Numerical modeling of free field vibrations due to pile driving using a dynamic soil-structure interaction formulation H.R. Masoumi, G. Degrande Department of Civil Engineering, K.U.Leuven, Kasteelpark Arenberg 4, B-31 Leuven, Belgium Abstract This paper presents a numerical model for the prediction of free field vibrations due to vibratory and impact pile driving. As the focus is on the response in the far field, where deformations are relatively small, a linear elastic constitutive behavior is assumed for the soil. The free field vibrations are calculated by means of a coupled FE-BE model using a subdomain formulation. First, the case of vibratory pile driving is considered, where the contributions of different types of waves are investigated for several penetration depths. The results show that, in the near field, the response of the soil is dominated by a vertically polarized shear wave, whereas in the far field, Rayleigh waves dominate the ground vibration and body waves are importantly attenuated. Second, the case of impact pile driving is considered. The results are discussed for different time steps and penetration depths. Finally, the computed ground vibrations are compared with the results of field measurements reported in the literature. Keywords: Dynamic soil-pile interaction, pile impedance, vibratory pile driving, impact pile driving, ground vibration, boundary element method, finite element method 1 Introduction Construction vibrations often affect adjacent and surrounding buildings in urban environments. The induced waves propagate in the soil and interact with nearby structures, which induces vibrations that may cause cracks in its walls or in the facade. Permanent settlement, densification and liquefaction may also occur in the soil due to such vibrations. The present study aims to develop a model to predict free field vibrations in the environment due to pile driving. Ground motions due to pile driving generally depend on (1) the source parameters (method of driving, energy released, and pile depth), (2) the interaction between the driving machine, the pile and the soil, and (3) the propagation of the waves through the soil. Three main driving techniques to install foundation piles and sheet piles can be distinguished: impact pile driving, vibratory pile driving, and jacking. Impact pile driving is known as the oldest technique of driving and produces a transient vibration in the ground. The required driving energy is provided using a gravity principle, in which a ram mass drops from a specific height and strikes the pile head with a downward impact velocity. As impact driving generates Corresponding author. E-mail: hamidreza.masoumi@bwk.kuleuven.be 1
high noise and vibration levels, it is not preferred in urban area and its use is more and more limited. During the last decades, the vibratory pile driving technique has been preferred to other driving techniques due to economical, practical and environmental concerns. Vibratory pile driving produces a harmonic vibration in the ground and is used to install piles in granular soils and with low operating frequency, also in cohesive soils. The vibratory motion is generated by counter-rotation of eccentric masses actuated within an exciter block. Jacking does not induce environmental disturbance but is very expensive. This technique is only used where sensitive environmental conditions are encountered. The vibration transmission through the soil is complex, resulting from the difficulty of modeling precisely the soil behavior in the near and far field. Research on pile driving problems has concentrated on two main directions. Setting up numerous field measurements, several authors have been focused on environmental effects (far field or external effects) [1, 2, 3]. Some efforts have been also made to develop models to assess the driving efficiency, investigating the driveability and the bearing capacity of driven piles (near field or internal effects) [4, ]. During pile driving, the transmitted energy through the soil is very high and cause plastic deformations in the near field. In the far field, however, reported data show that the induced vibrations cause deformations in the elastic range [6]. As the focus is on the response in the far field, where deformations