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This wood beam calculator designs rafters and joists in compliance with the BS EN1995-1 (Eurocode). The calculation will define the design capacities of the timber beams to meet flexural, shear, bearing, deflection and vibration requirements, taking into account a notch at the support.
This calculation has been written in accordance with:
  1. BS EN 1995-1-1 2004 A2 2014 Design of timber structures - General rules
  2. BS EN 1995-1-1 2004 A1 2008 UK NA to Design of timber structures - General rules
  3. GD6 Guidance Document 6 Vibration in timber floors (Eurocode 5) 2009 TRADA Technology Ltd
Typical timber roof frame with rafters


Calculation

Technical notes and assumptions:
  1. The calculator accepts all wood types (solid, glulam and LVL of softwood or hardwood)
  2. Input geometry is for rectangular and square sections only
  3. Timber member is assumed to be simply supported

Inputs

Loads



d
:5kN​



d (1)
:4kN m​

Note, V,Ed & M,Ed should be factored and per metre width


W, instant (for deflection)
:7.1kN​



W, creep (for deflection)
:3.0kN​

Note, W,instant & W,creep should be unfactored


Properties



Strength class
:C24​



Partial factor for materials
:Solid timber - grade stamped individually​



Service class SC
:1​



Geometry




h
:195mm​



b
:47mm​



lb
:50mm​



L
:3.6m​

Joist/rafter nomenclature



s
:600mm​

Note, if calculating for a single member, set s = 1000 mm
Joist/rafter nomenclature



Notch depth (h-h_ef)
:45mm​



x
:50mm​



i (h-h_ef)
:0mm​


Notch nomenclature



Modification Factors



Duration
:Long-term​



k, sys
:1.1​



Compression strength
:Discrete support, udl/PL>2h from support, solid softwood​

Note, select continuous support if sections are restrained along the compression flange by battens or floorboards


Vibration

Floorboard


Thickness t_floorboard
:18mm​



Surface density, kg/m2
:10.8​

Joists, others


Surface density, kg/m2 (1)
:8​



Herringbone strutting or blocking
:No​

Plasterboard


Thickness t_pboard
:12.5mm​



Surface density, kg/m2 (2)
:16​

Modulus of Elasticity


Floorboard E_fboard
:4930MPa​



Plasterboard E_pboard
:2000MPa​

Amplification factor for solid rectangular joist


k,amp
:1.05​

Equivalent length factor for simply supported joist


k,eff
:1.0​



Outputs



Designation
:195 x 47 C24​


Modification & Partial Factors



γ_M, material properties
:1.3​



k_mod, load and moisture content
:0.7​



k_h, depth
:1.0​



k_def, deformation
:0.6​



k_cr, crack shear resistance
:0.67​




k_n sheathing material
:5.0​



k_v notched beams
:0.6028307454196341​



k_strut
:1​



k_dist
:0.477936955936573​


ULS Checks

Bending


Bending capacity fm,y,d
:14.215384615384615​



Design bending stress σ_m,y,d
:8.057409039405767MPa​



Check
:< 14.22MPa OK​


Shear


Shear capacity fv,d
:1.4282451506865155​



Design shear stress τ_v,d
:0.9526833915528709MPa​



Check
:< 1.43MPa OK​

Bearing


Bearing capacity fc,90,d
:2.2211538461538463​



Design bearing stress σ_90,d
:1.276595744680848MPa​



Check
:< 2.22MPa OK​


SLS Checks

Deflection:


u_inst
:8.467116095537355mm​



span-to-deflection for u_inst
:Span / 420​




u_creep
:2.146592812953133mm​



u_final
:10.613708908490487mm​



span-to-deflection for u_fin
:Span / 330​

Note, the designer shall choose the deflection limit. A typical limit is span/250. Therefore any span-to-deflection ratio with the denominator greater than 250 is OK.
Vibration:


Fundamental frequency f1
:24.634178044324983Hz​



Check
:> 8Hz OK​




Step frequency limiting deflection a
:1.8mm​



Deflection under 1kN point load a
:1.5269086169927848mm​



Check
:< 1.8mm OK​



Explanation

Check out our Timber Design Standards - Eurocode 5 to get you started.

