sustainability

Article Control of Metro Train-Induced Vibrations in a Laboratory Using Periodic Piles

Meng Ma 1,2,* , Bolong Jiang 3, Weifeng Liu 1,2 and Kuokuo Liu 2

1 Key Laboratory of Urban Underground Engineering of Ministry of Education, Beijing Jiaotong University, Beijing 100044, China; wfl[email protected] 2 School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China; [email protected] 3 Rail Transit Digital Construction and Measurement Technology National Engineering Laboratory, China Railway Design Corporation, Tianjin 300308, China; [email protected] * Correspondence: [email protected]



Received: 30 June 2020; Accepted: 20 July 2020; Published: 21 July 2020


Abstract: Laboratories with sensitive instruments need a low-vibration environment. It is a challenge to control the train-induced vibration impact on these instruments when a newly planned metro line is adjacent to a laboratory building. An alternative method of mitigating train-induced ground vibrations involves installing measures along the transmission path. Recent research has highlighted the potential of periodic pile barriers with specifically designed band gaps for controlling environmental vibrations. This study performed in-situ measurements of ambient vibrations inside and outside a laboratory containing various types of sensitive instruments and located adjacent to a newly designed metro line. The vibration transfer function of the laboratory was then obtained. To help design and optimize the band gaps of periodic piles, a novel band gap performance evaluation function was proposed. Finally, numerical analysis was conducted to validate the mitigation effect of the designed periodic piles. The results showed that the band gap performance evaluation function can be used to optimize the mitigation effect of periodic piles. The proposed periodic piles clearly attenuated vibrations between 52.4 and 74.3 Hz, especially those at 63 Hz. A comparison of general vibration criteria (VC) curves revealed that vibration attenuation of one level can be obtained by the designed periodic piles.

Keywords: environmental vibration; metro train; sensitive instrument; vibration control; periodic piles; band gap theory

1. Introduction In recent years, the problem of train-induced vibrations have been paid increasing attention due to its potential impacts on the comfort of nearby residents [1,2], the long-term protection of historic buildings [3,4], and the operation of precision instruments [5,6]. For laboratories containing sensitive equipment such as high sensitivity weighing equipment, electronic microscopes, and magnetic resonance imaging instruments, controlling vibrations induced by rail and road traffic is a challenging issue. Previous studies have reported this problem; for example, Wolf [7] introduced a case study of the United States, where the planned Sound Transit Link Light Rail system was due to pass through the University of Washington. In Taiwan, the bullet train was designed to pass through Tainan Industrial Science Park [8]; as such, a honeycomb wave impeding barrier (WIB) was proposed to mitigate the resulting train-induced vibrations [9]. In Beijing, due to the impact of the planned metro line 4 on sensitive microscopes in a physics laboratory of Peking University, Gupta et al. [5] and Liu et al. [10] conducted numerical predictions and experimental analyses. Furthermore, Ding et al. [6] predicted train-induced vibrations in a laboratory in Beijing near metro line 8.

Sustainability 2020, 12, 5871; doi:10.3390/su12145871 www.mdpi.com/journal/sustainability Sustainability 2020, 12, 5871 2 of 20

Sustainability 2020, 12, x FOR PEER REVIEW 2 of 22 Sustainability 2020, 12, x FOR PEER REVIEW 2 of 22 Sustainability 2020, 12, x FOR PEER REVIEW 2 of 22 Generally,Generally, itit isis di difficultfficult to to control control vibrations vibrations in in a laboratorya laboratory with with sensitive sensitive instruments, instruments, especially whenespecially theGenerally, laboratory when theit is laboratory isdifficult located isto near located control a rail near vibrations transportation a rail transportation in a laboratory system. system. 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[28], et al. [18] Yuan et al. [30] et al. [18] FSTFST withbearing with rubber rubber bearing Yuan et al. [30] FST with rubber Yuan et al. [30] FSTFST +bearing +tunedtuned slab slab XuYuan [20], et ZhuXu al. [ 20[30]et], al. Xu [20], ZhuXu [et20 ], bearing damper/massdamperFST + tuned/mass block slab block XuZhu [20],[31,32], et Zhu al. [et31 al.,32 ], Xu al.[20],Zhu [32], Zhu et al. et [32], FST + tuned slab Xu [20], Zhu et al. Xu [20], Zhu et damper/mass block [31,32], al. [32], damper/mass block [31,32], al. [32], Other alternative measures include those taken along the transmission path, including Other alternative measures include those taken along the transmission path, including continuousOther [35,36]alternative and discontinuousmeasures include [37–39] those soil batarriers.ken alongTraditional the transmission measures in thepath, transmission including continuousOther [alternative35,36] and measures discontinuous include [37 those–39] soiltaken barriers. along Traditionalthe transmission measures path, in including the transmission pathcontinuous are continuous [35,36] and soil discontinuous barriers along [37–39]the railway soil balinrriers.e. Table Traditional 2 lists relative measures research in the on transmission continuous continuous [35,36] and discontinuous [37–39] soil barriers. 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Table 2 lists relative research on continuous vibrationbarriersvibration [14,40], mitigation mitigation stiff wave measures measures barrier along along [41], the hollow the propagatio propagation isolationn path, wall path, including [42], including subgrade open stiffeningtrench open trench[14,40], or WIB [soft-filled14 ,[9,43–40], soft-filled vibration mitigation measures along the propagation path, including open trench [14,40], soft-filled barriers46],barriers sheet [ 14[14,40], pile,40], wall stistiffff [47],wavewave and barrier heavy [41], [41 masses], hollow hollow along isolation isolation the walltrack wall [42], [48]. [42 subgrade ],In subgrade Table stiffening 2, almost stiffening or all WIB orresearch WIB[9,43– [ 9,43–46], barriers [14,40], stiff wave barrier [41], hollow isolation wall [42], subgrade stiffening or WIB [9,43– considered46], sheet pilethe at-grandwall [47], or and elevated heavy line, masses but notalong the theunderground track [48]. line.In Table Generally, 2, almost measures all research in the sheet46], sheet pile wallpile [wall47], and[47], heavyand heavy masses masses along along the trackthe track [48]. [48]. In Table In Table2, almost 2, almost all research all research considered transmissionconsidered the path at-grand are believed or elevated to only line, be butsuitab notle the for undergroundsurface railway line. lines, Generally, where Rayleighmeasures waves in the theconsidered at-grand the or at-grand elevated or line, elevated but not line, the but underground not the underground line. Generally, line. 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[58] etthe performed al. vibration [39] However, calculated an mitigation experimental whether the vibrationeffect periodic study of periodically re pile toduction svalidate could effect arranging mitigate the using calculated building pilesburied in band saturatedperiodicvibrations gaps barriersof soil. against soil-periodic Moreover, against train- arranging piles in saturated soil. Moreover, Ma et al. [58] performed an experimental study to validate inducedrailwaypiles.Ma et Albino al. vibrations. vibrations, [58] et performed al. [39] However, especially calculated an experimental whether from the an vibration periodicundergroun study repile toduction dsvalidate couldrailway effect mitigate the line, using calculated still building buried needs band periodicvibrationsto be gaps investigated. barriersof against soil-periodic against train- the calculated band gaps of soil-periodic piles. Albino et al. [39] calculated the vibration reduction inducedrailwaypiles. Albino vibrations.vibrations, et al. [39] However,especially calculated whether from the an vibration periodicundergroun repileductionds couldrailway effect mitigate line, using still building buried needs periodicvibrationsto be investigated. barriers against against train- effect using buried periodic barriers against railway vibrations. However, whether periodic piles could inducedrailway vibrations.Tablevibrations, 2. Research However,especially on continuous whetherfrom an vibrationperiodicundergroun mitigatipileds couldrailwayon measures mitigate line, alongstill building needs the propagation vibrationsto be investigated. path. against train- inducedmitigate Table buildingvibrations, 2. Research vibrations especially on continuous against from train-inducedan vibration undergroun mitigati vibrations,d railwayon measures especiallyline, alongstill needs fromthe propagation anto undergroundbe investigated. path. railway Measures along the Theoretical and In-Situ line, stillTable needsSketch 2. to Research be investigated. on continuous vibration mitigation measures along the propagation path. MeasuresPropagation along Path the NumericalTheoretical Study and MeasurementIn-Situ TableSketch 2. Research on continuous vibration mitigation measures along the propagation path. MeasuresPropagation along Path the NumericalTheoretical Study and MeasurementIn-Situ TableSketch 2. Research on continuous vibration mitigation measures along the propagation path. MeasuresPropagation along Path the NumericalTheoretical Study and MeasurementIn-Situ Sketch MeasuresPropagation along Path the ThompsonTheoreticalNumerical et andal. Study[14], Yang Measurement Sketch Open trench In-Situ Measurement Propagation Path ThompsonNumericalet al.et Study al.[40] [14], Yang Open trench Thompsonet al.et al.[40] [14], Yang Open trench Thompsonet al.et al.[40] [14], Yang Open trench Thompson et al. [14], Open trench et al. [40] Yang et al. [40] Soft-filled barriers/ stiff wave Thompson et al. [14], Yang Coulier et al. Soft-filledbarrier/ jetbarriers/ grouting stiff wall/ wave Thompsonet al. [40], et Coulier al. [14], et Yang al. Coulier[41] et al. Soft-filledbarrier/hollow jet barriers/ isolation grouting stiff wall wall/ wave ThompsonThompsonet [41],al. [40], Guo et et al.Coulier etal. [ 14al. [14],], [42] et Yang al. Soft-filled barriers/stiff Coulier[41] et al. Soft-filledbarrier/hollow jet barriers/ isolation grouting stiff wall wall/ wave ThompsonYanget [41],al. et[40], Guo al. et [Coulier40 etal.], al. [14], [42] et Yang al. wave barrier/jet grouting Coulier et[41] al. [41] Coulier et al. [41], Coulier et al. wallbarrier/hollow/hollow jet isolation isolationgrouting wall wall wall/ et[41], al. [40], Guo Coulier et al. [42] et al. Guo et al. [42] [41] hollow isolation wall [41], Guo et al. [42] Takemiya [9], Coulier et Subgrade stiffening beneath Takemiyaal.Takemiya [43], Thompson [9], [9], Coulier et al. et Gao et al. [46] SubgradeorSubgrade next to stiffening the sti fftrack,ening beneath WIB Coulieral.Takemiya[44], [43], etTang Thompson al. [9], [et43 Coulier al.], [45] et al. et Gao et al. [46] Subgradebeneathor next to orstiffening the next track, to thebeneath WIB Gao et al. [46] Thompsonal.Takemiya[44], [43], Tang Thompson et al.[9], et [ 44Coulieral.], [45] et al. et Gao et al. [46] Subgradeor nexttrack, to stiffening the WIB track, beneath WIB Tangal.[44], [43], et Tang al.Thompson [45 et] al. [45] et al. Gao et al. [46] or next to the track, WIB [44], Tang et al. [45] Dijkmans et al. Sheet pile wall Dijkmans et al. [47] Sustainability 2020, 12, x FOR PEER REVIEWSheet pile wall Dijkmans et al. [47] DijkmansDijkmans et[47] al. et [474 al.of] 22 Sheet pile wall Dijkmans et al. [47] Dijkmans[47] et al. Sheet pile wall Dijkmans et al. [47] Dijkmans[47] et al. Sheet pile wall Dijkmans et al. [47] [47] Heavy masses along the Heavy masses along the Dijkmans et al. [48] Dijkmans et al. [48] tracktrack

