Determination of Depth to Bedrock from Falling Weight Deflectometer Test Data

Transcription

1 68 TRANSPORTATION RESEARCH RECORD 154 Dtrmination of Dpth to Bdrock from Fallin Wiht Dflctomtr Tst Data JOSE M. ROESSET, KENNETH H. STOKOE II, AND CHIA-RAY SENG Dpth to bdrock can hav a sinificant ffct on th moduli of th surfac layr, bas, and subrad back-calculatd from Fallin Wiht Dflctomtr (FWD) tsts whn th calculations ar prformd usin a static analysis. On th othr hand, whn th back calculation is basd on dynamic analysis of th FWD tst, th rsultin moduli ar quit insnsitiv to th dpth to bdrock if this dpth is larr than about 3 m (1 ft). Whn only th pak dflctions at th svn rcivrs ar rcordd, thr is littl that can b don analytically to account for th dpth to bdrock. Howvr, whn th tim historis of th motions at th various rcivrs ar stord, on can stimat both th dpth to bdrock and th modulus of th sub rad rathr asily from th tim historis. Ths stimations ar most corrctly applid to tim historis of.12 sc or lonr rathr than th.6 sc convntionally rcordd. In this articl, th rsults of a numbr of paramtric studis simulatin th FWD tst on thr flxibl pavmnts and on riid pavmnt ar prsntd. From ths rsults, approximat xprssions wr dvlopd for (a) stimatin th dpth to bdrock, (b) computin th natural priod of vibration of th pavmnt systm, which is narly qual to that of th subrad layr, and (c) stimatin th wav propaation vlocity and modulus. Dflction basins causd by th dynamic loads imposd by th Fallin Wiht Dflctomtr (FWD) ar narly indpndnt of dpth to bdrock if this dpth is larr than about 3 m (1 ft). Dflction basins causd by th sam load applid statically ar, on th othr hand, affctd by th xistnc of bdrock to dpths of 15 m (5 ft) or mor. As a rsult, if th back-calculation procdur usd to dtrmin th lastic moduli of th surfac layr, bas, and subrad mploys a static analysis, th rsults will dpnd on th dpth to bdrock as shown by svral studis ovr th last dcad. Th objctiv of this articl is to discuss procdurs by which th dpth to bdrock and th modulus of th subrad can b stimatd dirctly from th FWD rsults, without back calculation, whn th tim history of th motions rcordd at th various rcivrs is providd rathr than only th pak dflctions. Analytical simulations of th FWD tst wr conductd usin th UTFWD proram (J,2). Th four pavmnt profils shown in Fiur 1 wr studid. Th main purposs of ths studis wr as follows: l. To dtrmin th dpth to bdrock at which rsonanc may occur for various stiffnsss of th subrad, includin unsaturatd and saturatd conditions; 2. To dvlop quations for stimatin this rsonant dpth to bdrock; 3. To dvlop a mthod for stimatin th dpth to bdrock basd on th fr vibrations of th pavmnt systm cratd in th FWD tst; and J.M. Rosst and K. H. Stako, II, Dpartmnt of Civil Eninrin, Univrsity of Txas at Austin, Austin, Tx C.-R. Sn, Unitd Gotchnical Consultants, Taipi, Taiwan. 4. To dvlop an approach for stimatin th stiffnss of th subrad layr basd on th phas shift btwn th first pulss in th dflction-tim rcords of th FWD tst at two masurmnt stations. Th dflction basins prsntd in this articl ar normalizd with rspct to th load. Thrfor, actual dflctions undr any load ar simply calculatd by multiplyin th normalizd dflction by th dsir load. MODEL PARAMETERS Only th subrad stiffnss and thicknss wr varid in this study. Th stiffnsss of th othr pavmnt layrs wr kpt constant as listd. 1. Continuously rinforcd concrt (CRC): shar.wav vlocity, Vs= 2,593 m/sc (8,5 fps) Youn's modulus, E = 37.4 MN/m 2 (5,425 ksi) 2. Asphalt concrt (AC): Vs = 915 m/sc (3, fps) E = 4.8 MN/m 2 (69 ksi) 3. Bas matrial: V, = 35 m/sc (1, pfs) E =.46 MN/m 2 (67 ksi) Th thicknss of ach of ths uppr pavmnt layrs varid btwn th four profils but it was kpt constant in ach profil. Th shar wav vlocity of th subrad layr in Profils l to 4 was varid from 15 to 45 m/sc (5-1,5 fps), and th corrspondin Youn's modulus varid from.11to.98 MN/m 2 (16 to 142 ksi). In addition, th dpths to bdrock of th four pavmnt profils (Fiur I) wr varid from 1.65 to 27 m (5.5-9 ft), which simply mans that th thicknss of th subrad was varid. Th stiffnss of th bdrock bnath th subrad was assumd to b infinit. To simulat an unsaturatd subrad, a Poisson's ratio of.33 was usd. Th matrial proprtis of th four pavmnt profils with unsaturatd subrad conditions ar ivn in Tabl 1. To simulat a saturatd subrad, th comprssion (P-wav) vlocity of th subrad was st qual to 1,525 m/sc (5, fps). This vlocity rprsnts typical fild conditions for uncmntd saturatd soils (3). Th shar wav (S-wav) vlocitis of th saturatd subrads wr varid from 15 to 45 m/sc ( 5-1,5 fps), as in th unsaturatd subrad condition. As a rsult, Poisson's ratio varid from.495 to.451 as th S-wav vlocity of th subrad varid from 15 to 45 m/sc (5-1,5 fps). Hnc, Youn's modulus (which is qual to th rsilint modulus) for th saturatd subrad varid from.12 to 1.7 MN/m 2 ( ksi). Th matrial proprtis of th four pavmnt profils with saturatd subrads ar ivn in Tabl 2.

