18th Engineering Mechanics Division Conference (EMD2007)
FREE FIELD VIBRATIONS DUE TO PILE DRIVING USING A HYBRID FREQUENCY-TIME DOMAIN APPROACH H.R. Masoumi, S. Franc¸ois, and G. Degrande Department of Civil Engineering, K.U.Leuven, B-3001, Leuven, Belgium
[email protected]
Abstract This work aims to develop a model to predict the vibrations induced in the environment due to pile driving. Modelling of the dynamic soil-pile interaction during pile driving requires to account for the non-linear constitutive behavior of the soil surrounding the pile. Therefore, the problem must be analysed in the time domain. The soil-pile system is decomposed into a finite domain (the pile and the surrounding soil) and an unbounded semi-infinite linear soil domain. The unbounded soil region is represented by an impedance matrix and is computed in the frequency domain by means of the boundary element method. Using a hybrid frequency-time domain formulation, the force-displacement relationship is obtained in the time domain. Newmark’s time integration scheme implemented in a FE code is used to solve the equation of motion of the coupled system. A non-linear analysis is performed due to an impact force on the pile head. The pile is assumed to behave linearly and the soil is modelled as an isotropic elastic, perfectly plastic solid which yields with the Drucker-Prager criterion. The computed ground vibrations are finally compared with experimental results reported in the literature.
Introduction In urban areas, vibrations generated by construction activities often affect surrounding buildings. These vibrations may also disturb sensitive equipment or people and cause cracks in walls or facades. In recent years, much progress has been made by developing models to assess the driving efficiency and the response of driven piles using a non-linear constitutive law for the soil (Holeyman, 2002; Liyanapathirana et al., 2001; Mabsout et al., 1995). These efforts are focused on the pile response, whereas the present work aims to predict the ground vibrations. Using a substructuring method, the soil-structure system is decomposed into two independent substructures: (a) the inhomogeneous or non-linear structure, and (b) the unbounded linear elastic soil. As it is assumed that the unbounded soil on the exterior of the interaction horizon up to infinity behaves linearly, this substructure can be analyzed in the frequency domain. The dynamic impedance of the unbounded soil is calculated by means of a boundary element formulation that employs the Green’s functions of a horizontally stratified soil in the frequency domain. Due to the non-linear constitutive behaviour of the soil surrounding the pile, the dynamic soil-pile interaction problem is non-linear and must be analysed in the time domain. Wolf (1988) has proposed several alternatives to formulate the soil-structure interaction forces in the time domain. In order to avoid the difficulties associated with the transformation of the dynamic-stiffness coefficients, a flexibility formulation is used in this work (Franc¸ois and Degrande, 2005). 1
Finally, the interaction forces in the time domain are added to the nodes of the soilstructure interface and the equation of the motion of the generalized structure is solved by means of Newmark’s time integration method. Numerical modeling The soil-pile system is decomposed into two subdomains: the bounded (generalised) structure Ωb that contains the pile and an irregular soil region adjacent to the pile that can behave non-linearly and the unbounded semi-infinite layered soil denoted by Ωext s . The interface between the bounded (structure) and unbounded (linear soil) subdomain is denoted by Σ, as shown in figure 1.
Gbs
Gss
ext
Ws Wb S
Gsoo
Figure 1: The geometry of the problem .
In a FE formulation, the equation of motion of the generalised structure can be defined as follows:
Mb1 b1 Mb2 b1
Mb1 b2 Mb2 b2
u¨ b1 (t) u¨ b2 (t)
+
fbint1 (t) fbint2 (t)
=
fbext (t) 1 ext fb2 (t)
−
0 Qb2 (t)
(1)
where the displacement vector of the structure ub is divided into the displacement vector ub1 corresponding to the nodes within the structure which are not located on the soilstructure interface Σ and the displacement vector ub2 corresponding to the degrees of freedom of the nodes on the interface Σ. Mb is the mass matrix of the structure and fbint denotes the vector of the internal forces. The vector fbext collects the external forces on Γbσ . The interaction forces Qb2 (t) depend on the tractions and displacements on the soilstructure interface Σ and are equal to the convolution integral of the dynamic soil stiffness matrix S(t) and the displacement vector ub2 (t): Qb2 (t) =
Z t 0
S(t − τ )ub2 (τ )d τ
(2)
As the unbounded soil domain behaves linearly, the dynamic stiffness coefficients of the soil can be calculated in the frequency domain by means of a boundary element formulation that uses the Green’s functions of a horizontally stratified soil. This formulation has been implemented in the computer program MISS (Mod´elisation d’Interaction SolStructure) by Clouteau (1999). This program is employed to compute the dynamic stiffness of the soil. The hybrid frequency-time domain approach Based on the stiffness formulation or the flexibility formulation, several alternatives to compute the dynamic stiffness coefficients in the time domain are discussed by Wolf Masoumi, H.R. , Franc¸ois, S. , and Degrande, G.
