The section that changes along the span

Every strength check in this collection so far has had the same two steps: find where the demand is largest, and compare it with the capacity there. The first step is usually easy, because the moment diagram is drawn and its peak is visible.

That procedure quietly assumes the capacity is the same everywhere. Taper the member and it is not, and the peak of the demand stops being the place to look — because at the section carrying the most moment there is also the most section to carry it with.

Where the two curves are closest #

The demand under a tip load is $M = Px$, measured from the free end. The capacity of a rectangular web of thickness $b$ and depth $d$ is $f_y b d^2/6$, and the depth is $d_0 + kx$ with $k$ the taper rate. So the utilisation is

$$u(x) = \frac{Px}{f_y\,b\,(d_0 + kx)^2/6} \;\propto\; \frac{x}{(d_0 + kx)^2}$$

Differentiate. The numerator of $u'$ is $(d_0 + kx) - 2kx = d_0 - kx$, so the maximum is at

$$x^* = \frac{d_0}{k} \qquad\text{where}\qquad d(x^*) = d_0 + k\cdot\frac{d_0}{k} = 2d_0$$

The governing section is where the depth has grown to exactly twice the depth at the free end. The load has cancelled. The length has cancelled. The web thickness, the yield stress and the taper rate have all cancelled. Whatever the member, whatever it carries, the worst place on a linearly tapered cantilever under a tip load is where it is twice as deep as its shallow end.

For the member drawn: $d_0 = 200$ mm, $k = 400/6000 = 0.0667$, so $x^* = 3{,}000$ mm — the exact middle, by coincidence of the numbers chosen. The utilisation there is 0.792 against 0.704 at the root.

The ratio between the two has a closed form too, and it is worth having because it says how badly the obvious answer misses:

$$\frac{u_{root}}{u_{max}} = \frac{4 d_0 (d_1 - d_0)}{d_1^2}$$

which is $8/9$ here. Checking the root rather than the governing station overestimates the capacity by 12.5%.

What the check would have been on a prismatic member is worth stating for contrast, because it is the habit this page is arguing with. One diagram, one peak, and the peak at the support: the diagram is an integral and its largest value is where the integration has been going longest. That is the right place to look only while the section is the same all the way along, and the moment it stops being the same the largest demand and the largest deficit part company.

When the interior station does not exist #

The result has a condition attached and it is the useful part of it.

$x^* = d_0/k$ lies inside the member only if $d_0/k \le L$, which is $d_1 \ge 2 d_0$. A taper shallower than two to one has no interior worst point at all — the utilisation is still rising at the root, and the root governs after all.

Read down the third column: once the interior station exists, it is at 400 mm however steep the taper gets. Only its position moves, sliding toward the tip as the taper steepens.

That is a genuinely useful piece of practice. A gently tapered member is checked at its root like any other. A steeply tapered one — a haunched portal rafter, a tapered mast, a cantilever canopy — has to be checked somewhere in the middle. The haunch at a portal frame’s corner is the commonest example on any steel building, and the place is found by a rule with no arithmetic in it.

A uniform load moves it to the root, and keeps it there #

Under a uniform load the moment is $wx^2/2$, so

$$u(x) \;\propto\; \frac{x^2}{(d_0 + kx)^2} = \left(\frac{x}{d_0 + kx}\right)^2$$

which is monotonic. There is no interior maximum, and the root governs for every taper.

Both results are checks on the same machinery rather than separate facts. The generator behind these figures walks 401 stations along the member, computes the moment and the capacity at each from the member’s own geometry, and reports the largest ratio; the closed form is then the test it has to pass. It does — the numerical maximum lands on $2.000\,d_0$ to three figures for every taper steep enough to have one.

So the governing station is a property of the load case and not of the member, which is a sentence worth sitting with. A tapered cantilever carrying both a tip load and its own weight has a governing station somewhere between the two answers, and it moves as the load ratio changes — which means it moves through the life of a structure being built in pieces.

The shape a fully stressed member would have #

If the utilisation is to be the same everywhere, the depth has to follow the moment:

$$f_y \frac{b d^2}{6} \propto M(x) \implies d \propto \sqrt{M(x)}$$

For a tip load, $M \propto x$ and the fully stressed depth goes as $\sqrt{x}$ — a parabola, lying on its side, sharp at the tip. For a uniform load, $M \propto x^2$ and the fully stressed depth goes as $x$ — a straight line through zero depth at the free end, which is a triangle.

That the triangle is exactly right for a uniform load explains a great deal of what tapered things look like. A retaining wall, a cantilever mast under wind, a chimney, a dam — all carry a load that grows toward the base as a moment growing with the square of the height, and all are triangular in elevation. They are not tapered because taper looks structural; they are the fully stressed shape of their own load case.

