PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Measurement of the bending stiffness of masonry walls by using
time-frequency analysis: Practical considerations and validation
Charlotte CRISPIN1; Christian MERTENS
1

Belgian Building Research Institute, Belgium

ABSTRACTThe bending stiffness is one of the parameters that determines the mechanical behaviour of a plate andtherefore influences the acoustic performance of building elements. Besides, it appears in the prediction
models for calculating the sound reduction index, R, or recently the vibration reduction index, Kij. Its
accurate determination is thus essential in the field of acoustics. The Atomic Energy Research Institute ofKorea has proposed an interesting measurement technique to determine this parameter in-situ. This technique,which uses the time-frequency analysis, has been successfully applied to thin plates. However, this paper
shows that corrections should be used for thick walls. Some practical considerations are given and somemeasurements results for masonry walls are presented.
Keywords: Material, Properties, Bending

1. INTRODUCTION
The vibratory analysis of a plate excited by a hammer hit shows that quasi-longitudinal waves(symmetric zero order mode, S 0, of the Lamb waves) and bending waves (Anti-symmetric zero ordermode, A 0 , of the Lamb waves) can settle in it. This article focuses on those latter. Indeed, they can beused to determine the dynamic properties of the plates.
The traveling velocity of the bending waves depends on the dynamic properties of the wall but also
depends on the frequency: the bending waves are said to be dispersive. This means that the highfrequency bending waves are faster than the low frequency.
The measurement of the traveling velocity of the bending wave’s energy and the adjustment of theresults on a theoretical curve lead to the determination of the interdependent dynamic properties ofplates.
For thin plates, the traveling velocity of the bending wave’s energy is easily measured withaccelerometers and the well-known relationships between the dynamic properties can be used without
special precautions. For thicker plates, the relevant signal is more difficult to obtain, and the simplified
formulas dedicated to the thin plates are no longer valid.

2. RELATIONSHIP BETWEEN THE DYNAMIC PROPERTIES
2.1 For thin plate
In most books on acoustics, the relationships between the dynamic properties of the plates, which
are proposed, are based on the theory for thin plates and are given by the well-known following
formulas:𝐶,
2 1.8𝑐 ℎ𝑓
(1)
𝑓

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Charlotte.crispin@bbri.be

√

(2)

And,
𝐵

(3)

With,𝐶,, the group velocity of the bending waves according to the thin plate theory;h, the thickness of the plate [m];f, the frequency [Hz];𝑐 , the quasi-longitudinal wavespeed [m/s]ρ, the density of the plate [kg/m³];
B, the bending stiffness [Nm];fc , the critical frequency [Hz];c0 , the wavespeed in air [m/s]Traditionally, the limit for the applicability of these formula is given by Cremer (1):𝜆6ℎ(4)At high frequencies, additional terms need to be added to take account the shear deformations andthe rotary inertia (2). These corrections result in a bending wavespeed that is lower than that given by
the equation for pure bending wave and the above equations are no longer valid.
2.2 For thick plateIn his article, Rindel (3) presents an equation for the effective phase speed, cph, in a thick plate.
This equation is a combination of bending and shear waves. The representation of this function as afunction of frequency shows at high frequency a convergence towards an asymptote given by the shear
wavespeed, c s, which cannot be accurately determined. Ross (4) proposes another formula. According
to him, in a uniform and monolithic plate, the phase velocity of the bending wave can be approximatedby:𝑐,.

𝑐.

(5)

And the group velocity is given by:𝐶,.

(6),The adjustment of this theoretical curve on a measured group velocity curve gives the quasilongitudinal wavespeed 𝑐 .
For masonry walls, the critical frequency is low enough to use the thin plate theory for its
determination and for the calculation of the bending stiffness, B, according to Eq.2 and Eq.3
respectively.

3. THE EXPERIMENTAL PROCEDURE FOR MEASURING THE TRAVELINGVELOCITY OF THE EFFECTIVE BENDING WAVE’S ENERGY
Different techniques were tested to measure the bending wavespeed or the longitudinal wavespeed(4, 5, 6, 8) in thick plates. In this article, it is the traveling velocity of the bending wave’s energy (thegroup velocity) which is determined and which is used for the calculation of the 𝑐 .
In their article, Y-C. Choi and Cie (7) present a suitable method to determine the group velocity ofthe bending waves by using a time-frequency analysis. The group velocity equals the distance between
two accelerometers divided by the time delay between these accelerometers. Being dispersive, thegroup velocity must be given as a function of the frequency.
𝐶 𝑓∆
∆

(7)

Where,Δt is the arrival time delay of the bending wave’s energy according to the frequency between twoaccelerometers;

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Δx is the distance between these two accelerometers.
The arrival time delay between the accelerometers is obtained by the Wigner-Ville analysis. This
analysis allows to obtain the energy distribution of a short signal as a function of time and offrequencies. The approach of Wigner-Ville is based on the instantaneous autocorrelation function andis defined as (9):𝑊 𝑡, 𝜔𝑠 𝑡𝑠∗ 𝑡𝑒𝑑𝜏(8)Thus, with this time-frequency analysis method, the arrival time delay can be obtained at eachfrequency with a very good time resolution.
The adjustment of the theoretical curve of C g,Ross [Eq. 6] on the measured group velocity curve [Eq.
7] provides then the quasi-longitudinal wave velocity 𝑐 .

