An analytical solution to predict axial load along fully grouted bolts in an elasto-plastic rock mass

Transcription

1 An analytal soluton to prdt axal load along fully groutd bolts n an lasto-plast rok mass by H. Jalalfar* T h n a l Synopss Nowadays, fully napsulatd rokbolts hav bom a ky lmnt n th dsgn of ground ontrol systms. Th man rason s that thy offr hgh axal rsstan to bd sparaton. In ths rsarh, th load transfr apaty of a fully groutd bolt s valuatd analytally n an lasto-plast rok mass ondton. Th rsarh onsdrd th fft of bolt nd-plat on load transfr apaty. Bolt and surroundng matrals wr assumd to b last and lastoplast matrals rsptvly. Th load transfr mhansm of a fully groutd bolt s a funton of paramtrs suh as bolt lngth, shar stffnss of ntrfas, n stu strss, prsn of fa-plat and dstan along th bolt. Ths fators wr analytally valuatd. Fnally, th load along th bolt was prdtd n dffrnt surroundng rok mass haratrsts. Kywords fully groutd bolt, axal load, lastoplast, analytal, numralload transfr. Introduton Th ntrfa shar strsss, rathr than th groutng matral tslf, ar of grat mportan n th ovrall rsstan of a rokbolt systm. Thr ar lmtatons to pull tsts n dtrmnng th rsstan of ntrfas, as strss dstrbuton n th systm s afftd by th gomtry of th bolt, borhol, and th mbdmnt matral proprts. Th haratrzaton of th bolt surfa has major fft on th load transfr apaty of a fully groutd bolt, baus surfa roughnss dtats th dgr of ntrlokng btwn bolt and rsn. In ths rsarh, to dfn th load dvlopd along th bolt, an analytal modl of a bolt mbddd n lasto-plast rok mass ondtons was dvlopd. Th modl was valuatd both wth and wthout an nd-plat. Fnally, dffrnt surroundng rok haratrsts wr ntrd n th modl and load transfr apaty along th bolt was prdtd. Load transfr apaty Durng rok movmnt, th load s transfrrd from th bolt to th rok va th grout by th mhanal ntrlokng aton btwn surfa rrgularts at th ntrfas. Whn axal sharng ours durng rok movmnt, th load s transfrrd to th bolt as th grout ntrfa shars1. Th ablty to transfr th load btwn bolt, grout, and rok dpnds on grout annulus, grout strngth, bolt profl haratrsts, th roughnss and strngth of th rok, and mhanal proprts of th bolt. Slppag may our at th rok/grout or grout/bolt ntrfas, both bng alld douplng. D-ouplng taks pla whn th shar strss xds th strngth of th ntrfa. Falur n a laboratory tst usually ours along th bolt/grout ntrfa. Howvr, f rok, nstad of a stl tub, s usd as an outr asng lmnt, thn th falur may our along th rok/grout ntrfa, dpndng on th strngth of th rok. If th rok s soft thn falur ours along th grout/rok ntrfa baus th mhanal ntrlok braks down at low loads and frtonal rsstan oms nto aount. In hard rok th mhanal ntrlok would b domnant. Kl2 rportd that whn th surfa frton of a borhol drass, slppag ours at th grout/rok ntrfa, and whn th lngth of th bolt and borhol xd a rtal lngth for a 21 mm damtr bolt n a 27 mm damtr hol, falur taks pla at th bolt. Ths has bn dmonstratd by laboratory tsts3. Fgur 1 shows th shmat rprsntaton of load transfr gnratd at th ntrfa togthr wth th * Dpartmnt of Ptrolum and Mnng Engnrng, Envronmntal and Enrgy Rsarh Cntr, Shahd Bohanar Unvrsty of Krman-Iran. Th Southrn Afran Insttut of Mnng and Mtallurgy, SA ISSN X/ Papr rvd Jul. 2009; rvsd papr rvd Jul N o t Th Journal of Th Southrn Afran Insttut of Mnng and Mtallurgy VOLUME 111 NOVEMBER

