3 ENERGY CALCULATION ENERGY CALCULATION Introduction

Transcription

1 ENERGY CALCULATION ENERGY CALCULATION 3.1 Introducton Energy changes determned n UDEC are performed for the ntact rock, the jonts and for the work done on boundares. The energy terms calculated here use the same general nomenclature as those used by Salamon (1984). Snce UDEC uses an ncremental soluton procedure, the equatons of moton are solved at each mass pont n the body at every tmestep. The ncremental change n energy components s determned at each tmestep as the system attempts to come to equlbrum. UDEC also keeps a runnng sum of each component. The UDEC model tself has fnte boundares that must be taken nto account n the energy analyss. The energy analyss requres that the outer boundary of the model be free to deform so that the work done at the boundary can be determned. Ths requres that ether a stress (force) or boundaryelement boundary be used.

2 3-2 Theory and Background 3.2 Energy Balance The total energy balance can be expressed n terms of the released energy (W r ), whch s the dfference between the work done at the boundary of the model and the total stored and dsspated stran energes: W r = W (U c + U b + W j + W p ) (3.1) where W r W U c U b W j W p = released energy; = total boundary loadng work suppled to the system; = total stored stran energy n materal; = total change n potental energy of the system; = total dsspated energy n jont shear; and = total dsspated work n plastc deformaton of ntact rock. A second calculaton of released energy can be made based on the knetc energy, mass dampng work, the work performed at vscous boundares, and the stran energy n excavated materal: where U k W k W v U m W r = U k + W k + W v + U m (3.2) = current value of knetc energy n the system; = total work dsspated by mass dampng; = work done by vscous (non-reflectng) boundares; and = total stran energy n excavated materal. Ths second form of the released energy s partcularly useful for dynamc problems snce the released knetc energy s easly calculated. The defntons of these ndvdual terms, and the method of ther calculaton, are gven n the followng secton.

3 ENERGY CALCULATION Calculaton of Indvdual Energy Components Total Boundary Loadng Work (W) The work done on the external boundares of the UDEC block structure s calculated from the boundary grdpont forces and dsplacements. Ether a stress (force) or boundary-element outer boundary must be used. For dynamc problems, the stress boundary s replaced by non-reflectng (vscous) boundares after ntal stress equlbrum s acheved. The work done at a grdpont s calculated:. where W g = work done at grdpont on the outer boundary; F x = x-orented force at the grdpont; u x = x-orented dsplacement at the grdpont; F y = y-orented force at the grdpont; and = y-orented dsplacement at the grdpont. u y W g = F x u x + F y u y (3.3) The total boundary work, W l, done durng tmestep l, s the sum of the work done for all grdponts on the boundary: W l = ngp =1 where ngp = the number of grdponts on the boundary. The boundary work s summed for all tmesteps: W g (3.4) W = nt l=1 W l (3.5) where nt = the number of tmesteps. W wll approach a constant value as the system approaches equlbrum. The rate of boundary loadng work, W, s smply the boundary work done per tmestep: W = W l+1 W l ( t) (3.6)

4 3-4 Theory and Background where t s the tmestep. Hstores of the total work, W, and ncremental work, W, are kept durng a smulaton Potental Energy (U b ) The change n gravtatonal potental energy s calculated from the grdpont gravtatonal forces and the dsplacements of the grdponts. The total potental energy s summed for ncremental dsplacements of all grdponts: U g = m [g x u x + g y u y ] (3.7) where U g = the potental energy of grdpont ; m = the mass of grdpont ; u x, u y = dsplacement components of grdpont ; and g x, g y = acceleratons n the x- and y-drectons (usually gravty). The total potental energy s found by summng the energy for all grdponts,, at a gven tmestep, k: The total U b s kept for all tmesteps: U b = ngp =1 U g (3.8) U b = nt j=1 U bj (3.9) where nt s the number of tmesteps; U b wll approach a constant value as the model approaches equlbrum.