Design Properties of Structural Timber Elements

Timber Types

Wood from trees is typically classified as either Hardwood or Softwood. You can be forgiven for thinking that Hardwood indicates the relative hardness of the wood, but this is often misleading as many hardwoods are relatively soft and vice versa. These are the main characteristic differences between Hardwoods and Softwoods:
Characteristic
Hardwood
Softwood
Durability
High-quality product, durable over time
Lower density, less durable over time
Strength
The dense cellular structure provides incredible strength properties
Weaker strength properties due to lower density
Workability
Due to higher density, hardwoods are harder to work with and assemble on-site
Easier to work with and can be used for a wide range of structural applications
Fire Resistance
High fire resistance
Typically, less fire-resistant than hardwoods


Strength Properties

EN 338:2016 outlines a system for strength classes for general use in European design codes applicable to all softwood and hardwood timber for structural use.
The strength classes are each designated by a number indicating the value of the edgewise bending strength in N/mm2. The designations are as follows:
  1. Softwood: C__
  2. Hardwood: D__
ES 338:2016 - Table 1 [3]


Service Class

EC5 Section 2.3.1 outlines a service class system to assign strength values and calculate deformations under the defined environmental conditions. The classes are characterised by the moisture content at 20⁰C temperatures and the relative humidity of the surrounding air.
Service class 1: The moisture content and relative humidity of the surrounding air only exceeds 65% for a few weeks per year. Most softwood will not exceed a 12% average moisture content in this class.
Service class 2: The moisture content and relative humidity of the surrounding air only exceeds 85% for a few weeks per year. Most softwood will not exceed a 20% average moisture content in this class.
Service class 3: Climatic conditions leading to higher moisture content than in service class 2.

Load duration classes

The load duration classes are defined by the effect of a constant load acting for a certain time period during the life of the structure. Actions are assigned to one of the load durations given below:
BS EN 1995-1-1:2004+A1:2008 Table 2.1 [2]

BS EN 1995-1-1:2004+A1:2008 Table 2.2 [2]



Modification & partial factors

Ultimate Limit State (ULS)

Bending

For this calculator, the design bending strength in the principal y-axis, fm,y,d is determined by:

fm,y,d=kmod ksys kh fm,kγmwhere:fm,y,d=design bending strength in the principal y-axiskmod=duration of load and moisture content factorksys=system strength factorkh=depth factorfm,k=characteristic bending strengthγm=partial factor for material propertiesf_{\mathrm{m}, \mathrm{y}, \mathrm{d}}=\dfrac{k_{mod}\,k_{sys}\,k_{h}\,f_{m,k}}{\gamma_m}\\\text{}\\\text{where:}\\f_{\mathrm{m}, \mathrm{y}, \mathrm{d}}=\text{design\ bending\ strength\ in\ the\ principal\ y-axis}\\k_{mod}=\text{duration\ of\ load\ and\ moisture\ content\ factor}\\k_{sys}=\text{system\ strength\ factor}\\k_h=\text{depth\ factor}\\f_{m,k}=\text{characteristic\ bending\ strength}\\\gamma_m=\text{partial\ factor\ for\ material\ properties}
The design bending stress σ_m,y,d is determined by:

σm,y,d=ME,dZy where:σm,y,d=design bending stressME,d=design bending momentZy=section modulus\sigma_{\mathrm{m},\mathrm{y},\mathrm{d}}=\frac{M_{\mathrm{E},\mathrm{d}}}{Z_y}\\\ \\\text{where:}\\\sigma_{\mathrm{m},\mathrm{y},\mathrm{d}}=\text{design\ bending\ stress}\\M_{\mathrm{E},\mathrm{d}}=\text{design\ bending\ moment}\\Z_y=\text{section\ modulus}

Shear

For this calculator the design shear strength is determined by:

fv,d=kmod ksys kh fv,kγmwhere:fv,d=design shear strengthkmod=duration of load and moisture content factorksys=system strength factorkh=depth factorfv,k=characteristic shear strengthγm=partial factor for material propertiesf_{\mathrm{v}, \mathrm{d}}=\dfrac{k_{mod}\,k_{sys}\,k_{h}\,f_{v,k}}{\gamma_m}\\\text{}\\\text{where:}\\f_{\mathrm{v}, \mathrm{d}}=\text{design\ shear\ strength}\\k_{mod}=\text{duration\ of\ load\ and\ moisture\ content\ factor}\\k_{sys}=\text{system\ strength\ factor}\\k_h=\text{depth\ factor}\\f_{v,k}=\text{characteristic\ shear\ strength}\\\gamma_m=\text{partial\ factor\ for\ material\ properties}
EC5 Section 6.1.7 outlines that the following:

τd=1.5Vbhef≤fv,d[6.60]where:τd=design shear stressV=shear force; volumeb=joist widthhef=joist effective depthfv,d=design shear strength for the actual condition\tau_{\mathrm{d}}=\frac{1.5V}{bh_{{\mathrm{ef}}}} \leq f_{\mathrm{v}, \mathrm{d}}\hspace{1cm}\text{[6.60]}\\\text{}\\\text{where:}\\\tau_{\mathrm{d}}=\text{design\ shear\ stress}\\V=\text{shear\ force;\ volume}\\b=\text{joist\ width}\\h_{\mathrm{ef}}=\text{joist\ effective\ depth}\\f_{\mathrm{v}, \mathrm{d}}=\text{design\ shear\ strength\ for\ the\ actual\ condition}
To account for spacing per metre between the joists the calculation has been modified as:

τd=1.5VE,db hef n kcr where:τd=design shear stressVE,d=design shear forceb=joist widthhef=joist effective depthn=number of joists per metrekcr=crack factor for shear resistance\tau_{\mathrm{d}}=\frac{1.5V_{\mathrm{E}, \mathrm{d}}}{b\,h_{{\mathrm{ef}}}\,n\,k_{\mathrm{cr}}}\\\ \\\text{where:}\\\tau_{\mathrm{d}}=\text{design\ shear\ stress}\\V_{\mathrm{E}, \mathrm{d}}=\text{design\ shear\ force}\\b=\text{joist\ width}\\h_{\mathrm{ef}}=\text{joist\ effective\ depth}\\n=\text{number\ of\ joists\ per\ metre}\\k_{\mathrm{cr}}=\text{crack\ factor\ for\ shear\ resistance}

Bearing

For this calculator the design bearing strength is determined by:

fc,90,d=kmod ksys kc,90 fc,90,kγmwhere:fc,90,d=design bearing strengthkmod=duration of load and moisture content factorksys=system strength factorkc,90=load configuration factorfc,90,k=characteristic compressive strength perpendicular to the grainγm=partial factor for material propertiesf_{\mathrm{c}, 90,\mathrm{d}}=\dfrac{k_{mod}\,k_{sys}\,k_{c,90}\,f_{c,90,k}}{\gamma_m}\\\text{}\\\text{where:}\\f_{\mathrm{c}, 90,\mathrm{d}}=\text{design\ bearing\ strength}\\k_{mod}=\text{duration\ of\ load\ and\ moisture\ content\ factor}\\k_{sys}=\text{system\ strength\ factor}\\k_{c,90}=\text{load\ configuration\ factor}\\f_{c,90,k}=\text{characteristic\ compressive\ strength\ perpendicular\ to\ the\ grain}\\\gamma_m=\text{partial\ factor\ for\ material\ properties}
The design bearing stress is determined by:

σ90,d=VE,dnblb where:σ90,d=design bearing stressVE,d=design shear forcen=number of joists per metreb=joist widthlb=bearing length\sigma_{90,\mathrm{d}}=\frac{V_{\mathrm{E},\mathrm{d}}}{nbl_b}\\\ \\\text{where:}\\\sigma_{90,\mathrm{d}}=\text{design\ bearing\ stress}\\V_{\mathrm{E},\mathrm{d}}=\text{design\ shear\ force}\\n=\text{number\ of\ joists\ per\ metre}\\b=\text{joist\ width}\\l_b=\text{bearing\ length}


Serviceability Limit State (SLS)

Deflection

Deflection is often the critical case for rafters and joists which are usually subjected to a uniform distributed load. Shear deflection in the usual case of longer-span rafters and joists is not normally significant but is included in this calculator.
Components of deflection

