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Prediction of train-induced vibrations inside buildings using transfer functions
ARTICLE IN PRESS
Soil Dynamics and Earthquake Engineering 27 (2007) 93–98 www.elsevier.com/locate/soildyn
Prediction of train-induced vibrations inside buildings using transfer functions Christoffer With, Anders Bodare Department of Civil and Architectural Engineering, Royal Institute of Technology (KTH), Brinellva¨gen 34, 100 44 Stockholm, Sweden Received 29 May 2006; accepted 4 June 2006
Abstract This article presents a study of the use of transfer functions to predict vibrations inside a building due to train-induced ground vibrations. It is proposed that knowing the transfer function of a building, the vibrations inside a similar building can be predicted given a known ground motion outside. A comparison has been conducted between predicted and measured vibrations inside a building in south-western Sweden due to freight and passenger trains. The transfer function was derived by using a stationary vibrating source. The average particle velocity (1 s-r.m.s.) inside the house was calculated with the transfer functions with an average error of 10%, 0.02 mm/s. Prediction was achieved with a standard deviation of 23%, 0.06 mm/s when no filtering of the data was used. Further work is needed to ascertain the accuracy of this method. r 2006 Elsevier Ltd. All rights reserved. Keywords: Train; Ground vibration; Prediction; Transfer function; Building
1. Introduction Vibrations inside buildings can cause irritation among residents and malfunctioning to sensitive equipment. In extreme cases, excessive train-induced vibrations can cause damage to buildings. It is therefore often of great interest to predict the magnitude of the expected vibrations inside houses when a new railway is proposed in a populated area or before a new house is built in the vicinity of a line. Accurately predicting train-induced vibrations inside houses can be quite challenging. The magnitude of the vibrations is known to be dependant on several factors such as the train load [1], the speed of the train, [2,3], rail joints, and roughness of wheels and rails [4]. The conditions at the site have a great influence on ground response. The material properties and geometry of the embankment as well as the soil under and beside the railway embankment are important factors that influence the character and size of train-induced ground vibrations [5]. Finally, the characteristics of the building itself have to be taken into account. Corresponding author. Tel.: +46 8 790 86 83; fax: +46 8 790 79 28
E-mail address: [email protected] (C. With). 0267-7261/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2006.06.003
Different models, empirical and numerical, have been developed to support designers and engineers, both in preliminary planning and in detailed proposals for embankments and countermeasures, [6–8]. No model is ideal and can meet the demand in each phase of the design process. Consequently, the emphasis of the models shifts as the project advances. In the beginning of a process, it is more important to get an overall view and recognize potential problems before focusing on detailed predictions. Furthermore, great detail and accuracy in predictions is likely to demand extended knowledge of site conditions. This kind of information may not be available in the beginning of a process and is time-consuming and costly to obtain. It is therefore often preferable to use simpler models in the beginning of a project, models that do not need a complete site investigation, in order to limit the problem and consequently reduce the number of sites. Thus, considerable returns can be expected by using the right model in each phase. This article has evaluated a method suggesting that a transfer function could be used to model the vibrations inside a building given the known vibrations on the ground outside. The transfer function must earlier have been calculated from a source for a similar building. The idea of
ARTICLE IN PRESS 94
C. With, A. Bodare / Soil Dynamics and Earthquake Engineering 27 (2007) 93–98
using transfer functions in this context was originally proposed in [1] but was not further investigated. Some field data was presented but usefulness or accuracy of the method is difficult to conclude. The transfer function method in [1] was used in the last sub-model of a larger model. Developed by Bahrekazemi, the model, EnVib-02, was intended to be used for prediction of ground borne vibrations on the railway embankment, in the vicinity of the track and inside buildings. EnVib-02 is a one-dimensional model comprising different sub-models that all represent different parts of that chain. The output from each sub-model is a result that can be viewed. The result from the previous sub-model is used as input for the next sub-model and so on. Load distribution from one type-train with a determined speed of the train and the shear wave velocity of the ground is used as the original input. The type-train and velocities can be altered. The most interesting part of EnVib-02 was the last sub-model, which could possibly be used by itself but could also be incorporated into other models. It has been suggested that the method of using transfer functions could be used when designing a new railway line to be located in a populated area or when new areas are under consideration for development near a railway line [1]. General discussions on the usefulness of transfer functions can be found in [9–11]. In specific, transfer functions were employed by Amini [12] to study the relationship between the duration of confining pressure on the shear moduli and damping, and also by Park [13] in studying the frequency dependency of wave speeds, bending and shear stiffness of beams. The aim of this paper has been to investigate a fast and straightforward way of using a transfer function to model the vibrations in a building.