(a) Analogy of phononic crystal and piles with a periodic arrangement.

(b) Illustration of the irreducible Brillouin zone for periodic structures with types of arrangement.

(c) Illustration of the band gap of a periodic structure.

Figure 1. Sketch of periodic structures with a band gap.

In this study, a case study was analyzed: a newly planned underground metro line was adjacent to a laboratory containing various types of sensitive instruments. To mitigate the train-induced vibrations, this study focuses on the feasibility of the propagation path solution using periodic piles. Firstly, in-situ measurements of the ambient vibrations were performed inside and outside the building to obtain the vibration transfer function from outside to inside the laboratory. Then, to help design and optimize the band gaps of the periodic piles, a novel band gap performance evaluation Sustainability 2020, 12, x FOR PEER REVIEW 4 of 22

Heavy masses along the Dijkmans et al. [48] track

Sustainability 2020, 12, 5871 4 of 20

(a) Analogy of phononic crystal and piles with a periodic arrangement.

(b) Illustration of the irreducible Brillouin zone for periodic structures with types of arrangement.

(c) Illustration of the band gap of a periodic structure.

Figure 1. Sketch of periodic structures with a band gap.

In thisthis study,study, a a case case study study was was analyzed: analyzed: a a newly newly planned planned underground underground metro metro line line was was adjacent adjacent to ato laboratory a laboratory containing containing various various types oftypes sensitive of sens instruments.itive instruments. To mitigate To themitigate train-induced the train-induced vibrations, thisvibrations, study focuses this study on the focuses feasibility on the of feasibility the propagation of the pathpropagation solution path using solution periodic using piles. periodic Firstly, in-situ piles. measurementsFirstly, in-situ ofmeasurements the ambient vibrations of the ambient were performed vibrations inside were and performed outside the inside building and tooutside obtain the vibrationbuilding to transfer obtain function the vibration from outsidetransfer to function inside the from laboratory. outside to Then, inside to helpthe laboratory. design and Then, optimize to help the banddesign gaps and of optimize the periodic the piles,band agaps novel of bandthe periodic gap performance piles, a novel evaluation band gap function performance is proposed. evaluation Finally, numerical analysis was performed to validate the mitigation effect of the designed periodic piles.

2. Problem Outline and Methodology In this case study, the newly planned metro line of Hefei city is located very close to the laboratory of the China Academy of Sciences (CAS). The CAS laboratory is located in an urban area west of S-S Road. The planned metro tunnel is a circular shield tunnel with an outer radius of 3 m and is located 14.8 m beneath the surface of S-S Road. The net distance between the outer walls of the laboratory Sustainability 2020, 12, x FOR PEER REVIEW 5 of 22

function is proposed. Finally, numerical analysis was performed to validate the mitigation effect of the designed periodic piles.

2. Problem Outline and Methodology In this case study, the newly planned metro line of Hefei city is located very close to the laboratory of the China Academy of Sciences (CAS). The CAS laboratory is located in an urban area west of S-S Road. The planned metro tunnel is a circular shield tunnel with an outer radius of 3 m Sustainabilityand is located2020 ,14.812, 5871 m beneath the surface of S-S Road. The net distance between the outer walls of5 of the 20 laboratory building and the west shield tunnel is only 27 m (Figure 2). In the laboratory building, various types of sensitive instruments are installed and plan to be installed in the future, including a buildingscanning andelectron the west microscope, shield tunnel e-beam is only lithograp 27 m (Figurehy, 2and). In an the laboratoryelectron paramagnetic building, various resonance types ofspectrometer. sensitive instruments To comprehensively are installed evaluate and plan the vibrat to beion installed level and in the mitigation future, includingeffects for aall scanning types of electroninstruments, microscope, the widely e-beam usedlithography, general vibration and an crit electroneria (VC) paramagnetic curves [59] resonancewere employed spectrometer. in this Toresearch. comprehensively evaluate the vibration level and mitigation effects for all types of instruments, the widely used general vibration criteria (VC) curves [59] were employed in this research.

Figure 2. Sketch of the location relationship between the laboratory building and metro tunnel. Figure 2. Sketch of the location relationship between the laboratory building and metro tunnel. To predict and evaluate the vibrations in the laboratory, a hybrid method of both measurement and numericalTo predict was employed.and evaluate In-situ the vibrations measurements in the were laboratory, performed a hybrid to determine method theof both ambient measurement vibration leveland numerical and transfer was function. employed. Vibration In-situ sensors measuremen were installedts were bothperformed outside to and determine inside the the laboratory ambient EXP EXP buildingvibration tolevel obtain and the transfer responses function. outside Vibration the building sensorsR outwere (installedfi) and inside both theoutside building and insideRin the(fi). outEXP i Then,laboratory the vibration building transfer to obtain function the responsesH(fi) was outside calculated the asbuilding follows: R (f ) and inside the building RinEXP(fi). Then, the vibration transfer function H(fi) was calculated as follows: R EXP( f ) inRfEXP ()i H( f ) = in i ,, (1) Hfi ()= EXP i R EXP(()fi) (1) outRfout i wherewhere ffii isis thethe ii-th-th centercenter frequencyfrequency ofof thethe one-thirdone-third octaveoctave bands.bands. Due to the complexcomplex buildingbuilding structure and dynamicdynamic soil–structure interaction, thethe laboratory building was not considered in the numerical model. It was assumed that vibration transfertransfer characteristics only depend on the building structure butbut areare independentindependent ofof thethe vibrationvibration source.source. Thus,Thus, thethe measuredmeasured transfertransfer functionfunction HH((ffii) can be employedemployed toto predictpredict vibrations vibrations inside inside the the building building if theif the metro metro train-induced train-induced ground ground vibrations vibrations are predictedare predicted by the by numericalthe numerical model: model:

PRE,m PRE,m RfHfRfPRE,m ()= ()PRE,m (), (2) Rin in( fi) ii= H( fi)Rout out ( fi i), (2)

where RinPRE,m(fi) is the predicted floor vibration response in the building, and RoutPRE,mPRE,m(fi) is the where Rin (fi) is the predicted floor vibration response in the building, and Rout (fi) is the predicted ground vibration response outside the building. The superscript “m” indicates that predicted ground vibration response outside the building. The superscript “m” indicates that responses responses are only induced by metro trains. are only induced by metro trains.

3. In-Situ Vibration Measurement

3.1. Sensor Arrangement Room 103 was selected as a typical laboratory containing sensitive instruments. Three measurement points were arranged in the room: B1 was at the center of the floor, B2 was also on the floor but close to the instrument workbench, and B3 was located on the workbench. Another nine measurement points, A1–A9, were arranged outside the building along S-S Road (Figure3). At each measurement point, both the vertical acceleration and vertical velocity responses were recorded by type LC0130 acceleration sensors and type 941B velocity sensors, which were connected to a synchronous data acquisition instrument of INV3060S type with 16 channels. To compare the vibration responses using VC curves, only the velocity responses are discussed in the following analysis. Two test Sustainability 2020, 12, x FOR PEER REVIEW 6 of 22

Room 103 was selected as a typical laboratory containing sensitive instruments. Three measurement points were arranged in the room: B1 was at the center of the floor, B2 was also on the floor but close to the instrument workbench, and B3 was located on the workbench. Another nine measurement points, A1–A9, were arranged outside the building along S-S Road (Figure 3). At each measurement point, both the vertical acceleration and vertical velocity responses were recorded by Sustainabilitytype LC01302020, 12acceleration, 5871 sensors and type 941B velocity sensors, which were connected6 of to 20 a synchronous data acquisition instrument of INV3060S type with 16 channels. To compare the vibration responses using VC curves, only the velocity responses are discussed in the following conditionsanalysis. wereTwo considered:test conditions background were considered: vibrations andbackground road traffi vibrationsc-induced and vibrations. road traffic-induced Background vibrationsvibrations. were Background measured vibrations in the early were morning measured when in the no carsearly or morning buses were when running no carson or S-Sbuses Road, were whereasrunning road on traS-Sffi Road,c-induced whereas vibrations road weretraffic-induced measured duringvibrations the eveningwere measured rush hour. during Each the test evening lasted 30rush s and hour. was Each repeated test lasted five times. 30 s and was repeated five times.