2 Rossr t al. 69 ACP Bas ACP Bas Subrad '''"'",'-"' Bdrock (a) Profil 7 (FM nar Paris, TX) Subrad Bdrock ( c) Profil 3 (Rout 7 nar Austin, TX) ACP Bas Subrad "' < Bdrock (b) Profil 2 (FM 795 nar Paris, TX) Subrad Bdrock (d) Profil 4 (IH 7 nar El Paso, TX) FIGURE 1 Idalizd cross-sction of th four pavmnt profils. DEFLECTION BASINS A typical xampl of th dflction basins cratd by th FWD tst as a function of dpth to bdrock is shown in Fiur 2. Ths rsults corrspond to Profil I with a soft, unsaturatd subrad. Th dynamic dflctions at all svn masurmnt stations (rcivrs) usd in convntional tstin vary only at shallow bdrock dpths of lss than 3 m (I ft) as shown in Fiur 2(a). An important obsrvation is that th dflction at th first masurmnt station (undr th load) rmains narly constant throuhout th ntir ran of bdrock dpths. Only at dpths lss than about 2 m (7 ft) is this dflction affctd. On th othr hand, th dynamic dflctions at shallow bdrock dpths ar influncd mor (rlativ to Station I) as th distanc from th load point incrass. Th static dflctions at all svn masurmnt stations for Profil I with an unsaturatd subrad ar shown in Fiur 2(b ). As can b sn comparin Fiur 2(a) and 2(b), static dflctions ar influncd mor by shallow bdrock than dynamic dflctions. In addition, thy ar influncd ovr a widr ran of bdrock dpths than dynamic dflctions. Ths two obsrvations ar critical points whn considrin th dynamics in th FWD tst. An asy way to viw th ffct of bdrock dpth on th FWD tst is in trms of dflction ratio; that is, th ratio of dynamic dflctions to static dflctions. Th dflction ratios for Profil l with an unsaturatd subrad ar shown in Fiur 2(c). In this cas, thr is a pak in th dflction ratios at a dpth of about -1.8 m (6 ft). Th dpth to bdrock corrspondin to th maximum dflction ratio is calld th rsonant dpth to bdrock (4). It is at this dpth whr back calculation usin a procdur that assums static loadin in th FWD tst would b most in rror. A typical st of plots showin th combind ffcts of sub rad stiffnss and dpth to bdrock on th dflction ratios masurd at

3 7 TRANSPORTATION RESEARCH RECORD 154 TABLE I Simplifid Equations for Estimatin Dpth to Bdrock with Unsaturatd Subrad Conditions a) FM137 (Profil 1) Unit S-wav P-wav Youn's Matrial Thicknss Poisson's Wiht Dampin Vlocity Vlocity Modulus Typ (in.) Ratio (pcf) Ratio (ksi) ACP , 5, Bas , 1, Subrad* h** , , 1, ,5 2, b) FM195 (Profil 2) Unit S-wav P-wav Youn's Matrial Thicknss Poisson's Wiht Dampin Vlocity Vlocity Modulus Typ (ln.) Ratio (pd) Ratio (fps) (fps) cksl) ACP , 5, Bas , 1, Sub rad h , , 1, ,5 2, c) Rout 1 (Profil 3) Unit S-wav P-wav Youn's Matrial Thicknss Poisson's Wiht Dampin Vlocity Vlocity Modulus Typ (ln.) Ratio (pct) Ratio (fps) (fps) (ksi) ACP , 5, Bas , 1, Subrad h , , 1, ,5 2, d) IH 1 (Profil 4) Unit S-wav P-wav Youn's Matrial Thicknss Poisson's Wiht Dampin Vlocity Vlocity Modulus Typ (in.) Ratio (pcf) Ratio (fps) (fps) (ksl) CRC ,5 13, ACP , 5, Bas , 1, Subrad h , , 1, ,5 2, Thr ar four diffrnt S-wav vlocitis for ach subrad. - Thicknss of subrad (h) was varid from 5.5 to 9 ft. all svn stations in th FWD tst is prsntd in Fiur 3. Ths rsults ar for Profil 1 with an unsaturatd subrad. As sn in th fiur, th rsonant dpth to bdrock incrass as th stiffnss of th subrad incrass. Also, it should b notd that Station 7 in th FWD tst is most affctd by shallow bdrock and Station 1 (at th cntr of th load) is last affctd. Dynamic and static dflction basins as a function of distanc from th sourc obtaind at th pak dflction ratio (th rsonant dpth to bdrock) ar shown in Fiur 4 for Profil 1 (flxibl pavmnt) with unsaturatd and saturatd subrad conditions, rspctivly. As can b sn, thr is littl diffrnc at th sourc btwn dynamic dflctions and static dflctions for Profil I with th softst subrad [Fiur 4(a)]. Th diffrncs btwn dynamic dflctions and static dflctions bcom larr as th distanc from th sourc incrass. This bhavior xplains why th dflction ratio at th narst station is th smallst and th dflction ratio at th farthst station is th larst, as illustratd in Fiurs 2(c) and 3. For th stiffst subrad condition, thr is littl diffrnc btwn th dynamic dflctions and th static dflctions, as shown in Fiur 4(b). For th saturatd subrad conditions at Profil I, th trnds ar similar to thos first dscribd for th unsaturatd subrad conditions, as shown in Fiur 4(c) and 4(d).

