The flange the prestress cannot use

Four inequalities and a wedge drew the four stress limits on a prestressed section — tension and compression at transfer, when the force is largest and only the beam’s own weight acts, and tension and compression in service, when a fifth of the force has been lost and the full moment arrives — as four lines on a plot of $1/P$ against the eccentricity, and found the prestresses that work as the wedge between them. Its section was a symmetric rectangle. It named the case it had not drawn: “the unsymmetric section, where the four lines lose their pairing and the transfer case starts to govern.”

That sentence contains the whole of this essay, once “pairing” is read the right way. The four lines pair up, and they pair by fibre.

Two fibres, two pairs #

Write the bottom fibre’s two limits side by side. At transfer the prestress $P$ at eccentricity $e$ and the self-weight moment $M_0$ must not crush it:

$$P\left(\frac1A + \frac{e}{Z_b}\right) - \frac{M_0}{Z_b} \le f_{ci}$$

and in service the reduced prestress $\eta P$ and the full moment $M_s$ must not crack it:

$$\eta P\left(\frac1A + \frac{e}{Z_b}\right) - \frac{M_s}{Z_b} \ge f_{ts}$$

with tension negative. Both contain the same bracket — the compression the prestress puts on the bottom fibre — and multiplying the first by $\eta$ and setting the two against each other removes $P$ and $e$ altogether:

$$M_s - \eta M_0 \le Z_b\,(\eta f_{ci} - f_{ts})$$

The same two steps on the top fibre’s pair give $M_s - \eta M_0 \le Z_t\,(f_{cs} - \eta f_{ti})$. Each fibre can accept a range of moment, set by its own modulus and its own pair of stress limits, and no choice of force or eccentricity can widen it. The Magnel wedge exists only when both fibres can span the moment range $M_s - \eta M_0$, and the eccentricity and force decide only where in the wedge the design sits.

For a symmetric section the two moduli are equal, and the two conditions differ only through their stress ranges — here $\eta f_{ci} - f_{ts} = 0.8 \times 18 + 1.6 = 16.0$ N/mm² at the bottom and $f_{cs} - \eta f_{ti} = 20 + 0.8 \times 2 = 21.6$ at the top. The bottom fibre is the tighter, but not by much. On an unsymmetric section the moduli differ as well, and whichever fibre is short of modulus closes the wedge on its own.

It is worth saying what the condition is, physically, because it is not a strength. The prestress puts a fixed compression on the bottom fibre, and the load takes some of it away. At transfer only the self-weight is there to take any away, so the soffit carries nearly all of it and must not crush; in service the whole moment is there, so the soffit carries what is left and must not crack. The difference between those two stresses is set by the moments alone — $(M_s - \eta M_0)/Z_b$ — and whatever prestress is chosen, it has to fit between the two limits. The load put on backwards described prestress as a load applied in advance, in the opposite direction to the one that will come; the fibre condition is the statement that the advance load cannot be tuned to two occasions at once unless the fibre is stiff enough in bending that the occasions differ by less than the limits allow.

The eccentricity’s job is different. The middle third located the kern, the zone inside which a compressive force leaves the whole section in compression, and the Magnel wedge’s cheapest point sits where the tendon is as far below the kern as the cover allows, because every millimetre of lever arm is prestress not bought. That is a question of economy. Whether there is any wedge at all is the fibre condition’s question, and the tendon cannot answer it.

Three sections, one depth #

Three sections of the same depth make the point. A symmetric I-section with 400 mm flanges. A tee with a 1,000 mm top flange and a 200 mm web, the shape a precast floor unit or a bridge beam with its deck takes. And a bulb-tee, a top flange of 800 mm with a 450 mm bulb at the soffit, which is what prestressed bridge beams actually look like. The tee has the most concrete, 44 per cent more than the I-section; its centroid is high, because the flange is at the top, so its top modulus is large and its bottom modulus the smallest of the three.

All three span 12 m and carry their own weight and 20 kN/m more.

The I-section has a region, and the cheapest point in it is where it always is, at the largest eccentricity the cover allows: 1,147 kN at 270 mm below the centroid.

The number can be checked against the limit that sets it. At 270 mm, the bracket $1/A + e/Z_b$ is $5.56 + 9.36 = 14.9 \times 10^{-6}$ per mm², so a unit of force puts 14.9 N/mm² per MN on the soffit. The service moment of 441 kN·m puts 15.3 N/mm² of tension there, of which 1.6 is allowed to remain, so the service force must supply 13.7 N/mm², which is 918 kN; with a fifth lost, 1,147 kN at transfer. At transfer that force puts 17.1 N/mm² on the soffit, and the self-weight’s 2.8 brings it to 14.3 against the 18 allowed. At the top fibre the force pulls 4.4 N/mm² of tension and the self-weight pushes back 2.8, leaving 1.6 of tension against the 2.0 allowed. Three of the four limits have room; the service tension at the soffit is the one that binds, which is the usual state of a section with a generous wedge.

