The kinematic interaction of a single pile with heterogeneous soil
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1 Earthquake Ground Motion: Input Definition for Aseismic Design 135 The kinematic interaction of a single pile with heterogeneous soil M. Maugeri, E. Motta, E. Raciti & D. Ardita Department of Civil and Environmental Engineering, University of Catania, Italy Abstract The behaviour of deep foundation under static loads has been widely investigated and the available calculation procedures can be considered suitable for the current engineering applications. However, pile behaviour under seismic loading is more complex and less known, because it involves the simultaneous action of inertial forces arising from the super-structure (inertial interaction) and of the soil deformations arising from the seismic waves (kinematic interaction). Italian code DM 14/01/2008 requires the dynamic soil-structure interaction in seismic foundation design to be taken into account, but it does not give any information about criteria for the evaluation of kinematic interaction strains. Experimental evidence and theoretical considerations by many authors show that the kinematic interaction alone may induce high stresses in piles, especially near an interface between a soft and a rigid soil layer. In this work pile behaviour due to kinematic interaction with soil will be examined. An approach based on a differential equation proposed in the relevant technical literature will be used. The analysis is focused on the response of a single pile in a heterogeneous three-layer soil profile. Keywords: deep foundations, seismic loads, dynamic soil-structure interaction, pile behaviour, numerical model. 1 Introduction Pile seismic response results from a complex soil pile superstructure interaction affected by non-linear phenomena, which take place in the soil near piles and by kinematic effects linked to ground shaking. Dynamic pile soil interaction is significant when piles are embedded in soil layers with highly discontinuous strength and stiffness. doi: / /14
2 136 Earthquake Ground Motion: Input Definition for Aseismic Design In 1977, Tajimi [1] accepted that pile design should take into account the dynamic behaviour. During the recent decades, a deep improvement in knowledge has been achieved, as many experimental observations during real earthquakes are now available and dynamic simulations on physical models as well as numerical analyses under various loading conditions have been developed. The results of those studies can be found in the technical literature [1 11]. To model the dynamic behaviour, the application of rigorous analytical tools would be desirable, but in design practice this is too onerous, especially when a seismic analysis is carried out in the frequency domain, as the pile response should be evaluated over a very high number of frequencies (thousands) that would be sufficient to cover the seismic signal frequency content. Nowadays, the practice level is lower since theoretical knowledge and theoretical developments have not yet been converted to simple calculation methods. In professional practice, the dynamic effects are usually neglected, because simple analysis methods are missing or such effects are considered negligible. So, piles are designed taking into account only the loads applied at the pile head. In this work, a simplified analytical model that allows the kinematic interaction with soil to be taken into account in design is presented for piles embedded in layered soils. The results of this study are shown through dimensionless plots for preliminary design estimations. 2 Mathematical model The proposed simplified model is based on the Beam on Dynamic Winkler Foundation (BDWF) method, according to which soil behaviour is represented by springs and dampers distributed along the pile (fig. 1). The soil around the piles is hypothesised to be under free-field conditions, so that seismic S-waves propagate vertically and are not influenced by pile presence. Based on the model proposed by Kavvadas and Gazetas [12] and on their application to a two-layer soil deposit, in this work an application to a three-layer soil profile is carried out. The analysis defines the free-field displacement of undisturbed soil u ff z,t and applies it to the pile through springs and dampers (representing the pile soil interface) to impose seismic actions on the pile. It is necessary to take into account both the pile with the soil layers parameters involved and the interface parameters, which are, however, functions of both soil and pile geometrical and physical parameters. One of the most critical aspects in modelling the soil pile system is the determination of the springs and dashpot mechanical parameters (stiffness k x and viscosity c x ), functions of frequency. To determine the soil free-field displacement, a one-dimensional S-wave propagation can be used, assuming linear hysteretic soil behaviour: u ff (z, t) = U ff (z)exp[i(t + ff )] = U ff exp(it) (1) where U ff (z) is the free-field soil displacement modulus; ff = tan 1 [2/(1 2 )] the phase displacement between seismic input at the bedrock and the soil response;
3 Earthquake Ground Motion: Input Definition for Aseismic Design 137 f / s the ratio of the excitation frequency to the fundamental natural frequency of the free (i.e. without piles) soil deposit in vertical S-waves; is the hysteretic damping ratio. Figure 1: BDFW model for a layered soil and a free head pile (from Kavvadas and Gazetas [12]). Each layer is characterized by a complex shear wave velocity: * s V V 1 2i (2) s where V s G / is the actual shear wave velocity. An acceleration with a frequency equal to the fundamental natural frequency of a homogeneous soil layer thick as the pile length has been used as a simplified seismic input. To determine the soil fundamental natural frequency, the Rayleigh method can be used. The differential equation governing the pile response is the following: 4 2 up up EI p p m ( ) 4 p S 2 x uff up z t (3) where E p I p is the bending stiffness, m p the mass per unit length, u p the pile displacement; S x represents features of the interface whereby free-field ground displacement transmits strains to the pile: S x k x ic x (4)
