Brick Wall Statics (2D): Wind Overturning, Middle Third, Bond Patterns and the Harmonic Overhang

Presets

Live readouts

Graphs

Experiments

Wall: Gust test applies the chosen wind for the gust duration and lets it go; Ramp raises the wind by 1 m/s every second until the wall fails. Both restart the wall first.

Numerical checks

running...

How to use

Choose a preset or an experiment. In wall mode tick Wind blowing and raise the speed, press Gust test to apply the wind for a limited time, or Ramp to find the failure speed. The picture shows the face in the chosen bond (red joints run straight through) and the magnified base section: the orange block is the compressive stress, the yellow dot the line of thrust, and the green band the middle third. In corbel mode set the number of bricks and the fraction of the harmonic limit: above 100% the stack tips about a joint. Space pauses, R resets.

The science

A wall is a stack of weights held in place by gravity. Dry or lightly mortared brickwork has almost no tensile strength, so its stability is a question of where the line of thrust passes through each joint. Per metre run of wall, with height h, thickness t and masonry density rho: W = rho g t h, M = q h^2 / 2, e = M / W = q h / (2 rho g t) q is the net wind pressure, q = (1/2) rho_air Cp v^2, and e is how far the resultant force on the base moves from the centre of the section toward the leeward toe.

Middle third, cracking, overturning. The base stress is sigma = W/t (1 +/- 6e/t) while the whole joint is compressed, which holds for e <= t/6 (the middle third). Beyond that the windward joint opens and the load is carried by a triangular block of length c = 3 (t/2 - e) with toe stress 2W/c. When e = t/2 the thrust reaches the toe and the wall rocks over it: q_over = rho g t^2 / h, q_crack = (W/t + f_t) t^2 / (3 h^2), q_slide = mu rho g t A mortar with flexural tensile strength f_t delays the first crack, but once the joint has opened only the weight is left, so mortar does not change the overturning pressure. Thin walls are slender in a very literal way: halving t cuts the overturning pressure to a quarter.

Toppling dynamics. The wall is a rigid block rotating by theta about the leeward toe with moment of inertia I = m (t^2 + h^2) / 3: I theta'' = q h^2/2 + W ( -(t/2) cos theta + (h/2) sin theta ) For a steady wind above q_over the torque is positive for every angle, so the wall never comes back. A short gust can be survived because the wall falls back if it has not passed the tipping angle atan(t/h) when the wind stops, which is why the gust duration matters.

Bonds. A bond pattern changes how head joints (the short vertical gaps) line up from one course to the next. Stack bond has no overlap at all: its vertical joints run straight through and the wall behaves like independent columns, which is why it is mainly decorative. Running bond overlaps by half a brick; one-third running overlaps by a third; Flemish and English bonds alternate headers and stretchers so that the wall holds together across its thickness. The red joints in the face view are those that continue through two or more courses. Bond affects horizontal bending strength and the cracking pattern; the free-standing overturning limit above is the same for all bonds.

The overhang puzzle. Stack N identical bricks of length L at the edge of a table, each pushed out as far as it can go. The top j bricks are stable on the brick below as long as their joint centre of mass lies over that brick. Working down from the top gives the harmonic series: overhang of brick k = L / (2k), total overhang = (L/2) (1 + 1/2 + 1/3 + ... + 1/N) ~ (L/2) (ln N + 0.5772) The series diverges, so any overhang is possible, but it grows only like ln N: one brick length needs 4 bricks, two lengths 31, three lengths 227. Equal steps do much worse: the best equal step is about L/N and the overhang never exceeds about one brick length. The stacks shown are one brick wide (brick on brick); two-dimensional counterweighted stacks do better, but need many more bricks to describe.

Honest simplifications. (1) The wall is rigid and its base is a single joint at ground level: no foundation settlement, no soft soil, no flexure of the wall between ground and top. (2) Wind is a uniform net pressure on the windward face with a single coefficient Cp; real wind is gusty, varies with height and wraps around the ends, and a wall of finite length is helped by its end returns. (3) After the thrust leaves the middle third the base stress block assumes a no-tension material and linear stresses; real brickwork also crushes locally. (4) The impact when a rocking wall falls back is plastic (it stops); the fallen wall is not followed after it reaches the ground. (5) Friction is a single Coulomb coefficient; sliding and rocking are not coupled. (6) The overhang stack is dry with perfect friction at the pivot edge. Real garden-wall design uses codes with safety factors and takes these formulas only as a start.

Checks run in the page. Max overhang from a greedy centre-of-mass construction against the harmonic sum (N = 4, 10, 30); a harmonic stack at 99.9% of its limit is stable and at 100.1% fails; the overturning pressure found by bisection on the rocking ODE against rho g t^2 / h; conservation of energy for the toppling wall integrated with RK4; and the integral of the base stress block against the weight and the wind moment.