5 ENERGY CALCULATION Knetc Energy (U k ) The knetc energy s determned for each grdpont at each tmestep, and s summed for all grdponts at that tmestep. A runnng total of the knetc energy s not kept; so, as the system approaches equlbrum, the knetc energy wll approach zero. The knetc energy s gven by U k = ngp where U k = knetc energy of all grdponts n a gven tmestep; m = mass of grdpont ; and u = velocty at grdpont. =1 1 2 m ( u ) 2 (3.10) The ncremental knetc energy s calculated so the user can examne the rate of change between tmesteps: U knc = U k n U kn 1 t (3.11) where U kn = knetc energy at tmestep n; U kn 1 = knetc energy at tmestep n-1; and t = tmestep. Knetc energy s also calculated for rgd blocks n a smlar fashon, based on the block mass and velocty Damped Energy (W k ) The mass-dampng work s the summaton of all energy absorbed by ether local dampng or adaptve global dampng (auto dampng), and s ntended for use prmarly wth statc analyss. For dynamc analyses, the work done on vscous boundares wll be much larger than the damped energy, and wll largely control the calculated value of the total released energy. The damped energy can most easly be seen by examnng a smplfed verson of the equaton of moton, u t = F m α u (3.12) where u = velocty of a grdpont of mass, m

6 3-6 Theory and Background F = the force sum at the grdpont; and α = dampng coeffcent, = 2πf γ, where γ = fracton of crtcal dampng; and f = natural frequency of the system (cps); the crcular frequency n radans per second s ω = 2πf The dampng force s gven by and the rate of damped energy change at a grdpont s F d = mα u (3.13) The damped energy over a tmestep at a grdpont, j, s Ẇ d = F d u = mα u 2 (3.14) W dj = αm u 2 dt = 2α tu k (3.15) where W dj s the energy damped at grdpont j, and U k s the knetc energy of the grdpont. Therefore, the damped and knetc energy components are related by the dampng coeffcent. The total mass dampng work s the sum of for all grdponts and tmesteps: where W d = total energy damped; nt = number of tmesteps; and ngp = number of grdponts. W d = nt =1 ngp The ncremental damped energy s also calculated as j=1 W dj (3.16) W dnc = 2α U k (3.17)

7 ENERGY CALCULATION Stran Energy Stored n the Rock Mass (U c ) The total stran energy stored n the rock mass s composed of two parts: the energy stored n the blocks, and that stored n the jonts. Each s calculated, and the total stored energy, U c, s determned as the sum of these two components Block-Stored Stran Energy (U cb ) The stran energy n the blocks s determned for all fnte dfference zones durng each tmestep. The total stored stran energy s calculated by summng the values for all blocks. The ncremental stran energy n each zone for a tmestep s gven by U cz = A 2 [ (σ11 + σ 11 )e (σ 12 + σ 12 )e 12 + (σ 22 + σ 22 )e ] 22 (3.18) where σ 11, σ 22, σ 12 = current zone stresses; σ 11, σ 22, σ 12 e 11, e 22, e 12 A = zone stresses from the prevous tmestep; = ncremental strans over the current tmestep; and = area of zone. The stran energy n a block s the sum of all zones wthn the block, U cb = U cz (3.19) The total stran energy n a gven tmestep s the sum for all blocks. A runnng total s kept for all tmesteps, so that the value of U cb wll approach a constant value wth tme as the system approaches equlbrum. The ncremental block stran energy s also calculated by U cbnc = U c b t (3.20) An excepton to the above calculaton method s used n the case of ntal equlbrum of the model pror to any excavaton. Normally, the n-stu stresses are set up n the model by freezng the stresses n each zone usng the INSITU stress command. By also applyng boundary stresses (whch are n equlbrum wth the nternal stresses) at the same tme, the model wll be at ntal stress equlbrum and wll not requre any tmesteppng. However, ths method of consoldatng the body under ntal stresses produces no stran n the body, and no apparent stran energy. Therefore, the alternate form of the stran-energy densty equaton s used to defne the ntal stran energy n the system:

