When there is no section to design

Every calculation on this site that begins by choosing a cross-section begins with the same assumption: that a plane cut through the member stays plane as the member bends, so that strain is linear across it and one number — a curvature — describes the whole face. The assumption is the whole of beam theory, and it is very good indeed. It is also false within about one depth of anything interesting.

The regions where the assumption fails have a name and a boundary. They are D-regions — D for discontinuity, or disturbed — and they extend about one member depth from every support, every concentrated load, every re-entrant corner, every change of section and every hole. Between them lie B-regions, where Bernoulli’s assumption holds and every method in the rest of this collection applies.

Where the section exists and where it does not #

The one-depth rule is Saint-Venant’s principle used as a boundary rather than as a reassurance. Saint-Venant says that two statically equivalent load systems produce the same stress field at a distance of the order of the loaded dimension. The usual reading of that is optimistic: it means moving a force is harmless far away. The pessimistic reading is the same sentence: it means the field near the load is decided by exactly how the load arrives, and by nothing a resultant records.

So the practical question is arithmetic. A member of span $L$ and depth $h$ has D-regions covering roughly $3h$ of its length; a B-region exists only if $L > 3h$, which is to say only if the member is at least three times as deep as it is — and a great many members are not. A pile cap, a corbel, a deep transfer beam, the panel zone of a moment connection, the region around a service hole: not one of them has a section anywhere in it.

The two kinds of region are not analysed separately and then hoped about. They meet at a boundary, and the boundary is where one method hands its answer to the other: the B-region’s section calculation produces an axial force, a shear and a moment at the station one depth from the discontinuity, and those three quantities are the loads applied to the D-region’s truss. The moment arrives as a couple — a compression at one face and an equal tension at the other, on the beam’s own lever arm — because a truss node cannot receive a moment.

So the section assumption is used right up to the point where it stops being true, and its last valid output is the input to what replaces it. That is why the one-depth rule matters as a number rather than as a caution: it says where to cut, and a cut taken too close hands the truss a set of resultants that were computed from a strain distribution the member does not have.

Which free body produced the number #

The model in the hero figure is not an approximation to a stress field. It is an exact equilibrium solution to a differently posed problem, and the free body says which one.

Cut the member on a vertical plane just to the left of mid-span and take the piece to the left. Across the cut passes: a compression of 960 kN in the inclined strut, and a tension of 750 kN in the horizontal tie. The applied load has not been reached; what is on the free body is the left-hand reaction of 600 kN.

Vertically: $960 \sin 38.7° = 600$ kN, which balances the reaction. Horizontally: $960 \cos 38.7° = 750$ kN, which balances the tie. Two equations, two members, and no section anywhere in the argument. The truss solver produces both from joint equilibrium, exactly as it does for every triangulated frame on this site, because a strut-and-tie model is a truss and is solved as one.

The lever arm is the only thing that was chosen rather than derived, and it is where the rest of this page goes.

The theorem that says a guess is enough #

The lower-bound theorem of plasticity says: if a stress field can be found that is in equilibrium with the applied loads and nowhere exceeds the material’s strength, then the structure will carry those loads. It says nothing about whether the field is the real one. It does not have to be.

That is an extraordinarily generous licence and it is the whole basis of strut-and-tie. The designer draws a truss, computes its forces, provides reinforcement for every tie and checks the concrete for every strut and node — and if all of those check out, the region is safe, whether or not the concrete has any intention of behaving that way.

A factor of 2.4 separates the extremes and all of them are safe. That sentence is either liberating or alarming depending on what a reader expects a structural calculation to be, and both reactions are appropriate.

It is liberating because it converts a problem with no closed-form solution into a problem with a great many solutions, any of which can be built. It is alarming because two competent engineers can produce reinforcement differing by a factor of two, both correct, and because a model that is admissible is not necessarily good.

What makes one model better than another #

Three things separate a good model from a merely admissible one, and none of them is an equilibrium consideration.

Ductility. The lower-bound theorem assumes the material can redistribute to reach the assumed field, and concrete’s capacity to do that is limited. A model far from the elastic stress field demands more redistribution than a model near it, and the demand has to be met by the reinforcement’s ability to yield and keep yielding without the concrete crushing first.

Crack width. A tie carries a real force in real bars, the bars strain, and the concrete around them cracks. A model that puts a large force in a tie at service load produces wide cracks whether or not the ultimate check passes, and the theorem is silent about it because it is a strength theorem.

