Vibration of buildings on pile groups due to railway traffic finiteelement boundary-element, approximating and prediction methods

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1 ie 010 Nottingham University Press Proeedings of the International Conferene on Computing in Civil and Building Engineering W Tizani (Editor) Viration of uildings on pile groups due to railway traffi finiteelement oundary-element, approximating and predition methods L Auersh BAM Federal Institute for Materials Researh and Testing, Berlin, Germany Astrat A finite-element oundary-element software for the dynami interation of flexile strutures and the soil has een extended for pile foundation. The oundary element method for the soil uses the Green s funtions of the layered half-spae whih have een generalised for interior loads. Pile groups of 10 to 0 piles of different arrays are analysed and ompared with single piles. Simplified models have een developed for a user-friendly, pratie oriented predition software for railway indued ground and uilding viration. Keywords: oundary elements, soil-struture interation, pile foundations, predition software 1 Comined finite-element oundary-element method ynami soil-struture interation is analysed y a omined 3-dimensional finite-element oundaryelement method (AUFEBEM, Auersh, 1988). Flexile strutures suh as eams, plates, walls, oxtype uildings, railway traks, single piles and pile groups are modeled y the finite element method whereas the homogeneous or layered soil is modeled y the oundary element method. A point load, a wheel-set load or and a wave field have een used as exitation. In this ontriution single piles and pile groups (Fig. 1) are presented. The asis is the point load solution (Green s funtion) for a layered soil. The Green s funtions are alulated y the following proedure. For eah layer, a dynami stiffness matrix is alulated in wave numer domain (Auersh, 008). All layer matries and the matrix of the underlying halfspae are assemled in a stiffness matrix K S of the whole soil ody, whih is inverted to a ompliane matrix N. The appropriate elements of this ompliane matrix, for example N zz (k,z 1,z ) for an interior soure at z 1 and an interior response at z, are integrated over the horizontal wave numer k to get the displaement at a horizontal distane r of the soure 1 F ( r, z kdk zz 1, z ) = N zz ( k, z1, z )J 0 ( kr) π 0 Similar formulas hold for the other four omponents of the Green s funtion F. These Green s funtions are used in the omined finite-element oundary-element method to estalish a oundary stiffness matrix of the soil (in frequeny domain, for n disrete points). A displaement matrix is ompiled diretly y using the point-load solutions u(x β ) = F(x β -x α ) p(x α ) u β = F βα p α () (1)

2 a Figure 1. Amplitude and phase of the a) vertial rigid, ) vertial flexile, ) horizontal ompliane of different pile groups grid, irle, parallel lines, ross and line, onrete piles of L = 0 m, R = 0.5 m, ontinuum soil with shear wave veloity v S = 00 m/s.

3 At the loading point, this would yield infinite values. Therefore, mean values on the ylindrial (in ase of a pile) or irular surfae (in ase of a surfae foundation) are alulated as the diagonal elements of the displaement matrix 1 u α = F( x x α ) p( x α ) da u α = F αα p α (3) Aα A α The fore matrix of the soil is the identity matrix, as 1) the Green s funtion has no stresses at the soil surfae and ) the stresses in the interior do not yield fores at another pile element (as far as the dimension is small ompared to the distane and wavelength). Thus, the inverse of the displaement matrix U = [F αβ ] α,β=1,n is the oundary stiffness matrix K S of the soil whih is added to the finiteelement stiffness matrix K B of the struture to yield the gloal dynami stiffness matrix K S +K B of the omined struture-soil system, and the FEBE method is omplete. As an example, five different pile groups, whih onsist of 10 to 16 onrete piles of 0 m length and with radius of 0.5 m, are examined. The soil is haraterised y its shear wave veloity v S = 00 m/s (G = N/m, ν = 0.33). The frequeny-dependent ompliane of the pile group is given as amplitude and phase for vertial exitation of rigid (Fig. 1a) and flexile (Fig. 1) piles, and for horizontal exitation of flexile piles (Fig. 1) whih are rigidly onneted in a pile ap. At zero frequeny, the horizontal ompliane is more than -times higher than the vertial ompliane. The stati omplianes of a single pile (3.8, 5.9 and m/n for the vertial rigid, vertial flexile, horizontal fixed situation, not displayed) are higher than the ompliane of the pile group, ut not y a fator of 10 to 16. That means that a pile is more ompliant in a pile group than if it is ating alone. All omplianes are dereasing with inreasing frequeny. The phase delay is also inreasing with frequeny and reahes values of 60 to 130. These effets are strongest for the vertial rigid and weakest for the horizontal flexile pile situation. The strong phase delay for the vertial rigid piles indiates a mass effet of the soil mass that is surrounded y the piles. There are some flutuations with frequeny whih an e explained y waves travelling from one pile to another. Approximating methods for soil-struture interation The results from detailed soil-struture models are ompared with the results of a Winkler soil, a simple soil model whih supports the foundation y spring and damper fores P = k u + u. Under ertain irumstanes there is a good orrelation etween the exat ontinuum soil and the simplified Winkler results. A suessful approximation has een found for strip foundations (Auersh, 1988), railway traks (Auersh, 006), and eams (Auersh, 008). In ase of a plate, the Winkler modulus depends on the ending stiffness of the plate (Auersh, 1996). It should e emphasized that the approximation depends on exat results and is restrited to ertain aspets, for example the ompliane of a struture-soil system. As demonstrated for railway traks (Auersh, 005), for whih the Winkler model is widely used without a theoretial founding, the Winkler model annot represent all theoretial or measured effets. In this ontriution, the approximation of pile omplianes is demonstrated. The orresponding exat results of single piles in a ontinuum soil are given in Figure (Auersh, 008; Lüddeke et al., 009). The Winkler parameters are hosen and modified aording to the rules whih have een found for eams on the soil (Auersh, 008). Although this has not een optimised yet, it works quite well representing results of the time onsuming FEBEM alulation (Fig. ) y a simple (expliit) and fast approximating model (Fig. 3). At first, the vertial prolem should e examined separately for finite and infinite piles. The ompliane is onsideraly higher and the phase delay is stronger for the finite piles. The vertial Winkler solution is therefore given for finite piles in Figure 3a. The situation is ompletely different for the horizontal and roking prolem (Fig. 3-d), where the loaded pile head ehaves similar for a

