Finite element modelling of airborne sound insulation in single and double cross laminated timber panels Maria Pettersson 1 Engineering Acoustics Luleå University of Technology, 971 87 Luleå, Sweden Fredrik Ljunggren 2 Engineering Acoustics Luleå University of Technology, 971 87 Luleå, Sweden

ABSTRACT Cross laminated timber (CLT) is manufactured with an odd number of layers of wooden boards glued together orthogonally. Compared to concrete or brick, CLT has low mass, which makes it subject to poorer sound insulation, particularly at lower frequencies. In this work, single and double CLT pan- els, with air and mineral wool respectively in the cavity, are modelled homogeneously in thickness using the finite element method. A 2D-3D Hybrid finite element model is introduced which rotates the panels to capture a diffuse-like sound field with reduced computational time. The goals are to predict the weighted sound reduction indexes R w and R w +C 50 – 3150 and to study the response as the material and dimensions alter. For single panels, the differences regarding R w and R w+ C 50 – 3150 are within ±2 dB compared to analytical calculations and laboratory measurements. The proposed model is suggested to be used for predicting R in one-third octave bands, R w and R w +C 50 – 3150 . For double CLT panels, the proposed model can provide an indication of the airborne sound insulation and serve as a tool for relative comparisons rather than providing exact results due to the lack of comparative data. 1. INTRODUCTION

Wood can be described as an orthotropic material with independent and unique mechanical properties in three mutually perpendicular axes; longitudinal, radial and tangential which makes sound insula- tion predictions of cross laminated timber (CLT) panels challenging. An upgrade on the predictions could optimize the design of CLT panels and further increase its use.

Finite element (FE) methods have become a main technique to analyze a variety of physical phe- nomena. CLT is normally modelled either by treating each layer separately or by modelling all layers homogeneously through thickness. When modelling layers separately, the material properties of wood are used for each perpendicular board layer. When modelling layers homogeneously, material properties are averaged and weighted in the parallel directions. Winter [1] studied single CLT panels using FE method and performed experimental modal analysis for comparison and validation at fre- quencies 31.5 - 8 000 Hz. At lower frequencies, using solid elements in the mesh, the differences on eigenfrequencies and eigenmodes between modelling layers separately or homogeneously was neg- ligible. Starting from 800 Hz, panels modelled with separate layers provided less modes. Both models

1 maria.1.pettersson@ltu.se

2 fredrik.ljunggren@ltu.se

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identified the modes in radial axis at similar frequency. Winter concluded that modelling CLT panels homogeneously in thickness simplifies the modelling and meshing inside the FE model since shell elements can be used.

Ljunggren [2] has developed an analytical method for calculating airborne sound insulation of CLT panels. For single panels, the prediction is based on a model originally provided by Sharp [3] for homogeneous materials but modified in such a way that the coincidence dip gets less pronounced. The modified model is also used as the input to predict airborne sound insulation of double panels. Ljunggren compared the outcome from the analytical model with other models and measurements. It was suggested that the analytical model can serve as an engineering tool in predicting airborne sound insulation. For single CLT panels, the deviation between predicted and measured weighted sound reduction index was consistently within ±2 dB but often within ±1 dB. Ljunggren also reported a lack of reference data for double CLT panels, which currently makes it difficult to quantify that model's uncertainty.

In this paper, single and double CLT panels, with air and mineral wool respectively in the cavity, will be studied using the FE method. The panels are modelled homogeneously in thickness to predict airborne sound insulation in the frequency range 20 - 3 150 Hz as the importance of frequencies below 50 Hz have been reported by Ljunggren et al. [4], [5]. The goals are to predict the weighted sound reduction indexes R w and R w+ C 50 – 3150 [6] and to study the sound reduction response as material con- figurations and dimensions of the panels are altered. 2. METHOD

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The multi-physics FE software Comsol versions 5.5-5.6 is used. CLT is modelled with "Solid me- chanics" and "Pressure acoustics, frequency domain" is selected for air and mineral wool (when pre- sent). The multi-physics-coupling which includes fluid load on the structure and structural accelera- tion as experienced by the fluid is applied. 2.1 Material properties The CLT panels are modelled homogeneously in thickness with orthotropic material properties ac- cording to Table 1. Air is modelled with its preset parameters, including atmospheric attenuation. For the mineral wool, the poroacoustics model "Delany-Bazley-Miki" [7] which describe fibrous materi- als, like mineral wool, including its bulk losses is used with flow resistivity 9 000 Pa/s/m 2 as input. Table 1: Material properties of the CLT panel. Density is 400 kg/ m 3 , Poisson's ratio is 0.3 and loss factors for Young's modulu s and shear modulus are 0.03 (3 %).

