Modeling the Behavior of Axially and Laterally Loaded Pile with a Contact Model

Transcription

1 Modeling the Behavior of Axially and Laterally Loaded Pile with a Contact Model Zakia Khelifi Department of Civil Engineering, University AbouBakr Belkaid, Algeria zakia_khelifi@yahoo.fr Abdelmadjid Berga Department of Civil Engineering, University of Bechar, Algeria bergaabdelmadjid@yahoo.fr Nazihe Terfaya Department of Mechanical Engineering, University of Bechar, Algeria t_nazihe@yahoo.fr ABSTRACT This work treats the behavior of a single pile subjected to two types of loading: axial and lateral, with taking into account the interaction between the soil and the pile. The zone of soilpile interface is modeled by a simple model of interaction; based on the Coulomb friction contact law. The numerical analysis is carried out in 2D using a finite element program. Comparisons are presented between modeled pile behavior and that predicted from experimental, analytical and analysis results, a reasonable good agreement is obtained between them. KEYWORDS: Soil-structure interaction; Piles; Interfaces; Frictional contact; Finite elements. INTRODUCTION In geotechnical engineering the loads are transferred between the structures and the soils mainly through surfaces of contact called "interfaces". These surfaces of contact are the siege of an important localization of deformation, play the part of a kinematics, a material and a discontinuity behavior, characterized by a high deformation gradients. Such loading conditions are traditionally simplified, either like prescribed loads (supposing a total flexibility of the structure), or like prescribed displacements (supposing the total rigidity of the structure). These simplifications lead to inaccurate previsions of the real behavior and are not possible only when surfaces in contact are a priori known. Contacts between the soils and the structures in several cases, involve a frictional sliding as well as the separation and the closing of surface during the

2 Vol. 16 [2011], Bund. N 1240 procedure of loading, and cannot be modeled by simple prescribed boundary conditions [1]. They can considerably affect the total load capacity of the structure. Therefore, the behavior of the piles under axial or lateral loadings is a typical example of problem involving an interface, because of the three-dimensional nature of the phenomenon and of its dependence of several key parameters, such as the behavior of pile, the soil, and the nature of the load and the effects of the interface. The traditional methods of calculation of a pile founded on empirical correlations lead to the determination of the limit bearing capacity [2],[3]. However, numerical calculation by the finite element method or the border elements offers an excellent alternative to study the interaction soilpile and the response of axially or laterally loaded piles. Among the approaches of the finite element method used to model the behavior of the pile, we quote the axisymmetric, threedimensional approach (a pile under axial and lateral load) and in plane deformation (a pile under lateral load) [4]. Numerical analysis by discretization can then bring a precious help to dimension the structure and to approach the mechanisms of deformation. The modeling of a single pile problem, from the simple aspect to the first access, is in fact complicated due to the modes and effects of the installation of the pile, and also, to the phenomena of interaction between the shaft pile and the soil. In this article, modeling of a drilled single pile axially and laterally loaded at the head is presented. The calculation has been carried out using the numerical program Plaxis 2D, based on the finite element method. The soil-pile interaction is modeled by the Coulomb friction contact law. PRESENTATION OF THE INTERFACE MODEL The behavior of the interface between the soil and the pile is modeled by the Coulomb friction contact model; because of its effectiveness and its apparent facility of application. This law was adapted by several authors, e.g., De Gennaro et al.,[2], Frank et al.,[5], Estephan [6]; Sheng et al., [7],[8]; Said [9]; Fischer et al., [10]. The theoretical description of the soil-pile interface behavior is based on the following experimental observations [5]: - Closing of the interface in normal compression - Separation for low normal tensile stresses - Slip beyond a shear stress (friction limit) From mathematical point of view, this is translated on the state of stresses to the interface, by the conditions of separation and friction - slip (contact and friction laws). Contact conditions between two surfaces are governed by kinematic constraints in the normal and tangential directions. The normal stress at contact is either zero, when there is a gap between the pile and the soil, or compressive when the pile is in contact with the soil. This constraint can be described as [11] [12]:

3 Vol. 16 [2011], Bund. N Condition of no contact (separation): σ n = 0 when x n > 0 (1) σ = 0 (Signorini condition) (2) n x n - Condition of unilateral contact: σ > 0 when 0 n x = (3) where x n is the gap, and σ n is the normal stress at contact. The compression is taken positive. The gap x n is never negative because the interpenetration of materials is not allowed. The frictional sliding at the pile soil interface is modeled by the Coulomb friction contact law. This law connects the normal and tangential components of the contact constraints, by an intermediary coefficient of friction, supposed constant. The Coulomb laws can be expressed as: - Adherent contact: - Slipping contact: n int r n τ < σ tan ϕ + (4) n int r C int r τ = σ tan ϕ + (5) C int r where q s = σ n tanφ intr + c intr is unit side friction limited with τ the tangential stress at contact; φ intr the angle of friction to the interface; c intr the cohesion of interface; μ = tanφ intr the coefficient of friction of the interface and σ n indicates the positive normal constraint in compression. The behavior of the interface between the pile and the soil is often represented by the Coulomb friction law, if this criterion is reached; the soil and the pile slips one compared to the other and only normal displacements remain continuous [4]. FINITE ELEMENT ANALYSIS OF SOIL-PILE SYSTEM A two dimensional finite element program Plaxis 2D was used in this work, in order to model a single pile drilled under axial and lateral loadings. In this program, modeling was carried out in plane and axisymmetric conditions with two degrees of freedom of translation per node. The soil was modeled by triangular elements with 6 or 15 nodes according to the example treated below; with an elastoplastic law behavior obeying the Mohr-Coulomb failure criterion. The pile was modeled as an elastic linear and isotropic material.

4 Vol. 16 [2011], Bund. N 1242 VALIDATION OF THE NUMERICAL MODEL First, the case of an axially loaded single pile will be discussed, for a better understanding of the interaction behavior between the pile and the soil. Then, the case of a laterally loaded single pile will be presented and analyzed [13]. Case I. Behavior of a Drilled Pile Under Axial Loading In this first example, we propose to analyze using finite elements method the behavior of a drilled pile under axial loading, based on the example of Frank et al., [14]. The results are then compared with those of Frank et al., [14]. The characteristics of the soil, the pile and the interface are given in Table 1. The boundary conditions are shown in Fig.1. Figure 1: Geometrical Characteristics and boundary conditions of the system Table 1: Constitutive parameters of materials Material Parameter Value Soil Young s Modulus E (MPa) 20 Poisson's ratio υ 0.3 Cohesion c 0 Friction angle φ( ) 35 Dilatancy angle ψ( ) 35 At-rest earth pressure coefficient (k 0 ) 0.43 Pile Young s Modulus E (MPa) Poisson's ratio υ 0.3 Interface Cohesion c intr 0 Friction angle φ intr 25

5 Vol. 16 [2011], Bund. N 1243 Our calculations allowed us to study the influence of the following parameters: - Young's modulus of the interface (E intr ) - Interface strength (R intr ) - Dilatancy angle of the interface (ψ intr ) - Thickness interface - Discretization (mesh) Geometry, Finite Element Mesh and Boundary Conditions The mesh used in our modeling is shown in fig.2. The mesh is relatively refined in the neighborhood of the interfaces, along the pile shaft and at the pile tip, where strong gradients risk is appearing. The pile-soil system was modeled in axisymmetric condition with 6 node triangular elements. Figure 2: (a) geometry; (b) mesh Initial Constraints In the modeling, the initial conditions determine the final solution. In the Plaxis program, this stage is carried out initially by the activation of the actual weight of the soil and the initial radial constraint (σ r = σ θ = k 0 γz), which define the initial state of stress (see fig. 3). In this example the installation effects are not taken into account.

6 Vol. 16 [2011], Bund. N 1244 Figure 3: Initial Constraints Calculation Phase Fig.4 shows the deformation of the single pile in the axisymmetric condition. Figure 4: Generation of 2D mesh for the single pile in axisymmetric condition From the modeling results of the compression loading test of the pile (dilating interface and elastoplastic soil), we can analyze the evolution of the calculated displacement field around the

7 Vol. 16 [2011], Bund. N 1245 pile (fig.5a). The displacement field is concentrated around the pile with a displacement of the pile head (approximately m) and under the pile tip. Fig.5b shows a formation of a bulb around the pile, characterized by vertical displacements directed downwards. This bulb is formed gradually around the inclusion during embedding; it is almost completely localized around the pile, in the interface layer, and below the pile base. Fig.5c illustrates an extension of horizontal displacements (negative) towards the surface of the pile; as well as an area of expansion of horizontal displacements (positive) below the pile tip and near the side surface of the pile. Figure 5: (a) Total displacements; (b) Vertical displacements; (c) Horizontal displacements Fig. 6 shows the stresses zones which concentrate below the pile tip (approximately 170 kn/m 2 for the pile is fixed).

