12 TIPOS DE CULTIVOS DE LA PAPA EN CHILE
12.1 E SQUEMA VII - T IPOS DE CULTIVOS DE LA PAPA EN C HILE
The investigations were performed by means of FE-analyses for plane strain conditions. The soil was assumed to be a linearly visco-elastic material, e.g. a Voigt-Kelvin-body. The considered cross section is a rectangular vertical plane, the upper boundary of which represents the free surface of the half-space. The system containing 5680 triangular elements is shown in fig. 1. The dimensions are 48 m ×6 m which corresponds to about 8λr×2λr, λr being the Rayleigh-wave length of the half-space material. The element density decreases from at the surface to near to the lower boundary (fig. 1).
Figure 1. FE-mesh with boundary conditions and levels.
At the left upper corner of the FE-mesh a concrete strip foundation of width/depth=1.8m/0.6m with the center at x=1.5m acts as the wave source. It is excited to steady-state harmonical vibrations in vertical direction by a force
P=Po eiωt (1)
with the amplitude po=333 kN/m2. Since the excitation is steady-state, the calculation is performed for quasi-static conditions with complex unknowns.
To simulate the infinite half-space at depth and at the sides of the considered FE-mesh, appropriate boundary conditions have to be applied. At the lower horizontal boundary this is best accomplished by the well-known “dash-pot” boundary condition by Lysmer/Waas [9].
For the vertical boundaries at the sides however, this condition can not be applied, because the longitudinal and the transverse displacements due to Rayleigh-waves are coupled. On the other hand, the R-wave boundary condition will initiate reflections if this boundary is not located far enough away from the wave source, that the theoretical far-field Rayleigh-wave can be developed. For the calculations presented here, the problem was solved by the so-called Influence- Matrix boundary condition (IM-BC) which consists essentially of a virtual lateral extension of the FE-field by about 6 Rayleigh-wave lengths. At the outer vertical boundary of this virtual area, the analytically calculated Rayleighwave boundary condition can now be used without the initiation of any reflections. The properties of the virtual area with respect to the propagation of waves are represented by the influence-matrix, which
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can easily be calculated by standard FE-procedure. If the IM-BC now is applied to the lateral boundaries of the original FE-grid it perfectly simulates the surrounding half-space. The good results, easy handling and considerable advantages with respect to calculation performance of this boundary condition are described in detail by Haupt [10, 11].
Material properties
Three different types of soil are considered. The type A material represents the unfrozen soil and is used as reference material. For the frozen soil types B and C two different shear moduli were selected, to take into account variations in the degree of freezing and temperature. The material properties and the resulting dynamic quantities are listed in table 1. Table 1: Material properties
quality dimension soil type concrete
A unfrozen B frozen C frozen Gdyn MN/m2 132 3300 825 9500 Poisson ratio – 0.25 0.25 0.25 0.33 density ρ t/m3 1 .937 1.937 1.937 2.548 R-wave velocity CR m/s 240 1200 600 1800 R-wave length λR m 6.0 30.0 15.0 45.0 Damping D % 2.5 2.5 2.5 1 .0 αs 1/m 0.0241 0.0048 0.0096 0.0013
The coefficient of attenuation for the shear-wave αs, which is given in the table 1, is calculated from:
(2) with f=frequency and cs=shear-wave velocity.
The frequency of the exciting force being f= 40Hz yields a Rayleigh-wave length of λr=240/40= 6, 0m in a homogeneous half-space of soil type A.
Although damping is presumably much smaller in a frozen soil than in a non-frozen soil, in this study the degree of damping D was assumed to be the same for all soil types. This assumption allows to observe the influence of the stiff top layer on the wave propagation without any additional effects of other parameters. For the same reason the density was taken the same in the frozen and unfrozen soil respectively.
Four different depth profiles of the wave velocity are considered. The profil 0 represents the homogeneous half-space of soil type A and is used as reference system. The profiles with a stiff top layer number I to III are shown in fig. 2.
Figure 2. Velocity profiles in terms of R-wave velocity.
