Conversion of Non-Linear Strength Envelopes into Generalized Hoek-Brown Envelopes

Transcription

1 Covero of No-Lear Stregth Evelope to Geeralzed Hoek-Brow Evelope Itroducto The power curve crtero commoly ued lmt-equlbrum lope tablty aaly to defe a o-lear tregth evelope (relatohp betwee hear tre, τ, ad ormal tre, ) for ol. I the Rocece lope tablty program Slde the crtero ha the form: τ b = a( + d) + c+ ta( θ w), () where a, b ad c are parameter typcally obtaed from a leat-quare regreo ft of data obtaed from mall-cale hear tet. The d parameter repreet the tele tregth of a materal, whle θ w kow a the wave agle. Aother popular tregth model ued lope tablty aaly the hear /ormal fucto. It cot of par of hear ad ormal tre value that defe arbtrary, o-lear hear/ormal tregth evelope for materal. Becaue o flow rule have bee derved or defed for the power curve ad hear/ormal fucto crtera, t curretly mpoble to ue them elato-platc fte elemet aaly. A a reult, whe uch a tregth model ext a Slde fle that mported to Phae 2, t coverted to a equvalet Geeralzed Hoek-Brow model. The Geeralzed Hoek-Brow crtero the mot wdely ued model for characterzg the tregth of rock mae, ad ha a well-defed platc flow rule. The ext ecto wll preet the equato of the Geeralzed Hoek-Brow crtero, ad wll outle the procedure for determg a Geeralzed Hoek-Brow crtero equvalet to a power curve or hear/ormal tregth model. The Geeralzed Hoek-Brow tregth crtero The o-lear Geeralzed Hoek-Brow crtero [] for rock mae defe materal tregth term of major ad mor prpal tree a: a = + m b + where the uaxal compreve tregth of the tact rock materal, whle GSI 00 = mexp 28 4 D, GSI 00 = exp 9 D, ad ( GSI /5 20/ a e e ) = (2)

2 m a tact rock materal property, GSI kow a the geologcal tregth dex, whle D termed the dturbace factor []. Ug relatohp developed by Balmer [, 2], a hear-ormal tre evelope equvalet to the Geeralzed Hoek-Brow prpal tre evelope ca be determed. The hear tre (τ ) ad ormal tre ( ) par correpodg to a pot o a prpal tre evelope ca be determed from the equato τ = d d d + d d ( ) ( ) 2 2 d + d = + d (). (4) For the Geeralzed Hoek-Brow crtero, the followg equato relate ad τ to ad : τ = + am b + a a 2 + am b + (5) a a + a = am b + (6) For a gve et of Geeralzed Hoek-Brow parameter ad a pefed value, ca be determed from Equato (5) through replacemet of wth the defto of the crtero (Equato ()).

3 Etmatg the parameter of a Geeralzed Hoek-Brow evelope equvalet to a Power Curve power GHB Fgure 2 how a power curve evelope, τ, ad a ew Geeralzed Hoek-Brow, τ, that approxmate the power curve. Both evelope are draw hear-ormal pace. For ay gve value, the quare of the error betwee the reduced ad approxmated evelope defed by the equato: 2 power GHB 2 ε = τ τ. (7) Orgal power hear evelope Approxmate GHB hear evelope 0.06 Dfferece (error) betwee the curve t max Fgure 2. Approxmato of a power curve wth a equvalet Geeralzed Hoek-Brow evelope hear-ormal pace. Notce the rego of error or dfferece betwee the two curve. The total error of the ft of fucto: GHB τ to power τ ca be obtaed through tegrato of the quared error

4 max t 2 Total error = (8) ε d over the rage t (the tele tregth) to a maxmum ormal tre value, max. Becaue the quared error fucto doe ot expltly relate to τ, the tegrato bet performed ug a umercal approach uch a gaua quadrature. The parameter of the bet-ft Geeralzed Hoek-Brow evelope to the power curve tregth evelope ca be obtaed through mmzato of the total quared error. Phae 2 doe th mmzato the Smplex techque, whch doe ot requre dervatve of the fucto beg mmzed. Procedure for computg equvalet Geeralzed Hoek-Brow parameter To reduce the uer of parameter to be determed, the curve-fttg procedure aume the dturbace parameter D = 0, ad etmate bet-ft value for the three parameter, m ad GSI. Th becaue, a ee from the equato that defe the Geeralzed Hoek-Brow crtero, the parameter,, ad a ca be calculated ug m ad GSI. Aumg D = 0 mplfe calculato ubtatally wth practcally o pealty to the accuracy of the curvefttg procedure. The tep for etmatg the Geeralzed Hoek-Brow parameter equvalet to a power curve evelope are the a follow: () Etablh the rage of mor prpal tree actg a lope. Sce the mmum tre take to be the tele tregth, t, t oly eceary to determe the maxmum value the lope. () Determe the correpodg value of ormal tre, max, ug Equato (5). () Mmze the quared error fucto over the rage [ t, max] ug a techque uch a the Smplex method. (The tegrato the quared error fucto performed ug the umercal gaua quadrature method.) The varable of the fucto are, m ad GSI. D aumed to have a fxed value of zero. (v) Ue m ad GSI to calculate the parameter m,, ad a. b

5 Determato of equvalet Geeralzed Hoek-Brow curve for Shear- Normal fucto The procedure for determg a Geeralzed Hoek-Brow curve that bet ft a hear-ormal fucto very mlar to thoe decrbed above for the power curve model. The prmary dfferece le the quared error fucto. Sce the hear-ormal fucto defed by a dcrete uer m of data pot, the quared error fucto tead of havg a tegral ue the ummato: m = 2, Total error = ε. (9) REFERENCES. Hoek E., C. Carraza-Torre, ad B. Corkum Hoek-Brow crtero 2002 edto. I Proceedg of the 5th North Amerca Rock Mechac Sympoum ad the 7th Tuellg Aoato of Caada: NARMS-TAC 2002, Toroto, Caada, ed. R.E. Hammah et al, Vol., pp Balmer G A geeral aalytcal oluto for Mohr evelope. Amerca Soety for Tetg ad Materal, vol. 52, pp