Analysis of Foundation-Layered Soil Interaction Using Propagating Wave-modes Analysis

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1 Aalysis of Foudatio-Layered Soil Iteractio Usig Propagatig Wave-modes Aalysis IGOR ŠPACAPAN & MIROSLAV PREMROV Faculty of Civil Egieerig Uiversity of Maribor Smetaova 7, Maribor SLOVENIA Abstract: - I the egieerig aalysis of the respose of the foudatio i layered soil, subjected to direct or seismic excitatio, parametric aalysis is the most desirable type of aalysis because it has potetials for better optimal desig. We are presetig a computatioal approach, which yields dyamic displacemets of a foudatio ad wave motio i layered soil. The computatioal approach yields wave-modes ad their amplitudes as parameters, ad the ifluece of each wave mode o the vibratio of the foudatio ad o the spreadig of waves ito the surroudigs. Computatio is accomplished i the frequecy domai. It uses the fiite elemet method where the radiatig coditios o the fictive boudary are satisfied exactly. We are presetig a brief outlie of the key formulas of the computatioal approach. Numerical results are partially compared to the exact oes, suggestig the efficiecy of the approach. Key-Words: - Layered soil-structure iteractio, Propagatig Wave-modes, Fictive boudary, FEM Itroductio I the aalysis of the soil-structure iteractio the soil behaves practically as a ifiite space (halfspace). Thus, the essetial pheomeo is the occurrece of the propagatig waves, which propagate i the directio away from the source of excitatio. Mathematically these waves satisfy radiatig (Sommerfeld) coditios, [] [], which imposes the crucial difficulty i the computig procedures. I the case whe a o-homogeeity ad/or variatio of the boudary coditios exist, the solutio of the wave equatio, which requires the fulfillmet of the radiatio coditios, is feasible oly umerically. Although a variety of umerical ad semi-aalytical methods are available for the aalysis of the soil-structure iteractio, oe of them is simple ad exact at the same time. If we glace over them, we could describe them roughly as follows. Boudary elemet methods satisfy radiatio coditios, but are ot simple to apply for complex cases, see for example [3] [4]. By fiiteelemet methods the radiatio coditios are satisfied oly by usig special elemets o the fictive boudary, or by evaluatig certai computatioal phases aalytically, see for istace [4]-[6]. Operator methods require the implemetatio of special operators o the fictive boudary, for istace [7]-[]. We ca classify the available methods superficially ad briefly as beig either a great deal sophisticated ad i certai cases exact, or simple ad cosiderably approximate. To aalyze the foudatio-layered soil iteractio we use the approach that yields exact results whe the layers rest over a rigid half-space, [3]. The results are cosiderably accurate also for the cases where the sub-soil is ot rigid, providig that the foudatio dimesios are approximately five times smaller tha the cumulative depth of the layers. The modelig of the soil ad the foudatio is performed by FEM, which is comfortable for complex cases. The computatio is i the frequecy domai ad the radiatig coditios are exactly satisfied o the fictive boudary. Whe eeded, the trasiet excitatio ca easily be aalyzed by trasformig the results from the frequecy to the time domai. The approach yields parametric aalysis of iteractio, which is particularly advatageous for practical egieerig aalysis. Problem Formulatio ad Outlie of the Computig Procedure We are cosiderig a two-dimesioal case of atiplae shear wave motio, yet the approach is valid for the case of geeral wave motio i parallel waveguides. The displacemets i a soil with parallel layers, which have o foudatio, are govered by the wave equatio i the frequecy

