This article courtesy of Dr. Mark R. Svinkin, to whom we are deeply grateful. It was originally presented at the 24th Annual Member's Conference of the Deep Foundations Institute in Dearborn, Michigan, 1416 October 1999.
Summary: Constructioninduced vibrations may be detrimental to adjacent structures and sensitive electronics operating nearby. Construction vibration sources have a wide range of energy and velocity, as a function of time, transmitted on the ground. Analysis of existing methods for predicting ground and structure vibrations shows that empirical equations provide calculations only of amplitudes of vertical soil vibrations with insufficient accuracy. This paper presents the application of the impulse response function concept to solve the geotechnical problem of predicting ground and structure vibrations before installation of vibration sources. Impulse response functions reflect real behavior of soil and structures without the investigation of soil and structure properties. A procedure is presented to compute predicted ground and structure vibrations. Good agreement is found between predicted and measured records.
1. Introduction
Sources of construction vibration, such as pile driving, dynamic compaction, blasting and operation of heavy construction equipment, may harmfully affect surrounding buildings and its effect ranges from serious disturbances of working conditions for sensitive devices and people to visible structural damage.
Considerable data have been collected, analyzed and published with respect to vibrations from construction and industrial sources, e.g. Barkan (1962), Richart et al. (1970), Wiss (1981), Mayne (1985), Massarsch (1992), Svinkin (1993), Dowding (1996), Woods (1997) and others.
Empirical equations employed for assessment of expected soil vibrations from construction and industrial sources usually only allow calculation of a vertical peak amplitude of vibrations and not always with sufficient accuracy. These equations cannot incorporate specific differences of soil conditions at each site because heterogeneity and spatial variation of soil properties strongly affect characteristics of propagated waves in soil from construction and industrial vibration sources.
Svinkin (1996a; 1997) has originated the Impulse Response Function Prediction method (IRFP) for determining complete time domain records on existing soils, structures and equipment prior to installation of construction and industrial vibration sources. The IRFP method has significant advantages in comparison with empirical equations and analytical procedures.
The purpose of this paper is to discuss various approaches and their accuracy to predict and calculate soil and structure vibrations before the beginning of construction activities.
2. Construction Vibrations
2.1. Sources of Vibrations
Impact hammers are common sources of construction vibrations. Maximum rated energy of the most commonly employed piling hammers varies from 5 to 200 kJ per blow. Two kinds of frequencies are observed on the pile acceleration and velocity records. Vibrations with high frequencies of about 300700 Hz are generated by the hammercushion system. Soil vibrations with such frequency content should be taken into account when pile driving occurs in close proximity to a building. Frequencies of natural longitudinal pile oscillations are in the range of 750 Hz, with predominance at the lower values. Measured maximum pile velocity and displacement values range from 0.9 to 4.6 m/s and 12 to 35 mm, respectively. Both parameters depend on pile type and hammer transferred energy. Displacement might be affected by soil conditions as well (Svinkin, 1992; 1996b).
Vibratory hammers for driving nondisplacement piles usually have low to moderate force amplitudes and operating frequencies above 20 Hz. Displacement piles are driven by vibratory hammers with frequencies of around 10 Hz and commonly along with much higher force amplitudes (Warrington, 1992). The soil resistance to pile penetration and the seismic effect of vibratory driven piles depend substantially on soil conditions, pile type and vibratory hammer model. A coincidence of the operating frequency with the soil layer frequency may generate large ground vibrations of the soil surrounding the pile. The use of vibratory hammers with variable frequency and force amplitude may minimize damage due to accidental ground vibration amplification.
For dynamic compaction of loose sands and granular fills, a large steel or concrete weight of 49.1 to 137.3 kN is usually dropped from a height of 15 to 30 m. Such dynamic impacts generate surface waves with a dominant frequency of 3 to 12 Hz. Thus, dynamic deep compaction is also a source of intensive low frequency ground vibrations (Mayne, 1985).