8 3-8 Theory and Background U cz = 1 ( σ 2 2E x + σy 2 2ν (σ y σ z + σ z σ x + σ x σ y ) + 2 (1 + ν) σxy Jont Stran Energy (U cj ) ) (3.21) The stran energy stored n the jonts s separated nto four component parts for elastc stran n shear (U js ), compresson (U jc ), tenson (U jt ) and energy dsspated n slp (U jf ). The calculatons of these components are performed dfferently for the two bult-n jont consttutve models (the Coulomb slp model and the contnuously yeldng model). In the Coulomb model, the jont normal and shear stffnesses are lnear, whereas the contnuously yeldng model allows nonlnear stffness. The energy s determned for each contact along all jonts n the model, although, at present, t s not possble to separate energy by jont. Coulomb Slp Model, Lnear Stffness If f n < 0, If f n 0, U jt = 1 2 (f n + f n )u n (3.22) If f s <f s max, U jc = 1 2 (f n + f n )u n (3.23) If f s f s max, U js = 1 2 (f s + f s )u s (3.24) U jf = 1 2 (f s + f s )u s (3.25) where f n, f s f n, f s u n, u s f s max = current normal shear force at a contact, compresson postve; = prevous normal shear forces at a contact; = ncremental normal and shear dsplacements at the contact over the current tmestep; and = shear stress at whch the Coulomb slp condton s met (f s max <f n tan φ + C). For the case n whch f s f s max and slp occurs, energy s dsspated n the Coulomb model by frcton (heat). (See Secton )

9 ENERGY CALCULATION 3-9 Contnuously Yeldng Model Nonlnear Stffness Allowed U jt = 0 no tenson n CY model (3.26) U jc = 1 2 (f n + f n )u n (3.27) U js = F 2 (f s + f s )u s (3.28) U jf = 1 F 2 (f s + f s )u s (3.29) where f n, f s f n, f s u n, u s F = normal shear forces at a contact; = prevous contact stresses; = ncremental, normal shear dsplacements over the current tmestep; and = the yeldng factor for the contnuously yeldng model as descrbed n Secton 2 n Consttutve Models. The total energy absorbed and dsspated by the jonts, U cj, s gven by the sum of all components: U cj = U js + U jc + U jt (3.30) Stran Energy Content of Excavated Materal (U m ) When rock s excavated, the stran energy that was stored n the excavated volume s released. UDEC allows excavaton of the rock blocks that form the openng through use of the DELETE command or assgnment of the null consttutve model (va ZONE model null or CHANGE cons 0). The null model does not delete the blocks, but forces n null blocks are prevented from beng passed to grdponts of adjonng blocks. The null zones can collapse due to deformaton of the openng, and can later be changed to a backfll materal. When a block s deleted or gven a null consttutve model, the energy sums are updated. The total stran energy n the excavated materal conssts of the stran energy n the blocks and the jonts. The stran energy n the blocks s calculated n the same manner as the stran energy descrbed prevously: U mb = nb =1 nz j=1 A 2E [ σ σ σ 2 3 2ν (σ 1 σ 2 + σ 1 σ 3 + σ 2 σ 3 ) ] j (3.31)

10 3-10 Theory and Background where U mb = block stran energy; A = area of zone; E = Young s modulus of the rock mass; ν = Posson s rato; σ 1, σ 2, σ 3 = prncpal stresses n zone centrod; nb = number of blocks n excavated materal; and nz = number of zones n the block. The total stran energy n the jonts boundng the excavated block(s) (U cj ) s gven as follows. Coulomb Jonts (Constant Stffness) nc U cj = + 1 [ f 2 n + f 2 ] s 2 k n =1 k s (3.32) where U cj = stran energy stored n the jonts; f n, f s = normal shear force n jonts; k n, k s = normal shear stffness of jonts; and nc = number of contacts. Contnuously Yeldng Jonts (Nonlnear Normal Stffness) U cj = nc =1 1 2 [ u en n [a n (1 e n )] 1 en e 1 + f 2 s k s n 1 ] (3.33) where e n u n a n f s k s nc = normal stffness exponent; = normal dsplacement (closure) of jont surfaces; = ntal normal stffness of jont; = shear force at contact; = shear stffness of contact; and = number of contacts. When a block s excavated, ts energy s removed from the total stran energy, U c, and added to the total for the excavated materal, U mb. The ntal or n-stu stran energy state for the rock mass s determned by usng the standard stran-energy densty functon, where the prncpal stresses

