Simple analytical solutions for underground circular and elliptical openings

Transcription

1 Simple nlyticl solutions fo undegound cicul nd ellipticl openings Autho: Pof. D. hbil. Heinz Konietzky, (TU Begkdemie Feibeg, Geotechnicl Institute) 1 Intoduction... Anlyticl solutions in D Intenl hydosttic pessue Cicul opening in infinite elstic spce nd nisotopic stess field Specil cse: Isotopic f field stesses Specil cse: Isotopic f-field stess nd hydosttic inne pessue Cylindicl tube unde inne nd oute hydosttic pessue Ellipticl opening in infinite elstic spce Elsto-plstic solution fo cicul opening in isotopic stess field Cicul inclusion Pimy inclusion Secondy inclusion Inclusion nd visco-elstic ock mss behviou Convegence-Confinement Method... 3 Litetue... 5 Edito: Pof. D.-Ing. hbil. Heinz Konietzky Lyout: Angel Giebsch, Gunthe Lüttschwge TU Begkdemie Feibeg, Institut fü Geotechnik, Gustv-Zeune-Stße 1, 9599 Feibeg sek_fm@ifgt.tu-feibeg.de

2 Only fo pivte nd intenl use! Updted: 19 Febuy 18 1 Intoduction Ove the lst decdes nd centuies lot of nlyticl solutions wee obtined fo elstic, elsto-plstic, visco-elstic nd visco-elsto-plstic poblems with simple geomety. Some of them, especilly those with openings (holes) in the -dimensionl hlf spce o infinite spce llow to estimte the stess-stin sitution fo geotechnicl stuctues (cicul tunnels, shfts, difts, boeholes etc.). Exemply, this chpte gives few vey simple -dimensionl nlyticl solutions. Fo some of the D solutions pol coodinte system (fig. 1) is moe suitble comped to Ctesin, lthough solutions e vilble fo both of them. The tnsfomtion of stesses between Ctesin (x-y-system) nd pol (-φ-system) coodintes is given by eq. 1 nd illustted by fig x y x - y cos f xy sinf 1 1 f x y - x - y cosf xy sinf 1 f - x - y sinf xy cosf (1) Also, in D we cn distinguish between plne stess: z xz yz nd z - x y nd E plne stin: nd ( ) if we conside the x-y-plne. z xz yz z x y Assuming tht the consideed plne is the x-y-plne nd the z-xis is pependicul to the x-y-plne, the following holds: Plne stess: xx xy ij yx yy nd xx xy ij yx yy zz () xx xy xx xy Plne stin ij yx yy nd ij yx yy (3) zz Consideing the Boltzmnn-Axiom (momentum equilibium) the following holds: nd xy yx xy yx Quite often the hoizontl pincipl stess q is given s fction of the pincipl veticl stess component p, wheeby q p. is lso clled ltel eth pessue coefficient (see fig. ). Pge of 5

3 Only fo pivte nd intenl use! Updted: 19 Febuy 18 Fig. 1: Illusttion of physicl units in pol coodintes. Fig.: Sketch fo vigin stess field (f-field stesses) The nisotopic f-field stess stte (vigin o pimy stesses) is given by the following equtions using pol coodinte system: o p 1 1 cos o p 1 1 cos (4) o p 1 sin In Ctesin coodintes the f-field stesses e: o x p, o y p nd. Anlyticl solutions in D.1 Intenl hydosttic pessue Hydosttic pessue q inside cicul opening of dius induces dil symmetic stess field with two pincipl stess components (dil nd tngentil stesses): o xy nd q (5) Pge 3 of 5

4 Only fo pivte nd intenl use! Updted: 19 Febuy 18 The vible ( ) mesues the distnce fom the cente of the opening. At the boundy of the opening ( ) the stesses ech mximum vlues: compession of mgnitude q s dil stess nd tension of mgnitude s tngentil stess.. Cicul opening in infinite elstic spce nd nisotopic stess field The elstic stess field ound cicul opening is given by the so-clled Kisch-solution, which consides n nisotopic f-field stess stte, chcteized by. Figue 4 illusttes the sitution nd Eq. 6 gives the coesponding equtions fo dil, tngentil nd she stesses. q Fig. 3: Illusttion of stess field ound cicul opening unde intenl hydosttic pessue. Fig. 4: Geometicl model of cicul hole in n nisotopic infinite spce Pge 4 of 5

