Hi friends,

In the last weeks, we learned all the basics of timber design.

Today we'll design the first timber element: a timber beam.

Also, I wanted to thank you guys for your support with Module #2.

I hope it'll help you in your engineerng journey and if you have any suggestions of making it better, please reach out!

After having received so much good feedback already about Module #1 and #2, I have already started with Module #3: Structural Design of Reinforced Slabs and Walls of a Residential Building.

Now to the topic of today's e-mail..

The 4 Steps To Verify Timber Beams According To Eurocode

Let's use the secondary beam of the canopy structure that we also used in the last episodes as an example to show the calculation steps.

If you prefer video, then you can watch this video tutorial.

Step #1: Define the material properties of the timber element

We first define the material properties of the timber beam. In this tutorial, we'll use C24 as the timber material.

For small structures like this which aren't exposed to big loads, C24 is enough.

We summarized all timber materials in ep. #2 about timber material properties or I also published it to the blog as a table (click here).

Here are the strength and stiffness properties of C24.

The partial safety factor is found in EN 1995-1-1 Table 2.3 as:

γ_M = 1.3

The beam is classified according to EN 1995-1-1 2.3.1.3 as service class 2 (assumption in this tutorial).

Then we'll verify the timber beam for a design load of load duration class short-term (EN 1995-1-1 Table 2.1) which leads to a modification factor (EN 1995-1-1 Table 3.1) of:

k_mod = 0.9

Step #2: Define the geometrical properties of the timber beam

The beam has the following cross-sectional properties:

• Cross-sectional height h=24cm
• Cross-sectional width w=12cm
• Moment of inertia strong axis I_y = (w ⋅ h³)/12 = 1.38 ⋅ 10⁸ mm⁴
• Moment of inertia weak axis I_x = (h ⋅ w³)/12 = 3.46 ⋅ 10⁷ mm⁴
• Torsional moment of inertia I_tor = 1/3 ⋅ h ⋅ w³ ⋅ (1 - 0.63 ⋅ w/h + 0.052 ⋅ w⁵/h⁵) = 9.49 ⋅ 10⁷ mm⁴
• Length (=span) s=5m

Step #3: Calculate the loads and internal forces acting in the timber element

First, we need to calculate the characteristic loads that act on the roof area of the canopy. The area loads are applied to the roof panels (could be OSB boards or trapezoidal steel sheeting). These panels then transfer the loads to the secondary beams which we verify in this tutorial. This video explains transfering vertical loads in detail (click → here ←).

We won't show how to calculate the loads and how to do the load transfer in this newsletter, as each calculation of the individual load is an article for itself.

But that's exactly what I teach in Module #2: Structural Design of a Residential Timber Roof.

In this email, we'll design the secondary timber beam for the following design line load.

Design line load:

p_d = 4.0 kN/m

From design load, we'll calculate the design bending moment:

M_d = p_d ⋅ s²/8 = 12.5 kNm

And the design shear force:

V_d = p_d ⋅ s/2 = 10 kN

Step #4: Verify the timber beam for bending

Characteristic bending resistance (see step #1):

f_m.g.k = 24 MPa

Design bending resistance:

f_m.g.d = k_mod ⋅ f_m.g.k/γ_M = 17.3 MPa

Design bending stress:

σ_m.d = M_d/I_y ⋅ h/12 = 10.8 MPa

Verification according to EN 1995-1-1 (6.11):

η = σ_m.d/f_m.g.d = 62.8%

Step #5: Verify the timber beam for shear

According to EN 1995-1-1 (6.13) shear is verified as:

τ_d ≤ f_v.d

Characteristic shear resistance (see step #1):

f_v.k = 4.0 MPa

Design shear resistance:

f_v.d = k_mod ⋅ f_v.k/γ_M = 2.88 MPa

The shear design shear stress is calculated as:

τ_d = 3/2 ⋅ V_d/A = 0.52 MPa

Verification according to EN 1995-1-1 (6.13):

