The pressure the openings decide

Most of it is suction added up the wind on a building’s faces, and it treated the inside in one section with a table. A sealed building stands at about −0.3, a door blown in on the windward face takes it to +0.7, and a door on the leeward face to −0.5. On the 600 m² roof those three numbers made uplifts of 108, 648 and 0 kN against a roof weighing 360 kN, and the essay drew the only conclusion available: the door goes first and then the roof. That is a good reason to look harder at where the three numbers come from, because the difference between the first two is the difference between a building that stays and one that does not, and nothing has yet said how large a door is a door.

A table has rows, and the air has no rows. The air inside the building is a free body, one of the simplest there is. Its weight is nothing, its inertia is small, and the only things acting on it are pressures. The inside of a building cannot go on filling up, and so whatever comes in through the openings on the windward side must leave through the openings elsewhere. That one sentence is a statics equation with one unknown.

One unknown, and the sum that finds it #

Take a set of openings of area $A_j$, in faces with external pressure coefficients $c_j$, and an internal coefficient $c_i$ that is not yet known. The flow through an opening goes as the area times the square root of the pressure drop across it, by Bernoulli, as at any orifice. Air coming in is positive, and air going out is negative. So

$$\sum_j A_j\,\mathrm{sign}(c_j - c_i)\,\sqrt{|c_j - c_i|} = 0.$$

The discharge coefficient and the air density are the same at each opening, so they cancel and do not appear. The sum falls steadily as $c_i$ rises, because raising the inside pressure lessens every inward flow and adds to every outward one, so it has exactly one root, between the largest and smallest external coefficient. That is the entire theory.

For a building, take the 30 by 20 by 24 m one of the earlier essay, at $q = 0.9$ kPa. It has coefficients of +0.8 on the windward wall, −0.5 on the leeward, −0.7 on the two sides and a mean of −0.5 on the roof. A building is not sealed, so give each face a leakage of a thousandth of its area, a reasonable figure for ordinary cladding: 0.72 m² on each of the windward and leeward walls, 0.96 m² on the sides, and 0.6 m² on the roof. That is 3 m² in total, the size of a double door, spread over 2,000 m² of skin.

Each curve crosses zero once, and that crossing is the answer. The curve is steep through the zero, since near that point the flows of a large area change quickly, and it gets flatter toward the ends. Leakage alone balances at −0.44, a little more suction than the −0.3 of the table. That difference matters, since a sealed roof’s uplift was $(c_i + 0.5)qA$. At −0.3 that gave the earlier essay’s 108 kN and here it gives 31 kN. The reason the building sits so low is simply that more of its skin is on the suction side, so three quarters of its leakage is drawing air out.

Note what does not appear. The total amount of leakage is not in the answer: double every area, and the balance is the same. Only the ratios among the openings matter. That is a useful robustness, since a building’s real leakage is known at best to within a factor of two or three, and a sealed coefficient that depended on it would be useless. The second curve, with a door of 4.8 m² added on the windward side, moves the crossing all the way to +0.60, and there the argument starts to bite.

How large a door is a door #

The roof’s uplift is the external suction plus whatever the inside adds to it. Walk the opening up in size:

A window of 1.8 m² is enough. This is a very different statement from “a door blown in”. An ordinary window, a metre by a little under two, is a quarter of one per cent of the windward wall. It increases the inside pressure until the 360 kN roof is carrying no net load, in a wind whose velocity pressure of 0.9 kPa is a good storm rather than a hurricane. At the door’s 4.8 m² the uplift is 593 kN, 233 kN more than the roof weighs, and most of the way to the earlier table’s 648. Beyond that the curve saturates: at 20 m² the inside has very nearly reached the external 0.8 and the uplift can only go a little further.

The hand check is short. At 1 m² of opening on the windward wall, the total windward area is 1.72 m². Try $c_i = -0.08$: inward flow is $1.72\sqrt{0.88} = 1.61$. Outward flow is $0.72\sqrt{0.42} + 0.96\sqrt{0.62} + 0.6\sqrt{0.42} = 0.467 + 0.756 + 0.389 = 1.61$. It balances, and so an opening of 1 m², a third of the total leakage, is already worth a shift from −0.44 to −0.08 in the coefficient. It is worth 196 kN on this roof.