are relatively small, a linear elastic constitutive behavior is assumed for the soil. In this study, a model for both vibratory and impact driving is proposed to predict free field vibrations. Using a subdomain formulation for dynamic soil-structure interaction, developed by Aubry et al. [7, 8], the dynamic soil-pile interaction problem is investigated. A coupled FE- BE model is presented, in which the pile (bounded domain) and the soil (unbounded domain) are modeled using the finite element (FE) method and the boundary element (BE) technique, respectively. The subdomain formulation has been implemented in a computer program MISS (Modélisation d Interaction Sol-Structure) [9]. The program MISS is used to determine the impedance functions as well as modal responses of the soil in the frequency domain. The dynamic impedance of the soil is calculated by means of a boundary element formulation that uses the Green s function of a horizontally stratified soil. Free field vibrations in the soil are evaluated due to impact and vibratory driving of a concrete pile in a homogeneous half space. Results are discussed for different penetration depths of the pile. 2 Numerical modeling 2.1 Problem outline The proposed model is based on the following hypotheses: (1) the soil medium is elastic with frequency independent material damping (hysteresis damping), (2) no separation is allowed between the pile and soil medium, (3) all displacements and strains remain sufficiently small, and (4) the pile is embedded in a horizontally layered soil. The domain is decomposed into two subdomains: the unbounded semi-infinite layered soil denoted by Ω ext s and the bounded structure (pile) denoted by Ω p. The interface between the soil and the pile is denoted by Σ (figure 1). 2.2 Governing equations First, the pile Ω p is considered. The boundary Γ p = Γ pσ Σ of the pile Ω p is decomposed into a boundary Γ pσ where tractions t p are imposed and the soil-pile interface Σ. The displacement
p s p s ext s Figure 1: Geometry of the subdomains. vector u p of the pile satisfies the following Navier equation and boundary conditions: divσ p (u p ) = ρ p ω 2 u p in Ω p (2.1) t p (u p ) = t p on Γ pσ (2.2) where t(u) = σ(u) n is the traction vector on a boundary with a unit outward normal vector n. Second, the exterior soil domain Ω ext s is taken into consideration. The boundary Γ ext s = Γ sσ Γ s Σ of the soil domain Ω ext s is decomposed into the boundary Γ sσ where tractions are imposed, the outer boundary Γ s where radiation conditions are imposed and the soil-structure interface Σ. A free boundary or zero tractions are assumed on Γ sσ. The displacement vector u s of the soil satisfies the Navier equation and the following boundary conditions: divσ s (u s ) = ρ s ω 2 u s in Ω ext s (2.3) t s (u s ) = on Γ sσ (2.4) The displacement compatibility and stress equilibrium condition are imposed on the interface Σ: u s = u p on Σ (2.) t p (u p ) + t s (u s ) = on Σ (2.6) According to the compatibility condition (2.), the displacement vector u s is termed as the wave fields u sc (u p ) that are radiated in the soil due to the pile motion u p, imposed on the interface Σ (figure 2): u s =u sc (u p ) in Ω ext s (2.7) The wave field u sc (u p ), radiated in the soil, satisfies the elastodynamic equation in Ω ext s and the following boundary conditions (figure 2): divσ s (u sc (u p )) = ρ s ω 2 u sc (u p ) in Ω ext s (2.8) t s (u sc (u p )) = on Γ sσ (2.9) u sc (u p ) = u p on Σ (2.1)
s s ext s Figure 2: The displacement field u s in term of the radiated wave fields u sc (u p ). 2.3 Variational formulation Accounting for the equilibrium of the tractions on the soil-pile interface Σ, the equation of motion of the dynamic soil-pile interaction problem can be formulated in a variational form for any virtual displacement field v imposed on the pile: ǫ(v) : σ p (u p )dω ω 2 v ρ p u p dω Ω p Ω p + v t s (u sc (u p ))dσ = v t p dγ (2.11) Σ Γ pσ In a FE formulation, the displacement field u p is approximated as u p = N p u p, where N p are the globally defined shape functions. The stress vector is defined as σ p = Dǫ p, with D the constitutive matrix for an isotropic linear elastic material. Substituting the strain-displacement relations and the linear constitutive equation, the virtual work equation (2.11) must hold for any virtual displacement field v and can be elaborated as follows: [ Kp ω 2 ] M p + K s up = f p (2.12) where the stiffness matrix K p is defined as follows: K p = B T p DB p dω (2.13) Ω p with the matrix B p = LN p. The mass matrix M p of the pile is: M p = N T p ρ p N p dω (2.14) Ω p and, the vector of external forces f p on the pile is defined as: f p = N T p t p dγ (2.) Γ pσ The dynamic stiffness matrix K s of the semi-infinite layered half space is equal to: K s = N T p t s (u sc (N p ))dσ (2.16) Σ
