DECOMPOSITION OF MEASURED GROUND VIBRATIONS INTO BASIC SOIL WAVES

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1 DECOMPOSITION OF MEASURED GROUND VIBRATIONS INTO BASIC SOIL WAVES D. Macjausas Department of Scence & Technology, Unversty of Luembourg, Luembourg S. Van Baars Department of Scence & Technology, Unversty of Luembourg, Luembourg ABSTRACT: Man-made vbratons from dfferent types of sources are usually measured on the surface of the ground or buldng. The measured sgnal s always the superposton of all travellng basc waves. For a homogeneous half space there are three basc waves the Compressonal (P-wave), Shear (S-wave) and Raylegh wave (R-wave). Dependng on the measurng equpment, only the acceleratons or veloctes n tme of the superposed wave can be measured, but not the dstrbuton of the ndvdual basc waves. Addtonal problems are that each of the basc waves has ts own velocty, besdes the body and surface waves have dfferent attenuaton laws. By usng the rules of superposton of harmonc waves and also the propagaton laws of the P-, S- and R-waves, t should be theoretcally possble to splt the measured superposed sgnal nto the basc waves, because mathematcally a system of equatons can be assembled whch descrbes the dsplacements at multple measurng ponts n tme. In ths paper ths problem has been solved for a homogenous, elastc and sotropc sol, whch s dsturbed by a harmoncally oscllatng dsc on the surface. A numercal smulaton was performed usng a fnte element method. The dsplacements n tme were recorded n 10 ponts on the surface and a system of superposed equatons was assembled and solved. The fndngs prove that each of the three basc waves has ts own phase shft wth the source, somethng whch was not nown before. 1 INTRODUCTION In urban areas where the nfrastructure s dense and constructon of new structures s near estng ones, the vbratons caused by human actvtes frequently occur. Generated waves n the sol may adversely affect surroundng buldngs. The most nown sources of the human nduced vbratons are traffc (trans, buses, trucs) and cvl constructon actvty (nstallaton of ples, sheet ples, tunnellng, demolshng of structures, etc.). Because of envronmental requrements the level of the vbratons should not eceed certan threshold values n order to protect people from dscomfort, the estng structures from damage and technologcal processes from dsturbance. Ths means that the level of vbraton must be predcted. Accordng to a research by Hölscher and Waarts (2003) the relablty of predcton methods s dsappontngly low, so t s evdent that the sol models for vbratons should be mproved frst n order to get more accurate predctons. Accordng to theory, the vbratons n a homogeneous half space are comple superposed waves, made up of body and surface waves, whch are the followng three basc waves the Compressonal (P-wave), Shear (S-wave) and Raylegh wave (R-wave). These three basc 462

2 waves have dfferent attenuaton, dfferent propagaton laws (wave speed) and dfferent dampng laws. The attenuaton and propagaton laws have been solved already, but the dampng laws are stll subject of study. Bolton & Wlson (1990) concluded from cyclc tests on sand that the stress-stran behavour of the sol s hysteretc and ts correspondng dampng parameters are stran dependent, though they are ndependent of the frequency up to 100 Hz. Van Baars (2011) notced that man-made vbratons result n shear deformatons wth an ampltude no greater than Van Baars used results from other researchers (Our & Ansal, 2007, for nstance) to demonstrate that fortunately the dampng rato s nearly constant up to ths magntude. Hysteretc dampng s n fact a dampng resultng from frctonal shear deformaton. Isotropc compresson causes probably lttle or no dampng. In P -waves there s a deformaton due to mostly sotropc compresson but also some shear deformaton, n S-waves there s only pure shear deformaton and n R-waves there s a combnaton of mostly shear deformaton but also deformaton due to sotropc compresson. Therefore the energy n the basc waves wll not only attenuate dfferently and propagate dfferently, but also damp dfferently. So, n order to have a better nsght nto the propagaton of vbratons n a sol medum, the measured sgnal has to be decomposed nto the basc waves and analysed separately. A man problem n the feld s that, not ndependent waves are measured, but only superposed veloctes or acceleratons. So n order to study the measurements, a way to decompose the superposed waves nto basc waves, should be found frst. 2 DECOMPOSITION OF WAVES In ths paper ths problem has been solved for a homogenous, elastc and sotropc sol, whch s dsturbed by a harmoncally oscllatng dsc on the surface, see Fgure 1. Fg. 1. Setch of an oscllatng plate and measurement ponts The dsplacement of each pont on the sol surface n drecton can be descrbed as a superposton of the dsplacements of the three basc waves n that pont: u r t u r, t u r, t u r t, p, s, r,, (1) where u p,, u s,, u r,, are the dsplacements of the P-, S- and R-wave respectvely. By tang nto account the propagaton laws of the general waves, Equaton (1) can be rewrtten for any measurement pont as follows:, sn sn cos u r t u t r u t r u t r p, p p s, s s r, r r, sn sn sn u r t u t r u t r u t r y p, y p p s, y s s r, y r r (2) 463

