Interface Formulation Problem in Geotechnical Finite Element Software

Interface Formulation Problem in Geotechnical Finite Element Software Adis Skejic M.Eng. Civil Engineering faculty Sarajevo, BiH [email protected]

ABSTRACT A serious problem has been discovered in the interface formulation in one of the most popular finite element software programs used in geotechnical practice. This paper analyses the problem of interface formulation which is not in agreement with the definition given in program manuals. Problem explained appears to have been solved in the most recent version, but the author finds it important to discuss the implications of this problem, because there are so many investigation conclusions given in the recent past without being aware of this problem. A simple example of sliding block on elastic soil is used for this investigation and the results with discussions are presented, with intimate details of the computational model.

KEYWORDS:

interface, finite element method, Plaxis 2D.

INTRODUCTION Very often in Geotechnical enginering practice using the advanced tool of the finite element technique, interface elements are used to model soil structure interaction. Modeling of discontinuities in rocks, soil – geogrid interaction modeling in reinforced soil, soil – pile interaction for pile capacity calculations are only few examples where interface plays a crucial role. Interface formulation given in programs manuals is not in agreemnet with program code, what makes the conclusions reached using the numerical model imprecise. After a brief interface formulation, details of practical example numerical model shall be presented, to prove the point of this article. Same problem was analysed with Plaxis version 2011, and problem was not found. To end this introduction, an interesting quotation given by Kulhawy (2011) is reminded: “Please do not use software if you do not understand what is it doing.

MATERIALS AND METHODS Numerical model of sliding block on elastic soil, is used to analyse the stress state on the discontinuity between block and soil medium. Plane strain model with 6-node triangular finite elements is employed. The soil – block contact is modeled by interface element. An isoparametric - 2035 -

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zero thickness interface element (Goodman et al. 1968, Ghaboussi et al., 1973; Carol and Alonso, 1983; Wilson, 1977; Desai et al. 1984, Beer, 1985). Interface is defined as thick layer connected with soil and structure elements with 2 degrees of freedom. (Figure 1)

Figure 1: Geometry and interface slippage criterion for 6 node element (Modified from Van Langen & Vermeer, 1990) The soil behaviour in this discontinuity can be different from soil behaviour, what it confirmed by many experimental investigations (Potyondy 1961, Desai, 1981, Acar et al., 1982, Desai et al. 1985, Boulon i Plytas, 1986, Boulon, 1989). Interface stiffness matrices formed according to constitutive low of its behavior, are assembled in stiffness matrice of particular problem as whole structure. After slippage occurs, volumetric as well as shear deformations occurs on interface. Volumetric strain magnitude, are controled by dilatancy angle (ψ) which defines the constitutive low flow rule.

Figure 2: Interface modeling and interface deformations definition The constitutive law of interface behavior is defined by:

σ = Dε e = D (ε − ε p ) where e nad p indexes defined elastic and plastic part of deformation respectively. For elastic deformations the stress and strain increment can be related by interface shear an normal stifness, ks and kn respectively.

 τn   ks σ  =  0  n 

0  εse    kn  εne 

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 Gi   τn   ti = σ    n 0 

 0  e   ε s  Eoed ,i  εne   ti 

Where Gi and Eoed,i are shear and oedometer modulus of interface. The relative thickness factor is taken as default value 0,1, and will not be analyzed in this article. Using the theory of plasticity terms (flow rule, consistency condition) the scalar multiplicator magnitude can be calculated, and elastoplastic stifness matrix can be derived as :

 Gi   τn   ti σ  =   n 0   τn   k s σ  =  0  n  

 0   εs  − μ ∂g Eoed ,i  εn  ∂σ  ti 

0 1 − k n  k s + kn ⋅ (tan ϕ ') ⋅ (tanψ

 k s2 − k s ⋅ kn ⋅ tan ϕ '  2 ')  − k s ⋅ k n ⋅ (tanψ ') kn ⋅ (tan ϕ ') ⋅ (tanψ

  εs     ')   εn 

For Plaxis 2D, the strength and stiffness of interface are defined as part of the strength and stiffness of soil adjusted to interface strength according to :

ci = Ri ⋅ c soil ϕ i = Ri ⋅ ϕ soil while in Reference manual, the last relation is defined as tan ϕ i = Ri ⋅ tan ϕ soil

Gi = Ri2 ⋅ G soil

Eoed ,i = 2Gi

1 −ν i ; ν i = 0.45 1 − 2ν i

where Ri is reduction coefficient. Also, for: ψi = 0 for Ri < 1, ψi = ψtla for Ri = 1 where: c – cohesion of soil material φ – soil friction angle ψ – soil dilatancy angle νi – Poisson ratio for interface And finally we can write the slippage criterion as :

f (σ ) = τ n − σ n ⋅ tan ϕ i' + c i' Next, the details of numerical model are presented. Refer to Figure 3 and Table

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Figure 3: Numerical model with material parameters shown in Table 1.

