Hi friends,

Just a quick reminder: It's 1 week out until the launch of Module #2: Structural Design of a Residential Timber Roof. I will send you an email with more information about the e-book on Saturday.

I already said it in last weeks e-mail, but here it is again: I am super excited to share this e-book with you. I am smiling while I am writing this.

Currently, I am working on the last few things that need to be done for the launch: Product page, description writing, pictures, launch e-mails, check-out and thank you e-mail, on-boarding video, etc. There's a lot of stuff that I am working on, so you will hopefully get the best buying experience possible.

​

But maybe I am also felling so great because I just returned from a 2 week Italy vacation with my girlfriend.

Anyways, I'll share more about the e-book on Saturday.

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

So today, I'll show you how to verify timber beams for lateral torsional buckling.

But what exactly is lateral torsional buckling?

Lateral torsional buckling is a phenomenon that can happen to steel, precast concrete and timber beams. It describes the behavior of a beam that is prone to lateral/horizontal buckling in the compression zone due to bending moments. At the same time the tension zone of the cross-section tends to remain almost undeformed. This results in a rotational and lateral deformation of the beam like in the following picture.

​

The support conditions influence the beams ability to buckle laterally.

→ Fixed supports provide a good torsional restraint, which leads to a greater lateral torsional buckling resistance.

→ Pinned supports lead to a smaller lateral torsional buckling resistance.

→ Lateral torsional movement restraints in the span of the beam also lead to a greater lateral torsional buckling resistance. Examples are beams that run transverse to the beams with a stiff connection.

Now, let’s get into the nerdy calculations..

The 4 Steps To Verify Timber Beams For Lateral Torsional Buckling

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.

Also let's assume that the beam is not held horizontally by some sheeting like trapezoidal sheeting or OSB boards.

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=20cm
• Cross-sectional width w=12cm
• Moment of inertia strong axis I_y = (w ⋅ h³)/12 = 8 ⋅ 10⁷ mm⁴
• Moment of inertia weak axis I_x = (h ⋅ w³)/12 = 2.88 ⋅ 10⁷ mm⁴
• Torsional moment of inertia I_tor = 1/3 ⋅ h ⋅ w³ ⋅ (1 - 0.63 ⋅ w/h + 0.052 ⋅ w⁵/h⁵) = 7.2 ⋅ 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 for shear in this tutorial. This video explains transfering vertical loads in detail (click → here ←).

We won't show how to calculate the loads in this newsletter, as each calculation of the individual load is an article for itself. I've written detailed articles and published video tutorials about loads, which you can follow to understand how to calculate these loads:

• ​Dead load​
• ​Live load​
• ​Snow load​

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

The design normal force is 0 kN:

N_d = 0 kN

Step #4: Verify the timber beam for lateral torsional buckling

First, let's calculate the bending stress (short term load):

σ_m.d = M_d/I_y ⋅ h/2 = 15.6 MPa

Compression stress (short term load):

σ_c.d = N_d/(w⋅h) = 0 MPa

Design bending resistance:

f_m.d = k_mod ⋅ f_m.k/γ_M = 17.3 MPa

Design compression resistance:

f_c.0.d = k_mod ⋅ f_c.0.k/γ_M = 15.1 MPa

Next, 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 = 4.9m

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) = 0.60

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

λ_rel.m < 0.75 → k_crit = 1

Now we only need the following formulas if there is also a normal force acting in the beam. In our case the normal force is 0 kN. But I'll still include the formulas to show you hwo to do it.

Buckling length out-of-plane:

l_z = 5m

Radius of inertia (weak axis):

i_z = √(I_z/w ⋅ h) = 0.035

Slenderness ratio:

λ_z = l_z/i_z = 144.34

Relative slenderness ratio (EN 1995-1-1 (6.22)):

λ_rel.z = λ_z/π ⋅ √(f_c.0.k/E_0.g.05) = 2.45

Factor for members within the straightness limits (EN 1995-1-1 (6.29)):

β_c = 0.2

Instability factor (EN1995-1-1 (6.28)):

k_z = 0.5 ⋅ (1 + β_c ⋅ (λ_rel.z - 0.3) + λ_rel.z²) = 3.7

Buckling reduction coefficient (EN1995-1-1 (6.26)):

k_c.z = 1/(k_z + √(k_z² - λ_rel.z²)) = 0.15

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

η = (σ_m.d/(k_crit ⋅ f_m.d))² + σ_c.d/(k_c.z ⋅ f_c.0.d) = 0.82

Final Words

Alright, this is how we design timber beams for lateral torsional buckling according to Eurocode.

I hope this helped.

Enjoy the rest of the week and your weekend.

I’ll see you on Saturday for the Module #2 information.

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)

#3: The reinforced concrete series (click here)

#4: The engineering mechanics series (click here)

​↓ Follow me on Social Media. ↓

​​