Modelling of non linear soil-structure interface behaviour using M. Boulon & P. G arnica

Transcription

1 Modelling of non linear soil-structure interface behaviour using M. Boulon & P. G arnica BEM Cecfecc P, France Abstract The soil-structure interaction problem is considered as a nonlinear contact problem between elastic bodies. The boundary element method is used for the numerical treatment and the non-linearities at the interface are modelized by means of a non-linear incrementally interface law. As a validation of the model for applications to pile problems numerical simulations of direct shear tests are presented and discussed. INTRODUCTION In the present practice of numerical simulation concerning geotechnical problems, the moderation of contact between solids is often necessary. If the non-linearities are localised in the thin transition layer (interface) between the structure and the soil then the soil-structure interaction problem can be considered as a non-linear contact problem between elastic bodies. The contact problem is formulated in terms of two coupled boundary integral equations (Andersson 982, Selvadurai 987), these equations are linked together using the contact conditions for tractions and displacements within the interface. We utilize a sub-structuration technique in order to obtain a reduced non-linear system of equations containing only the interface unknowns (tractions and displacements), which is solved by the iterative Newton-Raphson methods. The non-linear interface behaviour is modelized by a constitutive directionnally dependent interface model (Boulon & Plytas 986), based on a constitutive interpolation (Boulon & G arnica 990), allowing to model all principal features of soil-structure

2 540 Boundary Element Method XVI interface behaviour (Boulon 989), i.e., initial contractancy, dilatancy and also a damage effect produced by grain crushing of the granular material. In this paper, we present numerical simulations of a soil-structure direct shear tests (paths) as a validation of the model for application to pile problems (Garnica 993).. NUMERICAL TREATMENT Let consider two elastic solids O% and % on?r? in contact bounded by surfaces FI and F2, and let be F, their common contact surface. For each solid k (k =,2), the boundary element discretization of the classical integral equations for istotropic linear elastic bodies can be reduced to a matrix equation of the form j U* = g* * () Where H and Q_ are the boundary element influence coefficient matrices; U? and P* are the nodal boundary displacements and tractions respectively. Using equation () we can write for each solid k the matrix relation: W &"! r y : j m* g i ^ ) where the subscript s corresponds to the interface unknowns on the contact surface F,. If all boundary conditions are, for example, of the Dirichlet type (U^ imposed), the system (2) becomes: G a f P«* } r JT * i -fl? \ lhi>]^'> a Using the sub-structuration technique we can transform these equations in a reduced system only containing the interface unknowns (tractions P and displacements J) and we can write: where we note: *X* = C* (4) D =

3 Boundary Element Method XVI 54 Equations (4) are the starting point for the boundary element modelling of non-linear interface behaviour. 2. CONTACT WITH INTERFACE LAW From an equilibrium state, each loading step (say A...) will contain a number of increments (secant approach) for the local interface law integration. For a loading step the equation (4) becomes: and we are assuming equilibrium of the forces at the interface and compatible relative displacements between the solids from a constitutive point of view, that is: (5) We write now the interface law for second solid: Af (7) where Q is the transformation matrix from global to local axis, dg is a pseudo-elasticity matrix used for traction initialisation for the first equilibrium iteration and At is the correction term due to non-linearities. Let be T the banded matrix allowing for the contact conditions (6), the global system for the study of contact problem with interface law will finally be: ' \ (8) A* j Having a constant interpolation, for three dimensional cases, the square system (8) will contain nine equations by node. If At = 0 then system (8) is linear, otherwise (At ^ 0) we have a non-linear system that will be solved using the modified Newton-Raphson method (dg will be always the same). Stability of solution is directly dependent on the choice of dg which may lead to a very slow converging solution or even to the diverging case. 3. NON-LINEAR INTERFACE BEHAVIOUR The general principles of this constitutive equation were presented previously (Boulon 989). For a non viscous interface material the tangent constitutive relation (local axis) can be written as:

4 542 Boundary Element Method XVI where i and [U] are respectively the stress velocity and the relative velocity vectors, and d. is the tangent constitutive matrix: d = d(l[u], state) (0) For obtaining an incremental response for any incremental loading the main idea is to use a constitutive interpolation (Boulon & Ply t as 986). In the other word, if we know from laboratory tests (basic paths) the responses (r%) to N shear loadings (s,), then we can obtain, theoritically, a response (r) for any loading (s) using the some well defined interpolation functions W, (Boulon & G arnica 990): = The constitutive equation (9) is deduced from the previous interpolation by linearization of the response along a constitutive (loading) direction. This linearization can be performed by an analytical way using the Euler theorem related to the homogeneous functions, and then equation (9) can be expli cited as: dr dr d[w] d[u] where r and d^ are respectively the shear and normal stress velocities, [w] and [u] are the tangential and normal relative displacements velocities between the two boundaries of the interface. These components have an analytical expression derived from the interpolation functions W{ and from the expression of the basic paths. For integration of equation (9) we use an explicit scheme at first order versus time with a constant step size. The local integration is performed along the paths at prescribed relative displacement. W (2) 4. NUMERICAL SIMULATION OF A DIRECT SHEAR TEST This numerical test is the pulling out of a thin plate previously loaded over a granular material (Figure la), example. The boundary conditions correspond to a globally constant volume direct shear test. The stress state initialisation is performed by application of a displacement AC corresponding to 300 kpa of normal stress. The mesh system consists twenty-four plane traingular elements for each one of solids (Figure Ib). The mechanical parameters which used are: inclusion: G = 6.0^ kpa, v = 0.33 soil: G = 2.0* kpa, v = 0.33