CLT panel

Young’s modulus [MPa] Shear modulus [MPa] x y z x y z 70 mm (3 layer)

892 4 797.5 10 152 108 582 95

120 mm (5 layer)

2 304 4 650 7 859 119 565 83

2.2 Hybrid model The CLT panels represent a separating wall between two ordinarily sized bedrooms with a floor area of 3 x 4 m 2 and a height of 2.5 m. To reduce the simulation times, a 2D FE model is selected. A drawback of 2D models is that a fully diffuse-like sound field cannot be created. To overcome this issue, the rooms with its separating CLT panel are instead rotated 90 degrees including adjustment of the geometry and the material parameters to create a 2D-3D Hybrid FE model capable of predicting 3D effects in a simplified way. For each panel, two 2D models, denoted First and Second model, are analyzed, see Figure 1. In the First model, the two rooms measure 3 x 4 m and the length of the separating CLT panel is 4 m. The length direction of the panel is parallel to the z-axis, according to the figure in Table 1, and the outer board layers are vertically oriented. In the Second model, the rooms are rotated by 90 degrees and then measure 3 x 2.5 m where the length of the CLT panel is 2.5 m. The length direction of the panel is now parallel to the x-axis and the outer board layers are hori- zontally oriented.

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Length

Length

a) First model b) Second model Figure 1: Layout of the FE models: sending (green) and receiving (gray) rooms separated by a CLT wall (orange). Blue color represents the boundary condition “Perfectly matched layer”. 2.3 Sound field, Boundary condition The sound field is generated by the "Plane wave" option, which enables a sound wave to travel in axial direction. The pressure amplitude of the wave is 1 Pa (approximately 94 dB). The sound inci- dence of zero degree is defined as being perpendicular to the separating wall, see Figure 2. Four angles of incidence are used; 0, 30, 60 and 82.5 degrees, where the averaged response simulates the response of a diffuse sound field. In the subsequent evaluation, weighting factors are applied to the angles (1: 0 degree, 2: 30 degrees, 2: 60 degrees and 1: 82.5 degrees) to capture its share of the 90- degree sector within the 180-degree sector to simulate a diffuse sound field.

45

The boundary condition "Perfectly matched layer" is used encircling the actual model, visible in Figure 1. "Perfectly matched layer" is tuned by its geometry and mesh to absorb outgoing wave en- ergy in the frequency-domain to limit impedance mismatch causing reflections at the boundary to the actual model. With "Perfectly matched layer", the FE models are supposed to represent laboratory conditions where flanking transmission does not contribute to the sound transmission.

CLT Wall

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Figure 2: Sound incidence angels of the applied plane wave in degrees and corresponding weighting factors (purple). 2.4 Sound reduction index, R The sound reduction index, R is calculated as the incident boundary sound pressure level at the wall surface in the source room subtracted by the boundary sound pressure level in the receiver room. It is averaged and weighted with respect to angles of sound incidence, for the First and Second model respectively:

𝑅𝑅 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹,𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = −10𝑙𝑙𝑙𝑙𝑙𝑙 10 ( (10 (−𝑅 𝑅 0𝑑 𝑑 𝑑 𝑑 𝑑 𝑑 . /10) ) + (2 ∗10 (−𝑅 𝑅 30𝑑 𝑑 𝑑 𝑑 𝑑 𝑑 . /10) ) + (2 ∗10 (−𝑅 𝑅 60𝑑 𝑑 𝑑 𝑑 𝑔 𝑔 . /10) ) + (10 (−𝑅 𝑅 82.5𝑑 𝑑 𝑑 𝑑 𝑑 𝑑 . /10) )

6 )

(1)

𝑅𝑅 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 = −10𝑙𝑙𝑙𝑙𝑙𝑙 10 ( (10 (−𝑅 𝑅 𝐹 𝐹 𝐹 𝐹 𝐹 𝐹 𝐹 𝐹 𝐹 𝐹 /10) ) + (10 (−𝑅 𝑅 𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 /10) )

