MODELLING OF FRICTIONAL SOIL DAMPING IN FINITE ELEMENT ANALYSIS S. VAN BAARS Department of Science, Technology and Communication, Univerity of Luxembourg, Luxembourg ABSTRACT: In oil dynamic, the oil i often decribed a a vicou material. In a vicou material however, the diipated energy i aumed to be proportional to the wave frequency, which i abolutely not applicable to oil. It i therefore better to ue a concept of damping baed on dry particle friction. A non-vicou model baed on thi concept reult in a damping ratio that become contant for mall deformation for both and and clay, and i alo independent of frequency or hear train amplitude. Thi behaviour correpond with laboratory meaurement and require only one damping parameter which can be obtained from laboratory tet. Thi model i implemented in Plaxi a a Uer-Defined Soil Model to analye the problem of a trip footing ubjected to a dynamic load. The initial reult are rather remarkable.. For example, a Power Spectral Denity plot of the velocitie how that not all frequencie eem to be damped equally and at ome ditance the input frequency i not even preent. 1 INTRODUCTION The accuracy of prediction with Finite Element Model for Geotechnical Engineering can be rather good for tiffne and in trength calculation. In the field of oil dynamic however, the quality of the prediction i often diappointingly low. Therefore the hypothei will have to be evaluated that the current method of dynamic modelling neglect everal fundamental geotechnical apect uch a non-vicou damping, inhomogeneity, aniotropy, variable degree of aturation and other. Apect, which obviouly do not hinder the other well-known application (uch a ettlement and tability calculation), but might explain the unexpected deviation of the vibration prediction. In thi article the effect of the firt apect, i.e. the nonvicou damping, will be invetigated. In oil dynamic the implet way of calculating the energy diipation during the wave propagation i to decribe the oil a a Kelvin-Voigt material, which mean a a vicou material. For example in Plaxi the material damping i programmed a a Rayleigh Damping. Unfortunately thi method hold two large problem: 1. With vicou model like thee, the diipated energy i aumed to be proportional to the wave frequency, which i abolutely not applicable to oil (damping i rather contant with frequency). 2. Therefore there i no laboratory tet to obtain the two neceary material parameter of thi method; the ma factor α and the tiffne factor β.
Since a vicou model doe not reflect a correct oil behavior, it i better to ue a concept of damping baed on dry particle friction. 2 VISCOUS DAMPING One of the bet known way in oil dynamic for calculating the wave propagation and the energy diipation i to decribe the oil a a Kelvin-Voigt material. Thi model can be repreented by a parallel combination of a purely vicou damper and a purely elatic pring. Fig. 1 how a ma connected to thi pring and damper combination. du γ σ xz dz G η τ σ xz τ Fig. 1. Kelvin-Voigt model with ma Since thi vico-elatic model how harmonic hear train and tree of the form: γ = γ inωt τ = G γ inωt+ ωηγ co ωt (1) there will be an elliptical tre-train loop with an energy diipation per wave cycle of: leading to the damping ratio ζ, ξ or D of : t + 2 π / ω o γ 2 Δ W = τ πηωγ = (2) t to ΔW ζ = with W = W 1 ηω ξ = D = ζ = 4π 2G 1 2 2 Gγ (3) Thi indicate that for a Kelvin-Voigt material the diipated energy i proportional to the frequency of the loading, which i not the cae for oil. Solving thi problem by uing an equivalent vicoity uch a: 2G η = ξ (4) ω i not only cientifically incorrect (Kramer, 1996), but i alo unuable for numerical implementation. The problem i illutrated by conidering a uperpoition of wave with multiple frequencie, for which there can be no equivalent vicoity for all wave at the ame time.