FigureFigure 3. 3.Sketch Sketch of of sensor sensor arrangement. arrangement. 3.2. Experimental Results 3.2. Experimental Results Figure4 illustrates the average vibration responses measured in Room 103. The VC curves were Figure 4 illustrates the average vibration responses measured in Room 103. The VC curves were plotted to evaluate the current vibration level in the laboratory. Seven guideline curves, from VC-A to plotted to evaluate the current vibration level in the laboratory. Seven guideline curves, from VC-A VC-G, were suggested by Amick et al. (Table3)[ 59]. VC-A is a relatively loose limit curve appropriate to VC-G, were suggested by Amick et al. (Table 3) [59]. VC-A is a relatively loose limit curve for most optical microscopes, and VC-E is a challenging limit curve appropriate for the very sensitive appropriate for most optical microscopes, and VC-E is a challenging limit curve appropriate for the instruments. VC-F and VC-G are only used for evaluation of an extremely low-vibration environment. Sustainabilityvery sensitive 2020, 12instruments., x FOR PEER REVIEWVC-F and VC-G are only used for evaluation of an extremely7 low-of 22 A more detailed description of use for the VC curves can be found in Ref. [59]. vibration environment. A more detailed description of use for the VC curves can be found in Ref. [59]. Comparing the measurement responses with the VC curves, it can be observed that the -1 laboratory room was a low-vibration10 environment. Even during the evening rush hour, the traffic- VC-A induced vibration responses were below the VC-D curve. As the buildingVC-B foundation was specifically

designed according to the vibration10-2 isolation purpose, it providesVC-C quiet working conditions for most instruments under the VC-D vibration level; for example, theVC-D scanning electron microscope, VC-E transmission electron microscope, and E-beam system. ComparedVC-F with the background vibrations, 10-3 traffic-induced vibrations were larger at approximately 10 Hz.VC-G This was probably a contribution B1 background

caused by the natural frequencyVelocity/(mm/s) of B 2the back grflooround and the dominant frequency of the traffic-induced vibrations. Above 30 Hz, the10 vibration-4 B3 b aonckgro uthend workbench (B3) was larger than that on the floor (B1 B1 traffi c and B2). B2 traffi c B3 traffi c 10-5 100 101 102 1/3 octave band/Hz

Figure 4. Measured vibration responses in Room 103. Figure 4. Measured vibration responses in Room 103.

Table 3. Application and interpretation of vibration criteria (VC) curves.

Vibration Limit Criterion Curve between 1 and 4 Hz between 4 and 8 Hz between 8 and 80 Hz VC-A - 260 μg 50 μm/s VC-B - 130 μg 25 μm/s VC-C 12.5 μm/s 12.5 μm/s 12.5 μm/s VC-D 6.25 μm/s 6.25 μm/s 6.25 μm/s VC-E 3.1 μm/s 3.1 μm/s 3.1 μm/s VC-F 1.6 μm/s 1.6 μm/s 1.6 μm/s VC-G 0.78 μm/s 0.78 μm/s 0.78 μm/s

Figure 5 illustrates vibration responses outside the building (A1–A9) due to road traffic during the evening rush hour. Five vibration response samples were recorded for each measurement point, and the average response was plotted as a thick solid line. Then, the average vibration transfer function H(fi) between the ground outside the building (A1–A9) and the floor of Room 103 (B1 and B2) was calculated using Equation (1) and plotted in Figure 6. When H(fi) < 1, vibrations were attenuated through the building; otherwise, vibrations increased. The H(fi) of B2 was clearly larger than that of B1, especially below 8 Hz and at 50 Hz and 100 Hz. A comparison of Figures 4 and 5 suggests that the two dominant frequencies at 50 Hz and 100 Hz were likely caused by electrical interference from the equipment on the workbench.

10-1 Measured samples Averaged value

10-2

10-3 Velocity/(mm/s)

10-4

10-5 100 101 102 1/3 octave band/Hz

Figure 5. Measured vibration responses outside the building. Sustainability 2020, 12, x FOR PEER REVIEW 7 of 22

10-1 VC-A VC-B

-2 VC-C Sustainability 2020, 12, 5871 10 7 of 20 VC-D VC-E VC-F 10-3 Table 3. Application and interpretation of vibration criteriaVC-G (VC) curves. B1 background

Velocity/(mm/s) B2 background Vibration Limit Criterion Curve 10-4 B3 background between 1 andB1 traff 4i c Hz between 4 and 8 Hz between 8 and 80 Hz B2 traffi c VC-A - B3 traffi c 260 µg 50 µm/s 10-5 VC-B100 -101 130 µg102 25 µm/s 1/3 octave band/Hz VC-C 12.5 µm/s 12.5 µm/s 12.5 µm/s

VC-D 6.25 µm/s 6.25 µm/s 6.25 µm/s Figure 4. Measured vibration responses in Room 103. VC-E 3.1 µm/s 3.1 µm/s 3.1 µm/s VC-FTable 3. Application 1.6 µ mand/s interpretation of 1.6 vibrationµm/s criteria (VC) 1.6 curves.µm/s VC-G 0.78 µm/sVibration 0.78 µm/s Limit 0.78 µm/s Criterion Curve between 1 and 4 Hz between 4 and 8 Hz between 8 and 80 Hz ComparingVC-A the measurement responses- with the VC curves, 260 μ itg can be observed that50 the μm/s laboratory room wasVC-B a low-vibration environment.- Even during the 130 evening μg rush hour, the25 tra μffim/sc-induced vibration responsesVC-C were below12.5 the VC-Dμm/s curve. As the building12.5 μm/s foundation was specifically12.5 μm/s designed accordingVC-D to the vibration isolation6.25 purpose, μm/s it provides quiet6.25 working μm/s conditions for most6.25 μ instrumentsm/s under theVC-E VC-D vibration level;3.1 for μ example,m/s the scanning3.1 electron μm/s microscope, transmission3.1 μm/s electron microscope,VC-F and E-beam system.1.6 Compared μm/s with the background1.6 μm/s vibrations, traffic-induced1.6 μm/s vibrations were largerVC-G at approximately 100.78 Hz. μ Thism/s was probably a contribution0.78 μm/s caused by the natural0.78 μm/s frequency of the floor and the dominant frequency of the traffic-induced vibrations. Above 30 Hz, the vibration on theFigure workbench 5 illustrates (B3) was vibration larger than responses that on outside the floor th (B1e building and B2). (A1–A9) due to road traffic during theFigure evening5 illustrates rush hour. vibration Five vibration responses response outside samples the building were recorded (A1–A9) for due each to road measurement tra ffic during point, theand evening the average rush hour. response Five vibrationwas plotted response as a thick samples solid were line. recorded Then, the for eachaverage measurement vibration transfer point, andfunction the average H(fi) betweenresponse the was ground plotted outside as a thick the solid building line. Then, (A1–A9) the averageand the vibration floor of Room transfer 103 function (B1 and H(B2)fi) betweenwas calculated the ground using outside Equation the building(1) and plotted (A1–A9) in and Figure the floor6. When of Room H(fi) 103 < 1, (B1 vibrations and B2) waswere calculatedattenuated using through Equation the building; (1) and plottedotherwise, in Figure vibrations6. When increased.H(fi)

10-1 Measured samples Averaged value

10-2

10-3 Velocity/(mm/s)

10-4

10-5 100 101 102 1/3 octave band/Hz

Figure 5. Measured vibration responses outside the building. Figure 5. Measured vibration responses outside the building. SustainabilitySustainability2020 2020, ,12 12,, 5871 x FOR PEER REVIEW 88 ofof 2022

2 B1 B2

1 Transfer function/-

0 100 101 102 1/3 octave band/Hz

EXP Figure 6. The averaged vibration transfer function H (fi) between the ground outside the building Figure 6. The averaged vibration transfer function HEXP(fi) between the ground outside the building and the floor of Room 103. and the floor of Room 103. 4. Band Gap Analysis and Comparison for Periodic Piles 4. Band Gap Analysis and Comparison for Periodic Piles As the newly designed metro tunnel is located only 27 m from the laboratory building, operation of the metroAs the line newly will increase designed the metro building tunnel vibrations, is located which only may27 m a fromffect researchers’the laboratory use building, of the instruments. operation Toof mitigatethe metro train-induced line will increase vibrations, the measuresbuilding canvibr beations, taken which on the may track affect by inserting researchers’ elastic use elements, of the alonginstruments. the propagation To mitigate path train-in in theduced soil layers vibrations, and at measures the receiver, can using be taken an isolated on the track platform. by inserting In this study,elastic onlyelements mitigation, along measures the propagation along the path propagation in the soil layers path were and discussed;at the receiver, specifically, using an periodic isolated piles.platform. Existing In this research study, on theonly vibration mitigation mitigation measures effect along of periodic the propagation piles has indicated path were that discussed; band gap designspecifically, and optimization periodic piles. are keyExisting to mitigating research train-inducedon the vibration ground mitigation vibrations effect [60 of]. Toperiodic design piles optimal has periodicindicated piles that for band controlling gap design train-induced and optimization environmental are vibrations,key to mitigating a novel bandtrain-induced gap performance ground evaluationvibrations function[60]. To is design proposed optimal in the followingperiodic piles subsections. for controlling train-induced environmental vibrations, a novel band gap performance evaluation function is proposed in the following 4.1.subsections. Band Gap Performance Evaluation Function The main indices of band gaps include the lower bound frequency, upper bound frequency, and 4.1. Band Gap Performance Evaluation Function width of the band gap. However, these three indices exhibit different trends with various parameters, whichThe complicates main indices the of design band andgaps optimization include the lowe of ther bound band gap.frequency, Moreover, upper practical bound frequency, engineering and is particularlywidth of the concerned band gap. withHowever, the relationship these three betweenindices exhibit band gapdifferent and target trends isolation with various frequency parameters, range, whichwhich dependscomplicates not the only design on the and initial optimization band gap butof the also band on the gap. distribution Moreover, ofpractical all band engineering gaps. As an is example,particularly Figure concerned7 shows with two frequencythe relationship band gap between solutions band calculated gap and basedtarget onisolation di fferent frequency arrangements range, ofwhich periodic depends piles. not In theonly x-coordinate, on the initial X, band M and gapΓ representbut also on the the vertexes distribution of the reducedof all band Brillouin gaps. As zone. an Theexample, grey zoneFigure represents 7 shows the two frequency frequency band band gaps. gap Compared solutions with calculated the first bandbased gap on illustrated different inarrangements Figure7b, it exhibitsof periodic lower piles. bound In the frequency x-coordinate, and X, wider M and band Г represent gap width the in vertexes Figure7 ofa. the However, reduced FigureBrillouin7b probablyzone. The provides grey zone a represents better vibration the freque mitigationncy band e ff ectgaps. for Compared one possible with arrangement the first band of thegap periodicillustrated piles, in Figure especially 7b, belowit exhibits 80 Hz. lower bound frequency and wider band gap width in Figure 7a. However, Figure 7b probably provides a better vibration mitigation effect for one possible arrangement of the periodic piles, especially below 80 Hz. Sustainability 2020, 12, x FOR PEER REVIEW 9 of 22 Sustainability 2020, 12, 5871 9 of 20

