It does not buckle, it runs out of width

Every stability failure so far in this collection has been a bifurcation. A straight column discovers it can be bent. A flat plate discovers it can ripple. A beam in its strong plane discovers it can swing out of it. In each case the member could have carried tension, chose not to, and there is a load at which the choice becomes indifferent.

A masonry wall does none of that, and the reason is that it cannot be pulled.

Which free body produced the number #

A horizontal cut through the wall, carrying a resultant at some eccentricity $e$ from the centre.

The material takes no tension, so it bears only where the stress is compressive. For a resultant outside the middle third the bearing strip is $3(t/2 - e)$ wide, with a triangular stress block over it — the same construction the middle third is about. The load that block can carry, at a crushing strength $f$, is its area times its peak:

$$N = \tfrac{1}{2} f \cdot 3\left(\tfrac{t}{2} - e\right) = 1.5 f (t - 2e)$$

Divide by the squash load $ft$ and the reduction factor is

$$\Phi = 1 - \frac{2e}{t}$$

Exactly linear, zero when the resultant reaches the face, and it contains no material property, no modulus, no height, and nothing about buckling. It is a statement about where a resultant can sit on a strip that cannot be pulled.

What the kern actually costs #

Put $e = t/6$ into that and $\Phi = 2/3$.

A wall loaded at the edge of its middle third — the case every masonry text treats as the comfortable one, the case a designer aims for — has already given away a third of its capacity. And because the relationship is a straight line, there is no region where the cost is negligible: an eccentricity of $t/20$ costs 10%, one of $t/10$ costs 20%, and the penalty is proportional all the way down.

That is worth separating from what the kern is actually for. The kern boundary marks where the joint stops opening, which is a serviceability matter — cracks, water, appearance, and the softening described below. It has never marked where the capacity was intact, and reading it as though it did is the source of the second refutation above.

The bearing width is the physical thing that is running out. At $e = 0$ the whole 215 mm bears. At $e = t/4$ only 161 mm does. At $e = 0.45t$ it is 32 mm — a strip an inch wide carrying the whole wall, on a joint whose flatness nobody measured.

Where slenderness comes in, which is not as a critical stress #

A slender wall does bow, and the bow does matter. But it enters the arithmetic somewhere unexpected.

The eccentricity applies a moment. The moment bows the wall. The bow adds to the eccentricity at mid-height, which adds to the moment — the load that makes itself worse, in a member with no tensile strength. Codes carry that as an additional eccentricity rather than as a reduced allowable stress:

$$e_a = \frac{h_{ef}^2}{2400\,t}$$

and the capacity becomes $\Phi = 1 - 2(e + e_a)/t$. For this wall $h_{ef} = 3{,}000$ mm, $e_a = 17.4$ mm, and the reduction goes from 0.600 to 0.438 — a further 27% lost to a bow of seventeen millimetres.

Notice the form. Slenderness has not produced a stress limit; it has produced a length, added to another length, inside a rule that was already geometric. The whole calculation still contains no material property, right up to the point where the reduced capacity is multiplied by $ft$ at the end.

The $h^2$ is why the effective height matters so much. Fixing both ends rather than pinning them takes $h_{ef}$ from 3,000 to 1,500, which takes $e_a$ from 17.4 mm to 4.4 and the reduction from 0.438 to 0.559 — a quarter more capacity for a restraint that carries almost no force. That is the brace that need not be strong arriving in masonry, and it is why a wall built into a floor slab top and bottom is treated so differently from one that merely reaches it.

And Euler is nowhere near #

It is worth doing the buckling calculation, once, to see how far away it is.

For a 215 mm wall of masonry at $E \approx 1000f$, three metres tall and pinned:

$$P_E = \frac{\pi^2 E I}{h_{ef}^2} = 5{,}449 \text{ N per mm of run}$$

against a squash load of 1,290 and an allowed load, after the eccentricity and slenderness reductions, of 565. Euler is 9.6 times the load this wall is ever permitted to carry.

Nor does the gap close as the wall gets more slender. Running the same sums up the range:

The ratio dips and then rises again, because the allowed load is being driven to zero by the eccentricity rule faster than Euler’s load falls. At 6 m the wall has no capacity left at all by the eccentricity rule while Euler still gives it 1,362 N/mm.

The wall runs out of bearing width before it runs out of stability, at every slenderness anybody builds at. That is why no masonry code in the world contains a critical stress, and why a designer who reaches for a slenderness curve from steel design is answering a question this material does not ask.

Why the eccentricity is there in the first place #

$\Phi = 1 - 2e/t$ makes the eccentricity the whole design, so it is worth asking where it comes from. Four sources, and none of them is a load in the ordinary sense.

The floor bearing on it. A slab spanning onto a wall bears over part of the thickness and its reaction sits at the centroid of that bearing — typically at $t/3$ from the inner face for a slab that stops halfway through, which is an eccentricity of $t/6$ before anything else has happened.

A change of thickness. A wall that steps in at a floor level delivers the load from above at the old centreline into a section on a new one.

Lateral load. Wind on the face, or retained earth against a basement wall, produces a mid-height moment which divided by the axial force is an eccentricity.

The bow itself, which is the $e_a$ above and is a consequence rather than a cause.

The first two are geometric, are decided in a detail rather than in a calculation, and are the ones that actually govern most walls. A designer with a slender wall in trouble usually gains far more by changing how the slab bears on it than by changing anything about the wall.