3 where s the ampltude of the j wave n the drecton, s the angular frequency of the u j, harmonc wave, j s the j wave s number, j s the phase shft of a the j wave. In order to relate the measured vbratons of the dfferent ponts wth dfferent dstances, the attenuaton laws of the basc waves wll be used. The ampltudes of the body waves (Pand S-waves) attenuate proportonal to -1 and the surface wave (R-wave) attenuates proportonal to -1/2, where s the dstance from the as of symmetry, whch s the mddle of the dsc (Fg. 1). For relatvely small dscs follows = r. Now vbratons n any measurement pont = 1 n can be epressed as functons of ampltudes n any other pont, for eample the 1 st pont: u r t u t r u t r, sn sn 1 1, p,,1 p p s,,1 s s u cos t r r,,1 r r u r t u t r u t r, sn sn 1 1 y, p, y,1 p p s, y,1 s s u sn t r 1 r r 1 1 (3) The bac-calculated superposed sgnal can be used as a chec, as well as the theoretcal rato of the R-wave s ampltudes on the surface u r,,. 3 NUMERICAL SIMULATION In order to chec ths technque, a 2-dmensonal, aal symmetrcal numercal smulaton was performed wth the Fnte Element Method software Plas. The advantage of a numercal smulaton s that the materal dampng can be put to zero. The general forcedsplacement matr n Plas s based on the followng equaton: M u C u K u F (4) where [M] s the mass matr, {u} s the dsplacement vector, [C] s the dampng matr, [K] s the stffness matr and {F} s the load vector. Frst and second dervatves of dsplacement vector are velocty and acceleraton vectors respectvely. The dampng matr [C] represents the materal dampng. In Plas 2D, Raylegh dampng s used, where [C] s a functon of the mass and stffness matrces, accordng: C M K (5) where and are coeffcents, whch are ept zero n ths smulaton, to avod any type of materal dampng. The used geometry and mesh of the model can be found n Fgure 2. The model s 50 m n length and wdth. The 10 measurement ponts for dsplacement recordng were placed from 15 m to 24 m, at 1 m dstance from each other. The sol s modelled wth 15-node elements havng a Young s modulus E = MPa and a Posson s rato = The oscllatng 464

4 dsc has a radus of 0.2 m, whch s modelled wth a plate element wth stffness EI = 24 MNm 2 /m. Frst the dsc s loaded wth a statc load of 20 Pa and later a harmonc load of ±10 Pa at 10 Hz s ntroduced. Fg. 2. Geometry and mesh of the FEM model 4 RESULTS The dsplacements were frst used after about 3 cycles, so that the startng up effect of the harmonc load has vanshed. The tme wndow for the measurement was also selected such that are no reflectons yet from the absorbng boundares, snce they do not absorb perfectly. 4.1 Frst attempt The dea of the frst attempt was to use only the dsplacements whch were measured at a selected tme when the dsplacement reached a pea. It was assumed that the three basc waves were n phase wth the orgnal/superposed wave, smlar as n the analytcal soluton of Mller and Pursey (1955). Horzontal dsplacement [m] 1.E-06 8.E-07 6.E-07 4.E-07 2.E-07-2.E-07-4.E-07-6.E-07-8.E-07-1.E-06 u_bc u_plas Tme [s] Vertcal dsplacement [m] 3.E-06 2.E-06 2.E-06 1.E-06 5.E-07-5.E-07-1.E-06-2.E-06-2.E-06-3.E-06 uy_bc uy_plas Tme [s] Fg. 3. Frst attempt: bac-calculated results vs. Plas 465

