Roof calculation: how to calculate the angle of inclination of the roof, the length of the rafters and the area of ​​the roofing material

Recommendation points

• Units of the roof pitch angle
• Formulas for calculating the angle of inclination of the roof, the length of the rafters and the area of ​​\ u200b \ u200bthe roofing material
• Example
• Recommendations for roof slope depending on the purpose and material
• Determination of dynamic loads depending on the angle of inclination

When designing the roof rafters of a private house, you need to be able to correctly calculate the angle of inclination of the roof. How to navigate in various units of measurement, by what formulas to calculate and how the angle of inclination affects the wind and snow load of the roof, we will talk in this article.

The roof of a custom home can be very simple or surprisingly fancy. The slope angle of each slope depends on the architectural solution of the whole house, the presence of an attic or attic, the used roofing material, the climatic zone in which the personal plot is located. In a compromise of these parameters, an optimal solution must be found that combines the strength of the roof with the beneficial use of the under-roof space and the appearance of the house or complex of buildings..

Units of the roof pitch angle

The angle of inclination is the value between the horizontal part of the structure, slabs or floor beams, and the roof surface or rafters.

In reference books, SNiP, technical literature, there are various units for measuring angles:

• degrees;
• aspect ratio;
• interest.

Another unit for measuring angles – radians – is not used in such calculations..

What are degrees, everyone remembers from the school curriculum. The aspect ratio of a right-angled triangle, which is formed by the base – L, height – H (see the figure above) and the roof deck is expressed as H: L. If a ? = 45 °, the triangle is equilateral, and the aspect ratio (legs) is 1: 1. In the case when the ratio does not give a clear idea of ​​the slope, they talk about the percentage. This is the same ratio, but calculated as fractions converted to percentages. For example, with H = 2.25 m and L = 5.60 m:

• 2.25 m / 5.60 m 100% = 40%

The numerical expression of some units through others is clearly shown in the diagram below:

Formulas for calculating the angle of inclination of the roof, the length of the rafters and the area of ​​\ u200b \ u200bthe roofing material

To easily calculate the dimensions of the elements of the roof and rafter system, you need to remember how we solved problems with triangles at school, using the basic trigonometric functions.

How does this help in calculating the roof? We break complex elements into simple right-angled triangles and find a solution for each case using trigonometric functions and the Pythagorean theorem.

The easiest way to calculate a gable or gable roof. Ridge height and span – the values ​​are known, the angle and length of the rafters are easily determined.

More complex configurations are more common.

For example, you need to calculate the length of the rafters of the end part of the hip roof, which is an isosceles triangle. From the top of the triangle we lower the perpendicular to the base and get a right-angled triangle, the hypotenuse of which is the middle line of the end part of the roof. Knowing the width of the span and the height of the ridge, from the structure divided into elementary triangles, you can find the angle of the hip tilt -?, The angle of the roof -? and get the length of the rafters of the triangular and trapezoidal ramp.

Calculation formulas (length units must be the same – m, cm or mm – in all calculations to avoid confusion):

Attention! The calculation of rafter lengths using these formulas does not take into account the amount of overhang.

Example

The roof is four-pitched, hip. Ridge height (CM) – 2.25 m, span width (W / 2) – 7.0 m, roof end slope depth (MN) – 1.5 m.

Having received the values ​​of sin (?) And tg (?), You can determine the value of the angles using the Bradis table. A complete and accurate table with an accuracy of the minute is a whole brochure, and for rough calculations that are acceptable in this case, you can use a small table of values.

Table 1

Roof inclination angle, in degrees	tg (a)	sin (a)
five	0.09	0.09
ten	0.18	0.17
15	0.27	0.26
20	0.36	0.34
25	0.47	0.42
thirty	0.58	0.50
35	0.70	0.57
40	0.84	0.64
45	1.00	0.71
50	1.19	0.77
55	1.43	0.82
60	1.73	0.87
65	2.14	0.91
70	2.75	0.94
75	3.73	0.96
80	5.67	0.98
85	11.43	0.99
90	?	1

For our example:

• sin (?) = 0.832,? = 56.2 ° (obtained by interpolating adjacent values ​​for angles of 55 ° and 60 °)
• tg (?) = 0.643,? = 32.6 ° (obtained by interpolating adjacent values ​​for angles of 30 ° and 35 °)

Let’s remember these numbers, they will be useful to us when choosing a material.

To calculate the amount of roofing material, you will need to determine the coverage area. The area of ​​the gable roof slope is a rectangle. Its area is the product of the sides. For our example – a hip roof – this boils down to determining the areas of a triangle and a trapezoid.