11 ENERGY CALCULATION 3-11 are equal to the n-stu stresses. Ths s added to the boundary loadng work (W) for the ntal equlbrum, pre-mnng condton. The fnal values for stored stran energes are determned: U c = U c U mj U mb (3.34) U m = U m + U mj + U mb (3.35) U cb = U cb U mb (3.36) U cj = U cj U mj (3.37) where a refers to the values from prevous excavaton steps Frcton Work Done on Jonts (W j ) Energy s dsspated through frctonal heatng of jonts. Ths work done s exchanged from the elastc stran energy, and s rrecoverable. UDEC keeps track of the frctonal energy separately from the elastc (stored) jont energy terms (U jt, U jc and U js ). The frcton loss s calculated for lnear and nonlnear normal stffness as follows. Coulomb Jonts, Lnear Normal Stffness If f s f s max, U jf = nc =1 where U jf = frctonal energy at the contact durng a tmestep; 1 2 (f s + f s )u s (3.38) f s f s u s nc = current shear force at a contact; = prevous shear force at a contact; = ncrement n shear dsplacement; and = number of contacts. Contnuously Yeldng Jont (Nonlnear Normal Stffness) U jf = nc =1 1 F 2 (f s + f s )u s (3.39) where F = the yeld factor for the contnuously yeldng jont, and nc = number of contacts.

12 3-12 Theory and Background The total dsspated energy s kept by summng over all the tmesteps durng an excavaton step, W j = nt where W j = total dsspated frcton energy, and nt = number of tmesteps. =1 U jf (3.40) Vscous Boundary Work (W v ) Vscous boundares are used to dampen reflectons of ncdent stress waves. The energy damped from these stress waves s calculated from the boundary forces and deflectons at the boundary grdponts: W gj = f x u x + f y u y (3.41) where W gj = boundary work at a grdpont, j; f x, f y = boundary forces; and u x, u y = boundary dsplacements. The vscous energy for a tmestep s gven by W v = where nbp = number of boundary grdponts. nbp j=1 W gj (3.42) The total vscous work s summed for all tmesteps. The ncremental vscous boundary work s calculated by W v = W v t (3.43) where t = the tmestep.

13 ENERGY CALCULATION Energy Dsspaton n Blocks through Plastc Work (W p ) Several plastcty models that can descrbe the deformablty of the blocks are avalable n UDEC. Energy s dsspated through plastc work as the zones undergo rreversble deformaton. The stran n any zone can be dvded nto an elastc and a plastc part. The elastc stran can be determned, followed by the elastc stran energy as determned prevously. The plastc work s found by takng the dfference between the total stran energy and the elastc energy component. The elastc stran energy s gven by W e = A 2E [ σ σ σ 2 3 2ν (σ 1 σ 2 + σ 1 σ 3 + σ 2 σ 3 ) ] (3.44) The change n elastc stran energy between excavaton steps s gven by W e = W e W e (3.45) where W e s the current elastc stran energy, and W e s the prevous elastc stran energy. The total energy change can be found from the total stran and stress, W T = A 2 [ (σ11 + σ 11 )e ν (σ 12 + σ 12 )e 12 + (σ σ 22 )e 22 (3.46) The total plastc work dsspated durng an excavaton step s the dfference between the total and elastc energy change at any tmestep, W p = W T W e (3.47) and the total dsspated s smply the sum of W p for all blocks at each tmestep Energy Dsspated n Backfll Compresson UDEC keeps track of those blocks that have been excavated usng the null model method and replaced by backfll. In ths case, elastc stran energy may be stored n the fll, and energy may be dsspated va plastc deformaton. Ths would normally be the case for sandflls. Energy values are calculated as ether stored or dsspated for each zone (as prevously descrbed), and are added to the plastc work term.

