Where to put the supports, which is not at the ends

A beam with its supports at the two ends is the standard picture, and for a uniformly loaded beam it is close to the worst arrangement available.

Move the supports inward and two things happen at once. The span between them shortens, which reduces the sagging moment in the middle; and the overhanging ends begin to hog, which pushes a moment back over the supports. Somewhere between the two extremes the beam is working as evenly as it can, and the peak moment there is less than half what it was.

The two competing moments #

With the supports at the ends, the sagging moment at mid-span is $wL^2/8$ and there is no hogging anywhere. That is the entire budget spent in one place.

Move each support in by a distance $a$. The sagging moment falls, because the span between supports is now $L - 2a$ and sagging goes with the square of the clear span. The hogging moment over each support rises, because the overhang is a cantilever of length $a$ carrying $wa^2/2$.

One curve falls, the other rises, and the worst moment in the beam is whichever is larger. The best arrangement is therefore where the two are equal — the minimum of a maximum, which is the standard shape of an optimisation with two competing failure modes.

Setting the two expressions equal gives $a \approx 0.207L$, and the peak moment there is about $wL^2/46$ against $wL^2/8$ at the ends. A factor of 5.8 in bending moment, bought with nothing but the position of two supports.

That number, $0.2071L$, is $\left(1 - 1/\sqrt2\right)L/2$, and it turns up wherever the same question is asked. A ladder carried on two shoulders, a pipe on two trestles, the two lifting points on a precast beam — all the same calculation.

What the diagram does over a support #

The moment changes sign at the support region, and where it passes through zero is the point of contraflexure.

Contraflexure matters more in construction than in analysis. It is where the tension moves from the bottom of the beam to the top, so reinforcement has to be arranged to be continuous past it. It is also, for exactly that reason, the most convenient place to put a joint: a splice at the point of contraflexure is a splice where the moment is zero, so only shear has to be transferred.

That trick built the Gerber girder — a continuous beam with real hinges inserted at the contraflexure points, which converts an indeterminate structure into a determinate one with almost all of continuity’s benefit. Nineteenth-century railway bridges used it constantly, because a determinate structure is insensitive to the settlement of its piers and a continuous one is not.

Continuity does the same thing differently #

Moving supports inboard is one way of putting hogging into a beam. Making it continuous over its supports is another, and the effect on the moment diagram is very similar.

A beam built in at both ends has a mid-span moment of $wL^2/24$ and support moments of $wL^2/12$ — a peak a third of the simply supported value. The mechanism is the same as the overhang’s: hogging at the ends relieves sagging in the middle.

The difference is what it costs to know. The overhang case is solvable by statics, because the reactions follow from the two equations. The continuous case is not: the end moments depend on the stiffness of the beam and of whatever it frames into, and getting them requires a compatibility calculation.

So the two routes to the same benefit differ in what they demand. An overhang is free to analyse and awkward to build. Continuity is easy to build and needs an analysis method that did not exist until the 1930s.

The moving load, and why the optimum shifts #

The 0.207 figure assumes the load is uniform and permanent. Almost nothing is.

Put two point loads on the same span instead of a uniform one and the moment diagram is a different shape entirely — two straight segments meeting under each load rather than a parabola — with its peak in a different place and a different relationship to the overhangs. An arrangement optimised for one is not optimal for the other, and the arithmetic that gave $0.2071L$ used the uniform load in both of its free bodies.

A bridge carries a uniform dead load and a moving live load, and the worst case for the sagging moment is one live-load position while the worst case for hogging is another. The design has to satisfy both, so the beam is sized for an envelope of moment diagrams rather than for any single one.

Finding the worst position of a moving load is what influence lines do, and the answer is often surprising: for a continuous beam the worst hogging over a support comes from loading the two adjacent spans and leaving the next ones empty, which is a pattern nobody guesses.

The practical consequence is that the elegant 0.207 optimum applies to a beam carrying only its own weight — a pipe, a precast unit being lifted, a lighting truss. Anything carrying variable load ends up somewhere else, chosen by whichever case governs.

The same argument in a cantilevered structure #

Once the idea is visible it turns up at every scale.

Where the supports sit is a design decision, and it changes the reactions as well as the moments — which is often what the decision is actually about. Two supports 4.7 m apart under an 8 m beam carry the same total load as two supports 8 m apart, but each now delivers it through a shorter bearing and into a structure below that has to be somewhere else.

A cantilevered canopy is one half of the overhang problem. A slab spanning between beams with an edge that oversails is the same thing in a floor. The Forth Bridge is the argument taken to its conclusion — three balanced cantilevers with suspended spans between them, which is a Gerber girder at a scale of five hundred metres.