RESULTS
Presentation of the results
The function of an elastic wave propagating in x-direction in a soil body is generally given by the expression:
(3) with ω=2πf, t=time and x=horizontal coordinate. The last term represents a harmonic vibration, whereas the first two terms are functions of the distance. In the case of a wave generated by a wave source at the surface, the wave number k is not a constant but it depends on x: the phase function
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k(x). In the following presentation and discussion of the results the considerations are focussed on the amplitude function vo(x) and on the phase function k(x) at the surface of the half-space and at horizontal levels at different depths below the surface. These levels are (fig. 1)
No. 1 2 3 4
depth below the surface 0.0m 0.6m 1.2m 3.0m
Hence, in profile II the level 2 is situated at the interface between the top layer and the half-space below, in the profiles I and III this applies for level 3.
Figure 3. Amplitude and phase functions at profile III and amplitude at reference profile 0.
Wave length
A typical result of the calculations is shown in fig. 3. The amplitude function vo(x) at levels 1 and 2 for profile III and the corresponding curve at level 1 (surface) for the reference case (homogeneous half-space, material A) are plotted. Furthermore, in this diagram the curves k(x) for profile III are presented. Since in reality these two lines almost coincide, they are plotted separately for clearness. The slope of the phase function is a scale for the wave length: the steeper the slope, the smaller the wave length, e.g. the wave velocity.
The curves k(x) at the levels 1 (surface) and 4 (3.0m depth) respectively for the four considered profiles are together presented in fig. 4, again
they are plotted separately. It may be seen that the slopes of the curves for each case, e.g. the wave lengths, are the same at the surface and within the half-space. It can be shown that this is true also for greater depth, at least at some distance from the wave source [12].
The following wave lengths and wave velocities of the surface-wave ar resulting from the average slopes of the curves in fig. 4:
prof ile 0 I II III
λ [m] 6.08 9.23 7.94 7.86
CR [m/s] 243.2 369.2 317.6 314.4
The slight increase of the Rayleigh-wave velocity for the reference case as compared to the value in table 1 is due to the stiffening of the system by reducing the infinite number of degrees of freedom to a finite one in the FE-calculation. If one takes into consideration that in the homogeneous half-space about 36 % of the energy of the Rayleigh-wave is transmitted within a surface layer of thickness of 20 % of the Rayleigh-wave length—which in this case corresponds to 1.2m the increase in surface-wave length at profile I by a factor of 9.23/6.08=1.52 is surprisingly small. At the two other profiles, the increase is even smaller. The effect of the reduced wave velocity in the top layer at profile III equals about the smaller thickness of this layer at profile II.
Figure 4. Phase functions k(x) at all profiles, levels 1 and 4.
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Figure 5. Normalized amplitudes at profiles I to III, levels 1 and 2.
Amplitude functions
In fig. 3 it can be observed that the amplitude at the surface of the homogeneous half-space (profile 0) does not decrease monotonously with distance but that it shows some variation due to the interference of the Rayleigh-wave with body- waves. This phenomenon has been dealt with in detail in [10]. The amplitude functions in the cases of a half-space with stiff top layer do not show these interference pattern or at least they do to a much smaller extent.
In a plane strain system the attenuation of the amplitude of the Rayleigh-wave or surface-wave with distance is not caused by the propagation of the wave away from the source, if far enough from the wave source, but it is due to the damping only. Hence, if a reference point xo is defined, the amplitude at some point x is
(4)
If at profile 0 an averaging exponential function is drawn through the amplitude curve at the surface (fig. 3) from the values of this function at xo=12m and x=48m the coefficient of attenuation of the Rayleigh-wave is found to be
a=0.0139. This value is considerably smaller than that given in table 1 for the shear-wave. This can be explained by two reasons:
– It has been assumed that damping takes place only at shear deformations, which means that at pure compression no dissipation of energy occurs.
– The stiffening of the system due to the reduction of the degrees of freedom implies also a reduction of the energy dissipation.
Since all results of this study were obtained under the same conditions concerning damping, they can be compared with each other.