2
(1988). In a stiffness formulation, the inverse Fourier transform of the dynamic stiffness coefficients from the frequency domain to the time domain must be calculated and the asymptotic value for the frequency approaching infinity must be determined. The difficulties in determining the components of the dynamic stiffness can be avoided using a flexibility formulation. The dynamic flexibility coefficients of the soil in the frequency domain are the inverse of the dynamic soil stiffness coefficients and converge to zero when the frequency approaches infinity. The displacements of the interface nodes ub2 (t) are equal to the convolution integral of the dynamic flexibility matrix F(t) and the vector of interaction forces Qb2 (t): ub2 (t) =
Z t 0
F(t − τ )Qb2 (τ )d τ
(3)
Introducing a time domain interpolation function φ (t), the interaction forces Qb2 (t) are discretized as follows: ∞
Qb2 (t) =
∑ φ (t − k∆t)Qkb
2
(4)
k=1
where the interpolation function φ (t) is equal to: |t| 1 − ∆t 0
φ (t) =
if −∆t ≤ t ≤ ∆t if t ≥ ∆t or t ≤ −∆t
(5)
Substituting expression (4) into equation (3), the boundary displacement vector unb2 is written as: unb2 =
n
∑ Fn−k+1Qkb
2
(6)
k=1
where k
F =
Z k∆t 0
F(τ )φ ((k − 1)∆t − τ )d τ
(7)
Using the convolution theorem, the flexibility coefficient F1 is written as: F1 =
1 2π
Z ∞
−∞
ˆ ω )φˆ ′ (ω )d ω F(
(8)
with 1 − eiω ∆t + iω ∆t φˆ ′ (ω ) = ω 2 ∆t
(9)
The flexibility coefficient Fk (k ≥ 2) in equation (7) can be written as follows: Fk =
1 2π
Z ∞
−∞
ˆ ω )φˆ (ω )eiω (k−1)∆t d ω F(
(10)
with 2 − 2 cos(ω ∆t) φˆ (ω ) = ω 2 ∆t
(11)
The integral in equation (8) is evaluated by means of a trapezoidal rule, while equation (10) is evaluated by means of a Filon integration algorithm with an oscillatory kernel function. Masoumi, H.R. , Franc¸ois, S. , and Degrande, G.
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A modal reduction technique is applied where the vector of the interface displacements ub2 is decomposed as a linear combination of vibration modes ψ b2 m (m = 1, . . . , q): q
ub2
∑ ψ b m αsm = Ψb α s
≃
2
2
(12)
m=1
where the modes ψ b2 m (m = 1, . . . , q) are collected in a matrix Ψ b2 and the modal coordinates αsm (m = 1, . . . , q) are collected in a vector α s . The modal interaction forces fs are introduced as: fs = Ψ Tb2 Qb2
(13)
As the dynamic soil stiffness coefficients are usually computed in the frequency domain in terms of the interface modes, equation (6) can be represented in the following modal form: α ns =
n
∑ Fn−k+1fsk
(14)
k=1
where the modal interaction forces fsn can be written as: n−1
fsn = (F1 )−1 α ns − (F1 )−1 ∑ F n−k+1 fsk
(15)
k=1
Therefore, the equation of motion (1) can be written as:
Mb1 b1 Mb2 b1
Mb1 b2 Mb2 b2
u¨ nb1 u¨ nb2
+
n
fbint1 n fbint2
+
n ub1 0 0 = 0 Tq (F1 )−1 Tu unb2 ) ext n ( 0 fb1 + n fbext Qn−1 b2 2
(16)
where n−1
1 −1 Qn−1 ∑ F n−k+1fsk b2 = Tq (F )
(17)
k=1
ΨTb2 Ψ b2 )−1 , and Tu = (Ψ ΨTb2 Ψ b2 )−1 Ψ Tb2 . with Tq = Ψ b2 (Ψ
Numerical example The pile foundation and a part of the soil are modeled as a generalised structure embedded in a homogeneous half space. A circular concrete pile with a diameter d p = 0.50 m is considered. The pile has a length L p = 10 m, a Young’s modulus E p = 40000 MPa, a Poisson’s ratio ν p = 0.25, and a density ρ p = 2500 kg/m3 . The longitudinal wave velocity in the pile is equal to C p = 4000 m/s. Results are investigated for different penetration depths e p = 2, 5 and 10 m. The soil medium consists of a homogeneous half space with a Young’s modulus Es = 80 MPa, a Poisson’s ratio νs = 0.4, a material damping ratio βs = 0.05, and a density ρs = 2000 kg/m2 . The shear wave velocity in the elastic soil is equal to 120 m/s. The bounded subdomain is a cylinder with a diameter of 5.0 m and a height of 12.5 m. An axisymmetric model is considered for the pile and the soil (generalized structure) using Masoumi, H.R. , Franc¸ois, S. , and Degrande, G.
