The support that is not a point

Every free body in this collection has reactions on it, and every one of them is drawn as an arrow. The arrow is honest about what the equilibrium equations use — a force has a magnitude, a direction and a line of action, and three equations cannot ask for anything else.

What sits under the arrow is a bearing three or four hundred millimetres wide, delivering a pressure of some distribution over that width. The equations do not need the distribution. Almost everything else does.

Two lengths, then, where the arrow had one. The bearing is a dimension on a drawing and the loaded length is a consequence of the member, and this essay is about the difference between them: what the width does to the moment, to the shear, to the bars, to the rotation, and to the span itself.

Which free body produced the number #

Cut the beam at the centreline of a support and take the piece to the left. The forces on it are the applied load and the part of the reaction that lies to the left of the cut.

If the support is a point, that part is nothing — the whole reaction acts at the cut and has no lever arm about it. If the support is a bearing of width $b$ carrying a total reaction $R$ spread uniformly, then half the reaction lies to the left of the centreline, its resultant is at $b/4$, and it contributes

$$\frac{R}{2}\cdot\frac{b}{4} = \frac{Rb}{8}$$

of moment in the opposite sense to the hogging the beam is carrying. So the moment over a support is lower than a point-support analysis reports, by exactly $Rb/8$, and the reduction has no material, no span and no load arrangement in it.

The construction is the ordinary one: separate the beam at a station and sum whatever lies on the piece taken, including the part of a wide reaction that lies on the same side of the cut. A point support has no such part, which is the only reason the ordinary answer is what it is.

What the reduction is worth #

The size of that bite is easiest to read on a beam that has a hogging moment to take it out of, which means a continuous one.

On the two-span beam drawn there the interior reaction is 192 kN and the hogging moment 134.8 kNm. A 400 mm column takes 9.6 kNm off it, a 600 mm one 14.4 — 7% and 11%. Those are not large numbers and they are free, which is why every concrete design standard permits the reduction and why almost every hand calculation makes it.

Two things are worth noticing about it. The first is that the reduction is proportional to the reaction, so it is largest exactly where the hogging moment is largest, and the two effects partly cancel. The second is that it is a statics result and not an allowance: the moment at the face of the support genuinely is smaller, and nothing about the beam has been assumed.

The catch is what happens if the support is not spreading its reaction uniformly, which is the next question and the one statics cannot answer.

The distribution statics does not fix #

The clearest demonstration that the distribution is not the detailer’s to choose is to change the detail by a factor of four and watch how little moves.

Four fifths of the bearing has been taken away and a quarter of the loaded length has gone with it. Whatever is deciding that length, it is not the dimension on the drawing — and it follows that the pressure under a bearing is not the detailer’s to specify either.

A reaction’s total comes from equilibrium. Its distribution comes from compatibility, and depends on the relative stiffness of the beam and of what it is sitting on — the beam on the ground in miniature.

A stiff beam on a soft pad gets a nearly uniform pressure. A soft beam on a hard bearing gets a pressure concentrated at the edges, because the beam is trying to rotate and its underside lifts away in the middle of the bearing. A bearing at the end of a beam that is rotating gets a pressure concentrated at one edge, and the reaction’s line of action moves toward that edge — which shortens the span and changes the moment.

None of that is available from the three equations. It is why bearing design is a separate subject with its own conventions, and why the $Rb/8$ reduction above quietly assumes the answer.

Why the shear check moves away from the face #

The second consequence of a real support is about shear, and it is a load path argument rather than an arithmetic one.

Load applied within about one member depth of a support does not have to be carried by the shear mechanism at all. It arches directly into the bearing along a compression strut, in the way a deep member has no section to design. The shear the member has to carry through its web is therefore the shear at a distance $d$ from the face of the support, not the shear at the centreline, and on a heavily loaded short span the difference is substantial.

The distance is the member’s depth rather than the support’s width, which is the first result on this page that does not scale with $b$ at all. A 400 mm bearing under a 900 mm beam moves the shear station 200 mm out to the face and a further 900 to the check point, and on a short heavily loaded span that removes a substantial part of the shear the analysis reported at the node.

The bar that has to run further than the diagram says #

The third consequence is the one most likely to be got wrong, because it points the other way: near a support the tension chord carries more than the moment diagram says, not less.

A cracked concrete beam carries its shear as a truss, and the diagonal crack a designer cuts along passes through the tension chord as well as through the stirrups. The chord therefore carries $V\cot\theta/2$ of tension in addition to $M/z$ — half the shear, at the crack angle — and the whole force curve is the moment diagram shifted toward the support by $z\cot\theta/2$, which here is 585 mm.

