Span to the fourth, which is why spans are short

$$\delta = \frac{5wL^4}{384EI}.$$

The fourth power in that expression is the steepest relationship anybody meets in ordinary structural work, and it is worth taking apart, because each of the four is a different piece of physics and they compound.

Where each power comes from #

Two of the four powers are in the moment and two more come from turning moment into deflection.

The first two. A uniformly loaded beam has a total load proportional to $L$ and a lever arm proportional to $L$, so the bending moment goes as $L^2$. One power from more load, one from a longer arm.

The third and fourth. Curvature is $M/EI$, so it also goes as $L^2$. Deflection is the second integral of curvature, and integrating twice over a length $L$ multiplies by $L^2$ again. Two more powers, from the geometry of accumulating curvature.

$$L \times L \times L \times L.$$

Each factor is ordinary. The product is not, and it is the reason that a doubling of span — which sounds like a modest change — is a structural transformation rather than a scaling.

What it forces #

A designer who wants to double a span while keeping the same deflection has to increase $I$ by sixteen. Since $I$ goes as the cube of the depth for a solid section, that means a section 2.5 times deeper. The beam gets heavier, which increases $w$, which increases the deflection further.

The doubling is the memorable case and it is not the one that catches people out. A fourth power is steep enough that a change nobody would call a change is already a large one.

That is the practical shape of the law: the increments a designer treats as adjustments are, in the quantity that governs, not adjustments at all. The loop converges for ordinary spans. It does not converge forever, and the point at which it stops converging is what sets the practical limit of every structural form.

The crossover with strength is a symptom of the same arithmetic. Strength grows as $L^2$ and deflection as $L^4$, so the second overtakes the first at some span, and the span at which it does is a property of the section rather than the material.

Self-weight, and the span a form runs out at #

The fourth-power law assumes the load is fixed. For a long span it is not, because most of the load is the structure.

A beam scaled up uniformly by a factor $k$ has its volume, and therefore its weight, multiplied by $k^3$. Its section modulus — which sets what it can carry — grows only as $k^3$ as well, so at first sight the two keep pace.

They do not, because the moment from self-weight grows as $k^4$: the weight per unit length grows as $k^2$ and the span as $k$, and the moment goes as $wL^2$. So the demand outruns the capacity by one power of $k$, and a structure scaled up far enough cannot carry itself.

That is the square-cube law in structural form, and it is the reason nothing scales. Galileo put it in the Two New Sciences in 1638 with a drawing of two bones, one from a small animal and one scaled up, and pointed out that the large one had to be disproportionately thick.

The structural consequence is that every form has a span at which its own weight consumes its entire capacity. For a solid rectangular timber beam that limit is a few tens of metres. For a steel plate girder it is perhaps a hundred. For a truss it is several hundred, for an arch more, and for a cable more still — which is why the ranking of long-span forms is exactly the ranking of how efficiently each gets material away from its own centre of gravity.

The escape from the loop is depth: a chord force is the moment divided by the distance between the chords, so material moved apart carries the same moment at a smaller force, and every long-span form is a way of achieving depth without paying for solid material to fill it.

What actually gets built at each span #

The scaling argument shows up directly in what exists.

Up to about six metres, a timber joist or a small steel section, sized by strength.

Six to fifteen metres, a rolled steel beam, usually sized by deflection.

Fifteen to forty, a plate girder or a truss — the point at which a rolled section is no longer deep enough and depth has to be fabricated.

Forty to a hundred and fifty, a deep truss or a portal, where the structure is mostly air.

Beyond that, the form has to abandon bending altogether: an arch working in compression or a cable in tension, both of which are funicular shapes carrying load with no bending at all.

The sequence is not a matter of taste or of tradition. Each transition happens where the previous form’s self-weight has eaten the useful capacity, and the numbers have barely moved in a century because they depend on the ratio of material strength to material density — which has improved far less than absolute strength has.

Where the extra powers hide #

The fourth power is a property of a uniformly loaded beam, and other configurations differ.

A central point load gives $\delta = PL^3/48EI$ — the third power, because the load does not grow with span in that case. Comparing the two exponents is instructive: the fourth power in the uniform case includes one power that comes from there being more load, not from the geometry.

A cantilever under a uniform load gives $wL^4/8EI$, which is the same power with a coefficient forty-eight times worse than a simple span.

Continuity improves the coefficient without changing the power. A beam built in at both ends deflects at $wL^4/384EI$ — a fifth of the simple span — because the hogging at the ends curves the ends back. Redundancy buys a coefficient and never an exponent.

Getting the material further apart #

The escape from the fourth power is not a stronger material. It is a larger second moment of area, and every long-span form is a way of getting one.