4 Rosst t al. 71 TABLE2 Simplifid Equations for Estimatin Dpth to Bdrock with Saturatd Subrad Conditions a) FM 137 (Profil 1) Unit S-wav P-wav Youn's Matrial Thicknss Poisson's Wiht Dampin Vlocity Vlocity Modulus Typ (in.) Ratio (pct) Ratio {fps) _J!EL (ksl) ACP , 5, Bas , 1, Subrad* h** , , , 5, ,5 5, b) FM 195 (Profil 2) Unit S-wav P-wav Youn's Matrial Thicknss Poisson's Wiht Dampin Vlocity Vlocity Modulus Typ (in.) Ratio (pcf) Ratio (fps) _J!EL (ksl) ACP , 5, Bas , 1, Subrad h , , , 5, ,5 5, c) Rout 1 (Profil 3) Unit S-wav P-wav Youn's Matrial Thicknss Poisson's Wiht Dampin Vlocity Vlocity Modulus Typ (in.) Ratio (pd) Ratio (fps) (ksi) ACP , 5, Bas , 1, Subrad h , , , 5, ,5 5, 183. d) IH 1 (Profil 4) Unit S-wav P-wav Youn's Matrial Titlcknss Poisson's Wiht Dampin Vlocity Vlocity Modulus Typ (in.) Ratio (pcf) Ratio (fps) _J!EL (ksl) CRC ,5 13, ACP , 5, Bas i , 1; h , 17.7 Sub rad , , 5, ,5 5, Thr ar four diffrnt S-wav vlocitis for ach subrad. **Thicknss of subrad (h) was varid from 5.5 to 9 ft. Similar plots of dflction basins as a function of distanc from th sourc for Profil 4 (riid pavmnt) ar shown in Fiur 5 for unsaturatd and saturatd subrad conditions. Th amplituds of th dflction basins ar much smallr than thos obtaind with Profil I, bcaus th surfac layr of Profil 4 is much thickr and stiffr than that of Profil 1. Thr is also lss variation in th dflction basins with distanc from th FWD load than in Profil 1. This rspons can b most asily sn by comparin Fiurs 4(a) and S(a). This diffrnc occurs bcaus Profil 4 rprsnts a riid pavmnt, whras Profil 1 rprsnts a vry flxibl pavmnt. For saturatd subrad conditions [(Fiur S(c)], th trnds for th riid pavmnt ar similar to thos just dscribd for th unsaturatd subrad conditions [(Fiur 4(c)]. RESONANT DEPTH TO BEDROCK Th analytical simulations of th FWD tsts for th four pavmnt profils wr xprssd in trms of dflction ratios as a function of dpth to bdrock. Th rsonant dpth to bdrock, RDb, for ach pavmnt profil with ach subrad stiffnss was dtrmind as th dpth to bdrock corrspondin to th maximum dflction

5 72 TRANSPORTATION RESEARCH RECORD c c :1, a &;;;;.._ , c ; Dpth to Bdrock (ft) (a) Dynamic dflctions., V). L..11.m:;;..._. _....._ _ 2.S 2. ; 1.S Dpth to Bdrock (ft) 2 (b) Static dflctions Station 1...,._ Station 2...,_ Station 3 -o- Station 4 -c- Station S -o- Station 6 - Station Dpth to Bdrock (ft) ( c) Dflction ratios FIGURE 2 Dflction basins obtaind with FWD tstin for Profil 1 with unsaturatd subrad conditions [Vs of subrad = 5 fps (155 m/sc) and E = 16 ksi (.11 MN/m 2 )]. ratio. Th valus of RDb as a function of various subrad stiffnsss for th four pavmnt profils ar summarizd in Fiur 6. Th rsonant dpths to bdrock obtaind with th FWD tst varid from l.7 to 6.2 m (5.5-2 ft). Th valus of RDb wr plottd vrsus subrad stiffnss, and th rsults ar prsntd in Fiur 7. As sn in Fiur 7, th rsonant dpths to bdrock form two roups; on for flxibl pavmnts (Profils 1, 2, and 3) and th (a) Vs of Subrad = 5 fps (E= 16 ksi) ; ;l.2.l... l... l... L.l l... o.s-l. : I : j ; ; ; I I I I I I I I I I o ro w w (b) Vs of Subrad = 75 fps (E= 36 ksi) 3..! 2.5 ; : i i i i l i i 1 i r c 2.-: : : : : : : :=... I j... j...)... j I.5! : : : : I l : : ; I I I I I I I I I l o ro w w (c) Vs of Subrad = 1 fps (E= 63 ksi) '-! !--!-.---!I i: t-.=... = 1-1-! i- -.-!i jf !-.-..-;j. c o.s 1 1 1! I i I 1 I 1 T l I I I I I I I I o ro w w (d) Vs of Subrad = 15 fps CE= 142 ksi) l--+---! ,!-----! !I.Si 2. i! 1! 1!! 1! 