The 25° to 65° band. Outside it, a strut and a tie meeting at a node are nearly parallel, the node geometry becomes impossible to detail, and the redistribution demanded becomes unreasonable. The band in the figure is that limit, and the shallowest model in the family — 40% of the depth, at 22° — falls outside it.

So the practical rule is: follow the elastic stress field. Choose the model whose geometry is nearest to where the compression and tension actually run, because that is the model demanding least redistribution.

Where the compression actually runs #

The struts of the model are drawn as straight lines and the concrete’s compression is not straight. Between two node regions the compression spreads sideways, is widest in the middle, and narrows again at each end — a bottle-shaped strut — and the spreading generates a transverse tension across the strut’s own axis.

The standard estimate uses a 2:1 spread: the compression fans out at a slope of one across for two along, and the transverse tension needed to turn it back is a quarter of the strut force. Here that is 240 kN across each strut, and it has to be carried by reinforcement running perpendicular to the strut or the strut splits along its own length — which is a tension failure in a material chosen for its compression.

That is the failure mode the model does not contain, and it is the reason strut-and-tie detailing calls for a mesh of distributed reinforcement across the whole region rather than only for the bars the ties asked for. The truss says nothing about it; the concrete’s behaviour between the nodes does.

The quarter is a rule of thumb, and the same generator will compute the transverse tension properly on the one D-region where the geometry is simple enough to allow it: the block of concrete behind a post-tensioned anchorage, where a large force enters through a small plate and spreads to fill the section. Nothing in that calculation is elastic. An assumed spread of the longitudinal stress plus the two equilibrium equations fixes the transverse stress uniquely, and the test of whether the assumption was an equilibrium field rather than a sketch is whether the transverse stress closes to zero at the far face, where nothing is pushing.

That last comparison is the honest position of every number on this page. The field is a lower bound and the design tie sits deliberately above it, because the theorem licenses the field and says nothing at all about the crack that opens while the concrete finds it.

The nodes themselves are the other check. Where the two struts and the tie meet under the load, the compression is delivered over the bearing plate — 10.0 N/mm² over 300 mm here — and where the strut lands on a support it spreads over a width set by the bearing and the tie’s own cover. Spread over 812 mm the strut stress is 2.96 N/mm² against a limit of 15.8 for concrete cracked transversely, which is a comfortable margin and is the usual outcome: the concrete rarely governs, and the tie almost always does.

The bearing check is where that outcome is usually hidden rather than stated. Codes allow the strength under a small pad to be raised by the square root of the ratio of the area it spreads into to the area it is delivered over, and the enhancement is often read as a property of confined concrete. It is not. It is the same spreading argument again, one member long.

The cap of three that every code applies to that enhancement is not a material limit either. It is the angle past which the strut has spread so far that nobody is willing to believe it arrives, which is a statement about confidence in a drawing rather than about concrete.

The comparison with the beam calculation #

A beam calculation on the same member, taking the lever arm as $0.9 \times 0.95 h$, would have asked the tie for 702 kN. The model asks for 750 — 6.9% more.

That is a smaller discrepancy than the whole apparatus suggests, and it is worth being honest about why. For a single point load at mid-span the two methods almost agree, because the strut-and-tie model of that case is nearly a beam: one couple, one lever arm, one tie. The methods diverge sharply for load cases a beam calculation handles badly — a load applied near a support, a load hung from the bottom of a member, a reaction delivered indirectly, a hole in the wrong place — and diverge most for the thing a beam calculation cannot represent at all, which is the path the load takes.

A funicular polygon finds the shape a set of loads wants to take, and is a drawing rather than a calculation for exactly the reason this page has been arguing. A strut-and-tie model is the same drawing used the other way round: it chooses the shape the loads are going to be made to take, and then provides the steel that makes the choice true. Choosing the path is the whole of the design activity, and no section calculation contains a place to make that choice.

Which free body a hanging load has #

The clearest case where the two methods part company needs no arithmetic. Consider the same deep member with the load applied not on its top surface but hung from its bottom.

The bending moment is identical. The shear is identical. Every quantity a beam calculation produces is unchanged, because the two loadings are statically equivalent at any distance.

The strut-and-tie model is completely different. The load has to be lifted from the bottom of the member to the top before the arch can carry it, which requires vertical suspension reinforcement over the full depth — stirrups whose whole job is to hang the load up, carrying the full applied force. Leave them out and the member fails at a small fraction of the calculated capacity, and no bending check anywhere in the process objects.

A beam calculation cannot see where a load is applied through the depth, and that is the sharpest statement of what the section assumption throws away.