The tee has none. Its four lines cross in the wrong order and there is no force, at any eccentricity, that satisfies them all. Nothing about the tee looks inadequate: it is heavier, it is stiffer than the I-section, and it has more bending strength in every ordinary sense. It fails in the way the earlier essay singled out as unusual: the demand and the capacity are fine separately, and the problem has no solution. That essay put the cause in the section moduli being too small relative to the difference between the two moments. The fibre conditions say which modulus, and by how much.

Stiffer, heavier, and the wrong shape #

The tee’s second moment of area is its bottom modulus times the distance from the centroid to the soffit: $22.7 \times 10^6 \times 477 = 10.8 \times 10^9$ mm⁴, against the I-section’s $28.9 \times 10^6 \times 350 = 10.1 \times 10^9$. The tee deflects less under the same load, and in steel, where depth is the cheapest strength and a section is judged by the smaller of its moduli and the yield stress, the comparison would stop there and call the tee slightly better.

A prestressed section is not judged by its smaller modulus. Two strengths, depending which way up drew the unsymmetric section’s two moduli as two different bending strengths, one for each direction of moment, and the reason prestress makes both of them matter at once is that it reverses the sign of the stress at each fibre between two occasions. The top fibre must hold a range; so must the bottom; and each range is divided by its own modulus. The tee’s flange has made one fibre strong in the way that stiffness counts and left the other as it was.

Its kern tells the same story in a different register. The tee’s lower kern point is $Z_t/A = 187$ mm below its centroid, further down than the I-section’s 160, and its tendon can sit 397 mm below the centroid against the I-section’s 270. By every measure of the eccentricity available, the tee is the better prestressed section. None of them enters the fibre condition.

Each fibre has to earn its own prestress #

The fibre conditions say why at a glance. Every section’s top fibre has modulus to spare: the tee’s is nearly three times what its pair of limits needs. Every section’s bottom fibre is the tighter one, and the tee’s is short: 22.7 against 24.0. The tee’s flange puts its concrete where the prestress cannot use it. A wide top flange raises the moment of inertia and moves the centroid up, which raises the top modulus steeply and leaves the bottom modulus — the second moment divided by the now longer distance to the soffit — almost unchanged.

The pair that fails is the bottom fibre’s: the service limit wants enough prestress to hold the soffit out of tension under the full moment, and the transfer limit wants little enough that the soffit is not crushed when only the self-weight is there to relieve it. A tee’s small bottom modulus makes the soffit’s stress swing a long way between the two moments, and the swing is larger than the range between the stress limits. That is the precise sense in which the transfer case starts to govern: on the tee it is the transfer compression at the soffit, not the service tension, that the design cannot get past, since any prestress large enough for service crushes the soffit at transfer.

The fibre that does not notice the flange #

Widening the flange from nothing — a 200 by 700 mm rectangle — to 1,400 mm shows the two fibres parting company. The top fibre’s capacity climbs almost linearly with the flange, from 18.9 kN/m to 72.2. The bottom fibre’s barely moves: 13.8 kN/m for the rectangle, 18.9 at a 1,000 mm flange, 19.6 at 1,400 — seven times the flange for 41 per cent more load, and never reaching the 24.8 kN/m the symmetric I-section manages with less concrete than a 600 mm tee. The flange also adds weight, and the self-weight is part of the moment range the bottom fibre has to hold, which is why the curve flattens rather than climbing.

Where each section runs out #

Swept over the imposed load, all three sections need much the same prestress while they can be prestressed at all — the force is set by the moment and the lever arm, and the lever arms are similar — and they differ in where they stop. The tee’s region closes at 18 kN/m, the I-section’s at 24, and the bulb-tee’s at 32. The bulb-tee has less concrete than the tee and carries 75 per cent more, because it has put its extra concrete at the soffit, where the fibre that decides is. That is the reason prestressed bridge beams have bulbs: the bulb is not there for bending strength, which the deck provides, but for the bottom fibre’s modulus and for somewhere to put the strands.