Moving material outward raises $I$ without raising the weight, which is the only lever that acts on the denominator of the deflection formula. A truss is that principle applied until the middle of the section is mostly air. Take a rectangle apart into strips and the outer ones do nearly all of the work: the contribution of a strip goes as the square of its distance from the neutral axis, so the material near the middle is very nearly along for the ride, and every efficient section is that observation acted on.

The consequence for span is a cube against a fourth power. At constant material, the second moment of a fabricated section rises with the square of its depth and its strength with the first power, so depth buys stiffness faster than it buys anything else — and it is still one power short of what the span is spending. A rolled section runs out of depth long before a fabricated one does, which is the whole reason the second exists.

The transitions between structural forms — beam to girder to truss to arch to cable — are the points at which the available depth runs out and a different way of achieving it has to be found.

The free body behind the coefficient #

The exponent is geometry. The coefficient — the $5/384$ — is a free body, and it is worth doing the sum out, because the rule on this site is that a quoted number names the body it came from, and because this particular number needs two different bodies to produce it.

The first body is half the span. Cut a uniformly loaded simple beam at midspan and keep the left-hand piece. Three things act on it: the reaction $wL/2$ at the support, the share of the load it carries, which is also $wL/2$ and whose resultant sits a quarter of the span from the support, and on the exposed face a shear force and a bending moment. Taking moments about the cut,

$$M = \frac{wL}{2}\cdot\frac{L}{2} - \frac{wL}{2}\cdot\frac{L}{4} = \frac{wL^2}{8}.$$

The famous eighth is the difference between a half and a quarter. Nothing deeper is happening in it, and the two powers of $L$ are visible as one lever arm each.

The second body is the moment diagram itself. Curvature at every station is $M/EI$, and the deflection at midspan is how far the support has fallen away from the tangent drawn at midspan — which, since the tangent there is horizontal by symmetry, is the first moment of the $M/EI$ area between support and midspan, taken about the support.

The area of that parabolic segment is $wL^3/24EI$. Its centroid sits $5L/16$ from the support. The product is

$$\delta = \frac{wL^3}{24EI}\cdot\frac{5L}{16} = \frac{5wL^4}{384EI}.$$

So the five in the numerator is the position of the centroid of a parabola and nothing else, and the $384$ is $24\times16$. A coefficient that looks like a constant handed down from somewhere is two geometric facts multiplied together, and any change to the load pattern changes both of them — which is exactly why the cantilever’s coefficient is forty-eight times worse while its exponent is identical.

Worth naming: the cut face in the first free body carries a shear force, and the second calculation then ignores it entirely. Every standard deflection coefficient is a bending-only number obtained from a free body that was carrying shear at the moment it was drawn.

What the exponent costs #

A fourth power is not only a fact about arithmetic; it is a fact about money, and it arrives in three places.

Depth. Holding deflection constant while doubling the span needs sixteen times the second moment of area, and the cheapest route to $I$ is depth. In a building the depth is bought from the storey height, so a longer span raises the floor-to-floor dimension, which raises the height of the whole building, the area of cladding wrapping it, the length of every riser, and the volume being heated. A structural decision about one beam ends up in the facade budget, which is why the span-to-depth ratio is one of the first numbers argued over on a project and one of the last anybody is willing to move.

Self-weight fraction. As the span grows, more of what the structure carries is the structure. The useful fraction — the part of the capacity available for people, snow and machinery — falls, and it falls faster than the tonnage rises, because the tonnage is what caused it. A long-span roof can spend more than half its strength on itself, and the arithmetic gets worse in exactly the way a load that makes itself worse does: the response feeds the demand.

Ponding, which is the fourth power made vicious. A flat roof that sags collects water in the sag. The water is load. The load increases the sag, which enlarges the pool, which adds more water. On a stiff roof the series converges after a millimetre or two and nothing happens. On a flexible one it may not converge at all, and the collapse arrives during ordinary rain with no gust, no snow and no overload in any sense a code would recognise — the roof simply found a load that grew as fast as its resistance to it. Codes handle this by requiring either a fall to the drains or a stiffness check, and the stiffness check is the fourth-power law written as a stability criterion.

The general shape of all three is the same. A quantity that grows as $L^4$ against a capacity that grows as $L^2$ or $L^3$ does not merely get expensive; it gets expensive at an accelerating rate, and the span at which the accelerating rate becomes visible is where the structural form has to change.

None of that is visible on the structure. At span over three hundred and sixty, the sag of a ten-metre beam is under thirty millimetres, which is a shape no drawing can hold at true scale and no eye can find against a ceiling. That invisible sag is the shape a pool of water finds, and the reason ponding is argued about in arithmetic rather than by looking.

The deflection that arrives later #

There is a fifth factor, absent from the formula, that is larger than most of the ones inside it.