The penalty for a straight line where a parabola was wanted is small, and it is not smallest at either extreme:

The check that moves the other way #

The bending check migrates toward the root as the taper flattens. The shear check does the opposite, and for a reason that is almost too simple to state: the shear does not taper.

A tip load gives a shear that is the same at every station, so the shear stress is $V/(bd)$ and rises as $1/d$ toward the tip. On a member whose bending check has been carefully placed in the middle, the shear check belongs at the shallow end, and the two are as far apart as they can be.

This is the failure that catches tapered members in practice. A haunched rafter or a tapered cantilever bracket is drawn to suit its moment diagram, and the thin end — which nothing in the bending calculation is worried about — is where the web is asked for the most shear stress it will ever see, with the least web to supply it. The remedy is a minimum web thickness that has nothing to do with the moment.

Stiffer as well as lighter #

Removing material from the low-moment end sounds like a strength trade, paid for in deflection. It is not.

The tip deflection of the tapered cantilever above is 101 mm. A prismatic cantilever of the same mean depth — 400 mm, and therefore the same weight of web — deflects 161 mm. The taper is 37% stiffer at equal weight.

The reason is the same $d^3$ that decides everything about a section, applied where the curvature is being generated. Curvature is $M/EI$; near the root $M$ is large, and the taper has put the depth there; near the tip $M$ is small, and the material removed was contributing to a term that was small anyway. Both integrals reward the arrangement.

It is the same argument the ranking of sections makes at a single cut, taken along a member instead of across one. Four arrangements of one area span a factor of forty in second moment because of where the material sits relative to one axis; a taper moves material relative to the moment diagram instead, and improves the answer for exactly the same reason. A truss states it most plainly of all: the chord force is the moment divided by the depth, so a truss deepened where the moment is large has smaller chords for no more material, and every tapered truss, fish-belly girder and haunched frame is this arithmetic taken along the span.

The equivalent prismatic depth, and it is not the mean #

The deflection of a tapered member has no entry in any table, because every table’s coefficient was derived for a constant $EI$. The integral is not hard, and doing it once gives a number worth carrying.

For a cantilever of span $L$ under a tip load $P$, with the depth running linearly from $d_0$ at the tip to $d_1$ at the root and the width constant,

$$\delta = \frac{12P}{Eb\,k^3}\left[\ln\frac{d_1}{d_0} + \frac{2d_0}{d_1} - \frac{d_0^2}{2d_1^2} - \frac{3}{2}\right], \qquad k = \frac{d_1-d_0}{L}$$

which for the member drawn — 200 to 600 mm over 6 m, 12 mm web, 30 kN at the tip — gives 101.2 mm, the figure quoted above.

Now ask what prismatic cantilever deflects the same amount. Solving $PL^3/3EI$ for the depth gives $d_e = 467$ mm, against an arithmetic mean of 400 and a geometric mean of 346.

The equivalent depth is two thirds of the way from the shallow end to the deep one, not half. The reason is the $d^3$: the shallow end contributes to the deflection integral out of all proportion to its length, so the mean of the depths is much too generous a stand-in — but the curvature is largest at the deep end, where the moment is, and that end is weighted back in. The two effects do not cancel and the answer lands above the mean rather than below it.

The practical use is as a sanity check. A deflection estimated from the mean depth is 60% too large here; one estimated from the root depth is 55% too small; and the two-thirds rule lands within a few per cent for tapers up to about three to one, which covers nearly everything built. Beyond that the integral is worth doing, and it is four lines.

Taper the depth, not the width #

There is a second way to remove material from a low-moment region, and it is much less effective for a reason that reads straight off the exponents.

A member can be tapered in plan — narrowed rather than made shallower — which is what a triangular gusset, a base-plate rib and a tapered bracket in plan all are. The section modulus is $bd^2/6$, so:

• narrowing buys capacity linearly in $b$, and stiffness linearly in $b$;
• shallowing buys capacity as the square of $d$, and stiffness as the cube.

So the fully stressed profile differs in kind between the two. With the depth held, $Z \propto b$ and the moment is linear along a cantilever, so the fully stressed width is a straight line — a triangle, which is exactly the shape a gusset plate is cut to. With the width held, $Z \propto d^2$ and the fully stressed depth is a parabola, which is the profile the previous section drew.

