Kinematic internal forces in deep foundations with inclined piles

Kinematic internal forces in deep foundations with inclined piles ∗ L.A. Padr´on, A. Su´arez, J.J. Azn´arez, O.Maeso Instituto Universitario de Sistemas Inteligentes y Aplicaciones Num´ericas en Ingenier´ıa (SIANI) Universidad de Las Palmas de Gran Canaria Edificio Central del Parque Cient´ıfico y Tecnol´ogico Campus Universitario de Tafira, 35017, Las Palmas de Gran Canaria, Spain {lpadron,jjaznarez,omaeso}@siani.es 2015 Abstract This paper presents a parametric study that looks into the influence of pile rake angle on the kinematic internal forces of deep foundations with inclined piles. Envelopes of maximum kinematic bending moments, shear forces and axial loads are presented along single inclined piles and 2 ×2 symmetrical square pile groups with inclined elements subjected to an earthquake generated by vertically-incident shear waves. Inclination angles from 0 ° to 30 ° are considered, and three different pile–soil stiffness ratios are studied. Amplification factors for the kinematic bending moments with respect to the vertical configuration are also presented. These results are obtained through a frequency–domain analysis using a boundary element – finite element code in which the soil is modelled by the boundary element method as a homogeneous, viscoelastic, unbounded region, and the piles are modelled by finite elements as Euler-Bernouilli beams. The rotational kinematic response of the pile foundations is shown to be a key factor on the evolution of the kinematic internal forces along the foundations. 1 Introduction The interest in the seismic response of structures founded on inclined pile foundations, and in the behaviour of the raked piles themselves, has been growing very significantly in the last decade, giving rise to a significant number of both numerical and experimental publications on the topic. The motivations behind this interest and the main practical aspects of the issue are well described, for instance, by Giannakou et al. [1]. One of the more important aspects of the problem is understanding how dynamic internal forces evolve when inclining the piles, so that they can be detailed properly and all possible beneficial properties of raked piles can be exploited, even in seismically active regions. Several numerical [2,3,4,5,6,1,7] and experimental [8,9,10] studies have already looked into the seismic response of raked piles reaching very interesting conclusions for several particular cases. These results seem to confirm that inclined piles might develop kinematic forces at the pile-cap connection larger than those observed in their vertical counterparts, but they also suggest that batter piles might be beneficial in terms of the total response. There exist a need for further research in order to generate enough data so that general conclusions on the topic can be drawn. This paper aims at presenting the results of a parametric analysis studying the influence of rake angle on the kinematic internal forces of single inclined ∗This is the pre-peer reviewed version of the following article: L.A. Padr´on, A. Su´arez, J.J. Azn´arez, O. Maeso. Kinematic internal forces in deep foundations with inclined piles. Earthquake Engineering and Structural Dynamics, 44, pp. 2129-2135, 2015, which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/eqe.2559/abstract. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. 1 piles and 2×2 square symmetrical pile groups with inclined elements, subjected to an earthquake produced by vertically-incident shear waves. Three different pile-soil stiffness ratios are also considered. The results are presented as envelopes of maximum kinematic bending moments, shear forces and axial loads along the whole foundation. Amplification factors of the bending moments with respect to the vertical configuration are also presented, as well as the evolution of maximum kinematic bending moments at pile heads with rake angle. The rotational kinematic response of the pile foundations is shown to be a key factor on the evolution of the kinematic internal forces along the foundations. 2 Methodology The kinematic internal forces presented in this work have been obtained by making use of a coupled boundary element – finite element methodology developed in previous works for the dynamic analysis of pile foundations in the frequency domain. This soil – pile interaction model was first presented in Padr´on et al. [11], work where the model was validated for the computation of dynamic stiffness and damping functions for vertical piles. After applying the model to several other related problems [12,13,14], the formulation was generalized to inclined piles in Padr´on et al. [15] and validated for and applied to the computation of impedance functions for raked piles. The problem of impedance functions of end-bearing inclined piles was later treated in Padr´on et al. [16]. Recently, the model was also validated for and applied to the computation of kinematic interaction factors of deep foundations with inclined piles in Medina et al. [17]. The soil is modelled as a linear, elastic, zoned-homogeneous unbounded domain and discretized by the boundary element method, while