The relief that belongs to the pressure line

A tendon draped through a beam pushes up on the concrete wherever it curves and pulls along its own line where it is anchored, and read as a load it is simply another set of forces on the beam. In a continuous beam those forces try to lift the beam off its interior supports, the supports hold it down, and the prestress pushes back: reactions appear with no load applied, and the moment they make is the secondary moment. The tendon that can be moved showed that this division is bookkeeping — raise the tendon over the support without changing its drape, and the primary moment falls by exactly what the secondary moment rises by, because only the pressure line, where the prestress’s resultant actually acts, is real.

Both essays were about moment. A beam is also checked in shear, and the shear check has its own prestress term: the vertical component of an inclined tendon, $P \sin\theta$, carries part of the shear directly and is subtracted from the shear the links must resist. That term is the shear version of the primary moment — the tendon’s own force, read off the tendon’s own line. The question this essay asks is what the shear version of the secondary moment is, and whether a check that leaves it out is reading the right line.

A constant in each span #

Shear is the slope of the moment diagram — the diagram is an integral, read the other way — so it divides exactly as the moment does. The primary moment is $-P\,e(x)$, where $e$ is the tendon’s distance below the centroid; its slope is $-P\,e'(x)$, the tendon’s vertical component. The secondary moment is made by reactions alone, which are point forces at the supports, so between supports it is a straight line. Its slope is therefore a constant in each span:

$$V_{\text{sec}} = \frac{M_{\text{sec,right}} - M_{\text{sec,left}}}{L}.$$

In a two-span beam the secondary moment is zero at the two ends and some value $M_2$ over the middle support, so the secondary shear is $M_2/L$ in the first span and $-M_2/L$ in the second — the secondary reaction at each end support, seen from inside the span.

Take two 16 m spans, a 400 by 900 mm rectangle, prestressed with 2,400 kN on a parabolic tendon that runs from the centroid at each end to 320 mm below it at midspan and 300 mm above it over the middle support. The drape in each span is 470 mm, and the tendon’s equivalent upward load is $8Pd/L^2 = 35.3$ kN/m. The beam carries 45 kN/m.

The secondary moment over the middle support is 408 kN·m, so the secondary shear is $408/16 = 26$ kN, constant along the whole first span. Beside it the tendon’s own shear is large and steep — 209 kN of relief 0.8 m inside the end support, where the tendon falls at 87 mm per metre, and 299 kN 0.8 m inside the middle support, where it climbs to meet its high point.

The total — computed not by adding the two parts but from the equivalent load acting on the continuous beam, which is the honest way to get it — is their sum to the last kilonewton: −183 kN beside the end support, 324 beside the middle. Next to the end support the secondary shear opposes the tendon; next to the middle support it helps. That is the same sign pattern as the reactions: the end supports are pulled down onto the beam by a prestress that lifts it off the middle, so near the ends they push back against the tendon’s relief, and near the middle they add to it.

The slope that carries the shear #

There is a neater way to say the same thing, and it is the way that generalises. The total prestress moment at a section is $-P$ times the offset of the pressure line, because the pressure line is by definition where a force $P$ would have to act to produce that moment. Differentiate, and the total prestress shear is $-P$ times the slope of the pressure line.

So the relief a prestressed section actually receives is $P \sin\theta$ with $\theta$ read from the pressure line, not from the tendon. In a beam on two supports the two lines coincide and so do the two answers, which is why the rule is written the way it is. In a continuous beam the pressure line is the tendon shifted by the secondary moment over $P$ — upward here, since the secondary moment is sagging — and since that shift grows linearly from nothing at the end support to 170 mm over the middle one, the pressure line falls less steeply than the tendon all the way along the end span. Next to the end support it falls at 76.4 mm per metre against the tendon’s 87.0, and $2{,}400 \times 0.0764 = 183$ kN, where $2{,}400 \times 0.0870 = 209$.

The rule reads the wrong line, and it reads it with a precise error: the slope of a straight shift, which is the secondary shear.

Move the tendon and the two shears trade #

The theorem of linear transformation said that moving a tendon’s support ordinates without changing its drape leaves the total moment unchanged. Its derivative says the same of the shear, and the figure shows it happening. Lower the tendon over the middle support from 420 mm above the centroid to 105 mm above it, holding the drape at 470 mm in each span, and the tendon’s slope beside the end support steepens: its vertical component rises from 191 kN to 238. The secondary shear rises from 7 to 55 kN at the same rate. The total does not move from 183.