1 I -lhjmj112jmj111j111! I I 1 I I I I I I I o ro w w DDth to bdrock (ft) -+-Swionl --Swion2 -swion3 --Swion4... swions -Swion6 -Swion7 -+-Sralion l --Swion2 -swion3 -sw:ion4... swions -Swion6 -z-swion7 -t-swionl --Swion2 -swion3... swion4...swions -Swion6 -swion7 -t-swionl --Swion2 -Swion3... swion4... stations -Swion6 -swion7 FIGURE 3 Dflction ratio vrsus dpth to bdrock for FWD tsts at Profil 1 with unsaturatd subrad conditions. othr for th riid pavmnt (Profil 4). For flxibl pavmnts, a straiht lin was fittd to th data, which rsults in: RDb =.13\/, (1) For th riid pavmnt, th fittd lin was: RDb =.11 \!, (2) In Equations I and 2, RDb and V, can b xprssd in any consistnt st of units. It is asy to s that th rsonant dpth to bdrock can b prdictd from th subrad stiffnss. Equation 1 can also b xprssd in trms of Youn's modulus( ) as: RDb =.18 VE co.43 VE) (3) for flxibl pavmnts with th unit wiht of th subrad assumd to b 19 8 N/m 3 (11 lb/ft3) and Poisson's ratio of th subrad assumd to b.33. Likwis, Equation 2 can b xprssd as: RD1, =.15 VE co.36 VE) (4)

6 ..2 = tj Static Dflctions (B) c Unsaturatd subrad with S-wav vlocity = 5 fps (E = 16 ks/). -:;-.2 =.4.8.S! 1. tj (c) c Static Dflctions or '---.i.---.ii...--' Saturatd subrad with S-wav vlocity = 5 fps (E = 16 ks/)..2 =.4.8.S! 1. u Ii o Dynamic. Dflctions 1.6 Static Dflctions (b) FIGURE 4 conditions s 6 Unsaturatd subrad with s-wav vlocity = 1,5 fps (E = 142 ks/). -:;-.2 = tj I I I I I 1.8 Static Dflctions ,-.. --' (d) Saturatd subrad with S-wav vlocity = 1,5 fps (E = 142 ks/) Dflction basins as a function of distanc from th FWD sourc for Profil 1 with saturatd subrad. -:;-.2 = ro:: Static Dflctions (B) Unsaturatd subrad with S-wav vlocity = 5 fps (E = 16 ks/) (c)..2 =.4.8.S! 1. kl 1.2 = _ 1.8 Static Dflctions Saturatd subrad with S-wav vlocity = 5 fps (E = 16 ks/) (b)..2 = ii:: 1.4 CU.1.6 Cl c 1.8 Static Dflctions 2.i;a Unsaturatd subrad with vlocity = 1,5 fps (E = 142 ks/) 6 s-wav..2 =.4 c v 1.2 c! (d) 1.8 Cl Cl Static Dflct,lons 2. I.I , s 6 Saturatd subrad with S-wav vlocity = 1,5 fps (E = 142 ks/) FIGURE 5 conditions. Dflction basins as a function of distanc from th FWD sourc for Profil 4 with unsaturatd subrad

7 74 TRANSPORTATION RESEARCH RECORD :ii! u 2 "'C cu cc.s 15 a... 1 "' s Profil 1 8 Profil2 Profil 3 12 Profil 4 olm r::..&ju111r;;;;;;.j "-'-".c;.u... :;.&.I sot> (16) 1so (36) 1, (63) 1,soo (142) Vs of Subrad, fps (E, ksi) (a) Unsaturatd subrad conditions 25.:ii! 2 2 "'C 15 ' "'.,, Profil 1 9 Profil 2 Profil 3 Profil4 5 (16) 75 (36) 1, (63) 1,5 (142) Vs of Subrad, fps (E, ksi) (b) Saturatd subrad conditions FIGURE 6 Rsonant dpths to bdrock for FWD tstin at th four pavmnt profils with various subrad stiffnsss. for th riid pavmnt. In Equations 3 and 4, Eis in N/m 2 (or psf) and RDb in m (or ft). AMPLITUDE OF MAXIMUM DEFLECTION RA TIO Th amplitud of th dflction ratio at ach masurmnt station is an important indx of th potntial rror nratd in any static intrprtation procdur (4). Fiur 8 shows th maximum dflction ratios as a function of various sub rad stiffnsss for th four pavmnt profils. It should b notd that th maximum dflction ratio of th FWD tst always occurs at th farthst masurmnt station (Station 7). As can b sn in Fiur 8, th amplituds of th maximum dflction ratios of th four pavmnt profils dcras as th stiffnss of th subrad incrass for both unsaturatd and saturatd subrad conditions. This mans that th accuracy of back calculatd layr moduli basd on a static intrprtation mthod should improv as th subrad stiffnss incrass (4). 