2 An analytal soluton to prdt axal load along fully groutd bolts Fgur 1 Skth of bolt-grout-rok ntrfa onfguratons bolt profl onfguraton. Ths shows that mhanal ntrlokng ours whn rrgularts mov rlatv to ah othr (wdgs ar ratd). Durng sharng, surfa ntrlokng wll transfr th shar fors from on lmnt to anothr. Whn th shar fors xd th maxmum apaty of th mdum, falur ours and only frtonal and ntrlokng rsstan wll ontrol th load transfr haratrsts of th bolt. Efftv bondng btwn bolt, rsn, and rok an b attrbutd to adhson, frton, and mhanal ntrlokng, but thr rlatv mportan dpnds on th tst ondtons. Adhson s normally almost nglgbl, and ths was larly dmonstratd by sawng a olumn of rsn blok ast on a bolt along th axs. Eah ston of rsn dtahd lanly from th bolt wth th for appld, whh was also supportd by th rsults of bolt/rsn sharng tsts arrd out undr onstant normal stffnss4. Bondng strngth s almost zro whn normal strss s rdud. It should b notd that frton also dpnds on surfa roughnss. It s obvous that th onfnng prssur appld has a major nflun on th lvl of frton and ntrlokng aton at th bolt rsn ntrfa. Kasr t al.5 rportd that a mnngndud strss hang s on of th most mportant paramtrs ontrollng bond strngth. Bond falur mhansm Whn an axal load s appld to a bolt, t strths longtudnally and ontrats latrally baus of Posson s fft. Whn ontraton ours th bond braks at th ntrfa. Strthng and ontratng s alulatd n both pull and push tsts durng loadng. In pull and push laboratory tsts, slp and yld ourrd at th bolt/grout ntrfa and th bolt dd not fal ntrnally6. For a bolt to undrgo nkng t must b grppd frmly at both nds, but pullng a bolt rdus ts damtr longtudnally, rsultng n ontraton aordng to Posson s fft, whh would obvously afft ts load transfr apaty. Dbondng and rduton our aftr th load-dsplamnt urv has rsn lnarly. Basd on th numral analyss of6, thr s a hgh lvl of load ndud n th had, whh rdus xponntally along th lngth. Inrasng th load propagats th d-bondd ara furthr along th bolt and xpands t proportonally. From hr th load drass and th hng pont n th urv (load-dsplamnt), alld th maxmum barng apaty, s formd. Aftr th pak pont of th shar load, shar dsplamnt dpnds on ntrlokng phnomna, whh ar a funton of th profl spfatons, rsn grout proprts, and rsn thknss. Analyss of an last bolt surroundd by plast rok mass Th load transfr mhansm of a fully groutd bolt s a funton of th surfa ondton, baus surfa roughnss dtats th dgr of ntrlokng btwn bolt and rsn. Th shar strss at ntrfas, rathr than of th groutng matral, s of grat mportan n th ovrall rsstan of a rok bolt systm. In modllng a sngl dmnsonal rsn groutd anhor, Farmr7 advand a thortal soluton for a rular lastally anhord bar surroundd by an last grout onfnd n a rgd borhol. H drvd a homognous lnar dffrntal quaton dsrbng th dstrbuton of th for along th anhor. Th day funton was xponntal n form. Pull tsts on onrt, lmston, and halk yldd dffrnt rsults. Good orrlaton was obtand n onrt undr low axal loads, but n wakr lmston and halk th rsults wr nonsstnt. Farmr also found th shar strss n th rsn annulus was a funton of th rsn grout. Followng Farmr7, th qulbrum of a fully groutd rokbolt may b wrttn: [1] [2] whr; A b s th bolt ross-ston ara F x s th shar load du to th bond pr unt lngth undr last bhavour ondtons. Thn substtutng [3] nto [2]: In othr words, th shar for du to th bolt an b assumd as a lnar funton of th rlatv slp btwn th bolt and th rok8. Thn, whr: K = shar stffnss of th ntrfa (N/mm2) u r = rok dsplamnt along th bolt (mm), whh drass wth dstan from th surfa of th xavaton and dpnds on varous n stu paramtrs suh as, ntal strss, rok mass modulus, bolt lngth. By ombnng [5] and [4] th followng quaton for a dstrbuton of dsplamnt along th bolt was obtand. Moosav8 usd Equaton [6] for analyss of abl bolts, but h onsdrd both bolt and rok mass n an last stat. In ths modl th bolt and th rok mass ar onsdrd as last and lasto-plast rsptvly. [3] [4] [5] [6] 810 NOVEMBER 2011 VOLUME 111 Th Journal of Th Southrn Afran Insttut of Mnng and Mtallurgy