5 Only fo pivte nd intenl use! Updted: 19 Febuy 18 4 p cos 4 4 p cos 4 (6) 4 p sin 4 The Kisch-solution shows n inhomogeneous nd nisotopic stess field, wheeby the most citicl vlues (minimum nd mximum tngentil stesses) e eched t the sufce of the opening (deduced by setting fist deivtives of Eq. 6 to zeo): p1 1 p3 (7) p1 1 p3 1 (8) Figues 5 to 7 illustte the stess distibution ound the opening in fom of diffeent digms. Fig. 5: Tngentil stesses long the boundy of the opening fo diffeent vlues of. Pge 5 of 5

6 Only fo pivte nd intenl use! Updted: 19 Febuy 18 Fig. 6: Pincipl stess pthwy fo two selected points with exteme vlues. Pge 6 of 5

7 Only fo pivte nd intenl use! Updted: 19 Febuy 18 Fig. 7: Scled tngentil stesses vs. eth pessue coefficient t loctions of exteme vlues Induced dil nd tngentil displcements t the boundy of the opening ( e given by Eq. 9 nd 1. ) p u w cos E (9) 4 1 p u w 11 sin (1) E Fo plin stin holds: w 34 nd fo plin stess holds: w 1. Tnsfomtion into Ctesin coodintes leds to the following expessions: u x p 1 w cos (11) 4E Pge 7 of 5

8 Only fo pivte nd intenl use! Updted: 19 Febuy 18 u y p 1 w 1 3 sin (1) 4E The defomtions t the contou led to n ellipticl shpe of the opening. If we define u 1 w 1 the following expession is obtined: p 4E x u 3 1 cos cos eqution of ellipse in pmete nottion y u 3 sin sin (13)..1 Specil cse: Isotopic f field stesses In cse of isotopic pimy stess field ( Kisch-solution (Eq. 4) yields the following stess field (pincipl stess components): 1) nd vnishing inne pessue the p1 (14) p 1 (15) The obtined dil symmetic solution (stesses e independent on φ) gives mximum tngentil stesses t the contou of p (see lso Fig. 8). Fig. 8: Secondy stess stte ound cicul opening with isotopic f-field stesses of mgnitude p. Pge 8 of 5

9 Only fo pivte nd intenl use! Updted: 19 Febuy 18 Fig. 9: Pincipl stess vlues (fom B: f field - to A: boundy of opening) Figue 9 illusttes ll stess sttes in the pincipl stess digm. F wy fom the opening the vigin stess field with mgnitude p is eched (point B) nd t the boundy the exteme vlues ( long the line A-B. p nd ) e eched. Theefoe, ll stess sttes e locted.. Specil cse: Isotopic f-field stess nd hydosttic inne pessue Supeposition pinciple (vlid fo elstic stess fields) is used to deduce stess field unde considetion of isotopic f field stess p nd hydosttic pessue q t the boundy of the opening (Figue 1). Exteme vlues e obtined gin t nd 3 p q (16) 3 1 p q (17) Figue 1 illusttes ll stess sttes in the pincipl stess digm. F wy fom the opening the vigin stess field with mgnitude p is eched (point B) nd t the boundy the exteme vlues ( p q nd q) e eched. Theefoe, ll stess sttes e locted long the line A-B. Pge 9 of 5

10 Only fo pivte nd intenl use! Updted: 19 Febuy 18 Fig. 1: Cicul hole unde nisotopic f-field stess nd hydosttic inne pessue Fig. 11: Pincipl stess components vesus distnce fom boundy of opening Fig. 1: Pincipl stess vlues (fom B: f field to A: boundy of opening) in pincipl stess digm Pge 1 of 5

