Examine 3D Energy Balance Utility

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1 Exaine 3D Enegy Balance Utility Develoed fo the Canadian Rockbust Reseach Poga by The Rock Engineeing Gou Deatent of Civil Engineeing Univesity of Toonto Ail 99

2 Intoduction The uose of this oject is to develo the ethodology fo the calculation of enegy coonents fo a 3-D elastic analysis of excavations in an infinite edia; and then incooate this ethodology into an Exaine-3D utility. Exaine-3D is a thee-diensional bounday eleent oga. It uses the diect bounday integal ethod, with 3 noded and 6 noded tiangula confoing eleents, to calculate elastic stesses and dislaceents aound undegound excavations in ock. Closed fo integation techniques ae used fo singula and nealy singula influence functions, esulting in accuate dislaceents on, and close to, the ined suface. As will be seen, this is cucial fo the accuate calculation of cetain enegy coonents. Backgound As a esult of ining, a tansition between two equilibiu states occus. Duing this tansition, a tansfe of enegy takes lace within the ock ass. ok is done by the extenal (fa field) foces, enegy is stoed within the suounding ock ass, enegy is given u by the ined ateial, wok is done by all newly excavated faces and suot eleents, and enegy is eleased. Each of these coonents is at of the enegy balance fist intoduced by Cook (967) and late efined by alaon (97,98). This balance is defined as: + U Uc + s + [] whee, total wok done by the extenal (fa field) foces U stoed stain enegy in the ateial ined Uc incease in stoed stain enegy in the suounding ock ass s incease in stoed stain enegy in the backfill and othe suot eleased enegy As well, the eleased enegy is defined as follows: k + U [] whee, k The kinetic enegy eleased into the ock ass The kinetic enegy is the enegy eleased into the ock ass due to instantaneous excavation. This enegy is exessed as a sheical stess wave that oagates away fo the oening. Accoding to alaon (98) and Bady and Bown (985), it is equivalent to the wok

3 done by the induced dislaceents ove the excavated suface. By substitution of [] into [], and assuing an unsuoted excavation (s0), the following equation esults: Uc - k [3] alaon (98) esents a coehensive discussion of each of the enegy coonents and deives a set of basic atheatical definitions fo each te. It is these definitions that ae used to calculate the vaious enegy coonents within Exaine-3D. The following is a suay of the coonents as esented in alaon (98). ( ) () () i i ( Ti + Ti ) ui d [] i T u d k ( ) () i i ( ) ( ) i i U T u d [5] [6] whee, T ( ) i T u () i i ( ) i u () i i iitive stess o taction vecto acting on a suface induced stess o taction vecto acting on a suface iitive dislaceent vecto induced dislaceent vecto fa field suface enclosing all excavations uface ined The and Uc coonents can be calculated using the elationshis in [] and [3]. Moe in deth discussions of enegy changes as a esult of ining ae given in alaon (98), Bady and Bown (985) and Jaege and Cook (976). Results alaon (98) and Hedley (993) both deive a set of inte-elationshis between the vaious enegy coonents. As Hedley entions in his eot, the elationshis develoed by alaon ae incoect. Hedley then oceeds to deive a new set of elationshis based on the closed fo enegy coonent equations deived by Itasca fo a cicula tunnel unde hydostatic stesses. These equations whee then coaed to nueical esults using FLAC. The esults 3

