The load that has to be lifted

A beam carries a load down to its supports by making a triangle: the load pushes down at the top, a diagonal in compression runs from there towards the support, and a tie along the bottom holds the two ends apart. That is the truss inside every beam, and every shear calculation in this collection is a way of sizing its members.

The triangle has an assumption in it that nobody states, and it is not about concrete or steel or the angle of the diagonal. It is that the load arrived on the top.

Which free body produced the number #

Take the joint. Not the section, not the beam: the single panel point on the bottom chord where the load comes in.

Four members meet there in the model above: two chord members running along the bottom, and one vertical running up to the top chord. And — this is the whole essay — no diagonal. The struts in a beam run from the top chord down towards the nearer support, so on the left half they arrive at bottom-chord nodes that are further from the support than the top node they started at, and the node under the load is not one of them.

So vertical equilibrium at that joint has exactly two terms: the applied 600 kN downwards, and the vertical member. The vertical carries 600 kN in tension, and it is the only member that can.

Apply the same load to the corresponding top-chord node and two struts meet it, one from each side, and the vertical carries nothing at all. The load walks straight down the diagonals it was already in a position to use.

Everything else is identical, which is what makes it invisible #

The two models have the same reactions, 300 kN each. The same shear diagram. The same moment diagram — 1,667 kN in the bottom chord at mid-span in both, agreeing to machine precision, because the chord force is the moment over the lever arm and neither has changed.

A sectional calculation is a calculation about those quantities. It reads a shear at a section and converts it into a link requirement per unit length; it has no variable at all for which face the load arrived on, so the requirement that appears when the load is hung is not something a sectional check gets wrong — it is something the check has no way to express.

That is why it is written into codes as a separate clause about “indirect support” or “suspended loads”, in a different chapter from the shear rules, and why it is one of the most reliably missed requirements in reinforced concrete detailing.

How much steel, and where #

The shear at the panel is 300 kN. With a strut angle of $\cot\theta = 2.5$ and a lever arm of 810 mm, the sectional calculation asks for 0.30 mm² of link per millimetre of beam — which over the 324 mm the strut fan occupies is 96 mm².

The hanger carries 600 kN. At 500 N/mm² that is 1,200 mm², in the same place, and additional to the 96.

A factor of twelve and a half, arriving as links that are not per-metre but per-load, concentrated within about a lever arm either side of where the secondary beam frames in. It is not a small correction to a shear calculation; on this beam it is the shear reinforcement, and everything the sectional rules produced is a rounding on top of it.

The verticals in the two models say it more directly than the arithmetic does. Along the beam they read 0, 300, 300, 600, 300, 300, 0 with the load hung, and 0, 300, 300, 0, 300, 300, 0 with it on top. Every panel except one is doing the same job.

Where the load enters, between the two extremes #

Nothing forces a load to arrive at either face. A secondary beam framing into the web of a primary delivers its reaction somewhere in the middle of the depth, and a slab cast against the side of a downstand delivers it lower.

The requirement scales with how far the load still has to travel. A load entering at height $a$ above the soffit has already been lifted through $a$, and what remains to be hung is the fraction $1 - a/z$ of it — linear, because the strut that eventually carries it can only start from where the load already is.

Where it actually appears #

The arithmetic is simple and the detail is not, because the situations that produce it do not look alike.

A secondary beam framing into a primary. The commonest case, and the one the clause is written for. The secondary’s reaction enters the primary’s web at whatever level the two soffits align, which for beams of equal depth is the bottom.

A slab hanging from a downstand. The slab is cast against the side of the beam near the soffit and delivers a line load along it — so the hanger requirement is per metre rather than per load, and is easy to miss precisely because it looks like a distributed load rather than a point one.

A load hung from a soffit. Services, a runway beam, a suspended floor below. The load is unambiguously at the bottom and the requirement is unambiguously the whole of it, which is why this case is the one that gets detailed correctly.

And a support at the top of a member. The same argument upside down: a beam supported by hanging from something above, rather than bearing on something below, has to get its reaction down to the tension chord, and the tie required is the reaction.

None of those four depends on the load being near mid-span, which is worth showing rather than asserting, because the shear is largest where the requirement looks least like a shear problem.

A concentrated force crossing a surface the flexural model has no term for is the same class of question a punching check asks about a slab, and it gets the same kind of answer: a perimeter and a rule rather than a section.

