Four inequalities and a wedge

A prestressed beam is checked twice, against two different beams.

The first is the beam on the day it is stressed. The force is at its largest, nothing is on the member except its own weight, and the tendon is low — so the bottom fibre is in heavy compression and the top fibre may go into tension, which is the opposite of the state the beam is being built for.

The second is the beam in service, years later. The force has fallen by a fifth as the steel relaxes and the concrete creeps and shrinks; the full load is on; and now the top is in compression and the bottom is the fibre that may crack.

Four limits, then: a tension limit and a compression limit at each of the two moments. And they are simultaneous, which is what makes the design something other than a search.

Which free body produced the number #

Cut the section and write the stress at each face as the sum of three contributions: the axial effect of the prestress, its moment about the centroid, and the applied moment.

$$\sigma_{top} = \frac{P}{A} - \frac{Pe}{Z_t} + \frac{M}{Z_t}, \qquad \sigma_{bot} = \frac{P}{A} + \frac{Pe}{Z_b} - \frac{M}{Z_b}$$

with compression positive and $e$ measured downward from the centroid. Nothing here is new: it is the same superposition that every prestressed section is built on.

What is new is the rearrangement. Each of the four limits is a statement of the form $P\,g \geq R$ or $P\,g \leq R$, where $g$ is the bracket the force is multiplied by at that fibre and $R$ collects the stress limit and the applied moment. Divide through by $P$ and by $R$ and every one becomes a bound on $1/P$ that is linear in $e$.

That is the whole trick, and it is due to Magnel. A two-variable problem in which one variable appears in a product with the other becomes a two-variable problem in which both appear linearly, and a linear problem in two variables is a drawing.

The signs are the whole of the difficulty #

Written out, the four limits look symmetric. They are not, and the asymmetry is where a careless implementation goes wrong.

The bracket $g$ at the top fibre is $1/A - e/Z_t$, which is positive for a small eccentricity and negative past the kern. The right-hand side of the transfer top-fibre limit is $f_{ti} - M_0/Z_t$, which is negative for any ordinary beam because the permitted tension is small and the self-weight moment is not. Dividing an inequality by a negative quantity reverses it, and both of those quantities change sign somewhere in the range being plotted.

The consequence is that a line’s sense — whether the feasible side is above it or below it — is not fixed. It depends on the eccentricity being examined. An implementation that decides once which side of each line is allowed and applies that decision everywhere produces an empty region for a section that plainly works, which is exactly what the first version of this figure did.

The general lesson transfers well past prestressing: an inequality divided by a variable is not one inequality. It is one inequality on each interval where that variable holds its sign.

The region is a wedge, and it opens to the right #

The four lines on the figure make a region that is bounded on the left and open to the right, closing only when the eccentricity leaves the section.

That shape has a practical consequence worth stating on its own: the cheapest prestress is always at the largest eccentricity available. On the section drawn, at $e = 0$ the smallest acceptable force is 2,734 kN; at 200 mm it is 1,495; at 400 mm it is 1,029. The curve is monotonic and there is no interior optimum to find.

So the design decision is not “what force and what eccentricity” but “how low can the tendon go”, and the answer to that comes from somewhere the diagram cannot see: the cover, the duct diameter, the space for the anchorage and the shear links. The structural optimum is at the geometric limit, and a prestressed beam is a member whose most important dimension is chosen by detailing.

Losses move one pair of lines and not the other #

The two service limits are written at $\eta P$ rather than $P$, and $\eta$ — here 0.8 — is the fraction of the jacking force still present years later.

That factor moves the two service lines and leaves the two transfer lines alone, which is the geometric statement of a physical one: the losses are what stop the section being designed for one state. If nothing were lost, the transfer condition and the service condition would be the same beam at two loads, and the design would be a single check at the worse one.

The losses come from four places and they add: the steel relaxes at constant strain, the concrete creeps under sustained compression, it shrinks whether loaded or not, and the elastic shortening at transfer takes a share immediately. Every one of them is an imposed deformation rather than a load, and every one of them is estimated rather than known.

An error in $\eta$ therefore moves two of the four lines and can close the region from the right. That is a genuinely awkward sensitivity, because $\eta$ is the least reliable number in the calculation.

The size of it is worth drawing rather than asserting, because the diagram converts a guess about time into a force in kilonewtons.

