The load that arrives where the wind stops

Every other load in this collection is applied to a structure. Snow is placed on one.

The distinction sounds like pedantry and it decides the whole calculation. A wind pressure exists because air is moving past a building now; stop the wind and the load is gone. Snow fell, some time ago, over a region tens of kilometres across, and what is on a particular roof this morning is the history of every wind that has crossed it since. Nothing about that history is in the meteorological number, which is a ground snow load — a water equivalent measured on flat open ground, where nothing gets in the way and the wind has nowhere to put anything.

Which free body produced the number #

A column of snow, one square metre in plan, sitting on the roof. Two things touch it: the roof below, pushing up, and gravity. That is the whole free body, and it says the pressure on the roof is the weight of the column and nothing else — depth times unit weight, with no dynamics and no direction in it anywhere.

So the load is a depth, and the entire question is how deep the snow is here rather than somewhere else. Fresh snow is about 1 kN/m³ and drifted snow between 2 and 3, having been broken up, carried and packed by the wind that moved it; the figures here use 2.5. At that density the balanced layer on this roof — 0.48 kN/m² — is 190 mm deep, which is a small number, and the wedge against the parapet is 0.85 m, which is not.

Nothing in that pair required a wind speed. It required only that the snow which is no longer on the upwind roof is now somewhere, and the somewhere is where the wind stopped being able to carry it.

The drift is a conservation of mass #

Take the roof upwind of the parapet as the accounting region. Some fraction of the snow that fell on it has been scoured off and carried downwind — call it 30%, which is a judgement and the only one in this calculation. It ran along the roof and it stopped at the parapet, where the flow separated and dropped what it was carrying.

The drift is therefore a wedge whose area in section is the area the fetch lost. If the wedge runs out four times as far as it rises, which is roughly what drifts settle at, then

$$\tfrac{1}{2}\,\lambda h^2 = \beta L_u d_g \quad\Longrightarrow\quad h = \sqrt{\frac{2\beta L_u d_g}{\lambda}}$$

and the depth is a square root of the fetch. That single exponent is the most useful thing anyone knows about drifting. It means a roof twice as long upwind produces a drift 1.41 times as deep, not twice; and it means that the cheap intuition — a bigger roof is a proportionally bigger problem — is wrong in the direction that matters, because the problem grows more slowly than the building does.

The curve also stops. A drift cannot pile higher than the thing that is stopping it, so past a certain fetch the parapet is full, the wind carries the rest of the snow over the top, and more roof adds nothing at all. That is the one place in the whole of snow loading where the answer becomes independent of the weather, and it is worth knowing where it is: for a one-metre parapet in this arithmetic it is at 28 m of roof.

The intensity is not the quantity, and this is the trap #

Read the two numbers on the section together and they look contradictory. The peak load in the drift is 5.4 times the balanced load. The drift is 27% of the snow on the roof.

Both are true and neither is a rounding of the other. The wedge is short — 3.4 m of a 20 m roof — so a load several times the design intensity, over a sixth of the plan, is a modest fraction of the total weight. A structure that carries its roof snow as a global quantity, like a set of columns or a foundation, barely notices the drift, in the way a whole-body free body never notices anything internal. A member whose tributary area happens to sit inside the wedge carries five times what its neighbour four metres away does.

So the same event produces two entirely different design problems from one drawing. That pairing is exactly what happens with wind on a building, where the largest pressures sit on the faces that contribute nothing to the base shear — and the resemblance is not a coincidence. Both loads are decided by the shape of the building rather than by the intensity of the weather, and in both cases the quantity that governs stability and the quantity that governs a local member are different quantities.

The load that weighs less and does more #

The second half of the argument has nothing to do with drifts and everything to do with continuity.

Wind moves snow from one part of a roof to another. It does not create any. So an unbalanced case — one slope of a pitched roof cleared, the other loaded; one bay of a multi-bay roof scoured, its neighbour buried — weighs the same as the balanced case at most, and usually less. It is very easy to conclude from that fact that the balanced case governs, and on a simply supported member it does.

On a continuous member it does not.