For the 2D-3D Hybrid FE model, R is calculated as the mean value from the First and Second model:

2 )

(2)

3. RESULTS AND DISCUSSION

Results from the 2D-3D Hybrid FE model is validated by comparison with analytical calculations developed by Ljunggren [2] regarding sound reduction index, R and weighted sound reduction in- dexes R w and R w +C 50 – 3150 [6]. 3.1 Sound reduction, First and Second model In Figure 3, R from the First model (purple) and its angle dependence of the plane wave are presented for CLT 70 mm. At perpendicular sound incidence, the sound reduction follows the theoretical mass law without any coincidence effect since the panel acts as a piston without any bending wave velocity. At the critical frequency, sound waves propagate in the plane parallel to the panel and the bending wave velocity of the panel is equal to the velocity in air, whereby the sound reduction is reduced. The largest difference in sound reduction, about 15 dB, is found between zero degree and 82.5 degrees incidence at 3 150 Hz.

i g

The difference in sound reduction between the First and the Second model, Figure 3, relates to the different orientations of the panels. The First model provides increased sound reduction compared to the Second model, largest difference between the models is approximately 5 dB at 160 Hz.

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Figure 3: CLT 70 mm; Sound reduction index, R. 3.2 Hybrid model vs. Analytical calculation In Figures 4-5, R from the Hybrid model are compared to the analytical calculations for the studied CLT 70 mm and CLT 120 mm wall configurations of both single and double panels, Table 2 presents R w and R w +C 50 – 3150 . As for the denotation of double walls; CLT 70-45-70 mm consists of 70 mm CLT, 45 mm air and 70 mm CLT. CLT 70-M45-70 mm consists of 70 mm CLT, 45 mm mineral wool and 70 mm CLT.

Table 2: Wei g hted sound reduction indexes; R w (100 – 3 150 Hz) and R w +C 50 – 3150 (50 – 3 150 Hz).

Configurati o n Hybrid model Analytical calculation R w [dB] R w +C 50 – 3150 [dB] R w [dB] R w +C 50 – 3150 [dB] CLT 70 mm (single wall) 30 29 31 30 CLT 70-45-7 0 mm 41 36 42 40 CLT 70-M45-70 mm 50 44 46 43 CLT 70-145 -7 0 mm 53 50 50 47 CLT 70-M145-70 mm 64 57 55 51 CLT 120 mm (single wall) 36 35 35 34 CLT 120-45 -1 20 mm 58 55 49 46 CLT 120-M45-120 mm 67 56 52 49 CLT 120-14 5- 120 mm 67 65 57 53 CLT 120-M145-120 mm 77 71 62 58

ANGLES + First model: R (CLT 70 mm) First model: 0 deg. (CLT 70 mm) ~- First model: 30 deg. (CLT 70 mm) -+-First model: 60 deg. (CLT 70 mm) > First model: 82.5 deg. (CLT 70 mm) ONE-THIRD OCTAVE BAND [Hz]

MODELS “Hybrid model (CLT 70 mm) + First model (CLT 70 mm) “Second model (CLT 70 mm) SOUND REDUCTION INDEX, R [dB] ONE-THIRD OCTAVE BAND [Hz]

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“© CLT 70 mm HYBRID MODEL CLT 70-4 5-70 mm, CLT 70-145-70 mm -*=CLT-70-M45-70 mm, © CLT 70-M145-70 mm. R [4B] EDUCTION IDE SOUND RI 100 90 80 0 60 50 40 30 20 10 0 20 2s “0 282 FREQUENCY, ONE-THIRD OCTAVE BAND [Hz] ans

Figure 4: Single and double panels consisting of CLT 70 mm; Sound reduction index, R.

ANALYTICAL CALCULATION CLT 70 mm -© CLT 70-45-70 mm. CLT 70-145-70 mm -*= CLT. 70-M45-70 mm, CLT 70-M145-70 mm. 3 90 80 70 50 40 30 20 SOUND REDUCTION IDEX, R [dB] 10 a OREESRRSELEEEREEEE FREQUENCY, ONE-THIRD OCTAVE BAND [Hz]

HYBRID MODEL CLT 120 mm -© CLT 120-45-120 mm. ©CLT 120-145-120 mm -*CLT 120-M45-120 mm -=CLT 120-M145-120 mm. 120 lo S100 90 80 70 60 50 40 30 20 10 SOUND REDUCTION IDEX, R AaeSeezenss FREQUENCY, THIRD OCTAVE BAND [Hz]

Figure 5: Single and double panels consisting of CLT 120 mm; Sound reduction index, R.

