Effect of loadng frequency on the settlement of granular layer Akko KONO Ralway Techncal Research Insttute, Japan Takash Matsushma Tsukuba Unversty, Japan ABSTRACT: Cyclc loadng tests were performed both expermentally and numercally to study the effect of mpact loadng velocty on the settlement of a shallow granular layer lke a ballasted track. In the experment equal-szed sphercal balls were regularly stacked to make a granular layer to compare the result wth D Dscrete Element smulaton. It was found that resdual settlement after a certan loadng cycles ncreased wth ncreasng loadng velocty when we used the unfnshed pnballs wth rough surface. On the other hand, the experment wth polshed pnballs wth smooth surface gave an opposte tendency. The results of DEM smulatons mply that a volumetrc ncrease due to the strong mpact loadng accelerates the deformaton of the assemblage of unfnshed pnballs, whle t leads to consderable reversal of dsplacement durng the offloadng process n the polshed pnball assemblage. 1 INTRODUCTION Dfferental settlements of ballast layer causes hangng sleeper whch has a gap between sleeper and ballast layer. The hangng sleeper accelerates ballast deteroraton and ncrease waysde vbraton and nose. There are some methods for reducng the vbraton around the hangng sleeper, but the mechansm of the gap formaton has not been clarfed yet. Ths study s devoted to the hangng sleeper under the ral ont, whch s thought to be affected by mpact loadng by runnng vehcle. We developed an apparatus wth a hgh-capacty steppng motor to apply hgh frequency cyclc loadng. It can generate varous cyclc loadngs of varyng ampltude, duraton and nterval. The cyclc loadng tests employed two types of steel ball to avod the unevenness found n rregularly shaped ballast grans. Furthermore, a seres of smulatons by Dscrete Element Method were performed to show n detal the behavor of partcles durng cyclc loadng. EXPERIMENT Fg.1 shows an apparatus developed for hgh frequency cyclc loadng. These loadngs are appled by usng a steppng motor on the top of screw. Furthermore a box, 8 3 3 needs to be set up. Screw Load Cell for Measurng Steppng Motor Load Cell for Controllng Loadng Block Fg.1 Apparatus.1 Materal The box contaned two types Fg. Two types of steel balls for each case; pnballs and unfnshed pnballs as of pnballs shown n Fg.. These equal-szed pnballs have percent passng % 1 8 6 4 Pnball Unfnshed Pnball 1.98 11. 11. 11.4 11.6 11.8 11.1 Dameter of partcles Fg.3 Gran Sze Dstrbuton of pnballs
about 11 dameter but t s dspersed as shown n Fg.3. These pnballs are stacked up n a smple stagger pattern on a vertcal face as shown n Fg.4, but not staggered n depth to mtate twodmensonal condton.. Loadng Condtons Varous patterns of cyclc loadngs can be appled by hgh-capacty steppng motor, whch can control ampltude, duraton and nterval of cyclc loadng ndependently as shown n Fg.5. In ths study, the nterval and ampltude of cyclc loadngs are fxed for each tests of above mentoned materals. Only duratons are set for three patterns,.s, s and.s as shown n table1. The ampltude s set to 1kN..3 Results Fg.6 shows an example of the dsplacement-tme data from a test under.s duraton loadng. The dsplacement s represented by the settlement of the loadng block shown n Fg.1. Resdual Settlement, the plastc deformaton of the From the top From the front Loadng Block granular layer, wll be dscussed n ths study, so the relatonshp between Resdual Settlements and loadng cycles are plotted as open crcles n Fg.6. Fg.7(1) shows the one of the results from three tests n case1 usng unfnshed pnballs. It expresses that Resdual Settlement under cyclc loadng s the smallest under the.s duraton loadng and s the largest under the.s duraton loadng. It means that the faster loadng affect the larger deformaton for unfnshed pnballs, whch has rough surface. Fg.7() shows the one of the results from three tests n case usng well-polshed pnballs. It expresses that the Resdual Settlement s the smallest under the.s duraton loadng and s the largest under the.s duraton loadng. It means that the slower loadng affect the larger deformaton for smoothed pnballs. We carred out these tests tmes for each case usng those stacked-up pnballs. The results of each test are not so vared. Dsplacement of Loadng block. Dsplacement on loadng.15.1 Dsplacement off loadng Resdual Settlement. 4 6 8 1 tme s Fg.6 Resdual Settlement 34 balls Fg.4 Stacked Up Pnballs duraton, td nterval Fg.5 Loadng Pattern Ampltude Table1 Loadng Patterns Ampltude Materals Duraton Duraton,td Ampltude Materals Equal-szed,td Case Unfnshed 1kN 1 Equal-szed Case Pnballs Unfnshed 1.s Equal-szed Case Pnballs.s s Polshed s.s 1kN Equal-szed Pnballs Case.s Case Polshed Graded.kN 3 Pnballs Steel balls Interval Interval.5s.5s Resdual settlment Resdual settlment.16.1.16.1 4 Number of Cyclc loadng 5 (1) Unfnshed Pnballs 4 Number of Cyclc Loadng 5 () Polshed Pnballs Fg.7 Resdual Settlement from Experments under varous duraton cyclc loadngs