The dominant frequency of propagating waves from quarry and construction blasting ranges mostly between 10 and 60 Hz. Blasting energy is much larger than energy of other sources of construction vibrations. For example, the energy released by 0.5 kg of TNT is 5400 kJ (Dowding, 1996). Such energy is 50 to 1000 times the energy transferred to piles during driving and 15 to 80 times the energy transferred to the ground during dynamic compaction of soils.
2.2. Ground Vibrations
Sources of construction vibrations generate compression, shear and Rayleigh waves (Barkan, 1992, Richart et al., 1970). Rayleigh waves have the largest practical interest for design engineers because building foundations are placed near the ground surface. In addition, Rayleigh waves contain roughly 70 % of the total vibration energy and become predominant over other wave types at comparatively small distances from the vibration source. For example, pile driving from depths between 4 and 10 m will generate Rayleigh waves within 0.4 to 3 m of the pile, depending on the propagation velocities of Rayleigh and compression waves.
Soil vibrations are mostly vertical near the source of vertical impact loads, but as distance increases, vertical and horizontal soil vibrations become similar in magnitudes, and, for some locations at the ground surface, the amplitude of horizontal vibrations might be up to three times greater than that of vertical vibrations. Waves travel in all directions from the source of vibrations forming a series of fairly harmonic waves with the dominant frequency equal or close to the frequency of the source. Spectra of the radial and transverse components of horizontal soil vibrations may have a few maxima and the one corresponding the frequency of the source is not always the largest. In general, faster attenuation of high frequency components is the primary cause of changes of soil vibrations with distance from the source. However, some records can not be explained by this mechanism and the effect of soil strata heterogeneity and uncertainties of the geologic profile should be taken into account (Svinkin, 1996a).
The proximity of the frequency of horizontal soil vibrations to one of the building's natural frequencies may generate the conditions of resonance in that building. Moreover, vertical ground vibrations can cause dangerous structural settlements. Considerable data have been collected and published with respect to intolerable vibrations and settlements from construction and industrial sources, e.g. Barkan (1962), Richart et al. (1970), Wiss (1981); Lacy and Gould (1985); Massarsch (1992), Svinkin (1993); Dowding (1996) and others.
To prevent the unacceptable effect of construction vibrations, it is important to accurately predict expected ground and structure vibrations.
3. Empirical Equations
3.1. Golitsin Equation
As early as 1912, Golitsin (1912) suggested the following equation to calculate the amplitude reduction of Rayleigh waves, generated by an earthquake, between two points at distances r_{1} and r_{2} from the source as
(1)
Where
A_{1} = amplitude of vibrations at a distance r_{1} from the source
A_{2} = amplitude of vibrations at a distance r_{2} from the source
γ = attenuation coefficient.
The term (r_{1}/r_{2})^{0.5} indicates the radiation or geometric damping and the term exp[γ(r_{2}r_{1})] indicates the material damping of wave attenuation between two points.
Equation (1) has been originally derived to estimate the attenuation of low frequency Rayleigh waves with large wavelength for which the coefficient, γ, depends slightly on the soil upper layers properties. For such conditions, the coefficient, γ, changes reasonably in a narrow range in assessment of attenuation properties of soils.
Subsequently, in some studies, by Barkan (1962), Richart et al. (1970), Massarsch (1992), Woods (1997) and others, this equation was used for preliminary computation of ground vibrations from industrial and construction sources. Waves generated in the ground by construction sources have higher frequencies and smaller wavelength in comparison with waves from earthquakes and propagate mostly in the upper soil strata close to the ground surface. It is obvious that the coefficient, γ, is important for the accurate prediction of wave attenuation. Values of γ for various soil types can be found in the referenced publications. An experimental study to quantify of the coefficient, γ, was performed by Woods and Jedele (1985) who investigated soil damping for different construction operations at sites with soils of various stratifications. The observed data were approximated by average curves for frequencies of 5 and 50 Hz.
