Building materials: Influence of physical,mechanical and acoustic properties in sound prediction models

Abstract

Several models to compute the sound reduction index of a building partition, based on different approaches, can be found in the literature. Building construction involves a great variety of materials, and regardless of the chosen sound transmission simulation prediction approach, their elastic and physical properties, which are necessary as input data, have a strong influence on the accuracy of the predicted results. In this article, the influence of such properties is investigated, by means of a prediction model based on the method proposed by the recently updated standard EN ISO 12354. Moreover, the reliability of different prediction models and the consistency of their results have been tested, by simulating several building partitions made of various materials, both homogeneous and non-homogeneous.

Keywords

Prediction models material properties sound transmission sound insulation EN ISO 12354

Introduction

The acoustic design of buildings requires knowing numerous parameters, such as the acoustic properties of each single partition, related to either acoustic or mechanical excitations, and the parameters which govern the coupling between two building elements, in order to take into account the flanking sound transmission, as described, for example, in the series of standards EN ISO 12354.^1,2 Several models have been proposed to compute the sound insulation of a partition: analytical formulations have been developed, among others, by Cremer et al.³ and Fahy and Gardonio,⁴ for example, assuming a transversally infinite structure. This latter assumption is also used by the progressive impedance method (PIM)⁵ and by transfer matrix methods (TMM),⁶ powerful tools to simulate wave transmission through layered media. Otherwise, the vibroacoustic response of complex systems can be modelled by applying the statistical energy analysis (SEA).⁷ In the SEA approach, the investigated system is subdivided into elementary coupled subsystems. The power flow between them is computed from energy balance equations, derived in terms of modal density, internal and coupling loss factor and other mechanical and acoustic properties. SEA can be applied to compute the sound transmission in buildings, either by modelling the simple case of a partition between two rooms or by considering several paths involving airborne and structure-borne transmission.⁸ TMM and SEA represent computationally efficient prediction tools if compared to the finite element (FE) or the boundary element (BE) methods. However, they are especially suitable for the high-frequency range, characterised by a high modal density and modal overlap, but do not provide accurate results at lower frequencies, since they cannot take into account the modal behaviour. In order to increase the simulation accuracy also in the low-frequency range, where both the structural and the acoustic responses of a building partition are generally characterised by a low modal density, hybrid approaches have been developed combing finite element method (FEM) or wave-based (WB) approaches with SEA and TMM models.^9,10

Regardless of the approach chosen to model sound transmission through a building element, the reliability of the input data describing the mechanical properties of the materials has a strong influence on the accuracy of the predicted results. While the greater number of models has been developed for homogeneous isotropic elastic elements, building partitions are often constituted by different layers, involving orthotropic or anisotropic structures, non-homogeneous composites and visco-elastic materials. Building construction involves a great variety of materials, from mortar, bricks, concrete and wood to plasterboard and plywood panels, glass, resilient and fibrous or porous materials. This variety of structures and materials makes it even more crucial to be able to characterise the mechanical properties of buildings elements, in order to obtain accurate predicted results, and it also makes the application of the standard EN ISO 12354 to real buildings troublesome.^11–13

This article aims to provide the reader with a thorough analysis of the influence of the reliability of the material’s properties, used as input data in sound transmission modelling, on the predicted sound insulation of building elements, both considering homogeneous isotropic elements and composite, non-homogeneous or anisotropic systems. In the following section, a brief review of the different approaches to compute sound insulation is provided, highlighting advantages and drawbacks. Particular attention is paid to the models proposed in the recently updated standard EN ISO 12354,^1,2 since they have been used to investigate the influence of the physical and elastic properties of the materials on the prediction of the sound reduction index of a homogeneous monolithic element. The reliability and consistency of different prediction models have been tested by computing the sound reduction index of various building partitions.

