Special roofs — shape coefficients

Valleys of multi-span roofs, cylindrical roofs, and roofs abutting taller structures — EN 1991-1-3 §5.3.4-5.3.6.

Roof type

Snow from both slopes collects in the valley. Figure 5.4 gives two cases: undrifted, with μ₁ on each slope, and drifted, with μ₂ in the valley — evaluated at the MEAN pitch, not at either one.

Data

s_k — ground snow load [kN/m²]

α₁ [degrees]

α₂ [degrees]

The pitches of the two slopes meeting at the valley. The valley uses their mean, ᾱ = (α₁+α₂)/2.

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Tool information

What this page computes

The snow shape coefficients for the three configurations that EN 1991-1-3 treats separately from the single- or two-pitch roof:

§5.3.4 — multi-span roofs, with valleys;
§5.3.5 — cylindrical roofs;
§5.3.6 — roofs abutting or close to taller structures.

For each, the code gives two load cases: undrifted snow, case (i), and drifted snow, case (ii). The second is not "larger" — it is differently distributed, and that matters for structures where asymmetric loading governs.

The valley: the mean, not the maximum

On multi-span roofs, snow from both slopes collects in the valley. Figure 5.4 gives:

case (i): \(\mu_1(\alpha_1)\) and \(\mu_1(\alpha_2)\), each slope at its own pitch;
case (ii): \(\mu_1\) at the edges, and in the valley \(\mu_2(\bar\alpha)\) with \(\bar\alpha = (\alpha_1 + \alpha_2)/2\).

The mean pitch, not either one. Easy to get wrong, especially since with equal pitches the difference is invisible.

§5.3.4(4) requires special attention if one valley face exceeds 60°. The page flags it — snow sliding off the steep face has to end up somewhere, and where exactly is not computed here.

For valleys there is also the exceptional distribution of Annex B2, with a different cap and length. The two do not add: one belongs to the fundamental combination, the other to the accidental one.

The vault: everything depends on β

\(\beta > 60° : \mu_3 = 0 \qquad\qquad \beta \le 60° : \mu_3 = 0.2 + 10\,h/b\)

Above 60° at the springing, snow simply is not retained. Below, the coefficient grows linearly with the rise of the vault, up to a cap that is a nationally determined parameter — the recommended value is 2.0.

The plateau in Figure 5.5 begins exactly at \(h/b = 0.18\), because \(0.2 + 10 \cdot 0.18 = 2.0\). That is not a drafting coincidence; it is the point where the expression reaches the cap.

β is not given by the code as a formula. The page derives it from the geometry of the circular arc through the two springings and the crown — checkable: for a semicircle, \(h = b/2\), it comes out to exactly 90°. If your vault is not a circular arc, take β from the drawing.

The expressions apply to vaults without snow fences.

The third tier of the same situation

Here is the observation worth keeping. §5.3.6 deals with a roof next to a taller block — the same physical situation treated by §6.2 (an obstruction on the roof) and by Annex B3 (the accidental case). All three use the same base term, but with caps that double:

	§6.2	§5.3.6	Annex B3
situation	persistent	persistent	accidental
expression	(5.1), with \(C_e\), \(C_t\)	(5.1), with \(C_e\), \(C_t\)	(5.3), without
term	\(\gamma h/s_k\)	\(\gamma h/s_k\)	\(2h/s_k\)
cap	2.0	4	8
\(l_s\)	\(2h\), in [5, 15]	\(2h\), in [5, 15]	\(\min(5h, b_1, 15)\)

§5.3.6 and §6.2 have exactly the same term and the same drift length, but caps differing by a factor of two. The physical difference: §6.2 is an obstruction on the roof, §5.3.6 is a building block next to it — and the second additionally brings the snow that slides off the taller roof, through the term \(\mu_s\).

Do not take a value from one chapter and use it in another. A test guards the 2 / 4 / 8 ratio.

What the code does not give: μ_s

\(\mu_2 = \mu_s + \mu_w \qquad \mu_w = \frac{b_1+b_2}{2h} \le \frac{\gamma h}{s_k}\)

\(\mu_w\) is fully defined. \(\mu_s\) is not: below 15° it is zero, but above, the text only states that an "additional load … which may reach up to 50% of the maximum total load" is considered, determined per §5.3.3. That is not a formula, it is a design judgement — so on the page it is an input, not a result.

If the pitch exceeds 15° and you leave \(\mu_s\) blank, the page warns that the result is on the unsafe side.

What it does not cover

The interpolation required by NOTE 3 to (5.9), when \(b_2 \le l_s\) — the page flags the situation but does not interpolate.
The distribution along \(l_s\) on a vault: the ordinates of Figure 5.6 are given, at the springings and the quarter points, not the shape between them.
The effect of snow fences on vaults, which the code leaves to the National Annex.
Roofs of irregular geometry, where choosing the parameters requires judgement.

Expressions (5.4)–(5.9) and Figure 5.4 are transcribed from EN 1991-1-3:2005, pages 17–19, and each has a test that reproduces it by hand.

Snow — the coefficients for single- and two-pitch roofs, §5.3.2 and §5.3.3.
Snow drift — Annex B, the accidental case, compared against here.
Local effects — §6.2, the first tier of the same family.