2.4 Modal decomposition A modal decomposition is introduced, based on the modes ψ m (m = 1,...,q) of the pile with free boundary conditions. The modal decomposition for the displacement vector u p can be written as follows: u p = q ψ m α m in Ω p (2.17) m=1 Applying a finite element approximation of the modes ψ m N p ψ m, the structural displacement vector can be written as follows: u p q N p ψ m α m = N p Ψα (2.18) m=1 where the modes ψ m (m = 1,...,q) are collected in a matrix Ψ and the modal coordinates α m (m = 1,...,q) are collected in a vector α. Substituting the displacement vectors u p, u s and the virtual vector v = N p Ψα v in the variational equation (2.11) results into: [ Ψ T K p Ψ + iωψ T C p Ψ ω 2 Ψ T M p Ψ + Ψ T K s Ψ ] α = Ψ T f p (2.19) Using proportional damping, the equilibrium equation is also uncoupled, which is an advantage. The corresponding damping matrix C p is computed as follows: C p = M p Ψdiag(2ξ m ω m )Ψ T M p (2.2) and the equilibrium equation (2.19) can be elaborated as follows: ( Λqq + iωυ qq ω 2 I qq + Ψ T K s Ψ ) α = Ψ T f p (2.21) where the q-dimensional diagonal matrix Λ qq = diag(ωm 2 ) contains the squares of the eigenfrequencies ω m (m = 1,...,q). Proportional damping is represented by the q-dimensional matrix Υ qq = diag(2ξ m ω m ), where ξ m are the modal damping ratios. The term Ψ T K s Ψ = Σ (N pψ) T t s (u sc (N p Ψ))dΣ denotes the impedance matrix of the unbounded soil domain Ω ext s due to an imposed displacement field ψ m. This matrix is calculated using a boundary element technique. The solution of the dynamic soil-pile interaction problem in terms of the modal coordinates allows to compute the soil tractions at the interface and, subsequently, in the free field. A computation procedure is developed to solve the governing system of equations. This computation procedure can be partitioned into two parts. First, using the Structural Dynamics Toolbox (SDT) in MATLAB, the finite element model of the pile is made. In the second part, using the program MISS 6.3, the soil impedance Ψ T K s Ψ as well as the modal responses of the soil u sc (Ψ) are computed.
3 Numerical example 3.1 Problem outline The free field vibrations due to driving of a concrete pile in a semi-infinite medium are investigated. The pile has a length L p = 1, m, a diameter d p =.m, a Young s modulus E p = 4MPa, a Poisson s ratio ν p =.2, and a density ρ p = 2kg/m 3. The longitudinal wave velocity of the pile C p = 4m/s. The contributions of different types of waves are investigated for several penetration depths e p = 2m,m, and 1m (figure 3). F p L p e p d p Figure 3: Geometry of the problem. The pile is modeled using 8-node isoparametric brick elements. Only the axial modes of the pile are excited due to pile driving and considered in the modal superposition. The number of required modes depends upon the frequency content of the dynamic force. It is suggested to employ all modes with eigenfrequencies lower than 1. times the highest excitation frequency. Figure 4 shows the axial rigid body and flexible modes of the pile with free boundary conditions. The first axial mode is a rigid body mode and the second mode at 2.Hz corresponds to the first compression-tension mode. Mode 1:. Hz. Mode 2: 2. Hz. Mode 3: 4. Hz. Figure 4: The axial rigid body and flexible modes of the pile with free boundary conditions. The soil consists of a homogeneous half space with a Young s modulus E s = 8MPa, a Poisson s ratio ν s =.4, and a density ρ s = 2kg/m 3. The shear wave velocity of the half space is