2 domai, Equatio (), of course with distict wave umbers k for differet layers. u + k u = () The theoretical solutio is a liear combiatio of wave modes [], ad is preseted by Equatio () whe usig the co-ordiate system i Fig.. u(x, y) = ib x ib x f (y)(a e + Be ) () I Equatio () f are eige-fuctios, ad A, B are costats which are determied by lateral boudary coditios. They represet wave-modes ad amplitudes, respectively. Here, the amplitudes are called also weightig or modal factors. The b are distict wave umbers, which deped o the characteristics of the case uder cosideratio. Let us cosider wave motio i the segmet (a cell) betwee ay two cross-sectios, for istace cross-sectios or i Fig. (). The, accordig to Equatio (), each displacemet ad stress wavemode satisfy Equatio (3). = λ, =,,... τ, τ, (3) Idices ad stad for the values o crosssectios ad, respectively, ad stads for the - th wave-mode. Regardig Equatio, the meaig of λ is give by Equatios (4). ib x λ = exp (4) The costat i the expoet is give by Equatio (5). b ω k =. (5) ck,phase O the other had, due to the uiqueess of the solutio of wave equatio, Equatio (6) applies, where T - is the matrix of trasfer fuctios. τ, = T, =,,... (6) τ, After equatig the right had sides of Equatios (3) ad (6), we get the eigevalue problem, Equatio (7). u T = (7) τ ( λi) By modelig the cell with fiite elemets, the trasfer fuctio becomes the trasfer matrix, which is computed from the dyamic stiffess or from the flexibility matrix. Solvig Equatio (7) by stadard routies yields eigevectors represetig displacemets ad stress wave-modes. Evidetly, eigevalues with egative imagiary part, ad real eigevalues less the a uit, belog to radiatig modes. Whe the foudatio is preset, ad the excitatio as show i Fig., the displacemets o ay distat cross-sectio, for istace cross-sectio, must satisfy the radiatig coditios. Therefore, wave motio cosists of oly radiatig wave-modes, see Equatio (8), where A are the amplitudes, ad + sig stads for radiatig modes. u τ = τ + A (8) The excitatio displacemets u are related to the radiatig displacemets ad stresses o the cross sectio, called fictive boudary, by Equatio (9), where T - is the belogig trasfer matrix.. u = T (9) τ By solvig Equatios (8) ad (9) o radiatig displacemets u get Dirichlet s boudary coditios to solve the exterior problem as a iterior oe. Computed amplitudes A are the parameters of the wave motio showig exactly how much distict wave-modes cotribute to the displacemet ad stress field. Fially, it is obvious that various combiatios of excitatios ad boudary coditios ca be solved by the same approach as preseted above. 3 Numerical Example of Iteractio of Foudatio ad Two-layered Soil The aalyzed case is symbolically preseted i Fig. (). The soil cosists of two layers over rigid subsoil. The cotact betwee layers ad the sub-soil is cosidered as firm. Each layer is meters deep, the lower oe has the wave umber k =, while the upper oe is softer ad has k =. The shear module