The fibre condition, by hand #

The tee’s failure can be checked in three lines. Its self-weight is $0.26 \times 25 = 6.5$ kN/m, so $M_0 = 6.5 \times 12^2/8 = 117$ kN·m, and with the 20 kN/m imposed $M_s = 26.5 \times 12^2/8 = 477$ kN·m. The bottom fibre must hold a moment range of $M_s - \eta M_0 = 477 - 0.8 \times 117 = 383$ kN·m. Its modulus, 22.7 × 10⁶ mm³, times its stress range, 16.0 N/mm², is 363 kN·m. Short by twenty. The I-section, with $M_0 = 81$ and $M_s = 441$ kN·m, needs 376 kN·m of range and has $28.9 \times 16.0 = 462$.

The same three lines give the margin, which is what a designer actually wants from a check like this. The I-section has 86 kN·m to spare at its bottom fibre, about a fifth of its moment range; the bulb-tee, with $37.4 \times 16.0 = 598$ kN·m against about 380 needed, has 217, two and a half times the I-section’s margin. The tee is short by five per cent. A check that measures a margin in moment rather than stress is also one that responds to the load directly: each kN/m of imposed load on 12 m adds 18 kN·m to the range, so the I-section’s 86 kN·m of margin is worth about 4.8 kN/m, which is the step from the 20 kN/m it carries here to the 24.8 at which its region closes.

What a designer does about it #

The fibre condition says which remedies can work. Adding prestress cannot, and neither can moving the tendon, since neither appears in the condition. Raising the concrete grade at transfer can, because $f_{ci}$ is the limit that binds, which is why pretensioned beams are cast with high early strength and why the strength it had on the day of transfer is a design quantity rather than a site record. Adding self-weight moment at transfer can too — casting the deck before the strands are cut is not possible, but staging the prestress so that some is applied after the deck is, is common — because it reduces the moment range $M_s - \eta M_0$ the bottom fibre has to hold. And changing the section can, by moving concrete to the soffit.

What does not work is the instinct to add flange. A designer who finds a tee short of capacity and widens its flange, reasoning that a bigger section is a stronger one, has added to the fibre with capacity to spare and to the self-weight, and the section still has no Magnel region.

Where the model stops #

The limits are elastic stresses at two instants. Transfer and the full service moment are the two extremes of the life of a simply supported pretensioned beam; a post-tensioned beam stressed in stages, a continuous beam whose secondary moments enter both limits, or a composite beam whose deck is cast after transfer, has more than two instants, and each adds a pair of limits whose moment range is different.

The losses are a single fraction. Twenty per cent of the force is taken as lost between transfer and service. Losses vary along the member and with time, and a bottom fibre whose limits are nearly incompatible is sensitive to them: the moment range $M_s - \eta M_0$ grows as $\eta$ falls, so a section designed at 20 per cent loss and built at 25 has a narrower region than drawn. For the I-section the margin falls from 86 kN·m to 55, since the stress range $\eta f_{ci} - f_{ts}$ shrinks to 15.1 N/mm² at the same time as the moment range grows. Most of that loss is creep and shrinkage in the concrete, which belongs to the member rather than to the material, and whose share of the force depends on how early the beam was stressed.

The section is uncracked. The limits assume the soffit never cracks; a partially prestressed member, which is allowed to crack in service, replaces the service tension limit with a crack-width check and removes one line from the bottom fibre’s pair — which is often the practical way out for a tee.

What the pictures cannot show #

That the tee is usually not alone. A tee in a floor is one of many side by side, with a topping cast over them, and the topping makes each tee’s top flange part of a larger composite section whose centroid is higher still — the flange that is not all there is the reminder that how much of that flange works depends on the spacing. The composite section’s bottom modulus is the one that has to hold the service moment, while the precast tee alone has to hold transfer, so the bottom fibre’s two limits act on two different sections, and the pair condition becomes a condition on both. Two beams, or one beam four times as stiff showed that composite action multiplies a beam’s stiffness; for a prestressed tee under a topping it also moves the soffit’s service stress onto a larger bottom modulus, which relaxes the service half of the bottom fibre’s pair and leaves the transfer half exactly as severe as it was. And a topping cast on a beam that has already crept does not stay idle: the slab that shrinks onto a finished beam takes some of the soffit’s compression back with it.

Still open: the tendon zone along the member #

Everything here is one section, at mid-span, where the moment is largest. Towards the supports the moments fall, the moment range shrinks, and the bottom fibre’s condition loosens; near the supports the self-weight moment that relieved the soffit at transfer has gone, and a straight tendon at its mid-span eccentricity crushes the soffit or cracks the top. Whether a tee that fails at mid-span can be made to work by a tendon that deflects or debonds along its length, or whether the bottom fibre’s condition at mid-span is simply the governing statement for the whole member, is the question of the cable zone along the span — the four limits solved at every section at once.