2. Use of transfer function Determining the vibrations inside a building at an arbitrary distance from a railway line is not a trivial task. The simplest models apply coefficients derived through empirical knowledge [7]. These factors are then multiplied by a given signal or a critical magnitude of vibration on the ground outside the building. These are simple and fairly quick methods to obtain crude predictions. More advanced models use computers to calculate the cases in FEM programs given defined loads. These can be time-consuming methods but can also render a realistic description of the potential problem. The method presented here suggests the use of a transfer function to model the relationship between a signal on the ground outside a building and the resulting vibration inside. The transfer function is based on the ratio between crosspower spectrum and power spectrum as described in Eq. (1). The software used to process the signals was the MATLAB’s signal processing toolbox.
b1
a1
b2 Bedroom
Living room
a2 Living room
Bedroom
N
Kitchen Kitchen
Bedroom
Rönnvägen 12
Bedroom
Rönnvägen 10 0
5m
Fig. 1. One-floor semidetached house with basement where the survey was performed. The transducers on the ground were located at position a1 and b1. The transducers inside the house were located on the living-room floors in the centre of the room, a2 and b2. Signals recorded with transducers a1-2 were used to derive the transfer function using the RSMV as source. Transducers b1-2 were used to record data from trains later used to test the validity of the function derived.
A linear time-invariant estimated transfer function, Txy Eq. (1), can be modelled describing the relation between two signals. The transfer function is the ratio of the crossspectrum Pxy of the input signal x and the output signal y, and the power spectrum Pxx of x, f being the frequency. T xy ðf Þ ¼
Pxy ðf Þ . Pxx ðf Þ
(1)
A minimum of two geophones (or accelerometers) are needed to collect information in order to generate the transfer function. A transducer is used at the position of interest inside the house and another one is installed outside, Fig. 1. It is important, as will be shown, that the positions of the instruments used to sample these signals are chosen with care. Each building type has its own transfer function. In fact, different transfer functions might be needed to develop the response at different locations inside a building. This is especially the case if the building is relatively large compared to the distance to the track or has multiple floors. The transducer on the ground must not be in close proximity to the building or near other large objects like cellars or boulders that might influence the recordings. The idea is that the instrument outside the building records the motion at the site not influenced by the building. By doing so, a transfer function can be derived. This function can then be used elsewhere together with a recorded ground signal to predict the vibrations inside a house if it is built there. 3. Field test A building in Halmstad, Ro¨nnva¨gen 10–12, Fig. 1, south-western Sweden, was selected for investigation. Residents in the area had complained about excessive
ARTICLE IN PRESS C. With, A. Bodare / Soil Dynamics and Earthquake Engineering 27 (2007) 93–98
vibrations, especially from freight trains. A court ruled that the environmental impact from the railway had to be mitigated. Measurements presented here are part of the site investigations undertaken before the process of designing countermeasures started. The building is a two-family semidetached house with an almost symmetric layout. The transfer function was derived from signals recorded inside Ro¨nnva¨gen 10 which is the western part of the building and on the property, Fig. 1. The signals recorded were from ground borne vibrations induced by a special car, Rolling Stiffness Measurement Vehicle (RSMV) Fig. 2. The car, which is a modified freight car can be pulled by a standard engine car, [14], was not moving along the track at the time of measurements. Waves were induced by hydraulically forcing two masses, each 4000 kg, over one of the two axles. In this test were the masses were forced into motion in the frequency range 3–20 Hz. Ten pair or signals from a1–2, see Fig. 1, each 8 s long, was used to derive 10 transfer functions. The average of those 10 functions where then taken as a representative for the ground–building interaction. The observation position inside the building, Ro¨nnva¨gen 10, was located on the floor in the centre of the living room facing the railway (a2), 21 m from the centre of the railway track. The corresponding point outside the building was in line with the short side of the building 4 m from the house toward the railway (a1), 15 m from the centre of the railway track, Fig. 1. All instruments were well clear of the walls and heavy objects. Both geophones (Mark 4A-2 Hz) measured the vertical particle velocity. During a different field survey, instrumentation was installed on the other half of the building, Ro¨nnva¨gen 12, at corresponding positions. The transfer function derived from measurements carried out at Ro¨nnva¨gen 10 was used to predict the vibrations inside Ro¨nnva¨gen 12 using only the recordings from the ground surface in the garden (b1).
95
The predicted magnitudes of the vibrations (maximum 1 sr.m.s.) were then compared to those measured inside the house at Ro¨nnva¨gen 12 with transducer b2. The measurements were taken during normal traffic. Signals greater than eight seconds were divided into 8 s segments. The transfer function, derived earlier, was applied to each segment. The resulting predictions were then resolved into a single predicted signal. Analyses of the frequency content of the recorded signals from the passing trains on the ground indicate that, the significant proportion of the energy was contained at frequencies below 20 Hz. Therefore, the limitation, that the RSMV generate waves predominantly in the range 3–20 Hz, is believed not to be significant in this case. However, there was some noise predominantly at 50–80 Hz in the recordings made with transducers at position b1 which motivate the choice of a filter only allowing frequency below 20 Hz to pass undamped. The noise was seen in all recordings, both when vibrations from trains were measured and when background noise was measured. The source of the noise has not been successful identified; both electrical cables and water conduits have been ruled out as possible sources. The instrumentation used at this site has been used later producing no noise and is therefore believed to be accurate. Results from measurements are therefore presented for both the case in which signals were first recorded at Ro¨nnva¨gen 12 and used as they are, and when they had been filtered. 4. Site conditions, Ro¨nnva¨gen 10–12 The ground water table in the area is found at 1.5–2 m. Sand is dominating in the upper layer, the silt content increases with depth to 8–10 m where a silt layer with clay content is found [15], Fig. 3. The water content is about Helical auger, Cone penetration test, +11.2 Ballast Ballast / sand Ballast / slit
+5
+0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 Point resistance [MPa]
Fig. 2. Inside the RSMV, the weights in the rack above the rear axle is put in motion with hydraulic piston.