Fifteen points is not a pessimistic margin on a loss estimate. Relaxation, creep and shrinkage are three curves against time with no change of load behind any of them, and each carries its own scatter; a design that sits comfortably at 20 per cent and has no room at 35 has been designed against an assumption rather than against a section.

The failure mode is an empty region #

Raise the service moment and watch the region rather than the stresses.

The two service lines move toward the two transfer lines, the wedge narrows, and at some moment it closes. Past that point there is no force at any eccentricity that satisfies all four. The section has failed, and it has failed without any individual stress being dramatically exceeded — the transfer pair and the service pair have simply stopped overlapping.

That is an unusual kind of failure in this collection, where nearly everything fails by a demand exceeding a capacity. Here the demand and the capacity are fine separately and the problem has no solution, which is a statement about the section’s geometry: $Z_t$ and $Z_b$ are too small relative to the difference between $M_s$ and $M_0$.

The algebra says so directly. Subtracting the transfer top limit from the service top limit eliminates $P$ and $e$ entirely and leaves an inequality containing only the section moduli, the two moments, the losses and the stress limits — a condition on the section alone, before any tendon is considered. A section that fails it cannot be prestressed at any force whatever.

Why the limits are stresses at all #

Everything above is a serviceability calculation dressed as a strength one, and it is worth being explicit about that because it is unusual.

The ultimate strength of a prestressed beam is decided by a completely separate computation — a fibre-by-fibre equilibrium of a section with a bonded tendon at yield, which owes nothing to the four inequalities. A beam can pass all four comfortably and be short of ultimate capacity, or fail them and have strength to spare.

What the four limits control is cracking and the appearance of the member, and the reason they are given the leading role is history plus consequence. A prestressed member that cracks under service load has lost the property it was built for: the tendon is exposed to a crack width and an environment, and the section’s stiffness has dropped to the cracked value with the deflection that implies. Keeping the bottom fibre in compression, or nearly so, is what “prestressed” means operationally.

The state the four limits exist to avoid is not collapse. Once the bottom fibre cracks, the section’s stiffness falls by a factor of two or three and the neutral axis moves up; the prestress still works, the tendon is still nowhere near yield, and the beam is still safe against any strength criterion. What has gone is the arithmetic. Every serviceability number on this page was computed on an uncracked section with a single modulus, and none of them survives the crack: the deflection is larger than predicted, the stresses are redistributed, and the four inequalities are being applied to section properties the member no longer has. That is why the limits are stresses — they are the conditions under which the rest of the calculation remains true of the beam it describes.

A diagram that predates the computation #

Magnel’s construction is from the nineteen-forties, and it belongs to the same tradition as the funicular polygon and the force diagram: a drawing that is the calculation rather than a picture of it.

It survives the arrival of computers better than most of that tradition, and the reason is that it answers a question a solver does not. A numerical search returns one acceptable pair, chosen by whatever objective was supplied. The diagram returns the whole set — and with it, immediately and without further work, how much room there is, which limit is binding, which direction to move if a duct will not fit, and whether the section works at all.

That is the difference between an answer and an understanding of the answer’s neighbourhood, and it is why the figure is still drawn.

What the region is worth knowing beyond the answer #

A design that returns a single acceptable force answers the question and hides three things the region makes obvious.

Which limit is binding, and therefore which change helps. At the chosen point one of the four lines is touching and three are not. If it is the service bottom-fibre limit, more force helps and a lower tendon helps; if it is the transfer bottom-fibre limit, more force makes it worse and the answer is to stress in stages or to wait for the concrete to gain strength. Those are opposite actions, and nothing about the phrase “the section fails a stress check” says which is called for.

How much room there is. A point in the middle of a wide wedge is a design that will survive a duct being moved 20 mm, a concrete strength coming in low, or a loss estimate being optimistic. A point wedged into a corner is a design in which any of those closes the region. The distance to the nearest line is a tolerance, and it is a number a single answer does not carry.

Whether the problem was worth posing. A region that is wide open says the section is generous and the force could be cut, or the section made shallower. A region that is a sliver says the section is at its limit and the next revision of the load will not fit.

The same three observations are available from any feasible-region drawing — a $M$–$N$ interaction diagram carries them too — and it is the reason those drawings survive in a subject that could compute a point instead.

The fifth inequality, which is not a stress #

There is a constraint on the same diagram that none of the four lines represents, and on a large proportion of real sections it is the one that binds.