5 The correlaton factors are very low. The factor of the vertcal component even becomes negatve, whch means there s no good soluton. Also the rato of the R-wave s ampltudes u r,, = 3.73 whch s much greater than the theoretcal u r,, = Only for the selected tme (the pea of the dashed lne) the soluton almost ftted, but clearly not for the rest of the wave. 4.2 Attempt wth phase shfts A new attempt was done, but ths tme wth allowng the basc waves to have dfferent phases as the orgnal/superposed wave. Ths leads to nne unnowns (s ampltudes and three phase shfts). Snce the phases of the basc waves are now unnowns, the system of equatons s not lnear anymore, so the soluton had to be found by usng the least square method n an teratve way. In ths attempt not only the pea values of the tme-dsplacement graph, but all values of one cycle were used. One cycle T = 0.1 s. The calculatons were performed wth tme steps of t = s, whch gves 100 tme steps or ponts n a tme-dsplacement graph. Snce there are 10 measurement ponts wth each 2 equatons (horzontal and vertcal); there s a system of 2000 equatons. The horzontal and vertcal dsplacements of the basc waves were solved and are shown n Fgure 4. Horzontal dsplacement [m] 8.E-07 6.E-07 4.E-07 2.E-07-2.E-07-4.E-07-6.E-07-8.E-07 P-wave S-wave R-wave Tme [s] Vertcal dsplacement [m] 1.E-06 5.E-07-5.E-07-1.E-06 Tme [s] P-wave S-wave R-wave Fg. 4. Phase shfts: horzontal and vertcal dsplacements of the basc waves Ths tme a perfect ft of the wave dsplacements was found for the bac calculated wave wth the recorded wave of Plas. The correlaton factor R 2 s equal to for the combned dsplacements. Vertcal dsplacement [m] 2.E-07 P-wave 1.E-07 S-wave -1.E-07-2.E-07-2.E-07-1.E-07 1.E-07 2.E-07 Horzontal dsplacement [m] Vertcal dsplacement [m] 1.E-06 R-wave 5.E-07-5.E-07-1.E-06-1.E-06-5.E-07 5.E-07 1.E-06 Horzontal dsplacement [m] Fg. 5. Phase shfts: dsplacements on the surface of one cycle of the three basc waves 466

6 Fgure 5 shows the dsplacements on the surface of one cycle of the three basc waves n pont 1. Interestng s to note that both the P-wave and the S-wave do not act completely flat (horzontal for the P-wave and vertcal for the S-wave). Also nterestng s the rato of the R-wave s ampltudes s found to be u r,, = 0.655, whch s n a qute good agreement wth the theoretcal u r,, = Attempt wth flat waves The same attempt was done, but ths tme wth the assumpton that the body waves are flat, so the horzontal ampltude of the S-wave and the vertcal ampltude of the P-wave are zero. Also the rato of the theoretcal R-wave s ampltudes was used, so u r,, = Ths reduces the amount of unnowns bac to s (three ampltudes + three phase shfts). Horzontal dsplacement [m] 8.E-07 6.E-07 4.E-07 2.E-07-2.E-07-4.E-07-6.E-07-8.E-07 P-wave R-wave Tme [s] Vertcal dsplacement [m] 1.E-06 S-wave 5.E-07 R-wave -5.E-07-1.E-06 Tme [s] Fg. 6. Flat waves: horzontal and vertcal dsplacements of the basc waves Fgure 6 shows the dsplacements on the surface of one cycle of the three basc waves n pont 1. Table 1. Effect of the reducton of unnowns Wave Drecton Unnown 9 unnowns 6 unnowns Unts P h A p [m 10-6 ] S v A sy [m 10-6 ] R h A r [m 10-6 ] P v A py [m 10-6 ] S h A s [m 10-6 ] R v A ry from rato [m 10-6 ] P - p [deg] S - s [deg] R - r [deg] Rato of the R-wave ampltudes A r / A ry [ - ] mean(r 2 ) Forcng the body waves to be flat and the rato of R-wave s ampltudes to be as the theoretcal one, dd not change the correlaton factor very much. The combned mean 467