3 An analytal soluton to prdt axal load along fully groutd bolts In th abov quaton, u r an b rprsntd by an analytal funton of th gomtry of th tunnl and rok surfa movmnt. whr r o = tunnl radus. u ro s th total dformaton of th xavaton wall, and s wrttn as9 [7] [8] [9] [10] By dvdng th boltng to n qual stons (Fgur 2), and dfnng Δx = x - x = L, th xprssons +1 (nr o ) for th drvatvs of u x at x = x ar gvn as or Equaton [15] for = 2,, n an b wrttn as; [16] [17] [18] T h n a l N o t Paramtrs b, k, and f an b found from followng Equatons [9]. [11] [12] Ths n-1 quatons wth two boundary ondtons wll form a tr-dagonal systm of n + 1 lnar algbra quatons wth n + 1 unknowns, {u x (x )}, thus w hav [19] [13] whr v = Posson rato of rok mass Po = n stu strss = ohson ϕ = frton angl r = th boundary btwn th zon of plast and last dformaton By ombnng Equaton [8] nto Equaton [7] and thn Equaton [7] nto Equaton [6] and solvng, th followng numral mthod has bn dvlopd. It s notd that th numral mthod was usd as a powrful tool to solv th dvlopd analytal modl. In ths as th bolt was dvdd nto qual parts, and thn th load dstrbuton an b obtand by lnkng ths small stons togthr. Thus to solv th rfrn quaton (Equaton [6]), dmnsonlss quantts ar dfnd. A bolt undr tnson omprsss th rok, whh prvnts bd sparaton and frtonal fors dvlopng btwn th layrs, but ths dos not man that mor tnson rats bttr stablty10. Whn a bolt s pr-tnson loadd t would nflun th shar strngth of th jont wth fors atng both prpndular and paralll to th shard jont by ndung onfnng prssur. A gnral rul for dtrmnng maxmum pr-tnson s that t should not xd 60% of th bolt yld strngth or 60% of th anhorag apaty13. In ths rsarh th followng two ass ar xamnd; Cas 1 Bolt nstalld wthout fa plat F x = 0 at x = 0 and F x = 0 at x = L whr Dfnng th normalzd for F x = F x / (A b E b ), th abov boundary ondtons wll b quvalnt to: [20] [21] Ths an b wrttn as; [14] Kr o 2 A b E b Kr o 2, s a dmnsonlss quantty. By dfnng γ = Ab E b b wrttn as, t an [15] Fgur 2 Notaton for numral formulaton Th Journal of Th Southrn Afran Insttut of Mnng and Mtallurgy VOLUME 111 NOVEMBER

4 An analytal soluton to prdt axal load along fully groutd bolts Cas 2 Bolt nstalld wth attahd fa plat u x = u r0 at x = 0 and F x = 0 at x = L, or u x (x 1 ) = 1 and u x (x n+1 ) u x (x n ) = 0 Th load dvlopd along th bolt n both th abov ass was analysd and followng rsults wr obtand. Cas 1: Bolt wthout fa-plat Fgurs 3 to 8 show th dstrbuton of axal load dvlopd along th bolt and normalzd dsplamnt, wthout a faplat, nstalld n surroundng plast matrals. Thr dffrnt bolt lngths, namly 2.1 m, 5 m, and 10 m wr usd for th analyss. Th rsults ar n agrmnt wth th fndngs of Tang t al.11, who appld a gnralzd fnt lmnt thnqu. Th nput data for th surroundng matrals ar usd aordng to Strata Control Thnology s rport12. Th ntal strss and rok modulus of lastty ar onsdrd to b 25 MPa and MPa rsptvly. Fgur 3 shows th axal load profl along th bolt n dffrnt lngths. Wth nrasd lngth, th axal load s nrasd and also th pak pont of th load movs towards th nd of th bolt. In addton, th load s onntratd nar th xavaton surfa. Fgur 4 shows th normalzd dsplamnt as a funton of bolt lngth. It shows that wth an nras n lngth, dsplamnt of rok s rdud. Fgur 5 shows normalzd dsplamnt as a funton of lngth for a 2.1 m bolt, n 15 MPa ntal strss and wth dffrnt ntrfa shar stffnss valus. It an b sn that, wth an nras n ntrfa stffnss, dsplamnt