Moving a support is also often the cheapest available fix on a real project. When a beam is found to be overstressed, adding a support or moving one is nearly always less expensive than making the beam bigger, because the moment falls with the square of the clear span.

The two free bodies that give 0.207 #

The number was quoted above and is worth deriving, because it takes two different free bodies and neither of them is the whole beam.

The overhang. Cut the beam at the left-hand support and keep the piece outside it. That piece is a cantilever of length $a$ carrying a uniform load, held by nothing but the moment and shear on the cut face. Summing moments about the cut,

$$M_{\text{hog}} = \frac{wa^2}{2}.$$

Nothing about the rest of the beam entered that calculation, which is the whole convenience of an overhang: its moment is a local fact.

The half-beam. Now cut at mid-span and keep everything to the left. On that piece are the uniform load over half the beam — total $wL/2$, acting at $L/4$ from the cut — and the reaction $wL/2$ at the support, which stands a distance $L/2 - a$ from the cut. Taking moments about the cut,

$$M_{\text{sag}} = \frac{wL}{2}\left(\frac{L}{2}-a\right) - \frac{wL}{2}\cdot\frac{L}{4} = \frac{wL^2}{8} - \frac{waL}{2}.$$

The first term is the familiar simply supported value and the second is what the overhang gives back. Setting the two moments equal,

$$\frac{wL^2}{8} - \frac{waL}{2} = \frac{wa^2}{2} \quad\Longrightarrow\quad a^2 + aL - \frac{L^2}{4} = 0,$$

whose positive root is $a = (\sqrt2 - 1)L/2 = 0.2071L$. The peak moment there is $wa^2/2 = 0.02145\,wL^2$, against $0.125\,wL^2$ with the supports at the ends — a factor of $5.83$.

That the answer involves $\sqrt2$ and not some property of steel or timber is the tell: this is an arithmetic result about a loaded line held at two points, and every structure it applies to is borrowing it rather than possessing it.

What the overhang costs #

A factor of nearly six, free, invites suspicion. Four things are being paid.

Uplift, and the pattern that produces it. The optimum assumes the whole beam is loaded. Load only one overhang — a stack of material at one end, a vehicle parked on a cantilevered slab — and the far support can be pulled upward. A support that can only push then does nothing, the beam tips about the near support, and the arrangement that was optimal under one load case is a mechanism under another. The plank on two bricks is the domestic version and it fails the same way.

Shear and reaction concentration. Moving the supports inward does not reduce the total load, so the same force passes through two supports that are now closer together and carrying overhangs on both sides. The shear at the support faces rises, the bearing stress rises, and for a beam whose web or bearing detail was already marginal the change makes things worse where it made them better in bending.

Tension in the wrong place. Hogging puts the top fibres in tension. For a reinforced concrete beam that means top steel over the supports, for a timber beam it means the knot on the upper face is now the critical defect, and for a composite floor it means the slab — which is excellent in compression and cracks in tension — is on the wrong side. An arrangement chosen by a moment diagram can be undone by which face of the material the tension lands on.

Deflection at the tips. The clear span deflects less, which was the point. The overhanging ends, meanwhile, are cantilevers, and their tips move both from their own load and from the rotation of the beam at the support. A canopy edge that visibly droops is usually not weak; it is a correctly designed overhang doing arithmetic nobody plotted.

Optimal, and therefore critical everywhere #

There is a structural feature of the answer that is worth separating from the arithmetic, because it recurs whenever anything is optimised.

At $a = 0.207L$, the beam has two equally critical stations: mid-span and the two supports, all at the same moment. Below the optimum, mid-span governs alone and the supports have spare capacity. Above it, the supports govern alone and mid-span has spare capacity. Exactly at the optimum, nothing has spare capacity anywhere.

That is the general signature of a fully optimised design, and it is not entirely good news. A structure with one critical point fails there predictably, and the rest of it has margin that can absorb an error — a support built fifty millimetres out of position, a load heavier than assumed, a material weaker than specified. A structure that is critical everywhere simultaneously has spent that margin, and an error anywhere is an error at a critical point.

The trade shows up as soon as the design has to survive an assumption being wrong, which is the argument redundancy makes from the other direction: redundancy is deliberately unspent capacity, and optimisation is the systematic spending of it. The usual professional resolution is to sit deliberately off the optimum — a little shorter on the overhang than $0.207L$, so that mid-span still governs and the failure mode stays the one that was designed for.

There is one setting where the exact optimum is used without hesitation, and it is the one where the load really is uniform, permanent and fully known: the lifting points marked on a precast concrete unit. The unit’s only load while it hangs is its own weight, the slings are attached at the fifth points, and the moment the unit sees on the way to its final position is a twenty-third of what a two-end lift would have imposed. Handling stresses have cracked a great many precast elements, and the marks on the side are the countermeasure.