7 Ampltude multpler [Ͳ] correlaton factor for the horzontal and vertcal dsplacements s stll very hgh R2 = The ampltudes and phase shfts are shown n Table 1. For most of them the dfference s rather small, ecept for the S-wave s phase shft, whch was found to have a dfference of 25. The basc waves are shown n Fgure 7 for one cycle when the sol body oscllates harmoncally. As can be seen, the P- and S-waves are pror to the dsc loadng, but the Rwave s delayed. The values of the phase shfts shown n Fgure 7 are the ones of Table Ͳ0.2 Ͳ0.4 Ͳ0.6 Ͳ0.8 Ͳ1.0 Dsc Load PͲwave RͲwave SͲwave y RͲwave y Ͳ0.5 ǻijs ǻijr ǻijp Ͳ0.25 ǻijry 0 Perod Fg. 7. Phase shfts of the basc waves 5 ENERGY BALANCE Snce the wave s now decomposed, the energy balance can be checed. Snce there s no materal dampng, the emtted energy from the oscllatng dsc on the surface and the energy carred by the basc waves should be n balance (prncple of energy conservaton). 5.1 Total Energy Frst of all the energy emtted from the source was calculated. For dsplacement recordngs one pont on the dsc was selected. One pont s suffcent, because the dsc s stff enough, so ts own deformatons are neglgble Load [N] E-05-4.E-05-2.E-05 2.E-05 Vertcal dsplacement [m] Fg. 8. Total emtted energy per cycle E-05 6.E-05

8 The emtted energy per cycle from the oscllatng dsc was calculated from the load-dsplacement ellpse, whch represents the wor per cycle (Fg. 8). The total emtted energy was found to be Nm per cycle. 5.2 Energy n the basc waves The energy n the R-wave was calculated by usng the analytcal soluton of the R-wave s ampltudes n depth. These functons can be found n boos of sol dynamcs, le Kramer (1996), for nstance. In order to estmate the total energy n the R-wave per cycle E tot, r, the functons of the squared ampltudes (energy s proportonal to the squared of the ampltudes) are ntegrated over depth for one wave length: E tot y 2 2 2r r u u y 1 2, r dy (6) 2 where s the densty of the medum, s the angular frequency of the harmonc wave, û, û are the ampltudes of the R-wave n and y drecton respectvely, r s the radus, r s the y length of the R-wave. Unfortunately the analytcal ampltude functons of the P- and S-waves, used by Mller and Pursey (1955), est only for a very large radus, where the ampltudes are equal to zero on the surface. Ths s clearly not the case here. Therefore frst the ampltude functons of the P- and S-waves have to be constructed. Therefore the dsplacements were recorded n 19 addtonal ponts, placed n the sol volume (see Fg. 9 on the lef. By usng the ampltudes of the R-wave at the surface and the theoretcal ampltude functons n depth, the dsplacements of the R-wave were calculated for the same 19 ponts. These dsplacements n tme of the R-wave were subtracted from the recorded total dsplacements. The resdual dsplacements n and y drectons were projected onto the and y as (see Fg. 9 on the rgh, whch are the drectons of the moton of an ndvdual partcle of the P- and S-waves, respectvely. In ths way, for the P- and S-waves, the ampltudes of the dsplacements n and y aes were calculated. y m u u Coordnate system -y at any pont y' y ' Dsplacements n u-v coordnate system ', y', u, sn( ) u y, u, cos( ) u y, cos( sn( ) ) Fg. 9. Addtonal ponts for the ampltudes of S- and P-waves 469