2 u "'C cu c a Youn's Modulus of Subrad (ksi) o Profil 1 cu 1 "' a Profil 2.,, 5 Profil 3 Q.I Profil M , 1,25 1,5 Shar Wav Vlocity of Subrad (fps) FIGURE 7 Curv fittin of th rsonant dpth to bdrock for FWD tstin at th four pavmnt profils with various subrad stiffnsss.. Th stimatd amplituds of th maximum dflction ratios of th four pavmnt profils obtaind with saturatd subrad conditions ar nrally larr than thos obtaind with unsaturatd subrad conditions. This indicats that th back-calculatd layr moduli obtaind at pavmnt sits with unsaturatd subrad conditions basd on a static intrprtation mthod should b mor. E ::::s E ')( "' S.5 ';I Profil 1 E3 Profil 2 Profil 3 r2j Profil 4 o... -= i...,.a-=im- 5 (16) 75 (36) 1, (63) 1,5 (142) v, of Subrad, fps (E, ksi) 4 u 2.S cu E ::::s (a) Unsaturatd subrad conditions Profil 1 3.S E:3 Profil 2 Profil 3 3 r2i Profil S E ')( "' o.s LmC:L &.mm::::i...1&.1...&.m.._ (16) 75 (36) 1, (63) 1,5 (142) v, of Subrad, fps (E, ksi) (b) Saturatd subrad conditions FIGURE 8 Maximum dflction ratios for FWD tstin at th four pavmnt profils with various subrad stiffnsss.

8 Rosst t al. 75 accurat than thos obtaind at sits with saturatd subrad conditions (4). ESTIMATION OF DEPTH TO BEDROCK FROM FWD TEST Chan t al. (2) dvlopd a procdur for prdictin th dpth to bdrock basd on th natural priod of fr vibrations of th pavmnt systm immdiatly aftr FWD loadin. Fiur 9 illustrats th natural priods in th tim-dflction rcords obtaind from FWD tsts with four shallow dpths to bdrock at Profil 1 l !!! 1.5 c 1. ';i.5 u cu. 'W c Tim {sc) (a) Dpth to bdrock = S ft 2. i.s 1. Td I.5,..... ; lrl : Tim {sc) (b) Dpth to bdrock = 7.S ft..!!! j:i lrl o.o_ Tim {sc) '11l..5 (c) Dpth to bdrock = 1 O ft Td/ ,.1.15 Tim {sc) (d) Dpth to bdrock = 2 ft.2 FIGURE 9 Dflction-tim historis in FWD tstin and dampd natural priods (Td) for Profil 1 with various dpths to bdrock [Vs of subrad = 5 fps (155 m/sc) and E = 16 ksi (.11 MN/m 2 )]. for unsaturatd subrad conditions. It should b notd that a lonr tim intrval (.12 to.2 sc) than normally rcordd in FWD tstin (.6 sc) must b masurd to rcord th fr vibrations with confidnc. FWD quipmnt is oftn st to rcord 6 msc of motion a.1-msc intrvals. Incrasin t to.2 or.4 msc would incras th duration without loss in accuracy. Car may b ncssary to avoid drift but vn this miht b rlativly unimportant. Additional studis that ar prsntd in this articl hav bn conductd byond thos prformd in Chan t al. (2). In ths studis, various stiffnsss of th subrad layr and diffrnt subrad saturation conditions wr xamind. This work was prformd to provid a mor complt valuation of th stimation of bdrock dpth. Thr diffrnt drs of saturation for unsaturatd subrad conditions wr simulatd by usin Poisson's ratio of.2,.33, and.4 rprsntin dry, moist, and wt (but unsaturatd) conditions, rspctivly, whil kpin th shar wav vlocity constant. Th rmainin matrial proprtis ar th sam as thos in Tabl 1 (xcpt for th comprssion wav vlocity). Th matrial proprtis for th cas of saturatd subrad conditions ar shown in Tabl 2. Four diffrnt shallow dpths to bdrock of 1.5, 2.3, 3.1, and 6.1 m (5, 7.5, 1, and 2 ft, rspctivly) wr studid. Ths dpths wr slctd bcaus th rsonant dpth to bdrock for th FWD tst was always within 6 m (2 ft) of th pavmnt surfac as shown in Fiur 3, unlss a vry stiff subrad [V, > 458 m/sc (15 fps)] was ncountrd. Unsaturatd Subrad Conditions Th dpth to bdrock vrsus natural priod, Tct.. of th fr vibrations for an