8 Vol. 16 [2011], Bund. N 1246 Figure 6: Total stresses The shear and volumetric strains zones are displayed in Fig.7a and 7b respectively. The concentration of the deformations for the two types is below the pile tip (volumetric strains %, shear strains %). Figure 7: (a) Shear strains; (b) Volumetric strains

9 Vol. 16 [2011], Bund. N 1247 Influence of Young s Modulus of the Interface(E intr ) To investigate the influence of the Young's modulus of the interface on the behavior of pilesoil system, we took the following various values: - The stiffness modulus of the interface equal to that of the soil (E intr = E soil ) (Fig.8a). - The stiffness modulus of the interface equal to that of the pile (E intr = E pile ) (fig.8b). - The stiffness modulus of the interface equal to the average between the modulus of the soil and that of the pile (E intr = (E soil + E pile ) / 2) (Fig.8c). Fig.8 shows the evolution of the bearing capacity of the pile according to the variation of the stiffness modulus of the interface.

10 Vol. 16 [2011], Bund. N 1248 Figure 8: Influence of the Young's modulus of the interface on the pile-soil system: (a) E intr = E soil ; (b) E intr = E pile ; (c) E intr = (E soil + E pile ) / 2 From Fig.8 b and 8 c, we observe that the behavior of the pile-soil system is elastic considering the high value of E intr = E pile = MPa and E intr = E moy = MPa. On the other hand, the curve 8.a, has a better result for E intr = E soil = 20MPa. Thus, we conclude that the Young's modulus of the interface must be close to that of the soil. Influence of the Interface Strength In order to study the influence of the interface on the behavior of pile-soil system, we modelled the axially loaded pile, by introducing an interface between the pile and the soil with various strength values, varying from 0.5 to 1 (the case of associated φ intr = ψ intr (Fig.9) and no associated potential φ intr ψ intr (Fig.10)).

11 Vol. 16 [2011], Bund. N 1249 Figure 9: Influence of interface strength R intr (associated case) on pile-soil system The numerical curve load-displacement in Fig.9, seems to underestimate the bearing capacity of the pile Q t for values of R intr between 0.5 and 0.7, and to over-estimate it for R intr =1. Although, values of R intr between 0.8 and 0.9 are in good agreement with the results of Frank.R et al., [14]. For no associated case (φ intr ψ intr ), the curves of load-displacement are underestimated for values of R intr between 0.5 and 0.7 (Fig.10). But with values of R intr between (Fig.10), the curves of load-displacement are in good agreement. The case of R intr =1 is particularly remarkable, since the curve well approached the results of Frank et al. [14]. Figure 10: Influence of interface strength R intr (non-associated case) on pile-soil system

12 Vol. 16 [2011], Bund. N 1250 According to the results obtained and while comparing with those of Frank.R et al., [Ref, we can ascertain that the strength of interface influences the behavior of the pile-soil system, but in addition, we observed that the dilatancy of the interface (associated and non-associated case) practically, does not have an effect on the load-displacement curve, except for the case when R intr = 1. Influence of the Interface Thickness In Plaxis program, a "virtual thickness" is assigned with each interface. It is an imaginary dimension used to define the material properties of the interface. This virtual thickness is calculated as, the virtual thickness factor times the average element size. We suggest to study the influence of this parameter on the behavior of the pile-soil system. Fig.11 shows the influence of the thickness interface on the load-displacement curve. One can see that more the thickness is important more the elastic strain is. For a thickness equals to 1, the displacement of the pile reaches a value of 30 mm. Figure 11: Influence of the interface thickness Influence of the Discretization (Mesh) In this case we will present the effect of the mesh on the behavior of the pile-soil system. We used a coarse and fine mesh for modeling. Fig.12 shows a very weak influence of the used mesh (coarse or fine) on the calculation results.