C s = 12m/s. In order to account for the dissipation of the internal energy in the soil (material damping), complex Lamé coefficients parameters are used: µ s = µ s(1 + 2iβ s ) (3.22) λ s = λ s (1 + 2iβ s ) (3.23) where β s is the material damping ratio. In this example, a material damping ratio β s =.2 is used. The size of each boundary element should be sufficiently small with respect to the minimum wavelength λ min = C s /f max, where C s is the shear wave velocity in the soil and f max is the highest frequency in the dynamic excitation. It is proposed that the element size should be smaller than λ min /8. 3.2 Ground vibrations due to vibratory pile driving First, ground vibrations due to vibratory driving are considered. A standard hydraulic vibratory driver ICE 44-3V is selected. It operates at the frequency f = 2 Hz with an eccentric moment me =.7kgm, resulting in a centrifugal force F p = 8kN. The results are presented in the (r,z) plane, where the vertical coordinate varies from z =.m at the ground surface to z = 3m. The horizontal coordinate varies from the pile shaft at r =.m up to a distance r = 3. m from the pile center. 1 2 1 2 1 2 3 3 3 1 2 3 1 2 3 1 2 3 a. e p = 2m. b. e p = m. c. e p = 1m. Figure : The norm of the particle velocity in a homogeneous half space due to vibratory pile driving at 2Hz for different penetration depths (a) e p = 2m, (b) e p = m, and (c) e p = 1m. Figures shows the norm of the particle velocity due to vibratory pile driving at 2Hz for different penetration depths e p = 2m,m, and 1m. The characteristics of the propagating waves induced by vibratory driving can be classified as follows: (1) because of the soil-shaft contact, vertically polarized shear waves are generated which propagate radially from the shaft on a cylindrical surface; (2) at the pile toe, shear and compression waves propagate in all directions from the toe on a spherical wave front; (3) Rayleigh waves propagate radially on a cylindrical wave front along the surface. When the pile toe is near the surface, ground motions are influenced only by toe resistance, and the response of the soil around the pile shaft is dominated by Rayleigh waves. As the pile is driven, the contact area along the pile shaft increases, and vertically shear waves dominate the response around the pile shaft. In an elastic half space, both body waves and Rayleigh waves decrease in amplitude with increasing distance from the
1 PPV [mm/s] 1 1 1 1 1 1 Figure 6: Peak particle velocity (PPV) on the surface versus the distance from the pile due to vibratory pile driving at 2 Hz. The comparison is made for different penetration depths e p = 2m ( ), e p = m ( ), and e p = 1m ( ) with the results (dash-dotted line) reported by Wiss [2]. pile due to the geometrical damping. Figure shows how Rayleigh waves attenuate slower than body waves and propagate in a restricted zone close to the surface of the half space. Figure 6 illustrates the attenuation of the peak particle velocity (PPV) on the surface with the distance r from the pile. The maximum vibration amplitude in the near field occurs when the pile toe is near the ground surface (e p = 2m). However, reported data on pile driving show that ground vibrations are mostly independent of the pile penetration depth and depend on the soil properties [3, 1]. When the soil around the pile is assumed to deform in the elastic range, ground motions are affected by the impedance of the pile and the soil condition. The pile impedance affects ground vibrations in two opposite ways. Increasing the pile impedance, which occurs for increasing penetration depth, increases the force transmitted to the pile, but decreases the peak particle velocity [11]. In figure 6, the dash-dotted line represents approximate values obtained by field measurements during vibratory pile driving [2] and illustrates the relative vibration intensities. It is observed that the present model predicts more conservative ground vibrations in the far field, especially when the penetration depth is small. This reflects the fact that during actual driving, higher (plastic) strains are induced in the soil, leading to more material damping [3]. Depth [m] 1 Depth [m] 1 1 2 3 4 1 2 3 Particle velocity [mm/s] Particle velocity [mm/s] a. r =.1. b. r =. Figure 7: Peak particle velocity (PPV) versus the depth due to vibratory pile driving at 2Hz, at (a) r =.1 and (b) r = from the pile for different penetration depths e p = 2m ( ), e p = m ( ), and e p = 1m ( ).