unsaturatd sub rad with Poisson's ratio of.2 is shown in Fiur IO(a) and IO(b) for flxibl and riid pavmnts, rspctivly. Th sam st of plots for an unsaturatd subrad with Poisson's ratio of.33 is shown in Fiur IO(c) and (d). Ths two sts of subrad conditions can b considrd to rprsnt subrads on th dry sid of th optimum moistur contnt. For th cas of th subrad havin Poisson's ratio qual to.4 (subrad at or wt of th optimum moistur contnt), th linar rlationship btwn dpth to bdrock and th natural priod of ach profil is shown in Fiur l l(a) and (b). Thr is a linar (or narly linar) rlationship btwn dpth to bdrock and natural priod for ach stiffnss and stat of saturation of th subrad in ths fiurs. It should b notd that dpth to bdrock is dfind as th total dpth from th top of th pavmnt to th top of th bdrock. Th quations of th straiht lins for th flxibl pavmnts with diffrnt Poisson's ratios (v) of th subrad can b combind into on quation as: Th quations for th riid pavmnt with th subrad havin a ran in Poisson's ratios (v can also b combind into on quation as: (5) (6)

9 76 TRANSPORTATION RESEARCH RECORD 154 2,.._...,,_...,.-..m-""T"""-Glr---r-.,...--,-:illt-., :::::;, 15 1H----ml---1!f-r-_,_......, o V 5 5 fps; E 16 ksi av, 75 fps, E 36 ksl tlF t vs 1, fps, E "' 63 ksl Vs= 1,5 fps, E,. 142 ksi.s:: Q. QI c (BJ Flxibl pavmnts with Poisson's ratio of th unsaturatd subrad = ,.---ir-1---rr-'.:llV'"-r-r---r- ti 15H--+-lf--H''-+-l--f...-t-r:;..._+--t "O loh--+'l-.ip>:#»--t:;;iilllt--""1==1;:==;1;;:=::1:::=t==z===i-, '5 Q. cu 5 Vs= 5 fps, E = 16 ksi tl--lltjl.b;lll-+-r-.--t o Vs a 75 fps, E ksi a Vs= 1, fps, E = 63 ksi Vs= 1,5 fps, E"' 142 ksi (cj Flxibl pavmnts with Poisson's ratio of th unsaturatd subrad = a.---.c :::::;, 15 1n--r""7fP-::::t::=t::::::t==t::::::!::::::I.-. O Vs.. 5 fps, E,. 16 ksl.s:: 5 H-IP..l...,.a a Vs.. 75 fps, E.. 36 ksi Q. Vs a 1, fps, E = 63 ksl QI Vs"' 1,5 fps, E ksi c o::=:::::::lt:::::::::t====:::!:::::::::::::::==:::::z====::::r--' (bj Riid pavmnts with Poisson's ratio of th unsaturatd subrad =.2 2 ti 15 1-t--+--1'+--l'--t--+--,.Jtt!IC ' "O 1 t't--+-ill i s.s::. c 5t-t-P-::a-+-i I (dj Riid pavmnts with Poisson's ratio of th unsaturatd subrad =.33 FIGURE 1 Dpth to bdrock vrsus dampd natural priod for FWD tstin of pavmnt systms with subrads havin low to mdium dr of saturation ill!---CE>-T"O":.o"Clt"""T-9!"'"...,_.--, :::::;, 15 --Yt 'F t H-f-:J-t±=±=±::::±=:j=::!==:i.. o Vs= 5 fps, E 16 ksi 1! t--t-i D Vs= 75 fps, E = 36 ksl a Vs a 1, fps, Es 63 ksi QI Vs.. 1,5 fps, E = 142 ksi o ol.!5::::d::::::::=t:=::==:::::==::==:::::c==== (aj Flxibl pavmnts with Poisson's ratio of th unsaturatd subrad =.4 2n--r-r 'r-CllA---r-T7+r---,-..,.--, :::::;, ti 15 H--+-l*--+il''-+,,_,l--1111"' f--I "O 1 H---tF-,P::;-jth::;±::::;!;=:i=::±::::i=::l::::::J. o V 5 = 5 fps, E 18 ksi '5 5 H-tW:: ll-I a Vs= 75 fps, E = 4 ksl c. V 5 = 1, fps, E = 7 ksi cu Vs= 1,5fps,E=155 ksl i===t==:t::::::::::::::i:==:z:=::::z==':::i' (cj Flxibl pavmnts with a saturatd subrad 2n-"""T"-,-tll'-r-,SO-IOT-r--r ,..., :::::;, 15 tt--t--1r-#--t---::i-+--+:i i 1on-11rni:::::l:l:=::t=::±:::::l.._.S o Vs= 5 fps, E = 16 ksi -:; lllF a V 5 = 75 fps, E = 36 ksi c. Vs= 1, fps, E 63 ksi QI 111 O Vs= 1,5 fps, E = 142 ksl Ot!!:=::::±:::::::::±:=!::=t::==::::t====a::::==:;-' (b} Riid pavmnts with Poisson's ratio of th unsaturatd subrad =.4...,..2n--r-r-..---,-O-r--o--T-::-----r-..., ti 15 tt---t--t#--hll"'" t i "C 1 H--i-f :IJIEl I '5 5H-1::i1111"'' 't---t I Q. Q> C Ol!:::::::::t::::::!:i::::::!:::::t:::::!:::::C::::!:::::C:::!= (d} Riid pavmnts with a saturatd subrad FIGURE 11 Dpth to bdrock vrsus dampd natural priod for FWD tstin of th pavmnt systms with narly saturatd and saturatd subrads.