Nevertheless, there are some other factors that affect the coefficient, γ. Collected experimental data indicates (Svinkin, 1973; 1992) that the coefficient, γ, depends on physical parameters related to the vibration source (pile impedance, length and transferred energy to the pile, for example), frequency, distance from the source and variation of soil stratification at a site. Test data along the ground surface shows that for various pairs of widely separated points on the ground surface, values of γ can differ more than an order of magnitude and even change sign. Thus, the coefficient, γ, acceptable for small distances may be inadequate for long distances. On account of wave refraction and reflection from boundaries of diverse soil layers, an arbitrary arrangement of geophones at a site can yield incoherent results of ground vibration measurements because waveforms measured at arbitrary locations at a site might represent different soil layers. Coherent and consistent results for assessment of surface wave attenuation can be obtained on the basis of measurement of ground vibrations reflected from the same soil layer boundaries. Heisey et al. (1982) indicated certain requirements for a choice of appropriate spacing of the receivers in the application of the spectral analysis surface waves (SASW) method in the evaluation of soil properties.
One more important point. If it is possible to measure an amplitude of ground vibrations at the referenced distance r_{1} during construction activities, a ground vibration amplitude at a distance r_{2} can be measured as well. Besides, actual structure responses to ground vibrations can be measured without any prediction of expected vibrations. In the correct prediction of ground vibration amplitudes before the beginning of construction operations, the referenced amplitude is usually unknown. So, problems with uncertainty in assignment or determination of the coefficient, γ, and unknown referenced amplitude show that equation (1) cannot be used for predicting ground vibrations and the application of equation (1) to calculate of ground vibrations from construction and industrial sources may yield inaccurate results.
3.2. ScaledDistance Approach
The scaleddistance approach, ground velocitydistanceenergy relationship, was proposed by Attwell and Farmer (1973) to calculate the peak ground velocity at surface distance, D, from a source normalized with energy as
(2)
Where
W_{r} = energy of source or rated energy of impact hammer
k = value of velocity at one unit of distance.
Wiss (1981) reported an identical equation
(3)
Where the value of 'n' yields a slope in a loglog plot between 1.0 and 2.0 with an average value of 1.5. It was an important finding because a slope of amplitude attenuation for all tested soils was in the narrow range of 1 to 2. It turned out that the scaleddistance approach was very useful in the assessment of construction vibrations. Woods and Jedele (1985) and Woods (1997) gathered data from field construction projects at which the source energy was known or could be estimated and developed a scaled distance chart that correlated with ground types. Most of these data correlated with a slope of n=1.525 for soil class II and some of the data presented in that study showed n=1.108 for soil class III.
On the basis of the actual range of energy transferred to piles and the range of the measured peak particle pile velocity at the top of steel, concrete and timber piles, the results of Woods and Jedele (1985) were adapted by Svinkin (1992; 1996b) to calculate the peak ground velocity prior to the beginning of pile driving. The peak vertical ground velocity versus scaled distance from driven piles is depicted in Fig. 1. The reasonable pile velocity range for steel, concrete, and timber piles is 4.6 to 2.4, 2.4 to 0.9 and 4.6 to 1.5 m/s, respectively. The latter is actually the same as for steel piles. Values of 4600, 2400 and 900 mm/s have been marked as extreme left values on the slope lines. There are two areas constructed on the diagram: the upper area for steel and timber piles and the lower one for concrete piles with a slope, n=1.000, which is the upper limit for the peak particle velocity with the lower value for the rate attenuation. Data presented in Fig. 1 provide an opportunity to construct curves of the expected maximum peak ground velocity for various distances from pile driving sources and different magnitudes of transferred energy. The peak particle velocity at the pile head can be calculated in advance as (Svinkin, 1996b)
(4)
where
Z = ES/c is pile impedance
E = modulus of elasticity of pile material
S = pile crosssectional area
c = velocity of wave propagation in pile
W_{r} = energy transferred to the pile.
This new development of the scaleddistance approach eliminates the need to know in advance the factor, k, and enhances accuracy of predicted peak ground velocity before pile installation.
3.3. Pile Impedance
Heckman and Hagerty (1978) and Massarsch (1992) pointed out the important effect of the pile impedance on the peak ground velocity and showed that a reduction of the pile impedance from 2000 to 500 kNs/m could increase the peak ground velocity by a factor of 8 (Fig. 2). According to equation (4), the peak particle velocity of the source is inversely proportional to the square root of the pile impedance and, for the referenced impedance range, the expected amplification of the peak pile velocity and the peak ground velocity can only be 2. Equation (4) shows that pile length, velocity of wave propagation in the pile, and transferred energy also can affect the peak ground velocity by means of the wave source velocity.
