The flange that swings is in tension

The formula that reads four numbers tested the quarter-point formula for the moment-gradient factor on diagrams nobody had fitted it to, and every one of those diagrams belonged to a segment held at both ends by forks: free to rotate on plan and to warp, held against twisting. It closed on the member where that assumption does not merely weaken but disappears. A cantilever is held at one end, and what it is held by is not one condition but several, which a drawing of a beam projecting from a column shows as the same short line.

The usual treatment of a cantilever is to find it an effective length — the length of a simply supported span in uniform moment that would buckle at the same moment — and to read that length from a table indexed by the root, the tip and the load. This essay computes the critical moment directly for one beam under every combination the tables choose among, and asks what a table can and cannot hold. Two things come out that a table cannot say. The flange that needs holding at the tip is the tension flange, not the compression flange, which reverses the rule every brace on a span obeys. And the effective length is not a property of the supports: for the same root and the same load it moves by a factor of more than four as the cantilever changes length, because it depends on the section as well.

The flange that moves is not the flange that buckles #

The beam throughout is a 457 mm universal beam, the one the shape of the diagram, and not its peak used for every span, cantilevered 3 m and carrying a load at its tip. Its plastic moment is 522 kN·m. Under a downward tip load it hogs along its whole length: the top flange is in tension, the bottom in compression, and the compressive force in the bottom flange rises from nothing at the tip to its largest at the root. The bottom flange is the strut, and when the beam fails sideways it is the bottom flange that buckles.

The picture is of the buckled beam seen from above, with each flange’s sideways movement drawn separately, and it is not the picture intuition expects. The bottom flange does bow out, and furthest a little beyond half-way along — but by the tip it has come almost back to where it started. The flange that moves at the tip is the top one, the flange in tension, and it moves more than six times as far as the bottom flange moves anywhere.

The reason is that the section twists as it goes. Near the root the moment is large and the bottom flange is doing what a compressed strut does, bowing out between the root, which holds it, and the lightly loaded region further out. Near the tip the moment is small and neither flange is much stressed; what the tip does there is follow. And what it follows is a rotation about a point close to its bottom flange, because the web has carried the twist from the root outward and the tension flange, straightened by its own force near the root, has room to swing only where it is free — at the end. The tip section turns about its compression flange.

That is the observation every other result in this essay leans on, and it has an immediate consequence for where a restraint does any good. A restraint works in proportion to how far the point it holds would otherwise have moved. At the tip, the bottom flange would hardly have moved.

Four roots behind one line on a drawing #

The root is where a cantilever’s capacity is decided, and the word covers four quite different things. A beam can be built into a wall or welded to a stiff end plate, so that the root holds the beam’s sideways slope and also stops its flanges from sliding past one another — holds it against warping, in the language of the section that cannot stay flat. It can be connected in a way that holds the slope and lets the flanges warp. It can pass over a support and continue into a backspan, so that the support holds the beam’s position and twist but the slope is whatever the backspan allows. Or it can pass over a support that holds only its top flange, which is what a beam sitting on a wall or a stub column with nothing fixed to its bottom flange has.

The critical moment for the four, with the load at the shear centre, is 1,266, 521, 465 and 87 kN·m. With the load sitting on the top flange it is 350, 125, 203 and 37. From the stiffest root and the kindest load to the softest root and the worst load is a factor of thirty-four, for a beam whose section, length and load position along it are identical in every case.

The dashed line on the chart is the section’s plastic moment, and it divides the eight bars into two kinds of beam. The built-in cantilever with its load at the shear centre could not buckle before its section had yielded right through: its elastic critical moment is two and a half times its plastic moment. The root holding only its top flange would buckle at a sixth of the plastic moment, and at a fourteenth with the load on top. The same drawing — a beam, a short line across its end, an arrow at its tip — describes a member that stability never touches and one that it governs completely, and nothing on the drawing says which.