9 The total energy n the P- (or S-wave) per cycle E tot, p( s) can be calculated by ntegratng the squared functons of the ampltudes (energy s proportonal to the squared of the ampltudes) over a surface of a half ball for one wave length: / tot, p( s) p( s) p( s) sn E r u r d (7) where p(s) s the P- (or S-wave s) wave length, u p( ) s the ampltude of the P- (or S-wave), s the polar angle. The total amount of energy carred by the basc waves was summed and found to be Nm per cycle, whch s almost the same as the total emtted energy of Nm per cycle. 5.3 Dstrbuton of Energy n Waves Also the dstrbuton of the total energy n the basc waves was checed. Ths problem was solved analytcally by Mller and Pursey (1955), for a sol wth Posson s rato = 0.25, but ths has never been checed numercally. The percentages of the total energy dstrbuton n the basc waves can be found n Table 2. s Table 2. Dstrbuton of the total energy n the basc waves Soluton P-wave [%] S-wave [%] R-wave [%] Analytcal FEM As can be seen from ths table, the dstrbuton of the energy based on the analytcal soluton of Mller and Pursey fts reasonably well to the numercal soluton, despte the fact that Mller and Pursey were not aware of the phase shfts of the basc waves. The small dfferences could be eplaned by dfferent ntal condtons between the solutons. Frst, n the analytcal soluton no phase shfts were consdered. Second, a very small and fleble dsc wth nfnte small radus was used, whle n the numercal smulaton a rgd dsc was used. Thrd, numercal methods always have resdual errors. 6 CONCLUSIONS The eample n ths artcle dscusses a harmoncally oscllatng dsc on a homogeneous half space. The recorded superposed sol wave can be decomposed nto ts basc waves when multple geophones are used. From the recorded data a system of non-lnear equatons can be assembled wth s unnown parameters (three ampltudes and three phase shfts). These s parameters can be solved by usng an teratve way of the least square method. Ths leads to a decomposton nto the three basc waves, wth each ts own ampltude and phase shft. The superposton of only these basc waves descrbes very accurately the recorded superposed sol wave, proofng the estence of only three basc waves. The fndngs proof also that all three basc waves have phase shfts and these phase shfts are all dfferent from each other. Both facts were not nown before. Although t s mportant to notce the estng of the ndvdual phase shfts, t s stll not clear what causes them. An energy balance showed that the amount of emtted energy by the load on the dsc s the same as of the sum of energes of the basc waves. Ths s another type of evdence that only 470

10 three basc waves est. The dstrbuton of the energy over the three basc waves based on the analytcal soluton of Mller and Pursey (1955) fts reasonably well to the numercal soluton shown n ths artcle, despte the fact that Mller and Pursey were not aware of the phase shfts of the basc waves. The major part of the energy (more than 60 %) from a vertcally oscllatng dsc on the surface goes nto the R-wave. REFERENCES Journal Our, D.V. & Ansal A. (2007), Stffness degradaton of natural fne graned sols durng cyclc loadng, Sol Dynamcs and Earthquae Engneerng, Vol. 27(9), Boo Kramer, S.L. (1996), Geotechncal Earthquae Engneerng. Prentce Hall, New Jersey. Proceedngs Bolton M.D. & Wlson J.M.R. (1990), Sol stffness and dampng, Proceedngs of The Internatonal Conference on Structural Dynamcs, Eurodyn 90, Unversty of Bochum, 1, , Mller, G.F. & Pursey, H. (1955), On the Partton of Energy between Elastc Waves n a Sem-Infnte Sold, Proceedngs of the Royal Socety of London. Seres A, Mathematcal and Physcal Scences, Vol. 233(1192), Van Baars, S. (2011), Modellng of frctonal sol dampng n fnte element analyss, Proceedngs of The Second Internatonal Symposum on Computatonal Geomechancs, Croata, ISBN , Aprl Other Hölscher, P. & Waarts, P. H. (2003), Relablty of vbraton predcton and reducng measures, Fnal Report. Delft Cluster, Delft. 471