The node is the part that is checked

Where there is no section to design, a strut-and-tie model replaces the section with a truss drawn inside the solid. The lower-bound theorem then makes a promise, and the promise has a condition attached that is easy to read past: any model in equilibrium with the load is safe, provided every part of it can carry what the model asks of it.

The parts are struts, ties and nodes. Two of the three are straightforward.

A strut in the body of a member is wide, because a load that has travelled any distance has spread. The one drawn is a fifth of its permitted stress. A tie is a bar of steel with a yield strength that is measured, guaranteed and checked on site. Neither is where the model is in danger.

What a node is #

A node is the volume where two or more forces meet and turn. It is not a point, though every drawing shows it as one, and its size is what the calculation turns on.

Give it the shape that makes the arithmetic simplest: a polygon whose faces are perpendicular to the forces entering it, sized so that the stress on every face is the same. Such a node is called hydrostatic, and it has a property worth having — with equal stress on every face, the node is in equilibrium whatever the angles are, and the polygon closes automatically. The faces’ lengths then come out in proportion to the forces, which is a construction rather than a check.

Real nodes are rarely hydrostatic, because their dimensions are set by things outside the model: the width of a bearing plate, the depth of a bend in a bar, the cover to a layer of reinforcement. So the general node has unequal face stresses, its equilibrium has to be checked rather than assumed, and the highest of its face stresses is what is compared against a limit.

Three kinds, and one material with three strengths #

The permitted stress at a node depends on what is anchored in it, which is the part of the subject that surprises people.

A node with three struts and no tie — a CCC node, under a bearing plate at the top of a deep beam — carries the highest stress. Nothing crosses the compression field, so the concrete is in a triaxial state that is at least as strong as a cylinder test.

A node with one tie — a CCT node, over a support where the bottom bars are anchored — carries less. The bar crossing the field is a plane of weakness and, more importantly, the bar’s bond stress puts transverse tension into the concrete around it. A field cracked across itself is not as strong as an uncracked one.

A node with two or more ties — a CTT node — carries less again.

The usual factors are 1.00, 0.85 and 0.75 applied to the same basis. One material, three numbers, and the thing that decides which applies is a detailing decision rather than a material property.

That freedom is the reason the node limits matter so much. Every model in the band is admissible, so nothing in the analysis prefers one over another — and the thing that does the choosing, in practice, is which of them produces nodes that can actually be built.

Which free body produced the number #

The free body is the node itself: a small polygon cut out of the concrete with every strut and every tie severed just outside it.

That cut is the whole reason a node is a distinct object rather than a place on a drawing. Inside a strut the stress is one number over a large area; inside a node several forces at several angles have to be turned, and the state of stress is genuinely two- or three-dimensional. Cutting the node out asks the only question that can be asked of it: do the forces on its faces balance, and is the largest of the face stresses within the limit?

Two things about that free body decide everything.

Its faces are not free to be chosen. The face under a bearing plate is the plate. The face where a tie enters is set by the anchorage detail — a bend, a plate, a lapped bar. The struts’ faces are what is left. So the node’s geometry is an output of the detailing and an input to the check, which reverses the usual order.

It has no reserve. A strut that is overstressed can shed load sideways into the material around it, because a strut in a solid is a region of a stress field rather than a member. A node that is overstressed is at a boundary, has nothing beyond it, and is where the crushing actually happens in tests.

The node’s dimensions, which come from the detail #

A node check needs face areas, and the face areas are read off the reinforcement drawing rather than out of the analysis. Three numbers do most of the work.

The bearing width. The face under a plate is the plate, and the stress on it is the reaction divided by the plate area. Nothing about the model changes that number.

The depth of the tie band. The face where a tie enters is as deep as the bars are spread — one layer of bars gives a shallow face, three layers give a deep one, and the difference is a factor of three on the stress across it. That is why a heavily reinforced node is detailed with the bars in several layers rather than in one, and why the layers are then a dimension in the model, since the tie’s line of action is the centroid of the group.