4 138 Earthquake Ground Motion: Input Definition for Aseismic Design As a first approximation, the spring stiffness k x can be considered approximately frequency independent and expressed as multiple of the local soil Young s modulus E s : k x E s (5) where is a frequency independent coefficient assumed to be constant (i.e. the same for all layers and independent of depth), that will be called pile soil interaction coefficient. has been determined by Kavaddas and Gazetas using the finite element method [12]. The stiffness parameter c x in eqn (4) represents both radiation and material damping; the former arises from waves originating at the pile perimeter and spreading laterally outward and the latter from hysteretically-dissipated energy in the soil. Solving eqn (3), pile deformations (displacements and rotations), bending moment and shear force will be determined as functions of both depth z and time t. Horizontal pile displacements can be determined from the following equation: u p z,tu pp zexpit p U pp (z)expit) (6) where U pp z is the pile displacement modulus; p the phase difference between the free-field displacement and pile response in terms of displacement. Eqn (3) can be alternatively written as follows: IV 4 pp pp ff U U U (7) where is the excitation circular frequency and 2 m 4 p Sx (8) EI p p S x (9) EI p p Eqn (7) has the following general solution: D1 z z i z i z D 2 U ( ) e e e e pp z su ff ( z) D3 D 4 (10)
5 Earthquake Ground Motion: Input Definition for Aseismic Design 139 where D 1, D 2, D 3, D 4 are arbitrary constants to be evaluated on the basis of the compatibility equations and the boundary conditions, while s 4 4 q (11) q (12) V s By eqn (10), the following equation can be obtained: U i i pp ( z) z z z z e e e e U ff ( z) D1 U ( ) z z i z i z e e i e i e pp z D U ( ) 2 ff z s 2 z 2 z 2 iz 2 iz U pp ( z) e e e e D3 U ff ( z) 3 z 3 z 3 iz 3 iz e e i e i e D 4 U pp ( z) U ff ( z) (13) or, concisely, for a pile element in the domain of the soil layer j: U ( z) F ( z) D s U ( z) (14) pj j j j j The vector U j ( z) can be determined from the free-field displacement solution. In the case of a multi-layer soil profile with N layers (j = 1, 2,..., N), eqn (13) will be a system of 4N equations with 4N arbitrary constants D 1, D 2, D 3, D 4 that could be evaluated from the equations representing compatibility between pile and soil as well as the boundary conditions. The compatibility equations express that, at the (N 1) soil layer and pile interfaces, the pile deflection u p, rotation, bending moment M and shear force Q must be continuous: these compatibility requirements can be expressed by the following 4(N 4) equations (for an arbitrary interface j) U ( z ) U ( z ) (15) pj j p( j1) j Regarding the boundary conditions, it can be observed that at the pile top, in the case of a pile with free head, 0, t 0 (16) Q0, t 0 (17) At the pile end, in the case of a pile hinged at the bedrock:
6 140 Earthquake Ground Motion: Input Definition for Aseismic Design while in the case of a floating pile: Mz N, t0 (18) u p z N, t u g t (19) Mz N, t 0 (20) Qz N, t 0 (21) M and Q being pile bending moment and shear, respectively. Thus a set of 4N equations can be obtained and they can be solved for the constants D 1, D 2,, D N. Once these constants are evaluated, pile displacements, rotations, bending moments, shear forces can be obtained directly from eqn (13), since: Pile displacement: u pp z, t) (22) Pile rotation: z, t u (,) z t (23) Pile bending moment: Mz, te p I p u (,) z t (24) Pile shear: Q(z, te p I p u (,) z t (25) 3 Numerical analysis A system made of a fixed head, single pile, with length L and diameter d, embedded in a three-layer soil deposit, h 1, h 2 and h 3 thick (fig. 2(a) and (b)) has been studied. The soil deposit lies on a rigid bedrock and is subjected to vertically propagating S-waves that produce a horizontal harmonic motion of the kind: pp pp pp u g t U g expit) (26) The case of a floating pile and that of a pile hinged at the bedrock have been analysed. In the first case, M = 0 and Q = 0 are the boundary conditions at the pile end; in the second case the conditions u p = u g and M = 0 have been imposed. In the case of floating pile, h 1 h 2 h 3 /2, d L/30 and L 3h 1 have been assigned. In the second case, h 1 h 2 h 3 and once more d L/30 and L 3h 1 have been assigned. As regards the frequency of the bedrock displacement, the value of fundamental natural frequency calculated in the case of a homogeneous soil layer as thick as the pile length with the mechanical features of layer 1 has been assumed.
7 Earthquake Ground Motion: Input Definition for Aseismic Design 141 Figure 2: The calculation model. (a) Floating Pile. (b) Pile hinged at the bedrock. The soil has been hypothesized as a linearly hysteretic solid made of three layers, with respective Young s moduli E s1, E s2, E s3, damping ratios %, mass densities and Poisson s ratios A pile has been represented as a linearly elastic beam with mass density p For the elastic properties of the soil, E s3 /E s1 1 has been assigned and the cases E s1 /E s2 0.1, 0.5, 1.0, 2.0 and 10.0 have been analysed. 4 Preliminary results and conclusions Following Kavvadas and Gazetas [12], results have been displayed in dimensionless format. In particular, the dimensionless bending moment has been defined as M * M 4 d u p g 2 where ug ug is the maximum seismic acceleration at the bedrock. In fig. 3, the single pile response, in the case of a floating pile, is shown in terms of bending moment modulus for E s1 /E s2 = 0.1, 0.5, 1.0, 2.0, The results show that a peak of moment is recorded near mechanical discontinuities.