Fig. 3. Site investigations, result from a helical auger and cone penetration test, after [16,17].
ARTICLE IN PRESS C. With, A. Bodare / Soil Dynamics and Earthquake Engineering 27 (2007) 93–98
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20% at the top 5 m [15,16]. The mean shear velocity in the ground was assessed to 140 m/s by the Spectral Analysis of Surface Waves method (SASW) [18], for dominating frequencies. The assessment was difficult to perform as the profile is reversed in stiffness, see Fig. 3.
Recorded signals from the instruments outside and inside the house at Ro¨nnva¨gen 10 were used to derive a transfer function from the RSMV for the building (a1–2). A total of 10 signals were used, 8 s long, and with a sample rate of 200 samples per second. Fig. 4 presents the absolute values of the transfer function derived. The transfer function was then used together with the signals recorded on the ground at Ro¨nnva¨gen 12 (b1) to predict the vibration inside the building and then compared to recorded signal at b2. An example of unfiltered signals recorded at the site is presented in Fig. 5. The top signal is from the sensor outside the house (b1) and the middle is from the transducer inside the house (b2). Both signals were recorded when a freight train was passing by. The signal presented at the bottom of the figure is the predicted response inside the building using the transfer function in Fig. 4 and the top signal in Fig. 5. The peaks in Fig. 4 at 12, 14, and 17 Hz have wavelengths that can be compared to the dimensions of the living room (multiplied with two) and the short side of the building which is in the direction of the propagating waves generated from the RSMV. No significant multiplier was found for 7 Hz which could be related to the length of the building, which is parallel to the railway. Results from the site are presented in two tables. Table 1 lists four freight trains and five passenger trains together with the number of cars, speed of the train, measured particle velocity outside as well as inside the house. Transfer function
3.5
Value of transfer function [1]
3 2.5 2 1.5 1 0.5 0
0
2
4
6
8 10 12 Frequency [Hz]
14
16
18
20
Fig. 4. Calculated transfer function in the 0–20 Hz range using recorded signals from the RSMV.
Ground, measured
0
Particle velocity [mm/s]
5. Results
Train 1
2
-2
0
5
10
15
20
25
30
35
40
45
1
50
55
House, measured
0 -1
0
5
10
15
20
25
30
35
40
45
50
55
1 House, predicted
0 -1
0
5
10
15
20
25 30 time [s]
35
40
45
50
55
Fig. 5. Measured particle velocity on the ground and inside the house for train 1, and predicted response inside the house given the outside signal and transfer function.
One-second-r.m.s., as it is used in this report, is defined in [19] as the square root of the average of the square values over a certain time (1-s in this case) from a function in discrete time history. The reason for using 1 s-r.m.samplitude is the human response to vibration signals. People respond to the average vibration amplitude during 1 s rather than peak particle velocities [8]. Table 2 shows the difference between the predicted and measured magnitude of vibration (1 s-r.m.s.) in mm/s (calculated value minus measured value). The differences in percent (calculated value minus measured value over measured value), are also listed. The average vibration (1 s-r.m.s.) inside the house was calculated with the transfer functions with an error of 10%, 0.02 mm/, without filter and 5% 0.01 using the filter. The absolute error was 17%, 0.04 mm/s, 14%, 0.03 when filter was applied. The average magnitude of the vibrations outside the house on the ground surface was 0.65 mm/s and inside the house 0.22 mm/s. Seventy-five percent of the predicted values were predicted with an absolute error less than 30%, 0.08 mm/s, 19%, 0.05 mm/s when filter was applied. Standard deviation was calculated to be 23%, 0.06 mm/s, and 18% and 0.04 mm/s, respectively when the filter was used. It can be concluded from Table 2 that transfer functions on average underestimate the expected vibrations inside the building for freight trains. However, analysis of variance with a 5% level of significance cannot reject that the transfer functions from either category would not be of the same population. 6. Discussion and conclusions It has been suggested that a transfer function can be used to predict the vibrations inside a house [1].
ARTICLE IN PRESS C. With, A. Bodare / Soil Dynamics and Earthquake Engineering 27 (2007) 93–98 Table 1 Data of the seven trains in the study Train no.
Type
Numbers of cars
Speed of the train (km/h)
Ground vibration (mm/s)
House vibration (mm/s)
1 2 3 4 5 6 7 8 9
F P P F F P P F P
37 4 3 28 39 5 6 NA 2
52 120 65 66 57 114 94 57 48
0.57 1.11 0.56 0.74 0.51 0.76 0.86 0.50 0.24
0.23 0.31 0.10 0.24 0.19 0.28 0.28 0.29 0.07
97
built to predict the environmental impact for the user of the building. The average particle velocity (1 s-r.m.s.) inside the house was calculated with the transfer functions with an average error of 10%, 0.02 mm/s. Prediction was achieved with a standard deviation of 23%, 0.06 mm/s when the data was not filtered. Further work is needed to ascertain the accuracy of this prediction. Since the RSMV did not operate above 20 Hz during this experiment, a low-pass filter was used during a second set of analyses. The filter had some effect but not a major one. This is probably because the energy in the waves that reach the building is predominantly contained in the frequency range below 20 Hz.
