Simulation of wave propagation in irregular soil domains by BEM and associated small scale experiments

Simulation of wave propagation in irregular soil domains by BEM and associated small scale experiments L. Gaul, P. Klein and M. Pienge

Institute of Mechanics, University of the Federal Armed Forces Hamburg, D-2000 Hamburg 70, German), Refined approaches are developed for the calculation and measurement of waves propagating towards and from rigid or flexible, embedded or surface foundations and between adjacent foundations. The field equations of layered subsoil, foundation slabs and embedded structures are formulated by boundary integral equations and solved by boundary element discretization, after implementing boundary conditions and continuity requirements at the interfaces of substructure domains. Viscoelastic constitutive equations of differential operator or hereditary integral type are generalized by means of fractional order time derivatives in the boundary integral equations. Soil-structure-interaction experiments are performed on a lab scale. Conventional displacement measurement of surface waves by transducers is improved by applying interferometry and holography. Measured surface wave fields prove the necessity of layered boundary element models even for soil with homogeneous compactness. Key Words:

boundary elements, experiments, holography, interferometry, irregular soil domains, layered soil, viscoelasticity, wave propagation.

1. INTRODUCTION The prediction of surface wave propagation in soil is important because the waves effect the safety and comfort of people and the safety of structures interacting with subsoil. This includes vibration-sensitive installations in buildings. Surface waves are generated by active excitations, e.g. rotor unbalances of a turbogenerator in Fig. 2(a) or by passive excitation, e.g. incoming seismic waves in Fig. 2(b). The boundary element method (BEM) is well established to treat active and passive excitations of the semi-infinite subsoil domain by incorporating the radiation condition and mapping the domain behaviour to the boundary variables, e.g. Brebbia 3, Beskos ~, Klein 1°. boundary

r ' . - - - r ~ ÷ r ko bound~ b

Fb= F b

dornatJ halfspaee

lin t ~ ration

face

l-.k ----l-.k= F b k

Fig. 1. Notation for BE-domain-coupling

•

slab

Effects of nonhomogeneity, layering, excavations for embedded foundations, trenches and obstacles in subsoil have been treated by Leung et al. ~~, Gaul, Klein, Plenge s. The rheological properties of soil can be approximated in the range of small vibratory loads superimposed to static preloads by linear viscoelasticity, e.g. Gual et al. 4, Manolis and Beskos 12. In the present paper viscoelastic constitutive equations of differential operator or hereditary integral type are generalized by implementing fractional order time derivatives, as well as fractional integration in the boundary integral equations. This is efficiently done in a frequency BEM formulation with the aid of the correspondence principle, Gaul et al.4, while time BEM formulations suffer from the lack of analytical fundamental solutions including the influence of viscoelasticity. A formulation adopting the Laplace transform was given later by Xie et al tr. Calculations of the interaction between single, as well as adjacent structures (Fig. 2(c)) with soil, often take advantage of the substructure technique. Connecting transition elements have been formulated by Klein t°, such that frame-superstructures can be discretized by finite beam elements and coupled with boundary elements of a thick foundation slab (Fig. 2). When the BEM is used for modeling the soil medium, the usual assumption of a homogeneous, isotropic, and linear elastic or viscoelastic half-plane or half-space is made. However the idealistic soil model might be inadequate in many cases involving inhomogeneity,

200 Engineering Analysis with Boundary Elements, 1991, Vol. 8, No. 4

© 1991 ElsevierSciencePublishers Ltd.

Shmdation of wave propagation in irregular soil domahls : L. Gaul et al.

/I

I

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~.?.~.~..I foundation | ~ ? ~ ! ? ~ I . ........~..-.:.. :'.::..:'p...

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a)

b)

active excitation

passive excitation

c)

Fig. 2.

interaction between adjacent structures

Soil-structure-interaction

anisotropy, porosity effects etc., even in the linear range of material behaviour. Only few experimental results are available teaching us the necessary improvement to obtain more realistic linear soil models by incorporating some of these additional characteristics. This is why small scale soil-structure-interaction experiments are performed on a lab foundation (Fig. 3). The soil in a containment is equipped with embedded vibration transducers, represented by substructures in the BE model (Fig. 6). The transducer measurements have been compared with results measured by laser interferometer (Fig. 3) by Braun 2 and Plenge ~3. This governs the FRF of the transducer compared with the dynamics of the undisturbed soil. The pointwise measurement of surface wave displacements was extended towards field measurements of transient, as well as steady-state wave propagation, by adopting holographic interferometry with the double exposure technique, Gual, Plenge 7. Measured surface wave fields prove the necessity of layered BE-models even for soil with homogeneous compactness due to the influence of growing confining pressure with depth.