Concrete and timber creep. Held under sustained load, they go on deforming for years at constant stress, and the long-term deflection of a reinforced concrete beam is commonly two to three times the value the elastic formula gives. Timber under permanent load does something similar, by a factor between one and a half and two depending on moisture. Steel does not creep at ordinary temperatures, which is one of the quieter reasons steel floors are argued about in different terms from concrete ones.

Creep multiplies the coefficient. It does not touch the exponent, which is a fact about lever arms and integration and cannot be altered by what the beam is made of. But a design conversation that spends an afternoon on whether the coefficient should be $5/384$ or $1/185$ because of partial end fixity, and then applies a creep factor of $2.5$ chosen from a table, has spent its afternoon on the smaller number.

The order in which the effects should be argued about is therefore roughly the reverse of the order in which they usually are: span first, because it is a fourth power and nothing else comes close; then depth, because $I$ is a cube; then the long-term factor, which is a multiple; and only then the coefficient, which is a fraction between a fifth and one. Which limit governs at all is a question worth settling before any of them.

There is a real consolation in it. A deflection that arrives over ten years is invisible to everyone who uses the building, because nothing is there to compare it against — the eye reads a sag against a straight edge, and by the time the sag exists the straight edge has gone with it. What people notice is not deflection but change in deflection: the door that starts to bind, the crack in the partition, the floor that moves underfoot when someone walks across it. Those are the limits worth writing, and they are limits on increments and on rates rather than on totals.

Three different fourth powers, and which one is being quoted #

The exponent is not a property of beams. It is a property of a comparison, and the comparison has to say what is being held fixed. Three are in circulation and they give three different answers.

Same load per metre, same section. $\delta \propto L^4$. This is the curve at the top of the page, and it answers the question what happens to this beam if it is made longer — which nobody ever does, because a beam made longer is also made bigger.

Same total load, same section. Now $w = W/L$ and $\delta = 5WL^3/384EI$: the third power. This is the honest comparison when the load is a fixed quantity being carried further — a piece of plant, a stack of goods — rather than an intensity spread over an area.

Same load per metre, beam resized to the same stress. This is what actually happens as spans grow, and it is much gentler than either. Holding stress constant needs $Z \propto wL^2$, so a geometrically similar section has to scale linearly as $L^{2/3}$, giving $I \propto L^{8/3}$ and

$$\delta \propto \frac{wL^4}{L^{8/3}} = L^{4/3}, \qquad \frac{\delta}{L} \propto L^{1/3}.$$

The deflection ratio grows as the cube root of the span. That is the same result the span-to-depth formula gives from the other side — $\delta/L = (5/24)(\sigma/E)(L/d)$ with $\sigma$ fixed and $d \propto L^{2/3}$ — and it is the reason long-span structures are possible at all. A designer who took the fourth power literally would conclude that a forty-metre span deflects sixteen times as much as a twenty-metre one; it deflects about 2.5 times as much, and the difference between those two numbers is the depth that was added on the way.

None of which softens the first section’s argument. The exponent that matters for self-weight is the one that does not let the section grow to keep up, and that is where the square-cube law bites. What the three comparisons separate is the fourth power’s two jobs: it is a savage law about a beam that cannot change, and a mild one about a family of beams that can.

Where the model stops #

Constant load intensity. The fourth-power curve holds $w$ fixed while $L$ varies, which is the honest comparison for an imposed load and dishonest for self-weight.

Constant section. Real long-span members are deeper, so a real comparison would move along the curve and change $I$ at the same time.

Elastic material. The formula is a linear-elastic result throughout.

Bending only. Shear deflection is neglected, which is correct for slender beams and wrong for deep ones. For a truss, shear deformation of the web can contribute a quarter of the total movement, and for a deep short beam it dominates.

Small deflections. A cable’s deflection changes its geometry enough that no power law applies at all.

The figure at the top has an honest distortion: it plots relative deflection against relative span, which makes the fourth power visible and hides that the absolute values are tiny. A beam at a hundred times the deflection of a short one has still moved perhaps twenty millimetres, and the curve says nothing about whether any of the plotted points is acceptable. It is a statement about ratios, and the limits are statements about absolutes.

The ladder from here #

Later rungs: the standard deflection coefficients and their derivations. The unit-load method, which gets a deflection at one point without solving the whole beam. Maxwell’s reciprocal theorem. Shear deflection and when it matters. The square-cube law across forms. Span limits for each structural type. Self-weight fractions in long-span structures. Scaling in nature, and why large animals are not large small ones. And the structural index, which compares forms by how much they can span for a given weight.

Galileo’s two bones are the oldest drawing in structural engineering, and the argument they make has never been improved on. Everything since has been about how to arrange material so the limit arrives later.