The middle third

Most of this collection is about materials that can be pulled. Steel is as good in tension as in compression, reinforcement exists precisely so that concrete can be, and every bending calculation drawn so far has a tension side that carries stress.

Take that away — masonry, unreinforced concrete, a plate bearing on grout, a footing on soil — and a new question appears that has no analogue in the rest of the subject. The material cannot pull, so the resultant of everything pressing on the section has to land somewhere that leaves no part of it in tension. That somewhere is a region, and it has a name.

The vertical half-width is the middle third, arriving as a piece of section arithmetic rather than as a rule:

$$e_{\text{max}} = \frac{Z}{A} = \frac{bh^2/6}{bh} = \frac{h}{6}$$

which is a third of the depth, centred. The same $Z$ that decides how much moment a section can take decides how much eccentricity it can tolerate, and the second use is the one nobody teaches. Everything else in this essay follows from noticing that $Z/A$ is a property of a section, so it has a different value for every shape, and that the argument runs in every direction and not only the two the drawing has axes for.

Why it is a rhombus and not an ellipse #

The condition to be satisfied is that the stress at every point of the section stays compressive. For a resultant at $(e_x, e_y)$ that stress is

$$\sigma(x,y) \propto 1 + \frac{e_x x}{r_y^2} + \frac{e_y y}{r_x^2}$$

with $r^2 = I/A$ the radius of gyration squared. Each point of the boundary supplies one inequality, so the kern is the intersection of a half-plane per boundary point — and the intersection of half-planes is a polygon.

For a rectangle only the four corners bind, so the kern has four sides. Its vertices are on the axes at $b/6$ and $h/6$, and the sides run straight between them, which is why the diagonal is the tight direction: a resultant on the diagonal is being watched by a corner, and a corner is further from the centroid than a face is.

The practical consequence is the one the refutation states. A check that reads “the eccentricity is within b/6 across and within h/6 along, so it is inside the middle third” is checking a rectangle when it should be checking a rhombus, and the two differ by up to 38% on the diagonal. Biaxial eccentricity is not two uniaxial checks, which is exactly the same statement an interaction diagram makes about combined actions and is missed for exactly the same reason.

Every section has one and they are not alike #

The I-section is the surprise, and the reason is instructive. Its kern half-depth is $Z/A$, and an I-section is precisely the shape that maximises $Z$ for a given $A$ — putting the material at the extremes is the whole point of it. The kern is therefore enormous: ±176.6 mm on a 600 mm section, which is 58.9% of the depth against a rectangle’s 33.3%.

An I-section on a bearing that cannot pull is far more tolerant of an eccentric load than a solid one of the same depth. That is the opposite of what the phrase “middle third” leads people to expect, and it is worth stating as a rule: the kern is generous exactly where the section is efficient in bending, because both are measured by $Z/A$.

Across the section the same efficiency works against it. The I’s horizontal kern is ±39.0 mm against the rectangle’s ±66.7, because almost all its area is at the middle of its width. A section that is excellent in one direction is poor in the other, and here that shows up not as a stiffness but as a tolerance.

Outside the kern, the arithmetic changes kind #

The kern is not a strength limit. Nothing fails when the resultant crosses it. What happens is that part of the section stops working, the section that remains is smaller than the one that was designed, and every stress in it is computed from a different set of equations.

The factor of two at the kern boundary is worth remembering, because it is the same for every section: at the kern the stress block is a triangle from zero to $\sigma_{\max}$, whose average must be $N/A$, so the peak is $2N/A$ whatever the shape. Reaching the edge of the kern doubles the peak stress, and the design decision about whether to allow it is usually made on that basis rather than on the uplift itself.

Past it, the unknown is the contact length, and it is found by requiring the pressure block’s centroid to sit under the resultant. For a rectangle that gives $c = 3(B/2 - e)$ and a peak of $2N/Dc$ — an expression that goes to infinity as the resultant approaches the edge, which is the mathematics saying that a rigid body on a rigid surface balanced on its corner carries infinite stress.

Where the same idea keeps arriving #

The kern is one of the few pieces of arithmetic in this collection that appears identically in four fields, and it is worth collecting the appearances because the vocabulary hides the identity.

A whole body being blown over. Uplift at the windward edge begins exactly when the resultant leaves the base’s middle third, and the ratio between that height and the toppling height is √3.

A masonry arch. The line of thrust must stay inside the middle third of the ring for the joints not to open, which is the same condition applied section by section along a curve.

A base plate on grout. The plate is a section that cannot pull, the holding-down bolts are what happens when it does, and the moment at which they start working is the moment the resultant leaves the plate’s kern.

A prestressed beam near its ends. With no applied moment to help, the requirement that the top fibre stay in compression is exactly the requirement that the prestressing force land inside the kern.

Four fields, four vocabularies, one inequality. That is worth stating because it is how a subject is actually organised: the useful unit is not the topic but the condition, and the resultant must land inside Z/A is a condition that reaches from a Roman arch to a post-tensioned floor.