O2U i

+bi=P

~t 2.

(1)

Steady-state elastodynamics is represented by taking the Fourier transform of equation (1). With the transformed variables U,-(x, to)= F[u,(x, t)] this yields an elliptical problem for the equations in the domain

c*2U o j.ji - e~jkeu,,c~'Um.~j + B~/p + o~2Ui = 0

(2)

and transformed boundary conditions, whereby no initial conditions enter. The relaxation functions for plane dilatation Eo(t ) and shear G(t) are replaced by complex moduli E* = 2* + 2G* = Eo(1 + i~lo), G* = G(1 + itls)

(3)

and the complex propagation velocities of dilatational and distortional waves are

c~ 2 = E~/p, c~ 2 = G*/p. Conventional complex moduli derived from differential operator type constitutive equations with integer time derivatives D k -- dk/dt k k ~_N can be fruitfully generalized by replacing integer order time derivatives by those of fractional order D ~'. For example, the derivative of fractional order a of deviatoric stress so(t )

2. NUMERICAL TREATMENT OF SUBSOIL

D.so(t )

DYNAMICS BY BEM

The dynamics of the subsoil domain is governed by the equation of motion of a viscoelastic continuum with body forces b~(x, t) in terms of displacement coordinates ui(x, t) at location x

f'_

~.

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=

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(4) defined with the gamma function F(1 - :t) = [~: e-Xx-°dx, is the inverse operation of fractional integration attributed to Riemann and Liouville, Ross 14. Compared to the conventional approach, it has been shown that strong frequency dependence of actual

Enghwerin 9 Analysis with Boundary Elements, 1991, VoL 8, No. 4

201

Simulation of wave propagation in irregular soil doma#,s: L. Gaul et al.

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Fig. 3.

Lab foundation and measuring setup for e.xperimental static and dynamic SSI hn,estigations

viscoelastic material over many decades can be fitted with only few parameters by adopting the fractional derivative concept. Fractional operators give rise to a richer variety of functional families and hence the possibility of improved curve fitting, Torvik, Bagley zS. The defining equation (4) leads to the power law

FED~solt)] = (ie))~F[solt)] = (iog)~s*{o))

(5)

when Fourier transform is applied. This is why integer powers (ira)k only have to be replaced by those of fractional order (ic,))~L For example, the complex shear modulus G*(c,))with storage modulus G'(c9), loss modulus G"(o) and loss factor r/$(e)) is generalized according to

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qk(iO))~' S*~O9) = k=O -- G'(CO)+ iG"(co) G*(og) = 2e*.(o)) ~N pk(iCo)#' k=O

= G'(co)[1 + iq~(e))]

(6)

and related to the relaxation function G(t) in equation (I) by the inverse Fourier transform

a(t) = F - 1[G*(e))/(iu))].

(7)

Computations with the fractional derivative concept in frequency domain require a unique selection of complex roots in the power law (5), Gaul, Klein, Kempfle 6.

Engineering Analysis with Boundary Elements, 1991, I/ol. 8, No. 4

Simulation of wave propagation in irregular soil domains: L. Gaul et al.

• el30/.,'z ,e, looOHz .r +2ook~+

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Fig. 4. Measured decay o f vertical vibration amplitude vs. distance f r o m the exciting footing

ThermorheologicaUysimple materials allow to introduce the influence of temperature T in equation (1) by simply replacing time t by a reduced time #(t) = ~i ~b[T(x, r/)] dr/

(8)

which is based on the shift function ~b determined from experimental data corresponding to dilatational or shear deformation. The frequency co has to be replaced by a reduced frequency according to equation (8) in the complex moduli. Using the integral representation of equation (2), the displacement at the boundary point/~ may be written ciJ(~)UJ(~)

=

fr [u¢(x,

~)tj(X)-

T*(X,

~)Uj(x)]

+ j~ U~j(x, ~)(Bj/p)df2

U* = 4nG* [W(r, c~,

C~)~ij--X(r,

(9)

c~, c*)r3r )]

where

U~= Ir,, U&'fmPdFT~" T~= Ir,, T*n~dF contain the shape functions f~P(x) for the boundary elements (fmt~(x)= 1 for constant elements), U~ and t~ are the displacement and traction nodal values. After numerical integration, the matrix formulation of equation (10) for all n nodes and smooth boundary yields

1 Hu= Ut, H = ~ E + T , where u = { U t U 2 . . U ~.. u"}r, W = {U], U~, ~'3~'r'~r

(11)

BE-domains with different material properties such as the foundation slab ~ , and the halfspace f~b in Fig. 1 with associated BE-equations H~u~ = u~t~, Hbu~ = Ubtb

(12)

can be coupled by continuity requirements of displacements and tractions in the interface F k

u~k=uI=u k p~=_p~=pk.

