Buried Corrugated Thermoplastic Pipe: Simulation and Design

Transcription

1 Schafr and M c Grath Burid Corrugatd Thrmoplastic Pip: Simulation and Dsign Schafr, B.W., M c Grath, T.J. B.W. Schafr, Ph.D., Assistant Profssor, schafr@jhu.du Johns Hopkins Univrsity, Latrob Hall, Baltimor, MD tl. ()-, fax ()- T.J. McGrath, Ph.D., P.E., Principal, tjmcgrath@sgh.com Simpson Gumprtz & Hgr Inc., Broadway, Arlington, MA tl. ()-, fax ()- Submittd July to: Committ AC for prsntation at th Transportation Rsarch Board Annual Mting Word Count: Abstract Papr = + figurs@, + tabls@ = ABSTRACT Th objctiv of this papr is to dmonstrat a computational mthod for assssing th allowabl dpth of fill of a burid thrmoplastic profil wall (corrugatd) plastic pip and compar th rsults to th rcntly adoptd AASHTO dsign mthod. Th computational mthod is dmonstratd for a mm ( in.) diamtr HDPE profil wall pip, but is applicabl to all profil wall thrmoplastic pip that xhibit local buckling limit stats. Th computational modl compars strain dmands prdictd from a two-dimnsional plan strain finit lmnt modl of burid pip in th mbankmnt condition with strain capacity prdictd from a thr-dimnsional finit lmnt modl of a pipsoil sgmnt undrgoing thrust and/or positiv and ngativ bnding. Th strain dmands indicat th dominanc of thrust strains, as opposd to bnding strains, in th ovrall bhavior, particularly for intrmdiat to largr fill dpths. In th profil, th ultimat strain capacity is limitd by local buckling for thrust strains and/or positiv bnding (crst in comprssion), and inward radial movmnt of th crst for ngativ bnding (linr in comprssion). Dpth of fill prdictions btwn th nw AASHTO dsign mthod for thrmoplastic pip and th computational mthod agr within % of on anothr whn uniform soil distribution is considrd, and within % of on anothr whn a soft haunch and othr soft soils ar considrd in th pip-soil nvlop. INTRODUCTION Although thr hav bn significant advancs in th structural analysis of burid pips usd in dsign, for xampl, th SIDD mthod of dsign for concrt pip () and th proposd mthods for dsign of larg-span mtal and concrt culvrts () bhavior of burid thrmoplastic profil wall (corrugatd) pip has largly bn invstigatd through xprimntal mans or approximatd by simplifid dsign mthods. Considring th varity of thrmoplastic profil wall pip availabl combind with th varity of diffrnt installd conditions, it is prohibitiv to tst vry scnario. Thus, thr is a nd for computational mthods that captur th ssntial nonlinaritis of both th soil and th pip profil bhavior undr load, but do so in a mannr that rmains rlativly fficint. Such a procdur is rportd hrin for a mm ( in.) diamtr corrugatd HDPE pip and compard to th simplifid dsign mthod rcntly adoptd by AASHTO (,). BURIED PIPE SIMULATION Th approach takn hr for simulating burid profil wall plastic pip is to sparat th dmand and capacity calculation. For th dmand, CANDE () is mployd to dtrmin th appropriat global pip-soil intraction undr an mbankmnt condition and to prdict th applid strain dmands on th pip as dpth of fill incrass, using modls as shown in Figur. For th capacity, gnral purpos thr-dimnsional finit lmnt modls of a pipsoil sgmnt, as shown in Figur, ar mployd to dtrmin th ultimat strain capacity (limitd by local buckling and othr phnomna) undr all xpctd nd actions: positiv bnding, thrust, ngativ bnding, and combinations TRB Annual Mting Initial Submittal for Rviw

2 Schafr and M c Grath in btwn. With th ultimat strain capacity dtrmind from th pip-soil sgmnt modl th strain dmands from CANDE can b compard as dpth of fill incrass to dtrmin nominal and allowabl dpths of fill. SIMULATION OF BURIED PIPE STRAIN DEMANDS Applid strain dmands ar prdictd using finit lmnt modls dvlopd in CANDE. CANDE is a twodimnsional plan strain finit lmnt analysis program dvlopd xprssly for burid pip problms (). W mployd th nonlinar hyprbolic soil modls (, ) with th sam soil proprtis usd to dvlop th concrt pip SIDD dsign mthod (). Th modl is built by incrmntally adding pip and soil layrs, mimicking th actual construction procss. Additional surcharg loads may b addd on top of th fill to simulat furthr incrass in dpth of fill. Th modl prdicts strss