Figure 7 displays the variation of the peak particle velocity versus the depth at different dimensionless distances. As the strain level is proportional to the particle velocity [1, 2], the profile of the particle velocity around the pile can be interpreted as the variation of the shear deformation. Figure 7a shows how this profile is uniformly distributed along the shaft, except when the penetration depth is small. 3.3 Ground vibration induced by impact driving Ground vibrations due to impact driving are subsequently investigated. The hammer impact force is evaluated using a 2DOF model developed by Deeks and Randolph [12]. A BSP-37 hammer is used to drive a pile with an impedance Z p = 196kNs/m. The hammer cushion is steel with a stiffness k c = 1.6 1 6 kn/m. The specific ram and anvil masses for this hammer are m r = 686kg and m a = 8kg, respectively. The hammer stroke is h =.m. 6 3 (a) Force [MN] 4 2 (b) Force [kn/hz] 2 1 1 2 3 Time [ms] 1 2 3 4 Frequency [Hz] Figure 8: (a) Time history and (b) frequency content of the impact force. Figure 8 shows the time history and frequency content of the impact force. As the frequency content of the force is mainly dominated by frequencies below 3 Hz, the soil stiffness and modal responses of the soil are calculated in the frequency domain up to 3 Hz. Considering this excitation frequency, axial modes with eigenfrequencies up to 4 Hz are included in the model implementation. In order to obtain an accurate transform Fourier from the frequency domain to the time domain, the frequency step f must be sufficiently small with respect to C R /rmax, where rmax represents the largest distance from the pile center. Particle velocity [mm/s/hz] 8 x 1 4 6 4 2 1 2 3 Frequency [Hz] Figure 9: Frequency content of the particle velocity in a homogeneous half space at different distances r =. m (solid line), r =. m (dash-dotted line) and r = 3 m (dashed line) from the pile for the penetration depth e p = m.
Figure 9 display the frequency content of the particle velocity at different distances from the pile. It can be observed that the frequency content of ground vibrations reduces as the distance from the pile is increased. Such behavior is as expected because of material damping which results in a reduction of the frequency content in the far field. The ratio of the dominant frequency of ground vibrations to the natural frequency of the adjacent structures is one of the most important parameters to analyse. For most sandy and clayey soils, the principal frequency due to pile driving is generally reported between and 3Hz [2, 6]. e p = 2.m 1 1 1 2 1 2 2 1 2 2 1 2 e p =.m 1 1 1 2 1 2 2 1 2 2 1 2 e p = 1.m 1 1 1 2 1 2 2 1 2 2 1 2 a. t = 3 ms. b. t = 8 ms. c. t = 16 ms. Figure 1: The norm of the particle velocity in a homogeneous half space due to impact pile driving at time steps (a) t = 3ms, (b) t = 8ms, and (c) t = 16ms for different penetration depths e p = 2.m,.m, and 1.m. Figure 1 shows the norm of the particle velocity in a homogeneous half space due to impact driving at different time steps t = 3,8, and 16ms and for different penetration depths. The following observations can be made: (1) body waves dominate around the toe and propagate on a spherical wave front; (2) vertically shear waves dominate around the pile and propagate radially on a cylindrical wave front; and (3) Rayleigh waves propagate near the surface with a velocity slightly less than shear waves. Figure 11 displays the attenuation of the peak particle velocity (PPV) with the distance r