So the tendon’s vertical component is not a property of the beam. Two tendons with the same drape are, structurally, the same prestress — they have the same equivalent load, the same pressure line and the same total moment and shear everywhere — and a check that credits $P\sin\theta$ from the tendon gives them different relief. The second profile used in the earlier essay, 100 mm above the centroid over the support and 420 below at midspan, is one of them: its tendon gives 239 kN of relief beside the end support, its secondary shear takes back 56, and the beam is in exactly the same state as before.

The one profile for which the tendon’s slope is the whole answer is the concordant one, with no secondary moment and therefore no secondary shear. For this beam it would need the tendon about 470 mm above the centroid over the middle support — above the top face of the section — so it cannot be built. The secondary shear never quite reaches zero across the range of real tendons; it is smallest, 7 kN, with the tendon as high over the support as cover allows.

Where the omission errs, and which way #

The design question is the shear the links must carry, and that is the applied shear less the prestress relief. The applied 45 kN/m gives 234 kN 0.8 m inside each end support and 414 kN 0.8 m either side of the middle one. Subtracting the tendon’s whole shear leaves 51 kN at the ends and 90 at the middle. Subtracting only its vertical component leaves 25 at the ends and 115 at the middle.

So the omission is unconservative in one place and conservative in the other, and the unconservative place is not where anyone would look. The middle support is where the applied shear is largest, and the slope-only check is safe there by 25 kN. The end support is where the applied shear is smaller and the tendon’s relief nearly cancels it, and there the slope-only check finds half the shear that is actually present. Run the same check with the transformed tendon — same beam, same everything — and the slope-only figure at the end is −5 kN: the rule reports that the tendon carries more than the whole applied shear, and the links have nothing to do, while 51 kN remains.

That is the usual shape of a term left out of a statically indeterminate structure. The reactions that make the secondary shear sum to nothing, so a correction that helps at one support must hurt at another; a design governed by the large shear at the middle support never sees the small one at the end change sign.

A constant matters most where the rest cancels #

The secondary shear does not change with the applied load — it is made by the prestress and the supports, and the load does not enter it. So against the load the true net shear and the slope-only figure are two parallel lines 26 kN apart. What changes is the scale they are being compared with.

At 35.3 kN/m the applied load exactly balances the tendon’s equivalent load, the beam carries no net load at all, and the net shear is zero everywhere — that is what a balanced load means. The slope-only count does not see the balance. It reaches zero at 40.2 kN/m instead, and between the two it reports a shear of the wrong sign. Near any balanced state the error is a large fraction of the answer, because the answer is small and the error is not.

That is the general lesson of a constant correction. A term that is a ninth of the applied shear is half of the net shear when the net shear is what is left after a large cancellation, and prestressing is a design method built entirely on large cancellations. The end support of a continuous prestressed beam is often lightly linked for exactly this reason: the tendon’s slope seems to have done most of the work. A ninth of the applied shear is then not a rounding.

When it changes a design, and when it does not #

It is worth being exact about what the error buys in this particular beam, because the answer is: nothing. A 400 by 900 mm rectangle under 6.7 N/mm² of average prestress carries about 488 kN in shear on its concrete alone, by the expression EN 1992-1-1 gives for a section without links, and neither 51 kN nor 25 comes near it. Both answers call for minimum links. The rectangle is a fat web, and fat webs forgive a term of this size.

The term does not scale with the web, though, and that is where it starts to matter. The secondary shear is the secondary moment over the span, and the secondary moment is set by the prestress force and how far the tendon’s profile is from concordant — nothing about the width of the concrete enters it. Put the same tendon in an I-girder whose web is 150 mm thick and the concrete’s own share falls roughly in proportion to the web, to something under 200 kN, while the secondary shear stays at 26. Raise the load so the net shear beside the end support is a few hundred kilonewtons and the 26 is a tenth of the check, in the direction that is not conservative.

Two other places make it decisive at any web width. One is the check for cracking of the web, which in a prestressed beam is a principal-stress check made on the uncracked section under service load: there the shear enters directly, the margin is a fraction of the concrete’s tensile strength, and a shear understated by half beside the end support is a web that cracks when the calculation says it will not. The other is any temporary stage — transfer, with only the self-weight acting, or a propped construction stage — when the applied shear is small and the net shear beside the end support is close to zero or reversed. A correction that is fixed in size is decisive wherever the quantity it corrects is small, and in a prestressed beam the net shear beside an end support is small by design.

The middle span has none #

In a beam of three spans the pattern sharpens. With spans of 16, 20 and 16 m and the same tendon, the secondary moment is 319 kN·m over each interior support, so across the middle span it is a level line and the middle span carries no secondary shear at all: 20 kN in the end spans, zero in the middle. The tendon’s slope is exactly right in the middle span and wrong only beside the ends — which again is where the applied shear and the tendon’s relief come closest to cancelling.

