Where a bar may stop

A simply supported beam’s moment diagram is a parabola: largest at midspan and zero at the supports. The reinforcement that midspan needs is therefore not needed at the ends, and stopping some of it short is one of the few decisions in reinforced concrete that saves material a contractor can count.

It is also a decision surrounded by rules that look like conservatism and are not. Two of them, between them, take almost the whole of the saving, and both have arguments behind them that have nothing to do with caution.

The geometry, which is a square root #

Take a beam whose steel is sized so that its capacity $M_R$ just matches the midspan moment $M$. Curtail some of it, leaving bars able to carry a fraction $r$ of that. The remaining bars are sufficient wherever the moment is below $rM$, and the moment is above $rM$ between

$$\frac{x}{L} = \frac{1 \pm \sqrt{1-r}}{2}.$$

So the region that needs the curtailed bars is $L\sqrt{1-r}$ long. That square root is the whole of the geometry.

Halving the capacity — $r = 0.5$ — leaves a bar $\sqrt{0.5} = 0.707$ of the span: a shortening of 29 per cent, not 50. Dropping to a quarter leaves 0.866. Shortening a bar to half the span takes removing three quarters of the steel it represents.

The practical reading is that curtailment pays in the last layer rather than the first. Taking a beam from four bars to three shortens one bar by 13 per cent of the span; taking it from two to one shortens that bar by 29. Most schedules curtail in equal layers, which spends the drawing office’s attention on the cuts that are worth least.

The shift, which comes from shear #

The second obstacle is stranger, and it is the one worth understanding rather than remembering.

A cracked reinforced concrete beam is not a beam in the sense the moment diagram assumes. It is a truss: the shear is carried by diagonal compression in the concrete between the cracks and by tension in the links, and the bottom bar is the truss’s tension chord.

Resolve that truss at a node and the chord force at a section is not $M(x)/z$. The diagonal delivering the shear to the chord arrives at an angle $\theta$, so its horizontal component adds to the chord:

$$F = \frac{M(x)}{z} + \frac{V(x)}{2}\left(\cot\theta - \cot\alpha\right).$$

That extra term is exactly equivalent to reading the moment diagram a distance

$$a_l = \frac{z\,(\cot\theta - \cot\alpha)}{2}$$

further along. For vertical links and 45-degree struts it is half the lever arm — 270 mm on the beam drawn. The two constructions agree to 0.36 per cent, which is the second-order term of the parabola and nothing else.

The steel is designed for the moment somewhere else, and the distance is set by the shear’s diagonal rather than by the bending. That is why the shift appears in a clause about detailing and originates in a clause about shear, and why it is the rule most often applied without anybody being able to say where it comes from.

Which free body produced the number #

The shift’s free body is the one worth drawing, because it is the only place in this subject where a detailing rule has a two-line derivation.

Cut the beam along a diagonal crack at angle $\theta$, running from a point on the tension steel up to the compression zone. Draw the free body on the support side of it. Crossing the cut are the compression in the top, the tension in the bottom bar, the tension in every link the crack passes, and the diagonal compression in the concrete between them.

Take moments about the point where the compression resultant crosses. The applied moment is the one at the section where the crack reaches the compression zone; the bar’s tension acts at the section where the crack reaches the steel, which is a horizontal distance $z\cot\theta$ nearer the support. Moment equilibrium then gives the bar a force corresponding to a section further from the support than the one it is at — and half of $z\cot\theta$ is the shift, the half coming from the links sharing the transfer.

A bar is anchored to the moment at the far end of the crack that will cross it, and the crack is diagonal, so the far end is not overhead.

The anchorage, which is the larger of the two #

Past the shifted cut-off point the bar still has to develop its force into the concrete, over the development length bond requires — around 40 bar diameters, so 1,150 mm for a 25 mm bar in ordinary concrete.

Add the two and the tail is 1,420 mm at each end. On a 9 m beam that is 16 per cent of the span, before any steel has been saved.

Working the arithmetic through for the beam drawn: the outer layer’s theoretical cut-off is at 1,665 mm from the support; the shift moves it to 1,395; the anchorage moves it to 245; and the bar that could have been 5,670 mm long ends up 8,509 mm long in a 9,000 mm beam. The saving offered by the moment diagram is 19.7 per cent of the steel and what survives is 2.2 per cent.