ANALYTICAL CALCULATION © CLT 120 mm -© CLT 120-45-120 mm CLT 120-145-120 mm -*CLT 120-M45-120 mm, = CLT 120-M145-120 mm 120 gue = 100 90 80 70 60 50 40 30 20 10 SOUND REDUCTION IDEX, R Seseusegeesggs ii FREQUENCY, |E-THIRD OCTAVE BAND |[Hz| 1280 3180 o

For the double panels with air, increasing the cavity from 45 mm to 145 mm leads to increased sound reduction at frequencies up to approximately 1 250 Hz for both the Hybrid model and the analytical calculations. For higher frequencies, the smaller air cavity provides higher sound reduction in the Hybrid model whereas no difference regarding the cavity size is found in the analytical calculations.

When mineral wool is added to the cavity in the double walls, the Hybrid model reduces the sound reduction up to about 50 Hz for the 45 mm cavity and up to around 80 Hz for the 145 mm cavity. The analytical calculations increase the sound reduction from about 63 Hz. With 45 mm mineral wool in the cavity, the sound reduction increases, compared to the situation without mineral wool, at most around 500 Hz with the Hybrid model but between 1 250 - 3 150 Hz with the analytical calculations. In the case of 145 mm cavity, the sound reduction increases at most from 1 250 Hz upwards using the Hybrid model and more constantly between 80 – 3 150 Hz using the analytical calculation due to the mineral wool.

The Hybrid model with double panels, particularly the CLT 120 mm configurations, provide higher sound reduction compared to the analytical calculations in a range centered at around 200 Hz, which truly affect R w and R w +C 50 – 3150 . According to Table 2, the Hybrid model for double CLT walls show higher R w and R w +C 50 – 3150 compared to the analytical calculations, except for CLT 70-45-70 mm. 4. CONCLUSIONS

For single CLT panels, the results from the 2D-3D Hybrid FE model are well in line with analytical calculations and laboratory measurements. The differences between the Hybrid model, analytical cal- culations and laboratory measurements regarding the weighted sound reduction indexes R w and R w +C 50 – 3150 are all within ±2 dB. The proposed Hybrid model may be used to predict airborne sound reduction index in terms of R, R w and R w +C 50 – 3150 , for single CLT panels.

For double CLT panels, the proposed 2D-3D FE Hybrid model can provide an indication of the airborne sound insulation and serves as a tool for relative comparisons between different configura- tions. However, the absolute accuracy is questioned until verifying data has been obtained. 5. FUNDING

This work was financially supported by the Swedish Energy Agency (Energimyndigheten), project number 46771-1.

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6. REFERENCES

1. Winter, C. K. Frequency dependent modeling for the prediction of the sound transmission in tim- ber constructions, Technische Universität München, Ingenieurfakultät Bau Geo Umwelt, PhD thesis (2018). 2. Ljunggren, F. Sound insulation predictions of single and double CLT panels , Proceedings of ICA (23 rd ), pp. 242-248. Aachen, Germany, September 2019. 3. Sharp, B. H. Prediction methods for the sound transmission of building elements. Noise Control Engineering Journal, 11 (2) , 53-63 (1978). 4. Ljunggren, F., Simmons, C. & Hagberg, K. Correlation between sound insulation and occupants’ perception – Proposal of alternative single number rating of impact sound. Applied Acoustics, 85 , 57-68 (2014). 5. Ljunggren, F., Simmons, C. & Öqvist, R. Correlation between sound insulation and occupants’ perception – Proposal of alternative single number rating of impact sound, part II. Applied Acous- tics, 123 , 143-151 (2017). 6. European Committee for standardization, Brussels, Belgium. Acoustics – Rating of sound insula- tion in buildings and of building elements – Part 1: Airborne sound insulation. 7. Miki, Y. Acoustical properties of porous materials – Modification of Delany-Bazley models. The Journal of Acoustical Society of Japan, 11 , 44-55 (1990).

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