06 August 2021

Designing CLT in vibration – An extract from TRADA publication DTS

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The following is an extract from Chapter 5 of Designing timber structures: an introduction.

 

Vibration analysis is a complex area and requires significant thought. In the UK, vibration in building design does not consider structural failure (as you would in countries where there are earthquakes), but the experience of people in the building. As a result, vibration design is a serviceability limit state consideration.

 

Furthermore, because vibration is a function of human perception, it is very hard to determine what is acceptable. If you were to live in an old timber house and you could feel the floors vibrate, you may put it down to the age of the building and think no more of it. If it was a new building you may have a different reaction, expecting the building to perform differently.

 

Historically, we have designed timber floors to be ‘bouncy’ compared to steel and concrete buildings. This creates a challenge for us going forward as we start to consider the vibration of timber floors in more detail. Do we continue to achieve the same standard as we have done for many years in the UK, or do we try and enhance the design?

 

This section will explain the basics of vibration design, what we might consider and why, before giving some simple tools to use. It will not, however, go into substantial detail on the design for vibration; this could be a book in its own right.

 

We should also be aware just how important this area is. For other material types, vibration rarely governs design. But timber is strong and light. This leads to vibration often being the governing factor above deflection, bending and shear. As a result, for long spans we really need to design for vibration at the beginning of the design, not as an afterthought.

  

When designing for vibration we need to consider three things:

 

  • First, the natural frequency.
  • Second, what type of response we are concerned about (transient or resonant).
  • Third, the magnitude and whether it is an acceptable value. This is often considered by using a response factor. For example, if you are lying on a hospital bed, your body is much more likely to feel vibration than during a Zumba class and therefore the allowable acceleration/velocity needs to reflect this.

 

Natural frequency

 

The natural frequency is the frequency that the floor will choose to vibrate at, if it can. All structures have a variety of modes of vibration; if we consider our structure to be a simple beam element, the first three modes are shown below:

 

Figure 5.20 First three fundamental modes of a beam

 

When designing timber floors, we are typically only interested in the first mode, which has the lowest natural frequency. There are two reasons for this. First, the first mode will dominate the response. Second, as you will see in the next section, as the frequency goes up, so our awareness of the vibration decreases.

 

Therefore, for most timber floors we only need to design for the first modal frequency.

 

The first mode can be calculated in a number of ways.

 

The standard equation is:

 

 

Where:

l Span

EI Stiffness

m Mass (note not the weight) of the floor

 

We can rearrange the deflection equation:

 

 

In terms of EI/m (we need to convert m to ω by accounting for gravity as m is a mass and ω is a weight):

 

 

We can then substitute in the deflection equation and rewrite the equation as:

 

 

Where d is the deflection of the floor under the unfactored permanent action only.

 

For natural frequency calculations, the weight of the floor is taken as the weight that we can expect to be there when the vibration occurs. As a result, we normally only include the permanent loads, which include the self-weight of the floor, ceiling, services and finishes.

 

The equations above lead to natural frequencies that are lower than measured values on real floors. There are a number of reasons for this.

 

  • First, the frequency is very sensitive to the assumed permanent action (dead load). We normally overestimate permanent action to ensure our design is safe, but this will lead to a lower frequency.

 

  • Second, the above equations assume the ends are free to rotate (pinned) however in reality the ends of CLT floors (and joists) often show some degree of fixity, especially when considering vibration.

 

  • Third, there is no account for damping. Damping is the loss of energy and can occur for several reasons. Timber floors themselves demonstrate 1%–3% damping. In addition, the furniture and people stood on the floors will also damp out the vibrations.

 

  • Finally, the above equations don’t account for the transverse stiffness of the floor. We often assume our floors are one-way spanning, which is a good assumption for the design of bending, shear and deflection, even for CLT floors. But for vibration the transverse stiffness will have an impact, especially if the floor is supported on all four edges.

 

Figure 5.21 First mode of vibration for a CLT floor assuming one-way and two-way spanning

 

As a result, the above equations are often rewritten to account for all the different uncertainties in the design. Obviously, there are more precise solutions which can also be considered, but as a first approximation this is a good starting point[2]:

 

 

[2] Taken from Eurocode 5: timber design essentials for engineers, TRADA Technology, 2009

  

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