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

Vibration transmission between two reinforced concrete beams
with surface-to-surface contact conditions
Marios Filippoupolitis1,2; Carl Hopkins1,21

Institute for Risk and Uncertainty, University of Liverpool, UK
2Acoustics Research Unit, University of Liverpool, UKABSTRACT
To detect human survivors trapped in buildings after earthquakes by using structure -borne soundit is necessary to have knowledge of vibration transmission in collapsed and fragmented reinforced concrete buildings. In this paper, Statistical Energy Analysis (SEA) is used to model vibrationtransmission between two reinforced concrete beams with surface-to-surface contact conditions.
Experimentally validated Finite Element Models (FEM) of two elastically connected beams were used
to carry out a Monte Carlo simulation with 30 beam junctions. Coupling loss factors were determinedfrom the FEM data using Experimental SEA (ESEA) and these were compared against theoreticalmodels based on a lump spring connector. Results are shown for bending modes, torsional modes andthe combination of all modes. For only bending or torsional waves, close agreement was achieved
between FEM ESEA and the analytical model up to the frequency where half the bending or torsional
wavelength equaled the longest side of the contact area. When all wave types were combined,
reasonable agreement could still be achieved but only at low-frequencies.
Keywords: vibration transmission, statistical energy analysis, finite element methods

1. INTRODUCTION
Earthquakes have the highest rate of mortality among all the natural disasters. From 1970 to 2009,36% of fatalities that have occurred due to natural disasters are due to earthquakes (1). When victims
are trapped inside a collapsed building, the challenge is to detect and locate survivors within a periodthat will allow them to be rescued. Most documented live rescues are accomplished within the firstsix days (2). However, important variables affect the survivability including the structure t ype andvoid space formation, the cause of the structural collapse, the survival location in the building and thespeed and sophistication of available search and rescue capabilities (3). The prediction of vibrationtransmission in collapsed and fragmented reinforced-concrete buildings has the potential to inform
decisions about the possibility to detect trapped human survivors by using structure -borne soundpropagation. This research forms part of a funded project concerning an approach to search for humansurvivors using structure-borne sound propagation in collapsed and fragmented structures through thedevelopment, validation and use of theoretical models.
The aim of this paper is to assess the potential to use Statistical Energy Analysis (SEA) to mode lvibration transmission between two reinforced concrete beams when they are stacked on top of eachother (i.e. without any bond connecting the two beams) to make a surface-to-surface connection. Thisis carried out using numerical experiments with Finite Element Methods (FEM) to create an ensembleof beam junctions for a Monte Carlo simulation which will allow use of Experimental SEA (ESEA)
to determine Coupling Loss Factors (CLFs) between the beams.
The three main aspects to investigate are (a) whether it is possible to only consider one type ofwave motion (e.g. bending waves) or whether two or more types of wave motion can be considered
128140M.Filippoupolitis@liverpool.ac.uk
carl.hopkins@liverpool.ac.uk

simultaneously (e.g. bending and torsional waves), (b) the modelling of the contact condition in FEM
and (c) whether analytical models based on lump spring connectors can be used to model the contact
condition. The first aspect concerning the use of ESEA with multiple wave types is necessary because
in a collapsed structure it is not known whether one or more wave type will be excited at the surfaceto-surface connection. For coupled plates with wave conversion at the junction, Hopkins (4)previously showed that with ESEA it was not always possible to identify multiple wave types when
only bending waves are excited on one plate. The second aspect is partly addressed by previous work(7) using FEM of reinforced concrete beam junctions with a surface-to-surface connection that havebeen experimentally validated; these results indicated that a spring can be used to model the contact
condition in FEM. The latter finding provides the impetus to assess analytical models based on lumpspring connectors to calculate CLFs for SEA models.

2. METHODS
2.1 Finite Element Modelling
The junctions consist of two reinforced concrete beams, SS1 and SS2. SS1 is the lower beam of
the junction with 6.0 m length, 0.3 m width and 0.2 m depth. SS2 is the upper beam of the junction
with 5.0 m length, 0.2 m width and 0.3 m depth. Both beams are reinforced with four longitudinal
steel bars of 16 mm diameter. The transverse reinforcement of both beams consists of 8 mm diameter
stirrups placed at 200 mm centres along the beams.
FEM modelling was carried out using Abaqus v6.14. The solid element C3D20R (20 nodes) andthe beam element B32 (3 nodes) were selected from the element library of Abaqus (5) to model theconcrete and the steel bars respectively (see Figure 1). The mesh density fulfils the requirement for atleast six elements per wavelength in structural and vibroacoustic problems (6).
The beams were simply supported at both ends. The interaction between the two beams wasmodelled using an elastic contact in the normal direction. After model updating based on previous
experimental work (7) it was found that the contact stiffness followed a log-normal distribution witha mean value equal to 7.038E08 N/m. This value was used for the elastic contact in the FEM models.