stat in th soil and forcs in th pip. Th applid strain dmands on th pip ar dtrmind by convrting th forc and th momnt on th pip, from CANDE analysis, to strains using nginring bam thory and ignoring curvatur ffcts. For a particular point,, in th pip, th applid strains ar th sum of thrust strain and bnding strain: outsid fibr applid strain: ε out = P /(EA) + M c out /(EI) insid fibr applid strain: ε in = P /(EA) + M c in /(EI) whr: P = forc in th pip at point E = pip modulus of lasticity A = ara of th pip cross-sction M = momnt in th pip at point c out = distanc from th cntroidal axis to th xtrm outsid fibr (top of th crst) c in = distanc from th cntroidal axis to th xtrm insid fibr (bottom of th vally) I = momnt of inrtia of th pip cross-sction CANDE Modls Four modls of burid mm ( in.) diamtr corrugatd plastic pip wr dvlopd. Two soil distributions: uniform ML (silty soil compactd to % of maximum dnsity pr AASHTO T) Backfill rprsnting an idal situation with all backfill of th sam matrial and dnsity, including backfill in th haunch zon, and non-uniform ML which includs ML backfill around th pip, but with a soft haunch zon(cl, i.. a clay soil compactd to % of maximum dnsity pr AASHTO T) and CL in situ soil, s Figur ; and two pip moduli (E): short-trm, E pip =E i, (from, E i = MPa,, psi) and long-trm, E pip =E, i.., pip modulus of lasticity at yars (from, E i = MPa,, psi) wr considrd. Th finit lmnt msh, construction incrmnts, dpth abov th top of th pip for ach of th construction incrmnts, and th soil zons for th non-uniform soil cas usd in th pip-soil intraction studis in CANDE is shown in Figur. For th uniform soil cas all th soil zons of Figur ar modld as ML, which is an ASTM D, Class III backfill. Soil strsss Examination of th dvlopd soil strsss in th CANDE analysis for th four modls indicat: significant vrtical arching of th load around th pip, in th uniform ML soil cas at m ( ft) of fill, th vrtical soil strss immdiatly abov th pip is. kpa ( psi), whil th fr-fild soil strss at th sam dpth is kpa ( psi); lvatd horizontal soil strsss at th springlin and also for approximatly º on ach sid of th springlin as th pip pushs into th soil; and lvatd strsss in th stiffr soils (.g., ML vs CL, or in situ mdium stiffnss soil vs. CL) whn nonuniform soils ar prsnt in th mbankmnt. Pip Strain Dmands Strain dmands around th pip ar givn in Figur at. m ( ft) of fill. Rsults indicat that: thrust strains ar dpndnt on pip modulus, thus as th pip rlaxs thrust strains incras, TRB Annual Mting Initial Submittal for Rviw

3 Schafr and M c Grath bnding strains ar largly indpndnt of pip modulus. As th pip rlaxs, bnding strains ar constant, and soft haunchs and othr non-uniform soil support around th pip significantly incras bnding strain dmands. Th incras in thrust strains for th long trm condition (lowr pip axial stiffnss) highlights th importanc of th thrust capacity of th profil. Th rlativ insnsitivity of bnding strains to pip stiffnss indicats that control of bnding strains is achivd during th initial installation of th pip. Rsults nar th soft haunch indicat that poor ffort spnt in th haunch rgion can crat bnding strain dmands that xcd th xpctd primary strain dmands at othr locations in th pip. Pip Diamtr Chang As chang in diamtr is oftn usd to assss th prformanc of an installd pip, Figur is providd to giv th CANDE prdictions for diamtr chang as a function of dpth. Rsults indicat that: vrtical diamtr chang du to circumfrntial shortning is gratr for non-uniform soil support than uniform soil support, vrtical diamtr chang du to bnding is lss for non-uniform soil support than uniform soil support, and circumfrntial shortning dominats bhavior for larg dpths of fill (gratr than ~. m ( ft) in this analysis). SIMULATION OF BURIED PIPE STRAIN CAPACITY Applid strain capacity is dtrmind by using th gnral purpos finit lmnt packag ABAQUS (). Thrdimnsional modls of a sgmnt of th pip including local soil support ar analyzd undr a combination of varying thrust and nd momnts to dtrmin th pip buckling capacity at all locations (crown, shouldr, springlin, invrt, tc.). Th slctd pip-soil sgmnt is a dgr arc (Figur ) with on full corrugation of th pip, and soil support for this corrugation, modld. Symmtry is nforcd on ach sid of th pip-soil sgmnt. Radial support is providd for th outsid fac of th soil. Th dimnsions and thicknss of th pip ar rprsntativ of on manufacturr's mm ( in.) diamtr corrugatd HDPE pip in production in. Modl and Msh Siz Snsitivity of th modl to th º arc lngth and º lmnt spacing is invstigatd by ignvalu buckling analysis of th pip, without soil. Elmnt spacing btwn.