120 120

100 100

80 80

60 60 2nd band gap 2nd band gap Frequency/Hz Frequency/Hz 40 1st band gap 40 1st band gap

20 20

0 0 M Γ X M M Γ X M Bloch wave vector,K Bloch wave vector,K

(a) The first band gap with lower bound (b) The first band gap with higher bound frequency and wider band gap width frequency and narrower band gap width

FigureFigure 7.7. ComparisonComparison ofof twotwo bandband gapgap distributionsdistributions ofof periodicperiodicpiles. piles. The design and selection of periodic piles for vibration isolation depend on whether the center The design and selection of periodic piles for vibration isolation depend on whether the center frequency of the band gap is approaching the dominant frequency of the environmental vibration and frequency of the band gap is approaching the dominant frequency of the environmental vibration whether the band gap distribution is wide enough. Considering the two aspects, a comprehensive and whether the band gap distribution is wide enough. Considering the two aspects, a evaluation method for designing a periodic structure is proposed. Firstly, the difference between the comprehensive evaluation method for designing a periodic structure is proposed. Firstly, the center frequency of the band gap and the dominant frequency of the vibration indicates the closeness difference between the center frequency of the band gap and the dominant frequency of the vibration between them, which should be expressed in the evaluation method. Secondly, the impact of band indicates the closeness between them, which should be expressed in the evaluation method. width should be reflected in the evaluation method. Accordingly, a novel band gap performance Secondly, the impact of band width should be reflected in the evaluation method. Accordingly, a evaluation function Φ was proposed for designing the periodic structures, which is designed as: novel band gap performance evaluation function Φ was proposed for designing the periodic structures, which is designed as: X minωj+1 maxωj Max : Φ = − , (3) minminωωω−j+1 max+maxωj n jj+1 Max : Φ= ωD 2 ,  −minωω+ max n ω − jj+1 (3) D where n is the total number of band gaps; (max ωj) and (min2 ωj+1) represent the lower and upper bound frequencies of one band gap; (min ωj+1 max ωj) represents the band gap width; (min ωj+1 where n is the total number of band gaps; (max− ωj) and (min ωj+1) represent the lower and upper + max ωj)/2 represents the central frequency of the band gap; and ωD is the frequency of concern of bound frequencies of one band gap; (min ωj+1 −max ωj) represents the band gap width; (min ωj+1 + max the blocked objective. When the central frequency of the band gap is approaching the frequency of ωj)/2 represents the central frequency of the band gap; and ωD is the frequency of concern of the concern, ωD, the value of (ωD (min ωj+1 + max ωj)/2) approaches the minimum value. In contrast, blocked objective. When the −central frequency of the band gap is approaching the frequency of with increasing band gap width, Φ approaches the maximum value. The larger the Φ value, the better concern, ωD, the value of (ωD-(min ωj+1 + max ωj)/2) approaches the minimum value. In contrast, with the vibration mitigation effect for the frequency of concern. increasing band gap width, Φ approaches the maximum value. The larger the Φ value, the better the Different from other band gap evaluation methods as a single evaluation index, both the band vibration mitigation effect for the frequency of concern. gap width and the center frequency were comprehensively considered by Φ. This method avoids the Different from other band gap evaluation methods as a single evaluation index, both the band uncertainty of multi index analysis. In addition, the concerned frequencies of train-induced vibration gap width and the center frequency were comprehensively considered by Φ. This method avoids the are considered. Moreover, both the distribution of all band gaps and the initial band gap can be uncertainty of multi index analysis. In addition, the concerned frequencies of train-induced vibration considered comprehensively. Therefore, the band gap performance evaluation function, Φ, can be used are considered. Moreover, both the distribution of all band gaps and the initial band gap can be to design and optimize the vibration mitigation effects of periodic piles. considered comprehensively. Therefore, the band gap performance evaluation function, Φ, can be 4.2.used Band to design Gap Solution and optimize for Periodic the Pilesvibration mitigation effects of periodic piles.

4.2. BandTo calculate Gap Solution the distribution for Periodic Piles band gap of periodic piles, the finite element method (FEM) was employed to solve the dispersion relationship of periodic structures with various complex shapes based on theTo variational calculate principlethe distribution and weighted band gap residual of periodic method piles, [61]. the The finite infinite element degree-of-freedom method (FEM) of was the periodicemployed piles to cansolve be the condensed dispersion to the relationship master nodes of ofperiodic a single structures basic cell bywith applying various the complex Bloch-Floquet shapes periodicbased on conditions the variational (Figure principle8): and weighted residual method [61]. The infinite degree-of-freedom of the periodic piles can be condensed to the master nodesik a of a single basic cell by applying the Bloch- U(r + a) = e · U(r), (4) Floquet periodic conditions (Figure 8): Sustainability 2020, 12, x FOR PEER REVIEW 10 of 22

Sustainability 2020, 12, 5871 10 of 20 ika⋅ UU（）=（）ra+ e r, (4) wherewhere a isa isa periodic a periodic constant, constant, k isk ais Bloch a Bloch wave wave vector, vector, r isr ais position a position vector, vector, and and U Urepresentsrepresents displacement.displacement. Then, Then, the theband band gap gap of periodic of periodic piles piles was was calculated calculated in the in the basic basic cell cell domain domain and and the the Bloch-FloquetBloch-Floquet periodic periodic conditions conditions were were added added to tothe the boundary boundary of ofthe the basic basic cell. cell.

a

ikya UTL，fTL U T =e U B URT，fRT

基体

L U ax ik a U L 散射体基体 e = R U

UBR，fBR ULB，fLB U B FigureFigure 8. 8.TheThe periodic periodic boundary boundary of of the the basic cell,cell, andand the the generalized generalized displacement displacement and and force force of nodes. of nodes. As the continuous field is characterized by each node displacement and the interpolation function in eachAs basicthe cell,continuous the continuous field is problem characterized can be discretized.by each node The displacement relationship between and the node interpolation displacement functionand node in each force basic in a basiccell, the cell continuous is generated problem using an can element be discretized. stiffness matrix,The relationship and the adjacent between basic node cells displacementare connected and by node the equilibrium force in a basic equation, cell is forming generated a global using sti anffness element matrix. stiffness The governing matrix, and equations the adjacentfor the basic cell andcells the are whole connected domain by canthe equilibrium be expressed equation, as follows: forming a global stiffness matrix. The governing equations for the cell and the whole domain can be expressed as follows: Meüe + Keue = 0, (5) Meüe + Keue = 0, (5) Mü + Ku = 0, (6) Mü + Ku = 0, (6) where Me and Ke represent the mass and stiffness matrices of the basic cell, respectively; ue is the e e e wherenode M displacement and K represent vector; theM massand andK represent stiffness matr the massices of and the stibasicffness cell, matrices respectively; of the u whole is the domain,node displacementrespectively; vector; and u isM the and global K represent node displacement the mass vector.and stiffness Then, the matrices dispersion of the curve wholeω(k) domain, is obtained respectively;by sweeping and the u is Bloch the global wave node vector displacementk over the irreducible vector. Then, Brillouin the dispersion zone [61 ].curve ω(k) is obtained by sweepingThe calculation the Bloch wave was performed vector k over using theCOMSOL irreducible software, Brillouin which zone [61]. has the obvious advantage of addingThe calculation complex boundaries. was performed In theusing dispersion COMSOL calculation, software, which the periodic has the piles obvious were advantage assumed toof be addinginfinite, complex i.e., the boundaries. plane strain In assumptionthe dispersion was calculation, employed. the The periodic solid physical piles were module assumed and pressureto be infinite,acoustic i.e., module the plane were strain used assumption to solve the was in-plane employed. and anti-plane The solid band physical gaps, module respectively. and pressure The solver acousticmode correspondsmodule were to used the undamped to solve the eigenvalue in-plane mode.and anti-plane Based on band the plane gaps, strain respectively. assumption, The thesolver elastic modewave corresponds equation can to bethe decoupled undamped into eigenvalue a mixed modemode. in Based the plane on the and plane a shear strain mode assumption, out of the plane,the elasticas follows wave equation [62]: can be decoupled into a mixed mode in the plane and a shear mode out of the plane, as follows [62]: " # ∂ ∂ [ ( ) ] + ( ) ∂∂+ [ ( ) ] = ( ) 2 = T µ r Tui T µ r ui λ r T uT ρ r ω2 ui, i x, y, (7) ∇⋅∇ ·[]μμλρω()rrrr ∇∇ uu +∇⋅∇ · ()∂xi +∂xi[] () ∇⋅∇ · u =−− () uixy， = , , TTiTi∂∂ TTi (7) xxii 2 T [µ(r) Tuz] = ρ(r)ω uz, (8) ∇ · ∇ − 2 ∇⋅[μρω()rr ∇uu] =− () , (8) where λ and µ are Lamé constants; ρTTzzdenotes the mass density; u(r, t) is a displacement vector; uj(j = x, y, z) denotes the displacement component; is the Laplace operator; r = (x, y, z) is the coordinate where λ and μ are Lamé constants; ρ denotes the∇ mass density; u(r, t) is a displacement vector; uj(j = x, y,vector; z) denotes and t the is time displacement parameter. component; ∇ is the Laplace operator; r = (x, y, z) is the coordinate vector; andBased t is on time the parameter. theory of Bloch-Floquet, Equations (7) and (8) can be deduced as: Based on the theory of Bloch-Floquet, Equationsi(K (7)r ω andt) (8) can be deduced as: u(r, t) = e · − uK(r), (9) = i(Kr⋅−ωt ) ur(,t)e uK () r , (9) where uK(r) is the vector of the wave amplitudes, and K is the wave vector in the reciprocal space. where uConsideringK(r) is the vector theperiodicity, of the wave amplitudes, and K is the wave vector in the reciprocal space. Considering the periodicity, uk(r) =uk(r + a), (10) Sustainability 2020, 12, 5871 11 of 20 where a is the periodic constant vector. Substituting Equation (10) into Equation (9), the periodic boundary condition is obtained:

iK a u(r + a, t) = e · u(r, t), (11)