A beam that is not symmetric — unequal end spans, or a tendon stressed from one end with more friction loss toward the other — has different secondary moments over its interior supports, and its middle spans then carry secondary shears too. The rule is general: in every span the secondary shear is the difference between the secondary moments at its ends, over its length, and it is zero only where they are equal.

Two routes to one number #

The figures compute the total prestress shear twice, and the two routes are worth stating because each is a hand method. The first is the split: the tendon’s slope, $-P\,e'(x)$, plus the secondary constant, $(M_{\text{sec},R} - M_{\text{sec},L})/L$. The second takes the tendon as its equivalent load — 35.3 kN/m upward in each span, with forces at the anchorages and over the middle support where the tendon kinks — and finds the shear in the continuous beam from the total prestress moments at the supports, $(M_R - M_L)/L$ plus the equivalent load’s own share, exactly as for any continuous beam under any distributed load. The two agree everywhere to rounding.

By hand at the end support: the tendon’s slope there is $4d/L$ plus the chord’s slope, $4 \times 0.47/16 - 0.30/16 = 0.0988$, falling to 0.0870 at 0.8 m, and $2{,}400 \times 0.0870 = 209$ kN. The secondary moment over the middle support, 408 kN·m, is the total prestress moment there, 1,128 kN·m of sagging, less the primary moment $-P\,e = 2{,}400 \times 0.30 = 720$ kN·m that a tendon 300 mm above the centroid makes on its own, and $408/16 = 25.5$ kN. The applied shear at 0.8 m is $\tfrac{3}{8} \times 45 \times 16 - 45 \times 0.8 = 234$ kN. So the links carry $234 - 209 + 26 = 51$ kN.

The beam behind the numbers #

Elastic and uncracked. The secondary moment is computed for a beam of uniform stiffness, which is what the beam is in service. At the ultimate limit state a cracked beam redistributes, and how much of the secondary moment survives to collapse is a separate question that design rules answer with a partial factor and an allowance. The secondary shear inherits whatever answer is given for the moment it is the slope of.

The prestress is the same force everywhere. Real tendons lose force to friction along their length and at every deviation, so $P$ varies and the equivalent load is not uniform. That changes the numbers and not the structure of the argument: the secondary moment is still made only by reactions, so it is still straight between supports, and its slope is still a constant in each span.

A parabolic tendon with a sharp kink over the middle support. Real tendons are draped over a support on a short reverse curve, which spreads the kink force over a metre or two and rounds the primary shear’s jump there. The secondary shear does not notice: it depends on the support moments, not on how the tendon gets from one to the next.

The web, the anchorage and creep #

They cannot show where the shear goes once the web cracks. A prestressed web often does not crack at forty-five degrees, and the links needed for a given shear depend on the angle at which the truss model is drawn, which is a choice rather than a property. The shear to be carried is the input to that choice; this essay is about getting the input right.

They cannot show the anchorage. At the end support the tendon’s vertical component enters the concrete at the anchorage, as a concentrated force in a region that is not a beam at all, and the shear there is handled by the end block’s own reinforcement. The check section a small distance inside the support is the first place where beam theory applies, which is why the figures read the shear there and not at the support itself.

And they cannot show time. Creep relaxes the prestress and, in a beam made continuous after it was stressed, changes the secondary moment itself as the structure creeps toward the state it would have had if it had been built continuous. The secondary shear moves with it.

The pressure line’s slope, not the tendon’s #

Secondary shear is a constant in each span — the difference of the secondary moments at its ends over its length, which in an end span is the secondary reaction at the end support.

The relief a prestressed beam receives is P times the slope of the pressure line, not of the tendon. The two coincide only in a beam with no secondary moment.

Counting the tendon’s slope alone errs both ways. Beside the middle support of these two spans it overstates the shear by 25 kN; beside the end supports it understates it by the same amount, and there that is half of the 51 kN actually present.

Moving the tendon trades the two parts and leaves the sum. A check that credits the tendon’s slope gives two identical beams different relief, and can report that an end support needs no links at all.

Still open: the beam stressed one bay at a time #

Every beam here is stressed in one operation, with the whole tendon at its full force before any load arrives. A long continuous slab or beam is usually built and stressed bay by bay, each new length of tendon coupled to the last, and at each stage the structure is a different continuous beam with its own supports and its own secondary moments. What the finished beam carries is the sum of the stages, not the single-stage answer, and the secondary shear in the first span — which was stressed when it was the whole structure — need not be the same constant it would be if the beam had been stressed all at once. Whether the stage-by-stage secondary shear beside an end support is larger or smaller than the one-stage answer, and whether the creep that follows moves it back, is the question a bay-by-bay construction sequence asks of this one.