Three optimisations, three numbers, one rule of thumb #

The 0.207 figure minimises the peak bending moment. It is not the only thing a beam on two supports might be asked to minimise, and the other criteria have their own answers — which turn out to be almost the same answer, for a reason worth understanding.

The Airy points, at $0.2113L$ from each end, are where the supports make the bar’s two ends parallel: the slope at each tip is zero, so the end faces stay vertical however much the middle sags. That is what a length standard needs, because a bar measured between its end faces has to have those faces square whatever the bar is doing in between.

The Bessel points, at $0.2203L$, minimise the change in the bar’s overall length. Sagging shortens the horizontal distance between the ends; a support position can be chosen to make that shortening stationary, and it is a different position from the one that squares the ends.

And $0.2071L$ minimises the largest moment, which is the structural question.

Three criteria — no slope at the ends, no change in length, no moment larger than necessary — and the answers span $0.207$ to $0.220$, a range of 6% of the position. That is why “the fifth points” works as a rule of thumb for everything from a precast unit to a surveyor’s staff: whatever the beam is being asked to do well, the answer is near a fifth of the span in from each end.

The prices are not identical, though. Set the supports at the Bessel points and the hogging moment is $w(0.2203L)^2/2 = 0.0243\,wL^2$, which is 13% above the structural optimum; the Airy points cost 4%. So the moment criterion is the sharpest of the three, and a beam positioned for one of the metrological reasons is paying a real if modest structural premium — which is the usual shape of the thing when one object is optimised for two purposes.

The same question inside the section #

Moving a support and moving material in a cross-section are the same optimisation at two scales.

Four sections of identical area — a flat plate, a square, a rectangle on edge, an I-section — have capacities that differ by an order of magnitude, and not one of them uses more steel than another. Rearranging material within a section is the small-scale version of rearranging supports along a beam, and both are free in the sense that no more material is bought.

Getting material away from the neutral axis raises the capacity at constant weight; getting the supports away from the ends lowers the demand at constant everything. Structural efficiency is very largely those two moves, applied until something non-structural objects.

There is a second benefit that none of the moment diagrams above shows, because a moment diagram is one integration short of it. The deflected shape is the moment integrated twice, and an overhang improves it by both of the routes that integration has: the clear span is shorter, and the hogging at the supports curves the ends back the other way.

The deflection benefit is the larger one in practice. Deflection goes as the fourth power of the clear span, so shortening it by a fifth cuts the movement by more than half — which for a member governed by stiffness is worth more than the moment reduction.

Where the model stops #

Uniform load only. The 0.207 optimum is specific to a load uniform along the whole beam, including the overhangs. Load the middle only and the answer moves toward the ends.

Bending only. The optimisation minimises the peak moment and ignores everything else, including which limit actually governs. Deflection, shear at the supports, and the uplift that an overhang can produce at the far support are all unaddressed, and any of them can govern.

Uplift. If the overhang is long enough and loaded while the middle is not, the far support goes into tension — a reaction the support idealisation may not permit. A support that can only push then fails to hold, and the beam tips — which is how a plank on two bricks behaves when somebody stands on the end.

Which flange is in compression. Hogging over the support puts the top flange in tension and the bottom in compression, and the bottom flange is the one nothing is attached to. A floor slab restrains the top flange and leaves the compression one free over exactly the region where the hogging is largest, so an arrangement that improves the bending moment can worsen the stability check that goes with it. The moment diagram cannot see this, because it records magnitude and sign and not which face of the beam has something bolted to it.

Rigid supports at known positions. Real supports settle, and for a redundant arrangement that redistributes everything — which is one reason a determinate layout is sometimes chosen deliberately.

The figure at the top has a limitation worth naming: it plots the peak moments and not where they occur. Two arrangements with the same peak can put it in very different places, and a designer usually cares about both. The curve says how bad the worst point is, and says nothing about which point it is.

The ladder from here #

Later rungs: the optimum overhang derived properly. Points of contraflexure and where to splice. The Gerber girder, and determinacy by inserted hinge. Influence lines. Pattern loading on continuous beams. Moment envelopes. Cantilever construction and balanced cantilevers. Uplift, and the load cases that produce it. And the same optimisation applied to a slab spanning two ways, where the answer stops being a single number.

The 0.2071 ratio is the same number that appears in the design of a pipe rack, a launch cradle and a violin bass bar. It is not a structural constant so much as an arithmetic one, and it turns up wherever a uniformly loaded line is held at two points.