Table 1: Material parameters Soil & Interface Concrete block interface

Model Linear elastic MohrCoulomb

γ [kN/m3] 25,0

Eref [kPa] 3e7

c [kPa] -

φ [ᵒ] -

ν [-] 0,0

0,0

5e4

0,0

65

0,45

After generating initial stress state which is defined with self-weight of concrete block, prescribed displacement are applied at the left boundary, to cause slippage of concrete block. Shear and normal stresses are generated at the interface, and shear to normal stress ratios are analyzed to investigate the definition of interface strength. The stress distributions along interface, as well as plastic points are shown to prove that the slippage occurs on soil – block contact. Two different cases of reduction coefficient (Ri) and relatively high value of internal friction angle (φ = 65ᵒ) are used to show the difference between programs code and programs reference manual definition of interface strength. First Ri = 1,0, and then Ri = 0,2. The ratio of average stresses values is also compared to show the described difference.

RESULTS AND DISCUSSION The results of provided analysis, shows the shear and normal stress ratio for prescribed displacement of 2,0 cm, when plastic points occurs along full length of interface (figure 5). As it is said earlier, the strength of interface is actually defined as Ri·φ, and not Ri·tanφ, as it is written in programs manuals (figure 5). Of course for reduction coefficient equal 1,0 such a difference do net exists (figure 5).

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(a)

(b)

Figure 4: Slippage criterion difference in program’s code (Ri·φ) and reference manual (Ri·tanφ); (a) Ri = 0.2; (b) Ri = 1,0 As it can be seen, the slope of the line defined by normal and shear stresses on interface is defined with tangent of angle of internal friction of interface itself, and it is 0.2309, what is exactly tan(0.2 × 65), and not 0.2 × tan65. Finite element mesh as well as plastic points are shown on the picture below.

Figure 5: Finite element mesh and plastic points for Ri = 0.2 Finally, the results for normal and shear stress distribution along interface are shown.

(a) normal stress distribution

219.32 kPa

(b) shear stress distribution

470.34 kPa

Figure 6: Stress distribution along interface for prescribed displacement of 2.0 cm (Ri = 0.2)

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As a discussion part, the author would like to underline that explained problem does not exist in Plaxis version 2010 and 2011. The idea of this text is to show that some conclusions of investigations done with previous versions (like 8.1 and 8.5) of this very popular software may be questionable. This paper does not discuss the role of interface elements in practice of geotechnical engineering, but only inform about a particular problem found in a widely used software program.

CONCLUSIONS Interface formulation defined in Plaxis ver. 8.5 program manual does not agree with program’s code. The problem is definition of interface strength which is defined as Ri·tan·φsoil in manual, and Ri·φsoil in program’s code, according to calculation results shown in this paper. This problem becomes more obvious for higher values of internal friction angles (φ), and lower values of reduction coefficient (Ri). This problem should be on mind to everyone using named version of Plaxis software. Even for using newer version which solved this problem, suggested values of reduction coefficients for modeling any soil structure interaction problem investigated by doing back analysis with older versions, should be taken with caution. A very simple problem is analyzed in order to eliminate as many second order factors as possible.

REFERENCES 1. Binesh, S.M., Hataf, N. and Ghahramani A. (2010) “Elasto-plastic analysis of reinforced soils using mesh-free method”, Applied Mathematics and Computation, Elsevier Inc 2. Brinkgreve, R.B.J. (2002) PLAXIS – Finite Element Code for Soil and Rock Analyses: User’s Manual – Version 8, A.A. Balkema, Rotterdam, Netherlands 3. Coutinho, A.L.G.A., Martins, M.A.D., Sydenstricker, R.M., Alves, J.L.D. and Landau L (2003) “Simple zero thickness kinematically consistent interface elements”, Computers and Geotechnics 30 347–374 4. Grubić, N., Skejić, A. i Balić A. (2012) “Numerical modeling of interaction between stiff reinforcing elements and granular backfill under pullout conditions”, 7th International Conference on Computational Mechanics for Spatial Structures, IASS-IACM, Sarajevo 5. Li, J. & Kaliakin, V. N. (1993) “Numerical Simulation of Interfaces in Geomaterials : Development of Zero Thickness Interface Elements, Department of Civil Engineering, University of Delaware, Newark, Civil Engineering Report

6. Skejić, A., Balic, A., Grubić, N. (2011) “Uloga Interface elemenata pri numeričkom modeliranju armiranog tla”, Forth International conference Geotechnical aspect of engineering, Zlatibor. 7. Van Langen H. and Vermeer P. A. (1991) “Interface elements for Singular Plasticity Points“, International Journal for numerical and Analitical Methods in Geomechanics. Vol. 15, 301-3 15

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8. Van Langen, H. & Vermeer, P. A. (1990) “Automatic Step Size Correction for nonassociated Plasticity Problems”, International Journal for Numerical Methods in Engineering, Vol. 29, 579-598 9. Van Langen, H. (1991) Numerical Analysis of Soil Structure Interaction, PhD thesis, Delft University 10. Wood D.M. (2004) Geotechnical Modeling, Spon Press

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