Figure 1 – Example FEM model showing the surface-to-surface connection between beams SS1and SS2.
Table 1 shows the physical and mechanical properties of the materials used in the FEM model. The
critical damping, ζ, was set to be equal to 0.05.
Table 1: Material properties
Young’s modulus, E

Poisson’s ratio,[N/m2]

ν [-]

2287

34.7E09

0.2

7800200E09

0.3

Material

Density, ρ [kg/m3]

Concrete
Steel8141

Mode-based steady-state dynamic analysis was used to calculate the dynamic response of the beamjunctions up to 3200 Hz considering only the out-of-plane bending modes or only the torsional modesor the combination of all modes. Results are shown for 16 frequency bands with a bandwidth of 200 Hz
in the frequency range from 1 to 3200 Hz.
2.2 Driving-point mobilities
Driving-point mobilities of the individual beams SS1 and SS2 were calculated with FEM using
steady-state dynamic analysis at frequencies up to 3200 Hz, by using (a) only the out-of-plane bending
modes, (b) only the torsional modes or (c) all modes. The upper surface of beam SS1 and the lower
surface of beam SS2 were excited using a unit force (perpendicular to the surface) with random phase(i.e. rain-on-the-roof excitation) on 124 and 104 nodes respectively.
2.3 Monte Carlo simulation for ESEAThe experimentally validated FEM model was used as a basis for creating a sample of 30 beamjunctions using a Monte Carlo simulation with FEM. For convenience the angle between the twobeams was fixed at 41q so that the length of the longest side of the surface-to-surface contact area,L C,max , was constant. The relative position of the two beams was defined by the coordinates of thecentroid of beam SS2, C SS2(x, y). These were sampled from the uniform distributions C SS2(x)~U(-2.68,2.68) and C SS2(y)~U(-1.43, 1.43) with the rule that the black shaded area in Figure 2 which indicatesthe surface-to-surface connection area is constant and equal to 0.091 m2.
The beams were excited using rain-on-the-roof excitation (i.e. forces with unity magnitude andrandom phase); all the nodes of the lower surface were excited on beam SS1, and all nodes of theupper surface were excited on beam SS2.

Figure 2 – Relative beam positions in the ensemble of junctions.

2.4 Experimental Statistical Energy Analysis (ESEA)
Each beam represents one subsystem and the output from the FEM models was used to calculate
the subsystem energy and power input that apply to a SEA model of each beam junction. These FEM
data were then used in ESEA to determine coupling loss factors.
The ESEA matrix solution for two subsystems is given by (8)ଶ‫ۍ‬෍ ߟଵ௡‫ێ‬‫ێ‬௡ୀଵ‫ێ‬‫ ێ‬െߟଵଶ‫ۏ‬8142‫ې‬‫ܧ ۑ‬‫ ۑ‬൤ ଵଵଶ‫ܧ ۑ‬ଶଵ෍ ߟଶ௡ ‫ۑ‬‫ے‬௡ୀଵെߟଶଵܹ୧୬ሺଵሻ‫ܧ‬ଵଶ൨ൌ൦ ߱‫ܧ‬ଶଶͲͲܹ୧୬ሺଶሻ߱

൪(1)

where η ij is the coupling loss factor from subsystem i to j, η ii is the internal loss factor for subsystemi and Eij is the energy of subsystem i when the power is input into subsystem j, Win(i) is the power
injected into subsystem i, and Z is the angular frequency.
The energy associated with each subsystem is given by (8)‫ ܧ‬ൌ ݉‫ ݒۃ‬ଶ ‫ۄ‬௧ǡ௦

(2)where m is the mass of the subsystem and <v 2>t,s is the temporal and spatial average of the meansquare velocity of all the unconstrained nodes of the lower and top surfaces of subsystems SS1 andSS2 respectively.
For rain-on-the-roof excitation at P nodes, the power input,   is given by (8)௉ܹ୧୬ ൌ߱෍൫ ൛‫ܨ‬෠ ൟܴ݁ሼ‫ݓ‬ෝሽ െ ൛‫ܨ‬෠ ൟ‫݉ܫ‬ሼ‫ݓ‬ෝሽ൯௣ʹ(3)௣ୀଵwhere F is the force and ‫ݓ‬ෝ is the peak out-of-plane displacement associated with each node.
2.5 Analytical model based on lump spring connector
An analytical model for the surface-to-surface connection can be considered on the basis that when
the wavelength of the structure-borne sound is much larger than the length of the longest edge of thesurface-to-surface connection area, it can be modelled as a lump spring.
For N identical point connections between two beams, the coupling loss factor from beam i tobeam j can be calculated using [e.g. see (8)]ߟ௜௝ ൌܴ݁൛ܻ௝ ൟܰZ݉௜ หܻ ൅ ܻ ൅ ܻ หଶ௜௝(4)ୡ

where mi is the mass of beam i, the driving-point mobility of a thin beam of infinite extent (forexcitation of bending waves in the central part of the beam) is calculated using (8)ିଵܻ ൌ ቀሺͳ ൅ ݅ሻʹǤ͸͹ߩܵඥܿ୐ǡୠ ݄݂ቁ(5)and the mobility of the point connection, Y c, can be calculated using (8)
ܻୡ ൌ