The sequence in a real storm follows from the shape of that curve. A window breaks, usually from debris rather than from pressure, and it’s on the windward side because debris travels downwind. The inside goes up. The cladding that was designed for a sealed building now has the suction on its outer face and a push on its inner face, and it starts to go, somewhere on the side walls or the roof. Those new openings are in the suction zones, and they let air out, so the inside falls back part of the way. It is a remarkable feature of this equation that a second failure, on the far side of the building, relieves the first. That is, the roof’s uplift may be largest just before the second hole forms. This is why the failures seen after storms are so often a partial loss of a roof, stopping once a large enough piece of it has gone.

The sequence has a counterpart at the other extreme of the subject. Held up by the air inside describes a structure in which the internal pressure is the load path, deliberately kept up by fans against every leak. The balance there is the same sum with a fan as one more term. A pressurised roof is just the case in which someone has decided in advance which way the sum should come out, and a dominant opening in an ordinary shed is the same sum left to chance.

Reading the standard as a curve #

Having a curve means it can be compared with the rule. Eurocode 1 gives two treatments. For a building with no dominant face it gives a chart of $c_{pi}$ against the opening ratio, from +0.35 to −0.5. For a dominant face, defined as one with at least twice the openings of all the others combined, it sets the internal coefficient to a multiple of the external one: 0.75 of it at a ratio of two, and 0.9 at a ratio of three or more.

The standard’s two points sit about a tenth above the balance. So they are the balance, with a margin, and with the reasoning left out. What the figure adds is the whole range between those points, particularly below the ratio of two where a building is not officially dominant at all. At a ratio of one, with the windward opening equal to all the rest, the balance is already at about +0.11, a positive internal pressure in a building whose windward wall has only half of its openings. At a half it is still −0.30. Below two, the Eurocode’s chart for no dominant face runs up to +0.35, and that keeps this region covered, but the curve shows it is a region of rapid change. A designer who judges that a door is not quite dominant, on the basis of a guess at the leakage, is standing on the steepest part of the curve.

The curve also supplies a simple design rule that the table hides. To protect a building from a windward opening, what matters is not how strong the door is but how much other opening exists. A vent on the leeward side, or a louvre in the roof that opens under pressure, moves the ratio directly. This is how sheds with large doors are often designed, without anyone quite saying so, and it’s why a ridge vent is structurally useful as well as for ventilation.

The wall nobody thought was on the outside #

So far the inside has been one room. Most buildings have more than one room, and each partition is an opening of a different kind. Air gets through, but less readily, so there are now two unknowns, one for each room. Each room has its own balance, and each partition is one of the openings of both.

The solution gives something a table of building coefficients cannot show. At a leakage of 0.25 m² through the partition, about what a closed internal door and some gaps would provide, the windward room has reached almost the external pressure, and the leeward room has stayed at close to the sealed value. So the partition sees 1.07 kPa across it, with no wind touching it directly. The windward wall itself has only $0.8q = 0.72$ kPa of external pressure on it, and with the inside pressure balancing it, that wall is now under almost nothing. The load has walked straight through the opening and landed on a stud partition nobody designed for it. A plasterboard partition designed for a light horizontal load of a few hundred pascals will not stand this, and the window in the wall has become a door in the partition.

Then, as the partition’s leakage grows, the two rooms come together. By 5 m², the inside is one space with one pressure, and the load on the partition has gone. A partition is thus a load-attracting element in the sense of the stiffest path takes the load, with air flow in place of stiffness: the tighter it is, the more it carries, and a well-built tight partition is at more risk than a poor leaky one. And yet the tight partition is doing something useful, since it protects the leeward half of the roof. With the rooms at 0.78 and −0.41 the windward half of the roof carries the whole uplift of a dominant opening, and the leeward half carries nearly none, so a partition that holds is keeping half the roof on.

That is the structural argument for compartmentation under wind, and its price is the plain one in every load-sharing problem: a detail that stands up to a load gets the load.

The air is also a mass on a spring #

The balance is steady, and a broken window is sudden. Air can’t get to its new pressure instantly: it has to flow in through the opening, and it has mass. The plug of air in the opening is a moving mass $\rho L_e A$, with an effective length $L_e$ of about $0.89\sqrt{A}$. The air inside the room is a spring, since pushing more air in compresses it, with stiffness $\gamma p_0 A^2/V$ on the plug. Together they make a resonator, the same one that makes a bottle hum, with a frequency named after Helmholtz:

$$f_H = \frac{1}{2\pi}\sqrt{\frac{\gamma p_0 A}{\rho L_e V}}.$$

Holmes wrote this down for buildings in 1979, and it is the same kind of object as the water in a tank, where the liquid has a period of its own that has nothing to do with the structure around it. The only damping is the loss through the opening, which goes as the square of the flow.