º and.º indicats that lmnts vry º provid th bst compromis btwn fficincy and accuracy. Strain dmands ar rlativly uniform ovr a givn º arc (Figur ). Thus º is slctd to insur nough lngth to allow svral buckling wavs to form undr a varity of diffrnt momnts and thrusts without impding formation of th buckling du to nd ffcts, whil not so long as to rquir mor prcis tratmnt of th strain gradint around th sction. Initial Imprfctions Simulation of th paralll plat tst () indicats that introducing small imprfctions in th initial pip gomtry ar critical to th numrical succss of modls for buckling capacity of corrugatd pip. A suprposition of a crst buckling imprfction (similar to Figur (a)) and a linr buckling imprfction (Figur ()) was mployd. Maximum magnitud for imprfction of th linr is.t linr =. mm (. in.), maximum magnitud for imprfction of th crst is.t crst =. mm (. in.). Ths imprfctions includ th sam mod of dformation for th soil and th pip thrfor th initial stat for th soil and pip is matchd. Matrial Modl Matrial modls usd in th pip-soil sgmnt modl (Figur ) ar linar lastic. Th pip is modld using an initial modulus (E i ) and th soil is modld with a modulus, E, of. MPa (, psi), rprsnting a lowr bound valu for ML fill. Slction of linar lastic matrial modls is basd on a) dsir to focus on buckling phnomna, b) past succss with this approach in paralll plat modling (), and c) computational fficincis ndd for complx thr-dimnsional modls. Although th matrial modls ar linar lastic, significant gomtric nonlinaritis xist and ar capturd. Loading Loading of th pip-soil sgmnt is prformd by hoop displacmnt and rotation of th nds of th pip-soil sgmnt. Rotation is prformd about th lastic cntroid of th pip profil. Forc and momnt that dvlop at th nd of th pip du to th dformation and rotation ar rcordd and ar convrtd to strains in th sam mannr as TRB Annual Mting Initial Submittal for Rviw

4 Schafr and M c Grath compltd in th CANDE analysis. Both th pip and th soil ar displacd and rotatd at th nds of th sgmnt, insuring compatibility btwn pip and soil. Pip-Soil Contact and Analysis Mthod Th pip-soil intrfac is modld as frictionlss (unbondd) by using linar gap lmnts btwn ach nod of th pip and a corrsponding nod in th soil. Th analysis mploys an quilibrium path tchniqu (Riks, s ) that tracs th ntir load-displacmnt path of th pip. Buckling is obsrvd by th dformation of th pip and vntual loss of load carrying capacity. Nonlinar gomtry ffcts, initial imprfctions, nutral axis migration, and othr phnomna that dvlop during th loading history of th sgmnt prior to buckling ar includd th analysis. Rsults Nonlinar buckling analysis of th pip-soil sgmnt rsults in th buckling capacity curv providd in Figur. Each nonlinar analysis is valuatd with a diffrnt st of thrust and bnding strains on th pip-soil sgmnt. Th gap btwn th pip and th soil may b wid (.g., Figur (a)) although compatibility at th nds of th sgmnt, whr th displacmnts and rotations ar applid is always maintaind. Buckling of th sgmnt dos not occur suddnly, but rathr dvlops gradually ovr th duration of th loading. Undr positiv bnding th primary dformation of th pip is in th crst (Figur (a)) and contact btwn th pip and soil in th crst is largly lost. Undr incrasd thrust, but still significant positiv bnding (b), th dformations in th wb of th profil incras and contact btwn th pip and th soil is lost only in th inward buckling wavs. For larg amounts of thrust and small positiv bnding (c) th linr buckls as wll as th crst. For pur thrust (d) th ntir profil buckls, i.. crst, wb, vally and linr and contact btwn th pip and th soil is lost for th inward buckling wavs. Undr ngativ bnding (,f) th linr buckls and th crst xprincs larg inward radial dformations. Ths inward