It is worth setting the built-in result against the span every moment-gradient factor is measured from. A 3 m span with forks at both ends, in uniform moment, buckles at 802 kN·m. The built-in cantilever of the same length does better, at 1,266, because its root is stiffer than a fork in two ways at once and its moment falls to nothing at the tip. Free the root’s warping and it does worse, at 521. The whole of the difference between “better than a span” and “worse than a span” is a detail at the root that the engineer choosing an effective length has usually not yet drawn.

Hold the tension flange at the tip #

The tip is the other end, and the restraints on offer there are the ones a floor or a façade provides: something holding one flange sideways, something stopping the section twisting, or both.

Holding the bottom flange at the tip — the compression flange, the one the rule for spans says to hold — changes the critical moment by a factor of 1.00 with the load at the shear centre. At this length it buys exactly nothing, because in the buckled shape the bottom flange’s tip was not going to move; at 1.5 m and at 6 m it buys one and four per cent. With the load on the top flange it buys 28 per cent. Holding the top flange at the tip, the flange in tension, multiplies the critical moment by 2.47 with the load at the shear centre and 2.02 with it on the top flange.

The rule it reverses is the one the load that moves with the twist and the brace on the wrong flange established for spans: a brace on a span’s tension flange barely helps, because the span buckles by twisting about its tension flange, which stays nearly still while the compression flange swings out. The cantilever has the same mechanism in a different place. Along most of its length it too turns about its tension flange. At the tip it turns about its compression flange. So a brace belongs wherever the flange it holds would move furthest, and on a cantilever’s tip that is the top.

The last two rows settle a case the second of those essays had in mind when it called a cantilever braced by a slab along its top flange “braced on the wrong flange over its whole length”. Computed, that bracing multiplies the free cantilever’s critical moment by 2.83 with the load at the shear centre and 2.15 with it on the top flange — most of which the single tip restraint had already delivered. The same bracing along the bottom flange, the compression flange, multiplies it by only 1.70 and 1.35. On a cantilever the tension flange is the better flange to hold even along its length, because what is left free when the bottom flange is held is exactly the mechanism the unbraced cantilever was already using: the tip turning about its bottom flange. Holding the tension flange all along is not useless; it is merely redundant with a restraint at one point. What neither flange’s bracing can do is replace a restraint that stops the twist, which reaches 2,534 kN·m whatever the load’s height, because once the tip cannot rotate a load on top of it has no lever to act through.

A load on top of a cantilever #

A load applied above the shear centre moves sideways with the section as it twists, and in doing so it develops a lever arm about the shear centre that pushes the twist further. A load hung below does the opposite. That is load height, and on a span it is worth tens of per cent.

On a cantilever it is worth a factor of six. Built in, the critical moment is 2,147 kN·m with the load hanging from the bottom flange, 1,266 at the shear centre and 350 on the top: 6.14 times from one face of the section to the other. With the root free to warp the factor is 9.17. Between the two positions a designer might actually meet — a load delivered through the top flange by a beam bearing on it, and one delivered at the shear centre through a connection to the web — the cantilever loses 72 per cent of its critical moment built in and 76 per cent free to warp.

The reason it is larger here than on a span is where the load is. On a span the load sits somewhere near midspan, where the twist is largest but the section is held on both sides. On a cantilever the load sits at the tip, where the twist is largest and nothing holds the section at all, and the lever arm the load develops acts at the one place where the beam has no reply. That is also why the twist restraint in the previous figure removes the height effect entirely: it holds the one point where the height matters.

The comparison between this factor and the root’s is the practical point. The load’s height is worth more than the root: the gap between a built-in and a warping-free root, at a given height, is a factor of 1.9 to 2.8; the gap between the top and the bottom of the section, at a given root, is 6 to 9. A cantilever carrying a beam that bears on its top flange is a different member from the same cantilever carrying the same beam through a fin plate at mid-depth, and the difference is larger than any decision made at the root.

An effective length that will not stay put #

A table of effective lengths converts all of this into one number per case, so that the span formula can be used unchanged. The conversion is exact for a single beam: for any computed critical moment there is exactly one span length that buckles at it. The question is whether that length is a fixed multiple of the cantilever’s, so that one table serves every section and every length.