3 DEM SIMULATIONS 3.1 DEM Parameters In DEM, grans are modeled by dscrete rgd elements on whch contact forces affected by other attached elements and gravty act as shown n Fg.8. There are contact sprngs and dampers n the normal and tangental drecton and slders whch control slppage at contact ponts. DEM defne that those elements satsfy the equatons of motons whch are solved by tme steps. Table shows the parameters for DEM n ths study. Contact sprng coeffcent between elements and s defned as the equaton (1) 1*), whch s an approxmate equaton orgnally from Hertan Theory. 4b 1 Kn = 3π δ + δ 1 ν where, δ =, δ E π 1 ν =, b = 3 E π ----------------(1) *1) 3 (1 ν E ) r r P r + r In the equaton (1), E, the Young s modulus of the pnball s set as.5 1 11 N/m, and v, the Posson rato, s set as.8 *1). The pressure between attached elements s presumed to be about 36N consderng that 14 partcles attached to the loadng block receve.5kn at maxmum loadng. Ks s defned from the rato of the shear Young s modulus on the Young s modulus. The dampng coeffcent s defned n the equaton (), derved from the equaton of moton of mass pont havng sprng and damper. h c e b = exp π, h = ----() 1 h k m In the equaton (), e b, the coeffcent of resttuton of pnball s presumed to be.3 or.5 n ths case. k s set as the value of Kn to defne Cn. As for Cs, t s also defned from the equaton () n whch the value of k s Ks above descrbed. Frcton coeffcent s set to be.1,.3 and.6. As for steel balls, the frcton coeffcent s sad to be about.3. The smaller value.1 s presumed for the polshed pnball and the larger value.6 s presumed for the unfnshed pnball here. 3. Model Preparaton and Loadng patterns At the seres of DEM smulatons, crcle elements whch has same gran sze dstrbuton of real pnballs, as shown n Fg., are stacked up as same as the pnballs at the experments as shown n Fg.4. The patterns of cyclc loadngs are also appled as same as the experments. At these DEM smulatons, the loadng ampltude s set to be half value at the experments because there are two vertcal surfaces n depth at the experments. Fg.9 shows examples of the load-tme data from the DEM smulatons. 3.3 Results 3.3.1 Resdual Settlement Fg.1 shows the relatonshp between Resdual settlement of the loadng block and loadng cycles from those DEM smulatons. Fg.-(1), the upper three fgures shows the results usng the bgger dampng coeffcent and Fg.-(), the lower three fgures shows the results usng the smaller dampng coeffcent. Fg.-(a), the two fgures on the left, shows the results usng the frcton coeffcent of.1, Fg.-(b), two fgures on the mddle, shows the results usng frcton coeffcent of.3, and Fg.-(c), two fgures on the rght shows the results usng the frcton coeffcent of.6. Those express that the crcle assemblage whch has the frcton coeffcent of.6 deforms largely under the.s duraton loadng. Oppostely, the same crcle assemblage whch has the frcton coeffcent of.1 deforms largely under the.s duraton loadng. Ths qualtatve tendency s n good agreement wth the results from the experments. As for the quanttatve results, Resdual Settlement from the DEM smulaton s about 6-8 percent of those from the experments. The dfference seemed to be caused by the condtons of pnball assemblage at the experment whch s not completely twodmensonal condton. Loadng ampltude N 5 4 3 1 C s C n Fg.8 DEM model td=.s K s K n td=.s 1.5 1.6 1.7 1.8 1.9 tme s Fg.9 Examples of loadng-tme data from DEM Table Parameters for DEM Sprng Coeffcent (normal) N/m Sprng Coeffcent (shear) N/m 1.48 1 7 5.78 1 6 Dampng Coeffcent (normal) N Dampng Coeffcent (shear) N s/m Frcton Coeffcent 4.5 17.8.1.3 13 76.8.6
Resdual settlement Resdual settlement Loadng Cycles (a) µ=.1 (b) µ=.3 (c) µ=.6 (1) Cn=4.5 N s/m, Cs=17.8 N s/m Loadng Cycles Resdual settlement Resdual settlement Loadng Cycles Loadng Cycles Fg.11 Resdual settlement from DEM smulaton under varous duraton cyclc loadngs Resdual settlement Resdual settlement Loadng Cycles Loadng Cycles (a) µ=.1 (b) µ=.3 (c) µ=.6 () Cn=13 N s/m, Cs=76.8 N s/m 3.3. Dsplacement on-loadng and off-loadng Fg.11 shows the dsplacement-tme data from the DEM smulaton. Fg.-(1) shows the result usng smooth crcles wth the frcton coeffcent of.1 and Fg.