1 PPV [mm/s] 1 1 1 1 1 1 Figure 11: Peak particle velocity (PPV) versus the distance from the pile due to impact pile driving.the comparison is made for different penetration depths e p = 1.m ( ), e p =.m ( ), and e p = 1.m ( ) with the field measurements (dash-dotted line) reported by Wiss [2]. from the pile due to impact pile driving. The maximum vibration amplitude occurs when the penetration depth is small (e p = 2m). When no separation along the soil-pile interface is assumed (linear model), the peak particle velocity reduces for increasing penetration depth. In practice, during pile driving, a large portion of the energy is dissipated to overcome the soil friction, moving the pile downwards (non-linear behavior). Therefore, less energy is transmitted in comparison with the case where the pile has arrived at a deeper layer with a higher resistance. In figure 11, the results have also been compared with field measurements (dash-dotted line) reported by Wiss [2], in which the vibrations have been measured due to an actual impact pile driving with a diesel hammer with an energy of 48kJ. It can be observed that the predicted vibration levels attenuate slower than the field measurements. This reflects the fact that higher strain levels during actual driving result in a higher attenuation. 4 Conclusions A coupled FE-BE model has been developed to predict ground vibrations due to vibratory and impact pile driving. Using a subdomain formulation, a linear model for dynamic soil-pile interaction has been implemented. The pile is modeled using the finite element method. The soil is modeled as a horizontally layered elastic half space using a boundary element method. The characteristics of the propagating waves around the pile have been investigated. In the case of vibratory driving, the induced waves can be classified into: (1) the vertically polarized shear waves around the shaft; (2) the body waves around the pile toe, and (3) the Rayleigh waves on the surface. Propagating waves due to impact driving are more complex. It is observed that deformations around the shaft are dominated by the vertically shear deformation where the normal and the radial deformations are small compared to shear deformations. In the far field, however, body waves are attenuated and Rayleigh waves dominate the zone near the surface, independent of the penetration depth. Furthermore, it is observed that the frequency content of ground vibrations reduces and peak values occur at lower frequency as the distance from the pile is increased.
Acknowledgements The results presented in this paper have been obtained within the frame of the SBO project IWT 317 Structural damage due to dynamic excitation: a multi-disciplinary approach. This project is funded by IWT Vlaanderen, the Institute of the Promotion of Innovation by Science and Technology in Flanders. Their financial support is gratefully acknowledged. References [1] C.H. Dowding, Construction vibration, USA, 2. [2] J.F. Wiss, Construction vibrations: state of the art, ASCE Journal of the Geotechnical Engineering Division 17(GT2) (1981) 167 181. [3] G.W. Clough, J.L. Chameau, Measured effects of vibratory sheet pile driving, ASCE Journal of the Geotechnical Engineering Division 16(GT1) (198) 181 199. [4] E.A.L. Smith, Pile driving analysis by the wave equation, ASCE Journal of Soil Mechanics and Foundation 86( EM4) (196) 3 61. [] F. Rausche, G. Goble, G. Likins, Investigation of dynamic soil resistance on piles using GRLWEAP, Proceedings of the Fourth International Conference on the Application of the Stress-Wave Theory to Piles, A.A.Balkema, Netherlands, 1992, 137 142. [6] D.S. Kim, J.S. Lee, Propagation and attenuation characteristics of various ground vibrations, Int. Journal of Soil Dynamics and Earthquake Engineering 19 (22) 1 126. [7] D. Aubry, D. Clouteau, A subdomain approach to dynamic soil-structure interaction. In: V. Davidovici, R.W. Clough (eds.), Recent advances in Earthquake Engineering and Structural Dynamics, Ouest Editions/AFPS, Nantes, 1992, 21 272. [8] D. Clouteau, Propagation d ondes dans des milieux hétérogènes. Application à la tenue des ouvrages sous séismes, PhD thesis, Ecole Centrale Paris, 199. [9] D. Clouteau, MISS Revision 6.2, Manuel Scientifique, 1999. [1] M.R. Svinkin, Prediction and Calculation of Construction vibrations, The Wave Equation Page for Piling, 1999. [11] H.R. Masoumi, G. Degrande, Numerical modeling of free field vibration due to impact and vibratory pile driving using a dynamic soil-structure interaction formulation,, Internal Report BMW-2-2, SBO Project IWT 317, 2. [12] A.J. Deeks, M.F. Randolph, Analytical modeling of hammer impact for pile driving, Int. Journal for Numerical and Analytical Methods in Geomechanics 17 (1993) 279 32.