η = τ_d /f_v.d = 18%

Step #6: Verify the timber beam for lateral torsional buckling

First, we'll calculate the effective length of the beam according to EN 1995-1-1 Table 6.1:

l_ef = 0.9 ⋅ s + 2 ⋅ h = 4.98m

Critical bending stress (EN 1995-1-1 (6.31)):

Relative slenderness (EN1995-1-1 (6.30)):

λ_rel.m = √(f_m.k/σ_m.crit) = 0.64

Factor for reduced bending strength due to lateral torsional buckling (EN1995-1-1 (6.34)):

λ_rel.m > 0.75 → k_crit = 1.0

Utilization (EN1995-1-1 (6.35)):

η = (σ_m.d/(k_crit ⋅ f_m.d))² = 0.39

Step #7: Verify the timber beam for deflection

In the deflection analysis we check that the instantaneous w_inst and final deflections w_fin of the beams/rafters are less than limits we find either in Eurocode, the National Annex or values given by the client. EN 1995-1-1 Table 7.2 recommends values for w_inst, w_net.fin and w_fin which should not be exceeded for a simply supported beam.

This leads to the following limits.

Instanteneous deflection

w_inst = l/300 = 16.7mm

Final deflection

w_fin = l/150 = 33.3mm

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Verification of the instantaneous deflection

The instantaneous deflection is calculated with loads from the characteristic SLS load combination. In our case we only defined a design load in step #3. But let's assume that we have two characteristic loads that act on the beam - the dead load and the snow load.

Let's say the dead load = 2 kN/m and the snow load = 0.85 kN/m.

u_inst = u_g.k + u_s.k

The deflection of a simply supported beam due to a line load is calculated as:

w_max = 5/384 ⋅ ql⁴/(E_0.mean ⋅ I_y)

You can find many more deflection formulas here.

Deflection of the characteristic dead load:

u_g.k = 5/384 ⋅ g_k ⋅ l⁴/(E_0.mean ⋅ I_y) = 10.7 mm

Deflection of the characteristic snow load:

u_s.k = 5/384 ⋅ s_k ⋅ l⁴/(E_0.mean ⋅ I_y) = 4.5 mm

SLS load combination:

u_inst = u_g.k + u_s.k = 15.3 mm

Verification according to EN 1995-1-1:

η = u_inst/w_inst = 91.5%

Verification of the final deflection

The final deflection is calculated with loads from the characteristic SLS load combination and a long-term component which is found from EN 1995-1-1 2.2.3 (5).

u_fin = u_g.k ⋅ (1 + k_def) + u_s.k ⋅ (1 + k_def ⋅ ψ₂)

With

• ψ₂ = 0.5 (psi factor of snow)
• k_def = 0.6 (factor for solid timber in service class 1 (EN1995-1-1 Table 3.2))

This leads to a final deflection of

u_fin = u_g.k ⋅ (1 + k_def) + u_s.k ⋅ (1 + k_def ⋅ ψ₂) = 21.7 mm

Verification according to EN 1995-1-1:

η = u_fin/w_fin = 65%

Final Words

Alright, this is how we design timber beams according to Eurocode.

I hope this helped.

Enjoy the rest of the week and your weekend.

Let’s design better structures together,

Laurin.

P.S. If you want to learn more, here are a few ways I can help you:

#1: I teach you everything you need to know about load calculation. It's the most important fundamental of structural engineering. ​Without knowing the loads of a building you can't design the structural elements. Click → here ← to learn.

#2: Previous episodes of the timber design series:

• Ep. #1: Welcome to the timber design series (click here)
• Ep. #2: Timber material properties (click here)
• Ep. #3: Tension verification of timber (click here)
• Ep. #4: Compression verification of timber (click here)
• Ep. #5: Bending verification of timber (click here)
• Ep. #6: Compression perpendicular to the grain (click here)
• Ep. #7: Shear verification (click here)
• Ep. #8: Deflection verification (click here)
• Ep. #9: Torsion verification (click here)
• Ep. #10: Lateral torsional buckling (click here)

#3: The reinforced concrete series (click here)

#4: The engineering mechanics series (click here)

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