The analysis of soil vibration records, measured at the same distances from a few piles with different impedances and driven by the same hammer to the same pile penetration into the ground, was conducted. It turned out that a certain range represents the effect of the pile impedance on ground vibration velocity better than a single line (Fig. 2). A lower boundary of this range can be calculated using Equation (4).
3.4. Frequency of Vibration
Frequency and peak particle velocity are basic parameters for assessment of ground vibrations. Dowding (1996) underlying the importance of frequency because structural responses depend on the frequency of ground vibrations. The dominant frequency of expected ground vibrations can be determined prior to the beginning of pile driving (Svinkin, 1992).
4. Predicting Structure Vibrations  IRFP Method
4.1. Proposed Approach
The impulse response function prediction method (IRFP) is based on the utilization of the impulse response function technique for predicting complete vibration records on existing soils, buildings and equipment prior to installation of construction and industrial vibration sources (Svinkin, 1996a; 1997). The impulse response function (IRF) is an output signal of the system based on a single instantaneous impulse input. Impulse response functions are applied in the analysis of any complicated linear dynamic system with unknown internal structure for which its mathematical description is very difficult. In the case under consideration, the dynamic system is the soil medium through which waves propagate outward from sources of construction and industrial vibrations. The input of the system is the ground at the place of pile driving, dynamic compaction of soil, or installation of a machine foundation; the output is a location of interest situated on the surface or inside the soil, or any point at a building subjected to vibrations. Outcomes can be obtained, for example, as the vibration records of displacements or velocities at locations of interest.
Impulse response functions of the considered dynamic system are determined by setting up an experiment (Fig. 3). Such an approach (a) does not require routine soil boring, sampling, or testing at the site where waves propagate from the vibration source, (b) eliminates the need to use mathematical models of soil profiles, foundations and structures in practical applications, and (c) provides the flexibility of considering heterogeneity and variety of soil and structural properties. Unlike analytical methods, experimental IRFs reflect real behavior of soil and structures without investigation of the soil and structure properties. Because of that, the proposed method has substantially greater capabilities in comparison with other existing methods.
The following is a general outline of the method for predicting vibrations at a distance from an impact source.
 At the place in the field for installation of the impact source, impacts of known magnitude are applied on the ground (Fig. 3). The impact can be created using a rigid steel sphere or pearshaped mass falling from a bridge or mobile crane or a hammer blow on the tested pile. At the moment of impact onto the ground, oscillations are measured and recorded at the points of interest, for example, at the locations of devices sensitive to vibrations. These oscillations are the IRFs of the considered system which automatically take into account complicated soil conditions.
 Various ways are used to determine the dynamic loads on the ground from different vibration sources. For pile driving, dynamic loads are computed by the wave equation analysis. In the case of the operation of machines on foundations, these loads can be found using existing foundation dynamics theories. For dynamic compaction sites, loads from the source are easily calculated with known falling weights and heights.
 Duhamel's integral (Smith and Domney, 1968) is used to compute predicted vibrations which will arise from operating construction impact source.
4.2. Dynamic Loads Onto Ground
Machine Foundations
Dynamic loads at a machine foundation can be found using existing foundation dynamics theories, for example Barkan (1962) and Richart et al. (1970). It is known that the equation of vertical damped vibrations of foundations for machines with dynamic loads can be written as
(5)
with
(6)
where
b = viscous damping coefficient, kN/m/s
k_{z} = spring constant for the vertical mode of foundation vibrations, kN/cm
P(t) = exciting force, kN
M = mass of foundation and machine, t
f _{nz} = circular natural frequency of vertical vibrations of foundation, rad/s