dformations of th crst rsult in loss of pip to soil contact and rduc th bnding capacity. Simulation: Dpth of Fill Prdiction Th capacity of a pip may b limitd by buckling of th profil, or by gnral yilding. Exprinc in stub comprssion tsts () and paralll plat analysis () indicat that buckling, not yilding, is th mod of gratst concrn for many of th corrugatd plastic pip profils in currnt us. Th nonlinar buckling analysis curv (Figur ), combind with th strain limits discussd in () ar slctd as th basis for th limiting capacity curv. Any combination of strain dmands that fall within ths limits (buckling and yilding) ar sustainabl by th pip. In Figur, th strain dmands in a CANDE analysis with non-uniform ML backfill and long-trm pip modulus ar compard vrsus th capacity curv. Th strain dmands ar shown as lins manating from th origin; ach lin nds with a numbr, in dgrs, that rfrs to th initial location of th point on th pip. A labl of rfrs to th crown, rfrs to th springlin, and to th invrt. At. m ( ft) of fill abov th crown, Figur (a), strain dmands ar wll blow th capacity curv. Figur (b) through (d) show th ffct of incrasing th dpth of fill. At. m ( ft) of fill, th springlin (i.., º) as wll as ºs to ithr sid of th springlin (º and º) xprinc th gratst dmands. Thrust plays a significant rol in th bhavior vn at rlativly low fill dpths. At. m ( ft) of fill, dmand xcds capacity at th springlin (º). Expctd dformation at th springlin is givn in Figur (d,). Buckling of th linr and portions of th main profil may occur bfor. m ( ft) of fill, but byond. m ( ft) dformations ar too larg for th pip to carry additional strain without loss in load carrying capacity. As th dpth of fill incrass, dmand xcds capacity in additional rgions (i.., mor lins cross th capacity curv). Basd on th mthodology dscribd abov, dpth of fill prdictions ar givn in Tabl. Th four sparat CANDE analyss ar valuatd for nominal (ultimat) dpth of fill (no safty factors), and dsign dpth of fill (including safty factors). Rsults ar givn for th dpth of fill whn a sction of th pip rachs th buckling capacity curv of Figur and whn th strain limit is first rachd. Th limiting dpth of fill prdictions of Tabl ar basd on th first dpth of fill at which a point in th pip rachs a buckling or yilding limit stat. If th strain dmands at th faild location can b withstood, rdistribution of th forc would occur. Howvr, as Figur indicats th dmands for a full ºs on ithr sid of th springlin ar narly th sam as th springlin itslf. Thus, loss of carrying capacity at th springlin immdiatly portnds loss of capacity ovr a significant portion of th pip. TRB Annual Mting Initial Submittal for Rviw

5 Schafr and M c Grath Tabl Dpth of fill prdictions Dpth of fill, m, (ft) at limit of Backfill Soil Pip Modulus Buckling of Profil Strain Limit NOMINAL PREDICTION (ALL SAFETY FACTORS = ) Uniform E. (). () ML Non-Uniform E. (). () Uniform E i. (> ). (> ) Non-Uniform E i. (> ). (> ) DESIGN PREDICTION (WITH SAFETY FACTORS*) Uniform E. (). () ML Non-Uniform E. (.). (.) Uniform E i. (). () Non-Uniform E i. (). () * Safty factors: for buckling, for th yilding in thrust strain limit, and. for th combind yilding strain limit Pip Dflction CANDE rsults ar combind with th dpth of fill prdictions givn in Tabl to valuat th prdictd total vrtical dflction in th pip at th maximum dpth of fill. Th prdictd long-trm ultimat total vrtical dflction for a ML backfill is.% in uniform soil and.% in non-uniform soil. Th dsign dflction is.% in uniform soil and.% in non-uniform soil. Th rlativly small prdictions for th bnding dflction ar du to th larg lvls of prdictd thrust strains. Th thrust strains rsult in loss of capacity for th pip without larg vrtical dflction. Pip analyzd with initial modulus hav much highr lvls of dflction at th maximum dpth of fill. BURIED PIPE DESIGN Th nwly adoptd dsign mthods for profil wall thrmoplastic pip dsign (,) ar usd to valuat th sam mm ( in.) diamtr pip studid using th burid pip simulation mthods. Ths nw dsign mthods captur th following ssntial bhavior of th pip-soil systm: soil stiffnss (M s ) is a function of soil typ and dpth of fill, vrtical arching is a function of th ratio of soil to pip (hoop) stiffnss, dcrasd capacity du to local buckling is a function of buckling strain of an lmnt of th pip and th applid ultimat strain on th pip, bnding capacity of th pip is limitd by th total strain of th matrial lss th hoop strain th pip undrgos. Dsign: Dpth of Fill Prdiction For mm ( in.) diamtr pip, w xamind th allowabl dpth of fill for dsign and ultimat conditions. Th analysis is carrid out for four soil typs: SW, SW, ML, and CL. Th dsign condition mploys load factors (γ) of. on thrust,. on combind thrust and bnding, and rsistanc factors (φ) of. on pip prformanc,. on soil rsistanc. Ultimat capacity is xamind by stting all load factors (γ s) and rsistanc factors (φ s) to.. In all cass, th dpth of fill is limitd by th rducd capacity in local buckling. Th rsults ar summarizd in Tabl. TRB Annual Mting Initial Submittal for Rviw

6 Schafr and M c Grath Tabl Dpth of fill, m, (ft) for mm ( in.) pip as a function of soil typ Soil Typ Condition SW SW ML CL Considring Local Buckling Dsign. (.). (.).(.). (.) Ultimat. (.). (.). (.). (.) Ignoring Local Buckling Dsign. (.). (.). (.). (.) Ultimat. (>). (.). (.). (.) Th rsults indicat that: vn if local buckling is rmovd or rducd, through som improvmnt in dsign, burial dpths gratr than ft will rquir SW or bttr or ML or bttr soil, th factor of safty with rspct to dpth of fill (.g., in SW. m/. m =.) is gratr than th dsign factor of safty with rspct to th local buckling limit stat (.). Th allowabl vrtical dflction capacity of th pip wall is basd upon th availabl strain capacity - which is dtrmind from th -yar strngth and modulus of th HDPE. For th dsign condition th maximum bnding dflction is.% chang in diamtr, and th total dflction is.%. For th ultimat condition th bnding dflction is.% and th total dflction is.%. For a givn soil ths dflctions occur at th fill dpths shown in Tabl. Th bhavior of pip using this dsign mthod is drivn by thr ssntial quantitis: constraind soil modulus (M s ), vrtical arching factor (VAF), and local buckling capacity, i.., ffctiv ara (A ff ). As dpth of fill incrass M s incrass, VAF dcrass, and A ff dcrass all in a nonlinar fashion. Quality fills, such as ML or bttr, or SW or bttr provid significant incrass in M s with incrasd dpth of fill. Th pip hoop stiffnss of mm ( in.) diamtr pip is low nough that significant changs in VAF ar ralizd with quality fill. Th local buckling rduction also ss som bnfit with highr VAF, but sinc ffctiv ara is always calculatd at an ultimat limit stat this bnfit is limitd. DISCUSSION Dsign vs. Simulation Dpth of Fill Prdictions: Dsign vs. Simulation Th dpth of fill prdictions for ML backfill providd by th simplifid dsign mthod rportd in Tabl ar. m (. ft) for dsign and. m (. ft) for th ultimat condition. Th dpth of fill prdictions providd in Tabl, for uniform ML backfill, ar. m ( ft) for dsign and. m ( ft) for th ultimat condition. Considration of non-uniform ML backfill yilds a dpth of fill prdiction of. (. ft) for dsign and. m ( ft) for ultimat. Th agrmnt of th simplifid mthod with th comprhnsiv numrical analysis prsntd hrin shows that th simplifid dsign mthod can provid accurat dpth of fill prdictions for us in dsign. Strain Dmands Prdiction: Dsign vs. Simulation Th proposd dsign mthod mploys th shap factor approach for calculating bnding strains, via: c ε b = D f R D whr: D f = shap factor c = distanc from cntroid to fibr of intrst (usually outsid or insid of pip) R = cntroidal pip radius = vrtical diamtr chang du to bnding (rmov circumfrntial componnt) D = cntroidal diamtr TRB Annual Mting Initial Submittal for Rviw