11 Only fo pivte nd intenl use! Updted: 19 Febuy 18.3 Cylindicl tube unde inne nd oute hydosttic pessue The so-clled Lme-fomule descibes the pessue distibution inside tube with inne dius nd oute dius b unde supeimposed inne nd oute hydosttic pessue (Figue 1): b p q q 1 b (18) b p q q 1 b (19) Sevel specil cses cn be deduces if nd/o b ttin exteme vlues ( o )..4 Ellipticl opening in infinite elstic spce The tngentil stess σt t the contou long n ellipticl opening is given by the soclled Inglis-solution (see lso Fig. 13). This solutions equies, tht min xis of ellipse coincide with pincipl stess xes. f 1 f 1 f sin t p f 1f sin () The following geometicl eltions hold: y tn f tn, with cos, f sin nd f b x (1) Fig. 1: Illusttion of Lme fomule Pge 11 of 5

12 Only fo pivte nd intenl use! Updted: 19 Febuy 18 Fig. 13: Sketch of ellipticl opening in n infinite medium Exteme vlues fo stesses e obtined t nd : 1 f t p f () t f 1 1p (3) The potentil tnsition between compession nd tension fo tngentil stesses is locted t : sin f 1 f 1 1f (4) Two specil cses e inteesting fom the engineeing point of view: 1 1 f o : this leds to t 1 p const. f 1) (5) ) 1 1 o f : this leds to t. f 1 (6) The fist mentioned specil cse is clled pessue ellipse nd povides the smllest possible compession long the contou. The second specil cse is clled Fenneellipse nd descibes the sitution, whee the minimum tngentil stess eches zeo. Fig. 14 shows the tngentil stesses t the boundy fo the two loctions with exteme vlues ssuming diffeent xis tios f. Pge 1 of 5

13 Only fo pivte nd intenl use! Updted: 19 Febuy 18 Fig. 14: Scled tngentil stesses t contou of ellipticl opening with diffeent ellipticl xis tio vs. eth pessue coefficient The Inglis-solution cn be extended by dditionl considetion of intenl hydosttic pessue using the supeposition pinciple. The tngentil pessue t the contou of n ellipticl opening unde hydosttic inne pessue q is given by the following expession: f o f f f 1 sin t q q q f 1 f sin f 1 f sin (7) Exteme vlues e obtined fo nd f 1 q nd 1 t : t f q (8) The supeposition pinciple leds to the following expession fo the tngentil stesses long the contou of n ellipticl opening unde nisotopic f field stesses nd intenl hydosttic stess: Pge 13 of 5

14 Only fo pivte nd intenl use! Updted: 19 Febuy 18 t f 1 f 1 f sin p f f 1 f sin q f 1 f sin Agin, exteme vlues e obtined fo nd : 1 f t p 1 q f f (3) t f 1 1p 1 f q (31) Constnt minimum pessue long the contou is eched unde the following condition: 1 f q p (3) 1 f This leds to the genelized pessue ellipse, which esults in the following constnt mgnitude fo the tngentil stesses long the boundy: (9) f f t p q 1f 1f (33).5 Elsto-plstic solution fo cicul opening in isotopic stess field If limit stte (filue envelope) is eched, plsticity occus, which is combined with stess-edistibutions nd dditionl plstic defomtions. In the following we conside the simplest cse: cicul opening in n isotopic mteil unde isotopic f-field stess (Fig. 15). Due to the isotopic vigin stess stte, the boundy conditions nd the cicul opening, ottionlly symmetic solution is obtined. Cetin diffeent conditions hve to be consideed fo the elstic nd plstic pt. This leds to septe equtions in tems of stess nd stin fo the elstic nd plstic pt. But, due to the equilibium conditions nd the lw of continuity, noml stess nd dil displcements e identicl t the tnsition between plstic nd elstic pt, which is locted t R. The complete solution needs the ssumption of plsticity condition (filue citeion). We ssume the well-known Moh-Coulomb filue citeion (the supescipt st indictes plstic vlues): f * * F (34) whee: F unixil compessive stength 1 sin 1 sin fiction ngle Pge 14 of 5