Review of sound insulation prediction models for building partitions

During the last 60 years, many different approaches to investigate sound transmission through planar elements have been developed. One of the first analytic formulations to compute sound transmission through a building partition was derived by Cremer et al.,³ considering an infinite structure, with known wave mobility. By following a similar approach, Fahy and Gardonio⁴ derived the sound power transmission coefficient $τ$ , for the simple case of an infinite thin panel of thickness $h$ , excited by a plane sound wave with angular frequency $ω$ incident on the surface with an angle $θ$

τ (ω, θ) = \frac{{(2 ρ_{0} c_{0} \sec θ)}^{2}}{{(2 ρ_{0} c_{0} \sec θ + D ω^{- 1} η k_{0}^{4} \sin^{4} θ)}^{2} + {(ω ρ h + D ω^{- 1} η k_{0}^{4} \sin^{4} θ)}^{2}}

(1)

where $ρ_{0}$ and $c_{0}$ are the fluid density and the speed of sound, respectively, while $ρ$ represents the plate’s density and $D$ its bending stiffness, computed from its elastic modulus $E$ as

D = \frac{E h^{3}}{12 (1 - ν^{2})}

(2)

The logarithmic inverse of the sound transmission coefficient is defined transmission loss, TL, although in building acoustics it is usually called sound reduction index, $R$

R = - 10 \log τ

(3)

Based on these approaches, different formulations have been developed in order to consider also thick plates,¹⁴ orthotropic materials,¹⁵ and double walls, also considering the contribution of structural bridges.¹⁶ Analytical models are computationally efficient, although being usually developed for infinite structures they cannot take into account size effects and the influence of boundary conditions. The influence of the finite dimension of the structure on sound insulation was first studied by Sewell.¹⁷ Villot et al.¹⁸ proposed a correction term to obtain a more accurate prediction, based on a windowing approach. Several formulations of such term, known as finite size or non-resonant radiation efficiency, have been developed by different authors, reducing its computational cost.^19,20,21 The finite size radiation efficiency is often used to increase the low-frequency accuracy of the TMM simulation, a method largely used to compute sound transmission through multilayer structures, considering media of different nature: elastic solid, fluid and poroelastic materials. This method is particularly suitable to calculate sound insulation of building partitions, such as massive or lightweight walls, floating floors or suspended ceiling systems; moreover, it also allows to model double walls and lining systems taking into account the presence of structural connections.^22,23 Besides, it can also be applied to compute wave propagation through mechanically excited structures.²⁴ Alternatively, sound transmission in buildings can be calculated by using an SEA approach, which also allows to take into account different transmission path and excitation sources. It has been applied to compute the sound insulation of a single and double partition between two rooms.^25,26 The SEA framework has also been used by Craik and Smith²⁷ to compute sound transmission through studs in double walls. Based on a simplified SEA approach, the series of standard EN ISO 12354 have been developed in order to provide designers and acousticians with the tools to predict airborne and structure-borne sound transmission in buildings, considering both the direct and flanking paths and different kinds of excitation.²⁸ The calculation of the in situ acoustic performance, both the sound insulation of a partition between dwellings or a façade, and the impact sound insulation of a floor, is determined from the acoustics properties of the single buildings elements. These properties should be measured in laboratory conditions; however, very often this not being possible, it becomes necessary to calculate them. For each third-octave band centred at the frequency $f$ , the sound transmission coefficient $τ$ of a monolithic homogeneous building element can be computed by using the equations provided in Annex B of the standard EN ISO 12354-1¹

\begin{array}{l} τ = {(\frac{2 ρ_{0} c_{0}}{ω ρ h})}^{2} \frac{π f_{c} σ^{2}}{2 f η_{t o t}} & f > f c \\ τ = {(\frac{2 ρ_{0} c_{0}}{ω ρ h})}^{2} \frac{π σ^{2}}{2 f η_{t o t}} & f \approx f c \\ τ = {(\frac{2 ρ_{0} c_{0}}{ω ρ h})}^{2} (2 σ_{f}^{2} [1 - \frac{f^{2}}{f_{c}^{2}}] + 2 \frac{π f_{c} σ^{2}}{4 f η_{t o t}}) & f < f c \end{array}

(4)

where $σ$ is an acoustic descriptor known as radiation efficiency, or radiation ratio, for freely propagating waves, while $σ_{f}$ is the radiation efficiency for forced waves, $η_{t o t}$ is the total loss factor of the building structure and $f_{c}$ is the critical frequency, at which the structural wavelength equals the acoustic wavelength. The critical frequency can be computed for a thin plate as

f_{c} = \frac{c_{0}^{2}}{2 π} \sqrt{\frac{ρ h}{D}} = \frac{c_{0} \sqrt{3}}{π c_{L} h}