7 Schafr and M c Grath A tabl for th shap factor as a function of pip and soil stiffnss may b found in (). A shap factor btwn. and. is slctd for invstigation hr. Chang in th vrtical pip diamtr du to bnding, as dtrmind from CANDE analysis (Figur ), is usd for. Th maximum comprssiv strain in th outsid fibr du to bnding, ignoring th local paks in th soft haunch rgion, is usd for comparison (Figur ). Th comparison dmonstrats that th shap factor approach provids: a rliabl prdiction of initial strain dmands (i.., dpth of fill), a rliabl prdiction of strain dmands up to modrat fill dpths (< m, ft.), and a rasonabl and gnral mthodology for hand prdiction of bnding strain dmands. Sinc th shap factor tabl usd in this study was dvlopd for fibrglass pip, additional rsarch should b compltd to invstigat th applicability of th tabl to all typs of thrmoplastic pip undr additional installation and backfill conditions. Positiv Bnding - Maximum Buckling Capacity Curv vs. Strain Limits Th computational modl prdicts that th maximum positiv bnding strain is approximatly %, compard with % suggstd in th LRFD spcifications (). Th maximum bnding strain for () is basd, in part, on a study of th xpctd bnding strain at % vrtical dflction in th paralll plat tst. Analysis of mm to mm ( in. to in.) diamtr pip suggsts strain lvls btwn.% to.% strain, with th smallr diamtr pip undrgoing th highr strain lvls. Strain limits in () rcognizd th high strains xprincd in th smallr diamtr pip and thus suggstd a strain limit in bnding % highr than that usd in comprssion, i.., % strain. Howvr, using th shap factor approach mployd in () th prdictd strain at % dflction for th studid mm ( in.) diamtr pip is only %. Dformations in Figur (a) suggst that th soil provids littl or no support in th rgion whn th pip buckls in positiv bnding. In positiv bnding, th maximum buckling capacity curv is providing a strain limit that rcognizs a loss of soil support; th paralll plat tst is providing an uppr bound approximation that includs full soil support to th crst; and th strain limits of () ar providing an avrag strain capacity for all diamtr pip. With th currnt simplifications usd in th LRFD hand mthod, th avrag limiting strain of % in currnt us appars rasonabl. Simulation dtails Applid vs. Local Strain Dmands Applid strain dmands (P/(EA) + Mc/(EI)) diffr from th local strain dmands du to bnding of th pip as a plat from buckling dformations, shar lag ffcts that limit th ffctivnss of th linr to carry load, and any othr local or thr-dimnsional ffcts (). Comparison of th strain capacity to th dmand is compltd using applid strain dmands as CANDE s modling dos not incorporat ths ffcts. Howvr, th dtaild thrdimnsional ABAQUS analysis calculats local strain dmands at pak capacity, th total applid strain agrs wll with avrag strain in th cornr (Tabl ) of th crst and wb of th profil. An xcption to this conclusion is th cas of ngativ momnt and larg thrust (cas ()). In this cas, strain dmands basd on th initial gomtry ar problmatic, bcaus a larg amount of inward radial dformation occurs in th crst (for th stiffnd profil shown); howvr, at this location th strain dmands ar small. Pak strain dmands idntifid in th modl ar markdly highr than applid strains in this cas. In gnral, highly localizd strain dmands, at th xtrm fibrs through th thicknss of th plastic, and at juncturs of th diffrnt lmnts, must b rsistd by th matrial for th prdictd capacitis to b ralizd. This maks th us of a matrial with provn ductility an important critrion for rsin slction. TRB Annual Mting Initial Submittal for Rviw

8 Schafr and M c Grath Tabl Strain Dmands (%) from Nonlinar Buckling Analysis (at Ultimat) CANDE ABAQUS pip-soil modl applid crst/wb any location strain cornr min max condition ε total ε avrag ε tnsion ε comprssion (a) positiv momnt (b) positiv momnt and modrat thrust (c) positiv momnt and larg thrust (d) thrust () ngativ momnt and larg thrust (f) ngativ momnt Thrust Strains and Slip in Soil Strain dmands prdictd from CANDE ar primarily thrust strains, as opposd to bnding strains. For instanc, not in Figur, strain dmands ar ssntially horizontal lins, i.., thrust strain dmands gratly xcd bnding strain dmands. Th CANDE analysis is prformd assuming a prfct bond btwn pip and soil. If slip btwn th pip and soil is allowd, thrust strains will bcom mor uniform around th pip. Dcrass in th pak thrust strain dmand will rsult in incrasd prdictions for th allowabl dpth of fill and prcnt dflction. If th man thrust strain in th pip is usd as th dmand (to approximat soil compltly fr to slip) th allowabl dpth of fill (without safty factors) for an ML backfill incrass by. m ( ft) in uniform soil and. m (ft) in non-uniform soil and th dsign (with safty factors) allowabl dpth of fill for an ML backfill incrass