15 Only fo pivte nd intenl use! Updted: 19 Febuy 18 Fig. 15: Sketch of cicul opening with elstic nd plstic pt. The extension of the plstic zone R is given by the following expession: R p F F 1 (35) The dil stess t R is given by the following eqution: 1 * * F R p R q 1 F 1 1 (36) Inside the elstic egion ( R) tngentil nd dil stess components e given by the following expessions: 1 * F R p p q p p 1 1 (37) p p q p p 1 1 * F R 1 (38) Inside the plstic egion ( R) the following equtions e vlid: * F (39) 1 * F 1 F 1 (4) The dil displcements inside the elstic egion e given by: Pge 15 of 5

16 Only fo pivte nd intenl use! Updted: 19 Febuy 18 u * R p q R (41) E R p Unde considetion of ssocited plsticity (plstic potentil identicl to filue citeion), tht mens, diltion ngle identicl to fiction ngle), the dil displcements inside the plstic egions e given by the following expession: u 1 p 1 1 R E R (4) * The dil displcements t the contou due to the cetion of opening (induced displcements) e given by the following eqution: u 1 p 1 R 1 * E 1 R (43) Fig. 16 illusttes the development of pincipl stess components (dil nd tngentil components) s function of distnce fom the boundy of the opening. Fig. 17 illusttes ll stess sttes in the pincipl stess digm. F wy fom the opening the vigin stess field with mgnitude p is eched (point A). Fom tht point tngentil stesses e incesing until they ech the bsolute mximum vlue of p q* nd dil stesses decese until the ech q * t the tnsition between elstic nd plstic egion t R (point B). At the boundy (point C) the minimum vlues ( nd ) e eched. Theefoe, ll stess sttes e locted long the line A-B-C. p Fig. 16: Stess pths fo dil nd tngentil stess components s function of distnce Pge 16 of 5

17 Only fo pivte nd intenl use! Updted: 19 Febuy 18 Fig. 17: Stess vlues (fom A: f-field ove B: boundy elstic-plstic egion to C: boundy of opening) in pincipl stess digm.6 Cicul inclusion The poblem of inclusion is fundmentl mechnicl one in elsticity, but is lso of get pcticl impotnce fo ock mechnics, e.g. the intection of the lining of boeholes, shfts o difts with the suounding ock mss. Fig. 18 illusttes the poblem consideing tht n inclusion hs the shpe of cylindicl shell with inne dimete nd oute dimete b. Rock mss is chcteized by Young s modulus E nd Poisson s tio ν. The inclusion is chcteized by Young s modulus E nd Poisson s tio ν. Two diffeent ppoches cn be followed: Consideing tht ock mss nd inclusion exist (1. Phse) nd f-field stess is pplied ftewds (. Phse) pimy inclusion. Consideing tht ock mss exist, f-field stess is ledy pplied nd coesponding pimy defomtions hve occued (1. Phse); in next step (. Phse) cicul opening with dius b is ceted nd dil pessue is pplied to boundy to void ny displcement; in the lst step (3. Phse) the inclusion is inseted nd t the sme time the suppot pessue is deleted secondy inclusion. Fig. 18: Cylindicl shell s inclusion inside infinite ock mss unde isotopic f-field stess p. Pge 17 of 5