Then, the wave equations are expressed as:

(Ω(K) ω2M) u = 0, (12) − · where Ω is the stiffness matrix and M is the mass matrix. By solving for the eigenfrequencies of the Equation (12) for different values of K, the dispersion relation ω = ωn (K) and band gaps can be obtained. In general, when only the longitudinal waves were considered, the viscosity was ignored. The wave equation is expressed as follows [63]: ! 1 1 p = ω2p, (13) ∇ · ρ(r) ∇ − (r) 2(r) ρ cl in which, q cl = (λ + 2µ)/ρ, (14) where p is the pressure; cl is the longitudinal wave velocity; ρ is the medium density; λ and µ are Lamé constants; u is the displacement vector; r is a position vector; and ω is the angular frequency. If ρ, 2 1/ρ and 1/(ρcl ) in Equation (13) are replaced with uz, u, and ρ, Equation (13) becomes equivalent to Equation (8). Therefore, the dispersion relationship of the anti-plane shear mode can be obtained by solving the acoustic wave using Equation (13) instead of Equation (8).

4.3. Parameter Comparison and Selection for Periodic Piles To determine better design parameters for periodic piles in order to control train-induced ground vibrations, parameter comparison and selection were performed. As the material parameters for piles and the soil layers are known, the main design parameters include the lattice form of the basic cell, the pile radius, and the periodic constant. To calculate the band gap performance evaluation function, Φ, in Equation (3), the dominant frequency ranges of train-induced ground vibrations should be determined. Thus, in-situ measurements of ground vibrations were performed near Hefei metro line 1, where the embedded depth of the shield tunnel was similar to that of the planned tunnel near the laboratory. Figure9 illustrates the Fourier spectrum and the one-third octave bands of the measured ground vibrations 30 m from the central line of the metro running tunnel. According to the results, at a distance of 30 m from the tunnel center, train-induced ground vibrations could affect the normal operation of sensitive instruments. At approximately 63 Hz, the vibration responses exceeded the VC-C curve. Between approximately 44 Hz and 76 Hz, the vibration energy was obvious, and the responses exceeded the VC-D curve. Thus, for subsequent analysis, 44 Hz, 76 Hz, and 63 Hz were set as the lower, upper, and target frequencies. Sustainability 2020, 12, x FOR PEER REVIEW 12 of 22 Sustainability 2020, 12, 5871 12 of 20

5.0x10-6 10-1 VC-A VC-B 4.0x10-6 10-2 VC-C VC-D VC-E 3.0x10-6 VC-F 10-3 VC-G 2.0x10-6 velocity/(mm/s) Velocity/(mm/s)

10-4 1.0x10-6 30 m from the tunnel cent er

0.0 10-5 0 20406080100 100 101 102 Frequency/Hz 1/3 octave band/Hz

(a) (b)

FigureFigure 9. 9.Measured Measured metro metro train-inducedtrain-induced ground vibrations, vibrations, 30 30 m m away away from from the thetunnel tunnel center: center: (a) the (a)Fourier the Fourier spectrum, spectrum, and and(b) velocity (b) velocity in one-third in one-third octave octave bands bands together together with with VC VCcurves. curves.

BasedBased on on a geologic a geologic survey survey report, report, drilling drilling hole hole exploration exploration showed showed that that the the soil soil between between the the laboratorylaboratory and and the the planned planned tunnel tunnel can can be be simplified simplified into into three three layers layers from from top top to bottom.to bottom. The The detailed detailed soilsoil parameters parameters are are listed listed in Tablein Table4. As 4. As the the planned planned tunnel tunnel was was embedded embedded 14.8 14.8 m belowm below the the surface, surface, thethe periodic periodic piles piles can can be be regarded regarded as beingas being embedded embedded in thein the clay clay layer. layer. Therefore, Therefore, in thein the following following analysisanalysis for for each each basic basic cell, cell, parameters parameters specific specific to theto the clay clay layer layer were were employed. employed.

TableTable 4. Soil 4. Soil parameters. parameters.

Soil Depth Density Poisson’s ElasticElastic S-WaveS-Wave P-P-WaveWave Damping Soil Depth Density3 Poisson’s Damping Layers (m) (kg/m ) Ratio ModulusModulus (MPa) VelocityVelocity (m/s) VelocityVelocity (m /s) Ratio Layers (m) (kg/m3) Ratio Ratio Layer 1 0.9 1750 0.492 112.2(MPa) 141.1(m/s) (m/s) 1098 0.05 LayerLayer 2 1 34.1 0.9 1990 1750 0.474 0.492 576 112.2 311.8 141.1 1098 1462 0.05 0.04 Layer 3 - 2100 0.472 1239.9 458.7 2038 0.03 Layer 2 34.1 1990 0.474 576 311.8 1462 0.04 Layer 3 - 2100 0.472 1239.9 458.7 2038 0.03 Two types of basic cell shape were considered: square and hexagonal. Additionally, two types of pilesTwo were types considered: of basic a cell concrete shape pilewere and considered: a steel pipe square with and a concrete hexagonal. pile inside.Additionally, By combining two types theof basic piles cell were shape considered: and pile type,a concrete a total pile of fourand typesa steel of pipe lattice with form a concrete were analyzed. pile inside. For By each combining lattice form,the twobasic groups cell shape of parameters and pile type, were a calculatedtotal of four for types the periodic of lattice constant form were and analyzed. pile radius, For as each shown lattice in Tableform,5 two. In thegroups calculation, of parameters the concrete were calculated material had for athe Young’s periodic modulus constant of and 3 pile1010 radius,N/m2, aas mass shown × densityin Table of 2500 5. Inkg the/m calculation,3, and a Poisson’s the concrete ratio ofmaterial 0.2, whereas had a theYoung’s steel materialmodulus had of 3 a × dynamic 1010 N/m elastic2, a mass modulusdensity of of 2.1 2500 10kg/m11 N3,/ mand2, aa densityPoisson’s of ratio 7890 of kg 0.2,/m3 ,whereas and a Poisson’s the steel ratiomaterial of 0.275. had a For dynamic each case, elastic × themodulus band gaps of 2.1 were × 10 calculated11 N/m2, a using density COMSOL. of 7890 kg/m The band3, and gapa Poisson’s performance ratio of evaluation 0.275. For function, each case,Φ the, forband the eight gaps cases were were calculated calculated using using COMSOL. Equation The (3). band The gap final performance calculation evaluation results of Φ function,are listed Φ in, for Tablethe5 .eight The maximumcases were calculatedΦ was 68.31 using forcase Equati 6, whichon (3). involvedThe final acalculation hexagonal results lattice, of a periodicΦ are listed constant, in Table a, of5. 3.8The m, maximum and a concrete Φ was pile 68.31 radius, for caseR, 6,of which 1 m. The involved bandgap a hexagonal distribution lattice, for casea periodic 6 is illustrated constant, ina, of Figure3.8 m, 10 and. a concrete pile radius, R, of 1 m. The band gap distribution for case 6 is illustrated in Figure 10.

SustainabilitySustainability 2020, 12 2020, x FOR, 12, xPEER FOR REVIEW PEER REVIEW 13 of 2213 of 22