4
Mode 5.
Mode 6.
Mode 7.
Mode 8.
Figure 2: Some axisymmetric modes of the interface. −3
2 Displacement [m]
2
Force [MN]
1.5 1 0.5 0 0
(a)
20
40 60 Time [ms]
80
x 10
1.5 1 0.5 0 0
100
(b)
20
40 60 Time [ms]
80
100
Figure 3: Time history of (a) the hammer impact force and (b) the pile head displacement for e p = 10 m, computed by the hybrid method (solid line) and by a solution in the frequency domain (dash-dotted line).
4-node plane axisymmetric finite elements. The size of the finite element mesh is limited to 0.15 m in the horizontal direction and 0.20 m in the vertical direction. The boundary element method is applied to compute the impedance of the unbounded domain. The BE mesh on the interface corresponds to the two-dimensional axisymmetric FE mesh. A rigid body mode in the z−direction and 49 flexible modes are selected. Figure 2 illustrates some flexible modes of the interface with free boundary conditions. The dynamic response of the generalised structure due to an impact force (a single blow) is investigated where a BSP-357 hammer is used to drive the pile in the soil. The force is evaluated using a 2DOF model developed by Deeks and Randolph (1993). The hammer cushion is a steel plate with a stiffness kc = 1.6 × 106 kN/m, and the ram mass mr = 6860 kg. The pile impedance is equal to Z p = 1960 kNs/m. The impact velocity is v0 = 1 m/s, resulting in an impact with a transferred energy of about 3.4 kJ. The impact force is applied at the center of the pile head (figure 3a). Newmark’s method is used to solve the equation of motion (16) with a time step ∆t = 1 × 10−3 s. In order to improve the stability of the integration, the Newmark parameters β = 0.45 and γ = 0.70. Figure 3b shows the pile head displacement due to a single blow in a linear analysis for penetration depth e p = 10 m. Results of the hybrid method show a good agreement with those obtained by the frequency domain solution (Masoumi et al., 2007). Masoumi, H.R. , Franc¸ois, S. , and Degrande, G.
5
−3
−3
x 10
20
8
Displacement [m]
Displacement [m]
10
6 4 2 0 0
25
(a)
50 Time [ms]
75
x 10
15 10 5 0 0
100
25
(b)
50 Time [ms]
75
100
Figure 4: Time history of the pile head displacement in (a) a linear analysis and (b) a non-linear analysis for different penetration depths e p = 2 m (− ⋄ −), e p = 5 m (− ◦ −) and e p = 10 m (− ∗ −).
0 150
100
Depth [m]
PPV [mm/s]
5
10
50
15
0 0
(a)
5
10
15
20
r [m]
20 0
25
(b)
5
10 15 PPV [mm/s]
20
25
Figure 5: PPV versus (a) the distance from the pile and (b) the depth at the distance r = 20 m for different penetration depths e p = 2 m (− ⋄ −) , e p = 5 m (− ◦ −) and e p = 10 m (− ∗ −), for linear analysis (dashed lines) and non-linear analysis (solid lines).
In order to take into account the non-linearity of the soil around the pile, the soil is idealized as a Drucker-Prager elasto-plastic material. The soil is a medium sand with friction angle φ = 25o and cohesion c = 15 kPa. Furthermore, a perfectly bonded interface is assumed between the pile and the surrounding soil. Figure 4 shows the time history of the displacement at the pile head for both linear and non-linear analysis. The calculation is performed for different penetration depths e p = 2, 5 and 10 m. It is observed that the pile head displacement decreases with increasing penetration depth for both linear and non-linear cases. It should be mentioned that no continuous penetration is considered and results are obtained due to a single impact in each penetration depth. Figure 5a illustrates the attenuation of the peak particle velocity (PPV) at the surface with the distance r from the pile for different penetration depths. The slope of the attenuation curve can be interpreted as an attenuation coefficient that represents the effect of both geometrical and material damping in the soil. The comparison between the vibrations computed by a non-linear analysis with those of a linear analysis shows a reduction with Masoumi, H.R. , Franc¸ois, S. , and Degrande, G.