P ¼ passenger train, F ¼ freight train.
Acknowledgements Table 2 Difference between calculated and measured 1 s-r.m.s particle velocity inside the house in mm/s and percent Train no.
mm/s
%
Filter mm/s
Filter %
1 2 3 4 5 6 7 8 9
0.00 0.01 0.01 0.08 0.01 0.02 0.15 0.09 0.01
0 3 11 32 5 6 53 30 13
0.01 0.05 0.01 0.02 0.00 0.01 0.06 0.10 0.00
7 19 13 12 2 5 31 34 2
Mean Std.
0.02 0.06
10 23
0.01 0.04
5 18
Collecting necessary data in the field can be time consuming but the post-processing is fairly fast. Whether the accuracy of this method and the time required to obtain results are fast enough must be judged in the individual case, both factors can be addressed. In this investigation a transfer function has been calculated from recordings made from a RSMV. There might be a possibility to improve the accuracy of the model by creating different transfer signals for a smaller range of frequency and then combining them into one. However, the downside of this would be the increase of post-processing that would be needed. The aim of this paper, however, has been to investigate a fast and straightforward way of using a transfer function to model the vibrations in a building. This survey indicates the strength of transfer functions in this context. Further and more extended investigations are needed in order to fully make use of this concept. It has been shown that, for a building at a site, it should be possible to derive a transfer function using a stationary source. This transfer function could then be used together with any ground signal believed to be representative of the conditions likely to occur after a planned railway has been
The author would like to thank the Swedish Rail Administration (Banverket) for financing this report. We are indebted to the Lindgren families at Ro¨nnva¨gen 10 and 12, who let us use their houses. The comments from Dr. S. Brunsberg and the reviewers have been valuable.
References [1] Bahrekazemi M. Train-induced ground vibration and its prediction. PhD. thesis, Royal Institute of Technology, 2004. [2] Krylov VV. Generation of ground vibration boom by high-speed trains. In: Krylov VV, editor. Noise and vibration from high-speed trains. Thomas Telford Publishing; 2001. p. 251–83. [3] Kaynia AM, Madshus C, Zackrisson P. Ground vibrations from high-speed trains: predictions and countermeasures. J Geotech Geoenviron Eng 2000;126:531–7. [4] Jones CJC. Use of numerical models to determine the effectiveness of anti-vibration systems for railways. Proceedings of the Institute of Civil Engineers. Transportation 1994:43–51. [5] Adolfsson K, Andre´asson B, Bengtson P-E, Bodare A, Madshus C, Massarch R, Wallmark G, Zackrisson P. High speed lines on soft ground. Evaluation and analysis of measurements from the West Coast Line. Technical report, Banverket, Sweden, 1999. [6] Paolucci R, Maffeis A, Scandella L, Stupazzini M, Vanini M. Numerical prediction of low-frequency ground vibrations induced by high-speed trains at Ledsgaard, Sweden. Soil Dyn Earthquake Eng 2003;23:425–33. [7] Madshus C, Bessason B, Ha˚rvik L. Prediction model for low frequency vibration from high speed railways on soft ground. J Sound Vibrat 1996;193(1):195–203. [8] US Department of Transportation. High-speed ground transportation noise and vibration impact assessment. Office of Railroad Development, report 293630-1, 1998. [9] Bendat JS, Piersol AG. Engineering application of correlation and spectral analysis. New York: Wiley-Interscience; 1980. [10] Stoica P, Moses R. Introduction to spectral analysis. Englewood Cliffs, NJ: Prentice-Hall; 1997. [11] Oppenheim AV, Willsky AS, Nawab SH. Signals and systems. Englewood Cliffs, NJ: Prentice-Hall; 1997. [12] Amini F. Effect of confining pressure on dynamic soil properties using improved transfer function estimators. Soil Dyn Earthquake Eng 1993;12:145–7. [13] Park J. Transfer function methods to measure dynamic mechanical properties of complex structures. J Sound Vibrat 2005;288:57–79.
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[14] Berggren E. Dynamic track stiffness measurement—a new tool for condition monitoring of track substructure. Lic. thesis, Royal Institute of Technology, 2005. [15] Wallmark G. Possible countermeasures against train-induced vibrations in Furet (In Swedish: Mo¨jliga a˚tga¨rder mot spridning av vibrationer fra˚n ta˚gtrafiken i omra˚det Furet), Halmstad, Km 149-150 (G Bld 36), Banverket, Tekniska avdelningen, Geoteknik, PM 198905-12, 1989. [16] Andre´asson B. Report Geotechnical investigations—Halmstad (Rapport Geotekniska underso¨kningar—Halmstad), Furet. WSP report, 2005 [in Swedish].
[17] Andre´asson B. Furet, Halmstad—The vibration problem—the vibration phenomena (In Swedish: Furet, Halmstad—Vibrationsproblemet—Vibrationsfenomenet). WSP, PM, 2005. [18] Stokes KH, Wright SG, Bay JA, Roesset JM. Characterisation of geotechnical site by SASW method. In: Woods RD, editor. Technical review: geotechnical characterisation of sites (ISSMFE Technical Committee 10). Oxford Publishers; 1994. [19] International Organization for Standardization. ISO 2631-1. Vibration and shock—evaluation of human exposure to whole-body vibration—part 1: general requirements, 1997.