(13)

Unknown displacements and tractions including those at the interface are obtained from the combined equations

u~

o

7

oII ,

U~J~t~J (14)

J tk

after imposing displacement and/or traction boundary conditions on parts of the boundaries F~ and F~.

where the potentials W and X depend on the distance r = (riri) 1/2, r i = x i - ~i and complex wave velocities. Only boundary integrals remain in equation (9) if the body forces B~ vanish. The integrals are discretized into the sum of integrals extending over the boundary elements. This yields for a boundary node -'JC~!~'t)I~'d-- = -- ijUc~tIJ'j -- -- uT:'~t l~~'--

Interference pattern o f surface ware propagation

t= {tttm,.t~..tn} r, t~= {t~, t~, t~} r

dF

where x are the boundary points to which the integral extends. U~(x) and t~(x) are displacement and traction components at x, U~(x, g, co) and r~(x, g, co) stand for displacement and traction components of the fundamental solution at the field point x, when a unit point load is applied at the load point g following the i direction, and %j is a coefficient that depends on the geometry of the boundary at g(c~j= 6u/2 for smooth boundary). One can solve the integral equation for a sufficient number of co and numerically invert U,(x, co) to obtain the timedependent displacement, KleinL°. Viscoelastic material properties enter the fundamental solution

1

Fi~. 5.

(10)

3. EXPERIMENTAL STUDIES ON WAVE PROPAGATION Experimental investigations on soil-structure-interaction (SSI) are carried out at the Institute of Mechanics under the guidance of the first author. A schematic illustration and photo of the lab foundation consisting of a soil box,

Engineering Analysis with Boundary Elements, 1991, Vol. 8, No. 4

203

Sinlulation of wave propagation in irregular soil domains: L. Gaul et al. which is equipped with conventional transducers and optoetectronic wave measurement facilities, is shown in Fig. 3. An experimental program has been started to measure interaction phenomena, as well as surface wave propagation for homogeneous and inhomogeneous soil, Plenge 13. Figure 4 depicts the decay of vertical vibration amplitudes w related to the amplitude d'o at the excited footing with increasing distance from the footing x for 5 excitation frequencies. The expected amplitude decay proportional to 1/x/x for homogeneous soil is only fulfilled in an average, It is interesting to note that deviations are superimposed which behave like amplitudes of damped vibrations. After proper scaling of the distance x with respect to the wavelength 2s = c J f of the generated shear wave for 5 excitation frequencies, it could be shown, that the local minima and maxima of the amplitude decay coincide. The measured amplitude oscillations can not be explained by reflections from the wall of the soil box. This was shown by 3-D BE-calculations of the discretized entire soil box and proved by experiments as well. Even though the soil compactness is homogeneous, interference effects are created by the increase of soil-stiffness due to growing confining pressure with depth. Thus faster running waves from greater depth interfere with slower ones closer to the surface and create local maxima and minima of the amplitude decay. Similar effects have been measured by Haupt 9, who studied the effect by a 2-d FE-model. In the paper at hand, an interpretation based on a 3-d BE-calculation is given. The interference effects in Fig. 4 were measured by embedded acceleration pickups at the indicated locations. Optoelectronic measurements were applied as well, e.g. by laser-interferometer (Fig. 3) and double exposure holographic interferometry (Fig. 5). 4. CALCULATION OF SURFACE WAVES IN LAYERED VISCOELASTIC SOIL BY BEM A first approach to simulate the measured amplitude decay (Fig. 4) by 3-d BE-calculation is based on a cubic domain f~ describing an acceleration pickup embedded in a homogeneous viscoelastic halfspace domain f~b. Propagating waves are generated by harmonic vertical excitation force with complex amplitude ~ , acting on a rigid square of dimensions 2a x 2a at the surface. Rigid body motion is easily implemented in equation (14) by coupling those displacements of u~ which are located in the square. The excitation force is the vertical resultant of the surface tractions. Real and imaginary part of the complex compliance ft_.aG/F: = f_.__+ i9::

(15)

govern the space- and time-dependent displacement-field of the surface wave by u._(x, t)= [f_.__(x)cos c:gt- 9_._.(x)sin cot] F../(aG).

(16)

Figure 6(a) depicts the real and imaginary part of the compliance (15), as well as the amplitude (f~: + 9~:)z/-'. From the amplitude decay it can be concluded that the homogeneous haifspace is no proper model to explain the measured results in Fig. 4. The variation of soil moduli with depth due to increasing confining pressure can be modeled by layered

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BE domains. Proper truncation of the surface- and interface-discretization has been analyzed by Klein 1°. Only one layer with underlying halfspace can already explain the measured amplitude decay (Fig. 4). The calculated amplitudes in Fig, 6(b) indicate that the locations of local maxima and minima fit quite well with the measured results.

