Vibration Design of Cross-Laminated Timber Plates

Technical Article on the Topic Structural Analysis Using Dlubal Software

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

Image 01 - Vibration Analysis (Source:

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

Example: Beam Structure

The advantages and disadvantages of member and surface design are explained on a practical structural component. The ground plan of a building has the dimensions of 8.44 m x 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 was initially created and analyzed 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

Image 02 - Floor Plan

Image 03 - Load Data from RX-TIMBER DLT

The calculation performed in RX-TIMBER DLT gives the result of the required cross-section of 14/32 cm.

The simplified vibration design in RF-TIMBER Pro with the load combination of LC1 + LC2 provides 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. The vibrations are thus kept mathematically over a value of 8.0 Hz. Find more information in [3].

Formula 1

fe  17.893ww  17.893²fe² = 17.893²8²wlimit,8Hz  5 mm

Image 04 - Loads

A cross‑section of 14/62 cm would be required to comply with the simplified vibration design in RF‑TIMBER Pro.

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

Image 05 - Flowchart from [3]

First, the detailed analysis checks whether the natural frequency is f0 ≤ fmin.

Image 06 - Mode Shape from RF -DYNAM Pro - Natural Vibrations

fmin = 4.5 Hz < f0 = 4.99 Hz

Second, you can check whether the acceleration is a ≤ alimit. For this, the 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], Ch. 2.2.4, the acting force variable in time and location with Fdyn = 0.4 F(t) applies.

Image 07 - Time History 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], 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.

Image 08 - Analysis of Time History in RF-DYNAM Pro - Forced Vibrations

Image 09 - Damping Measure in RF-DYNAM Pro - Forced Vibrations

alimit = 0.1 m/s > a = 0.077 m/s²

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 an 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.

Image 10 - Acceleration a from RF -DYNAM Pro - Forced Vibrations:

Image 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 the total length of 10.47 m, and the span width of 5.99 m (Span 1) and 4.48 m (Span 2) is defined. The plates with the 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 a of 21.4 mm in the characteristic/quasi-permanent combination. Also in this case, the simplified vibration design is exceeded. Therefore, the procedure of the previous chapter will be repeated for the plate structure.

Image 12 - Cross-Laminated Timber Cross-Section

Image 13 - Plate Geometry

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.

Image 14 - Span Direction of Plates (Main Span Direction Red)

Image 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 Mode Shape No. 1 of the plate structure, the maximum failure mode also results in the mid‑span of the first panel.

Image 16 - Combination for Vibration Analysis

Image 17 - Natural Vibration

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 alimit ≤ 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.

Image 18 - Time History Diagram for Two-Dimensional 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.

Image 19 - Determining Equivalent Stiffness

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.

Summary

The 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.

Literature

[1] Blaß, H.J.; Ehlbeck. J.; Kreuzinger H.; Steck G.: Explanations for DIN 1052: 2004-08, 2nd ed.). Cologne: Bruderverlag, 2005
[2] Allgemeine bauaufsichtliche Zulassung Z-9.1-599, dated 13th of January, 2012
[3] Hamm, P.; Richter, A.: Bemessungs- und Konstruktionsregeln zum Schwingungsnachweis von Holzdecken. In: Fachtagungen Holzbau 2009. Leinfelden-Echterdingen, 26th of November, 2009. Published by: Landesbeirat Holz Baden-Württemberg e.V., Stuttgart. pp. 15-29.

Author

Dipl.-Ing. (FH) Bastian Kuhn, M.Sc.

Dipl.-Ing. (FH) Bastian Kuhn, M.Sc.

Product Engineering & Customer Support

Mr. Kuhn is responsible for the development of products for timber structures and provides technical support for our customers.

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  • Updated 11/26/2021

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RFEM 5
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