The strut width that follows. With the other two faces fixed, the strut’s face is whatever the geometry leaves, which is $a_1\sin\theta + u\cos\theta$ for a bearing $a_1$ and a tie band $u$ at a strut angle $\theta$. That expression is the whole of the node’s geometry, and it is what makes a shallower strut a wider one.

There is an awkward consequence in that last line. Flattening the strut raises its width and lowers the node stress, and it also lengthens the tie and increases its force — so the two checks pull opposite ways, and the model that minimises steel is not the model that eases the node. A strut-and-tie design is a compromise between two members meeting at a place neither of them is, and the place is the only one with no reserve.

A node, in numbers #

Take the support node of the deep member above: a reaction of 1,200 kN on a 400 mm plate, a member 500 mm wide, a tie of 3,448 mm² in two layers spread over 200 mm, and a strut at 38.7°.

The bearing face carries 1,200 kN over 400 × 500 mm — 6.0 N/mm², comfortably inside the CCT limit of about 15 for this concrete. The tie face carries the tie’s 1,500 kN over 200 × 500 mm, which is 15.0 N/mm² and is at the limit. The strut face is 400·sin 38.7 + 200·cos 38.7 = 406 mm wide, carrying 1,921 kN over 406 × 500 — 9.5 N/mm².

Three faces of one node, three quite different stresses, and the one that governs is the face nobody draws: the horizontal one where the reinforcement is anchored. Spreading the same steel over 300 mm rather than 200 drops it to 10.0 N/mm² and widens the strut face at the same time, which is a detailing change with no analysis behind it and a 33 per cent effect on the check.

The bearing enhancement is a tie in disguise #

The commonest node in practice is the one under a bearing plate, and the strength quoted for it looks like a property of concrete and is not.

The concrete under a small pad on a large block is not stronger. It is confined, and the confinement is supplied by the ring of material around it — which is only there if that material can carry the hoop tension the spreading creates. The enhancement is the tie, not the concrete, and a designer who takes the √(A₂/A₁) and omits the reinforcement has taken a strength and left its mechanism behind.

The cap of three is the same argument again. Past a spread ratio of nine the struts are so steep that nobody believes them, and the number is a limit on the model rather than on the material.

The tie has to be anchored inside the node #

A tie’s force is developed by bond over a length, and the length is usually longer than the node.

The force that splits what it pushes on is the same phenomenon inside the node: a compression that turns generates a transverse tension, and the tension has to be tied. It follows that a node is not merely a place where the concrete has to be strong enough — it is a place that needs reinforcement of its own, in a direction none of the model’s members run.

That figure carries the practical difficulty. A tie of 3,448 mm² is nine 25 mm bars, each needing 906 mm to develop, in a node perhaps 400 mm across. The bar cannot be straight, so it is bent, hooked, welded to a plate or looped — and every one of those details changes the node’s own geometry and therefore the check made on it.

The anchorage is where the model touches the drawing, and it is why strut-and-tie models are drawn by people who know what the reinforcement will look like. A model whose ties cannot be anchored is in equilibrium and is not a design.

Nodes in a web, which are the ones nobody draws #

Everything above is about a node with a plate on it. The other kind is much more numerous and never appears on a drawing at all.

Every place a stirrup meets a chord in that picture is a node — a smeared one, distributed along the member rather than concentrated, and checked in aggregate through the limit on strut stress rather than one at a time. The truss analogy in a web is a strut-and-tie model whose nodes have been averaged, and the averaging is legitimate because there are hundreds of them and each carries a small share.

That contrast is the useful one. A node is dangerous when it is unique. A web has so many that a local overload redistributes; a bearing has one, and it does not.

What a node check actually forbids #

Read as a design rule rather than as a verification, the node limits forbid three specific arrangements, and each of them is a thing people want to do.

A very small bearing. The face stress is the load over the plate, so halving the plate doubles it. Bearing plates are almost always sized by the node and not by the member.