݅Z
݇

(6)

where U is the density of the solid beam, S is the cross-sectional area of the beam, f is frequency,
h is the depth of the beam, c L,b is the phase velocity of the beam for quasi-longitudinal waves, k is thedynamic stiffness of the point connection acting as a spring (N/m).

3. Results
3.1 Bending and torsional wavelength
To assess the potential frequency range of application for the lump spring model, the length of thelongest side of the surface-to-surface contact area, L C,max, is compared with the bending and torsional
wavelength. The lump spring model is expected to be valid at frequencies where λ B/2 >> LC,max forbending motion and λ T/2 >> L C,max for torsional motion.
Figure 3 shows the bending wavelength, λ B and the half-wavelength, λ B/2 for beam SS1 and SS2.
L C,max is equal to λ B/2 for beams SS1 and SS2 at 1720 Hz and 2580 Hz respectively. For torsional
waves, L C,max is equal to λ T/2 for beams SS1 and SS2 at 2360 Hz.

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Figure 3 – Bending wavelength, λB of beams SS1 and SS2. The red line indicates the length of the longest
side, LC,max=0.45 m of the contact area.

Figure 4 – Torsional wavelength, λT of beams SS1 and SS2. The red line indicates the length of the longest
side, LC,max=0.45 m of the contact area.
3.2 Assessment of the FEM driving-point mobilities
Figure 5 and 6 compare the spatial average of the driving-point mobilities of beams SS1 and SS2
from FEM using either bending or torsional or all modes against the driving-point mobilities of a thin
beam of infinite extent for bending wave excitation (Eq. 5). Results are shown for both the real partand the magnitude that are used in Eq.(4) to calculate the CLF.
The lowest local bending mode frequency for isolated beams SS1 and SS2 is 9.7 and 21.4 Hz
respectively. For bending wave excitation of beams SS1 and SS2, the ratio Y B,FEM /Y B,Inf is below 3 dB
between the 100 and 3100 Hz frequency bands, both in terms of the real part and the magnitude of thedriving point mobilities.
The lowest local torsional mode frequency for isolated beams SS1 and SS2 is 164.4 and 172.8 Hz
respectively. For torsional wave excitation, the ratio Y T,FEM /Y B,Inf is below 3 dB between the 300 and
3100 Hz bands. However, in the 100 Hz band the ratio is up to 7 dB.
For all modes (i.e. the combination of bending, torsional and longitudinal modes) of beams SS1and SS2, the ratio Y A,FEM /Y B,Inf is below 6 dB between the 100 and 3100 Hz frequency bands, both interms of the real part and the magnitude of the driving point mobilities.
For beam SS1 (which is thinner in the direction of bending wave motion) the infinite beam mobility
is a reasonable estimate for both bending and torsional modes and whilst it is also reasonable forbending waves on SS2, it is less reasonable for torsional modes. For both beams the infinite beamtheory does not give a reasonable estimate for the combination of all modes. However, due to thedifferences between the FEM and the infinite beam mobilities, both will be used to calculate the CLF
in Section 3.3.

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Figure 5 – Ratio of the real part of the driving-point mobilities (FEM to infinite beam theory) for beam SS1(left) and beam SS2 (right).