Sustainability 2020, 12, x FOR PEER REVIEW 13 of 22 Table Table5. Calculated 5. Calculated band gapband performance gap performance evaluation evaluation function function Φ for Φeight for cases.eight cases. Sustainability 2020, 12, x FOR PEER REVIEW 13 of 22 Table 5.Periodic CalculatedPeriodic band Pile gap Pileperformance Lower Lowerevaluation Upper functionUpper Φ for eight cases. SustainabilityCaseSustainability Case2020 Lattice, 12 2020,Lattice 5871 , 12, x FOR PEER REVIEW BandwidthBandwidth 13 of 2013 of 22 TableConstant 5. CalculatedConstant a bandRadius a gapRadius performanceFrequency Frequency evaluation Frequency functionFrequency Φ for eight cases. Φ Φ No. No. Form Form Periodic Pile Lower Upper (Hz) (Hz) Case LatticeTable 5. Calculated(m) bandR, r gap (m) performance (Hz) evaluation function(Hz) Φ for eightBandwidth cases. ConstantPeriodic(m) a RRadius,Pile r (m) FrequencyLower(Hz) FrequencyUpper(Hz) Φ 1 CaseNo.1 TableLatticeForm 5. Calculated 4.0 band4.0 gap R performance= 1.4 R = 1.4 evaluation 36.5 36.5 function 72.8Φ for 72.8 eight cases. 36.3Bandwidth (Hz) 36.3 4.36 4.36 ConstantPeriodic(m) a RRadius,Pile r (m) FrequencyLower(Hz) FrequencyUpper(Hz) Φ CaseNo. LatticeForm Bandwidth(Hz) 1 Periodic(m)4.0 RPile R, =r (m)1.4 Lower(Hz) 36.5 Upper(Hz) 72.8 36.3 4.36Φ Lattice Constant a Radius Frequency FrequencyBandwidth CaseNo. No. Form Constant Radius R, Frequency Frequency (Hz) Φ 2 21 Form 4.0 (m) 4.0 R = 1.5R R, =r (m)1.51.4 38.0 (Hz) 38.036.5 78.9 (Hz) 78.972.8 (Hz) 40.9 40.936.3 8.98 8.984.36 a (m) r (m) (Hz) (Hz) 1 4.0 R = 1.4 36.5 72.8 36.3 4.36 2 4.0 R = 1.5 38.0 78.9 40.9 8.98 1 2 4.0 4.0 R = 1.5 RR = =1.4 1.5 36.5 38.0 72.8 78.9 36.3 40.9 4.36 8.98 3 3 4.0 4.0 49.8 49.8 78.5 78.5 28.8 28.8 25.03 25.03 2 4.0 r = 1.2 Rr == 1.21.5 38.0 78.9 40.9 8.98 3 4.0 49.8 78.5 28.8 25.03 2 4.0 R r= =1.5 1.2 38.0 78.9 40.9 8.98 R = 1.4R = 1.41.5 4 43 4.0 4.0 50.7 50.749.8 72.4 72.478.5 21.7 21.728.8 15.27 15.2725.03 r = 1.2RRr= =1.5 1.21.5 3 4.0 Rr == 1.21.4 49.8 78.5 28.8 25.03 34 4.0 r = 1.2 49.850.7 78.572.4 28.821.7 25.0315.27 r = 1.2 5 5 4.1 4.1 R = 1.2R R= =1.4 1.21.4 50.8 50.8 76.6 76.6 25.8 25.8 36.37 36.37 44 4.04.0 50.750.7 72.472.4 21.721.7 15.27 15.27 rR=r =1.2 1.21.4 45 4.04.1 R = 1.2 50.7 50.8 72.4 76.6 21.7 25.8 15.27 36.37 r = 1.2 6 5 65 3.8 4.1 3.84.1 R = R1 R= R = 1.2= 1.2 1 52.4 50.8 52.450.8 74.3 76.6 74.376.6 25.8 21.9 21.925.8 36.37 68.31 68.3136.37 56 4.1 3.8 R R = = 1.2 1 50.852.4 76.674.3 25.821.9 36.3768.31 6 3.8 R = 1 52.4 74.3 21.9 68.31 6 3.8 R = 1.2R R = = 1.2 1 52.4 74.3 21.9 68.31 7 7 4.0 4.0 R = 1.2 46.8 46.8 81.3 81.3 34.6 34.6 33.23 33.23 7 6 4.0 3.8 r = 1.1Rr R == = 1.11.2 1 46.8 52.4 81.3 74.3 34.6 21.9 33.23 68.31 7 4.0 r = 1.1 46.8 81.3 34.6 33.23 R = 1.2r = 1.1 RR= =1.2 1.2 8 887 4.2 4.24.24.0 41.841.8 41.846.8 73.7 73.7 73.781.3 34.634.6 34.6 6.076.07 33.236.07 r = 1.0rR=r ==1.0 1.01.11.2 78 4.04.2 46.841.8 81.373.7 34.6 33.236.07 Rr == 1.11.01.2 8 4.2 41.8 73.7 34.6 6.07 100 100 Rr = 1.01.2 8 90 4.2 41.8 73.7 34.6 6.07 10090 r = 1.0 80 80 10090 70 7080 10090 60 60 52.4~74.3Hz52.4~74.3Hz 70 9080 50 5060 52.4~74.3Hz 8070 40 40

Frequency/Hz 50 Frequency/Hz 7060 52.4~74.3Hz 30 30 4050 52.4~74.3Hz Frequency/Hz 60 20 20 30 5040 Frequency/Hz 10 1020 4030 Frequency/Hz 0 0 10 M 30 20 M Γ Γ X X M M 0 2010 Wave vector,KWave vector,K M Γ X M 100 Wave vector,K FigureM 10. Band Band Γ gap gap distribution. distribution. X M 0 Figure 10. Band gap distribution. M Γ Wave vector,K X M 5. Controlling Metro Train-Induced EnvironmentalFigure 10. Band Vibrations gap distribution. by Periodic Piles 5. Controlling5. Controlling Metro Metro Train-Induced Train-Induced Enviro EnvironmentalnmentalWave Vibrations vector,K Vibrations by Periodic by Periodic Piles Piles Figure 10. Band gap distribution. 5.1. Tunnel-Soil-Periodic5. Controlling Metro Pile Train-Induced FE Model Environmental Vibrations by Periodic Piles 5.1. Tunnel-Soil-Peri5.1. Tunnel-Soil-Periodic Pileodic FE Pile Model FE Model Figure 10. Band gap distribution. To5. validateControlling the e Metroffect of Train-Induced periodic piles on Enviro controlling-metronmental Vibrations train-induced by Periodic environmental Piles vibrations, 5.1. Tunnel-Soil-Periodic Pile FE Model a coupledTo5. Controllingvalidate tunnel-soil-periodicTo validate the Metro effectthe Train-Induced effectof pile periodic dynamicof periodic piles Enviro FE model pilesonnmental controlling-metro wason controlling-metro built Vibrations using MIDAS bytrain-induced Periodic /train-inducedGTS software Piles environmental (Figureenvironmental 11). vibrations,Consideringvibrations,5.1. Tunnel-Soil-PeriTo a thecoupledvalidate a space coupled tunnel-soil-periodicthe betweenodic effecttunnel-soil-periodic Pile FE theof Model laboratoryperiodic pile piles dynamic buildingpile on dynamic controlling-metro andFE model theFE tunnel,model was builtwas fourtrain-induced builtusing rows using ofMIDAS/GTS pilesenvironmental MIDAS/GTS were software (Figure 11). Considering the space between the laboratory building and the tunnel, four considered.5.1.softwarevibrations, Tunnel-Soil-PeriTo The validate(Figure a center coupled 11).the ofodic theConsidering effect tunnel-soil-periodicPile four FE of rowsModel periodic the of pilesspace piles pile was between on dynamic located controlling-metro the at laboratoryFE a distancemodel buildingwastrain-induced of 13.5 built m and fromusing the environmental both MIDAS/GTStunnel, the four rows rowsof piles of werepiles wereconsidered. considered. The center The center of the of four the rows four rowsof piles of waspiles located was located at a distance at a distance of 13.5 of m 13.5 m outersoftwarevibrations, tunnelTo wallvalidate (Figure a andcoupled the11).the buildingConsidering effecttunnel-soil-periodic of wall. periodic the The space pilepiles pile lengthbetween on dynamic controlling-metro was the set laboratoryFE to 25model m, which buildingwastrain-induced built was and largerusing theenvironmental than MIDAS/GTStunnel, the four from fromboth theboth outer the outertunnel tunnel wall andwall the and building the building wall. waThell. pile The length pile length was set was to set 25 tom, 25 which m, which was was embeddedvibrations,rowssoftware of depth piles (Figure a of werecoupled the 11). tunnelconsidered. Consideringtunnel-soil-periodic bottom The (20.8 centerthe m). space of A pilethe tunnel-soil between fourdynamic rows the model ofFElaboratory piles withoutmodel was was buildinglocated piles built was at and alsoausing distance the calculated MIDAS/GTStunnel, of 13.5 four m largerlarger than thethan embedded the embedded depth depth of the of tunnel the tunnel bottom bottom (20.8 m).(20.8 A m). tunnel-soil A tunnel-soil model model without without piles waspiles was for comparison.softwarefromrows ofboth piles (Figure the Rayleigh were outer 11). considered. tunnel dampingConsidering wall The was and thecenter employed the space building of the between for four wa the rowsll. the soil The oflaboratory model. pilepiles length was The building located twowas keyset at and to frequenciesa distance25 the m, tunnel, which of in 13.5 fourwas m also calculatedalso calculated for comparison. for comparison. Rayleigh Rayleigh dampin damping wasg employedwas employed for the for soil the model. soil model. The two The key two key the Rayleighrowslargerfrom ofboth than piles damping the the were outerembedded calculationconsidered. tunnel depth wall were The ofand thecenter set the astunnel 1buildingof and the bottom 100four wa Hz. rows (20.8ll. TheThe ofm). dampingpilespile A tunnel-soil length was ratiolocated was andmodel set at othertoa distance without25 dynamicm, which pilesof 13.5 was m frequenciesfrequencies in the in Rayleigh the Rayleigh damping damping calculation calculation were wereset as set 1 and as 1 100 and Hz. 100 The Hz. damping The damping ratio andratio and parametersfromalsolarger calculatedboth than for eachthe the outer embedded soilfor comparison. layertunnel are depth wall shown ofRayleighand the inthe tunnel Table building dampin4 bottom. To gwa solve was (20.8ll. The theemployed m). problempile A tunnel-soillength for of thewas wave soilmodel set reflection model.to without25 m, The atwhich piles thetwo waswaskey other otherdynamic dynamic parameters parameters for each for soileach layer soil layerare shown are shown in Table in Table 4. To 4.solve To solvethe problem the problem of wave of wave boundaries,largerfrequenciesalso calculated than thespring-damper the in embeddedthe for Rayleighcomparison. depth elements damping ofRayleigh the were calculationtunnel applied dampin bottom onwereg thewas (20.8 set boundaries, employed m).as 1A and tunnel-soil for which100 the Hz. were soilmodel The model. widely damping without The used pilesratio two in wasandkey the similaralsootherfrequencies calculated researchesdynamic in theforparameters [3 ,Rayleigh64comparison.,65]. The for damping dynamiceach Rayleigh soil calculation elastic layer dampin modulus,are weregshown was set massemployed in as Table 1 density and 4. for100 To and the Hz.solve Poisson’ssoil The themodel. damping problem ratio The of ratio thetwoof wave andkey 3 tunnelfrequenciesother lining dynamic structure in the parameters were Rayleigh 42 GPa, fordamping 2400each kgsoil calculation/m layer, and are 0.3, were shown respectively. set in as Table 1 and 4. 100 To Hz.solve The the damping problem ratio of wave and other dynamic parameters for each soil layer are shown in Table 4. To solve the problem of wave Sustainability 2020, 12, x FOR PEER REVIEW 14 of 22 Sustainability 2020, 12, x FOR PEER REVIEW 14 of 22 reflection at the boundaries, the spring-damper elements were applied on the boundaries, which reflectionwere widely at theused boundaries, in the similar the researches spring-damper [3,64,65]. elements The dynamic were applied elastic moondulus, the boundaries, mass density which and wereSustainabilityPoisson’s widely ratio2020 used, of12, the 5871in the tunnel similar lining researches structure [3,64, were65]. 42 The GPa, dynamic 2400 kg/m elastic3, and mo 0.3,dulus, respectively. mass density 14 and of 20 Poisson’s ratio of the tunnel lining structure were 42 GPa, 2400 kg/m3, and 0.3, respectively.