4 ovided good ageeent between the closed fo solutions and the nueical esults. But thee still eains the question of whethe these inte-elationshis can be used fo any ining geoety unde any fa field stess state. To test the hyothesis that the enegy inte-elationshis develoed by Hedley hold fo all ining conditions, two sile closed fo calculations wee efoed. The fist, found in aendix A, calculates the enegy coonents fo a shee unde hydostatic fa field stesses. To ake the calculations soewhat easie, the shee is excavated in one ste. The solution fo, k and Uc fo a sheical cavity, can also be found in Bady and Bown (985). The second solution, found in aendix B, involves a cicula tunnel with a biaxial stess field. Both of these solutions have been tested and veified using Exaine-3D, with the esults being tabulated in aendix C. It is iediately clea fo both solutions, that the enegy inte-elationshis given by Hedley, ae not univesally alicable. Hedley s equations (below) ae coect fo a cicula tunnel in a hydostatic stess field, but they do not aly to all ining situations. Fo these esults, clealy, the inte-elationshis deend on the geoety, and the fa field stess state. As a esult, these enegy inte-elationshis ae deendent on the ining conditions. Given below is a table of the elationshis fo a cicula tunnel and a sheical cavity in a hydostatic stess field, and a cicula tunnel in a biaxial stess field. Cicula Tunnel (Hedley 993)* heical Cavity* k + U k + U + ν 3 ( )( ν) ν νk + U U c k + U + ν U c 3 ( )( ν) ( ν) ( v) * hydostatic fa field stesses ( ν) 3 ( + ν) Cicula Tunnel Biaxial Loading AK+,BK- Kstess atio ( A + B ( 3 ν ))( k + U) ( ν)( A + B ) ( A + B ( 3 ν ))( k + U) ( ν)( A + B ) A B ( ν)( + ) ( A + B ( 3 ν )) Fo the above discussion, it is clea that the cicula tunnel enegy inte-elationshis can not be used to calculate the enegy coonents fo abitay thee-diensional ining geoety s. In Exaine-3D, the k, U, and thus the enegy coonents can be calculated diectly fo the induced dislaceents, fa field stesses and volue of excavation. The enegy coonents ae calculated based on total excavation in one ste. As a esult, the stain enegy in the ateial ined is due only to the stain caused by the fa field stesses. Thus, if the volue ined is known, then U can be calculated fo the following equation fo stain enegy density in Jaege and Cook (976).

5 U { 3 ( 3 3 ) } E σ σ σ ν σ σ σ σ σ σ [7] whee, σ, σ, σ 3 E youngs's odulus ν oisson atio incial stesses olue ined Fo equation [], the eleased enegy can then be calculated. This leaves the calculation of, the wok done by the extenal (fa field) foces, and Uc, the incease in stain enegy in the ock ass. To calculate within Exaine-3D, an exteio bounday would have to be laced at a eote location to all excavations. Accuate dislaceents would then have to be calculated on this bounday and integated ove the suface of this bounday to yield the aount of wok (see equation []) The oble with doing this is that the dislaceents tend to zeo as you get fathe fo the excavations. Machine accuacy difficulties can ceate obles in accuately calculating dislaceents on this bounday. This is not an insuountable oble, but one that ight equie odifications to Coute-3D. In the case of Uc, the bounday integal eesentation would still equie this integation ove the eote bounday. Othewise, a volue discetization could be efoed. The incease in stain enegy density could then be calculated within the volue, and integated ove the volue to yield the incease in stain enegy in the ock ass. tesses would have to be calculated on a dense gid that extends a consideable aount fo the oenings. This would equie a geat deal of coutational effot, while calculating the bounday integal would be uch oe efficient. In the case of both a sheical oening and the cicula tunnel, if the excavation is done in one ste, then the following elationshis exist between,k, and Uc. k [8] Uc k [9] In the case of a sequential excavation, the enegy tansfe ocess is oe colicated. Enegy equations eoted in Hedley (993) fo the excavation of an annulus of ateial aound a cicula tunnel, indicate that the enegy elationshis ae not as sile as indicated by equations [8] and [9]. In the cuent ileentation of the enegy utility, the calculation of the enegy balance is based on a one stage ocess. In view of this, the utility fo calculating the enegy coonents uses equation [8] and [9] fo calculating and Uc. 5