10 Rosst t al. 77 In Equations 5 and 6, Db and V, can b xprssd in any consistnt st of units. Equations 5 and 6 can b xprssd in trms of Youn's modulus of th subrad instad of its shar wav vlocity assumin a valu of th mass dnsity as bfor. Usin a dnsity of 2 k/m 3 (or a unit wiht of 11 pcf in British units) th xprssion substitutin for V, bcoms: v - V s - 63 VI+u (7) Saturatd Subrad Conditions For saturatd subrad conditions, four diffrnt valus for Poisson's ratios wr usd (.495,.489,.479, and.451), which corrspond to th four stiffnsss of th sub rad that rsult in th comprssion wav vlocity qualin m/sc (5, fps) (s Tabl 2). Fiur 11 (c) and 11 (d) show th dpth to bdrock vrsus natural priod in this cas. Th quation for th flxibl pavmnt is: D _ V_.Td b Th fittd quation for th riid pavmnt can b xprssd as: D _ V,Tt1 b Th xprssions in trms of th Youn's modulus can b obtaind aain usin Equation 7. ESTIMATION OF SUBGRADE STIFFNESS FROM FWD TESTS To us th abov quations, a ood stimat of th stiffnss of th subrad is rquird. Th stiffnss of th subrad can b stimatd by in situ sismic tstin, by dynamic laboratory tsts on undisturbd sampls, or possibly vn by xprinc. Howvr, a mor convnint and accurat way to stimat subrad stiffnss was dvlopd in this study. It was obsrvd that on could masur th offst tim (T ) of th first puls in th tim-dflction rcords as shown in Fiur 12. Th offst tim is rlatd to th Raylih wav vlocity of th subrad. With th assumptions discussd in th followin sction, and knowin th distanc btwn two masurmnt stations, th shar wav vlocity can b dtrmind. Thr ar svral assumptions that must b mad to us th offst tim mthod. First, th subrad should b abl to b approximatd as a uniform matrial. Scond, th wavlnth should b lon nouh so that th surfac, bas, and sub-bas layrs hav littl ffct on th Raylih wav vlocity. Gnrally, this mans that th wavlnth should b at last l tims th total thicknss of th surfac, bas, and sub-bas layrs for untratd bass and sub-bass (5). Third, th bdrock nds to b dp nouh so that it has littl ffct on th Raylih wav vlocity. This condition is usually mt if th bdrock dpth is ratr than.5 tims th Raylih wavlnth in th subrad. Fourth, it is assumd that nar-fild ffcts ar small and that thy can b inord. Fifth, th first pulss of Stations 5 and 7 wr usd to masur th offst tim bcaus th dflctions obtaind at th stations away from th sourc should bttr rprsnt th proprtis of th subrad. Finally, th diffrnc (8) (9) lim (sc) FIGURE 12 Offst tim of th first pulss btwn Stations 5 and 7 for FWD tstin at Profil 1 [V, of subrad = 5 fps (155 m/sc), E = 16 ksi (.11 MN/m 2 ) and dpth to bdrock = 1.6 m (5 ft)]. btwn th Raylih wav vlocity and th shar wav vlocity is lss than 1 prcnt if Poisson's ratio of th matrial is ratr than.2 (3) so that it can b inord. (Not: ths assumptions may not apply in many tst situations.) Tabl 3 illustrats an xampl of comparisons btwn th stimatd shar wav vlocity of th subrad and th actual shar wav vlocity for Profil 1 with an unsaturatd subrad havin a Poisson's ratio qual to.2. Good stimations of th shar wav vlocity of th sub rad ar mad in th cas of th softst subrad. Howvr, as th stiffnss of th sub rad incrass and as th stiffnsss and thicknsss of th uppr pavmnt layrs incras, thr ar cass in which stimations cannot b mad. This