Figure 11. Tunnel-soil-piles coupled FE model. Figure 11. Tunnel-soil-piles coupled FE model. To calculatecalculate thethe movingmoving traintrain loads,loads, the MATLAB-basedMATLAB-based vehicle-trackvehicle-track coupled model was employedTo calculate [66]. [66]. In In thisthe this 2Dmoving 2D model, model, train vehicles vehiclesloads, includ the include eMA fourTLAB-based fourwheelsets, wheelsets, twovehicle-track bogies two bogies and coupled one and car one body,model car which body,was employedwhichwere connected were [66]. connected In by this Kelvin 2D by model, Kelvin elements vehicles elements to model includ to model thee four primary the wheelsets, primary and secondary andtwo secondarybogies suspensions. and suspensions. one car body,The rail The which was rail werewasmodelled modelledconnected by an by by Euler an Kelvin Euler beam elements beam with with infiniteto infinite model length, length,the primary which which andwas was secondaryperiodically periodically suspensions. supported supported Theby by fastenersrail was modelledmodeled byby Kelvinan Euler elements beam (Figurewith infinite 1212).). The length, Herzia Herzian whichn springs was were periodically used to modelsupported the contactby fasteners forces modeledbetween wheelsby Kelvin and elements rails. A more (Figure detailed 12). The description Herzian springs of this modelwere used can beto model found inthein Ref.Ref. contact [[66,67].66,67 forces]. between wheels and rails. A more detailed description of this model can be found in Ref. [66,67]. v x v z Mc Jc x M J φc z c c zc φc kt ct zc φ kMt t Jctt t φ kw Mt J t cw t zt ough R (x) kw cMw w z zw kH t 8 Rough(x) Mw zw 8 kH 8 8 L L Figure 12. Vehicle-track coupled model. Figure 12. Vehicle-track coupled model. Tables6 6 and and7 7list list the the parameters parameters for for the the ballastless ballastless track track and and the the vehicle. vehicle. The The running running train train speedTables was 60 6 km/h. kmand/h. 7 Track Tracklist the irregulari irregularity parametersty was for simulated the balla stless according track to and the Class the vehicle. 5 track Theirregularity irregularity running power train speedspectral was density 60 km/h. function Track determinedirregularity forwas U.S.U. simulatedS. railways, according whichwhich isis to expressedexpressed the Class 5 asas track follows:follows: irregularity power spectral density function determined for U.S. railways, which is expressed as follows: Ω22 kAvc Ω= kAvΩc2 S(ΩS()) = 22Ω 2,, (15) ΩΩ+Ω2 kA2vc2 S()Ω= ( ()+ c) , ΩΩΩ+Ω22Ω Ωc 2 (15) ()c v c in whichwhich kk == 0.25, AAv = 0.2095,0.2095, and and ΩΩc== 0.8245.0.8245. By By inputting inputting the the trai trainn parameters, parameters, track track parameters, in which k = 0.25, Av = 0.2095, and Ωc = 0.8245. By inputting the train parameters, track parameters, and track irregularity, thethe dynamicdynamic forces underunder each fastener were calculated and applied to the FE and track irregularity, the dynamic forces under each fastener were calculated and applied to the FE model. A A typical typical time time history and and Fourier spectrum of the dynamic forces under one fastener are model. A typical time history and Fourier spectrum of the dynamic forces under one fastener are illustrated in Figure 1313.. illustrated in Figure 13. Table 6. Track parameters. Table 6. Track parameters. ParameterParameter Category Category * * Value Value Parameter Category Category * * Value Value Parameter Category * Value Parameter Category * Value8 Rail mass per unit length, m 121.28 kg/m Fastener stiffness, kr 1.2 × 108 N/m Rail mass per unit length, m 121.28 kg/m Fastener stiffness, kr 1.2 10 8N/m Rail mass per unit length, m 121.28 kg/m11 2 Fastener stiffness, kr 1.2 ×× 104 N/m Elastic modulus modulus of of rail, rail,E E 2.059 2.059 10× 1011 N /N/mm2 Fastener damping, damping,c cr 6 6 ×10 104 Ns Ns/m/m 11 2 r 4 Elastic modulus of rail, E 2.059× × 10 5 −N/m5 4 4 Fastener damping, cr 6 ×× 10 Ns/m InertiaInertia momentmoment of of rail rail reaction, reaction,I I 6.434 6.43410 × −10m m Fastener spacing, spacing,L L 0.60.6 m m × −5 4 Inertia momentRailRail loss loss factor,of factor, railη reaction, η I 6.4340.01 0.01× 10 m Fastener spacing, L 0.6 m Rail loss factor, η * all* all track track parameters parameters0.01 correspond correspond to to two two rails. rails. * all track parameters correspond to two rails. Sustainability 2020, 12, 5871 15 of 21

Table 7. Train parameters.

Parameter Category Value Parameter Category Value Secondary suspension 5.8 × 105 Sustainability 2020Vehicle, 12, 5871 body mass, Mc 4.3 × 104 kg 15 of 20 stiffness, kt N/m Secondary suspension 1.6 × 105 Bogie mass, Mt 3.6 × 103 kg Table 7. Train parameters. damping, ct Ns/m Primary suspension stiffness, 1.4 × 106 3 ParameterWheelset Category mass, Mw 1.7 Value × 10 kg Parameter Category Value kw N/m 4 5 VehicleVehicle body body mass, massMc inertia 4.3 10 kg SecondaryPrimary suspension suspension stiffness, kt 5.8 10 N/m 1.7 ×× 1036 kg·m2 5 × 10×4 Ns/m5 Bogie mass,moment,Mt Jc 3.6 10 kg Secondarydamping, suspension cwdamping, ct 1.6 10 Ns/m × 3 × 6 Wheelset mass, Mw 1.7 10 kg Primary suspension stiffness, kw 1.4 10 N/m Bogie mass inertia moment Jt 9.62× × 103 kg·m2 Semi fixed distance b 6.3× m Vehicle body mass inertia moment, J 1.7 106 kg m2 Primary suspension damping, c 5 104 Ns/m c −8 w ×5.1473 × 10· 2 × BogieWheel-rail mass inertia contact moment constantJt G 9.62 10 kg m SemiSemi fixedwheelbase distance a b 1.16.3 m m × 2/3· Wheel-rail contact constant G 5.147 10m/N8 m/ N2/3 Semi wheelbase a 1.1 m × − TrainTrain length lengthl l 19 19 m m Static Static wheel-rail wheel-rail force force P0 7 ×7 10410 N4 N 0 ×

4 4x10 104

4 3x10 103

4 2x10 102

4 1

Load/N 1x10 10 Load/(N/Hz)

0 100

-1 -1x104 10 012345678910 0 10 20 30 40 50 60 70 80 90 100 110 120 Time/s Frequency/Hz

(a) (b)

FigureFigure 13. (a )13. Time (a) Time history history and and (b) ( Fourierb) Fourier spectrum spectrum of of the the dynamicdynamic supporting supporting force force under under a fastener. a fastener.

5.2. Ground5.2. Ground Vibration Vibration Analysis Analysis By applyingBy applying the train the train loads loads to the to FEthe model,FE model, the the ground ground vibrations vibrations at at a a distance distance ofof 3030 mm fromfrom the tunnelthe center tunnel were center calculated. were calculated. Nine location Nine pointslocation were points analyzed were analyzed (corresponding (corresponding to the measuredto the pointsmeasured A1–A9 illustrated points A1–A9 in Figure illustrated3). Figure in Figure 14 shows 3). Figure the statistical14 shows root-mean-squarethe statistical root-mean-square (r.m.s.) values of the ground(r.m.s.) values vibration of the responses ground vibration with and responses without with periodic and without piles. periodic With periodic piles. With piles, periodic the vibration piles, theSustainability vibration 2020 velocity, 12, x FOR energy PEER REVIEW was clearly attenuated, with a maximum attenuation of 15%. 17 of 22 velocity energy was clearly attenuated, with a maximum attenuation of 15%.

-5 8x108x10-5 WithoutWithout pilespiles -5 7x107x10-5 WithWith pilespiles

-5 6x106x10-5

-5 5x105x10-5

-5 4x104x10-5

-5 3x103x10-5 Velocity r.m.s./(m/s) Velocity r.m.s./(m/s) -5 2x102x10-5

-5 1x101x10-5

00 A1A1 A2 A2 A3 A3 A4 A4 A5 A5 A6 A6 A7 A7 A8 A8 A9 A9 DetectingDetecting pointpoint

Figure 14. Statistical root-mean-square (r.m.s.) values of ground vibration responses with and without Figure 14. Statistical root-mean-square (r.m.s.) values of ground vibration responses with and periodic piles. without periodic piles.