Soil Dynamics and Earthquake Engineering 27 (2007) 93–98 www.elsevier.com/locate/soildyn
Prediction of train-induced vibrations inside buildings using transfer functions Christoffer With, Anders Bodare Department of Civil and Architectural Engineering, Royal Institute of Technology (KTH), Brinellva¨gen 34, 100 44 Stockholm, Sweden Received 29 May 2006; accepted 4 June 2006
Abstract This article presents a study of the use of transfer functions to predict vibrations inside a building due to train-induced ground vibrations. It is proposed that knowing the transfer function of a building, the vibrations inside a similar building can be predicted given a known ground motion outside. A comparison has been conducted between predicted and measured vibrations inside a building in south-western Sweden due to freight and passenger trains. The transfer function was derived by using a stationary vibrating source. The average particle velocity (1 s-r.m.s.) inside the house was calculated with the transfer functions with an average error of 10%, 0.02 mm/s. Prediction was achieved with a standard deviation of 23%, 0.06 mm/s when no filtering of the data was used. Further work is needed to ascertain the accuracy of this method. r 2006 Elsevier Ltd. All rights reserved. Keywords: Train; Ground vibration; Prediction; Transfer function; Building
1. Introduction Vibrations inside buildings can cause irritation among residents and malfunctioning to sensitive equipment. In extreme cases, excessive train-induced vibrations can cause damage to buildings. It is therefore often of great interest to predict the magnitude of the expected vibrations inside houses when a new railway is proposed in a populated area or before a new house is built in the vicinity of a line. Accurately predicting train-induced vibrations inside houses can be quite challenging. The magnitude of the vibrations is known to be dependant on several factors such as the train load [1], the speed of the train, [2,3], rail joints, and roughness of wheels and rails [4]. The conditions at the site have a great influence on ground response. The material properties and geometry of the embankment as well as the soil under and beside the railway embankment are important factors that influence the character and size of train-induced ground vibrations [5]. Finally, the characteristics of the building itself have to be taken into account. Corresponding author. Tel.: +46 8 790 86 83; fax: +46 8 790 79 28
E-mail address: [email protected] (C. With). 0267-7261/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2006.06.003
Different models, empirical and numerical, have been developed to support designers and engineers, both in preliminary planning and in detailed proposals for embankments and countermeasures, [6–8]. No model is ideal and can meet the demand in each phase of the design process. Consequently, the emphasis of the models shifts as the project advances. In the beginning of a process, it is more important to get an overall view and recognize potential problems before focusing on detailed predictions. Furthermore, great detail and accuracy in predictions is likely to demand extended knowledge of site conditions. This kind of information may not be available in the beginning of a process and is time-consuming and costly to obtain. It is therefore often preferable to use simpler models in the beginning of a project, models that do not need a complete site investigation, in order to limit the problem and consequently reduce the number of sites. Thus, considerable returns can be expected by using the right model in each phase. This article has evaluated a method suggesting that a transfer function could be used to model the vibrations inside a building given the known vibrations on the ground outside. The transfer function must earlier have been calculated from a source for a similar building. The idea of
ARTICLE IN PRESS 94
C. With, A. Bodare / Soil Dynamics and Earthquake Engineering 27 (2007) 93–98
using transfer functions in this context was originally proposed in [1] but was not further investigated. Some field data was presented but usefulness or accuracy of the method is difficult to conclude. The transfer function method in [1] was used in the last sub-model of a larger model. Developed by Bahrekazemi, the model, EnVib-02, was intended to be used for prediction of ground borne vibrations on the railway embankment, in the vicinity of the track and inside buildings. EnVib-02 is a one-dimensional model comprising different sub-models that all represent different parts of that chain. The output from each sub-model is a result that can be viewed. The result from the previous sub-model is used as input for the next sub-model and so on. Load distribution from one type-train with a determined speed of the train and the shear wave velocity of the ground is used as the original input. The type-train and velocities can be altered. The most interesting part of EnVib-02 was the last sub-model, which could possibly be used by itself but could also be incorporated into other models. It has been suggested that the method of using transfer functions could be used when designing a new railway line to be located in a populated area or when new areas are under consideration for development near a railway line [1]. General discussions on the usefulness of transfer functions can be found in [9–11]. In specific, transfer functions were employed by Amini [12] to study the relationship between the duration of confining pressure on the shear moduli and damping, and also by Park [13] in studying the frequency dependency of wave speeds, bending and shear stiffness of beams. The aim of this paper has been to investigate a fast and straightforward way of using a transfer function to model the vibrations in a building.