13 Vol. 16 [2011], Bund. N 1251 Figure 12: Influence of the Mesh Diagram of side Friction It is useful to represent the diagram of the shear stresses at the interface, in comparison with those calculated analytically by the following formula: τ ( z) = C + k Hσ V ( z) = k σ ( z) tgϕ = k intr tg in tr (with Cintr 0) H H V γztgϕ int r int r ϕ = (6) Figure 13: Diagram of the shear stresses at the interface

14 Vol. 16 [2011], Bund. N 1252 From Fig.13, we can observe that the shear stress at the interface follows the Coulomb friction law. The discrepancy in the neighborhood of the lower extremity of the pile, is due to the influence of the pile tip. Case II. Study of a Laterally Loaded Pile at the Head Bouafia et al., [15] have presented an experimental study of a laterally loaded pile at the head. The tests were carried out on centrifuged models in L.C.P.C (Laboratoire Central des Ponts et Chaussés, France). The results obtained in their experiments, will be compared with those obtained by the program Plaxis. The pile is drilled in a sand soil, with a diameter of 0.5 m and a driving depth D = 5 m as shown in Fig.14, with an applied lateral load Q = 900 kn, and an eccentricity load e = 2.25 m. The characteristics of the pile and the sand soil are given in the Table 2. Figure 14: Pile-Soil System Table 2: Constitutive parameters of materials Material Parameter Value Sand Poisson's ratio υ 0.3 Cohesion c 0 Friction angle φ( ) 42 Dilatancy angle ψ( ) 42 At-rest earth pressure coefficient (k 0 ) Pile Flexural rigidity EI (MPa) 628 Poisson's ratio υ 0.3

15 Vol. 16 [2011], Bund. N 1253 Geometry, Finite Element Mesh and Boundary Conditions For the mesh, a model of finite element in plane strain condition was used, with 6 node triangular elements for the whole pile-soil. The geometry and the mesh of the pile-soil system are given in the Fig.15. In order to realize the mesh of the pile, we have to add part of virtual sol with a null strength along the part of the eccentricity (the higher layer in red). Figure 15: Geometry and Mesh Calculation Phase Fig.16 shows the deformed mesh of the pile-soil system, where we can observe the bending of the pile and some uplift of soil due to the deflection of the pile. Figure 16: Deformed Mesh

16 Vol. 16 [2011], Bund. N 1254 Study of the Influence of Parameters The numerical solution depends on several parameters such as: - Young' modulus of the sand massive E soil (unknown Young' modulus) and of the interface; - Friction angle of the interface (φ intr ). - Influence of the Young's modulus of the sand massive and the interface In this case, we propose to study the influence of the Young's modulus of the soil, since it is an unknown parameter. For that, we take E soil = E intr for various values as shown in Fig.17. Figure 17: Influence of the Young's modulus Fig.17 shows that for a value of E soil = E intr = 9MPa, the shape of the curve obtained by Plaxis is nearer to that of Bouafia et al., on the elastic-plastic part, and digress (spreads) in the neighbourhood of the ultimate state. The same remark is reported for the bending moment profiles represented in Fig.18. In the elastic part, the gaits for various values of E soil = E intr are in accordance with those of Bouafia et al [15], and spread in the neighbourhood of the ultimate state.

17 Vol. 16 [2011], Bund. N 1255 Figure 18: Comparison of the bending moment profile for different values of E soil = E intr - Influence of the friction angle of the interface To show the influence of the interface on the behavior of the soil-pile system, we studied the effect of the variation of the friction angle of the interface φ intr. In Fig.19, we observe that, for a value of φ intr =17, the numerical curve load-displacement is practically overlapped with that obtained in experiment [16], for a maximum head displacement reaching the value of 2400 mm. Figure.20, presents the bending moment profiles, for various values of φ intr. We can observe that the bending moment decreases with decreasing the friction angle of interface (φ intr ). This is due to the slip which is favored compared to the bending of the pile. On the other hand, when φ intr is high enough (φ intr equal to 25 and 30 ), it is the bending of the pile which is most favored.

18 Vol. 16 [2011], Bund. N 1256 Figure 19: Influence of the friction angle of the interface CONCLUSIONS The purpose of this work was to bring a contribution to the modeling of the behavior of the pile-soil system which conditions in a significant way the behavior of many works in civil engineering. It comes out from this study that the numerical modeling of the works in geotechnics requires a rather precise knowledge of the behavior of the soil and of the contact zones, between the solids of different nature. Thus, three various nonlinearities must in theory be considered related respectively to the soil, the pile and their interaction. Our research presents numerical and analytical results obtained in the nonlinear case, in plane and axisymmetric deformation of a laterally and an axially loaded pile. In the case of an axially loaded pile, in comparison with the numerical results obtained by Plaxis and those of Frank. R et al. [14], we found out that the modeling of the pile-soil system is strongly influenced by the resistance of the interface. By varying the Young's modulus of the interface, we concluded that, it must be near to that of the soil. Therefore, the thickness of the interface has a big influence on the behavior of the pile-soil system, which it must be taken into account in the modeling of the system. The effect of the mesh does not have an important influence on the modeling of the behavior of the pile-soil system. The discrepancy in the diagrams of the side friction obtained by Plaxis and that calculated analytically, was due to the effect of the pile tip on the behavior of the system. According to the example of Bouafia et al., we noticed that the numerical solution depends on several parameters such as the Young's modulus of the sand soil (E soil ), of the interface (E intr ) and the friction angle of the interface (φ intr ). We observed that for values of E soil = E intr the numerical results are satisfactory. Moreover, the friction angle of interface has a big influence on the modeling.