3 FRICTIONAL DAMPING The hear diplacement u can be ued to relate the hear velocity v to the hear train γ u = u in( ωt+ kx) u v co( ) = = uω ωt+ kx v = uω t u γ = = uk co( ) ˆ ωt+ kx γ = uk x v v 2π v v ˆ γ = uk = k= = = ω 2π f λ λ f c (5) According to the Dutch vibration tandard, the human amplitude perception threhold i at a velocity of.4 mm/ and the plater damage threhold for houe i at 4. mm/. A trong hear wave with a lateral velocity amplitude of even v = 1 mm/ in clay with a wave peed of c = 1 m/, will therefore reult in a hear deformation amplitude no greater than v γ = =.1%. c Variou reearch reult (Kokuho, Yohida, and Eahi,1982; Vucetic and Dobry, 1991; Wang and Kuwano, 1999; Okur and Anal, 27) have hown that the damping ratio i nearly contant up to a deformation of thi magnitude ( ˆ γ <.1% ). D = 3% Fig. 2. Damping veru hear train amplitude (Okur and Anal) Thee reult are for both and and clay, and alo for low movement (viz..5 Hz or 1. Hz). Thi almot contant damping ratio i about D = 3% (or ζ = 38%), ee Fig. 2. The added dahed line how the reult of the theory to be preented below. Since the damping in thi range i related not to frequency, but to hear train, it i better to ue the concept of damping baed on dry particle friction. Shear deformation will alway be accompanied by ome dry hear movement between two adjacent particle. The energy aborbed in thi dry friction i independent of particle velocity. A poible dynamic oil
model that embodie thi concept can be baed on a force-diplacement curve that ha an exponential ditribution. Then the hear tre can be written in the form: ( γ ) X τ = G (6) with G and X a material parameter; G with the dimenion of a tre and X dimenionle. The zero tree and train are defined from a turning point in the cycle, ee alo Fig. 3, o the peak-to-peak hear train amplitude i γ ' = 2 γ. τ τ γ W t W r ΔW γ γ Fig. 3. Energy lo of tre-train cycle. The area of the ellipe repreenting the energy diipation per cycle i defined by Δ W. The area below the upper line i (including ellipe and the triangle below) i defined by: W t = τ γ = γ ' 1 ( X + 1) G γ ' ( X + 1) (7) The energy diipation of the ellipe i therefore: Δ W = 2 W τγ ' ' t 2 X = G γ ' Gγ ' X + 1 = 1 X (1 + X ) G γ ' 1 + X ( + 1) ( X+ 1) (8) The peak energy tored in the cycle i:
W 1 1 1 1 1 = τγ= ( τ ')( γ ') = τ ' γ ' = 2 2 2 2 8 1 8 G γ ' (1 + X ) (9) With thi we find a damping ratio of: ΔW 1 X 8 ζ ζ = = 8 X = W 1 + X 8 + ζ (1) Thu, by calculating X with eq.(1) and by uing a certain input damping ratio D or ζ, eq. (6) will alway reult in a damping ratio which i equal to the input ratio. So, the choen exponential ditribution i uch that the damping ratio i contant for all hear train amplitude. In order to verify thi contant damping behaviour, eq. (6) ha been ued to calculate the tre-train cycle for a range of different hear train. The power i aumed to be X =.91 (o D = 3% or ζ = 38% according to eq.(1)). The damping wa calculated for each cycle. The reult are plotted a a dahed line in Fig. 2. It how that the damping ratio, reulting from eq. (6), are indeed independent from the hear train amplitude. Becaue of the exponential behaviour of thi damping model, the correponding hear modulu i not automatically comparable with the hear modulu of the normal Mohr- Coulomb model. So, in order to make the hear moduli comparable, there ha to be a correction factor, which depend on the maximum hear train: f τ MC Gγ ' = = = γ ' X τ DMC G γ ' ( 1 X ) (11) Another way of writing the damping model could therefore be: π D π D ( ) ( ) X 1 X 2 2+ with: f ' ' τ = fg γ = γ = γ (12) In that cae the hear modulu G i more comparable with the Mohr-Coulomb model, but the correction factor f i for the relative large damping of granular material not fully contant, a can be een in the table below. Table 1. Correction factor veru maximum hear train γ f D = 3.% f D = 3.6%.1%.44.37.1%.35.29.1%.29.23 4 FRICTIONAL DAMPING THEORY VERSUS MEASUREMENTS Beide the correct modelling of the damping ratio, it i alo important to verify the total tre-train path. The tre-train path of Fig. 4 ha been meaured by Okur and Anal (27). The meaured damping ratio of thi cycle i D = 3.6% (o, X =.89).