For the whole building, 14,400 m³ with a 4.8 m² door, $L_e$ is about 2 m and the frequency works out at 0.72 Hz. For a small room of 120 m³ with a 2 m² window it is 6.3 Hz.

So a suddenly broken window on an otherwise closed room causes the pressure to go past its steady value and swing back, as a step load does on any mass on a spring. A lightly damped system would reach nearly twice the steady value. The quadratic loss does a good deal to stop that, and the amount it does depends on how far the air has to move. In the small room, air has little distance to go and little to lose on the way, so the pressure overshoots to 1.35, 69 per cent over the steady 0.8, and it does so within a tenth of a second. In the big building, the plug has to travel so far to fill the volume that the loss is large, and the overshoot is 5 per cent, spread over a second and a half.

A room is therefore worse off than a hall, though its opening is smaller. Standards’ coefficients don’t include this overshoot, and they don’t need to for roofs and walls of massive construction, where a tenth of a second is short compared with their own response. For glazing and light cladding, a tenth of a second is plenty, and an impulsive pressure inside a small room is the likely reason that one broken window so often takes the other windows in the same room with it.

The steady wind is not steady either, and the second reading of the resonator is what it does to gusts.

Below the resonance the inside simply follows the outside, ratio one, which is the justification for treating internal pressure as quasi-static. Above it the inside can’t keep up, and small gusts are filtered out entirely. In between lies the peak. The wind is a spectrum put most of a gust’s energy well below 1 Hz, so a large building’s resonator, at 0.72 Hz with a large opening, sits right on the tail where the wind still has useful energy. The curve then says that the inside’s fluctuating pressure can exceed the outside’s by a factor of up to 1.5 for small gusts, and that large gusts are damped more strongly: at a fluctuation of 0.2 the peak is 1.14. This is one of the few places where a structure’s response to wind has an amplitude-dependent gain, in which big storms are better damped than small breezes.

There is a further interaction that the figure leaves out. A light roof moves when the pressure changes, and a roof that gives adds volume, softening the air spring and bringing the frequency down, towards the wind’s energy. Vickery showed that the effect can be significant for flexible roofs on large buildings, which are the ones most exposed to it.

What one coefficient per face hides #

One external coefficient on each face. A real windward wall has about 0.8 at its centre and less near its edges, and the side walls have a large suction near the windward corner and little at the far end. An opening close to that corner is a much stronger drain than the mean suggests, and the balance is only as good as the coefficient at the opening itself.

The discharge coefficient is one number. It depends on the shape and on the opening’s angle to the wind, and a louvre or a grille will behave differently from a window. An opening’s cross-flow also changes matters when the wind blows across rather than at it.

Areas are fixed. Doors open further when pushed, cladding laps open under suction, and gaps at the head of a wall widen. So a building’s leakage depends on the load, and the balance is then a coupled problem with the cladding’s own stiffness. The figure takes it as constant.

The air is treated as a uniform pressure. At these frequencies that is a good assumption in a room, and a poorer one in a long building with a door at one end, where a pressure wave takes a noticeable fraction of a second to arrive.

The roof’s uplift uses a mean external coefficient. The local peaks at eaves and corners are much larger, and those are where failures start. The balance sets the part of the load added inside, and the corner still has its own load on top. Equivalent in work, not in resultant is the reminder of what a smeared load can and can’t stand for.

The wind is steady across a storm. A shift in the wind direction moves an opening from windward to the side, and a vortex street can sit over a side-wall opening and drive it at its own frequency. The inside can then go through its whole range in a few seconds, from one extreme to the other, which makes it a reversal, whereas the figures here treat each direction as a separate state.

Still open: the corridor between rooms #

Two rooms joined by one partition have one difference to carry. A real floor has rooms on both sides of a corridor, with doors of different sizes and different leakage to each, and the corridor connects them. The balance extends to any number of rooms and is the same kind of problem as a pipe network, or as a resistor network with a nonlinear law. What it would decide is which partition, among many, ends up carrying the largest difference. It is not obvious whether that is the first partition behind the broken window or a tighter one further in. Finding out needs a real floor plan with measured leakage values, and few of those have been published.