piles are modelled as Euler-Bernoulli linear– elastic beams by the finite element method. Soil and pile are assumed to be perfectly bonded. Pile cap is considered perfectly rigid and not in contact with the soil. Six-noded triangular elements or nine-noded quadrangular quadratic elements are used to discretize the soil surface, while three-noded beam elements are used for the piles. One of the main advantages of the formulation lies in the fact that the interface between pile and soil does not need to be discretized. For the problem at hand, the boundary integral equation is written in terms of diffracted fields and only for the nodes in the ground surface and for the pile nodes as internal points. This set of equations, together with the FEM equations for the piles, leads to a non-singular system of linear equations after imposing compatibility, equilibrium and boundary conditions. Once the solution is known, and as part of the post-processing, internal forces can be computed at the extreme nodes of each finite element as F(ω) = (K−ω2M)u(ω) + Qq(ω) (1) where Mand Kare the mass and stiffness matrices, ωis the circular frequency of the excitation, u(ω) and q(ω) are the vectors of displacements and pile–soil interaction forces along the piles (primary variables of the model), and Qis the matrix that transforms nodal interaction force components into equivalent nodal forces. More details on the formulation can be found in the references given above. Once the complex frequency response function F(ω), corresponding to all internal forces at extreme nodes of all pile elements, is computed along the whole frequency range of interest for a vertically–incident S wave field that produces unitary horizontal free-field ground motion (uff ), the standard frequency domain method approach [18] is used to obtain the evolution of the internal forces in the time domain f(t) for a certain input accelerogram. Such time functions are used to produce the envelopes presented in section 3. 2 3 Results 3.1 Geometrical parameters and problem definitions Two different cases are studied: a single inclined pile and a square 2 ×2 symmetrical pile group with battered elements. In both cases, the foundation is embedded in a homogeneous half-space. Rotation of the pile head is prevented for the single pile (ϕg= 0). Regarding the pile group, the pile cap is assumed to be massless, infinitely rigid, free to rotate and move horizontally and vertically, and not in contact with the soil. Fixed connection between piles and cap is considered. The main geometrical parameters of the pile foundations are illustrated in figure 1: piles diameter d= 0.6 m, piles length L= 12 m, and center-to-center pile head separation s= 3 m, which corresponds to the slenderness and center-to-center pile separation dimensionless ratios L/d = 20 and s/d = 5. Besides, four rake angles are studied: θ=0 ° (vertical pile), 10 ° , 20 ° and 30 ° . Piles are assumed to be reinforced concrete elements of solid circular cross-section with Young’s modulus Ep= 3 ×1010 Pa and a density of ρp= 2500 kg/m3. Soil and pile Poissons ratios are νs= 0.4 and νp= 0.25. Three different soils are considered for the half-space, with shear wave velocities cs= 350, 250 and 110 m/s, respectively, and a density of ρs= 1750 kg/m3, which corresponds to a ρs/ρp= 0.7 soil-pile density ratio, and pile-soil Young’s modulus ratios of Ep/Es= 50, 100 and 500 (softest soil). Soil hysteretic damping coefficient is defined as β= 5% in all cases. Figure 1: Problem description. The seismic input motion is defined by a synthetic free-field ground surface horizontal accelerogram generated to be compatible with the type 1 elastic response spectrum for ground type C and 5% damping according to part 1 of Eurocode 8 [19]. Vertically–incident shear waves generating horizontal displacements parallel to the vertical plane containing the inclined piles (see figure 1) are assumed. The magnitude of the design ground motion in terms of reference peak ground acceleration is ag= 0.375 g, while the duration of the signal is t= 20 s. Results have been obtained for three different synthetic accelerograms of these characteristics, being the conclusions of the study completely equivalent for all of them. For this reason, and for the sake of clarity and intelligibility of the plots, results are presented for one specific accelerogram. Stability and convergence analysis of the meshes have been performed in order to ensure the accuracy of the obtained results. 