| Name | Result |
|---|---|
| Rafter length | |
| Roof angle | |
| Roof height H | |
| Design snow load | |
| Design wind load | |
| Design permanent load | |
| Total design load per rafter | |
| Design line load on the rafter | |
| Maximum bending moment M | |
| Deflection limit | |
| Allowable deflection | |
| Calculated deflection in span | |
| Required rafter section h×b |
About Rafter Length Calculation
This rafter length calculator performs roof load and size calculation for a timber rafter, including rafter length, roof slope, rafter spacing effects, snow, wind and permanent loads, bending moment, deflection and the required rectangular rafter dimensions. The calculation uses a structural model with two supports and one cantilever roof overhang and can be used to estimate the required rafter size and load capacity.
The rafter section is checked against both bending strength and deflection limits. Load combinations include permanent, snow and wind actions, while timber properties are based on the selected C16, C24 or C30 strength class.
Guidelines and recommendations
Roof geometry
Roof slope. The horizontal distance B is measured between the support axes. The height H is the vertical difference between the rafter support levels. If the height is specified, the roof angle α is calculated from the right triangle:
α = arctan(H / B)
If the roof angle is specified instead of the height, the roof height is calculated in the opposite direction:
H = B · tan(α)
Span and overhang. The rafter length between the supports and the cantilever overhang length are calculated separately. The horizontal overhang S does not affect the roof angle.
Ls = B / cos(α)
Lo = S / cos(α)
The geometric length along the roof slope is the sum of these two parts:
Lg = Ls + Lo
Rafter stock length. The result includes not only the geometric length along the slope but also the additional length required for the vertical end cuts. After the rafter depth h has been determined, the required stock length is calculated as:
L = Lg + h · tan(α)
The addition h · tan(α) accounts for the total difference between the upper and lower rafter edges caused by the two vertical end cuts.
Snow load
Characteristic snow load. The input value Sg,k is the characteristic ground snow load taken from national design data for the project location. The calculator converts it to the characteristic snow load on the roof using the shape coefficient μ. The exposure and thermal coefficients are both taken as 1.0 in the calculation model.
Sr,k = μ · Sg,k
The shape coefficient depends on the roof slope:
- for
α ≤ 30°:μ = 0.8 - for
30° < α < 60°:μ = 0.8 · (60 − α) / 30 - for
α ≥ 60°:μ = 0
This relationship represents the uniform snow load case for a pitched roof: the coefficient remains constant up to 30°, then decreases linearly to zero at 60°.
Permanent and wind loads
Permanent load. The characteristic permanent load Gk is the sum of the selected roof covering weight and the additional permanent load from the remaining roof layers.
Gk = Groof + Gadd
Wind load. The specified characteristic wind load Wk is used in the calculation model as a load acting normal to the roof slope. The input value should be taken from national design data for the project location.
Design factors. For the ultimate limit state strength calculation, the calculator uses a factor of 1.35 for permanent loads and 1.50 for variable snow and wind loads:
Gd = 1.35 · Gk
Sd = 1.50 · Sr,k
Wd = 1.50 · Wk
Load combinations
Most adverse load combination. Snow and wind are not simply added as two simultaneous maximum loads. The calculator checks two combinations: snow as the leading variable action and wind as the leading variable action. A combination factor of ψW = 0.60 is used for accompanying wind and ψS = 0.70 for accompanying snow.
For the combination with snow as the leading action:
pS = 1.35 · Gk · cos(α) + 1.50 · Sr,k · cos2(α) + 1.50 · 0.60 · Wk
For the combination with wind as the leading action:
pW = 1.35 · Gk · cos(α) + 1.50 · Wk + 1.50 · 0.70 · Sr,k · cos2(α)
The larger value is used for the subsequent strength calculation:
pULS = max(pS, pW)
Load projection. Permanent load from self-weight acts vertically and is converted to the component producing rafter bending using cos(α). Snow load is specified per horizontal roof projection, so cos2(α) is used when converting it to the load that bends the inclined rafter. In the adopted model, wind load is already treated as acting normal to the roof slope.
Line load and load on one rafter
Line load. The design area load is converted to a uniformly distributed line load using the rafter spacing D. If pULS is expressed in kN/m² and D in millimetres:
q = pULS · D / 1000
The resulting q is expressed in kN/m. For unit conversion, the calculator uses 1 kN/m² ≈ 101.97 kg/m².
Tributary area for one rafter. For the reference value of the total load on one rafter, the geometric slope length without the end-cut allowance is used:
A = Lg · D / 1 000 000
Here Lg and D are entered in millimetres, and the resulting tributary area A is expressed in m².
Bending moment
Structural model. The rafter is treated as a beam on two supports with a cantilever overhang on the left and a uniformly distributed load along its length. The calculation uses the inclined support span Ls and the inclined overhang length Lo.
