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 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.

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.

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.

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.