For our example, the area of ​​one end triangular slope at CN = 2.704 m and W / 2 = 7.0 m (the calculation must be performed taking into account the elongation of the roof outside the walls, we take the overhang length – 0.5 m):

• S = ((2.704 + 0.5) (7.5 + 2 x 0.5)) / 2 = 13.62 m²

The area of ​​one side trapezoidal slope at W = 12.0 m, H_from = 3.905 m (trapezoid height) and MN = 1.5 m:

• L_to = W – 2 MN = 9 m

We calculate the area taking into account the overhangs:

• S = (3.905 + 0.5) ((12.0 + 2 x 0.5) + 9.0) / 2 = 48.56 m²

The total area covered by four slopes:

• S_? = (13.62 + 48.46) 2 = 124.16 m²

Recommendations for roof slope depending on the purpose and material

An unexploited roof can have a minimum slope of 2-7 °, which makes it immune to wind loads. For normal snow melting, it is better to increase the angle to 10 °. Such roofs are common in the construction of outbuildings, garages..

If the under-roof space is supposed to be used as an attic or attic, the slope of the single or gable roof must be large enough, otherwise the person will not be able to straighten up, and the usable area will be “eaten” by the rafter system. Therefore, it is advisable to use in this case a sloping roof, for example, an attic type. The minimum ceiling height in such a room should be at least 2.0 m, but it is desirable for a comfortable stay – 2.5 m.

Options for arranging the attic: 1-2. Classic gable roof. 3. Roof with a variable angle of inclination. 4. Roof with outriggers

Taking this or that material as roofing, it is necessary to take into account the requirements for the minimum and maximum slope. Otherwise, there may be problems requiring repair of the roof or the whole house..

table 2

Roof type	Range of permissible mounting angles, in degrees	Optimal roof slope, in degrees
Roofing roofing with topping	3-30	4-10
Roofing roof, two-layer	4-50	6-12
Zinc roof with double standing seams (made of zinc strips)	3-90	5-30
Roofing paper, simple	8-15	10-12
Sloping roof covered with roof steel	12-18	15
4-groove tongue and groove shingles	18-50	22-45
Shingle roof	18-21	19-20
Tongue shingles, normal	20-33	22
Corrugated board	18-35	25
Corrugated asbestos-cement sheet	5-90	thirty
Artificial slate	20-90	25-45
Slate roof, two-layer	25-90	30-50
Slate roof, normal	30-90	45
Glass roof	30-45	33
Roof tiles, two-layer	35-60	45
Grooved Dutch roof tiles	40-60	45

The slope angles obtained in our example are in the range of 32-56 °, which corresponds to a slate roof, but does not exclude some other materials.

Determination of dynamic loads depending on the angle of inclination

The structure of the house must be able to withstand static and dynamic loads from the roof. Static loads are the weight of the rafter system and roofing materials, as well as the under-roof equipment. This is a constant.

Dynamic loads are variable values ​​that depend on climate and season. To correctly calculate the loads, taking into account their possible compatibility (simultaneity), we recommend that you study SP 20.13330.2011 (sections 10, 11 and Appendix G). In full, this calculation, taking into account all possible factors for a specific construction, in this article cannot be stated.

The wind load is calculated taking into account the zoning, as well as the features of the location (leeward, windward side) and the angle of inclination of the roof, the height of the building. The calculation is based on wind pressure, the average values ​​of which depend on the region of the house under construction. The rest of the data is needed to determine the coefficients that correct the relatively constant value for the climatic region. The greater the angle of inclination, the more severe wind loads the roof experiences.

Table 3

Construction area	I	II	III	IV	V	VI	Vii	VIII
Estimated snow load	0.8 (80)	1.2 (120)	1.8 (180)	2.4 (240)	3.2 (320)	4.0 (400)	4.8 (480)	5.6 (560)

The snow load, in contrast to the wind load, is related to the angle of inclination of the roof in the opposite way: the smaller the angle, the more snow is retained on the roof, the lower the probability of the snow cover converging without the use of additional means, and the greater the load the structure experiences.

Table 4

Snow region	Cities	Snow load kgf / m3
Single slope	Gable
0-25 °	25-30 °	20-39 °
1	Kaliningrad, Donetsk, Vilnius, Rostov-on-Don, Astrakhan	50	40	65
2	Riga, Minsk, Kiev, Belgorod, Volgograd	70	55	90
3	Moscow, Smolensk, Bryansk, Kursk, Voronezh, Saratov, Tambov, Ulyanovsk	one hundred	80	125
4	Arkhangelsk, Vologda, Petrozavodsk, Nizhny Novgorod, Samara	150	120	190

Take the issue of determining loads seriously. The calculation of cross-sections, construction, and therefore, the reliability and cost of the rafter system depends on the values ​​obtained. If you are not confident in your abilities, it is better to order the calculation of loads from specialists.