The force that splits what it pushes on

A post-tensioned beam is stressed by pulling a tendon against the concrete and locking it off against a steel plate. On the beam drawn here the force is 1,200 kN and the plate is 200 mm deep in a section 700 mm deep. The bearing stress under the plate is 20 N/mm²; a metre further along the member the stress is 5.7 N/mm² and uniform.

Between those two stations the force has spread out, and spreading is not free. A compression field that starts narrow and ends wide has flow lines that bow outward, and a bowing compression needs something pulling inward to keep it in equilibrium. Nothing outside the beam is pulling. So the concrete does it, in tension, along the axis of the tendon — and concrete’s tensile strength is a tenth of its compressive one.

End blocks split. They split along the line of the tendon, in the first half-metre, and the reinforcement provided to stop them is designed for a force that appears in no equilibrium equation written on the member as a whole.

Which free body produced the number #

Take the half of the end block above the tendon’s axis, bounded by the loaded face, the top face and a section far enough along that the stress has become uniform.

Along the axis, symmetry says the shear is zero. The top face is free. At the loaded end the plate delivers $P/2$ over the top half of its depth, so that force’s resultant acts at $a/4$ above the axis. At the far end the same $P/2$ leaves uniformly over the top half of the section, with its resultant at $h/4$.

Sum forces along the member and nothing is learned: $P/2$ in, $P/2$ out. Sum moments and something is: the two resultants act at different heights, so their moments about any point differ by $(P/2)(h - a)/4$, and there is nothing left to balance it except the transverse stress on the axis.

That difference is the whole of the anchorage-zone problem, and it is a moment rather than a force — which is why the answer depends on the lever arm it is divided by.

What that free body has produced is a load path rather than a stress field, and it is licensed by the lower-bound argument this region belongs to: any equilibrium system the material can sustain is safe, whether or not the concrete has any intention of behaving that way. The anchorage zone is the smallest and the most inevitable of the regions that argument was written for. It is a disturbed region by construction, because the force enters through a plate.

The coefficient that is a lever arm #

Choose the tie’s position at half a section depth from the loaded face — which is a claim that the flow has spread by about that much by then — and the tie force follows from the node:

$$T = \frac{P}{2}\cdot\frac{(h-a)/4}{h/2} = \frac{P}{4}\left(1 - \frac{a}{h}\right)$$

For the block drawn: 214 kN, from an applied 1,200. That is Guyon’s expression, which has been in every prestressed-concrete text since the nineteen-fifties, and the derivation above is the whole of it. The 0.25 is $\tfrac12 \times \tfrac14 \times \tfrac{1}{1/2}$ — three geometric fractions, none of them measured.

Move it the other way and the arithmetic runs the other way with it. Nothing about the concrete changed in that direction either.

So the coefficient is not a measurement of anything. It is the reciprocal of a lever arm somebody chose, and the choice is a claim about how far the flow has spread by the station where the reinforcement is put.

What an actual equilibrium field says #

The strut-and-tie model is a lower bound and makes no claim to be the stress field. It is worth asking what a field that is one says, and the calculation needs no elasticity at all.

Assume how the longitudinal stress spreads — a block starting at the plate’s width and opening out to the full section over about a depth — and normalise it at every station so that it carries exactly $P$. That normalisation is what makes the assumption admissible rather than decorative. Then the two-dimensional equilibrium equations,

$$\frac{\partial\sigma_x}{\partial x} + \frac{\partial\tau}{\partial y} = 0, \qquad \frac{\partial\tau}{\partial x} + \frac{\partial\sigma_y}{\partial y} = 0$$

give the shear and the transverse stress by integrating down from the free face, with nothing else supplied.

What comes back has the shape every photoelastic study of an end block shows. Directly under the plate the concrete is in transverse compression — the plate is confining it. About a third of a depth in, the transverse stress crosses zero. It peaks in tension at 2.74 N/mm² a little under half a depth in, and decays to nothing by one depth.

The number is a property of the assumption #

Integrate the tensile part and it comes to 174 kN, against the strut-and-tie model’s 214.

Then change the assumed spread — keep the same section, the same plate and the same force, and only alter how fast the flow is imagined to open out. A gentle spread gives 96 kN. A fast one gives 198. All three satisfy equilibrium to machine precision, all three have zero stress on the free faces, and all three integrate to nothing across the axis. The bursting force is not a property of the block.

This is not a failure of the calculation; it is the reason the design is done the way it is. The tensile resultant depends on a flow nobody can observe, so the sensible thing is to pick a lower-bound tie, provide reinforcement for it, and detail that reinforcement so it works wherever the tension actually turns out to be. Guyon’s coefficient sits above all three computed values, which is what a design number should do.