The two models in that figure are the same truss. Every node in either of them is a small free body with three or four forces meeting at it, and the detailing of the node — the bearing area, the bend radius of the bar, the anchorage beyond it — is the physical realisation of that free body. What separates them is one panel point that touches no strut, and a member built to the left-hand drawing while loaded like the right-hand one has no path for the load at all.

The same idea in the members that are not concrete #

Nothing in the theorem mentions concrete, and two other places on this site are the same argument in different clothing.

A steel connection is a D-region. Every argument in the connections field is a strut-and-tie argument with the ties in bolts and the struts in bearing: a bolted end plate has a compression zone at one flange and a tension zone at the other, the panel zone carries the difference as shear, and every dimension of it is a lever arm. The component method used there is the same model with springs added so that it produces a stiffness as well as a strength.

A masonry arch is a D-region everywhere, and Heyman’s safe theorem for it predates the concrete version by twenty years. The thrust line that has to fit inside the masonry is a strut-and-tie model with no ties in it at all, which is what a material with no tension available reduces to.

And the whole plastic method for steel frames is the same theorem with the members as the medium: a collapse mechanism is an upper bound, a statically admissible moment diagram a lower one, and only the second is safe to build on. What differs between the three is how much redistribution the material will supply before something brittle happens, and that is a materials question rather than a statics one.

Where the model stops #

A lower bound is not a prediction. The model says the region will carry the load. It says nothing about what the region will do at service load, where the cracks will be, how wide they will be, or how much the member will deflect. Every one of those is decided by the elastic field the model was allowed to ignore.

Ductility is assumed and is finite. The theorem needs the material to redistribute, and concrete crushing is brittle. A model that demands large redistribution is licensed by a theorem whose hypothesis it may not satisfy, and the 25°–65° band is a crude guard against exactly that.

Nothing here sizes anything. The strut strength used, $0.6(1-f_{ck}/250)f_{ck}$, is a representative value for a strut cracked transversely; nodes have their own and different limits depending on how many ties enter them, and the numbers vary between codes.

Anchorage is not in the model and often governs. A tie carrying 750 kN has to develop that force beyond the node, in a length that a deep member’s geometry may not provide. The truss diagram shows a line ending at a point; the reinforcement ends in a bend, a plate or a lap, and the detailing of that end is where D-regions are most often got wrong.

And the model is two-dimensional. A pile cap, which is the most common D-region of all and is where a structure meets the ground, spreads its load in two directions at once, and the planar truss has to be replaced by a three-dimensional one whose statics is six equations rather than three.

What the pictures cannot show #

The struts are drawn as lines of no width, and then a width is quoted for them in the caption — 812 mm, which is most of the depth of the member. The line and the number describe different objects, and a reader who takes the line seriously will imagine a thin diagonal where the real compression fills most of the region.

Nothing in the drawings shows a crack, and a D-region designed this way is fully cracked at service load by intention: the tie is a tension member in a material with no tensile strength, so its force exists only because the concrete around it has split. The clean lines of the model are the state of a member that has already failed in the sense every other page on this site would use the word.

And the family figure draws a continuum of models as a smooth curve, which suggests that a designer might choose 73.4% of the depth. Nobody does. The lever arm is set by where the reinforcement can physically go — a bar layer above the cover, a bearing plate of a chosen size, a node that has to fit — so the real choice is among a handful of buildable geometries, and the curve’s job is to show that the answer is insensitive to which of them is picked.

The ladder from here #

Later rungs on this anchor: node types and their strength limits, and why a node with two ties entering it is worth less than one with none. Bottle-shaped struts derived properly, with the transverse tension from the spread and the reinforcement that carries it. Three-dimensional models for pile caps and anchor blocks. The corbel, the dapped end and the half joint, which are the three D-regions with the worst failure records and the clearest models. Load-path methods and topology optimisation, which produce strut-and-tie models automatically from an elastic field and raise the question of what “the” model even means. The upper-bound counterpart — yield-line and mechanism methods — and the reason it is unsafe to design on. And the history: Ritter and Mörsch had a truss analogy for shear in 1899, it was regarded as conservative for eighty years, Schlaich and his co-workers generalised it into a design method in 1987, and the generalisation is one of the few genuinely new ideas in reinforced concrete since the war.

The most useful thing to carry away is not the method. It is the boundary: the question “what is the section here?” has no answer over a large fraction of most structures, and a discipline that teaches sections first tends to leave that fact for later and sometimes for never.