= effective damping constant, rad/s.
An expression derived from equation (5) for a dynamic load applied to the soil is
(7)
The dynamic force transmitted from the machine foundation to the soil depends on foundation and machine masses, the damping constant, the natural frequency of vertical foundation vibrations and vertical foundation displacements as a function of time.
Vibration displacements of the machine foundation can be assigned by sampling an arbitrary function or analytically as a damped sinusoid
(8)
with
(9)
where
I_{F} = impulse force transmitted from machine to foundation, kNs
Φ = modulus of damping, s/rad
k_{z¢ }= coefficient of vertical subgrade reaction, kN/m^{3
f}_{nd} = circular natural frequency of vertical damped vibrations of foundation, rad/s
S_{1} = contact area between foundation and soil, m^{2}.
The modulus of damping, Φ, ranges in a relatively narrow range and is slightly dependent on soil conditions (Savinov, 1979). For instance, values of Φ range from 0.004 to 0.008 s/rad for the foundation contact area less than 10.0 m2. Coefficient, k_{z¢ }is determined according to Barkan (1962). Also, it is possible to use other approaches for determining values of α and k_{z¢ }.
Pile driving
Equation (7) can be used to determine the dynamic loads transferred from the pile to the surrounding soil. In this case, M is the pile mass. The effective damping constant, α, is chosen from the range of damping constants for foundations with the smallest contact areas. A frequency of the hammerpilesoil system is calculated by an equation (Svinkin, 1992) which takes into account pile material, the ratio of wave velocity in the pile to pile length and the pile weight to ram weight ratio.
Pile displacements as a function of time are computed by the wave equation analysis, using for example, the GRLWEAP Program (GRL and Associates, 1995). Computed displacement records at the pile top, middle, and bottom are presented in Fig. 4 for a 457 mm x 457 mm prestressed concrete pile with a length of 20.4 m. These three records are very similar and displacements at the pile top can be taken as a function z(t). For comparison, pile displacements at the same points, obtained by dynamic measurement at the pile top and computed with the CAPWAP program (GRL and Associates, 1993) for the pile middle and bottom are shown in Fig. 4. Both sets of curves were derived for the same pile capacity. It can be seen that measured results confirm the reasonableness of the use of the wave equation analysis to compute pile displacements for vibration predicting.
Dynamic Compaction
For dynamic compaction of granular soil, loads from the source are calculated with known falling weights and heights.
4.3. Computation of Predicted Vibrations
For each single output point, the considered input  soil medium  output system is a one degree of freedom system and predicted displacements can be written as follows
(10)
where F(τ) = the resultant dynamic force transmitted to the ground; x,y = coordinates of the output point under consideration at the ground or the structure; h_{z}(x,y,tτ) = impulse response function at the output point under consideration; = variable of integration.
With substitution of expressions (8) and (9) equation (10) becomes
(11)
Examples of predicted results are shown in Fig. 5, 6 and 7. Measurements and prediction of vertical and horizontal ground surface displacements were made at diverse distances from the foundation under a sizeable drop hammer with a falling weight of 147.2 kN and a maximum drop height of 30.0 m. The soil at the site consisted of about 1.6 m of loose sand followed by about 6.8 m of medium density sand and 1 m of sandy clay underlain by about 10 m of slightly moist sand. The water table was about 6 m below the ground surface. The Rayleigh wave velocity was 270 m/sec. A layout of the machine foundation, the place of impact on the ground and geophones is displayed in Fig. 5.
Predicted and measured vertical and horizontal components of ground surface vibrations at eight locations are shown in Fig. 6. It can be seen that good agreement is matched in time domain vibration records, except for horizontal vibrations at two locations close to the foundation. This can be explained by the different wave paths from the foundation under the operating machine and the place for impact on the ground. The distance between these two sources was 18.7 m. Lack of coincidence of the two dynamic sources slightly affected the predicted ground vibrations at a distance from the machine foundation. Agreement of predicted and measured vibration displacements is quite satisfactory. The differences between the peak predicted and measured vibration amplitudes are less than 30 % at distances larger than 43.0 m from the foundation (Table 1). For some individual points amplitudes actually coincide.
Table 1. Peak Measured and Predicted Vibration Amplitudes
Distance from source
(m)

