# 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 - 1 - Vibration Design (Source: [3])

Usually, the design is also carried out on a biaxial tensile structures such as cross-laminated timber slabs on a single-axis equivalent member. Therefore, a member is analyzed first to explain the theoretical background.

#### Example: Member structure

The advantages and disadvantages of member and surface design are explained in a practical component. The ground plan of the building is 8.44 mx 10.83 m. At 5.99 m in the longitudinal direction of the building, there is a load-bearing inner wall. As shown in Figure 2, a timber beam ceiling is designed first, calculated with the RX-TIMBER DLT program. In addition to the line loads shown in Figure 3, a single load results from the change at the end of the stairwell.

LC1 = 6.9 kN

LC2 = 5.6 kN

Figure 03 - Load Data from RX-TIMBER Continuous Beam

The calculation in RX-TIMBER DLT results in a required cross-section of 14/32 cm.

In the simplified vibration analysis in RF-TIMBER Pro, the load combination LC1 + LC2 results in a maximum deformation of 23.8 mm. The two-span beam can be converted into a restrained single-span beam system, thus permitting the following deformation deformation limits. The vibrations are thus calculated below a value of 8.0 Hz. held. Further information can be found 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, a cross-section of 14/62 cm would be necessary.

With RF-DYNAM Pro - Natural Vibrations and RF-DYNAM Pro - Forced Vibrations, it is possible to perform a more detailed analysis, which takes into account the regulations in [3].

Figure 05 - 5 - Flowchart from [3]

In the closer examination, it is first analyzed whether the natural frequency f _{0} ≤ f _{min} .

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

f _{min} = 4.5 Hz <f _{0} = 4.99 Hz

In the next step, it is analyzed if the acceleration a ≤ a is _{limit} . For this purpose, a periodic function of 2 Hz is defined in RF-DYNAM Pro - Forced Vibrations. Converted to ω with 2Hz ∙ 2π = 12.566 rad/s. According to [3], Section 2.2.4, the acting time-location-variable force can be applied with F _{dyn} = 0.4F (t).

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

In the following, a load case with a concentrated load of 1 kN (man load) is defined, which is selected for analysis in RF-DYNAM Pro - Forced Vibrations. The concentrated load is defined at the location of the selected maximum eigenvalue. According to [1], Lehr = 0.01 is used as the damping measure as the damping measure. The acceleration extends for 5 seconds at 2 Hz. The root mean square value (see Figure 10) is 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

a _{limit} = 0.1m/s> a = 0.077 m/s²

The design for the root mean square value is thus provided. However, there is a slight overshoot of 0.16 m/s² at t = 0.85 s. According to [3], it is possible to consider the screed as an additional stiffness and mass in the calculation. The cross-section is defined under the assembled cross-sections in RFEM. The connection between screed and timber cross-section does not transfer any stiffnesses (shearless bond). The structural height of the screed is 8 cm. More information about the assembled cross-sections can be found in the RF-TIMBER Pro manual.

Even with the composite cross-section, the limit value of the acceleration at t = 0.35 s is slightly exceeded with 0.13 m/s². In the following, the calculation is continued with the root mean square value.

Figure 11 - Composite Cross-Section

#### Example of Tensile Structure

The example according to the floor plan in Figure 2 is converted into a cross-laminated timber panel with the cross-section CLT 240 L7s-2 (according to [2] ). The plates in the lower part are defined identical to the member structure as continuous beam with a total length of 10.47 m and a span of 5.99 m (panel 1) and 4.48 m (panel 2). The 3.38 m long slabs are connected to the continuous slabs (see Figure 13). The connection flexibility of the plates is not taken into account here, since it is assumed that the shorter plates are placed on the continuous plates, so that there is no compliance. Only for the rotation is a line hinge with the degree of freedom φ _{x} = 0 kNm/rad/m defined on all plate edges. The clamping direction of the plates is explained in Figure 14.

The design is carried out in RF-LAMINATE and the stiffnesses calculated in this way result in a deformation of 21.4 mm in the characteristic/quasi-permanent combination. Again, the simplified vibration design would be exceeded. Therefore, the procedure of the previous chapter is repeated for the slab structure.

Figure 12 - Cross-Laminated Timber Cross-Section

The design in RF-LAMINATE is described in the manual.

For a more accurate design of the tensile structure with RF-DYNAM Pro - Natural Vibrations and RF-DYNAM Pro - Forced Vibrations, a combination with LC1 + LC2 is created.

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

Figure 15 - Deformation in Characteristic/Quasi-Permanent Situation

In RF-DYNAM Pro - Natural Vibrations, this combination results in a natural vibration of 4.8 Hz. The maximum mode of failure in the middle of the field of the first field is also obtained for the first mode shape of the tensile structure.

Figure 16 - Combination for Vibration Design

Here, too, a concentrated load of 1 kN is applied and superimposed with the same function as the member structure. In Figure 18, a root mean square value of 0.0469 m/s² is determined for 5 seconds. Even the maximum acceleration is almost within the _{limit criterion} of a _{limit} ≤ 0.1 m/s². The limit value is only slightly exceeded at 0.12 m/s². Furthermore, the stiffness and mass of the cross-section are increased in RF-LAMINATE with an 8 cm thick screed. For this, the stiffness of the cross-laminated timber board is represented by an equivalent orthotropic timber cross-section.

Figure 18 - Time Course Monitor of Plate Structure

The determination of the stiffness matrix for this composite cross-section is determined without considering the shear bond between screed and cross-laminated timber slab.

Figure 19 - Determination of Equivalent Stiffness

With this method, it is finally possible, as shown in Figure 20, to also perform the maximum value of the acceleration below the limit criterion.

#### Summary

By performing a biaxial analysis of the structural component, it was possible to reduce the cross-section from 64 cm to 22 cm thickness of the cross-laminated timber panel while fulfilling the vibration design according to Eurocode 5.

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