Figure 6 – Ratio of the magnitude of the driving-point mobilities (FEM to infinite beam theory) for beam
SS1 (left) and beam SS2 (right).
3.3 Comparison of coupling loss factors from lump spring theory and FEM ESEA
Figure 7 to 9 show the coupling loss factors, η 12 and η 21 from FEM ESEA for the 30 beam junctions
along with two prediction models based on the analytical lump spring model. The FEM ESEA results
are shown in terms of a mean value with 95% confidence intervals. The analytical lump spring modelis calculated using (a) the driving point mobilities, Y B,Inf, of thin beams of infinite extent for excitationof bending waves in the central part of the beams and (b) the spatial average of the FEM driving point
mobilities, Y FEM , over the surface of beams SS1 and SS2.
For bending waves only in the FEM model (Figure 7), there is reasonable agreement between thecoupling loss factors from FEM ESEA and the analytical model up to the 1700 Hz frequency band
(where λ B/2 >> LC,max), with differences up to 5 dB. Above 1700 Hz, the differences increased up to12 dB where the contact area between SS1 and SS2 cannot be considered as a point connection. Using
FEM mobilities instead of the infinite beam mobilities for bending wave motion did not significantly
improve the agreement.
For torsional waves only in the FEM model (Figure 8), there is reasonable agreement between
FEM ESEA and the analytical model up to the 2300 Hz frequency band (where λ T /2 >> LC,max), withdifferences up to 5 dB. Note that in two low frequency bands (100 Hz and 300 Hz) the spread of theFEM ESEA coupling loss factors was ± 5 dB. Above 2300 Hz, the differences were between 5 and 12
dB where the contact area between SS1 and SS2 cannot be considered as a point connection. The
infinite beam mobilities are intended for bending rather than torsional motion but are shown forreference. Using FEM mobilities instead of the infinite beam mobilities for torsional motion did not
significantly improve the agreement.
For the combination of bending, torsional and longitudinal modes (Figure 9), reasonable agreement
was achieved between the FEM ESEA and the analytical model between 100 and 900 Hz, withdifferences up to 5 dB. The analytical model is not expected to be valid above 1700 Hz (where λ B/2>> LC,max) but the combination of all modes seems to reduce the frequency where reasonableagreement is obtained from 1700 to 900 Hz. The infinite beam mobilities are an approximation as they
intended for bending rather than all wave motion; hence it is a coincidenc e that the infinite beam
mobilities show better agreement than the FEM calculated mobilities with the analytical model above
1100 Hz.

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Figure 7 – Comparison of FEM ESEA (bending modes) and the analytical model (LSC) coupling loss
factors η12 and η21.

Figure 8 – Comparison of FEM (torsional modes) and the analytical model (LSC) coupling loss factors η12and η21.

Figure 9 – Comparison of FEM (combination of all modes) and the analytical model (LSC) coupling loss
factors η12 and η21.

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4. CONCLUSIONS
Finite element models were used to calculate the driving-point mobilities of reinforced concretebeams for bending modes only, torsional modes only and the combination of all modes in thefrequency range up to 3200 Hz. These mobilities were in reasonable agreement with the theoretical
driving-point mobilities of a thin beam of infinite extent for bending wave excitation but not for thecombination of all modes.
An ensemble of 30 random beam junctions was generated for Monte Carlo simulations with FEM
that allowed ESEA to be used to determine coupling loss factors between the two beams. These werecompared with coupling loss factors calculated using an analytical model based on a lump spring
connector. For only bending waves or torsional waves, close agreement was achieved between
FEM ESEA and the analytical model up to the frequency where half the bending or torsional
wavelength equaled the longest side of the contact area. Above this frequency the interaction between
the two beams cannot be considered anymore as a point connection. The inclusion of the FEM drivingpoint mobilities in the prediction model (instead of the infinite beam model) does not significantly
improve the agreement. When all wave types are combined, reasonable agreement can still be achieved
at low-frequencies.

ACKNOWLEDGEMENTS
The authors are grateful for the funding provided by the EPSRC and ESRC Centre for Doctoral
Training in Quantification Management of Risk & Uncertainty in Complex Systems and Environments
at the University of Liverpool.

REFERENCES
1. Bobrowsky, P. T. Ed., Encyclopedia of Natural Hazards, Springer Netherlands, (2013).
2. Macintyre, A., Barbera, J. A. and Smith, E. R. Surviving Collapsed Structure Entrapment afterEarthquakes: A “Time-to-Rescue” Analysis, Prehospital and Disaster Medicine, 21 (1), 4–19, (2006).
3. Macintyre, A., Barbera, J. A. and Petinaux, B. P. Survival Interval in Earthquake Entrapments: ResearchFindings Reinforced During the 2010 Haiti Earthquake Response, Disaster Medicine and Public HealthPreparedness, 5 (1), 13–22, (2011).
4. Hopkins, C. Experimental statistical energy analysis of coupled plates with wave conversion at thejunction. Journal of Sound and Vibration 2009;322:155-166.
5. Hibbitt, D., Karlsson, B. and Sorensen, P., Abaqus 6.14 Documentation and User Manual, Dessault
Systemes Simulia Corp, Providence, Rhode Island (2014).
6. Atalla, N. and Sgard, F., Finite Element and Boundary Methods on Structural Acoustics and Vibration,
CRC Press, Taylor & Francis Group, Boca Raton (2015).
7. Filippoupolitis, M., Hopkins, C. Investigation of support conditions between stacked reinforced concretebeams using experimental modal analysis and finite element methods, Proceedings of the 9th InternationalCongress on Computational Mechanics, Chania, Greece, 4–6 June, (2018).
8. Hopkins, C., Sound Insulation, Butterworth-Heinemann, Oxford (2007).

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