14 3-14 Theory and Background Volume of Excavated Materal (V m ) When a block s deleted or assgned a null consttutve model type, the volume s added to the value V m. The area s calculated by usng the mass and densty of the zones whch comprse t: A z = mass/densty where A z = area of a zone. The volume of the deleted blocks s then equal to V m = A z (3.48) for all zones n the excavated blocks.

15 ENERGY CALCULATION Method of Operaton n UDEC The energy calculatons n UDEC are ntated usng the SET energy on command. From ths pont on, the energes descrbed n the prevous sectons are calculated n an ncremental fashon at each tmestep from the stress, force, dsplacement and stran changes. All energy values are summed from ths pont, wth the excepton of the knetc energy, U k, whch s kept as an ncremental value. Therefore, the magntude of the energes upon prntout wll be the sum for the problem snce tmesteppng began, and wll nclude that computed for all excavaton steps. Several ponts are noted: 1. The U m component of energy s calculated mmedately as blocks are deleted. 2. The mass-scalng opton n UDEC must be dsabled (by specfyng MSCALE off), as t artfcally adjusts masses of grdponts to speed convergence. 3. If boundary element couplng s used to represent the outer boundary, stress unts must be n terms of MPa. Two addtonal commands have been added to UDEC for plottng and prntng these energy components. The command HISTORY energy wll keep tme hstores of all energy components, and PRINT energy wll provde a summary lstng of all energy components n table form. Secton 3.5 presents an example that llustrates the energy montorng calculatons n UDEC.

16 3-16 Theory and Background 3.5 Energy Calculatons: Excavaton of a Crcular Hole n an Infnte Elastc Medum Salamon (1984) solved the problem of the stored and released energy n creaton of a crcular tunnel n an nfnte, elastc rock mass subjected to hydrostatc stresses. Here, UDEC s used to generate energy components for ths example, and the results are compared to the analytcal soluton. Salamon assumes an nfnte rock mass, such that the tractons and dsplacements nduced by the excavaton of the hole become vanshngly small as the dstance from the openng becomes large. However, the followng must be consdered n ths problem. (1) As the boundary approaches nfnty, the nduced tractons and dsplacements approach zero, but the area of the surface over whch these tractons act approaches nfnty. (2) For a fnte boundary, the tractons and dsplacements are not zero, and ther dot product (work) s a scalar. Therefore, the work done by external forces cannot be canceled lke the tractons. Snce the UDEC model s of fnte sze, the nduced tractons and dsplacements are not zero, and the outer boundary of the model must be taken nto account n determnng stored stran energy and boundary work components. The followng secton revews the Salamon soluton and the changes necessary for determnaton of these two components Dervaton of Analytcal Soluton to Cylndrcal Tunnel n an Infnte Medum The dervaton of the energy equatons s gven for the analytcal soluton to a cylndrcal tunnel n an nfnte medum. Consder the 2D secton shown n Fgure 3.1, whch has a Stage I excavated radus of a and Stage II radus of c. The boundary s located a dstance R from the center of the tunnel. The tunnel s assumed to be suffcently long that there are no end effects. Solvng the problem requres a few defntons. Frst, the volume of rock to be mned s V m = π(c 2 a 2 ) per unt length of tunnel. The stress dstrbuton at any pont around the tunnel s gven by the radal stress, σ r (p), and tangental stress, σ (p) t, as follows (Jaeger and Cook 1979). σ (p) r σ (p) t ( ) = p 1 a2 r 2 ( ) = p 1 + a2 r 2 (3.49)

17 ENERGY CALCULATION 3-17 R S S m S o c a V o V m V Fgure 3.1 Cross secton through an nfnte length tunnel n whch r s the dstance from the tunnel center to the pont of nterest; p s the vrgn hydrostatc compresson stress; and a s the Stage I excavated radus. Assumng plane stran condtons, the strans are related to stresses by ɛ (p) r ɛ (p) T = p [ (1 2ν) a2 2G r 2 = p [ (1 2ν) + a2 2G r 2 ] ] (3.50) The nduced stresses that result from gong from Stage I to Stage II equlbrum are gven by subtractng stresses n Eq. (3.49) at excavated radus c from those at a. These stresses are gven by