Both spans loaded gives a span moment of $9wL^2/128$. One span loaded gives $49wL^2/512$ — a third larger, for half the weight. The mechanism is the neighbouring span: when it is loaded, it pushes the shared support into hogging and that hogging reaches back into the span being designed, holding its sagging moment down. Clear the neighbour and the help goes away. The support moment halves at the same time, so nothing is uniformly worse and both arrangements have to be carried.

This is the same structure of argument as the imposed load envelope, and it deserves the same warning. The set of cases is not ordered. There is no worst one.

No single arrangement produces the envelope of all of them, which is what makes a load that can be present or absent a set of states rather than a state. The envelope is the outer boundary of a family, and no member of the family reaches it everywhere.

Why the pattern cases are the ones that get missed #

There is a practical reason the unbalanced case is skipped more often than any other in structural design, and it is worth naming because it is not laziness.

A balanced load is a number. It goes into a load schedule, it multiplies a tributary area, and every member gets the same one. An unbalanced case is a decision about geometry — which bay, which slope, which side of the ridge — and there are as many of them as there are ways to divide the roof. Nothing in the load schedule prompts anyone to make that decision, and a computer model handed the balanced case produces a complete, plausible, internally consistent set of results in which nothing is wrong except that the worst case was never run.

The check that finds it is the same check that finds every other pattern problem: draw the influence line for the quantity being designed, and put load only where it does harm.

Where the influence line is positive, load makes the quantity worse; where it is negative, load helps. An unbalanced snow case is nothing more than obeying that instruction, and it is worth noticing that the ratio it produces is a property of the arrangement rather than of the member.

That is a general instrument and it makes the snow question mechanical rather than imaginative. Load where the influence line is positive, leave the rest bare, and the arrangement that governs falls out without anybody having to guess which one it is.

Where the load acts, which is not where its total acts #

There is a third consequence of the wedge shape, quieter than the other two.

A triangular load and a uniform load of the same total do not produce the same moment diagram, because their resultants sit in different places. Replacing a distributed load by its resultant is exact for the reactions and wrong for everything internal, which is the point the distributed load essay makes in general and which a drift makes concrete: the wedge’s centroid is a third of its length from the wall, so its effect on the member nearest the wall is larger than its weight suggests, and on the second member smaller.

The reactions computed from the wedge and from a single force at its centroid are identical to the last figure. The peak moment is not, and the difference is largest at the section the substitution was made about — which on a drift is the section nearest the wall, where the member is usually smallest.

The roof next door, which is where this is worst #

There is one arrangement that combines every effect above and it is common enough to be worth naming on its own: a low roof beside a tall building.

The tall wall is an obstruction of unlimited height, so the drift never saturates and the square-root curve runs as far as the fetch allows. The low roof is downwind of a large area of high roof, which supplies the fetch. And a steep upper roof sheds its own snow onto the lower one by sliding, which adds a quantity the mass balance above does not contain at all, arriving as a single event rather than accumulating.

The result is that the highest snow load on almost any project is on its lowest roof, at the point where it meets something taller, and that the member carrying it is usually a modest one chosen from a table of spans. Nothing about the geometry looks severe. A canopy, a link corridor, a plant enclosure roof at the base of a tower: each is a small structure whose load case is decided by a large one standing beside it.

The design consequence is uncomfortable and worth stating plainly. The load on the low roof depends on a building that may not be in the same ownership, may be built later, and may be extended upwards. A roof designed correctly for an open site becomes wrong on the day a neighbour builds tall, and nothing in the low building has changed. Of the loads a structure carries, that dependence on somebody else’s geometry belongs almost uniquely to snow.

The wind that built the drift is not the wind the roof is checked for #

A drift is physical evidence that a wind has blown. That makes snow the one load in this collection whose worst case carries an implicit statement about a second load, and the two do not combine in the obvious way.

Snow moves at modest wind speeds — a few metres a second lifts loose dry snow, and the transport rate rises steeply above that. A design wind, by contrast, is a fifty-year event, and at those speeds a roof does not accumulate anything: the snow is scoured off and gone. So the deepest drift and the strongest wind are not merely independent events that happen to be unlikely together; they are close to mutually exclusive, because the conditions that build a drift are conditions the design wind destroys.