2. Use of transfer function Determining the vibrations inside a building at an arbitrary distance from a railway line is not a trivial task. The simplest models apply coefficients derived through empirical knowledge [7]. These factors are then multiplied by a given signal or a critical magnitude of vibration on the ground outside the building. These are simple and fairly quick methods to obtain crude predictions. More advanced models use computers to calculate the cases in FEM programs given defined loads. These can be time-consuming methods but can also render a realistic description of the potential problem. The method presented here suggests the use of a transfer function to model the relationship between a signal on the ground outside a building and the resulting vibration inside. The transfer function is based on the ratio between crosspower spectrum and power spectrum as described in Eq. (1). The software used to process the signals was the MATLAB’s signal processing toolbox.
b1
a1
b2 Bedroom
Living room
a2 Living room
Bedroom
N
Kitchen Kitchen
Bedroom
Rönnvägen 12
Bedroom
Rönnvägen 10 0
5m
Fig. 1. One-floor semidetached house with basement where the survey was performed. The transducers on the ground were located at position a1 and b1. The transducers inside the house were located on the living-room floors in the centre of the room, a2 and b2. Signals recorded with transducers a1-2 were used to derive the transfer function using the RSMV as source. Transducers b1-2 were used to record data from trains later used to test the validity of the function derived.
A linear time-invariant estimated transfer function, Txy Eq. (1), can be modelled describing the relation between two signals. The transfer function is the ratio of the crossspectrum Pxy of the input signal x and the output signal y, and the power spectrum Pxx of x, f being the frequency. T xy ðf Þ ¼
Pxy ðf Þ . Pxx ðf Þ
(1)
A minimum of two geophones (or accelerometers) are needed to collect information in order to generate the transfer function. A transducer is used at the position of interest inside the house and another one is installed outside, Fig. 1. It is important, as will be shown, that the positions of the instruments used to sample these signals are chosen with care. Each building type has its own transfer function. In fact, different transfer functions might be needed to develop the response at different locations inside a building. This is especially the case if the building is relatively large compared to the distance to the track or has multiple floors. The transducer on the ground must not be in close proximity to the building or near other large objects like cellars or boulders that might influence the recordings. The idea is that the instrument outside the building records the motion at the site not influenced by the building. By doing so, a transfer function can be derived. This function can then be used elsewhere together with a recorded ground signal to predict the vibrations inside a house if it is built there. 3. Field test A building in Halmstad, Ro¨nnva¨gen 10–12, Fig. 1, south-western Sweden, was selected for investigation. Residents in the area had complained about excessive
ARTICLE IN PRESS C. With, A. Bodare / Soil Dynamics and Earthquake Engineering 27 (2007) 93–98
vibrations, especially from freight trains. A court ruled that the environmental impact from the railway had to be mitigated. Measurements presented here are part of the site investigations undertaken before the process of designing countermeasures started. The building is a two-family semidetached house with an almost symmetric layout. The transfer function was derived from signals recorded inside Ro¨nnva¨gen 10 which is the western part of the building and on the property, Fig. 1. The signals recorded were from ground borne vibrations induced by a special car, Rolling Stiffness Measurement Vehicle (RSMV) Fig. 2. The car, which is a modified freight car can be pulled by a standard engine car, [14], was not moving along the track at the time of measurements. Waves were induced by hydraulically forcing two masses, each 4000 kg, over one of the two axles. In this test were the masses were forced into motion in the frequency range 3–20 Hz. Ten pair or signals from a1–2, see Fig. 1, each 8 s long, was used to derive 10 transfer functions. The average of those 10 functions where then taken as a representative for the ground–building interaction. The observation position inside the building, Ro¨nnva¨gen 10, was located on the floor in the centre of the living room facing the railway (a2), 21 m from the centre of the railway track. The corresponding point outside the building was in line with the short side of the building 4 m from the house toward the railway (a1), 15 m from the centre of the railway track, Fig. 1. All instruments were well clear of the walls and heavy objects. Both geophones (Mark 4A-2 Hz) measured the vertical particle velocity. During a different field survey, instrumentation was installed on the other half of the building, Ro¨nnva¨gen 12, at corresponding positions. The transfer function derived from measurements carried out at Ro¨nnva¨gen 10 was used to predict the vibrations inside Ro¨nnva¨gen 12 using only the recordings from the ground surface in the garden (b1).
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The predicted magnitudes of the vibrations (maximum 1 sr.m.s.) were then compared to those measured inside the house at Ro¨nnva¨gen 12 with transducer b2. The measurements were taken during normal traffic. Signals greater than eight seconds were divided into 8 s segments. The transfer function, derived earlier, was applied to each segment. The resulting predictions were then resolved into a single predicted signal. Analyses of the frequency content of the recorded signals from the passing trains on the ground indicate that, the significant proportion of the energy was contained at frequencies below 20 Hz. Therefore, the limitation, that the RSMV generate waves predominantly in the range 3–20 Hz, is believed not to be significant in this case. However, there was some noise predominantly at 50–80 Hz in the recordings made with transducers at position b1 which motivate the choice of a filter only allowing frequency below 20 Hz to pass undamped. The noise was seen in all recordings, both when vibrations from trains were measured and when background noise was measured. The source of the noise has not been successful identified; both electrical cables and water conduits have been ruled out as possible sources. The instrumentation used at this site has been used later producing no noise and is therefore believed to be accurate. Results from measurements are therefore presented for both the case in which signals were first recorded at Ro¨nnva¨gen 12 and used as they are, and when they had been filtered. 4. Site conditions, Ro¨nnva¨gen 10–12 The ground water table in the area is found at 1.5–2 m. Sand is dominating in the upper layer, the silt content increases with depth to 8–10 m where a silt layer with clay content is found [15], Fig. 3. The water content is about Helical auger, Cone penetration test, +11.2 Ballast Ballast / sand Ballast / slit
+5
+0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 Point resistance [MPa]
Fig. 2. Inside the RSMV, the weights in the rack above the rear axle is put in motion with hydraulic piston.