5. CONCLUSIONS The BE-program PROBEM developed at the Institute of Mechanics, Gaul, Zastrau 8, Klein L° allows to model soil irregularities by the substructure technique. Generalized viscoelastic constitutive equations are implemented. Measured surface wave fields prove the necessity of layered BE-models even for soil with homogeneous compactness. Detailed 3-d models including embedded pickups lead to the deviation between the pickupresponse and the response of the undisturbed soil at the pickup-location. Displacement measurements by laser-interferometer, Braun-', Gaul, Plenge 7, allow to detect deviations with accelerometer measurements and lead to the important FRF of the pickup.

Engineerin 9 Anah'sis with Boundary Elements, 1991, Vol. 8, No. 4

Simulation o f wave propagation in irregular soil domains." L. Gaul et al. REFERENCES 1 2 3 4

5

6 7

8

Beskos, D. E. Boundary element methods in dynamic analysis, Applied Mechanics Reviews. 1987, 40~1), 1-23 Braun, M. Zur Messung und Randelementberechnung yon Oberfl~chenwellenfeldern an Modellgrtindungen. Master thesis, Institute of Mechanics, Univ. Fed. Armed Forces Hamburg, 1988 Brebbia, C. A., TeUes, J. C. F. and Wrobel, L. C. Boundary Element Techniques, Springer-Vedag Berlin, 1984 Gaul, L., Klein, P. and Plenge, M. Dynamic boundary element analysis of foundation slabs on layered soil. Computational Mechanics Publ. Southampton, Springer-Verlag Berlin, Boundary Elements X, Geomechanics, WaL,e Propagation and Vibrations, 1988, 4, 29--44 Gaul, L., Klein, P. and Plenge, M. Continuum-boundaryelement and experimental models of soil-foundation interaction. A. A. Balkema, Rotterdam, Numerical methods in geomechanics, Innsbruck, 1988, 1649-1661 Gaul, L., Klein, P. and Kempfle, S. Impulse response function of an oscillator with fractional derivative in damping description. Mechanics Research Communications, 1989, 16(5), 297-305 Gaul. L. and Plenge, M. Theoretische und experimentelle Untersuchung der Struktur-Baugrund-Wechselwirkung im Hinblick auf die Erkennung verdeckt verlegter Minen im Erdreich. Proc. Syrup. Neue Methoden zum Aufsparen ton Minen, 1990, 19-40 Gaul, L. and Zastrau, B. Rechnergestiitzte (FEM, BEM, MKS) und experimentelle (CAT) Spannungs-, Verformungs- und Bewegungsanalyse statisach und dynamisch beanspruchter

9 10

I1

12 13 14 15

16

Strukturen. In Technologieangebote wiss Einrichtungen Hamburgs. Ed. Arbeitskreis Technologief~rderung in Hamburg: 8.04. 1989 Haupt, W. Bodendynamik. Vieweg and Sohn, BraunschweigWiesbaden 1986 Klein, P Zur Beschreibung der dynamischen Wechselwirkung yon Fundamentstrukturen mit dem viskoelastischen Baugrund durch dreidimensionale Randelement formulierungen. PhD thesis, Institute of Mechanics, Univ. Fed. Armed Forces Hamburg, 1989 Leung, K. L., Vardoulakis, I. G., Beskos, D. E. and Tassoulas, I. L. Vibration isolation by trenches in continuously nonhomogeneous soil by the BEM, ./. of Soil Dyn. Earthquake En#g, 1991, 10, 4 Manolis, G. D. and Beskos, D. E. Boundary Element Methods in Elasto-Dynaraics, Unwin-Hyman, London, 1988 Plenge, M. Ein Beitrag zur Untersuchung des dynamischen Verhaitens geschichteter Baugriinde. PhD thesis, Institute of Mechanics, Univ. Fed. Armed Forces Hamburg, 1990 Ross, B. Fractional calculus. Mathematics Maga:ine, 1977, 50(3), 115--122 Torvik, P. J. and Bagley, D. L. Fractional derivatives in the description of damping, materials and phenomena. The role of damping in vibration and noise control, ASME DE, 1987, 5, 125-135 Xie, K., Royster, L. H. and Ciskowksi, R. D. A boundary element method formulation for fractional operator modeled viseoelastodynamic structures. Proc. 1lth Int. Conf. on BEM, Cambridge, Springer-Verlag Berlin, 1989, 55--64

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