18 Only fo pivte nd intenl use! Updted: 19 Febuy 18 Elstic solution of this poblem is bsed on the ssumption tht displcement continuity t the boundy between ock mss nd inclusion exist nd tht noml pessue q cts long the line linking ock mss nd inclusion..6.1 Pimy inclusion Fo the elstic cylindicl shell the following holds fo the dil displcement consideing plne stess: 1 u b b b b b E (44) By substituting the tngentil nd dil stess components by the expession fo thickwlled tubes unde inne nd oute pessue ccoding to Lme the following expession cn be obtined: b u b m q, (45) E whee m b b. Fo the ock mss the following expession holds fo the dil displcement: b u b p q E 1 (46) Equte expession (44) nd (46) nd solve the eqution fo q leds to the following expession: q E m E E 1 p (47).6. Secondy inclusion The genel pocedue is simil to the one explined in Chpte.6.1, but the pimy displcements hve to be subtcted. Theefoe, the equivlent expession to 46 is s follows: b b 1 1 u b p 1 q p b p q E (48) E E Eq. (48) is set equl to Eq. (45): q E m 1 E E 1 p (49) Consideing Eq. (47) nd (49) the following conclusions cn be dwn: Pge 18 of 5

19 Only fo pivte nd intenl use! Updted: 19 Febuy 18 q q pimy secondy fo secondy inclusion: if E o E, then p q fo pimy inclusion: if E o E, then q 1 p if o E, then q Pimy inclusion is typicl fo cut-nd-cove technology whees secondy inclusion epesents the sitution of suppot in undegound mining o tunnelling..6.3 Inclusion nd visco-elstic ock mss behviou The bove mentioned expessions cn be extended by consideing heologicl, tht mens time-dependent, ock mss behviou. Using the so-clled Volte o Coespondence Pinciple, elstic constnts hve to be eplced by visco-elstic opetos. Simplifying Poisson s tio fo ock mss nd inclusion e set identicl. Unde this ssumption Young s modulus hs to be eplced by the following expessions: Kelvin model (pllel connection of sping nd dshpot with viscosity η): d E E1 E dt (5) Mxwell model (seies connection of sping nd dshpot with viscosity η): d E E dt d (51) 1 dt Insetion of expessions 5 nd 51 into Eq. (49) nd ssuming tht ν=.5 leds to diffeentil eqution fo q, which cn be solved unde considetion of initil conditions, so tht the following finl expessions cn be obtined: fo Kelvin model: q t 1 3 p m fo Mxwell model: E E 1e t (5) q t p1e 1 t E 1 m 3 E (53) Pge 19 of 5

20 Only fo pivte nd intenl use! Updted: 19 Febuy 18 whee E E 1 m 3 E nd E m 3 E. Both ppoches delive non-line q-t-esponse. Fo t e obtined: fo Kelvin model: the following esults p q 1 m 3 E E fo Mxwell model: p q p (54) (55).7 Convegence-Confinement Method The Convegence-Confinement Methods (CCM) ws developed fo tunnel dimensioning nd consist of two pts: the Gound Rection Cuve (GRC) the Suppot Chcteistic Cuve (SCC) CCM ssumes cicul tunnel unde isotopic f-field stess. The ock mss itself is descibed by eithe isotopic nd homogeneous elstic o isotopic nd homogeneous elsto-plstic mteil behviou. The GRC eltes the suppot pessue cting t the tunnel boundy to the dil displcements t the tunnel boundy. The GRC fo isotopic line elstic ock mss esponse is illustted in Fig. 19. Rdil displcement is zeo in cse the dil pessue t the tunnel contou q equls the f-field stess p. Rdil displcement u inceses with decesing dil pessue until the mximum displcement u is eched fo vnishing dil pessue. Fig. 19: Gound Rection Cuve fo line elstic ock mss: dil pessue vs. dil displcement. Pge of 5

21 Only fo pivte nd intenl use! Updted: 19 Febuy 18 Fig. : Suppot Chcteistic Cuve fo line elstic suppot esponse: suppot pessue vs. dil displcement. The SSC descibes the being pessue due to the convegence of the tunnel. If one gin ssumes line elstic esponse fo the suppot coesponding SCC cn be illustted s shown in Fig.. The displcement vlue u descibes the initil gp between the tunnel contou nd the suppot nd k is the stiffness of the suppot. The intesection between GRC nd SCC descibes the opetion point of the system nd llows the detemintion of both, the suppot lod nd the tunnel convegence. The woking point cn be detemined by the following equtions: q u u fo SCC nd k u q p p fo GRC (56) u Detemintion of the intesection delives the following vlues fo the woking point: k u u q* p k u p (57) k u u u* 1 p ku u (58) Pge 1 of 5