6 Conclusions o fa a good basis has been deived fo the calculation of enegy coonents though ining. Both theoetical and actical asects of enegy theoy have been exloed, and a sile odel has been oosed fo the calculation of enegy changes fo esults obtained fo Exaine-3D. To futhe add to the value of the wok esented hee, the following ites should be added to the enegy calculation algoith. Addition of an oute bounday fo the calculation of total wok done by extenal foces. This is equied to check that validity of equations [8] and [9] fo actical obles. Addition of staging fo calculation of inceental enegy changes. These two enhanceents would geatly add to both the theoy and actical alication of enegy theoy in ining. To ake these additions, a eiod of two to fou weeks would be equied. UTILITY MANUAL - PROGRAM ENCOMP The utility, enco, was witten to coute the enegy coonents associated with an Exaine-3D analysis. The oga equies the.ex3/.es files fo an analysis. The ex3 file need not contain any stess lanes o gids fo calculation of stesses and dislaceents in the ock ass. By default, Coute-3D wites all the bounday dislaceent infoation needed fo calculation of the enegy coonents. The oga calculates,k,,uc,u, and, the excavated volue. The esults ae inted on the sceen and witten to a file with a.egy extension. 6

7 Refeences Cook, N.G.., 967. Design of Undegound Excavations, 8th U.. Rock Mech. y., Minnesota, Bady, H.G. and Bown, E.T., 985. Rock Mechanics fo Undegound Mining, London: Geoge Allen and Unwin, Hedley, D.G.F., 993. Notes on the New Enegy Balance, Reseach Reot, MRD/CRRP, 8. Jaege, J. and Cook, N.G.., 976. Fundaentals of Rock Mechanics, nd edn., London: Chaan and Hall. alaon, M.D.G., 98, Enegy Consideations in Rock Mechanics: Fundaental Results, J..Af. Inst. Min. Metal., ol. 8, No. 8, alaon, M.D.G., 97, Rock Mechanics of Undegound Excavations, Poc. 3d Cong. Int. oc. Rock Mech., Denve, Coloado, ol., Pat B,

8 APPENDIX A Analytical olution of the Enegy Coonents fo the Excavation of a heical Cavity A-

9 R a E young's odulus ν oisson atio volue ined πa 3 3 σ T i a σ a 3 i induced stess / taction T u u i fa field stess io to excavation ( + ν) a E ( + ν) a E 3 3 induced dislaceent Calculation of wok done by extenal R i T u d + T u d 3 3 ( + ν) a a + a d + ( ν) 3 ER R ER πr ( + ν) a πr a ( + ν) a 3 ER R ER ( ν) a E 3 R 3 ( + ν) E 3 i i i 3 3 as R (fo alaon 98) 3 d A-

10 Calculation of enegy stoed in ateial excavated U { σ σ σ 3 ν( σσ σσ 3 σ σ 3) } E { 3 6ν } E 3 E ( ν ) Calculation of kinetic enegy T u d k ( + va ) d E πa ( + v) a E 3 ( + v ) E i Calculation of enegy eleased + U k 3 ( + v ) 3 ( v ) + E E 9 ( v ) E Calculation of enegy in ock ass U + U c 3 ( + v ) 3 ( v ) 9 ( v ) + E E E 3 ( + v ) E k A-3

11 Calculation of / fo a shee 6 ( + v) 9 ( v) ( + v) 3 ( v) 3 ( + U )( + v) k ( v) Calculation of Ucf(k,U) fo a shee U c 3 ( + v) 9 ( v) ( + v) 3 ( v) U c 3 ( + U )( + v) k ( v) Calculation of f() fo a shee 6 ( + v) 9 ( v) ( + v) 3 ( v) ( v ) ( + v) 3 A-

12 APPENDIX B Analytical olution of the Enegy Coonents fo the Excavation of a Cicula Tunnel with Biaxial Loading B-

13 K R y a x K G shea odulus E (+ ν) E young's odulus ν oisson atio AK+ BK- volue ined / unit length πa u u i i θ a a A+ B ( v) cosθ G a G B v a ( ) + sinθ u u cosθ u x i i y a a a A B v G G B v a + ( ) cosθ cos θ ( ) + sinθ sinθ u u sinθ + u θ θ sinθ cosθ a a a A+ B v + + G G B v a ( ) cosθ sin θ ( ) sin θ cosθ Tactions in catesian xy sace: K { x y 0 cosθ } σ ij j { K cosθ sinθ} Ti T T n 0 sinθ Tactions in ola θ sace: { θ } K + B Ti T T cos θ sin θ sinθcosθ σ ij nj B sinθcosθ K cos θ sin θ 0 { K cos θ sin θ B sinθcosθ} ( A + B cosθ) B sin θ B-