by. m (. ft) in uniform soil and. m (. ft) in non-uniform soil vs. thos of Tabl. Ovrall, othr ffcts, such as strains from th construction procss, countr-act such bnficial ffcts as dcrass in th pak thrust strain du to circumfrntial slippag at th pip-soil intrfac. Tst Mthods and th Capacity Curv Altrnativ to th dtaild finit lmnt analyss, tst mthods could b mployd for dtrmining th capacity curv. Th paralll plat tst prdicts th maximum bnding strain that th pip profil will maintain whn supportd against a rigid foundation (th stl platns) this provids an upprbound approximation of th positiv bnding strain capacity. Th stub comprssion tst (), or soil cll hoop tsts, can provid th capacity in pur thrust. Ths two valus along with rasonabl strain limits could b usd to form a basic capacity curv. Factor of Safty Th analysis for factor of safty indicats that, givn installation in accordanc with th dsign conditions, th ultimat dpth of fill xcds th allowabl dpth of fill by mor than th factor of safty of about targtd by AASHTO, and traditionally applid to all flxibl pip. From an altrnat point of viw, looking at backfill or compaction not mting spcifid lvls, a pip dsignd for SW backfill with a dsign dpth of fill. m (. ft) would not rach an ultimat condition as long as th backfill proprtis xcdd thos of ML. This is a substantial allowanc for non-conforming installation. Givn th substantial uncrtaintis in th soil and installation, and to a lssr xtnt in th pip prformanc, systmatic mthodologis for dtrmining th rliability (and rsulting factor of safty) for th pip-soil systm ar still ndd. CONCLUSIONS A computational mthod was prsntd for dtrmining th allowabl dpth of fill of mm ( in.) diamtr profil wall (corrugatd) HDPE pip and compard with rcntly adoptd AASHTO dsign mthods for th sam pip. Th mthod rlis on strain dmands from a two-dimnsional plan strain modl of th pip in an mbankmnt condition undr incrasing fill dpths and dtrmination of th ultimat strain capacity from a thr-dimnsional modl of a pip and soil sgmnt undr xpctd nd actions: thrust and/or bnding. Th dvlopd soil strsss, dirct pip strain dmands, and chang in pip diamtr as dpth of fill is incrasd all indicat th prominnt rol of thrust strains (as opposd to bnding strains) in th bhavior of ths pip in th mbankmnt condition. Analysis of TRB Annual Mting Initial Submittal for Rviw

9 Schafr and M c Grath th pip-soil sgmnt undr incrasing thrusts and/or bnding indicat a bounding capacity curv that xists du to local buckling of th pip profil. For th xamind pip this capacity curv is in gnrally good agrmnt with th strain limits mployd in th nwly adoptd AASHTO dsign mthod. Furthr, comparison of th dpth of fill prdictions btwn th nw AASHTO dsign mthod and th simulations indicats that agrmnt is rmarkably good: dpth of fill prdictions ar within % of on anothr whn uniform soil distribution is considrd, and within % of on anothr whn a soft haunch and othr soft soils ar considrd in th mbankmnt. Th simulation modl prsntd provids a gnral mans for assssing capacity of burid pip, and srvs to provid additional validity to th rcntly adoptd AASHTO spcification mthods. ACKNOWLEDGEMENTS Th authors would lik to gratfully acknowldg th financial support of Hancor, Inc. in dvloping th work prsntd hrin. In addition, Dan Valntin, formrly of Simpson Gumprtz & Hgr, Inc. aidd in th calculations for th dsign dpth of fill by th rcntly adoptd AASHTO mthod. REFERENCES. Hgr, F.J., (), "Nw Installation Dsigns for Burid Concrt Pip," Piplin Infrastructur Procdings of th Confrnc, Amrican Socity of Civil Enginrs, Nw York, NY, pp. -.. McGrath, T.J., Moor, I.D., Slig, E.T., Wbb, M.C. and Talb, B., () "Rcommndd Spcifications for Larg Span Culvrts," Tansportation Rsarch Board, National Cooprativ Highway Rsarch Program (NCHRP). Rport (numbr will b availabl by Sptmbr).. McGrath, T.J., Sagan, V.E. (). LRFD Spcifications for Plastic Pip and Culvrts. Tansportation Rsarch Board, National Cooprativ Highway Rsarch Program (NCHRP) Rport.. AASHTO (). AASHTO-LRFD Bridg Dsign Spcification, with nd Edition with Intrim Spcifications. CANDE- Culvrt Analysis and Dsign computr program Usr Manual. Publication No. FHWA-RD--. Rsarch, Dvlopmnt, and