The input loads W_inst (instantaneous) and W_creep (long-term) are the equivalent UDL converted to point loads applied to the middle of the beam. The calculation of deflection for instantaneuous u_inst and long term u_creep are given below. The final deflection u_fin is taken as the sum of u_inst and u_creep to account for both effects:

uinst=(1+15.4(L/h)2)⋅(5winst32Eb(Lh)3)ucreep =0.6(1+154(L/h)2)⋅(5wcreep32Eb(Lh)3)ufin=uinst+ucreep where:uinst=instantaneous deflectionwinst=instantaneous load for deflectionucreep=long term deflectionwcreep=long term load for deflectionl=joist lengthh=joist heightb=joist breadthE=Elastic modulusu_{\text {inst}}=\left(1+\dfrac{15.4}{(L / h)^2}\right) \cdot\left(\dfrac{5 w_{\text {inst}}}{32 E b}\left(\dfrac{L}{h}\right)^3\right)\\u_{\text {creep }}=0.6\left(1+\dfrac{154}{(L / h)^2}\right) \cdot\left(\dfrac{5 w_{\text {creep}}}{32 E b}\left(\dfrac{L}{h}\right)^3\right)\\u_{fin}=u_{inst}+u_{creep}\\\ \\\text{where:}\\u_{inst}=\text{instantaneous\ deflection}\\w_{inst}=\text{instantaneous\ load\ for\ deflection}\\u_{creep}=\text{long\ term\ deflection}\\w_{creep}=\text{long\ term\ load\ for\ deflection}\\l=\text{joist\ length}\\h=\text{joist\ height}\\b=\text{joist\ breadth}\\E=\text{Elastic\ modulus}\\
Table NA.5 in BS NA EN 1995-1-1 outlines that the limit of final deflection is given for a member of span l between two supports:
  1. Span/250 with plasterboard finishes
  2. Span/150 without plasterboard finishes
Table NA.5 limiting values for deflections of individual beams [4]



Vibration

Unlike some international standards like the Australian Standards, EC5 Section 7.3 provides commentary about vibration requirements. The fundamental frequency for residential floors should be more than 8Hz in domestic residential situations. The fundamental frequency f1 is approximately calculated as:

f1=π2ℓ2(EI)ℓm(7.5) where:m=mass per unit areal=floor span(EI)l=equivalent plate bending stiffness of the floor about an axis perpendicular to the beam directionf_1=\dfrac{\pi}{2 \ell^2} \sqrt{\dfrac{(E I)_{\ell}}{m}}\hspace{1cm}\text{(7.5)}\\\ \\\text{where:}\\m=\text{mass\ per\ unit\ area}\\l=\text{floor\ span}\\(EI)_l=\text{equivalent\ plate\ bending\ stiffness\ of\ the\ floor\ about\ an\ axis\ perpendicular\ to\ the\ beam\ direction}

The EC5 formula for fundamental frequency is based on two-way spanning panels but most floors span one-way in the UK. The formula has been adjusted as outlined in GD6 Guidance Document 6 Vibration in timber floors (Eurocode 5).

Table NA.6 in BS NA EN 1995-1-1 outlines that the limit of a deflection of a floor under 1 kN point load as:
Table NA.6 Limits for a and b [4]

As per above, the step frequency limiting deflection in this calculator, a in millimetres is taken as:

a=min⁡(1.8,16500l1.1) where:l=spana=\operatorname{min}\left(1.8,\frac{16500}{l^{1.1}}\right)\\\ \\\text{where:}\\l=\text{span}

Related Resources

  1. Timber Design Standards - Eurocode 5
  2. Wood Beam Calculator to AS 1720.1
  3. Wood Column Calculator to AS 1720.1

📚References

[1] European Union. (2005). Eurocode: Basis of Structural Design (BS EN 1990:2002+A1:2005). SAI Global. https://www.saiglobal.com/
[2] European Union. (2008). Eurocode 5: Design of Timber Structures (BS EN 1995-1-1:2004+A1:2008). SAI Global. https://www.saiglobal.com/
[3] European Union. (2016). Structural Timber - Strength Classes (EN 338:2016). SAI Global. https://www.saiglobal.com/
[4] European Union. (2008). UK National Annex to Eurocode 5: Design of Timber Structures (NA to BS EN 1995-1-1:2004+A1:2008).