For one case it nearly is. A built-in cantilever with its load at the shear centre has an effective length between 0.70 and 0.82 of its length over the whole range from 1.5 to 9 m, and a table that prints 0.8 would serve it well. For the other three the number moves. Built in with the load on top, it falls from 1.80 to 1.01 as the cantilever lengthens. Free to warp with the load at the shear centre, from 1.70 to 0.92. Free to warp with the load on top, from 6.13 at 1.5 m to 1.32 at 9 m — a factor of more than four, for the same root, the same load and the same section, at different lengths.

The cause is the section’s two ways of resisting twist. St Venant torsion resists the rate of twist and does not care how long the beam is; warping resists the rate of change of the rate of twist and matters most in a short beam, where the twist has to change quickly — the same division that makes a short cantilevered I-section in torsion a warping member and a long one a St Venant member. The balance between them is measured by a single number, $K = \sqrt{\pi^2 E I_w / G J L^2}$, which for this section is 2.3 at 3 m and falls to 0.8 at 9 m. A root that holds warping reaches the warping resistance; one that does not, cannot. So the effective length is a function of $K$ as well as of the supports, and a table of dimensionless factors — which is what a table of effective lengths has to be — has chosen a $K$ without saying which.

That is the precise sense in which the tables cannot agree. Two tables built from calculations on different sections, or at different lengths, describing their roots in the same words, will print different factors, and both will be correct for the beam they were computed on. The figure shows how far apart they could be for the case where it matters most.

What a warping restraint is worth, and how stiff it has to be #

Between a root free to warp and a root that holds warping completely there is every real connection. An end plate welded across both flanges holds warping partly; how much depends on the plate’s thickness and on what it is bolted to.

The curve has the shape of every partial restraint: nothing at one end, nearly everything at the other, and a transition two decades wide in between. Measured against the beam’s own warping stiffness over its length, $EI_w/L$, the root has to be about fifty times stiffer to deliver nine tenths of what full warping restraint is worth — 56 times with the load at the shear centre, 37 with it on the top flange. A root of the same order as the beam’s own warping stiffness delivers about a fifth of the benefit, and one ten times stiffer about seventy per cent.

That is a demanding requirement and it answers a question the tables do not ask. A built-in root is a claim that the connection is fifty times stiffer in warping than the cantilever it holds. A heavy end plate welded to a column flange that is itself restrained may be; a thin plate bolted to a column web is almost certainly not, and the column web itself twists. The difference between the two is the difference between 1,266 and 521 kN·m, and the effective-length table asks the designer to choose between them by name.

A cantilever that is really the end of a backspan #

The commonest real cantilever is not built into anything. It is a beam passing over a column and projecting beyond it — a canopy, a balcony, the overhang of a roof — and its root is the support where it passes over.

What such a root lends the cantilever is whatever the backspan will give, and the backspan gives less the longer it is. With a support that holds the twist, a backspan half the cantilever’s length makes the root nearly as good as built in, 950 kN·m against 1,266; one twice as long leaves 465, a little less than the warping-free root; one four times as long leaves 185. The backspan is then not restraining the cantilever at all — the two buckle together, the long backspan in reverse curvature, and the critical moment is a property of the pair.

With a support that holds only the top flange the cantilever is in poor shape at any backspan: 238 kN·m at half its length, 87 at twice, 47 at four times. Continuous over a support of that kind, the root is effectively a pin about which the whole section can rotate as the compression flange swings, and the flange it holds is the one that would not have moved much there anyway.

So “continuous over a support” is two conditions, not one, and each of them is a curve rather than a number. A table that assigns it one effective length has chosen a backspan.