That number is the reason many engineers stopped curtailing bottom steel in ordinary beams and simply run it through. It is not laziness; it is the arithmetic.

The forty diameters is worth looking at rather than quoting, because the straight line it comes from is an assumption and the assumption has a limit.

The flattening is the part that matters for detailing. Beyond that ceiling a bar cannot be rescued by making it longer, which is why the devices that shorten anchorage all work by changing the mechanism rather than the length. The reason is visible in the bond stress itself.

It is worth separating the two obstacles, because they behave differently. The shift is a mechanical requirement: the chord really does carry that force at that section, and no detailing choice removes it. The anchorage is a transfer requirement, and it can be shortened — by a hook, by a mechanical anchorage, by welding a plate to the bar end, or by using smaller bars. On the beam drawn the anchorage is 81 per cent of the tail, so every one of those devices attacks the larger half. A beam whose curtailed bars end in hooks recovers most of what the arithmetic above took away.

When it starts to be worth it #

The tail does not grow with the span and the saving does, so there is a length past which the calculation turns positive.

The saving offered by the geometry is a fixed fraction of the steel and therefore scales with the span. The tail is $a_l + l_{bd}$, which depends on the lever arm and the bar diameter — both of which grow slowly with span — so it is close to a fixed length. Their ratio is the whole economics.

On the 9 m beam the tail is 16 per cent of the span and the saving is 2. On an 18 m beam with the same bar size the tail is 8 per cent and the saving is around 12. On a 30 m beam it is 5 and 15. Curtailment is a long-span decision, and the schedules that curtail everything and the schedules that curtail nothing are each right about half the building.

There is a corollary about bar size that runs the other way. The development length is proportional to the diameter, so a beam reinforced with many small bars has a shorter tail than one reinforced with few large ones — and small bars also allow finer curtailment steps. A beam designed for curtailment is a beam with more, smaller bars than one designed for placing speed.

Continuous beams, where it is compulsory #

Everything so far is a simply supported beam, where curtailment is optional. Over a support it is not optional, because the moment changes sign.

The top steel over an internal support is needed where the moment is hogging and is useless where it is sagging, so it must stop somewhere; the only question is where. The same two rules apply, with one extra difficulty: the point of contraflexure is not a property of the beam but of the loading arrangement, and it moves.

That is a real trap. The point of zero moment under a uniform load on every span is not the point of zero moment under load on alternate spans, and the envelope of the two is what the steel has to cover. A top bar curtailed at the contraflexure point of the full-load case is short for the pattern-load case, by an amount that can be a substantial fraction of the span.

The staircase and the curve #

Drawing the resistance as a staircase against the demand as a curve is the whole check, and it makes two things visible that a table of cut-off points does not.

The first is that the staircase must envelope the curve everywhere, not merely at the cut-off points. It is easy to satisfy the check at each step and violate it between two of them, and the place it happens is where the shift has moved the demand curve steeply — near the supports, where the moment is changing fastest.

The second is the vertical distance between them, which is unused capacity. On the beam drawn the staircase sits a long way above the demand over most of the span, because the layers are equal and the parabola is not. The steel is being sized in three steps to match a shape that has no steps in it, and the mismatch is inherent rather than a sign of poor detailing: a bar is either there or it is not.

That is worth setting against the way a tapered member solves the same problem, by changing the section continuously instead of the steel in jumps. The two are the same optimisation with different granularity, and the reason the continuous one is rare is fabrication rather than mechanics.

The bars that must not be curtailed at all #

Three requirements override the arithmetic entirely, and each has a different reason.

A proportion of the bottom steel must reach the support. A quarter of the midspan area, typically, anchored past the bearing. The reason is not bending — the moment there is zero — but the shift itself: the truss demands a chord force at the support equal to $V\cot\theta/2$, which is not small, and something must carry it.

Steel required for robustness is continuous by definition. Tie forces are about a structure that has lost a member, and a tie curtailed where the moment diagram allows is a tie that is not there when its own load case arrives.