Read the two figures together and the anchorage zone is the same phenomenon as any other stress concentration: a local disturbance that decays over a distance of the order of the feature that caused it. What is unusual here is only that the feature is the plate, so the disturbance decays over the section depth rather than over a hole’s radius — Saint-Venant’s principle with a number attached to it, and the number is the one the reinforcement has to reach past.

The same number as a deviation force #

There is a second derivation of Guyon’s coefficient that uses no prism, no node and no assumed flow, and it is worth having because it says which variable the answer really depends on.

A compression that changes direction pushes sideways. That is the load that comes from changing direction — the same statement that makes a curved tendon press on its duct — and it applies to a compression field just as it applies to a cable: the transverse force is the compression times the angle it turns through.

Here the compression turns because its resultant has to move. Half the force enters at $a/4$ above the axis and leaves at $h/4$, so over a spreading length $\ell$ it has been deviated through an angle

$$\theta \approx \frac{(h-a)/4}{\ell}$$

and the transverse force required to turn it is $(P/2)\theta$, giving

$$T = \frac{P\,(h-a)}{8\ell}$$

Put $\ell = h/2$ into that — the same claim about the spreading length the strut-and-tie model made when it placed its tie — and it returns $\frac{P}{4}(1 - a/h)$ exactly. Two derivations, one from a node and one from a curvature, agreeing to the last symbol.

What the second one makes obvious is that $\ell$ is the whole answer. The bursting force is inversely proportional to the length over which the flow is allowed to open out, which is the analytic form of the finding above: the gentle assumed spread gave 96 kN and the fast one 198, and the ratio between them is very nearly the ratio of their lengths inverted. Nothing else in the problem has that leverage — halving the plate depth changes the answer by a quarter, and halving the spreading length doubles it.

It also says what a detailer can do about it. Anything that lets the compression turn more gradually reduces the tension: a larger plate, a trumpeted anchorage, a flared end block, or simply more length before the section is asked to be uniform. And anything that forces it to turn quickly — an anchorage close to a change of section, a second tendon crowding the first — raises it, at a rate nothing in the member’s own analysis reports.

The steel is a fixed fraction of the force #

One consequence of an answer proportional to $P$ and to nothing else deserves stating on its own, because it is unusual.

The bursting force is $\tfrac{P}{4}(1 - a/h)$. Divide by the reinforcement’s design strength and the area required is

$$A_s = \frac{P(1 - a/h)}{4 f_{yd}}$$

For the block here — 214 kN at 435 N/mm² — that is 492 mm², which is four 12 mm links or a modest spiral. Against an anchorage force of 1,200 kN, delivered through a plate at twenty newtons per square millimetre, the whole countermeasure is under five hundred square millimetres of steel.

Two things follow. The concrete grade does not appear. A stronger concrete raises the permissible bearing stress under the plate and does nothing whatever for the bursting reinforcement, because the tension is set by the force spreading rather than by any strength. A designer who upgrades the concrete to solve an end-block problem has solved the wrong half of it.

And the quantity is small enough that there is no reason to be near it. The links are cheap, the consequence of omitting them is a member split along its tendon before it has carried any load, and the calculation that produces the number is a lower bound on a quantity that varies by a factor of two with an assumption. That combination — cheap countermeasure, catastrophic omission, uncertain magnitude — is the standing argument for generous detailing rather than accurate analysis, and the anchorage zone is where the argument is least contested.

Two other tensions in the same region #

The bursting tension along the axis is the largest of the anchorage zone’s problems and not the only one.

Spalling. On the loaded face itself, between two anchorages or above a single eccentric one, the concrete is pulled apart parallel to the face. It arises from the same spreading — the flow lines have to leave the face and turn — and it produces cracks that open at the end face and run in a few tens of millimetres. It is small, it is close to the surface, and it is why anchorage zones carry a mesh right at the end face as well as spirals behind the plates.

Equilibrium of the whole end block. An anchorage placed off the centroid puts the section into bending as well as compression, and the end block has to carry that too. With several tendons at different levels — normal, since the profile wants the force low at mid-span — the end block is a small deep beam with several point loads on one face, and it is designed as one.

The eccentricity a span wants is not the one its end can take, which is why tendons are lifted toward the centroid at the anchorage. That lift is a detail whose whole purpose is to make the end block’s job possible, and it has nothing to do with the bending the tendon was put there to produce.

It is worth separating this from a limit it resembles. The kern is the same section’s answer to a different question — where a force may act without producing tension anywhere on the section — and an anchorage placed well inside it still bursts the block. Bursting is caused by the force spreading, not by the force being eccentric, so the two limits look alike and constrain different things.

Why concrete in particular #

Every material spreads a concentrated force this way, and steel end blocks exist too. The reason the anchorage zone is a concrete problem is arithmetic about ratios.