The crack that says it was missed #

The failure mode is specific and recognisable. Without the hanger, the only thing carrying the load up to the compression zone is the concrete’s tensile strength across a horizontal plane at the level the load entered.

So the crack is horizontal, or nearly so, starting at the point of application and running along the beam — and it opens under service load rather than at failure, because the tension it is resisting is the full applied load rather than a fraction of it. Below the crack the beam is a bracket carrying the load on nothing; above it, the beam is intact and its own calculations remain valid.

The beam does not look overloaded. Its moment is fine, its shear is fine, its deflection is fine, and there is a crack running along it at soffit level near every point where something is hung from it.

What is left once that crack has opened is a different mechanism with a lower answer. Reinforcement crossing a crack that already exists carries load by clamping and friction rather than by tension across intact concrete, so a beam repaired after the fact is not the beam that was designed — it is a shear-friction calculation on a plane the design never contained.

The steel version is a stiffener, not a bar #

The same equilibrium happens in steelwork and produces a completely different-looking answer.

A secondary beam bolted to the web of a primary near its bottom flange delivers its reaction into the web at that level. The web has to carry that force up to the top flange, and a web is a plate: what it offers is a strip of itself in tension, over a width the load can spread into, and the check is whether that strip is adequate rather than whether reinforcement has been provided.

Where it is not — where the reaction is large, or the web is thin, or the connection is close to the bottom flange — the answer is a stiffener: a plate welded between the flanges at the connection, taking the force directly from the incoming beam to the top flange. It is the hanger, made of plate, and it is put in for exactly the reason the bars are.

The vocabulary hides the identity. One trade calls it suspension reinforcement and the other calls it a bearing stiffener, and neither set of rules mentions the other; the free body at the joint is the same in both.

Why the strut-and-tie model finds it and nothing else does #

This is a clean demonstration of what a lower-bound model is for.

A strut-and-tie model is a complete equilibrium system: every force is followed from where it enters to where it leaves, through members that are drawn. Nothing can be carried by a path that has not been drawn, and so nothing can be carried by a path that does not exist.

A sectional model is not complete in that sense. It computes a resultant at a cut and converts it into reinforcement by a rule, which is exact for the region and the loading the rule was derived for and silent about everything else. Its silence is not conservative — there is no factor of safety on a requirement that has not been noticed.

That is the argument for drawing the model rather than applying the clause, and it generalises well past this case: anywhere a load enters a member somewhere other than where the standard picture assumed, the requirement is found by asking how the force gets from where it is to where the model needs it, and by drawing a member to carry it.

That completeness is what a region with no section in it has to be designed with, and it comes at a price worth stating: the model is not unique, and the theorem does not ask it to be.

A hanger requirement survives every one of those choices. Move the lever arm anywhere in the band and the tie force changes by a factor of two, and the vertical under a bottom-chord load is still the whole of the load, because it is fixed by which node the force entered at rather than by the geometry of the truss above it. That is the difference between a quantity a model estimates and a quantity a model either contains or does not.

What it costs to get it right, which is nothing #

The last thing worth saying about this requirement is how cheap it is.

The hanger on this beam is 1,200 mm² — six sixteen-millimetre links, over a length of a few hundred millimetres, at a cost of a few kilograms of steel in a member that already contains several hundred. There is no impact on the concrete section, no impact on the depth, no impact on the reinforcement anywhere else in the beam, and no impact on anything the analysis produced.

It is a requirement whose entire difficulty is knowing it exists. That puts it in a small and interesting category — with the hanging load’s mirror image at a corner joint, where the reinforcement detail decides whether a joint reaches a fraction or the whole of the moment its members can carry, and with the reinforcement across a construction joint. Each of them is a few bars, each is invisible in the analysis, and each converts a member that works into one that does not.

The lesson generalises past concrete. Wherever a model produces a force at a point and a detail delivers it somewhere else, there is a transfer between the two that nothing has been asked to carry — and the cost of carrying it is almost always trivial next to the cost of the member it is attached to.

A joint reinforced without regard to which way the moment is opening it reaches a fraction of the capacity of the members meeting there, and the fix there is also a few bars, bent the other way. The two details are the same kind of object: cheap, invisible in the analysis, and decisive.

The same argument at the support: a dapped end #

Turn the essay round. The load has been arriving in the wrong place; now let the reaction arrive in the wrong place.