s rdud. Usng Equatons [8] to [10], th valu of u ro n 15 MPa and 25 MPa ntal strss s 6.3 mm and 10 mm rsptvly. Fgur 6 shows th load dvlopd along th bolt nrasng wth an nras of ntal strss at a onstant stffnss. Fgurs 7 and 8 show load dstrbuton along th bolts 2.1m and 10 m long rsptvly at 25 MPa ntal strss for dffrnt valus of rok modulus of lastty. It shows that softr roks gnrat hghr load along th bolt. From th fgurs t s notd that th maxmum load dvlopd along th bolt s los to th bolt fa-plat n th long bolt and almost ntralzd n normal lngth (2 m), whh s n agrmnt wth th fld rsults and bolts nstalld n jontd roks. Whn rokbolts ar nstalld n th tunnl, th load gnraton ntats at th bolt/grout/rok strutur. Th full lngth of th bolt an xprn loadng. In ralty, whn adjant rok bloks ar shard, du to jont roughnss dlaton ours and ths gnrats tnsl fors n th bolt. Th rsults obtand from th analytal dvlopd modl wr vrfd wth th rsults obtand from th fld nvstgaton as shown n Fgur 913. Th load dvlopd on th bolt s wth rspt to th rtratng longwall fa postons. For xampl, th load dvlopd long th bolt was montord Fgur 4 Normalzd dsplamnt vrsus bolt lngth for a bolt wthout a plat wth 25 MPa ntal strss and 15 GPa modulus of surroundng rok Fgur 5 Normalzd dsplamnt vrsus bolt lngth for a bolt wthout a plat, wth 15 MPa ntal strss and 15 GPa modulus of surroundng rok at dffrnt k (shar stffnss, N/mm 2 ) valus Fgur 3 Axal load along th bolt vrsus bolt lngth, wth 25 MPa ntal strss and 15 GPa modulus of surroundng rok, no fa plat Fgur 6 Load dvlopd along th bolt vrsus bolt lngth n as of a bolt wthout a plat, wth 15 GPa modulus of surroundng rok at dffrnt ntal strsss 812 NOVEMBER 2011 VOLUME 111 Th Journal of Th Southrn Afran Insttut of Mnng and Mtallurgy

5 An analytal soluton to prdt axal load along fully groutd bolts Fgur 7 Load dvlopd along th bolt vrsus bolt lngth n as of a bolt wthout a plat, wth 25 MPa ntal strss and dffrnt modulus of surroundng rok at k=10 (shar stffnss, N/mm 2 ), L = 2 m and dstrbuton of th bolt ntrfa dsplamnt for dffrnt bond strngth, rok mass modulus of lastty, and bolt lngth n varous ntal strsss wr analysd. Fgurs 10 and 11 show rsptvly th axal load and dstrbuton of th bolt dsplamnt n two dffrnt bond stffnss ondtons. It an b sn that th bond has sgnfant nflun on th dvlopmnt of load along th bolt lngth and th dsplamnt. Fgur 12 shows th dstrbuton of axal load for dffrnt valus of rok modulus and dffrnt T h n a l N o t Fgur 10 Load dvlopd along th bolt vrsus bolt lngth n as of usng nd plat wth 25 MPa ntal strss and dffrnt k (shar stffnss, N/mm 2 ), at Er = 5 GPa Fgur 8 Load dvlopd along th bolt vrsus bolt lngth n as of a bolt wthout plat, wth 25 MPa ntal strss and dffrnt modulus of surroundng rok at k=10 (shar stffnss, N/mm 2 ), L = 10 m Fgur 11 Normalzd dsplamnt vrsus bolt lngth n as of usng nd plat wth 25 MPa ntal strss and dffrnt k (shar stffnss, N/mm 2 ), at Er = 5 GPa Fgur 9 Load transfrrd on th bolt nstalld at th Mtropoltan Collry wth th tm lapsd (aftr Jalalfar 13 ) from th ntal longwall poston 255 m ahad of th st to whn th longwall fa passd th st by 260m. As Fgur 9 shows, th maxmum load s approxmatly at th mddl of th bolt, whn th bolt s nstalld through th roof, whh vrfs th dvlopd analytal approah. Cas 2: Bolt wth fa-plat Usng Equaton [17] and boundary ondtons n as 2 (usng an nd-plat), th axal load bult up along th bolt Fgur 12 Axal load vrsus bolt lngth n as of usng nd plat wth 25 MPa ntal strss and dffrnt rok modulus and bolt lngth, k=10 (shar stffnss, N/mm 2 ) Th Journal of Th Southrn Afran Insttut of Mnng and Mtallurgy VOLUME 111 NOVEMBER