The peak transverse tension here is 2.74 N/mm² under a bearing stress of 20 — about a seventh. In a material whose tensile and compressive strengths are the same, a seventh is nothing. Concrete’s tensile strength is between a twelfth and a fifteenth of its compressive strength, so the ratio the geometry produces and the ratio the material can survive are the same order of magnitude. The transverse direction is exactly as critical as the longitudinal one, in a member being loaded in one direction only.

The reinforcement is designed for a resultant, not a stress #

Two consequences follow from designing this region with a tie force rather than a stress field, and both are practical.

The first is that the reinforcement can be spread over the region where the tension is, rather than concentrated where the model drew its tie. A spiral behind the plate, links through the first depth, a mesh at the face: between them they cross every plane the tension could be acting on, and the total area is set by the tie force.

The second is that the design is insensitive to exactly which admissible field the concrete chooses. That is the lower-bound theorem’s whole promise — the structure is safe if some equilibrium field within the material’s strength exists — and the anchorage zone is the case where the promise is most obviously being cashed, because three equally defensible fields give answers a factor apart.

The same substitution of a resultant for a stress is what the bearing clause of every code is doing one line earlier, in a place nobody reads as an anchorage-zone calculation at all.

The cap of three that every code puts on that enhancement is not a property of concrete either. It is the angle past which nobody is willing to believe the strut arrives, which is the same kind of statement as the choice of lever arm three sections above: a decision about how much of a drawing to trust, wearing the clothes of a material constant.

The region that was found by watching it fail #

The end block is one of the few parts of structural engineering whose theory arrived after the cracks did.

Early post-tensioned members were anchored with whatever plate the system supplied, into an end detail chosen for constructability, and they split. The cracks were longitudinal, they appeared on stressing rather than under load, and they appeared in members whose flexural design was faultless — because the flexural design contains no term that knows about them. The response was reinforcement chosen by eye, then by test, and finally by the symmetric-prism argument that produced the coefficient above.

What makes the sequence worth recording is that the region is created by a design decision rather than encountered. A pre-tensioned member has no plate and no end block; the force arrives through bond over a transmission length and the splitting problem is different. A reinforced member has no anchorage zone at all. The end block exists because somebody chose to introduce a very large force through a very small area, which is exactly the choice a bearing, a base plate or a support makes on a smaller scale, and the tension behind it is the price of that choice rather than a property of the beam.

It is also the clearest case on this site of a region where the section stops being the unit of design. Everything else in a prestressed beam is designed section by section; the last half-metre at each end is designed as a body, with a free body drawn round it and moments taken.

Where the model stops #

The block is two-dimensional. A real anchorage spreads in both transverse directions, and the section is usually not rectangular: an I-beam’s anchorage sits in an end block cast solid, and the transition from that block back to the thin web is a second D-region behind the first.

The plate is rigid and the bearing uniform. Neither is quite true, and the difference matters most where the bearing stress is highest — under a plate sized so that the concrete beneath it, locally confined by the spiral, is being worked at several times its cylinder strength.

And the force arrives all at once. Tendons are stressed in sequence, so the end block sees one anchorage loaded, then two, then all of them. The worst case for spalling between two anchorages is often the intermediate one, and no single figure on this page corresponds to a real moment in the construction.

What the pictures cannot show #

The transverse stress is drawn as a curve along a line. In the block it is a field over an area, and the tension is spread across a region roughly one depth long and half a depth deep. The crack that eventually forms does not follow the peak; it forms where the concrete happens to be weakest along a plane that is in tension over most of its length.

Nor can the figures show the confinement. The concrete immediately under a plate is triaxially held by the spiral around it and by the mass of concrete surrounding it, and its local strength is far above the value any cylinder test gives. The bearing check that permits 20 N/mm² on a 30 N/mm² concrete is relying on that, and confinement is not drawn anywhere here.

The assumption the figure rests on #

The equilibrium field assumes the longitudinal stress is distributed across a growing width with a fixed shape, and normalises it to carry $P$ at every station. That is enough to make the field admissible and not nearly enough to make it real: the true distribution depends on the elastic properties, on the section’s shape, on the plate’s stiffness and on whether the concrete has already cracked. What survives the assumption is the resultant on the axis being zero and the tension being of the order of a fifth of the applied force — and the design uses only the second of those, with a margin.

The ladder from here #

Later rungs on this anchor: the anchorage zone with several tendons at different levels, where the general zone is a deep beam and the local zones sit inside it. Bearing under a rigid plate on confined concrete, where the permissible stress is several times the cylinder strength and the reason is the spiral. Pre-tensioned members, where there is no plate at all and the force enters by bond over a transmission length, with a different and larger splitting problem. Bursting behind a bearing on a bridge pier, which is the same figure with the prestress removed. And the general question of how far a designer may go in choosing an equilibrium field, which is the lower-bound theorem’s practical edge and is where this region’s design has lived for seventy years.