# Vibration Design of Cross-Laminated Timber Plates

### Technical Article

For wide-span ceilings, the vibration design of cross‑laminated timber plates is often governing. The advantage of the lighter material of timber over concrete turns into a disadvantage because a high mass material is advantageous for a low natural frequency.

Figure 01 - Vibration Design (Source: [3])

For biaxial plate structures too, such as cross‑laminated timber plates, the design is usually performed on a uniaxial equivalent member. To explain the theoretical background, we will analyse a member first.

#### Example: Beam Structure

The advantages and disadvantages of member and surface design are explained in a practical structural component. The ground plan of a building has the dimensions of 8.44 m × 10.83 m. At 5.99 m in the longitudinal direction of the building, there is a structural interior wall. As you can see in Figure 02, a timber beam floor has been initially created and analysed in the RX‑TIMBER Continuous Beam program. In addition to the uniform loads displayed in Figure 03, a concentrated load results from the transition at the end of the staircase well.

LC1 = 6.9 kN

LC2 = 5.6 kN

Figure 03 - Load Data from RX-TIMBER Continuous Beam

The calculation performed in RX-TIMBER Continuous Beam gives the result of 14/32 cm for the required cross‑section.

The simplified vibration design in RF‑TIMBER Pro with the load combination of LC1 + LC2 gives the maximum deformation of 23.8 mm. The two‑span beam can be converted into a fixed single‑span beam, so the following limiting values of the deformation are available. Thus, the value of vibrations is computationally kept lower than 8.0 Hz. Find more information in [3].

$$\begin{array}{l}{\mathrm f}_\mathrm e\;\approx\;\frac{17.893}{\sqrt{\mathrm w}}\\\mathrm w\;\approx\;\frac{17.893²}{\mathrm{fe}²}\;=\;\frac{17.893²}{8²}\\{\mathrm w}_{\mathrm{limit},8\mathrm{Hz}}\;\approx\;5\;\mathrm{mm}\end{array}$$To comply with the simplified vibration design in RF‑TIMBER Pro, the cross‑section of 14/62 cm would be required.

You can perform more precise design in RF‑DYNAM Pro - Natural Vibrations and RF‑DYNAM Pro - Forced Vibrations, taking into account the requirements mentioned in [3].

Figure 05 - Flowchart from [3]

First, the detailed analysis checks whether the natural frequency is f_{0} ≤ f_{min}.

Figure 06 - Mode Shape No. 1 from RF-DYNAM Pro - Natural Vibrations

$${\mathrm f}_\min\;=\;4.5\;\mathrm{Hz}\;<\;{\mathrm f}_0\;=\;4.99\;\mathrm{Hz}$$Second, you can check whether the acceleration is a ≤ a_{limit}. For this, the periodic function of 2 Hz is defined in RF‑DYNAM Pro - Forced Vibrations. Converted to ω with 2 Hz ⋅ 2π = 12.566 rad/s. According to [3], Chap. 2.2.4, the acting force variable in time and location with F_{dyn} = 0.4 F(t) applies.

Figure 07 - Time Course in RF-DYNAM Pro - Forced Vibrations

In the next step, a load case is defined with the concentrated load of 1 kN (maintenance load), which is selected for design in RF‑DYNAM Pro - Forced Vibrations. The concentrated load is defined on the location of the selected maximum eigenvalue. According to [1], the Lehr's damping of ξ = 0.01 is used. The acceleration extends with 2 Hz over 5 seconds. The root mean square (see Figure 10) is then calculated with 0.077 m/s².

Figure 08 - Time History Analysis in RF-DYNAM Pro - Forced Vibrations

Figure 09 - Damping in RF-DYNAM Pro - Forced Vibrations

$${\mathrm a}_\mathrm{limit}\;=\;0.1\;\mathrm m/\mathrm s\;>\;\mathrm a\;=\;0.077\;\mathrm m/\mathrm s^2$$Thus, the analysis for the root mean square has been done. However, the limiting value has been slightly exceeded in the case of t = 0.85 s of 0.16 m/s². According to [3], it is possible to consider a screed as additional stiffness and mass in the calculation. The cross‑section is defined under the composite cross‑sections in RFEM. The connection between the screed and the timber cross‑section transfers no stiffnesses in this case (connection without shear). Structural height of the screed is set to 8 cm. More information about the composite cross‑sections is available in the manual of RF‑TIMBER Pro.

Even when using the composite cross‑section, the limiting value of acceleration limit is slightly exceeded in the case of t = 0.35 s with 0.13 m/s². A further calculation applies the root mean square.

Figure 11 - Composite Cross-Section

#### Example: Plate Structure

The example of the ground plan shown in Figure 02 is converted to a cross‑laminated timber plate with the cross‑section CLT 240 L7a‑2 (according to [2]). The panels in the lower part are defined in the same way as the beam structure: the continuous beam has a total length of 10.47 m, and a span width of 5.99 m (Span 1) and 4.48 m (Span 2) is defined. The plates with a length of 3.38 m are connected to continuous plates (see Figure 13).

The connection rigidity of the plates is not considered in this case, since it is assumed that the shorter plates are placed on the continuous plates, so there is no rigidity. Only for the rotation is a line release with the degree of freedom φx = 0 kNm/rad/m to be defined on all plate edges. The stress direction of the plates is illustrated in Figure 14.

The design is performed in RF‑LAMINATE and the result of the calculated stiffnesses is 21.4 mm in the characteristic/quasi-permanent combination. Also in this case, the simplified vibration design is exceeded. Therefore, the procedure in the previous chapter will be repeated for the plate structure.

Figure 12 - Cross-Laminated Timber Cross-Section

The design process in RF‑LAMINATE is explained in the manual.

In order to achieve a more precise calculation of the plate structure in RF‑DYNAM Pro - Natural Vibrations and RF‑DYNAM Pro - Forced Vibrations, a combination with LC1 + LC2 is created again.

Figure 14 - Stress Direction of Plates (main stress direction is red)

Figure 15 - Deformation in Characteristic/Quasi-Permanent Situation

The result of the calculation with this combination in RF‑DYNAM Pro - Natural Vibrations is the natural vibration of 4.8 Hz. In the case of the Mode Shape No. 1 of the plate structure, the maximum failure mode also results in the mid‑span of the first panel.

Figure 16 - Combination for Vibration Design

Also in this case, the concentrated load of 1 kN is defined and superimposed with the same function as in the case of the member structure. Figure 18 shows the root mean square of 0.0469 m/s² at 5 seconds. Even the maximum acceleration is almost within the limit criterion of a_{limit} ≤ 0.1 m/s². The limit value is slightly exceeded with 0.12 m/s². For further analysis, the stiffness and mass of the cross‑section will be increased by a screed with a thickness of 8 cm in RF‑LAMINATE. For this, the stiffness of the cross‑laminated timber plate is represented by an equivalent orthotropic timber cross‑section.

Figure 18 - Time Course Monitor of Plate Structure

The stiffness matrix of this composite cross‑section is determined without considering the shear coupling between the screed and the cross‑laminated timber plate.

Figure 19 - Determination of Equivalent Stiffness

By using this method, we finally succeeded in achieving the maximum value of the acceleration below the limit criterion, as you can see in Figure 20.

Figure 20 - Acceleration for Equivalent Cross-Sectional of Plate Structure

#### Summary

Biaxial design of a structural component allows you to reduce a cross‑section from 64 cm to 22 cm of thickness of a cross‑laminated timber plate while the vibration design according to Eurocode 5 is fulfilled.

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