(5)

where $c_{L}$ is the phase velocity of longitudinal waves propagating along the plate, which can be related to material density and elastic properties as

c_{L} = \sqrt{\frac{E}{ρ (1 - ν^{2})}}

(6)

In each of the three frequency ranges for which equation (4) is defined, the sound transmission is governed by different properties of the material. Their influence will be discussed in detail in the following section. The sound insulation contribution due to additional layers ∆R can be taken into account as described in Annex D of the standard.

Review of impact sound insulation prediction model

The floating floor system is the most used technology to reduce the noise in dwellings due to impact sources. Their acoustic performance is evaluated in terms of improvement of impact sound insulation ∆L. The Cremer-Vér mass-law model,³ which allows to calculate the noise reduction of the floating floor, starting from the mass–spring–mass resonance of the system, was recently extend by Schiavi.²⁹ This alternative approach, based on the force transmissibility theory, allows one to accurately compute the sound insulation provided by the floating floor. Part 2 of standard EN ISO 12354-2² provides a calculation model, based on a simplified SEA approach, to compute both the standardised impact level for a homogeneous floor $L_{n}$ and the reduction of the impact sound pressure levels due the floating system ∆L. In Annex B, the standardised impact level generated by a standardised tapping machine, described in Annex E of standard UNI EN ISO 10140-5,³⁰ is given by

L_{n} \approx 155 - 30 \log (ρ h) + 10 \log T_{s} + 10 \log σ + 10 \log (\frac{f}{1000})

(7)

where $T_{s}$ is the structural reverberation time of the floor expressed in seconds. Annex C provides the equations to predict the reduction of impact sound of floating floors

Δ L = C \log (\frac{f}{f_{0}})

(8)

The constant $C$ depends on the floor materials: $C = 30$ for calcium–sulphate or sand/cement floor screeds; $C = 40$ for asphalt or dry floating floors. The mass–spring–mass resonance frequency $f_{0}$ of the floating floor system $f_{0}$ can be determined as

f_{0} = 160 \sqrt{\frac{s^{'}}{ρ_{s} h_{s}}}

(9)

The dynamic stiffness $s^{'}$ of the resilient layer can be measured according to standard EN 29052,³¹ while $ρ_{s}$ and $h_{s}$ represent the density and the and thickness of the floor screed, respectively.

The influence of material properties

Prediction of sound insulation of building partitions

It is well known that the surface mass of a partition, its stiffness and damping properties and also its size have a significant influence on its acoustic performance in different ranges of frequency. Some examples are reported recalling such important aspects related to sound insulation. The transmission loss of a homogeneous plasterboard panel has been computed by means of an in-house implemented code, based on the EN ISO 12354-1:2017 model, investigating the influence the physical and mechanical properties. The influence of the panel thickness is numerically investigated by simulating a plasterboard with well-defined properties, such as density, elastic modulus, loss factor and Poisson’s ration, but varying thickness. As shown in Figure 1, reducing the panel’s thickness significantly reduces the sound transmission governed by the surface mass up to the critical condition, given in equation (5), which is linearly shifted towards a higher frequency. Moreover, in Table 1, the thickness influence is summarised in terms of weighted single number sound reduction index $R_{w}$ and critical frequency $f_{c}$ , associated with the curve of transmission loss presented in Figure 1. Reducing the panel’s thickness from 18 mm to 9.5 mm leads to a 3 dB lower $R_{w}$ and shifts the critical frequency three third-octave bands down. In the region below the critical condition, the sound insulation is governed by the mass per unit of surface of the partition $m^{'} = ρ h ({kg / m}^{2}),$ according to the mass law. In this region, the sound insulation clearly increases as the panel’s thickness is increased. An analogous effect is shown in Figure 2, which offers a comparison between the sound reduction of the plasterboard panel calculated for different material’s densities. Below the critical condition, in the mass-law region, the sound insulation increases as the density increases (Table 2). However, a smaller influence on the critical condition is highlighted, compared to the variation in the structure’s thickness. The stiffness and damping properties of the element govern the sound transmission behaviour at and above the critical condition. As shown in Figure 3, reducing the elastic modulus of the material extends the frequency region governed by a mass-law sound transmission, significantly increasing the critical frequency. By shifting the critical frequency from the third-octave band centred around 4000 Hz to the 1250 Hz one, the single number sound insulation index of the partition $R_{w}$ is significantly reduced, as shown in Table 3. The total loss factor $η_{t o t}$ , describing the global damping of the element, is most commonly evaluated from the structural reverberation time $T_{s}$ , although other experimental methods exist, based on wave propagation or wave-imaging techniques and statistical approaches.^32,33 In particular, the total loss factor takes into account all the damping mechanisms of a specific structure: the internal losses, the losses due to radiation and the losses at the boundary. While the former two are mainly dependent on the material and its physical characteristics, the latter is strictly dependent on the boundary condition. The loss factor influences the sound insulation in the frequency region around and above the critical condition. In fact, as shown in Figure 4, increasing the loss factor reduces the dip associated with the critical condition and increases the sound insulation above the first coincidence. As reported in Table 4, increasing the loss factor from η₁ = 0.01 to η₂ = 0.06 increases the single number index $R_{w}$ of 1 dB.