A bar curtailed where the moment diagram permits it is a bar that stops half a metre too soon, in the region where the shear is largest. That is the shift rule, it is a consequence of a support having a region rather than a point around it, and it is one of very few results in the subject where the honest answer is worse than the simple one.

The support that is also a restraint #

A wide support does something else no arrow shows: it restrains rotation.

A beam bearing on 400 mm of wall, with mortar or a bedding compound in the joint, is not free to rotate about a point, whatever the free body drawn for it assumed. As the beam deflects, the pressure moves toward the inner edge and a couple develops — small, uncontrolled, and reliably present. It is why a “simply supported” precast unit cracks over its bearing, and why a nominally pinned connection is neither pinned nor rigid.

The magnitude is bounded by the reaction and the width: the largest couple a bearing can deliver is $R b/2$ if the whole reaction moves to one edge. On the beam above that is 38 kNm — a quarter of the hogging moment the analysis reported, arriving in a member designed as though it were zero.

The counterpart of that restraint is a movement. An 11 m beam 900 mm deep at an ordinary deflection rotates 33 milliradians at its end, and rotating about the bearing carries the top of the section 29.7 mm sideways. Something has to allow that movement — a joint, a gap, a slotted hole — and if nothing does, the beam has an axial force in it that no analysis contains. A stiff restraint against a small movement is exactly the combination that produces a large force, which is the point the movement nobody applied makes in general.

The other reason the width matters, which is the perimeter #

Where the support is a column rather than a bearing, the width does something arithmetically different: it becomes a perimeter.

A check made on a perimeter rather than on a section behaves quite unlike a beam shear check. Making the column larger adds four straight lengths and does nothing to the corner arcs, which are 2,765 mm long regardless and 63% of the whole perimeter at the size drawn — so the resistance grows as roughly the 0.44 power of the column size rather than in proportion to it. Doubling a column’s plan dimension buys 36% more punching resistance.

How far the disturbance reaches #

All of the above is local, and there is an exact statement of how local.

The difference between a point reaction and a distributed one is a self-equilibrating load system: the two have the same resultant and the same moment, so they differ by a set of forces summing to nothing. Saint-Venant’s principle says such a difference dies away, and the version with a number in it says it dies away over the wavelength of the self-equilibrating component — about the depth of the member, with 1.4% left at one depth.

So the whole of this essay is about a region one member depth long at each end of every member, and about nothing outside it. That is a comforting boundary and a deceptive one, because a great many of the things that go wrong in structures happen inside exactly that region.

The support that moves the span #

There is one more quantity the arrow hides, and it changes an answer by more than any of the effects above.

The span used in a calculation is a distance between two lines of action, and a bearing’s line of action is wherever its pressure resultant happens to sit. For a beam on two pads of width $b$, an analysis usually takes the span as centre to centre, and the pressure resultant sits nearer the inner edge of each pad as the beam rotates — so the effective span is shorter than the drawing says, by something up to $b/2$ at each end.

On a 4 m span sitting on two 400 mm bearings, that is up to 400 mm off an 4,000 mm span: 10%, and the moment goes as the square, so 19%. On a 20 m span it is a fifth of a per cent and nobody cares. The rule of thumb is the same as everywhere else in this essay — the correction is a bearing width divided by a span, so it matters exactly when the member is short.

Precast concrete is where it bites, because precast members are frequently short, heavily loaded, and sitting on bearings that are a substantial fraction of their span. It is also where the correction runs the wrong way for the designer: the shorter effective span reduces the moment and increases the shear, and a short deep precast unit is governed by shear.

The width the bearing needs for itself #

Everything above treats the bearing’s width as given and asks what it does to the beam. There is a check running the other way, and on a precast member it is usually the one that sets the dimension.

The reaction has to be delivered through the contact area at a stress the material can take. For a dry bearing of concrete on concrete the permissible value is a fraction of the design strength — around 0.4 $f_{cd}$, so about 8 N/mm² on an ordinary mix — and the 192 kN reaction on a 300 mm wide beam therefore needs

$$\frac{192{,}000}{300 \times 8} = 80\ \text{mm}$$

of contact. That is the effective bearing length: the length actually in contact and actually carrying pressure. It is not the length on the drawing.

Three allowances sit between the two, and each is a deduction. The edge of a concrete member spalls — the corner is unconfined, the pressure is highest there, and a chamfered or damaged arris carries nothing — so an allowance of 15 to 25 mm comes off at each end. Construction tolerance takes another 15 or 20, because a unit set 20 mm off its intended position has 20 mm less bearing at one end and nobody measures it afterwards. The nominal bearing is the sum: 80 mm of contact plus two spalling allowances plus a tolerance is 150 mm on a drawing, and a detailer who writes 100 has provided 40.