4. EXPERIMENTAL RESULTS
4.1 Aluminium plate
A hammer hit is applied on a suspended aluminium plate with the dimensions 1.45 m x 1.35 m x0.0025 m (figure 1). The density of the plate is 2153 kg/m³. Two accelerometers are spaced of 0.1 m
and the impact is given at 0.15 m from the accelerometer 1 on the same line. The signals are recorded
by a data acquisition system with a sample frequency of 192000Hz.
Time signals
Aluminium plate
0.8
Acc [m/s²]

0.6
0.4
0.2
0‐0.2
‐0.4
‐0.6
‐0.80.0257

Figure 1 – Picture of the Aluminium plate

Acc10.0262
Time [s]Acc20.0267

Figure 2 – Time signals for the two accelerometers

Figure 2 shows the acceleration signals of the two accelerometers. The Wigner-Ville analysis
applied to both signals gives their frequency distribution of the energy as a function of time (figure3). As expected, and according to the dispersion law of the bending waves, the highest frequencies
arrive first followed by the medium frequencies and finally by the lowest frequencies. On these graphs,
the peak magnitudes of the energy distributions are gathered (red curves). The two curves are thendisplayed on the same graph (Figure 4) in order to determine the time delay at each frequency.

Figure 3 – Frequency distribution of the energy as a function of time for the two accelerometers1288

The group velocity can be then calculated by applying [Eq.7]. The determination of the first
dynamic property, the quasi-longitudinal wavespeed , 𝑐 , is reached by the adjustment of thetheoretical curve Cg,Ross [Eq. 6]. The results are shown figure 5. The Cg,thin [Eq.1] of the thin plate
theories is also presented in this figure.
For the aluminium plate, the thin plate theory gives nearly the same results than the thick plate
theory and the relationships Eq 1 to Eq 3 can be used for the determination of the dynamic properties.
The calculated quasi-longitudinal wavespeed, 𝑐 , is 3632 m/s; the critical frequency, f c , is 7019 Hz
and the bending stiffness, B, is 37 Nm,.
This procedure was also successfully applied for the determination of the properties of theplasterboards.

8000
7500
7000
6500
6000
5500
5000
4500
4000
0.0002

Acc1

The group velocity
Aluminium plate
800

Acc2

600
Cg [m/s²]

Frequency [Hz]Frequency distribution of the peak magnitudeenergy as a function of time

400
Cg (thin plate theory)
Measuring dataCg (Ross approximation)

200

t1t20.0004

00.000600.0008

1000 2000 3000 4000 5000 6000 7000 8000

Time [s]Frequency [Hz]Figure 4 – Frequency distribution of the peakFigure 5 – The measured group velocity compared toenergy as a function of time for the two

the theoretical expressions

accelerometers
4.2 Gypsum block walls
The wall is composed of gypsum blocks (figure 6). Its dimensions are 2.4m x 4m and the thickness
is 0.07 m. The block density is 1066 kg/m³.
This partition wall is in fact a composition of a double wall for which the airborne sound insulation
was measured (figure 7). The result of this test is important because it shows us that the criticalfrequency falls in the third octave band of 500 Hz. This data will allow us to validate our results.
For this case, a horizontal line of 8 accelerometers spaced of 0.05 m is used. The impact is given
at 0.10 m from the accelerometer 1 on the same line. The averaged time delay obtained between allof these accelerometers gives a best accuracy on the results.
The signals are recorded by a data acquisition system with a sample frequency of 204800Hz.
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R [dB]

7060
504020Figure 6 – Picture of the gypsum block50
6380100
125
160
200
250
315
400
500
630
800
1000
1250
1600
2000
2500
3150
4000
5000

30f [Hz]wall and the horizontal line ofFigure 7 – The sound reduction index ofaccelerometers

the double gypsum wall

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The exploitable signals are in the first milliseconds because the standing waves set in very quickly.
A time window is then used to isolate the relevant signals. A low-pass and a high-pass filters are alsoapplied. The frequency range of interest is that which gives the wavelengths between 3h and L/3(where h is the thickness and L, the width of the walls). In this case:0.21 m<λinterest<0.8m
Or,
400 Hz < finterest<5000 Hz
A hammer with a hard head is used to give sufficient energy in this frequency range. Figure 8 gives
the Wigner-Ville analysis for the first four accelerometers. The distribution curve appears quite clearlyon the graphs.