The positive bending moment within the span is calculated with the influence of the cantilever overhang included:
Ms = q · (Ls2 − Lo2)2 / (8 · Ls2)
This value is used when Lo < Ls. The moment at the support due to the cantilever overhang is:
Mo = q · Lo2 / 2
The larger absolute value is used for sizing the rafter section:
M = max(Ms, Mo)
Timber strength and rafter sizing
Design bending strength. For timber strength classes C16, C24 and C30, the characteristic bending strengths are taken as 16, 24 and 30 N/mm² respectively. The calculator applies a modification factor kmod = 0.70 and a material factor γM = 1.30.
fm,d = kmod · fm,k / γM
- C16:
fm,d = 8.62 N/mm² - C24:
fm,d = 12.92 N/mm² - C30:
fm,d = 16.15 N/mm²
The required section modulus is determined from the maximum bending moment:
Wreq = M / fm,d
For a rectangular rafter section:
W = b · h2 / 6
If the ratio r = h/b is specified, the minimum rafter depth required for strength is calculated as:
h = (6 · Wreq · r)1/3
If the rafter width b is specified directly:
h = (6 · Wreq / b)1/2
Deflection check
Modulus of elasticity. Mean modulus of elasticity values of 8000 N/mm² for C16, 11000 N/mm² for C24 and 12000 N/mm² for C30 are used for the deflection calculation.
Serviceability load combination. The factors 1.35 and 1.50 are not applied for the deflection check. Two serviceability load combinations are checked:
pSLS,S = Gk · cos(α) + Sr,k · cos2(α) + 0.60 · Wk
pSLS,W = Gk · cos(α) + Wk + 0.70 · Sr,k · cos2(α)
pSLS = max(pSLS,S, pSLS,W)
Allowable deflection. The limit is based on the inclined span between the supports. With the selected coefficient k equal to 200, 250 or 300:
flim = Ls / k
A larger denominator means a stricter stiffness requirement.
Calculated deflection. The second moment of area of the rectangular section is:
I = b · h3 / 12
Deflection is calculated over the full support span with the uniformly distributed load on the cantilever overhang included. For coordinate x measured from the left support, the following relationship is used:
f(x) = q / (24 · E · I) · [−6 · Lo2 · x2 + 2 · (Ls2 + Lo2) · x3 / Ls − x4 − Ls · (Ls2 − 4 · Lo2) · x]
The maximum absolute deflection is determined numerically along the span using a step of Ls / 1000. After the preliminary section has been selected, the actual deflection is recalculated. If it exceeds the allowable value, the rafter depth h is increased in 1 mm increments until f ≤ flim.
How the final rafter size is selected
Two independent conditions. The rafter section must satisfy both bending strength and deflection requirements. The minimum depth required for strength and the minimum depth required for stiffness are calculated separately, and the larger value is selected.
If an h/b ratio is specified, the width is calculated from this ratio and both dimensions are rounded upward. If the width is specified directly, it remains fixed while the depth is increased until both conditions are satisfied. The final rafter dimensions are rounded upward to whole millimetres.
Practical guidelines
Rafter spacing. In low-rise roof construction, rafter spacing of about 600-900 mm is commonly used, although the actual spacing depends on the span, loads, roof build-up and selected rafter section.
Section proportions. For rectangular timber rafters, h/b ratios of approximately 1.5 to 3 are common. Increasing the rafter depth is generally much more effective than increasing the width when deflection governs, because the second moment of area depends on h3.
Climatic loads. Characteristic ground snow load and wind load values should be taken for the specific project location from national design data and the relevant National Annexes to the Eurocodes.
European standards
The calculation model and notation follow the general European structural design approach. Load combinations are related to EN 1990 “Eurocode - Basis of structural design”. Snow actions are covered by EN 1991-1-3 “Eurocode 1: Actions on structures - Part 1-3: General actions - Snow loads”, while wind actions are covered by EN 1991-1-4 “Eurocode 1: Actions on structures - Part 1-4: General actions - Wind actions”.
Strength classes C16, C24 and C30 are related to EN 338 “Structural timber - Strength classes”. Design properties of timber members and strength and serviceability checks are related to EN 1995-1-1 “Eurocode 5: Design of timber structures - Part 1-1: General - Common rules and rules for buildings”. National Annexes may define different climatic data and other nationally determined parameters.
FAQs
Why does the roof overhang S not affect the roof angle?
The roof slope is defined by the triangle between the two supports, so it is determined by the horizontal span B and the height H. The overhang S is an extension of the already inclined rafter beyond the left support. Changing the overhang increases the rafter length and cantilever moment but does not change the roof angle.
Why is the required rafter length greater than the geometric slope length?
The geometric length is calculated from the support span and roof overhang along the slope. After the rafter depth has been selected, the calculator also accounts for the vertical end cuts, which require an additional length of h · tan(α). The required stock length therefore depends on both the roof geometry and the calculated rafter depth.
Why is the total design load not simply the sum of snow, wind and permanent loads?
Snow and wind are variable actions, so the calculator checks separate combinations with snow leading and wind leading. The accompanying action is reduced by a combination factor ψ, while vertical loads are also transformed according to the roof slope. The more adverse of the two combinations is used for the rafter calculation.
How does roof slope affect the required rafter size?
As the roof angle increases, the inclined support span and the transformation of permanent and snow loads change. Above 30°, the snow shape coefficient μ also begins to decrease, reaching zero at 60° in the adopted calculation model. A change in roof slope can therefore affect the load, bending moment and deflection at the same time.
Why do C16, C24 and C30 produce different rafter sizes?
These timber strength classes have different characteristic bending strengths and different moduli of elasticity. Higher bending strength reduces the required section modulus, while a higher modulus of elasticity reduces deflection for the same rafter dimensions. The final rafter size is still governed by whichever condition requires the larger section: strength or stiffness.