A very flat strut. A strut arriving at a shallow angle presents a small face to a node whose other faces are fixed, so the stress on that face is high. This is why the admissible band above stops at 25°, and why a model that reaches for a long lever arm to save steel usually fails at the node it saves it at.

Too many ties in one place. Anchoring two ties in one node drops the limit to 0.75 and doubles the transverse cracking, and it is what happens at a corner where a beam and a column meet with reinforcement from both entering the same volume.

Each of those is a case where the model is perfectly satisfactory and the node is not, which is the sense in which the node is the part that is checked.

Why the freedom in the model collapses here #

The family of admissible models is genuinely wide — a factor of 2.4 on the tie force across the band drawn above — and it is worth asking why designers do not simply take the model that needs least steel.

They do not because the cheapest model is the one with the longest lever arm, and a long lever arm means a flat strut, and a flat strut arrives at the support node at a shallow angle. The face it presents there shrinks as $\sin\theta$, so the node stress rises as the tie force falls, and the two curves cross somewhere in the middle of the band.

The node is therefore the thing that picks the model. The analysis offers a continuum and says every point on it is safe; the node check deletes both ends of the continuum, and what is left is a narrow range that every experienced designer arrives at by habit and every code encodes as a permitted range of strut angles.

That is a general shape worth recognising, because it recurs. A lower-bound theorem gives permission rather than an answer, and something else always has to supply the choice. In plastic frame analysis the chooser is rotation capacity; in the variable-angle truss for shear it is web crushing; here it is the node. In each case the theorem’s freedom is real and the thing that closes it is a local check the theorem does not contain.

Where the model stops #

The node limits are empirical. The 1.00, 0.85 and 0.75 come from tests on nodes of particular shapes, and they are a summary of behaviour rather than a mechanical derivation. A node of unusual geometry is outside the evidence they were fitted to.

Hydrostatic nodes are a convenience. Real nodes are not, their faces are set by the detailing, and the true stress distribution across a face is not uniform. The check is made on an average and the peak is somewhere else.

Three dimensions are treated as two. Almost every model drawn is planar, and a node under a bearing on a wide member spreads in both directions at once. The transverse tension in the second direction is real, is usually smaller, and is usually omitted.

Nothing here is a serviceability check. A node at its limit is a volume of concrete carrying very high compression next to a bar carrying very high bond, and the crack widths around it are not what any of this computes.

And the model is assumed to be the one the structure uses. The lower-bound theorem guarantees safety if the structure can reach the state assumed, which needs enough ductility for the concrete to redistribute into it. A node crushing at 20 per cent above the model’s stress has removed exactly that ductility, which is why the node limits are quietly the thing that makes the theorem apply at all.

A node’s capacity is a bearing stress over an area the model has to define, which is the same difficulty three times as strong under a smaller pad is about — the strength depends on how much material surrounds the loaded area, and a truss model of a region has to decide that before it can check anything.

The ladder from here #

Later rungs on this anchor: the dapped end and the half joint, where the load enters at the wrong face and has to be hung back into the full depth. Three-dimensional models for pile caps and anchor blocks, where the truss has no plane to be drawn in. The tie’s anchorage worked properly, with the bend radius, the hook and the loop all changing the node’s own geometry. Bottle-shaped struts, and the transverse tension a spreading compression field puts into the concrete along its length rather than at its ends. Load-path methods and topology optimisation, which produce models automatically from an elastic field and raise the question of what the model even means. And the upper-bound counterpart — yield line and mechanism methods — which reach the same structures from the unsafe side.

The node limits arrived last and were the reason the method took eighty years to become a design tool. Ritter and Mörsch had the truss analogy for shear in 1899 and it was regarded as conservative for most of a century, because nobody could say how much stress the concrete at a joint could take. Schlaich and his co-workers generalised the analogy in 1987, and what made the generalisation usable was not the truss — which was old — but a set of node strengths, which were new.