3 of the upper layer is two times greater tha the lower layer. Excitatio is give by liearly distributed displacemets with the amplitude a uit, as suggested i Fig. (). The frequecy of excitatio is a uit. The rigid foudatio dimesios are 4mx5m with the material desity two times greater tha that of the layers. The fictive boudary is 3m from the excitatio cross-sectio, which makes the cosidered sectio 3 meters log. The width of the cell is,3 meters ad is chose arbitrarily. Fiite elemets are simple liear oes. The mesh has 46x4 odal poits, which ca be observed i figures. The solutios of Equatio (7) are preseted i Figures () ad (3), some of the eigevalues i Fig. (), ad some of the eigevectors i Fig. (3). Compariso to exact values demostrates excellet coicidece. All eigevalues that belog to radiatig wave modes are situated o the real axis betwee zero ad oe, they are dimiishig stadig waves, or have a egative imagiary part, propagatig waves. The eigevalues are marked by umbers, separately for stadig ad propagatig wave-modes ad accordig to the directio of propagatio. Some of these umbers occur also i Fig. (3) to see to which eigevalue a wave-mode belogs. For the case of absece of the foudatio, a aalytical solutio is computed i order to verify the umerical results. I the left graph i Fig. (4) radiatig displacemets are preseted, while the graph o the right shows the absolute values of modal weightig factors (amplitudes) of costituet wave-modes it represets the spectrum of displacemets o the fictive boudary. It shows that the third mode is domiat, while the stadig modes have vaished. Both figures demostrate a cosiderably good agreemet betwee aalytical ad umerical results. The displacemet field of foudatio-soil iteractio is preseted i Fig. (5), ad the absolute values of displacemet profiles i Fig. (6). Fist three graphs show the displacemets i the cross-sectios of excitatio, foudatio site ad fictive boudary, respectively. The fourth graph presets maximal displacemets occurrig outside of the aalyzed segmet at various cross-sectios. 4 Coclusio The preseted theory ad examples demostrate that the computig approach is cosiderably simple. Already by simple fiite elemets ad a rather coarse mesh we get excellet results. A advatage of the approach is that various egieerig aspects ca be aalyzed with the aid of wave-modes ad their weightig factor. Ufortuately, due to the limited legth of this paper, oly some advatages are preseted. Refereces: [] Achebach, J.D. Wave Propagatio i Elastic Solids, North Hollad Publishig Compay: Amsterdam, New York ad Oxford, 973. [] Baldock, G.R., & Bridgema, T., The Mathematical Theory of Wave Motio, Joh Wiley & Sos: N.Y, Chichester, Brisbae, Toroto, 98. [3] Brebbia, C.A. & Telles, J.C.F & Wrobel, L.C. Boudary Elemet Techiques, Spriger-Verlag: Berli, N.Y., Tokyo, 984. [4] Beer, G. & Watso, J.O. Itroductio to Fiite ad Boudary Elemet Methods for Egieers, J.Wiley & Sos, N.Y., Brisbae, Toroto, 99. [5] Wolf, J.P. & Chomig Sog, Fiite elemet modellig of ubouded media, Joh Wiley & Sos, Chichester, New York, Toroto, 995. [6] Wolf, J.P. & Chomig Sog, The Semi- Aalytical Fudametal-Solutio-Less Scaled Boudary Fiite-Elemet Method to Model Ubouded Soil, EUROMECH Colloquium 44, Boudary Elemets for Soil, Uiversity of Cataia, -3 Jue. [7] Egquist, B. & Majda, A. Absorbig boudary coditios for the umerical simulatios, Mathematical Comp.,977; 3(39): [8] Bayliss, A. & Turkel, E. Radiatio Boudary Coditios for Wave-Like Equatios, Commuicatios o Pure ad Applied Mathematics 98, XXXIII: [9] Feg, K., FEM ad Natural Boudary Reductio, Proc. of the Iter. Cogress of Mathematicias, Warszava, August 6-4, 983. [] Keller, J.B. Exact No-reflectig Boudary Coditios, Joural of Computatioal Physics 989; 8:7-9. [] Givoli, D. & Keller, J.B. No-reflectig Boudary Coditios for Elastic Waves, Wave motio 99; :6-79. [] Premrov, M. & Umek A. & Špacapa I., A iterative FEM for solvig elasto-dyamics i ifiite domais, Zeitschrift für agewadte Mathematik ud Mechaik,, 8, suppl. 3, p [3] Špacapa I., & Premrov, M. Aalysis of Wave Propagatio i Waveguides by FEM, Iteratioal Coferece CMEM X, Alicate,, WIT Press, Bosto.

4 y-axis atio rigid (4.5x4.5m give u ) o Fictive boudary m h =m µ =µ, ρ =ρ h =m µ,ρ x-axis () Cross-sectio Aalysed Sectio(3m) () () Cross-sectios Figure. Dispositio of layers ad foudatio. clamped. IMAGINARY Figure. Eigevalues. Explaatio of the graph - eigevalues belog to: radiatig modes (umbers o patches,. oly few leadig.4 of them.6 are marked),.8 exact values (filled REAL circles, oly few leadig are preseted), computed stadig wavemodes (void circles), computed propagatig wavemodes ( + sigs)

5 8 6 height 5P 4P 4 m P 3P P ampitude Figure 3. Firs five propagatig wave-modes. Exact values -solid lies, computed - dashed lies. HEIGHTS m imag real Excit WEIGHTING FACTORS (ABSOLUTE) - DISPLACEMENTS 5 CONSECUTIVE NUMBER OF MODE Figure 4. Displacemets o the fictive boudary ad their weightig factors. Exact values - solid lies, computed - dashed lies.

6 Real displacemets x (meter) Logitudial dir. 5 5 y (meter) Figure 5. Displacemets field. EXCIT FOUND. FICT.B. EXTREM m Figure 6. Absolute displacemets profiles