6 An analytal soluton to prdt axal load along fully groutd bolts Th dstrbuton of bolt dsplamnt s narrowr wth nrasng bond strngth and bolt lngth A lowr valu of th plast zon rdus th valu of bolt load gnraton. Softr roks gnrat hghr loads along th bolt. Rfrns Fgur 13 Axal load vrsus bolt lngth n as of usng nd-plat n dffrnt ntal strss ondtons wth 5 GPa rok modulus, k=10 (shar stffnss, N/mm 2 ) 1. SERBOUSEK, M.O. and SIGNER, S.P. Lnar load-transfr mhans of fully groutd roof bolts. Rport of Invstgatons US Burau of Mns, Pttsburgh, PA, KILIC, A. and ANIL, M. Th ffts of grout proprts to th bolt apaty. 16th Mnng Congrss, Ankara Turky, pp AZIZ, N. Bolt surfa profls- an mportant paramtr n load transfr apaty apprasal. Prodngs of th Ffth Intrnatonal Confrn on Ground Control and Mnng Construton, Prth, Australa AZIZ, N. A nw thnqu to dtrmn th load transfr apaty of rsn anhord bolts. 3rd Australan Coal Oprators Confrn, Wollongong, NSW, Australa, pp Fgur 14 Axal load vrsus bolt lngth n as of usng nd-plat n dffrnt plast zon radus wth 5 GPa rok modulus, k=10 (shar stffnss, N/mm 2 ) 5. KAISER, P.K., YAZICI, S., and NOSE, J. Efft of strss hang on th bond strngth of fully groutd abls. Intrnatonal Journal of Rok Mhans and Mnng Sns & Gomhans Abstrats, vol. 29, no. 3, pp AZIZ, N. and JALALIFAR, H. Invstgaton nto th transfr mhansm of loads n groutd bolts. Australan Gomhans, vol. 40, no. 2, pp bolt lngths. It shows that a hghr rok modulus of lastty gnrats a lowr axal load along th bolt. Ths trnd drass xponntally towards th bolt nd for both bolt lngths. Fgur 13 shows th axal load dstrbuton along th bolt n dffrnt ntal strss ondtons. It rvals that th surroundng roks wth hghr ntal strss ndu a hghr axal load along th bolt. As Fgur 14 shows, th axal load rdud wth drasng radus of th plast zon around th tunnl. Conluson From th axal load dvlopd along th last bolt surroundd by lasto-plast matrals n a rular tunnl, t an b nfrrd that bond strngth, rok mass modulus, and ntal strss hav a sgnfant afft on th load dstrbuton lvl. Also, whn a bolt s not anhord at both nds, th pak maxmum axal load appars n th mddl of a bolt 2.1 m n lngth. Howvr, nrasng th bolt lngth shfts th pak poston losr to th surfa of th xavaton. From th abov both ass analyss, t an b nfrrd that: Hghr valus of k gnrat hghr axal loads. Axal load nrass wth gratr lvl of th ntal strss Hghr valus of rok modulus of lastty ndu hghr valus of axal loads 7. FARMER, I.W. Strss dstrbuton along a rsn groutd rok anhor. Intrnatonal Journal of Rok Mhans and Mnng Sns & Gomhans Abstrats, vol. 12, pp MOOSAVI, M. Load dstrbuton along fully groutd abl bolts basd on onsttutv modls obtand from modfd Hok lls. Ph.D.thss, Faulty of Mnng Engnrng, Qun s Unvrsty, Ontaro, pp. 9. STILLE, H., HOLMBERGE, M., and NORD, G. Support of wak rok wth groutd bolts and shotrt. Intrnatonal Journal of Rok Mhans and Mnng Sns & Gomhans Abstrats, vol. 26, no. 1, pp PENG, S. and GUO, S. An mprovd numral modl of groutd bolt-roof rok ntraton n undrground opnngs. Intrnatonal Symposum on Rok support n Mnng and Undrground Construton, Sudbury, Ontaro, Canada, pp TANG, B., MITRI, H.S., and BOUTELDJA, M. Fnt lmnt modllng of rok anhors. Journal of Ground Improvmnt, vol. 4, no. 2, pp TARRANT, G. Assssmnt of support rqurmnts assoatd longwall rorntaton. Rport No. MET2431. Strata Control Thnology, Wollongong, NSW, Australa, pp. 13. JALALIFAR, H. A nw approah dtrmnng th load transfr mhansm n fully groutd rok bolts. PhD Thss, Unvrsty of Wollongong NSW, Australa, NOVEMBER 2011 VOLUME 111 Th Journal of Th Southrn Afran Insttut of Mnng and Mtallurgy