8 142 Earthquake Ground Motion: Input Definition for Aseismic Design Figure 3: Dimensionless bending moment for different soil layer strains for a floating pile. For floating piles, the maximum bending moment value is recorded where the difference between mechanical properties is higher (E s1 /E s2 = 0.1; 10.0). For piles hinged at the bedrock, the maximum moment along the pile seems to be always lower than that observed at the pile top in the case of homogeneous soil layer. However, this can be attributed always to the input frequency adopted, coinciding with the fundamental natural frequency of the case of a homogeneous soil layer. In fig. 4, for the only case of floating pile, the dimensionless moment modulus versus depth is shown, for various intermediate soil layer thickness, that is, for h 2 /h 1 = 0.1, 0.2, 0.3, 0.5. Regardless of the stiffness ratio between various layers, for an increase of the intermediate layer thickness, a raise in the maximum excitation has been observed. This is even more evident in fig. 5, where, for the case of a floating pile, the ratio M het /M hom, of the maximum bending moment along the pile in a heterogeneous soil profile to that of the case of a homogeneous soil layer, with varying intermediate layer thickness is shown.
9 Earthquake Ground Motion: Input Definition for Aseismic Design 143 Figure 4: Dimensionless bending moments for various intermediate layer thicknesses.
10 144 Earthquake Ground Motion: Input Definition for Aseismic Design Figure 5: Floating pile: the dimensionless ratio of maximum bending moments M het (in a heterogeneous soil profile) to M hom (in a homogeneous soil profile) versus the ratio h 2 /h 1 for various intermediate layer thicknesses. Some final remarks of this preliminary study can be summarised as follows: As observed by other authors, in this study it has also been highlighted that, in the presence of mechanical discontinuities due to stratigraphical discontinuities, peak in bending moment values are recorded. Among the analysed cases and in the case of floating pile, the maximum excitations in the presence of mechanical discontinuities appeared higher than in the case of homogeneous soil, regardless of higher or lower rigidity of the intermediate layer. Moreover, as regards floating piles, an effect of bending moment amplification, in comparison to the maximum bending moment of the pile embedded in a homogeneous soil deposit, with the intermediate layer thickness rising has been observed. This result seems to be independent of the higher or lower rigidity of the intermediate layer in comparison to the external ones. Finally, it must be underlined that the excitation frequency can sensitively condition results, depending on being near or far from the fundamental natural frequency. However, this study is still at its preliminary phase, so it is not possible to derive final general remarks. References [1] Tajimi, H., Seismic effects on piles. State-of-the-art Report 2, Int Conf. on Soil Mechanics and Foundation Engineering, Proc. of Specialty Session 10, pp , [2] Dennehy, K. & Gazetas, G., Seismic vulnerability analysis and design of anchored bulkheads, research report, Rensselaer Polytechnic Institute, Troy, NY, chapters 5-6, 1985.
11 Earthquake Ground Motion: Input Definition for Aseismic Design 145 [3] Flores-Berrones, R. & Whitman, R.V., Seismic response of end-bearing piles. Journal of the Geotechnical Engineering Division, ASCE, 108(4), pp , [4] Gazetas, G., Seismic response of end-bearing single piles. Soil Dynamics and Earthquake Engineering, 3(2), pp , [5] Kagawa, T. & Kraft, L.M., Lateral pile response during earthquakes. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 107(12), pp , [6] Kaynia, A.M. & Kausel, E., Dynamic behaviour of pile groups. Proc. of the 2 nd Int. Conf. on Numerical MethodsiIn Offshore Piling, Austin, pp , [7] Kobori, T., Minai, R. & Baba, K., Dynamic behaviour of a pile under earthquake-type loading. Proc. of 1 st Int. Conf. on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, Rolla 2, pp , [8] Masayuki, H. & Shoichi, N., A study on pile forces of a pile group in layered soil under seismic loading. Proc. of the 2 nd Int. Conf. on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, St Louis 3, [9] Penzien, J., Soil-pile foundation interaction. Earthquake Engineering, ed. R.L. Wiegel, Prentice-Hall: New York, chapter 14, [10] Tajimi, H., Dynamic analysis of structure embedded in elastic stratum. Proc. of the 4 th World Conf. on Earthquake Engineering, Santiago, pp , [11] Wolf, J.P. & Von Arx, G.A., Horizontally travelling waves in a group of piles taking pile-soil-pile interaction into account. Earthquake Engineering and Structural Dynamics, 10, pp , [12] Kavvadas, M. & Gazetas, G., Kinematic seismic response and bending of free- head piles in layered soil. Géotechnique, 43(2), pp , 1993.
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