Tchnology, Turnr-Fairbank Highway Rsarch Cntr, Fdral Highway Administration, U.S. Dpartmnt of Transportation. Duncan, J.M. and Chang, C-Y. () Nonlinar Analysis of Strss and Strain in Soil, Journal of th Soil Mchanics and Foundations Division, ASCE, Vol., No. SM: -.. Hgr, F.J., Lipins, A.A., and Slig E.T., (), "SPIDA: An Analysis and Dsign Systm for Burid Concrt Pip," Advancs in Undrground Piplin Enginring Procdings of th Intrnational Confrnc, Amrican Socity of Civil Enginrs, pp. -.. McGrath, T.J., and Schafr, B.W. () Paralll Plat Tsting and Simulation of Corrugatd Thrmoplastic Pip Submittd for prsntation at th Transportation Rsarch Board Annual mting, Washington D.C.. ABAQUS Vrsion.. () Hibbitt, Karlsson & Sornsn, Inc. Pawtuckt, RI.. AWWA (). AWWA M Manual for Fibrglass Pip Dsign. TRB Annual Mting Initial Submittal for Rviw

10 Schafr and M c Grath - surcharg load.... ML..... ML. CL, pip. CL. ML.... in situ Mdium Stiffnss INC. DEPTH(FT) Figur CANDE modl for strain dmand calculation showing construction incrmnts and soil zons Figur ABAQUS modl for strain capacity calculation showing pip and local ring of soil TRB Annual Mting Initial Submittal for Rviw

11 Schafr and M c Grath axial strain (%) outs id fibr bnding strain (% ) DEPTH =. FT Uniform ML, E pip =E initial Uniform ML, E pip =E NON-Uniform ML, E =E pip initial NON-Uniform ML, E =E pip thta (dg. =crown) thta (dg.) o =crown Figur CANDE prdictions for applid axial and bnding strains around th circumfrnc of th pip (not, axial strain rfrs to circumfrntial hoop strain, not longitudinal ffcts) vrtical diamtr chang (%ID) VERTICAL DIAMETER CHANGE Uniform ML, E pip =E in itial Uniform ML, E pip =E NON-Uniform ML, E pip =E in itial NON-Uniform ML, E pip =E - VERT. DIAM CHANGE DUE TO CIRCUMFERENTIAL SHORTENING vrtical diamtr chang (%ID) horizontal diamtr chang (%ID) vrtical diamtr chang (%ID) - HORIZONTAL DIAMETER CHANGE VERT. DIAM CHANGE DUE TO BENDING - Figur Vrtical and horizontal diamtr chang vrsus dpth of fill basd on CANDE dmands TRB Annual Mting Initial Submittal for Rviw

12 Schafr and M c Grath outsid bnding strain (%) (a) (b) (f) thrust strain (%) Figur Rsult of pip-soil sgmnt analyss undr varying amounts of applid bnding and thrust, nvlop of th rsults is th buckling capacity curv for th studid pip diamtr (Figur provids dformations at pak capacity for points (a) (f) idntifid in th figur) (a) pak positiv bnding capacity (c) pak positiv bnding capacity undr larg thrust () pak ngativ bnding capacity undr modrat thrust () (c) (d) (b) pak positiv bnding capacity undr modrat thrust (d) pak thrust capacity (f) pak ngativ bnding capacity Figur Cut-away viw of pip-soil dformation at pak capacity (idntifid in Figur ) TRB Annual Mting Initial Submittal for Rviw

13 Schafr and M c Grath (a) (c) ) (% in a g str n d i n b d - u tsi o outsid bnding strain (%) ) (% in a g str n d i n outsid bnding strain (%) b d u tsi o - OUTSIDE STRAIN AT DEPTH OF FILL =. FT. Buckling Yilding thrust strain (%) - - OUTSIDE STRAIN AT DEPTH OF FILL =. FT. Buckling Yilding thrust strain (%) (b) (d) ) (% in b d u tsi o a g str n d i n - ) (% in b d u tsi o - OUTSIDE STRAIN AT DEPTH OF FILL =. FT. Buckling Yilding thrust strain (%) a g str n d i n - - OUTSIDE STRAIN AT DEPTH OF FILL =. FT thrust strain (%) Figur Strain dmands vs. capacity as dpth of fill Incrass (Applid strain dmands shown hr ar for th CANDE modl with non-uniform ML backfill, Figur, and th long-trm pip modulus, E ) outsid fibr strain (%) outs id fibr s train (%).. Uniform ML, E pip =E initia l Uniform ML, E pip =E in itia l. Shap Factor D f = (AWWA) Shap Factor D f = (AWWA)... NON-Uniform ML, E pip =E initia l NON-Uniform ML, E pip =E initia l Shap Factor D f = (AWWA) Shap Factor D f = (AWWA) outsid fibr strain (%) outs id fibr s train (%) outsid bnding strain (%) outsid bnding strain (%).. Uniform ML, E pip =E Uniform ML, E pip =E. Shap Factor D f = (AWWA) Shap Factor D f = (AWWA)... NON-Uniform ML, E pip =E NON-Uniform ML, E pip =E Shap Factor D f = (AWWA) Shap Factor D f = (AWWA) Figur Comparison of shap factor approach for applid strain dmands with CANDE modls Buckling Yilding TRB Annual Mting Initial Submittal for Rviw