18 3-18 Theory and Background σ () r ( c 2 a 2 ) = p σ () t = p r 2 ( c 2 a 2 r 2 ) (3.51) The nduced strans are determned n the same manner, and are gven by ɛ () r = p ( c 2 a 2 2G r 2 ɛ t () = p ( c 2 a 2 2G r 2 ) ) (3.52) Dsplacements at any pont around the tunnel durng Stage I are gven by Jaeger and Cook (1979): u (p) r u (p) t = 0 = p ( a 2 2G R ) (3.53) Smlar expressons are used for Stage II dsplacements, except that a 2 s replaced by c 2. Gven the above defntons, t s now possble to solve for the energy terms. The equaton relatng the change n potental energy to the work done by the body n gong from Stage I to Stage II s gven by (after Salamon 1984, Eq. (18a)) W + U m = U c + W r (3.54) where W = work done by external and body forces when actng through the nduced dsplacements; U m = stored stran energy n the mned rock volume V m at Stage I; U c = change n stored stran energy n the unmned rock volume V ; W r = released energy, = U m + W k ; and W k = knetc energy dsspated by dampng n the unmned rock and supports.

19 ENERGY CALCULATION 3-19 The stored stran energy n the volume of rock mned, U m, at Stage I can be defned n terms of a stran energy densty functon, φ. The stran energy densty functon s defned as φ = 1 2 σ j ɛ j (3.55) whch, accordng to Jaeger and Cook (1979), can be expressed n terms of the stress tensor as φ = 1 2E [ σ 2x + σ 2y + σ 2z 2ν (σ yσ z + σ z σ x + σ x σ y ) + 2(1 + ν)(τ 2xy + τ 2zx + τ 2zy ) ] (3.56) For the case of two-dmensonal plane stran, two of the three shear stress components are zero (τ xz = τ yz = 0), and the stran energy densty functon n Eq. (3.56) reduces to φ = 1 2E [ σ 2x + σ 2y + σ 2z 2ν (σ yσ z + σ z σ x + σ x σ y ) + 2 (1 + ν)(τ 2xy ) ] (3.57) The stored stran energy n the mned volume of rock, U m, at Stage I s the ntegral of the energy densty functon over the entre volume, V m ; that s (Salamon 1984, Eq. (14)), U m = φdv (3.58) V m By substtutng Eqs. (3.49) and (3.50) nto Eq. (3.58), the stored stran energy n the rock to be mned s 2π c [ U m = φ I dv = r V m 0 0 [ = p2 r 2 a4 (1 2ν) θ 2G 2 2r 2 θ [ ] = p2 V m 2G (1 2ν) + a2 c 2 σ (p) r ɛ (p) r ] c a 2π 0 + σ (p) t ɛ (p) t ] dr dθ (3.59) Ths term s the same as gven by Salamon (1984), and s ndependent of the boundary radus. The work done by external and body forces, W, s often referred to as the gravtatonal or potental energy change. Changes n potental energy s the sum of external work, Wext, plus the

20 3-20 Theory and Background work done by body forces, W body. The work done by external forces, Wext, can be expressed n terms of surface tractons, T. The work done by the body forces, W body, s expressed n terms of force per unt volume, X. Ths work s gven as (Salamon 1984, Eq. (7)) n whch T () u () W = Wext + W body (3.60) ( = + 1 ) 2 T () u () ds + X u () dv = T T (p) ; = u u (p) ; S T (p) T (p) = boundary tractons at Stage I; = boundary tractons at Stage II; T T () = nduced tractons from Stage I to II; u (p) = boundary dsplacements at Stage I; = boundary dsplacements at Stage II; u u () X = nduced dsplacements from Stage I to II; = body forces per unt volume n volume V at Stage II; and X (p) = body forces per unt volume n volume V at Stage I. Gven the denttes n Eq. (3.60) for T and u, the work done by external forces can be rewrtten as V Wext = 1 2 S ( )( ) T + T (p) u u (p) ds (3.61) The change n potental energy, W, s a functon only of the work done by the external forces, Wext, as t has been assumed that there are no body forces; hence, W body s zero. Substtutng relatons for tractons, as gven by stresses n Eq. (3.49), and dsplacements n Eq. (3.53), the change n potental energy s