The two checks are coupled, and the coupling is the part worth carrying. The effective bearing is both what carries the pressure and what defines where the reaction acts — so an allowance made for spalling shortens the contact, raises the pressure, and moves the resultant inward, which shortens the effective span. On a short deep unit that is a helpful direction for the moment and an unhelpful one for the shear, and the shear is what governs.

It also explains why bearing failures are so often failures of position rather than of capacity. The calculation is a division and it passes comfortably; the unit arrives 25 mm out, the spalling allowance has already been spent, and the remaining contact is a third of what was assumed — with everything else on the drawing unchanged and correct.

The steel version of the same question is worth following through, because there the resistance can be traced to the dimensions that supply it and the answer is not the one a shear check would suggest.

Nearly the square, from a thickness that appears once in the yield expression. A millimetre added to the web is worth far more at a bearing than the same millimetre is worth to a shear check on the same member, which is why a bearing stiffener is so often cheaper than the web that would otherwise be needed. The obvious companion move is to thicken the flange instead, since the flange is what spreads the load, and it does almost nothing.

The 0.19 power is close enough to nothing that a flange chosen for bending is a flange chosen for this, and no flange is ever thickened to improve a bearing. Two dimensions, then, and the one that spreads the load is not the one that carries it — a separation that is invisible until the two terms are drawn apart, because both of them plainly involve the flange.

What ties the pair together is the same construction the columns and plates on this site are checked by.

Twenty-eight per cent is the whole of why the arithmetic at the start of this section is not the end of it. The bearing length decides a yield resistance; the web’s slenderness then decides how much of that resistance is available; and a support width chosen to satisfy a bearing pressure has settled only the first of the two. The width under the arrow is an input to a check the arrow cannot see, and the check has a buckling curve in it.

Where the model stops #

The reduction assumes the reaction is uniform. Everything about $Rb/8$ depends on a pressure distribution that compatibility, not statics, decides — and the section above says compatibility gives a different answer for every combination of stiffnesses. The reduction is therefore a good approximation and not an identity.

A wide support is not a wide member. A beam framing into a deep transfer girder has a support several metres wide, and none of the arithmetic above survives that: the “support” is a structure with its own deflected shape, and the beam is continuous with it rather than sitting on it.

Nothing here is about the support itself. The pressures discussed are what the beam applies to the bearing, which is what the bearing has to carry, and a bearing that crushes redistributes everything that was assumed about it.

And the width is not always known when the analysis is run. A column size is an output of a design that has the beam moments as inputs, so the face-moment reduction is another small fixed point: assume a width, size the column, check the width. It converges in one pass and it is worth noticing that it is an iteration at all, because the direction of the error is not obvious — a larger column reduces the beam’s hogging moment and increases its own.

And the reaction itself was a decision before it was a number. The same floor panel divided two ways gives entirely different beam loads, and every reaction on this page follows whichever division somebody chose. A width correction of seven per cent applied to a reaction that was allocated rather than computed is an accurate adjustment to an approximate quantity, which is a shape worth recognising wherever it appears.

What the picture cannot show #

A moment diagram is drawn as a continuous curve running to the ends of a line representing the beam. The line has no thickness and the beam has a depth; the curve runs to a point and the beam sits on a bearing. Everything in this essay lives in the gap between those two statements, and none of it is visible on the diagram because the diagram is a plot of a function of one variable.

Nor does the drawing show that the reaction is a response. It is the only force on a free body that is not applied by anybody: it is whatever the support has to do, at whatever distribution the two stiffnesses settle on, and it is the one arrow in the picture whose value was computed rather than specified.

The generalisation #

The habit worth carrying is to ask what a simplification has thrown away, and where.

Reducing a bearing to a point is one of a family of moves this subject makes constantly: a distributed load becomes a resultant, a connection becomes a node, a column becomes a line, a slab becomes a plane. Each is exact for the quantity it was made for — global equilibrium — and each fails within about one member depth of where it was made.

It is the same trade a distributed load makes when it is replaced by its resultant, and the same one a rigid joint makes when it becomes a node. The general rule is the one Saint-Venant supplies. Two load systems with the same resultant differ only locally, and “locally” means about the size of the region over which they differ. So the simplification is safe everywhere except near itself, and every check that lives near a support is a check the simplification cannot be used for.

That is why the region above a support carries such a disproportionate share of the subject’s detailing rules — the shift rule, the shear at $d$, the face-moment reduction, the punching perimeter, the bearing pressure, the rotation capacity. Each of them is a patch on the same idealisation, applied in the one place it does not hold.