14 Calculation of wok done by extenal R T T θ i T u d ; d dθ i i T u d + T u d i θ π a u d ( A B ) ( ) GR A B v a + + cos θ ( ) cosθ R Rd θ 0 a { ( ) ( )} { } G A B v a π + ( R ) π A + B ( v) 8G ; R π i a uθ d ( B sin θ ) ( ) + θ θ GR B ( v a )sin R 0 Rd a π { + ( )} { } G B v a ( R ) B ( v) 8G i θ { A B ( v) } { B ( v) } { ( 3 )} G 8G G A B v Calculation of enegy stoed in ateial excavated U { σ σ σ 3 ν( σσ σσ 3 σ σ 3) } E σ K σ σ v( σ + σ ) va 3 { ( } K + + v A v va + KAv + K E ( ( ν ) ) G A + 8 B B-3

15 Calculation of kinetic enegy T Tθ T u d ; d dθ k i i i i θ i θ T u d + T u d π a u i d ( A + B ) ( ) G A+ B v cos θ ( 3 ) cos θ d θ 0 a { ( ) ( )} G A B v G A B π + ( 3 ) π + ( v 8 3 ) π i a uθ d ( B ) sin θ ( 3 ) 0 G B ( v )sin θ d θ a { 3 ( )} 8 3 G B v π B ( ) ( v) G G A B B v v G { 3 } k + ( 8 3 ) + ( 8 3 ) 8G A + B ( v ) Calculation of enegy eleased + U k { ( 3 )} { ( ) } G A B v + + G A v B { ( ) ( )} G A v + B v Calculation of enegy in ock ass U c k { ( 3 )} { ( 3 )} G A B v + G A + B 8 v { ( 3 )} G A B + v k 8 B-

16 Calculation of / fo a shee G G { A + B ( 3 v) } { A ( v) + B ( v) } A + B ( 3 v) ( va ) + B note: ( v) ; fo B 0, K o v 05. B-5

17 APPENDIX C uay of Equations and Tabulated Results C-

18 uay of Enegy Equations fo a heical Cavity a E young's odulus ν oisson atio volue ined πa ( ν) E U 3 ( ν) E U c 3+ ( ν) E 9 ( ν) E k 3+ ( ν) E C-

19 uay of Enegy Coonents fo a Cicula Tunnel with Biaxial Loading K a K G shea odulus E (+ ν) E young's odulus ν oisson atio AK+ BK- volue ined / unit length πa { A B ( 3 ν )} + G U 8G { A (- ν)+ B } U c 8G { A +B (3- ν) } G {( - ν)( A + B ) } k 8G { A + B (3- ν) } C-3

20 Coaison of Enegy Coonents fo a heical Cavity (in MJoules) shee with, a MPa ν0.5 E0000 MPa Analytical olution Exaine-3D U Uc k /..06 note: Exaine-3D calculates using the elationshis k. can also be calculated using the elationshi v, whee v is the voluetic closue and is the fa hydostatic stess. Exaine-3D also calculates this elationshi and fo a sheical cavity 0.09 MJ. Coaison of Enegy Coonents fo a Biaxial Loaded Cylinde (in MJoules) cylinde with, a MPa ν0.5 E0000 MPa length0.0 K0 K K Exaine- Analytical Exaine- Analytical Exaine- Analytical 3D 3D 3D U Uc k / /k note: Exaine-3D calculates using the elationshis k. C-

EN40: Dynamics and Vibrations. Midterm Examination Tuesday March

EN40: Dynamics and Vibrations. Midterm Examination Tuesday March EN4: Dynaics and Vibations Midte Exaination Tuesday Mach 8 16 School of Engineeing Bown Univesity NME: Geneal Instuctions No collaboation of any kind is peitted on this exaination. You ay bing double sided