And the two are worth different amounts. Removing a unit of material from the width removes one unit of capacity and one of stiffness; removing it from the depth removes half a unit of capacity and a third of a unit of stiffness, because the material being removed was near the neutral axis where it was contributing least. A depth taper is two to three times more efficient than a width taper, per unit of material saved, and the ratio is the exponent.

Which is why a member is shaped in elevation and a plate is shaped in plan. A beam has a depth to spend and a gusset does not — its “depth” is the direction the load runs in, and the only free dimension is the one across it.

There is one arrangement that gets both, and it is worth naming because it looks like decoration and is not. A haunch cut from a rolled section and welded under a rafter tapers the depth; a bracket plate cut to a curve tapers the width; and a fabricated cantilever made from plate can taper both at once, with the depth following a square root and the width following a straight line. The compound shape is the fully stressed profile of a member free in two dimensions, and the reason it is rare is that it has two curved cuts and no straight edge to set out from.

The catalogue has no tapered members in it #

There is a practical asymmetry worth naming, because it decides where tapered members are actually found.

A rolled section is prismatic by the nature of the process: hot steel is pushed through a set of rolls whose gap is fixed, and what comes out has one cross-section for its whole length. Every tapered member in steel is therefore fabricated — cut from plate and welded, or made by slitting a rolled section along a diagonal and re-welding the two halves the other way round, which is how tapered beams are commonly produced and which leaves a member whose depth varies and whose flanges came off the same roll.

That means the choice to taper is a choice to leave the catalogue, and it is made when the saving is worth the fabrication. Three places where it usually is:

Portal frames, where the haunch at the eaves is short, the moment there is much the largest in the frame, and the alternative is a heavier section for the whole rafter. The haunch is often made from a cutting of the same rafter section, so the taper costs one diagonal cut and two welds.

Long cantilevers, where the moment ratio between root and tip is unbounded and a prismatic member is mostly carrying nothing. A canopy, a grandstand roof, a balcony edge.

Anything site-assembled from plate, where the member is being cut out anyway and a sloping line costs no more than a straight one — crane gantries, bridge girders with varying depth over the piers, masts.

Where it usually is not: short spans, repetitive members, and anything where the connection detail at each end has to be repeated. A taper turns every member into a one-off, and one-offs are what fabrication cost is made of.

Where the model stops #

Bending stress in a tapered member is not $M/Z$. The flanges are not parallel, so the flange force has a component perpendicular to the axis, and that component carries part of the shear — reducing the web’s share at one end and increasing it at the other, depending on which way the taper runs relative to the moment. For tapers of a few degrees the correction is a few per cent; for a steeply haunched rafter it is not.

The web is treated as the whole section here. A real tapered member has flanges, and with flanges the tidy result disappears: at 600 mm² of flange the governing depth moves from $2.00 d_0$ to $2.58 d_0$ and the closed form no longer exists. What survives is the shape of the argument, not the number.

Local buckling has been ignored. A tapered web has a $d/t$ that grows toward the root, so the deepest section may be the most slender one, and a member whose bending check is satisfied everywhere may have a section that cannot reach its own strength at exactly the end where the moment is largest — the same trap as a section that is strong on paper and slender in fact.

And the tapered member is assumed straight and laterally restrained. A tapered beam’s resistance to failing sideways varies along its length with the section, and the elastic critical moment of a tapered member is not the prismatic one with an average section substituted.

What the pictures cannot show #

The elevations are drawn with the depth exaggerated against the span — a 6 m member 600 mm deep is a span-to-depth ratio of ten, which drawn honestly would be a thin wedge with the argument invisible inside it.

The utilisation curves are smooth, and a real check is made at a handful of sections chosen by somebody. A curve with a broad maximum, like the tip-load one here, forgives a poor choice of station; a curve with a sharp one does not, and nothing in the figure says which is which except its own shape.

And the fully stressed profile is drawn as though it were an available alternative. A parabolic web plate can be cut, at a price, and a beam whose depth follows a curve has flanges that are not straight lines — which is why almost every tapered member ever built is a straight taper approximating a curve it will never be.

The ladder from here #

Later rungs on this anchor: the haunched portal frame, where the taper is local to the corner and the governing station is inside the haunch. The stepped member — a plate girder with flange curtailment — which is the same problem with a discontinuous capacity and a governing section at every step. Tapered members in lateral-torsional buckling, where the varying section changes the critical moment and no equivalent uniform moment factor quite works. The fully stressed design idea taken seriously, and why it produces members nobody builds. Fabrication: what a taper costs in cutting, welding and handling against what it saves in steel, which is the calculation that decides whether any of this is used. And the deep version: a structure fully stressed everywhere is a structure with no reserve anywhere, which is an argument against the whole idea.