Fig. 3. Site investigations, result from a helical auger and cone penetration test, after [16,17].
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20% at the top 5 m [15,16]. The mean shear velocity in the ground was assessed to 140 m/s by the Spectral Analysis of Surface Waves method (SASW) [18], for dominating frequencies. The assessment was difficult to perform as the profile is reversed in stiffness, see Fig. 3.
Recorded signals from the instruments outside and inside the house at Ro¨nnva¨gen 10 were used to derive a transfer function from the RSMV for the building (a1–2). A total of 10 signals were used, 8 s long, and with a sample rate of 200 samples per second. Fig. 4 presents the absolute values of the transfer function derived. The transfer function was then used together with the signals recorded on the ground at Ro¨nnva¨gen 12 (b1) to predict the vibration inside the building and then compared to recorded signal at b2. An example of unfiltered signals recorded at the site is presented in Fig. 5. The top signal is from the sensor outside the house (b1) and the middle is from the transducer inside the house (b2). Both signals were recorded when a freight train was passing by. The signal presented at the bottom of the figure is the predicted response inside the building using the transfer function in Fig. 4 and the top signal in Fig. 5. The peaks in Fig. 4 at 12, 14, and 17 Hz have wavelengths that can be compared to the dimensions of the living room (multiplied with two) and the short side of the building which is in the direction of the propagating waves generated from the RSMV. No significant multiplier was found for 7 Hz which could be related to the length of the building, which is parallel to the railway. Results from the site are presented in two tables. Table 1 lists four freight trains and five passenger trains together with the number of cars, speed of the train, measured particle velocity outside as well as inside the house. Transfer function
3.5
Value of transfer function [1]
3 2.5 2 1.5 1 0.5 0
0
2
4
6
8 10 12 Frequency [Hz]
14
16
18
20
Fig. 4. Calculated transfer function in the 0–20 Hz range using recorded signals from the RSMV.
Ground, measured
0
Particle velocity [mm/s]
5. Results
Train 1
2
-2
0
5
10
15
20
25
30
35
40
45
1
50
55
House, measured
0 -1
0
5
10
15
20
25
30
35
40
45
50
55
1 House, predicted
0 -1
0
5
10
15
20
25 30 time [s]
35
40
45
50
55
Fig. 5. Measured particle velocity on the ground and inside the house for train 1, and predicted response inside the house given the outside signal and transfer function.
One-second-r.m.s., as it is used in this report, is defined in [19] as the square root of the average of the square values over a certain time (1-s in this case) from a function in discrete time history. The reason for using 1 s-r.m.samplitude is the human response to vibration signals. People respond to the average vibration amplitude during 1 s rather than peak particle velocities [8]. Table 2 shows the difference between the predicted and measured magnitude of vibration (1 s-r.m.s.) in mm/s (calculated value minus measured value). The differences in percent (calculated value minus measured value over measured value), are also listed. The average vibration (1 s-r.m.s.) inside the house was calculated with the transfer functions with an error of 10%, 0.02 mm/, without filter and 5% 0.01 using the filter. The absolute error was 17%, 0.04 mm/s, 14%, 0.03 when filter was applied. The average magnitude of the vibrations outside the house on the ground surface was 0.65 mm/s and inside the house 0.22 mm/s. Seventy-five percent of the predicted values were predicted with an absolute error less than 30%, 0.08 mm/s, 19%, 0.05 mm/s when filter was applied. Standard deviation was calculated to be 23%, 0.06 mm/s, and 18% and 0.04 mm/s, respectively when the filter was used. It can be concluded from Table 2 that transfer functions on average underestimate the expected vibrations inside the building for freight trains. However, analysis of variance with a 5% level of significance cannot reject that the transfer functions from either category would not be of the same population. 6. Discussion and conclusions It has been suggested that a transfer function can be used to predict the vibrations inside a house [1].
ARTICLE IN PRESS C. With, A. Bodare / Soil Dynamics and Earthquake Engineering 27 (2007) 93–98 Table 1 Data of the seven trains in the study Train no.
Type
Numbers of cars
Speed of the train (km/h)
Ground vibration (mm/s)
House vibration (mm/s)
1 2 3 4 5 6 7 8 9
F P P F F P P F P
37 4 3 28 39 5 6 NA 2
52 120 65 66 57 114 94 57 48
0.57 1.11 0.56 0.74 0.51 0.76 0.86 0.50 0.24
0.23 0.31 0.10 0.24 0.19 0.28 0.28 0.29 0.07
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built to predict the environmental impact for the user of the building. The average particle velocity (1 s-r.m.s.) inside the house was calculated with the transfer functions with an average error of 10%, 0.02 mm/s. Prediction was achieved with a standard deviation of 23%, 0.06 mm/s when the data was not filtered. Further work is needed to ascertain the accuracy of this prediction. Since the RSMV did not operate above 20 Hz during this experiment, a low-pass filter was used during a second set of analyses. The filter had some effect but not a major one. This is probably because the energy in the waves that reach the building is predominantly contained in the frequency range below 20 Hz.
P ¼ passenger train, F ¼ freight train.