22 Only fo pivte nd intenl use! Updted: 19 Febuy 18 Fig. 1: Illusttion of woking point s intesection of GRC nd SCC fo line elstic mteil esponse of ock mss nd suppot. Fig. : Illusttion of intection between GRC nd SCC in cse of nonline behvio Fig. 1 shows tht being pessue of the suppot inceses with incesing stiffness of the suppot, educed gp between suppot nd tunnel contou nd decesing stiffness of the ock mss nd vice ves. Considetions egding GRC nd SCC cn be extended by including nonline behviou fo ock mss nd suppot. Typicl cuve pogession is shown in Fig.. Suppot is epesented by elstic pefect plstic behviou. Rock mss is epesented by elsto-plstic behviou ccoding to Moh-Coulomb theoy. Plstic ock mss behviou stts, wheneve the extension of the plstic zone eches the dius of the opening ( R, whee q q nd u u ) 1 1 Extending the solution given in chpte.5 one cn show which suppot pessue q would be necessy to void the development of plstified zone. The dil pessue inside the plstified zone unde ction of intenl pessue q is given s follows: F * 1 q (59) Pge of 5

23 Only fo pivte nd intenl use! Updted: 19 Febuy 18 Accoding to the Moh-Coulomb filue citeion (34) t the tnsition between elstic nd plstic egion the dil stess is defined s: * p R 1 F (6) Equlizing (59) nd (6) fo = R llows to detemine the dius of the plstic zone unde considetion of suppoting pessue q: R 1 1 p F F 1 1 q 1 F (61) Plstifiction of ock mss stts wheneve the following condition holds ( R ) : p F q 1 (6) A moe genel solution consideing non-line Hoek-Bown filue citeion is given by Cnz-Toes & Fihust (). Futhe comments e given fo exmple by Kinth-Reumye et l. (9) nd Oeste (9). Simple nd esy to hndle softwe tools wee developed like Rocksuppot (15). Such tools llow to conside nonlineities of ock mss nd suppot. They lso delive dt sets to convet suppot effects of nchos, shotcete etc. into equivlent suppot pessue q. Fig. 3 illusttes diffeent potentil constelltions of intection between ock mss nd suppot. This digm shows, how stiffness nd instlltion time o gp between suppot nd ock mss influence the ch foming. Pge 3 of 5

24 Only fo pivte nd intenl use! Updted: 19 Febuy 18 Fig. 3: Illusttion of potentil intection of GRC with diffeent SCC Pge 4 of 5

25 Only fo pivte nd intenl use! Updted: 19 Febuy 18 3 Litetue Aednoy, B.S. & Looyeh, R. (11): Petoleum ock mechnics, Gulf Pofessionl Publishing, 35 p. Cnz-Toes, C. & Fihust, C. (): Appliction of the convegence-confinement method of tunnel design to ock msses tht stisfy the Hoek-Bown filue citeion, Tunneling nd Undegound Spce Technology, 15: Hudson, J.A. (Ed.), (1993): Compehensive ock engineeing, Pegmon Pess, Volume 1-5 Jege, J.C. et l. (7): Fundmentls of ock mechnics, Blckwell Publishing, 475 p. Kinth-Reumye, S. (9): The convegence confinement method s n id in the design of deep tunnels, Geomechnics nd Tunneling, (5): Obet, L. & Duvll, W.I. (1967): Rock mechnics nd the design of stuctues in ock, John Wiley & Sons, 65 p. Oeste, P. (9): The convegence-confinement method: oles nd limits in moden geomechnicl tunnel design, Ameicn J. of Applied Sciences, 6(4): Piseu, W.G. (7): Design nlysis in ock mechnics, Tylo & Fncis, 56 p. Rocksuppot (15): Thei, Z. (4): Fühe duch die äumliche Elstizitätstheoie, ICONEON, 159 p. Pge 5 of 5