FigureFigure 15 illustrates 15 illustrates the typicalthe typical Fourier Fourier spectra spectra for for ground ground vibrationsvibrations with with and and without without periodic periodic piles andpiles the and corresponding the corresponding amplitude amplitude attenuation attenuation factor, factor,A Ar,r, which which is defined defined as: as: A ()f Af()A=w(wf ) , A ( f )r = (),(16) (16) r Aw/o f Aw/o( f )

where Aw(f) and Aw/o(f) are the Fourier spectra of ground vibrations with and without piles, respectively. Vibrations between 52.4 Hz and 74.3 Hz were effectively attenuated, which correlates with the calculated design band gaps illustrated in Figure 10. The maximum vibration attenuation within the band gaps was 80%. Figure 16a illustrates the ground vibration velocities, averaged for points A1–A9, in one-third octave bands (RoutPRE(fi)), with and without periodic piles. Without piles, the ground vibration responses 30 m from the tunnel center exceeded the VC-C curve, which is similar to the measured results illustrated in Figure 9b. Below 10 Hz, the measured ground vibrations were substantially larger than the calculated vibrations because road traffic and other environmental vibrations contributed to the final measured results.

-5 1.6x10 1.0

1.4x10-5 With piles Without piles 0.8 1.2x10-5

-5 1.0x10 0.6

8.0x10-6 0.4 -6 Velocity/(m/s) 6.0x10

-6 4.0x10 0.2

2.0x10-6 52.4-74.3 Hz Amplitude attenuation factor Ar 0.0 0.0 0 1020304050607080 0 1020304050607080 Frequency/Hz Frequency/Hz

(a) (b)

Figure 15. (a) Typical Fourier spectrum of ground vibrations with and without periodic piles, and its (b) amplitude attenuation factor. Sustainability 2020, 12, 5871 16 of 21

Figure 14. Statistical root-mean-square (r.m.s.) values of ground vibration responses with and without periodic piles.

Figure 15 illustrates the typical Fourier spectra for ground vibrations with and without periodic piles and the corresponding amplitude attenuation factor, Ar, which is defined as: A ()f Sustainability 2020, 12, 5871 Af()= w , 16 of 20 r () (16) Aw/o f where Awherew(f ) and AwA(f)w /ando(f ) areAw/o the(f) Fourierare the spectraFourier ofspectra ground of vibrationsground vibrations with and with without and piles, without respectively. piles, Vibrationsrespectively. between 52.4Vibrations Hz and between 74.3 Hz 52.4 were Hz e ffandectively 74.3 Hz attenuated, were effectively which attenuated, correlates withwhich the correlates calculated design bandwith the gaps calculated illustrated design in Figureband gaps 10. Theillustrated maximum in Figure vibration 10. The attenuation maximum vibration within the attenuation band gaps within the band gaps was 80%. Figure 16a illustrates the ground vibration velocities, averaged for was 80%. Figure 16a illustrates the ground vibrationPRE velocities, averaged for points A1–A9, in one-third points A1–A9,PRE in one-third octave bands (Rout (fi)), with and without periodic piles. Without piles, octavethe bands ground (Rout vibration(fi)), responses with and 30 without m from periodic the tunnel piles. center Without exceeded piles, the VC-C the groundcurve, which vibration is responsessimilar 30 mto fromthe measured the tunnel results center illustrated exceeded in Figure the 9b. VC-C Below curve, 10 Hz, which the measured is similar ground to the vibrations measured results illustratedwere substantially in Figure larger9b. Belowthan the 10 calculated Hz, the measuredvibrations because ground road vibrations traffic and were other substantially environmental larger than thevibrations calculated contributed vibrations to becausethe final measured road traffi results.c and other environmental vibrations contributed to the final measured results.

-5 1.6x10 1.0

1.4x10-5 With piles Without piles 0.8 1.2x10-5

-5 1.0x10 0.6

8.0x10-6 0.4 -6 Velocity/(m/s) 6.0x10

-6 4.0x10 0.2

2.0x10-6 52.4-74.3 Hz Amplitude attenuation factor Ar 0.0 0.0 0 1020304050607080 0 1020304050607080 Frequency/Hz Frequency/Hz

(a) (b)

SustainabilityFigure 15.2020Figure(a, 12) Typical15., x FOR(a) Typical PEER Fourier REVIEWFourier spectrum spectrum of ground of ground vibrations vibrations with with and and withoutwithout periodic periodic piles, piles, and and its its18 of 22 (b) amplitude(b) amplitude attenuation attenuation factor. factor.

10-1 (a) 10-1 (a) VC-A VC-A VC-B VC-B 10-2 VC-C 10-2 VC-C VC-D VC-D VC-E VC-E VC-F -3 VC-F 10 10-3 VC-GVC-G Velocity/(mm/s) Velocity/(mm/s)

-4 -4 10 10 B1 with piles B 1B 1w iwt h/o p pilileess B1 w/ o pil es

10-5 -5 10 100 101 102 0 1 2 10 1/310 octave band/Hz 10 1/3 octave band/Hz

10-1 10-1 (c) (b) VC-A VC-A VC-B VC-B

10-2 VC-C 10-2 VC-C VC-D VC-D VC-E VC-E VC-F VC-F 10-3 10-3 VC-G VC-G Velocity/(mm/s) Velocity/(mm/s)

10-4 10-4 B1 wit h pil es B2 wit h pil es B1 w/ o pil es B2 w/ o pil es

10-5 10-5 100 101 102 100 101 102 1/3 octave band/Hz 1/3 octave band/Hz

Figure 16. One-third octave vibration velocities (a) on the ground with and without periodic piles, Figure 16. One-third octave vibration velocities (a) on the ground with and without periodic piles, (b) (b) in the laboratory room for point B1, and (c) in the laboratory room for point B2. in the laboratory room for point B1, and (c) in the laboratory room for point B2.

Based on the vibration power superposition, the vibration responses due to road traffic and metro trains can be determined as follows:

22 PRE()=+ PRE,m () EXP,r () , (17) Rfiniii R in f R in f

where RinPRE,m(fi) is the floor vibration in Room 103 induced by metro trains, which can be determined using the predicted ground vibration and measured transfer function in Equation (2); and RinEXP,r(fi) is the floor vibration in Room 103 induced by road traffic measured during rush hours. The floor vibration in Room 103 induced by different types of traffic were then plotted in Figure 16b,c together with the VC curves. The final floor vibrations were attenuated within the designed band gaps, providing a better environment for more sensitive instrument installation. A comparison of VC curves revealed that vibration attenuation of one level was obtained by employing the periodic piles design. However, due to finite pile length and pile rows, the vibration mitigation effect was still limited compared with other vibration source solutions, such as floating slab tracks.

6. Conclusions To control the impacts of metro train-induced vibrations on a laboratory containing sensitive instruments, band gaps for periodic piles were designed based on a novel band gap performance evaluation function, Φ. The vibration mitigation effect of the periodic piles was validated by numerical calculation. The main conclusions of this study are as follows: • The band gap performance evaluation function can be used to optimize the mitigation effect for periodic piles. • With the designed periodic piles, vibrations between 52.4 Hz and 74.3 Hz were effectively attenuated, especially at 63 Hz, which agrees with the calculated design band gaps. Sustainability 2020, 12, 5871 17 of 20

Based on the vibration power superposition, the vibration responses due to road traffic and metro trains can be determined as follows: r h i2 h i2 PRE( ) = PRE,m( ) + EXP,r( ) Rin fi Rin fi Rin fi , (17)

PRE,m where Rin (fi) is the floor vibration in Room 103 induced by metro trains, which can be determined EXP,r using the predicted ground vibration and measured transfer function in Equation (2); and Rin (fi) is the floor vibration in Room 103 induced by road traffic measured during rush hours. The floor vibration in Room 103 induced by different types of traffic were then plotted in Figure 16b,c together with the VC curves. The final floor vibrations were attenuated within the designed band gaps, providing a better environment for more sensitive instrument installation. A comparison of VC curves revealed that vibration attenuation of one level was obtained by employing the periodic piles design. However, due to finite pile length and pile rows, the vibration mitigation effect was still limited compared with other vibration source solutions, such as floating slab tracks.

6. Conclusions To control the impacts of metro train-induced vibrations on a laboratory containing sensitive instruments, band gaps for periodic piles were designed based on a novel band gap performance evaluation function, Φ. The vibration mitigation effect of the periodic piles was validated by numerical calculation. The main conclusions of this study are as follows:

The band gap performance evaluation function can be used to optimize the mitigation effect for • periodic piles. With the designed periodic piles, vibrations between 52.4 Hz and 74.3 Hz were effectively • attenuated, especially at 63 Hz, which agrees with the calculated design band gaps. A comparison of VC curves revealed that vibration attenuation of one level can be achieved using • the periodic piles design. However, due to finite pile length and pile rows, the vibration mitigation effect was still limited. As this study only considered a propagation path solution using periodic piles, the mitigation effects and feasibility of both periodic piles and isolated tracks should be further investigated.

Author Contributions: Conceptualization, M.M. and W.L.; methodology, M.M.; software, B.J. and K.L.; validation, M.M., B.J.; formal analysis, M.M.; investigation, M.M. and B.J.; resources, W.L.; data curation, M.M. and W.L.; writing—original draft preparation, M.M.; writing—review and editing, B.J.; visualization, K.L.; supervision, M.M. and W.L.; project administration, W.L.; funding acquisition, M.M. and W.L. All authors have read and agreed to the published version of the manuscript. Funding: The research was supported by Beijing Natural Science Foundation (Grant No. 8202040) and National Natural Science Foundation of China (Grant No. 51978043). Acknowledgments: The authors would like to appreciate Linlin Du from SINOMACH Academy of Science and Technology Co., Ltd. and Jialiang Chen, Linfeng Li and Chao Niu from Beijing Jiaotong University to participate the in-situ measurement. Conflicts of Interest: The authors declare no conflict of interest.

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