Figure 8 – Frequency distribution of the energy as a function of time for the first four accelerometersaligned horizontally

1000

2000
Frequency [Hz]The group velocity
Gypsum blocks ‐ Horizontal direction

1200

Acc4Acc3

Acc2

Acc1

2500

Frequency distribution of the peak magnitude energy
as a function of time

800
Cg [m/s²]

1500
1000
500

t100.0006

t2

t3

t4

600
400

Cg (thin plate theory)
Measuring data
Cg (Ross approximation)

200
00.00070.0008Time [s]0.00090.001

0200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]Figure 9 – Frequency distribution of the peakFigure 10 – The measured group velocity compared

energy as a function of time for the first fourto the theoretical expressions for the horizontal

accelerometers aligned horizontallydirectionThe peaks of the bending wave energy of each graphs are gathered and they are presented on thesame graph (figure 9). At each frequency, the goup velocity is evaluated between different1290

accelerometers. The light grey curves show these results on figure 10. The red curve represents theaverage of these curves. The adjustment of the theoretical curve according the thin plate theory gives
inconsistent results because this plate can no longer be considered as a thin plate.
The adjustment of the Cg,Ross gives: 𝑐 ≈ 1699 m/s. The calculated fc , according to Eq. 2, is then
536 Hz and the bending stiffness is B=87958 N.m;
This critical frequency falls in the third octave band of 500 Hz observed on the spectrum of soundreduction index, R (figure 7).
The figures below present the results for the vertical direction. The adjustment of the Cg,Ross (figure13) gives: 𝑐 ≈ 1867 m/s. The calculated fc is then 488 Hz and the bending stiffness is B=106156
N.m;This critical frequency falls yet in the third octave band of 500 Hz observed on the spectrum ofsound reduction index, R.

Figure 11 – Frequency distribution of the energy as a function of time for the first four accelerometersaligned vertically
The group velocity
Gypsum blocks ‐ Vertical direction

Frequency distribution of the peak magnitudeenergy as a function of time
3500

1200
1000
Cg [m/s²]

2500

Acc3

1000

Acc2

1500

Acc4

2000
Acc1

Frequency [Hz]

3000

800
600
Cg (thin plate theory)
Measuring data
Cg (Ross approximation)

400
200

500
00.000200.00030.00040.00050.0006Time [s]

0500

1000

1500

2000

2500

3000

Frequency [Hz]Figure 12 – Frequency distribution of the peakFigure 13 – The measured group velocity compared

energy as a function of time for the first fourto the theoretical expressions for the verticalaccelerometers aligned verticallydirection1291

4.3 Brick wall
This procedure was also tested for a wall composed of perforated bricks (h is 0.14 m and ρ is 900kg/m³) an with the same dimensions than the gypsum block wall. The results are summarised at figures
15 and 16 only for the diagonal direction. The measurements were carried out on the plastered wall
(Figure 14).

Figure 14 – Picture of the brick wall and the diagonal line of accelerometres

Figure 15 – Frequency distribution of the energy as a function of time for the first four accelerometersaligned diagonally
The group velocity
Brick wall ‐ Diagonal direction

1000
900
800
700
Cg [m/s²]

600
500400300Cg (thin plate theory)
Measuring data
Cg (Ross approximation)

200
10000200

400

600

800

1000 1200 1400 1600 1800 2000 2200
Frequency [Hz]Figure 16 – Frequency distribution of the peak energy as a function of time for six accelerometers

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The adjustment of the Cg,Ross (figure 16) on the results gives: 𝑐 ≈ 900 m/s. The calculated fc , isthen 506 Hz and the bending stiffness is B=166736 N.m. This estimated fc is consistent with the result
obtained on the spectrum of the sound reduction index.
The procedure described here for the determination of the dynamic properties is applicable as longas the size of the inhomogeneities (the size of the holes in the blocks) is small compared to generated
wavelengths.

5. CONCLUSIONS
The adjustment of a theoretical curve of the group velocity, Cg , on measurement results allows toobtain the dynamic properties of masonry walls. The theoretical expression of Cg coming from thethin plate theories gives aberrant results because the masonry walls are relatively thick compared to
the wavelength of the structural waves and they exhibit more complicated dispersions than thin plate.
It is the Ross’s approximation which is more relevant for thick plate, that should be used to estimate
theoretically Cg and cph .
Experimentally, the group velocity is reached by the measurement of the arrival time delay of thebending wave’s energy according to the frequency between two accelerometers whose the distancethat separate them is known. This is the Wigner-Ville analysis, which is used to obtain, with relatively
good accuracy, these arrival time delays. Practically, the exploitable signals for masonry walls, are in
the first milliseconds because the standing waves set in very quickly. A time window is then used toisolate the relevant signals. A low-pass and a high pass filters are also applied.
This procedure is applicable as long as the size of the inhomogeneities in the masonry blocks issmall compared to generated wavelengths.

ACKNOWLEDGEMENTS
This study was done in the framework of the project A-light II (Integrating lightweight concepts
in acoustical standardization). The authors are grateful for the financial support from the Federal
Public Service Economy of Belgium.

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