-() shows that of usng rough crcles wth the frcton coeffcent of.6. As for the smooth crcle assemblage, the dsplacements at maxmum loadng are same at two cases under the.s or the.s duraton loadng. After that, durng off-loadng, the deformaton reverses largely under the.s duraton loadng. Then Resdual settlement s small under the short duraton loadng for the case usng smooth crcles. The other sde, the dsplacement of rough crcle assemblage at maxmum loadng s larger under the.s duraton loadng. The dfferences between those maxmum dsplacements under the.s and the.s duraton loadng are almost constant at every cycle. However, the reversble dsplacement under the.s duraton loadng decreases gradually. The reversble dsplacement s large at the 1st and the nd cycles, so Resdual settlement s smaller than under the. duraton loadng. Then the reversble dsplacement under the.s duraton loadng decreases wth loadng cycles. So Resdual settlement under short duraton loadng ncreases gradually and goes over that of under long duraton loadng at the latter stage. 3.3.3 Partcle Movement Fg.1 (1) and () show partcle movement between before loadng and at the maxmum loadng and after loadng at the 3rd cycle. Fg.-(1) shows the Dsplacement Dsplacement. -.3 - - -.1. -.3 - - -.1.5 tme s 1 1.5 (1) =.1 td=.s eb=.5 µ=.1 eb=.5 µ=.6.5 tme s 1 1.5 () =.6 Fg.11 Dsplacement-tme data from DEM result under the.s duraton loadng wth smooth crcles, Fg.-() shows the result under the.s duraton loadng wth rough crcles and Fg.-(3) shows the result under the.s duraton loadng wth rough crcles. The length of arrows ndcates 1 tmes of real dsplacement. Fg.-(1) shows that the smooth crcles under short duraton loadng deforms and reverses dramatcally. Fg.-() shows that the rough crcles under short duraton loadng deforms largely on-loadng, but reverses gradually off-loadng, compared wth the rough crcles under long duraton loadng as shown n Fg.-(3) on loadng.
Depth m Depth m Depth m 5 5..1.15. poston m.5.3.35 5 5 Loadng Block..1.15. poston m.5.3.35 5 5 Loadng Block Loadng Block..1.15. poston m.5.3.35 Loadng depth m depth m 5 5 (1) µ =.1, duraton =.s..1.15..5.3.35 poston m 5 5 () µ =.6, duraton =.s depth m Fg.1 Partcle movement at the 3rd cycles Loadng Block Loadng Block..1.15..5.3.35 poston m 5 5 Loadng Block..1.15..5.3.35 poston m Unloadng (3) µ =.6, duraton =.s 3.3.4 Contact and slppng pont Fg.13 shows the rato of the number of contact ponts and slppng pont on the ntal contact ponts at the case of rough crcles. It shows that the number of contact pont decrease durng on-loadng or offloadng, especally under the. duraton loadng. The other hand, the numbers of slppng ponts ncrease durng on-loadng or off-loadng. 3.3.5 Energy Balance Fg.14 shows energy balance *) durng cyclc loadng wth the.s and the.s duraton loadng usng rough crcles. It s calculated by DEM smulaton and the balance express error of the calculaton. At the case of the.s duraton loadng, the dampng energy and potental energy are large compared wth the case of the.s duraton loadng. From that results, volumetrc deformaton may be predomnant at the gran assemblage under the. duraton loadng, fast loadng n other word. As for the gran assemblage under the.s duraton loadng, slow loadng, the shear deformaton may be predomnant. The rato of the number of contact or slppng ponts on the ntal contact ponts 1.8.6.4. contact pont at slppng pont at td= s.5 1 1.5 tme s Fg.13 the rato of the number of contact or slppng ponts on the ntal contact ponts
4 CONCLUSION Cyclc loadng tests were performed expermentally usng the equal-szed sphercal balls and also smulated by DEM. The fndngs are as follows. (1) The Resdual settlement of ralway ballast layers s rapd at frst and decrease gradually under cyclc loadng. The resdual settlement vs loadng cycle curve gven by experment and smulaton n ths study shows the same tendency as ballast layers do. () However, ths tendency s slght n the case of granular layer composed of stacked-up equal-szed pnball, n ths study, compared wth real ballast layers. (3) As for rough pnballs, the resdual settlement s large at frst under slow loadng, but the later settlement ncrease mnmally compared wth the same under fast loadng. (4) The smooth pnball assemblage deforms on loadng and reverse off-loadng dramatcally under fast loadng, but rough pnballs doesn t reverse as much as smooth polshed pnballs durng off-loadng. (5) Ths tendency seems to relate to that volumetrc deformaton s predomnant under fast loadng and shear deformaton s predomnant under slow loadng. REFERENCES 1) The Socety of Powder Technology ; Funta smulaton Nyumon (n Japanese), 1998 ) T.Matsushma et.al ; Gran-shape Effect on Peak Strength of Granular Materals, Computer method and Advances n Geomechancs, pp361-366,1 Energy N m Energy N m 1.E- 5.E-3.E+ -5.E-3-1.E- -1.5E- 1.E- 5.E-3.E+ -5.E-3-1.E- -1.5E- StranE StranE knetc E workload by cyclc loadng.5 1 1.5 tme s knetc E loss by nter-patcle Dampng E frcton Dampng E workload by cyclc loadng potental E (1) loss by nter-patcle frcton potental E.5 1 1.5 tme s () Fg.14 Balance of Energy Balance Balance