Acknowledgements Table 2 Difference between calculated and measured 1 s-r.m.s particle velocity inside the house in mm/s and percent Train no.
mm/s
%
Filter mm/s
Filter %
1 2 3 4 5 6 7 8 9
0.00 0.01 0.01 0.08 0.01 0.02 0.15 0.09 0.01
0 3 11 32 5 6 53 30 13
0.01 0.05 0.01 0.02 0.00 0.01 0.06 0.10 0.00
7 19 13 12 2 5 31 34 2
Mean Std.
0.02 0.06
10 23
0.01 0.04
5 18
Collecting necessary data in the field can be time consuming but the post-processing is fairly fast. Whether the accuracy of this method and the time required to obtain results are fast enough must be judged in the individual case, both factors can be addressed. In this investigation a transfer function has been calculated from recordings made from a RSMV. There might be a possibility to improve the accuracy of the model by creating different transfer signals for a smaller range of frequency and then combining them into one. However, the downside of this would be the increase of post-processing that would be needed. The aim of this paper, however, has been to investigate a fast and straightforward way of using a transfer function to model the vibrations in a building. This survey indicates the strength of transfer functions in this context. Further and more extended investigations are needed in order to fully make use of this concept. It has been shown that, for a building at a site, it should be possible to derive a transfer function using a stationary source. This transfer function could then be used together with any ground signal believed to be representative of the conditions likely to occur after a planned railway has been
The author would like to thank the Swedish Rail Administration (Banverket) for financing this report. We are indebted to the Lindgren families at Ro¨nnva¨gen 10 and 12, who let us use their houses. The comments from Dr. S. Brunsberg and the reviewers have been valuable.
References [1] Bahrekazemi M. Train-induced ground vibration and its prediction. PhD. thesis, Royal Institute of Technology, 2004. [2] Krylov VV. Generation of ground vibration boom by high-speed trains. In: Krylov VV, editor. Noise and vibration from high-speed trains. Thomas Telford Publishing; 2001. p. 251–83. [3] Kaynia AM, Madshus C, Zackrisson P. Ground vibrations from high-speed trains: predictions and countermeasures. J Geotech Geoenviron Eng 2000;126:531–7. [4] Jones CJC. Use of numerical models to determine the effectiveness of anti-vibration systems for railways. Proceedings of the Institute of Civil Engineers. Transportation 1994:43–51. [5] Adolfsson K, Andre´asson B, Bengtson P-E, Bodare A, Madshus C, Massarch R, Wallmark G, Zackrisson P. High speed lines on soft ground. Evaluation and analysis of measurements from the West Coast Line. Technical report, Banverket, Sweden, 1999. [6] Paolucci R, Maffeis A, Scandella L, Stupazzini M, Vanini M. Numerical prediction of low-frequency ground vibrations induced by high-speed trains at Ledsgaard, Sweden. Soil Dyn Earthquake Eng 2003;23:425–33. [7] Madshus C, Bessason B, Ha˚rvik L. Prediction model for low frequency vibration from high speed railways on soft ground. J Sound Vibrat 1996;193(1):195–203. [8] US Department of Transportation. High-speed ground transportation noise and vibration impact assessment. Office of Railroad Development, report 293630-1, 1998. [9] Bendat JS, Piersol AG. Engineering application of correlation and spectral analysis. New York: Wiley-Interscience; 1980. [10] Stoica P, Moses R. Introduction to spectral analysis. Englewood Cliffs, NJ: Prentice-Hall; 1997. [11] Oppenheim AV, Willsky AS, Nawab SH. Signals and systems. Englewood Cliffs, NJ: Prentice-Hall; 1997. [12] Amini F. Effect of confining pressure on dynamic soil properties using improved transfer function estimators. Soil Dyn Earthquake Eng 1993;12:145–7. [13] Park J. Transfer function methods to measure dynamic mechanical properties of complex structures. J Sound Vibrat 2005;288:57–79.
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[14] Berggren E. Dynamic track stiffness measurement—a new tool for condition monitoring of track substructure. Lic. thesis, Royal Institute of Technology, 2005. [15] Wallmark G. Possible countermeasures against train-induced vibrations in Furet (In Swedish: Mo¨jliga a˚tga¨rder mot spridning av vibrationer fra˚n ta˚gtrafiken i omra˚det Furet), Halmstad, Km 149-150 (G Bld 36), Banverket, Tekniska avdelningen, Geoteknik, PM 198905-12, 1989. [16] Andre´asson B. Report Geotechnical investigations—Halmstad (Rapport Geotekniska underso¨kningar—Halmstad), Furet. WSP report, 2005 [in Swedish].
[17] Andre´asson B. Furet, Halmstad—The vibration problem—the vibration phenomena (In Swedish: Furet, Halmstad—Vibrationsproblemet—Vibrationsfenomenet). WSP, PM, 2005. [18] Stokes KH, Wright SG, Bay JA, Roesset JM. Characterisation of geotechnical site by SASW method. In: Woods RD, editor. Technical review: geotechnical characterisation of sites (ISSMFE Technical Committee 10). Oxford Publishers; 1994. [19] International Organization for Standardization. ISO 2631-1. Vibration and shock—evaluation of human exposure to whole-body vibration—part 1: general requirements, 1997.