The wall as an arch, which is the other half of the same result #

There is a second reading of $\Phi = 1 - 2e/t$, and it connects this essay to the masonry arch rather than to the column.

A wall carrying a load at eccentricity $e$ is a structure whose thrust line enters at $e$ from the centre and has to reach the ground somewhere inside the section. The capacity reduction is the statement that a thrust line closer to the face has less section on either side of it to spread into. Slenderness moves the line further out at mid-height. Restraint holds it. A pinnacle — or in this case, a floor load applied on the far side, or the wall’s own weight above — pulls it back.

Every device in the buttress essay is available here, and they are usually available for free. A wall carrying substantial load from above is being helped by it: that load arrives at the centre of the wall and dilutes the eccentric one, exactly as a pinnacle does. So a wall lower down a building is not merely stronger because more of it is in compression; it is less eccentrically loaded, and the two effects compound.

The consequence is the one that surprises people about masonry buildings. The critical wall in a load-bearing masonry structure is very often the top storey, where there is least load from above to straighten the thrust line and the eccentricity from the roof bearing is at its most relative. That is the opposite of the intuition a column-based training produces.

The one case where it really is a buckling problem #

There is a corner of masonry where the linear rule is not enough, and it is worth naming because it is where the material’s own stiffness finally enters.

Very slender walls of high-strength units — thin-joint blockwork at $h_{ef}/t$ above about 27, which the table above shows the linear rule taking to zero — do have a genuine instability, and the more careful treatments handle them with a full second-order analysis on a cracked section rather than with an added eccentricity. The section’s stiffness is then a function of how much of it is bearing, which is itself a function of the load, so the problem becomes nonlinear in a way the additive rule cannot represent.

That treatment is what the $e_a$ formula is a linearisation of, and the reason the formula is written with 2,400 in the denominator rather than something derived is that it was fitted to the results of doing it properly. It is a curve fit standing in for an analysis, in exactly the way the Whitmore section is, and it should be read as one.

What the linear rule hides, which is a reserve #

There is one respect in which $\Phi = 1 - 2e/t$ is pessimistic, and it is worth stating because it explains why old masonry survives conditions the arithmetic says it should not.

The triangular stress block assumes the material stays elastic right up to its crushing strength. Masonry does not: it has a rounded stress-strain curve, and at high stress the block is closer to a parabola than a triangle. A parabolic block over the same bearing width carries about a third more load than a triangular one at the same peak stress, so the true reduction factor is above the straight line — and it is furthest above it exactly where the line is lowest.

That reserve is not usually claimed, and there is a good reason. Mobilising it means driving the toe of the wall to its crushing strain, which is where masonry has almost no ductility and where the joint’s workmanship stops being a detail. A wall relying on it has no warning left.

But the reserve is why walls at eccentricities the rule rules out have stood for centuries, and it is the physical content of the large partial factors mentioned above. The linear rule is a serviceable lower bound on an ultimate capacity that is genuinely higher and genuinely unreliable, which is a different kind of conservatism from the sort that comes from rounding a number down.

Where this model stops #

The wall is assumed to be a strip. Everything here is per unit length of a wall spanning vertically. A wall restrained on its vertical edges as well spans two ways, the effective height is reduced by a factor that depends on the panel proportions, and the whole calculation moves.

Self weight is ignored in the eccentricity. It should not be: the wall’s own weight arrives at the centre of the section and therefore reduces the average eccentricity of everything below mid-height. On a tall lightly loaded wall that is a real effect and it is conservative to ignore, which is why it usually is.

The crushing strength is treated as a constant. Masonry’s strength depends on the unit, the mortar, the joint thickness and the direction of loading, and it is the least well-determined number in the calculation — which matters little, since it enters only as a multiplier at the very end.

Concentrated loads are a different problem. A beam bearing on a small area produces a local stress far above the average and a spreading problem underneath it, which is the dispersion question rather than a slenderness one, and which is checked on a bearing area rather than on a strip.

What the picture cannot show #

The figure draws a smooth reduction against a smooth eccentricity, as though the wall were a homogeneous strip. It is a stack of units in mortar, and the joints are where the reality lives: the bearing area that is “running out” is a bed joint, its flatness is a workmanship question, and a joint that is slightly hollow bears on its edges rather than across its width regardless of what the resultant is doing.

That is the honest reason masonry capacities carry large partial factors while the arithmetic above is exact. The equation is a clean statement about a strip; the strip is an idealisation of a thing built by hand out of parts, and the scatter between them is larger than any of the refinements this essay has discussed. It is the size-effect argument in its practical form: the specimen a strength was measured on and the wall it is applied to are not the same object.

The generalisation #

The habit worth carrying is the question of what a member runs out of.

A steel column runs out of stability: it has plenty of section left when it goes. A short concrete column runs out of strength: the material reaches its limit with the geometry intact. A masonry wall runs out of width — the bearing area shrinks under a resultant that is moving towards its edge, and both the shrinking and the moving are geometry.

Those are three different failure modes wearing one word, and the design variable is different for each. Stability is bought with restraint and with second moment. Strength is bought with material. Width is bought by moving the resultant, which is why the most effective intervention on a slender masonry wall is nearly always a detail at the floor bearing rather than anything done to the wall itself.