19 Vol. 16 [2011], Bund. N 1257 Modeling by finite elements offers a better alternative method to study the pile-soil interaction and the response of the laterally and axially loaded pile, as it was shown with the results obtained by Plaxis. The behavior of the piles under lateral and axial loads is a complex problem of soil-pile interaction, because of the three-dimensional nature of the phenomenon and its dependence on several parameters. In spite of important progress were realized in the field of modeling of this problem, but several points still remain suspense suspended, such as the laws of behavior of the interface between an adherent soil and a pile, the laws of behavior of the natural soils, and the initial state of the constraints in the soil before the penetration of the pile. We can also observe that the empirical methods used for calculation of the piles under lateral and axial loads are inoperative. Indeed, the experimental studies carried out under good conditions allow studying these complex problems. REFERENCES 1. Daichao Sheng., Peter Wriggers., Scott W Sloan, "Application of frictional contact in geotechnical engineering", International Journal of Geomechancis, 2007, Vol 7, No 3. ASCE. 2. Vincenzo De Genaro., Roger Frank,"Modélisation de l'interaction sol-pieu par la méthode des éléments finis", Bulletin des laboratoires des ponts et chaussées, Juillet-Aout- Septembre 2005, pp Roger Frank, "Calcul des fondations superficielles et profondes", Techniques de l Ingénieur et Presses de l ENPC, Paris, Philippe Mestat., Michel Prat, "Ouvrages en interaction. Emploie des éléments finis en Génie Civil", AFPC, Hermes Science Publications, Roger Frank., Philippe Mestat, "Aspects expérimentaux et numériques du frottement latéral des pieux", Editions Scientifiques et Médicales Elsevier, 2000, pp Roger Estephan, "Contributions aux méthodes de calcul des groupes et des réseaux de micropieux", Thèse de doctorat, Ecole Nationale des Ponts et Chaussées (CERMES), France, Daichao Sheng., K. Dieter Eigenbrod., Peter Wriggers, "Finite element analysis of piles installation using large-slip frictional contact", Computers and Geotechnics 32, 2005, pp Daichao Sheng., Peter Wriggers., Scott W Sloan, "Improved numerical algorithms for frictional contact in pile penetration analysis", Computers and Geotechnics 33, 2006, pp Iman Said, "Comportement des interfaces et modélisation des pieux sous charges axiales", Thèse de doctorat, Ecole nationale des ponts et chaussées, France, Kathrin A. Fischer., Daichao Sheng., Andrew J. Abbo, "Modeling of pile installation using contact mechanics and quadratic elements", Computers and Geotechnics 34, 2007, pp Zhi-Qiang Feng, "Contribution à la modélisation des problèmes non linéaires: contact, plasticité et endommagement", Thèse de Doctorat, UTC, France, 1991.

20 Vol. 16 [2011], Bund. N Nazihe Terfaya, "Contribution à la modélisation des problèmes de contact et de frottement bidimensionnel. Génération de maillage et programmation orientée objet de la méthode des éléments finis", Mémoire de Magister, Université de Béchar, Algérie, Zakia Khelifi, "Modélisation du comportement d'un pieu isolé sous charges latérales par un modèle de contact", mémoire de magistère, université de Béchar, Roger Frank., Athanasios Barbas, "Contribution à l'utilisation de la méthode des éléments finis en mécanique des sols dans le domaine de l'élastoplasticité", Rapport de recherche N 116, Laboratoire des Ponts et Chaussées, 1982, 132p. 15. Ali Bouafia., Jacques Garnier., Daniel Levacher, "Comportement d'un pieu isolé chargé latéralement dans le sable", Colloque International, Fondations Profondes, Paris Mars ejge