Vertical

Horizontal

Measured
(μm)

Predicted
(μm)

Error
(%)

Measured
(μm)

Predicted
(μm)

Error
(%)

25

450

330

27

180

510

+183

33

351

216

38

227

396

+74

43

270

232

14

238

252

+6

57







162

144

11

132

55

60

+9







200

30

30

0

65

59

9

266

28

36

+30

25

29

+16

Spectrum analysis of predicted and measured time histories revealed that both records have similar frequency domain curves with the same dominant frequency. Moreover, predicted records are slightly dependent on the parameters in equation (7) for determination of the dynamic force transmitted from the source to the soil (Svinkin, 1999).
Predicted vibration curves in Fig. 7 at a distance of 266.0 m from the machine foundation were computed with various values of initial parameters in Eq. (7), Table 2. In spite of the change of the computed natural foundation frequency in the range of 23.863.5 rad/s and the damping constant from 8.5 to 60.5 rad/s, the shapes of measured and predicted records are almost the same and their spectra show the same dominant vibration frequency. An increase of the computed natural frequency of foundation vibrations with respect to the measured frequency leads to an increase of the largest amplitude by 1030 % for both vertical and horizontal predicted soil oscillations. Spectra of these oscillations show a stability of frequency composition for even very long duration soil oscillations. Thus, variations of predicted soil oscillations do not exceed measurement errors even with a 2.7 times increase in the computed natural frequency of the foundation.
TABLE 2. Parameters of FoundationSoil System
Record No.

k_{z}
(kN/m^{3})

α
(rad/s)

Φ
(s/rad)

f_{nz}
(rad/s)

M
t

2

Experimental time domain foundation displacement

3

34433

8.5

0.03

23.8

9614

4

67885

60.5

0.03

63.5

2650

5

39240

35.0

0.03

48.3

2650

5. Conclusions
 Construction operations such as pile driving, dynamic compaction and blasting are widespread sources of ground and structure vibrations. These vibration sources have a wide range of energy and velocity, as a function of time, transmitted on the ground. Constructioninduced vibrations may harmfully affect surrounding buildings. It is important to accurately predict vibrations of ground, structures, and sensitive devices prior to the beginning of construction activities to avoid the undesirable effect of generated vibrations.
 Empirical equations provide only calculation of a vertical amplitude of ground vibrations and not always with sufficient accuracy. For pile driving, the scaled distance approach with calculated peak particle velocity of the source is probably the most appropriate method for predicting upper limits of the peak particle velocity of ground vibrations. The effect of pile impedance on ground vibrations is exaggerated in some publications. Other parameters of the hammerpilesoil system like pile length, velocity of wave propagation in the pile, and transferred energy to the pile can affect the peak ground velocity as well.
 The impulse response function prediction method (IRFP) is used to solve a geotechnical problem in predicting time domain ground and structure vibrations prior to the beginning of construction activities or installation of machine foundations.
 The proposed approach uses the impulse response function technique for a considered dynamic system: ground at the place for the source of vibrations  soil medium  output locations of interest on the ground or in any structure receiving vibrations. Experimental impulse response functions reflect real soil behavior and take into account uncertainty in the geologic environment. Such an approach does not require routine soil boring, sampling, and testing at the site where waves propagate from the vibration source. Different ways were shown to determine dynamic loads onto the ground from machine foundations, pile driving and dynamic compaction of granular soil. An algorithm is presented to compute predicted vibrations, and examples of predicted results are demonstrated for vertical and horizontal ground displacements. There is quite satisfactory agreement between predicted and measured records.
 The proposed approach provides the method for determining and monitoring of ground, structures and sensitive devices vibration levels before the start of construction or industrial vibration activities.
6. Acknowledgement
The writer wishes to thank the reviewers for their constructive reviews of the paper.
7. References
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