Normalized amplitudes
If the curves vo(x) in the cases of profiles I to III are normalized on the corresponding amplitude functions of the reference case 0, the curves in fig. 5 can be plotted. First of all it is to state that the vibration amplitudes of the
foundations at the stiff top layer profiles are considerably reduces as compared to the reference case. The ratios vf/vr are 0.45 (profil I), 0.68 (profile II) and 0.59 (profile III). The index f refers to the profiles I to III, the index r to the reference profile 0. This reduction is easy to understand by keeping in mind the stiffening of the foundation-subsoil system due to the top layers.
Furthermore, attention is drawn to the attenuation of the normalized surface-wave amplitudes with distance. If again an averaging exponential function is plotted through the normalized amplitude curves and the values of this functions at xo=12m and x =48m are considered, a reduction is found by a factor, which is given in line 1 of table 2. However, at equal values of D, the coefficient of attenuation a decreases with increasing wave length. The values of a, calculated by equation (2) and using surface-wave lengths found at the velocity profiles I to III are presented in line 3 of the table. It follows from this, that the normalized amplitudes should increase between the two points by the factor given in line 4 of table 2. Hence, the attenuation of the amplitudes found from the FE-calculation is about 2.6–3.2 times greater than the one corresponding to the above assessment.
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Table 2: Attenuation of surface-wave amplitudes with distance
Profile I Profile II Profile III
1 0.39 0.35 0.44
2 λf/λr 1.52 1.31 1 .29
3 αf 0.0092 0.0106 0.0108
4 1.19 1.13 1.12
5 Zeile 4/ Zeile 1 3.1 3.2 2.6
This unexpected result may possibly be explained by the following process: Part of the energy transferred by the top layer is continuously emitted downwards into the half-space. By this the amplitude at the surface is more attenuated than it should be expected from pure material damping effects. On the other hand, the amplitudes in the subsoil then should be increased as compared to theory. This, in fact, is the case as may be seen from fig. 6. It shows the amplitude of the vertical vibration component depending on depth at x=42m and x=48m respectively for profile I.
For comparison the amplitude in the reference case (profile 0, material A) is presented as it is obtained from the FE- calculation. It coincides very closely with the analytically calculated function. However, this amplitude is much greater, by a factor of about 10. The different scales of the plots should be noticed.
The discussed result is not necessarily in contradiction to the experience that vibrations are transmitted better and to a greater distance in a frozen soil than in an unfrozen soil. In reality, the damping in the frozen soil will be much smaller than assumed in this study. There are also indications that in the stiff top layer the horizontal vibration component is transferred as a body-wave rather than as a surface-wave [12].
Figure 6. Amplitude of the vertical component depending on depth, profiles 0 and I.
CONCLUSION
The investigations described in this report are dealing with:
– the influence of a stiff top layer on the propagation velocity of surface-waves; – the attenuation of the wave amplitudes with distance at the surface.
The findings will help for a better understanding of wave propagation processes in the soil. However, the detected phenomena still demand for more detailed explanations and further investigations will be necessary.
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REFERENCES
1. Ewing, W.M., Jardetzky, W.S., Press, F. Elastic Waves in Layered Media, Intern. Series on Earth Science, McGraw- Hill, New York, Toronto, London, 1957.
2. Rücker, W. Schwingungsausbreitung im Untergrund, Bautechnik 66, 1989.
3. Chow, N., Le, R., Schmied, G. Ausbreitung von Erschütterungen im inhomogenen Boden, Bauingenieur 65, 1990. 4. Haupt, W. Ausbreitung von Wellen im Boden. Chapter 3, Bodendynamik, Grundlagen und Anwendung, Vieweg- Verlag, Braunschweig, Wiesbaden, 1986.
5. Rao, H.A.B. Nondestructive Evaluation of Airfield Pavements (Phase I), Techn. Rep. No. AFWLTR-71–75, Air Force Weapons Lab., Albuquerque, N.M., 1971.
6. Vinson, T.S. Parameter Effects on Dynamic Properties of Frozen Soil, Proc. ASCE, No. GT 10, Oct. 1978. 7. Finn, W.D.L., Yong, R.N. Seismic Response of Frozen Ground, Proc. ASCE, No. GT 10, Oct. 1978.
