor slab or a backspan usually holds it, and the figures here assume one does.

The components are EN 1993-1-8’s springs. They are calibrated for a joint in bending about the beam’s strong axis, where a bolt row is pulled by the flange’s force. Pulling it by the flange’s turning in plan loads one bolt of the row rather than both, and the effective length of the plate around a single bolt is less certain. The column’s twist dominates the answer, and it does not depend on those calibrations.

The column is held against twisting at the floors. A column whose floors do not hold its twist — a column braced only in one direction, or in an open storey — is a weaker root, as the top-storey case shows.

The cantilever is elastic and its section perfect. Every number is an elastic critical moment or a design resistance read from it, as in the restraint that beats the gradient, and the comparison between details is what matters.

Still open: the root that is a backspan #

Every root here is a column. Many cantilevers are not stopped at a column but run over it into a backspan, and then the cantilever’s root is the backspan’s end, which warps as the backspan does and holds the cantilever’s flanges as well as the backspan’s own warping stiffness lets it. A long, slender backspan is a weak warping spring and a short, heavy one a strong one, and the formula that reads four numbers has no term for either. Whether a backspan of a given length holds a cantilever’s root better or worse than a boxed column — and whether the column the backspan passes over adds its own twist to the chain or is bypassed by it — is the question the column on the other side of the joint puts to every cantilever that does not stop at its support.