Reading it backwards, as a way of sizing things #

The kern is usually met as a check: compute the eccentricity, compare, worry. Read the other way it is a sizing rule, and in that direction it is one of the fastest pieces of arithmetic in structural engineering.

A footing under a column carrying $N$ with a moment $M$ has an eccentricity $e = M/N$, fixed before anything is chosen. Requiring the resultant to stay inside the kern then gives the base length directly:

$$\frac{M}{N} \le \frac{B}{6} \quad\Longrightarrow\quad B \ge \frac{6M}{N}$$

so a column with 900 kN and 300 kNm needs a base at least 2.0 m long if the whole of it is to stay in contact. That is a dimension arrived at in one line, with no soil property, no concrete grade and no reinforcement in it — and it is usually within a size or two of the final answer, because the eccentricity is the thing that decides base plan dimensions and the bearing pressure is the thing that decides them only when the moment is small.

The same reading sizes a wall thickness for a given thrust eccentricity, a base plate for a given column moment, and the prestress a beam needs at a section where the tendon must be near the centroid. In each case the quantity that goes in is a ratio of two actions, which is often known long before either action is, and that is why the rule survives as a design instinct: eccentricity over depth is a dimensionless number an engineer can carry in their head, and a sixth is the value at which the arithmetic changes.

The rule is older than the mechanics #

The middle third predates every piece of theory in this essay. It appears in eighteenth- and nineteenth-century masonry practice as a rule about joints — keep the thrust within the middle third and the joint will not open — and it was stated, argued over and applied long before Navier’s bending theory gave it the derivation above in the 1820s.

That order of events explains the rule’s peculiar status. It was arrived at empirically, from the observation that arch and wall joints open on the side away from the thrust, and it was then found to be exactly $Z/A$ for a rectangle — which is a much stronger statement, because it generalises to every other shape and to sections nobody had built with. The empirical rule and the derived one agree for the case that generated it and diverge everywhere else, which is the usual fate of a good rule of thumb.

There is a second reason it stuck. Masonry design in the nineteenth century had no way to compute stresses in a curved structure, and the middle third gave a geometrical criterion — draw the thrust line, see whether it stays in the middle third of the ring — that could be executed with a straightedge. That is drawing as calculation in its purest form, and it is why the rule outlived several generations of the theory that justified it.

Heyman’s twentieth-century reworking made the modern position clear: the middle third is sufficient for a masonry structure to be safe and it is not necessary, because a joint that opens is a hinge and a structure with hinges can still be perfectly stable. The rule is conservative, it was known to be conservative, and it was kept because the alternative required knowing how much hinge rotation the structure could take — which is the same trade the safe theorem makes everywhere it is used.

The one section whose kern contains nothing #

There is a case that reads as a paradox and is not. A thin ring — a chimney, a silo, a tubular pile — has almost all of its area at the extreme fibre, so its $Z/A$ is close to $R/2$, and the kern is a disc of half the radius. That is by far the most generous kern of any shape, and it is why a chimney can take an enormous wind moment without uplift.

The opposite extreme is a section whose material is concentrated at its centroid, for which $Z/A$ goes to zero. A cable is the limiting case: its kern is a point, it cannot take any eccentricity at all, and that is precisely the statement that a cable can carry no bending. The kern is a measure of how much bending a no-tension section can absorb before it stops being a section, and it runs from a point for a cable to half the radius for a thin tube.

Between them, the ordinary shapes: 33% of the depth for a rectangle, 25% of the diameter for a solid circle, 59% for an I-section, and the limits of both extremes reached by things that are barely sections at all.

What the picture cannot show #

Every figure here assumes a rigid section on a rigid support. A real base plate bends, a real footing is flexible relative to the soil under it, and the linear pressure distribution the whole calculation rests on is an idealisation whose error grows with the flexibility. The kern boundary is robust — it depends only on where the resultant is — but the peak pressure past it is not.

The material is assumed to have unlimited compressive strength. In practice the peak pressure runs into a bearing limit long before the contact length approaches zero, and what actually happens beyond the kern is a plastic block of limited stress and finite length rather than the triangle drawn. The base-plate figure uses that treatment; the others use the elastic one, and they disagree once the eccentricity is large.

Nothing here has a safety factor in it. The kern is a geometrical boundary and not a limit state: designs routinely and deliberately operate outside it, with the consequences computed rather than avoided. What the boundary is good for is knowing which calculation applies, and that is a question with a yes-or-no answer.

Where the ladder goes #

The first rung is the arch, where the same condition applied continuously along a curve becomes a thrust line and a collapse mechanism — and where the interesting question is not whether a line exists but how many can.

The second is the section that has left its kern and is working on a reduced area: the cracked no-tension section, whose depth of compression is itself an unknown and which behaves nonlinearly under a load that is applied linearly.

The third is the one this essay has kept close to the surface. Everything above is about a material property — the inability to be pulled — expressed as a geometric region. That translation is what makes the rule usable by somebody with a drawing and no calculation, and it is why the middle third has survived as a rule of thumb for four hundred years while the mechanics behind it was being rewritten twice.