3.2 Comparison results Now, in order to validate the methodology described in section 2when used to compute internal forces in piles, envelopes of shear forces and bending moments along vertical piles are presented in this section against results obtained by the authors using a standard Beam-on-Dynamic-WinklerFoundation formulation (BDWF). This BDWF, developed in the line of the works by Kavvadas and Gazetas [20] or Mylonakis [21], assumes a vertical pile embedded in a homogeneous, isotropic and linearly elastic half-space, with perfect bonding between soil and pile. Pertinent values 3 of springs and dashpots are needed in its definition. References with suggestions about such values are abundant. In this respect, two groups could be made: a) references providing values tuned by comparison against results from more rigorous models (see for instance Kavvadas and Gazetas [20] or Makris [22]; or b) references providing values obtained directly from theoretical models as, for instance, the Novak-Baranov plane-strain elastodynamic theory. This is the case of the expressions proposed by Novak et al. [23]. An interesting revision regarding this topic can be found in Mylonakis [21] and Mylonakis [24]. In this paper, the coefficients proposed by Novak et al. [23] are used. Figure 2presents the envelopes of bending moments and shear forces along the vertical single pile defined above. The results obtained from the BEM–FEM coupling model are shown together with the results from the BDWF formulation described above. The dots on the curves corresponding to the BEM–FEM model represent the nodal values actually computed. The results are presented for the three pile-soil Young’s modulus ratios that where defined above and that are going to be used in the next section. In general, the results obtained by both methodologies are in good agreement with each other. The envelopes of bending moments are exceptionally close taking into account the inherent differences between the two models. Along the pile, discrepancies both in bending moments and shear forces tend to grow for increasing contrast between soil and pile Young’s moduli, with the BEM–FEM methodology yielding sligthly lower values in most instances, with the exception of the bending moments around the tip, where the BEM–FEM formulation yields larger values. In terms of bending moments for all stiffness ratios, and shear forces for Ep/Es= 50, the most significant differences appear mostly near the tip of the pile, where the three–dimensional nature of the problem is inherently considered by the BEM-FEM model, which will also be able to include pile inclination in a rigorous manner. 3.3 Kinematic internal forces of deep foundations with inclined piles This section presents results that allow to evaluate the influence of rake angle on the kinematic internal forces developed along the inclined piles subjected to vertically-incident S waves. The configurations under study are those defined in section 3.1. Figures 3,6and 7present envelopes of kinematic bending moments, shear forces and axial forces, respectively, corresponding to a single inclined pile and a 2 ×2 pile groups with battered elements. First, second and third columns in figures 3and 6correspond to pile-soil stiffness ratios of Ep/Es= 50, 100 and 500, respectively, while the first row corresponds to the single pile configuration and the second row corresponds to the 2 ×2 pile group. In case of figure 7, first and second columns correspond to axial compression and traction forces, respectively, for a single inclined pile, while third and fourth columns show the same information for a 2 ×2 pile group. First, second and third rows in figure 7represent results corresponding to pile-soil stiffness ratios of Ep/Es= 50, 100 and 500, respectively. Figure 3illustrates how the kinematic bending moments along the single pile decrease monotonically for increasing rake angles, irrespective of the pile-soil stiffness ratio. In the case of the pile group, the same trend is observed along the deepest two thirds of the foundations, while a completely different behaviour is observed in the shallowest part of the piles. There, the bending moments increase significantly at the head of the piles in the group when the piles are inclined. These differences between the maximum bending moments corresponding to the two deep foundations under study are due to the different rotational behaviours of single pile and pile group with inclined elements. As shown by Medina et al. [17], rake angle has very little effect on the rotational kinematic interaction factors of a single pile, while its influence on the rotational kinematic interaction factors of a pile group is very significant in the configuration studied in this work (direction of shaking parallel to the vertical plane containing the inclined piles). In this case, the magnitude of the kinematic rotation at the pile cap increases significantly when inclining the piles, and becomes out of phase with the horizontal free-field ground motion. This evolution in the kinematic rotation, together with the kinematic restriction imposed by the rigid cap, induces the increase in bending moments at the pile head observed in figure 3for the pile 4 −12 −10 −8 −6 −4 −2 0 2 4 6 8 Depth (m), bending moments M (kN⋅m) −12 −10 −8 −6 −4 −2 0 0 1 2 3 4 Depth (m), shear forces V (kN) BDWF BEM−FEM 0 5 10 15 M (kN⋅m) 0 2 4 6 8 V (kN) 0 10 20 30 40 50 M (kN⋅m) 0 5 10 15 20 V (kN) Ep/Es= 500Ep/Es= 100Ep/Es= 50 Figure 2: Envelopes of shear forces and kinematic bending moments corresponding to a single vertical pile in a homogeneous half-space submitted to vertically-incident shear waves. Comparison results. group. Remember that rotation is prevented at the head of the single pile configuration studied here. In order to gain a better insight into the magnitudes of the amplification factors affecting the envelopes of kinematic bending moments when inclining the piles, figure 4presents