8. Czajkowski, R.L., Vinson, T.S. Dynamic Properties of Frozen Silt under Cyclic Loading, Proc. ASCE, No. GT 9, Sept. 1980.
9. Lysmer, J., Kuhlemeyer, R.L., Finite Dynamic Model for Infinite Media, Proc. ASCE, No. EM 4, Aug. 1969. 10. Haupt, W. Verhalten von Oberflächenwellen im inhomogenen Halbraum mit besonderer Berücksich-tigung der Wellenabschirmung, Veröff. des Inst. f. Bodenmech. und Felsmech., Universität Karls-ruhe, Heft 74, 1978.
11. Haupt, W. Numerical method for the computation of steady-state harmonic wave fields. Proc. Dyn. Meth. in Soil and Rock Mech. (DMSR 77), Vol. I, A.A.Balkema, Rotterdam, 1978.
12. Haupt, W. Erschütterungsabschirmung in gefrorenem Boden, Veröffentlichung des Grundbauinstituts der LGA, Heft 43, 1985.
Wave Transmission at a Multimedia Interface
R.S.Steedman (*), S.P.G.Madabhushi (**)
(*) Geotechnics and Special Projects Division, BEQE, Science Park, Cambridge, CB4 4 WE, U.K. (**) Cambridge University Engineering Department, Trumpington Street, Cambridge, CB2 1PZ, U.K.
ABSTRACT
In the physical or numerical modelling of a dynamic soil structure interaction problem the semi-infinite extent of the soil medium must be simulated. In the physical modelling of a dynamic problem, for example using a geotechnical
centrifuge, this can be achieved using an energy absorbing boundary made from a clay-like material. In the numerical modelling using finite element techniques the free field condition of the soil deposit can be simulated by superposition of two solutions carried out in a narrow boundary region with ‘Neumann’ and ‘Dirichlet’ boundary conditions. This
scheme, often termed as the Smith-Cundall boundary, involves an interface between three media. Stress waves impinging on such an interface are partially reflected back into the main mesh and are partially transmitted into the connecting boundary regions. In this paper a relation for achieving complete transmission of the incident stress waves into any number of media connected by the interface is derived. The validity of this relation is demonstrated for interfaces joining three and five elastic media.
INTRODUCTION
The tectonic plate movements during an earthquake induce stress waves in the overlying soil layers. These stress waves, usually in the form of compression waves (P waves) and shear waves (S waves) are propagated through the soil medium in all directions. In analysing soil-structure systems subjected to earthquake vibrations it is important to simulate the semi-infinite extent of the soil medium. In dynamic centrifuge tests the technique of using an energy absorbing boundary made from a clay-like
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commercial sealant (Duxseal) to simulate the free field conditions has been developed at the Princeton University (Coe et al, 1983) and has now been widely adopted. The performance of duxseal as an absorbing boundary was evaluated recently by Steedman and Madabhushi (1991).
In numerical analyses using the finite element method non reflecting boundaries need to be used to simulate the semi- infinite extent of the soil. Smith (1973) proposed the superposition of two solutions for the same boundary value problem with ‘Neumann’ and ‘Dirichlet’ boundary conditions. Cundall et al (1981) have suggested that multiple analyses and superposition can be carried out in two narrow boundary regions attached to a main mesh. This technique, often termed as the Smith-Cundall boundary involves an interface between three media as shown in Figure 1.
Figure 1 Two independent overlapping boundary zones connected to main mesh (after Wolf, 1988)
Stress waves impinging on such an interface will be partially reflected back into the main mesh and are partially transmitted into the boundary regions. However, the reflected portion may interfere with the oncoming stress waves in the main mesh resulting in erroneous solutions. This can be avoided if the incident waves are completely transmitted into the two boundary regions. In this paper a relation between the properties of the main medium and the boundary media is derived for complete transmission of stress waves to occur. Continuity of stress conditions and displacement conditions along the interface must be assumed.
FORMULATION
The mathematical formulation is first established for an interface between two elastic media and then it is extended for an interface connecting more than two media.