Table 1.

Influence of panel thickness: single number weighted sound reduction index $R_{w}$ and critical frequency $f_{c}$ computed for the plasterboard wall panel with different thicknesses $h_{i}$ .

	$h_{1}$	$h_{2}$	$h_{3}$	$h_{4}$
$R_{w}$ (dB)	27	28	29	30
$f_{c}$ (Hz)	3666	2786	2321	1935

Table 2.

Influence of the material’s density: single number weighted sound reduction index $R_{w}$ and critical frequency $f_{c}$ computed for the plasterboard wall panel with different densities $ρ_{i}$ .

	$ρ_{1}$	$ρ_{2}$	$ρ_{3}$	$ρ_{4}$
$R_{w}$ (dB)	26	27	28	29
$f_{c}$ (Hz)	2484	2601	2786	2959

Table 3.

Influence of the material’s elastic modulus: single number weighted sound reduction index $R_{w}$ and critical frequency $f_{c}$ computed for the plasterboard wall panel with different elastic moduli $E_{i}$ .

	$E_{1}$	$E_{2}$	$E_{3}$	$E_{4}$
$R_{w}$ (dB)	29	27	26	25
$f_{c}$ (Hz)	4232	2677	1893	1338

Table 4.

Influence of the material’s internal loss factor: single number weighted sound reduction index $R_{w}$ and critical frequency $f_{c}$ computed for the plasterboard wall panel with different loss factor $η_{i}$ .

	$η_{1}$	$η_{2}$	$η_{3}$	$η_{4}$
$R_{w}$ (dB)	27	27	28	28
$f_{c}$ (Hz)	2786	2786	2786	2786

Prediction of the improvement of impact sound insulation of floors

The improvement of impact sound insulation of floors is required from a resilient layer inserted between the slab and the floor screed, in order to create a mass–spring–mass system. The design tool provided in Annex B of the standard EN ISO 12354-2:2017, given in equation (8), allows the prediction of the improvement of the impact sound insulation provided by the floating floor, based on the resonance frequency of the mass–spring–mass system, given in equation (9). The surface mass of the floor screed $m^{'}$ and the dynamic stiffness of the resilient layer $s^{'}$ govern the system resonance. In fact, Cremer demonstrated that above the resonance, the improvement of impact sound insulation is independent from the elastic properties of both the structural slab and the floor screed, but only depends on the mass–spring–mass system achieved by means of the floating floor. In Figure 5, the impact sound insulation improvement of a floating floor with a surface mass $m^{'} = 90 {kg / m}^{2}$ and different values of dynamic stiffness of the resilient layer is shown. By changing the dynamic stiffness from $s^{'} = 10 {MN / m}^{3}$ to $s^{'} = 160 {MN / m}^{3}$ , the resonant frequency of the system is shifted from the third-octave band centred around 50 Hz to the 160 Hz band, reducing the frequency range in which the acoustic performance of the floor is enhanced. An analogous comparison of the improvement of impact sound insulation can be made considering different values of the surface mass of the floor screed and the same dynamic stiffness, as shown in Figure 6. It is quite intuitive that as the surface mass increases, the resonant frequency lowers down. The influence of different combinations of floor screed mass and resilient layer dynamic stiffness is shown for a broader range of values in Figure 7.