happns bcaus th first pulss in th dflction-tim rcords wr distortd and th paks could not b dtrmind. SUMMARY Analytical simulations of th FWD tst wr conductd usin th computr proram UTFWD (I,2). Four pavmnt profils ranin from flxibl to riid wr studid. Th.stiffnss of th subrad layr and th dpth to bdrock (thicknss of th subrad) wr th only paramtrs varid to simulat typical rans in pavmnt systms that could b tstd usin th FWD. Equations for stimatin th rsonant dpth to bdrock, RDb, wr dvlopd for both th flxibl and riid pavmnts. Dpths to bdrock at which rsonanc ffcts may occur wr found to vary from 1.7 to 6.2 m (5.5-2 ft), dpndin on th stiffnss of th subrad. Saturatd subrad conditions av th sam trnd in rsonant dpth to bdrock with subrad stiffnss. Equations for stimatin th dpth to bdrock basd on th natural priod of fr vibrations of th pavmnt systm immdiatly aftr FWD load application hav bn prsntd. In ths quations, ffcts of stiffnss and dr of subrad saturation wr takn into account. On important aspct in applyin ths quations is that complt tim historis rcordd for.12 sc or lonr ar dsird.

11 78 TRANSPORTATION RESEARCH RECORD 154 TABLE 3 Estimatd Shar Wav Vlocity of Subrad from Offst Tim of th First Pulss for FWD Tstin at Profil 1 Units: Vs (fps), E (ksi) Dpth to Actual Vs of Subrad (E of Subrad) Bdrock 5 (16) 75 (36) To Estimatd Error** To Estimatd Error Vs Vs (ft) (s c) (fps) (%) (sc) (fps) (%) % na*** na na % % % % % % Dpth to Actual Vs of Subrad (E of Subrad) Bdrock 1 (62) 15 (142) T Estimatd Error To Estimatd Error Vs Vs (ft) (s c) (fps) (%) (sc) (fps) (%) 5 n a na na na na na 7.5 n a na na na na na % na na na % % *Poisson's Ratio of Subrad quals.2. ** Error=[ (Estimatd Vs/ Actual Vs )-1] * 1% *** Offst Tim is not availabl bcaus th first pulss in FWD tsts ar distortd A mthod to stimat th stiffnss of th subrad from th offst tim of th first pulss of th dflction-tim rcords at two masurmnt stations in th FWD tsts has bn proposd. Th shar wav vlocity of th subrad can b stimatd dividin th offst tim into th distanc btwn ths two stations. At prsnt, it sms that this approach is mor appropriat in cass in which th stiffnss of th subrad is soft to modrat, V,. = 155 to 233 m/sc (V, = 5-75 fps) [ =.11 to.25 MN/m 2 (E = ksi)] and th bdrock dpth is 3.1 m (1 ft) or mor. ACKNOWLEDGMENTS W thank th Txas Dpartmnt of Transportation for partial support of th study undr projct Th support of th Advancd Tchnoloy Proram of th Stat of Txas is also sincrly apprciatd. REFERENCES I. Chan, D. W., J.M. Rosst, and K. H. Stoko, II. Nonlinar Effcts in Fallin Wiht Dflctomtr Tsts. Transportation Rsarch Rcord 1355, TRB, National Rsarch Council, Washinton, D.C., 1992, pp Chan, D. W., V. Y. Kan, J.M. Rosst, and K. H. Stoko, II. Effct of Dpth to Bdrock on Dflction Basins Obtaind with Dynaflct and FWD Tsts. Transportation Rsarch Rcord 1355, TRB, National Rsarch Council, Washinton, D.C., 1992, pp Richart, F. E., Jr., R. D. Woods, and J. R. Hall. Vibrations of Soils and Foundations. Prntic Hall, Enlwood Cliffs, N.J., Sn, C.R., K. H. Stoko, II, and J.M. Rosst. Effct of Dpth to Bdrock on th Accuracy of Backcalculatd Moduli Obtaind with Dynaflct and FWD Tsts. Rsarch Rport Cntr for Transportation Rsarch, Univrsity of Txas at Austin, Austin, Tx Aouad, M. Evaluation of Flxibl Pavmnts and Subrads Usin th Spctral-Analysis-of-Suifac-Wavs (SASW) Mthod, Ph.D. dissrtation. Univrsity of Txas at Austin, Austin, Tx Publication of this papr sponsord by Committ on Soil and Rock Proprtis.