The arithmetic for the built-in case #

The built-in cantilever with its load at the shear centre has a critical moment of 1,266 kN·m at its root, which is a tip load of 422 kN over its 3 m. Its effective length, the span that buckles at the same moment in uniform moment, comes out at 2.35 m, or 0.78 of its length. The span formula can be run by hand at that length to show the conversion does what it says:

$$M_{cr} = \frac{\pi}{L_e}\sqrt{E I_z\,G J + \frac{\pi^2 E I_z\,E I_w}{L_e^2}}.$$

With $E I_z = 3.05 \times 10^{12}$ N·mm², $G J = 3.01 \times 10^{10}$ N·mm² and $E I_w = 1.48 \times 10^{17}$ N·mm⁴, and a round $L_e = 2{,}340$ mm, the first term under the root is $9.15 \times 10^{22}$ and the second $8.13 \times 10^{23}$, so the root is $9.51 \times 10^{11}$ and the moment $1.34 \times 10^{-3} \times 9.51 \times 10^{11} = 1.28 \times 10^{9}$ N·mm — 1,277 kN·m, against 1,266 at the unrounded length.

The arithmetic also shows where the answer comes from. The second term is nearly nine times the first: at this effective length the section is resisting twist almost entirely by warping, which is why the root’s warping restraint matters so much. Free that restraint and the effective length jumps to 1.27 times the length, where the warping term is only 3.4 times the St Venant term and the critical moment falls to 521.

A thin-walled beam, solved whole #

The calculation behind every figure keeps both of the buckling displacements — the sideways movement of the shear centre and the twist about it — as separate unknowns along the beam, rather than eliminating one as the forked-end calculation of the restraint that beats the gradient could. That is what a cantilever demands: a root that holds the slope but not the warping, or a tip held at one flange, is a condition on the two displacements separately. The beam is divided into short lengths, each carrying a cubic in each displacement; the energy of lateral bending, St Venant torsion and warping is balanced against the work the moment does through the coupled sideways bending and twist, and against the load’s lowering as its point of application swings with the section. The smallest load at which the balance can be broken is the critical one.

With the warping stiffness set to zero, the same calculation returns Timoshenko’s constants for a narrow rectangular cantilever — a tip load of $4.013\sqrt{E I_z G J}/L^2$ and a uniform load totalling $12.85\sqrt{E I_z G J}/L^2$ — and for a span held by forks in uniform moment it returns the closed form above to four figures. The flange that buckles out in each case is checked against the sign of the moment: under hogging it is the bottom one.

Elastic, perfect and doubly symmetric #

The critical moments are elastic. Where one exceeds the plastic moment the beam fails by yielding, and the figure’s bar says only that stability is not the question. Where one is well below it, the design resistance is a reduction from it along a column-type curve, and the ratios between cases survive that reduction only roughly.

The beam is perfect. A real cantilever starts slightly bent and twisted, and its restraints carry a force from the start; the tip restraint on the top flange is then a brace with a stiffness requirement, like the brace that need not be strong on a span.

The section is doubly symmetric and keeps its shape. A section with unequal flanges has a Wagner term that depends on the sign of the moment, and a cantilever’s hogging puts the larger flange in tension or compression depending on which way up it was rolled. And the web is assumed stiff enough to carry the twist from one flange to the other; at a support where only the top flange is held, the web can distort instead, and the calculation above is then an upper bound on a mode it does not include.

The root and tip conditions are ideal. Each is either fully present or fully absent except where the figure varies a spring, and the backspan carries no load of its own.

What the pictures cannot show #

That the numbers are for one section. The effective-length figure is the argument that they cannot be transferred to another at a fixed ratio: a lighter section with a lower warping constant sits further to the right on it at every length, and a deeper one further to the left. The shape of every result survives — the tension flange swinging at the tip, the load’s height outweighing the root — but the factors are this beam’s.

Still open: what an end plate is worth as a warping spring #

The warping curve says a root must be about fifty times the beam’s own warping stiffness to count as built in, and stops there, because the stiffness of a real connection against warping is not a number any table gives. An end plate resists warping by bending in its own plane as the two flanges try to slide in opposite directions, and it does so only if the column behind it does not twist first. Whether an ordinary flush end plate is closer to the free end of that curve or to the fixed end — and so whether an ordinary bolted cantilever is closer to 521 kN·m or to 1,266 — depends on the plate, the bolts and the column together, and is the calculation that would turn the root from a name into a stiffness.