The calculated improvement of impact sound insulation should be subtracted from the normalised impact level of the bare floor in order to obtain the impact level transmitted in the receiving room. However, it should be pointed out that the formulation provided in the standard, reported in equation (7) of this article, is only valid for homogeneous massive floors. For lightweight structures, such as timber floors and walls,^34–39 more complex mechanisms are involved. Timber floors are not covered by this model, although a formulation to compute the normalised impact level on timber floor has been proposed in a recent study.⁴⁰

Reliability of prediction models

The market offers a variety of commercial software, based on the approaches previously described, specifically developed for building acoustics design; each one implemented with a database of material properties. The sound reduction index of five building elements was predicted by means of four commercial software packages, based on different approaches: software A implements Cremers analytical model,³ software B is developed within the PIM framework,⁵ software C is based on Sharps model for single-leaf wall¹⁶ and software D implements the first version of the EN 12354:2002 model. Besides, two in-house implemented codes have been used: one based on the EN ISO 12354:2017 and the other on the TMM approach. In Figures 8 –12, the sound transmission loss, averaged over the results obtained from these six software packages, is compared to the experimental sound reduction index of five different structures, for which the assumptions of homogeneity and isotropy are increasingly less suitable. The collected experimental data were measured according to the standard EN ISO 10140-2;⁴¹ which specifies the procedure to determine the airborne sound insulation of building elements by measuring the sound pressure levels in two adjoining reverberant rooms: one designed as source room, while the other as receiving room. The five parts of the series of standards EN ISO 10140 provide all the specifications and the requirements related to the measurement procedures of airborne and impact sound insulation of building elements and to the test facilities, the size of the samples and their boundary and mounting conditions. More specifically, the mounting conditions are accurately described for different kinds of building elements, such as walls, floors, doors, windows, shutters, glazing and other technical elements. These laboratory conditions try to be representative of the ones obtained in real buildings while preventing the flanking transmissions of energy, and they can hardly be associated to the standard ideal boundary conditions used in analytical or numerical simulations. The variation in the degree of restraint, which is provided by the in situ mounting conditions against the one obtained in laboratory, might have some influence on the transmission loss at the lower frequencies, which are also affected by a higher uncertainty, due to the dimensions of the sample and the sound field diffusiveness. Together with the average predicted sound reduction index, a shaded area is provided to represent the spread of the numerical results. The physical and mechanical properties of the materials used as input data for each structure are shown in Table 5.

Table 5.

Physical and mechanical properties of the partitions investigated with the prediction models.

	Single glazed glass	Single-leaf plasterboard	Concrete wall	Solid bricks wall	Hollow bricks wall
$S$ (m²)	2.6	10	10	10	10
$h$ (m)	0.004	0.0159	0.200	0.270	0.120
$ρ$ (kg/m³)	2500	750	2300	1500	900
$E$ (Pa)	$7 \times 10^{10}$	$2.5 \times 10^{9}$	$2 \times 10^{10}$	$4.8 \times 10^{9}$	$4.8 \times 10^{9}$
$η$ (–)	0.02	0.06	0.005	0.01	0.005
$ν$ (–)	0.22	0.3	0.15	0.3	0.15

In Figure 8, the simulated sound insulation of a single glazed 4-mm-thick glass is compared with the experimental results. For such a homogeneous material, all the prediction software packages provide results with satisfying accuracy, correctly computing the critical frequency within the 3150 Hz band. However, the spread of the results increases in the low frequencies, since some of the software cannot take into account the influence of the finite dimension of the structure.

Similar results are found for a single-leaf plasterboard wall. As shown in Figure 9, all the tested software allows for an accurate approximation of the experimental sound reduction index. The critical frequency falls within the band centred on 2500 Hz, as correctly predicted by all the packages. At low frequencies, a smaller discrepancy is found compared to the glass element, since as the size of the structure increases, its influence on the sound transmission is reduced.