5 Boundary Element Method XVI 543 interface: ^ - 6.0* kpam, t, = 5.0^ kpam, t, = 5.0^ kpam and INTERFACE NODES Figure. Constant normal stress test: geometry, boundary conditions and raesn. It is further assumed that over each element of the surface, the tractions and displacements are constant. Then the sub-structurated system contains thirty-six equations. The stress state initialisation is performed only by the application of the normal stres and taking zero tangential stress at &e interface nodes. The displacement loading step (Ay) is Q-* mm and the computation is going until 20 mm. The evolution of normal stress (3) shows all the main features of soilstructure interface behaviour as observed in laboratory (Boulon 989), i.e., initial contractancy, diiatancy and a new contractancy due to grain crushing because of hi%h normal stress. This damage phenomenon is considered in Che interface law as a function of the interface energy level. We can note that since the prescribed boundary conditions are globally corresponding to a constant volume path, the shear path at the interface is in fact at prescribed normal stiffness. This normal stiffness is directly depending on rhe contrast of shear modulus between the soil and the inclusion (figure 4).

6 544 Boundary Element Method XVI 800. NODE 2 NODE 22 NODE 23 NODE TANGENTIAL RELATIVE DISPLACEMENT (mm) Figure 2: Local evolution of normal stress during shearing. o o_ LJ Of NODE 2 NODE 22 NODE 23 NODE 24 w NORMAL STRESS (kpa) Figure 3: Local shear paths.

7 Boundary Element Method XVI 545 a) 2500 ^^ D CL _y ^^ () 500 ' () Ld C \ 000 (n _j<2 500 CE Oz! I : : -! ^ ^, * ^-^" i Gsoil = 2E4 kpa Gsoil = 5E4 kpa Gsoil =!E5kPa Gsoil = 2E6 kpa NORMAL RELATIVE DISPLACEMENT (mm) b) 2000 O CL (^ 500 CO UJ () ^ 500 O ^^* % **^"^ *. '*' " ^ v^\\ "'^ y^ Xv^ TANGENTIAL RELATIVE DISPLACEMENT (mm) Gsoil = 2E4 kpa Gsoil = 5E4 kpa Gsoil =!E5kPa Gsoil = 2E6 kpa Figure 4: Constant volume test: Influence of the shear modulus of soil (Gsoil).

8 546 Boundary Element Method XVI CONCLUSION The numerical tests studied in this paper show that the boundary element method can be successfully applied to examine non-linear phenomena at soil-structutre (geomaterial) interfaces, mainly if solids are linear elastic. But the numerical modelling of soil-structure interaction in the static range of load is highly dependent on the type of constitutive law used for simulating the contact between the surrounding deformable bodies. Interface behaviour for the contact between the solids is modelized here by a non-linear incremental interface law which is able to modelise the main features related to interface behaviour, but any other interface model can be employed. The model used is interesting because the non-interpenetration of solids is automatically taken into account by the oedometric interface characteristics identified from laboratory tests. So, it is not necessary to introduce a fictitious value for interface thickness, like for elastoplastic models. Another advantage it's that we don't need to separate the contact boundary I\ in the classical way (i.e., bonded contact, contact with friction and debonded contact). REFERENCES. Andersson T. Boundary elements in two dimensional contact and friction, Ph.D Thesis, Linkoping University, Sweden, Boulon M. 'Basic features of soil-structure interface behaviour'. Computers & Geotechnics, 7, pp. 5-3, Boulon M., Plytas C. 'Soil structure directionally dependent interface contitutive equation. Application to the predictio of shaft friction along piles'. Proceedings of the 2nd International Conference on Numerical Models in Geomechanics, Ghent, Belgium, Boulon M., Garnica P. 'Constitutive interpolation and soil structure directionally dependent interface law'.proceedings of the 2nd European Conference on Numerical Methods in Geomechanics of second ENUMGE, Santander, Spain, Cruse T.A. 'Numerical solutions in three dimensional elastostatics', Intrernational Journal of Solids and Structures, 5, , G arnica, P. Simulation numerique du frottement entre solides par equations integrales aux frontieres et modele d'interface non-lineaire. Application aux pieux, These de Doctorat, Universite Joseph Fourier, Grenoble, France, Selvadurai A.P.S. 'Boundary element modelling of interface phenomena'. Topics in Boundary Element Research ed. Brebbia), Volume 4, Springer Verlag, 987.