Steel in a plastic hinge region is not curtailed near the hinge. Moment redistribution requires rotation, rotation requires the section to yield and go on yielding, and a curtailment inside that region turns a plastic hinge into an anchorage failure.

What a curtailment failure looks like #

The failure mode is specific and identifiable, which is unusual for a detailing defect.

A bar stopped too early does not fail in bending. It fails in bond, at the cut-off point, and what appears is a diagonal crack starting at the end of the curtailed bar and running up into the beam — because the sudden loss of tension chord there is exactly the discontinuity a diagonal crack forms at. The crack is often mistaken for a shear crack, which in a sense it is, and blamed on the links.

The distinguishing feature is its location: it starts at the bar end, which is a position recorded on the bending schedule and nowhere else. A diagonal crack whose foot coincides with a curtailment point is a curtailment failure, and it is one of the few cracks whose cause can be identified from a photograph and a bar bending schedule.

The remedy in codes is to require additional links over a length either side of every curtailment, which is an admission that the cut-off is a discontinuity even when it is correctly placed.

Steel plate girders, where the same rule wears different clothes #

Curtailing a flange plate on a welded girder is the same decision with two differences worth noting.

The geometry is identical — the moment diagram is the same parabola and the same square-root law applies — and the shift is not. Steel beams are not trusses in the reinforced concrete sense; the web carries shear by shear rather than by diagonal compression in struts, so the chord force is $M/z$ and the cut-off point is where the moment diagram says.

But something replaces the shift, and it is worse. A curtailed flange plate ends at a transverse weld, which is a fatigue detail of low category placed at a point where the stress is by definition close to the capacity of what remains. So the plate is extended past the theoretical cut-off, not to develop a force, but to reach a section where the stress range is low enough for the detail category to accept.

One material extends the bar to develop a force and the other extends the plate to reduce a stress range, and both arrive at a tail of comparable length. The rules look unrelated and the effect on the drawing is the same.

The check that could refuse it #

The claim that the shift and the truss chord force are the same statement is one this page can be wrong about, and it is worth writing down what would show it.

Compute the chord force two ways along the beam: once as $M(x)/z + \tfrac12 V(x)(\cot\theta - \cot\alpha)$ straight from the truss, and once by reading the moment diagram $a_l$ further along. If the two constructions were merely similar the difference would grow with the shear, and it would be largest at the supports where the shear is largest.

They differ by 0.36 per cent of the peak chord force, everywhere, and the residual is a parabola rather than something proportional to the shear. That is the second-order term: $M(x + a_l) = M(x) + a_l V(x) - a_l^2 w/2$, and the last term is the whole of the disagreement. So the shift is not an approximation to the truss result; it is the truss result, exactly, up to a term of order $a_l^2$.

If the demand curve had come from a load pattern for which the moment diagram is not smooth — a point load, where $V$ jumps — the two would differ at that section by a finite amount, and the correct construction is the truss one. The shift is the convenient form of a statement about shear, and where the shear is discontinuous it is the shear that is right.

Where the model stops #

The moment diagram is a parabola. The square-root law is exact for a uniform load on a simple span and approximate for anything else. A point load makes it a triangle, for which the bar length is $L(1-r)$ — linear, and curtailment is worth much more.

The shift assumes a constant strut angle. A variable-angle truss has a shift that varies along the beam, and the constant value is the one at the section being checked.

Anchorage length is a design value, not a measurement. It depends on cover, on bar spacing, on transverse reinforcement and on whether the bar is in a good bond zone, and the factors between best and worst case span more than two to one.

And nothing here prices the drawing. A beam with four curtailment points has more bar marks, more chances of an error and more site queries than one with none, and the saving computed above is in kilograms rather than in money.

Where the ladder goes #

Later rungs on this anchor: curtailment in continuous beams against a moment envelope, and the pattern loading that sets it. The shift for variable strut angles and inclined links. Curtailment interaction with shear at the cut-off, and the additional links that follow. Flange curtailment in plate girders and the fatigue category at the transverse weld. Curtailment in slabs, where the same arithmetic runs in two directions. Anchorage at supports, and the quarter of the steel that has to arrive. Lapping as the inverse operation. And the question underneath: whether a saving of two per cent of the steel is worth a discontinuity in the tension chord at all.