the ratio between the maximum kinematic moment in the inclined pile to the maximum kinematic moment in the vertical pile, both at the same depth. Envelopes of bending moments are multiplied by factors below unity for the deepest parts of the piles in the group, and for all depths in the single pile, reaching reduction factors of up to 0.25 for 30 ° . At the head of piles in a group, on the contrary, bending moments are amplified by factors of up to 2.6 in some cases. Both figures 3and 4show that, at pile heads, kinematic bending moments decrease monotonically with the rake angle in the case of single piles. In the case of piles in a group, on the contrary, the evolution is not monotonic. Such evolutions can be seen in figure 5that, in order to analyse these trends, presents the maximum bending moments at the pile heads divided by the pile-soil stiffness ratio as a function of the batter angle. The differences between single pile and pile group configurations are well illustrated in this figure. The monotonic decrease in the 5 −12 −10 −8 −6 −4 −2 0 5 10 15 Depth (m), 2x2 pile group M (kN⋅m) −12 −10 −8 −6 −4 −2 0 Depth (m), single pile θ= 0° θ=10° θ=20° θ=30° 0 10 20 30 M (kN⋅m) 0 20 40 60 80 100 M (kN⋅m) Ep/Es= 500Ep/Es= 100Ep/Es= 50 Figure 3: Envelopes of kinematic bending moments corresponding to a single inclined pile and a 2 ×2 pile group with battered elements submitted to vertically-incident SH waves. Pile-soil stiffness ratios considered: Ep/Es= 50, 100 and 500 6 −12 −10 −8 −6 −4 −2 0 0.5 1 1.5 2 2.5 Depth (m), 2x2 pile group M/Mθ=0° −12 −10 −8 −6 −4 −2 0 Depth (m), simple pile θ=10° θ=20° θ=30° 0.5 1 1.5 2 2.5 M/Mθ=0° 0.5 1 1.5 2 2.5 3 M/Mθ=0° Ep/Es= 500Ep/Es= 100Ep/Es= 50 Figure 4: Ratios relating the envelopes of kinematic bending moments of inclined piles to those of vertical piles. Single piles and 2 ×2 pile groups submitted to vertically-incident SH waves. 7 0 50 100 150 200 250 300 0.0 5.0 10.0 15.0 20.0 25.0 Mmax/(Ep/Es) (N m) θ (°) Ep/Es=50 Ep/Es=100 Ep/Es=500 0.0 5.0 10.0 15.0 20.0 25.0 30.0 θ (°) Figure 5: Maximum kinematic bending moment at pile head as a function of rake angle. Single piles and 2 ×2 pile groups submitted to vertically-incident SH waves. case of single piles is completely different from the significant increase in the case of piles in a group, that show a maximum moment for angles around 20 ° . The largest variations are observed for the smallest soil–pile stiffness ratios, that is, for the stiffer soil. Figure 6shows that kinematic shear forces also decrease monotonically for increasing rake angles along the deepest parts of the piles. Along the shallowest parts, on the contrary, shear forces increase significantly with the batter angle in most cases both for single piles and for pile groups, although the increases are much more important in the case of pile groups. In such configuration, as expected, shear forces at pile heads go from zero to very significant values but, in most cases, maximum values do not occur at pile heads but at depths of a few diameters. At such depths, the magnitude of the increase in shear forces observed when inclining the piles decreases when the pile–soil stiffness ratio increases, in such a way that, for Ep/Es= 500, maximum shear forces correspond to the vertical case at all depths. Finally, figure 7presents the evolution of the axial forces along the piles. As expected, both compressive and tensile kinematic axial forces increase monotonically with the rake angle at all depths. The kinematic axial forces are null in the case of a single vertical pile. In the case of the pile group, they are not identically zero due to the kinematic restriction imposed by the pile cap. In all cases, the increase in the kinematic axial forces with the inclination of the piles is very significant, with maxima ocurring at depths depending on the pile–soil stiffness ratio and rake angle but always deeper than L/2. 4 Conclusions The influence of pile rake angle on the kinematic internal forces along deep foundations with inclined elements is investigated in this paper making use of a time–harmonic linear–elastic boundary element – finite element coupling formulation in which the BEM is used to model the soil as a homogeneous viscoelastic half-space and the FEM to model the inclined piles as Euler-Bernoulli beams. Envelopes of maximum kinematic bending moments, shear forces and axial load, corresponding to different rake angles and soil properties, are presented for single inclined piles and for 2 ×2 pile groups with batter elements and subjected to vertically-incident SH waves. Fixed pile-cap connection is considered in the study. In the case of the single pile, rotation is assumed to be prevented at the pile head. The three main conclusions drawn from the analysis are: 8 −12 −10 −8 −6 −4 −2 0 1 2 3 4 5 6 Depth (m), 2x2 pile group V (kN) −12 −10 −8 −6 −4 −2 0 Depth (m), single pile θ= 0° θ=10° θ=20° θ=30° 0 2 4 6 8 10 V (kN) 0 5 10 15 20 25 V (kN) Ep/Es= 500 Ep/Es= 100Ep/Es= 50 Figure 6: Envelopes of kinematic shear forces corresponding to a single inclined pile and a 2 ×2 pile group with battered elements submitted to vertically-incident SH waves. Pile-soil stiffness ratios considered: Ep/Es= 50, 100 and 500 9