Why a one-brick wavy wall stands

Overturning

Wind pushes. Weight pushes back. Take a straight wall of height H and thickness t, and a steady pressure p on the face. The wind's force on a unit length is pH, and it may be taken as acting halfway up, so the overturning moment is pH²/2.

The wall weighs ρtH, with ρ the weight of a cubic metre of brickwork. A rectangular base stays in compression only while the resultant falls inside the middle third. That allows the weight an eccentricity of t/6, and a resisting moment of ρtH × t/6. Set the two moments equal and the pressure the wall can meet is ρt²/(3H).

The comparison on this site uses ρ = 19 kN/m³ and a breeze of 0.15 kN/m², about 35 miles an hour. One wythe is 102.5 mm thick. At the Suffolk height of 1.65 m, a straight single wythe can meet only 0.04 kN/m². The height-to-thickness ratio is 16.1. The same breeze, on that thickness, keeps the whole base in compression only up to about 0.44 m, a ratio near 4.3. The garden wall is several times that. Two wythes, at 205 mm, raise the allowable pressure by four, because both the weight and the lever arm double. Here that is 0.16 kN/m², just clear of the breeze. The single straight wall falls. The double one stands, narrowly.

What the sine adds

Write the plan as z = A sin(2πx/λ). A straight wall keeps every brick within t/2 of the centre line, about 51 mm. The sine carries brick out to the amplitude. The second moment of the section grows with the square of that offset, which is the section modulus story in one sentence: the same bricks, moved off the centre, resist bending as if the wall were much deeper.

There is a second way to see it, and it is the one a mason would trust. Each bay is a shallow arch laid on its side. Wind along the run is the load, and the curve carries it in compression toward the next crossing. Brickwork is good at compression and poor at tension. The arch lets it do the thing it is good at. In this model the Suffolk curve can meet about 1.06 kN/m², against 0.16 for the straight double wall and 0.04 for the straight single wall. The figure is a model of that idea. It is not a calculation a building official would stamp.

How deep, for the length

The Brick Industry Association's note on garden walls gives a rule of thumb for a minor wall about 4 inches thick, built from circular arcs rather than a sine. The radius of each arc should be no more than twice the height above the ground, and the depth of the curve no less than half that height. The rule is limited to garden walls. It is not a code clause for a wall that holds people or soil.

On the Suffolk bay the amplitude is 0.82 m and the height is 1.65 m, so the depth is half the height, which is the line that note draws. A sine is a different curve. It is nearly straight where it crosses the centre, and tightest at each crest. Builders who set out these walls in arcs of constant radius were avoiding that flat crossing. The American Brick Institute notes, as a Warwick survey of wall shapes records them, describe durable serpentine walls as deep, with a depth not less than half the height.

The brick count

The bricks do not walk the straight run. They walk the arc,

L = ∫ from 0 to R of √(1 + (2πA/λ · cos(2πx/λ))²) dx

Take the Suffolk bay on the front page. The run is 14 m, the amplitude 0.82 m, the wavelength 4.4 m, and the height 1.65 m. The integral comes to 18.05 m, about 29 percent longer than the straight run. The English module is 225 mm along the wall and 75 mm up. That height is 22 courses. The arc holds 80 bricks in a course, and 22 courses make 1,760 bricks. The straight wall, two wythes thick, holds 62 bricks in a course and then doubles, which is 2,728 bricks. The curve saves 968 bricks, 35 percent, on this bay and in this model.

The saving is not a constant. A Library of Congress caption on Jefferson's walls puts the saving at about a quarter. That is one comparison, on one geometry. Bend the wave harder and the arc eats the advantage. The slider below keeps the run, the wavelength, and the height fixed, and moves only the amplitude.

Where the walls are

England has a crowd of them, and Suffolk kept the most that people still point to. The Dutch name is slangenmuur, a snake wall, and the hollow of a bay was a place to ripen fruit out of the wind. Thomas Jefferson used the same plan at the University of Virginia, which he set out from 1817 and which was chartered in 1819. The garden walls between the pavilions are one wythe, about 6 to 7 feet high. He did not invent the form. A paper in his own hand works the saving in bricks, set against a straight wall that needed two thicknesses.

The model leaves out cut bricks, a thicker footing, spilled mortar, and any pier a builder still added at a gate. A wall that keeps people off a drop, or that holds soil, is an engineer's wall.