A dapped end — a half joint, a nib, a beam notched down at its end so it can sit inside the depth of what supports it — puts the reaction into a reduced section at the bottom of the member. The full-depth beam beyond the notch is an ordinary beam whose truss expects its reaction at a bottom node, which is where it is; what is missing is the step. The reaction enters the nib, and the nib’s own compression chord is 400 mm up on a member that is 900 deep. The force has to be lifted from the nib into the full section, and there is nothing in the beam’s shear calculation that lifts it.

The hanger required is the whole reaction — 300 kN, 600 mm² at 500 N/mm² — in full-depth links immediately behind the notch, and it is entirely separate from the nib’s own reinforcement. The nib is a short cantilever bracket and needs its own strut-and-tie or shear-friction steel; the hanger is what connects the bracket to the beam. Two requirements at one detail, from two different mechanisms, and the second is the one that is left out.

There is a second requirement at the same detail that is usually read as a material property rather than as a tie, and it is worth putting beside the hanger because it is the same argument again.

So the nib carries three requirements rather than two, and every one of them is a tie somebody has to draw before it can be provided.

What makes half joints notorious is where the crack starts. The re-entrant corner at the notch is a stress raiser with nothing to relieve it, so a diagonal crack initiates there under service load and runs upward into the beam — and the geometry that produced it also traps water and chlorides at exactly that face, so the reinforcement crossing the crack is the reinforcement in the least inspectable and most corroded part of the structure. A generation of bridges was built with half joints and a subsequent generation has spent a great deal on them.

The general statement covers both halves of this essay at once. A truss inside a beam has nodes where its designer put them, and any force entering or leaving anywhere else has to be taken there first. A load on the soffit needs lifting to the top chord; a reaction at a nib needs lifting to the full section. In both cases the sectional calculation is correct about everything it computes and silent about the transfer, and in both the transfer is a few links whose absence is a crack.

Where the model stops #

The truss is one of many. The strut angles were chosen, and a different choice moves the strut fan and the panel widths. What does not move is the vertical at the loaded node, because that follows from the load having no diagonal available, not from the angle the diagonals happen to make.

The hanger is drawn as a member and is really a region. How far either side of the load the links may be counted is a detailing rule, and the rules differ: some allow a spread of half the effective depth each way, some tie it to the width of the incoming beam. Nothing here derives that spread.

Struts are assumed to be able to carry what they are given. A strut arriving at a node with three members meeting is subject to a node-stress check the model above does not perform, and at the node under a hung load that check is often the one that governs the width of the incoming member.

And the beam is reinforced concrete throughout. In steelwork the same argument becomes a question about stiffeners: a secondary beam framing into the web of a primary near its bottom flange puts a tension into that web which has to reach the top flange, and the answer is a stiffener rather than a bar. The mechanism is identical and the vocabulary is not.

What the pictures cannot show #

The two models in the first figure are drawn side by side and look nearly the same, which is the point and also the difficulty: the difference between them is one member’s force, and a reader who is not looking for it will not see it.

Nor can any of these figures show what the requirement looks like on a drawing. Hanger reinforcement is a handful of extra links, drawn at a closer spacing over a few hundred millimetres, in a bar-bending schedule that already has several hundred entries. It is not a distinctive detail; it is a local change of spacing that an experienced detailer puts in without being asked and an automated one omits without being noticed.

The assumption the figure rests on #

The whole essay assumes the compression chord is where the load has to get to.

That is true for a beam in ordinary bending, and it stops being true in exactly one case worth naming: a member in which the load is applied below the neutral axis of a section that is already in tension there — a hanging member, a tension tie carrying load transversely, the bottom chord of a truss with a load on it. There, the load is not being lifted into a compression zone, it is being introduced into a tension field, and the requirement is a bond and anchorage question rather than a suspension one.

The distinction matters because the two get confused in the same detail. A load hung from the bottom chord of a truss needs the chord checked for local bending and the connection checked for its own capacity — and needs no hanger at all, because there is no compression zone above it to reach.

A truss is the structure where the same load needs no lifting. Its bottom chord is in tension along its whole length and its diagonals arrive at bottom-chord nodes by design, so a load applied there has a path waiting for it. The beam’s problem is that its truss is implicit, and an implicit member is one nobody checked.