As shown in Figure 10, it is possible to predict the sound insulation provided by a concrete wall with accuracy. In fact, a good approximation of the massive behaviour of such partition is given with small spread up to the 2000 Hz band. At higher frequencies, however, the results show a significant deviation from the experimental sound reduction index, even up to 10 dB.

The deviation between experimental and predicted sound insulation significantly increases when the simulated structure is not homogeneous, but made of distinct elements, or constituted by a certain number of layers made of different materials. This is shown in Figures 11 and 12, by comparing the measured and calculated sound insulation of solid and hollow brick walls, respectively. For both of these structures, the prediction models provide results with low accuracy, with a deviation from the experimental data up to 20 dB in the higher frequencies. Moreover, a huge spread is found between the results obtained by the different models. The prediction of sound insulation of such inhomogeneous structures still represents one of the biggest challenges in building acoustics. As shown by this analysis, the existing models represent a useful design tool as long as the investigated partition can be considered as a homogeneous and isotropic structure. However, they clearly fail when these two assumptions are not fulfilled. In this case, it is either possible to use more sophisticated approaches, such as FEM or boundary element method (BEM), which require a great computational effort, or a complete SEA modelling. Otherwise, it is possible to adopt homogenisation techniques, which allow to apply the investigated simple approaches to a complex structure as well, treated as an equivalent homogeneous material. This latter approach, which is often used to investigate sandwich elements,^42,43 requires a frequency-dependent characterisation of the mechanical properties of the structure.⁴⁴ It has been successfully applied in building acoustics in order to investigate sound insulation through complex layered elements, like thermal–sound insulation cladding systems,²² and to investigate sound radiation and sound transmission through orthotropic cross-laminated timber panels.⁴⁵ Moreover, it is necessary to stress that it is not always straightforward to compute the improvement of sound insulation ∆R, which represents the enhancement of the acoustic performance provided by resilient mounted wall linings and cladding systems coupled with the base element of building partitions. While the TMM or PIM approaches allow to easily add the layers of different materials composing the simulated structure, analytically and statistically based methods present some difficulties in the computation of the improvement of sound insulation due to additional layers. Annex D of the standard EN ISO 12354-1, for example, only provides the references for the experimental measurement of the sound insulation improvement and a method to evaluate the weighted sound reduction index improvement ∆R, for external and internal linings based on the resonance frequency of the system. However, it does not allow for an accurate computation of the frequency-dependent improvement provided by such treatments.

In this study, we have investigated only the influence of mechanical properties of elastic structures. However, the reader should be aware that the materials used to increase the acoustic performance of building elements are usually visco-elastic layers, characterised by frequency-dependent elastic properties, and porous or fibrous materials, for which both the fluid phase and the solid frame need to be characterised. Therefore, the prediction of the increase in sound insulation and impact sound insulation is still challenging, and in order to obtain accurate results, there is a need for a thorough characterisation of the material’s physical and mechanical properties.

Conclusion

The influence of the physical and elastic properties of a building partition on the predicted sound reduction index has been investigated. The effects of varying the material’s properties, such as the surface mass, the elastic modulus or the loss factor, has been highlighted by the results computed with the prediction model proposed by the standard EN ISO 12354, but can be generalised to all the other prediction approaches. Particular attention has been paid, for example, to the mass effect for those partitions with a critical frequency falling in the highest frequencies. Moreover, the importance of the elastic and damping properties, within the frequency region characterised by the first coincidence, has been stressed. Different prediction models have provided reliable and consistent results in the prediction of the sound reduction index of homogeneous isotropic partitions, although they failed in the prediction when inhomogeneous walls were tested, with an increasing discrepancy between simulated results and experimental data, especially at high frequencies. This highlighted the need for either more sophisticated prediction models, which, however, would require a significant computational effort or the application of homogenisation techniques to describe the building partition as an equivalent homogeneous element characterised by frequency-dependent properties.

Footnotes

Acknowledgements

We would like to acknowledge Ing. Miller Tartari for his Master’s Dissertation work at the University of Ferrara, form which part of the results used to compare different sound transmission prediction models have been taken.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Andrea Santoni

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