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1000
PPV [mm/s]
PPV [mm/s]
1000
100
10
1
(a)
5
10
25
r [m]
50
100
10
1
(b)
5
10
25
50
r [m]
Figure 6: The envelope of the PPV versus the distance from the pile due to the impact driving with a transferred energy from 3.4 kJ to 19.2 kJ, computed by (a) a linear analysis and (b) a non-linear analysis and compared with experimental results (dash-dotted line) reported by Wiss (1981).
a factor of 3 to 5 due to the plastic deformation of the soil in the non-linear analysis. It is also observed that the maximum vibration amplitude in the linear case occurs when the pile is near the surface (e p = 2 m); in the non-linear case, however, the PPV seems to be less dependent on the penetration depth. Figure 5b displays the variation of the PPV versus the depth at r = 20 m from the pile. In the linear case, the PPV shows an exponential decrease along the depth (as is typical for Rayleigh waves); in the non-linear case, however, the PPV decreases slightly along the depth. Figure 6 displays the envelope of the PPV versus the distance from the pile due to impact pile driving with a transferred energy from 3.4 (the lower bound) to 19.2 kJ (the upper bound). The predicted vibrations are compared with results of field measurements reported by Wiss (1981) for impact driving with a diesel hammer at a rated energy of 48 kJ. Considering a transfer coefficient of about 0.40, this energy is equivalent with a transmitted energy of 19.2 kJ. In the linear case, the computed vibrations are conservative even when a low impact is considered. In the non-linear case, however, results of the present model show a good agreement with those of field measurements and almost the same attenuation coefficient as in experimental results is observed. Conclusion A coupled FE-BE model has been developed to predict ground vibrations due to impact pile driving. A hybrid frequency-time domain approach has been proposed to solve the problem in the time domain. In a linear analysis, results of the present method show a good corresponding with of the frequency domain solution. The free field vibrations due to impact pile driving are successfully simulated assuming a non-linear constitutive behavior of soil around the pile. Results are discussed for different penetration depths. The envelope of the predicted ground vibrations for a transferred energy from 3.4 to 19.2 kJ shows a good agreement with experimental results reported in the literature.
Masoumi, H.R. , Franc¸ois, S. , and Degrande, G.
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Acknowledgements The results presented in this paper have been obtained within the frame of the SBO project IWT 03175 ”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 D. Clouteau (1999). MISS Revision 6.2, Manuel Scientifique. Laboratoire de M´ecanique des Sols, Structures et Mat´eriaux, Ecole Centrale de Paris. Deeks, A.J. and Randolph, M.F. (1993). ‘Analytical modeling of hammer impact for pile driving’. International Journal for Numerical and Analytical Methods in Geomechanics 17:279–302. Franc¸ois, S. and Degrande, G. (2005). ‘Non-linear dynamic soil-structure interaction in the time domain: response of a structure with a disk foundation’. Tech. Rep. BWM2005-03, Department of Civil Engineering, K.U.Leuven. Research assistantship FWO Flanders. Holeyman, A.E. (2002). ‘Soil behavior under vibratory driving’. In Proceedings of the International Conference on Vibratory Pile Driving and Deep Soil Compaction, Transvib 2002, pp. 3–19, Louvain-la-Neuve, Belgium. Keynote lecture. Liyanapathirana, D.S. and Deeks, A.J. and Randolph, M.F. (2001). ‘Numerical modelling of the driving response of thin-walled open-ended piles’. International Journal for Numerical and Analytical Methods in Geomechanics 25:933–953. Mabsout, M.E. and Reese, L.C. and Tassoulas, J.L. (1995). ‘Study of pile driving by Finite Element method’. Journal of Geotechnical Engineering, Proceedings of the ASCE 121(7):535–543. Masoumi, H.R. and Degrande, G. and Lombaert, G. (2007). ‘Prediction of free field vibrations due to pile driving using a dynamic soil-structure interaction formulation’. Soil Dynamics and Earthquake Engineering 27(2):126–143. Wiss, J.F. (1981). ‘Construction vibrations: state-of-the-art’. Journal of Geotechnical Engineering, Proceedings of the ASCE 107(GT2):167–181. J.P. Wolf (1988). Soil-structure-interaction analysis in the time domain. Prentice-Hall, Englewood Cliffs, New Jersey.
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