The dahed cycle in the figure i the reproduced tre-train path baed on the propoed concept of dry particle friction formulated in eq. (6). The reproduced data fit almot exactly the meaured data. D = 3.6%; G = 37.4 MPa; Fig. 4. Cyclic behaviour and damping; propoed method veru data from Okur and Anal (27) 5 FRICTIONAL DAMPING AND FINITE ELEMENT MODELLING In order to tudy the effect of thi frictional damping, a finite element calculation with thi model ha been made. Verruijt et al. (28) ha howed that the numerical reult of the dynamical calculation with Plaxi give the ame reult a the analytical olution found by Verruijt for the elato-dynamic trip load problem. Thi i an important validation. Therefore a imilar type 2D-calculation with a trip load on a half-pace ha been made. The oil i aumed to behave a a Mohr-Coulomb material with a Young Modulu of E = 1. kpa and a Poion ratio of ν =.25. Fig. 5. Cyclic trip load on a half-pace
The load i a inuoidal load with amplitude of 1 kpa applied to a trip of 1 m width. The frequency i 5 hz and there are 3 load cycle. Thi mean there i alo tenion. Therefore the oil i programmed to be trongly coheive in order to avoid platicity. Fig. 6. Cyclic load of three cycle After the three load cyle, the half-pace i free to move. The calculation i made both with the normal Mohr-Coulomb model and with the Damping Mohr-Coulomb model, with a damping of D = 3.6%, which i a normal value for frictional oil material. Thi frictional damping model i implemented in Plaxi with the help of the Uer-Defined Soil Model option of Plaxi. Thi mean that for the Damping Mohr-Coulomb model a Dynamic Link Library (DLL) ha been programmed and compiled, in thi cae in Delfi, and added to the ource code of Plaxi. Of coure for value of D = %, exactly the ame value are found for thi model a for the Mohr-Coulomb model of Plaxi. At a ditance of 1 m from the center, the abolute velocitie were recorded numerically. Thi can be found in Fig. 7. The firt wave which arrive are the compreion wave. After thee the other wave (hear wave and urface wave) pa by. Fig. 7. Abolute velocity at 1 m ditance
Several concluion can be drawn from the figure: The inuoidal input load i not found back in the output recording; thee wave are trongly deformed, both for the undamped and damped wave. The compreion wave in the tart do not create the larget velocitie. Since thee compreion wave have mainly a compreion component and hardly a hear component, there i hardly any damping. The wave after the firt compreion wave have mainly a hear component and are trongly damped. Fig. 8. Power Spectral Denity plot of velocity Fig. 8 how the Power Spectral Denity plot, obtained through a Fourier tranformation of the recorded velocitie of Fig. 7. Thi PSD plot how ome remarkable point: The effect of the damping i very trong, but epecially the larger peak are damped. The damping doe not damp all frequencie equally. Although the input frequency i 5 Hz, the output ignal how a gap (relatively low average velocity) for thi frequency of 5 Hz (ee graph near arrow in Fig. 8.). In fact it i even le than the average velocity of the urrounding frequencie between 4 and 55 Hz. The reaon for thee finding i unknown but, ince the damping i implemented correctly, it might lead to an explanation of the unexpected deviation of the vibration prediction. That mean that the firt tep of thi reearch, finding unexpected vibration behavior, i ucceful. 6 CONCLUSION Baed on the meaured contant damping ratio at mall hear deformation, it mut be concluded that a vico-elatic model like Kelvin-Voigt i inappropriate for a correct modeling of the energy diipation of granular material. The propoed pring-lider ma model however, correctly reproduce the cyclic tre-train behaviour and the correponding damping, independent of frequency or hear train amplitude. A non-vicou damping model
baed on thi dry-friction concept correpond with laboratory meaurement and require only one damping parameter which can be obtained from laboratory tet Thi model i implemented in Plaxi a a Uer-Defined Soil Model. Firt reult of a dynamical trip footing are remarkable. For example a Power Spectral Denity plot of the velocitie how that the damping doe not damp all frequencie equally and the input frequency i not found back at ome ditance, there i even a gap at thi frequency. That mean that the firt tep of thi reearch, modelling and finding unexpected vibration behavior, are ucceful. Reference Journal Salvadori, C.K. & Martin, S.H. (1989), Coupled ue of reduced integration and nonconforming mode in improving quadratic plate element. Int. J. Num. Meth. Eng., Vol. 28(4), 199-1928. Kokuho, T.,Yohida, Y. & Eahi, Y. (1982), Dynamic propertie of oft clay for wide train range. Soil and Foundation, Volume 22, Iue 4, 1 18. Vucetic, M. & Dobry, R. (1991) The effect of oil platicity on cyclic repone. ASCE Journal of Geotechnical Engineering, Vol. 117, No. 1., 89-17. Wang, G. X. & Kuwano, J. (1999) Modelling of train dependency of hear modulu and damping of clayey and, Soil Dynamic and Earthquake Engineering, Volume 18, Iue 6, Augut 1999, pp. 463-471 Okur, D.V. & Anal (27) Stiffne degradation of natural fine grained oil during cyclic loading, Soil Dynamic and Earthquake Engineering, Volume 27, 843 854 Verruijt, A. & Brinkgreve, R.B.J. & Li, S., Analytical and numerical olution of the elatodynamic trip load problem, Int. J. Numer. Anal. Meth. Geomech. 28; 32:65 8 Book Kramer, S. L. (1996) Geotechnical Earthquake Engineering, Prentice Hall, New Jerey. ISBN-1: 133749436, ISBN-13: 978133749434, pp176-177