That is why load combinations reduce the accompanying variable action rather than adding two full design values. A drift at its fifty-year depth alongside a wind at its fifty-year speed is a combination with a return period far longer than either, describing weather that does not occur, and designing for it is not conservatism but a misreading of what the two numbers are.

The combination that does need care runs the other way, and it is the pattern this essay has already met twice. For a light roof the governing wind case is uplift, and snow is a gravity load resisting it. So the dangerous state is the strong wind with the roof bare — maximum uplift, minimum weight holding it down — and any calculation that carries snow along as a helpful permanent presence has done the same thing as counting a crane’s hook load as ballast. Snow is a variable action, its minimum value is zero, and zero is the value that governs every check in which it helps.

Which gives the pair of rules the whole section reduces to. Where snow is the action, take it at its worst arrangement and take the wind accompanying it as a fraction. Where snow resists, take it as absent. And notice that the two rules put snow at two extremes of the same winter, which is exactly what makes it an envelope rather than a state at a level above the one this essay has been arguing about.

Where the model stops #

Three places, and the first is the one every number above rests on.

The moved fraction is a judgement. Nothing here computes how much snow the wind takes off a roof; 30% is a plausible value and the honest description of the whole calculation is that a measured ground load and an assumed scour fraction are combined by arithmetic that is exact. Every published drifting rule contains the same free parameter, dressed as a coefficient fitted to case records rather than as an assumption, and the fitting is what turns a mass balance into a design rule.

Snow slides, melts and refreezes. A slippery pitched roof sheds its snow at somewhere around 15° to 30°, so the load falls off a steep roof rather than accumulating — and lands on whatever is below it, which is how a low roof beside a tall one acquires the highest snow load anywhere in a project. A snow layer that partly melts and refreezes has a much higher density than the one that fell, and ice at the eaves can dam meltwater into a pond, which is a different load with a different failure mode behind it.

The obstruction is not always a parapet. A step in roof level, a plant enclosure, a row of photovoltaic panels, a wall of an adjacent building, a solid balustrade added by a later tenant: all of them are obstructions, and the last of those is the one that matters most, because it arrives after the structure was designed. A drift load is one of very few load cases that a change of use can create out of nothing without anybody applying for anything.

What the picture cannot show #

The section shows a wedge with a clean triangular face, standing still. Neither is true.

Drifts are built and destroyed repeatedly through a winter. The one that governs is the deepest that ever existed, which nobody observed, and the profile is a succession of layers of different densities laid down in different storms, each with its own crust. A single triangle at a single unit weight is a stand-in for a stratified deposit whose density varies by a factor of two through its own depth.

Nor does the drawing show the plan. Snow does not pile in a two-dimensional section; it piles into a corner, and the deepest snow on any roof is nearly always at a re-entrant corner where two obstructions meet and the flow separates in both directions. The section is the easy case, drawn because it is the one with an arithmetic.

What none of the figures needs to show is what happens after the depth is known. Whatever the snow turns out to be, everything downstream of it is the ordinary integration — load to shear to moment to a section — and not one of the difficulties in this essay lives in that part. The whole of the trouble is in deciding the number that goes in at the top.

The generalisation #

The habit worth carrying out of this is about where a load’s shape comes from.

Most loads in structural design inherit their shape from the structure: a floor load is uniform because floors are used uniformly, a wind pressure follows a face because faces are flat. Snow inherits its shape from the flow field around the building, which is a property of the shape of the roof and of nothing the designer chose deliberately. The consequence is that the interesting quantity is never the intensity in the schedule; it is the geometry of everything standing above roof level.

That makes snow the clearest available example of a rule this collection keeps arriving at from different directions. A load case is not a number. It is a statement about a structure, and two structures given the same number are given different loads. Earth pressure makes the point one way, by depending on how much the wall moves; silo pressure makes it another, by depending on the friction of the wall it acts on. Snow makes it geometrically, and more visibly than either: the load is literally in a different place on two roofs that received identical weather.

And the corollary is the one that closes the argument about patterns. A structure with more supports than equations has bought robustness at the price of caring where its load is, because the internal forces in a redundant structure depend on the arrangement of the load and not merely on its total. Continuity, which buys a moment over the support, is what makes the unbalanced case govern. It is the same purchase, read from the load’s side rather than the structure’s.