4 a d Figure. Amplitude and phase of the a) vertial, ) horizontal, ) roking, and d) oupling ompliane of onrete piles (L = 10 m, R = 0.5 m) on different ontinuum soils with v S = 100, 150, 00, 300m/s.

5 a a a d d Figure 3. Amplitude and phase of the a) vertial, ) horizontal, ) roking, and d) oupling ompliane of infinite onrete piles (R = 0.5 m) on different Winkler soils with k = 1,.5, 4, N/m and = 0.8, 1., 1.6, Ns/m, a) vertial finite pile of L = 10 m, and with lower Winkler parameters k = G, = 4ρv S R.

6 finite and an infinite pile. The three omplianes of the infinite horizontal-roking prolem of Figure 3- an e given expliitly as u = α EI EI 1/ 4 3 / 4 1/ 1 / EI EI 1 / 1/ 3 / 4 1/ 4 P M with the dynami support parameter = + ' iω m' ω. The influene of the soil, whih is presented in Figures and 3, an also e derived from this formula. The strongest influene is found for the horizontal ompliane (u/p ~ k -3/4 ), the weakest influene of the soil is on the roking ompliane (α/m ~ k -1/4 ). The frequeny dependeny is ruled y the same laws, and the phase asymptotes an e otained as ϕ = -135 for the horizontal, ϕ = -90 for the oupling, and ϕ = -45 for the roking ompliane. All these phenomena are in good agreement etween the exat and approximating model. (4) 3 Predition software for railway indued ground and uilding virations The simple approximating models are integrated in a user-friendly, pratie-oriented predition software (Gersterger et al., 006), whih yields a quik response, and needs only a minimum input easily otainale for speialists and non-speialists. The properly oordinated modules for emission, transmission, and immission inlude the multi-eam trak model on Winkler soil (Auersh, 006), the layered dispersal soil model (Auersh, 005), and the soil-wall-floor model of a uilding (Auersh, 009). To onlude, omputational models should e as detailed as neessary, if the orret soilstruture interation has to e evaluated, and as simple as possile, if the oserved rules have to e implemented in a predition tool for railway indued uilding virations. Referenes AUERSCH, L, Wehselwirkung starrer und flexiler Strukturen mit dem Baugrund insesondere ei Anregung durh Bodenershütterungen. BAM-Researh Report 151/Thesis, Berlin. AUERSCH, L, ynami plate - soil interation - finite and infinite, flexile and rigid plates on homogeneous, layered or Winkler soil. Soil ynamis and Earthquake Engineering 15, AUERSCH, L., 005a. Simplified methods for wave propagation and soil-struture interation: The dispersion of layered soil and the approximation of FEBEM results. Pro. 6 th Int. Conf. on Strutural ynamis (EUROYN 005), Millpress, Rotterdam, pp AUERSCH, L., 005. ynamis of the railway trak and the underlying soil: the oundary-element solution, theoretial results and their experimental verifiation. Vehile System ynamis 43, AUERSCH, L., and 3-dimensional methods for the assessment of allast mats, allast plates and other isolators of railway viration. International Journal of Aousti and Viration 11, AUERSCH, L.; 008a. ynami interation of various eams with the underlying soil finite and infinite, half-spae and Winkler models. European J. Mehanis A/Solids 7, AUERSCH, L., 008. Stati and dynami soil-struture interation of finite and infinite eams and plates, traks and piles. Pro. 7th Int. Conf. on Strutural ynamis (EUROYN 008), Southampton, pp. 1-1 (C-ROM). AUERSCH, L., 009. Building response at foundation, walls, olumns and floors due to ground viration simplified alulation ased on experiene with detailed models and measurements. Pro. 16th Int. Congress on Sound and Viration, Krakow, pp. 1-8 (C-ROM). GERSTBERGER, U., AUERSCH, L., MEINHART, C., RÜCKER, W., 006. Ein einfah handhaares Prognoseprogramm für Shienenverkehrsershütterungen. VI-Berihte 1941 Baudynamik, VI Verlag, üsseldorf, pp LÜECKE, F., AUERSCH, L., BAESSLER, M., RÜCKER, W., 009. ynamishes Verhalten von Gründungen von Offshore-Windkraftanlagen. VI Berihte 063 Baudynamik, VI-Verlag, üsseldorf, pp

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