The tendon has to fit inside the beam. The eccentricity is a distance from the centroid to the tendon’s own centre, and that distance is limited by the geometry: the cover to the duct, half the duct’s diameter, the links passing outside it, the clearance for a vibrator, and whatever the anchorage hardware needs at the ends. On a 700 mm deep section with 40 mm cover, 10 mm links and a 70 mm duct, the tendon’s centre cannot get closer than about 90 mm to the soffit — so with the centroid at mid-depth the eccentricity cannot exceed 260 mm, whatever the stresses would like.

On the Magnel diagram that is a vertical line, at $e = e_{\max}$, and everything to the right of it is unbuildable. It is not a stress condition, it carries no material property, and it is drawn on the same axes as the four that do.

The consequence is a different design action from any of the four. If the wedge is closed because two stress limits have crossed, the remedies are a stronger concrete, a different loss estimate, or stressing in stages. If the wedge is open but lies entirely beyond $e_{\max}$, none of those helps: the section is asking for a lever arm it does not have, and the only answers are a deeper section or a second tendon at a different level.

Which is why the geometric limit is worth drawing rather than checking afterwards. A designer who solves the four inequalities, finds a comfortable answer, and then discovers the tendon does not fit has to start again from a different section — and the diagram would have said so before any arithmetic.

There is a second geometric constraint of the same kind and it applies from the other side. The tendon must also stay far enough below the top face for the same reasons, so there is an upper vertical line too — irrelevant at mid-span, where nobody wants the tendon high, and decisive over an interior support of a continuous member, where the profile does.

It binds hardest at the ends. There the applied moment is zero, so the stress limits want the tendon near the centroid; and the anchorage hardware, the bursting reinforcement and the end block’s own congestion also want it near the centroid. The two agree, which is convenient — and it is the reason the tendon is lifted at the ends for two independent reasons that are usually credited to one.

Where the model stops #

The section is uncracked and linear-elastic throughout. Every stress above comes from $P/A \pm Pe/Z \mp M/Z$, which needs a homogeneous section with a single modulus. A composite member built in stages, or one whose slab is cast later, has different section properties for different parts of the load, and the four limits become six or eight written on different sections.

The losses are a single factor. They are not uniform along the member: friction along a curved duct means the force at mid-span is lower than at the anchorage, so $\eta$ is a function of position and the diagram is different at every station.

And the two load cases are the extremes. A beam that is stressed, stored, transported and only then loaded may have a worse case than either — a young concrete, full prestress, and a support position it was never designed for.

The tendon’s own position is measured from the same centroid the section moduli were computed about, and that centroid moves as soon as the section is not the one drawn. A precast beam with an in-situ topping has one centroid before the topping is cast and another afterwards, and the eccentricity that appears in all four inequalities is the distance to whichever of them applies to the load case in hand. Getting that wrong is not a small error: on a beam of this depth the two centroids can be eighty millimetres apart, which is a quarter of the eccentricity the whole design is bought with.

What the picture cannot show #

The diagram is drawn at one section, and a beam has a continuum of them. The mid-span section governs the choice of force, but the region at a support is a different shape entirely — no applied moment, full prestress, and a tendon that has been lifted toward the centroid to keep the top fibre out of tension. A tendon profile is the answer to the whole family of diagrams at once, and this figure is a single frame of it.

Nor can it show the ultimate limit state, which is where the beam’s safety is actually decided. Everything on the page is about the beam at working load.

The assumption the figure rests on #

The section moduli are constants, and for a symmetric section $Z_t = Z_b$ so the four lines come in two parallel pairs. Real prestressed sections are almost never symmetric: a tee or a girder with a wide top flange has $Z_b$ well below $Z_t$, the lines lose their symmetry, and the wedge tilts. The consequences are not cosmetic — an unsymmetric section changes which limit binds first, and can make the transfer condition rather than the service one the one that fixes the force.

The ladder from here #

Later rungs on this anchor: the unsymmetric section, where the four lines lose their pairing and the transfer case starts to govern. The tendon zone along the member, which is the diagram solved at every station at once. Partial prestressing, where some tension is deliberately allowed at service and the four limits become three plus a crack-width calculation. Friction and the force that varies along the duct. And the continuous member, where the secondary moments join the applied ones on the right-hand side of every inequality and the tendon’s own profile changes them.