21 ENERGY CALCULATION 3-21 ( W = Wext = S 2π [ ( = p 0 = p2 2G ( T (p) 1 a2 R 2 2 c2 + a 2 R 2 ) T () u () ) + 12 ( c 2 ( p) a 2 ) V m ds (3.62) )] ( p c 2 a 2 ) Rdθ 2G R R 2 Ths equaton s dfferent than Salamon s, but t s derved drectly from the surface ntegral rather than by makng substtutons relyng on the assumpton that U (p) = U m. The change n stored stran energy n the unmned rock, U c, at Stage II s the dfference n ntegrated energy densty over the volume, V, at the equlbrum states of Stage I and Stage II. It s wrtten as U c = V [ ] φ II φ I dv (3.63) The change n stored stran energy n the unmned rock, U c, s obtaned from Eq. (3.63). The stored stran energy at Stage II s gven by U = V = p2 2G = p2 2G 2π R [ ( 1 φ II dv = r σr p 0 c 2 [ r 2 ] c4 (1 2ν) θ 2 2r 2 θ R c [ ] (1 2ν) + c2 R 2 (R 2 c 2 )π ɛ p r + σ p t 2π 0 ɛ p t )] dr dθ (3.64) and the stored stran energy at Stage I s gven by U (pp) = V = p2 2G 2π R [ ( 1 φ I dv = r σr p 0 c 2 [ r 2 ] a4 (1 2ν) θ 2 2r 2 θ R c [ (1 + 2ν) + a4 c 2 R 2 = p2 2G (R2 c 2 )π ɛ p r + σ p t 2π 0 ] ɛ p t )] dr dθ (3.65)

22 3-22 Theory and Background Note that Salamon does not work out U or U (pp), but the equaton for U (pp) s the same as Salamon s (1984) Eq. (II.5). Subtractng Eq. (3.65) from Eq. (3.64) gves ( U c = p2 1 2G (c2 + a 2 ) c 2 1 R 2 ) V m (3.66) Ths expresson does not agree wth Salamon s (1984) Eq. (II.12). To check ths result, U c can be calculated as gven by Salamon (1984): The nduced stored stran energy s gven by U () = V φ () dv = = p2 2G (c2 a 2 ) U c = U () + 2U (p) (3.67) 2π 0 R c ( 1 c 2 1 R 2 [ ( 1 r 2 ) V m σ r ɛ r + σ t ɛ t )] dr dθ (3.68) The stored stran energy nduced by Stage I forces on the dsplacements that occur n Stage II s gven by U (p) = V = p2 2G a2 φ (p) dv = 2π 0 ( 1 c 2 1 R 2 ) R c V m r [ ( 1 σr p 2 ɛ r + σ p t ɛ t )] dr dθ (3.69) U c s obtaned by substtutng Eqs. (3.69) and (3.70) nto Eq. (3.68). The released energy, W r,asgvenbyeq. (3.56), gves The knetc energy, W k, wll be W r = W + U m U c (3.70) = p2 G (1 ν) V m ( W k = W r U m = p2 1 a2 2G c 2 ) V m (3.71)

23 ENERGY CALCULATION UDEC Energy Calculaton The analytcal soluton descrbed n Secton 3.4 s compared to the UDEC model results for the case of the ntal excavaton (Stage I) and enlargement (Stage II) of a crcular tunnel. The radus of the ntal excavaton s 1 m, and the enlargement produces a tunnel of 2 m radus. The outer radus of the UDEC model s 10 m. A hydrostatc compressve stress of 100 MPa exsts pror to excavaton. The elastc materal has a shear modulus of GPa and Posson s rato of 0.2. Fgure 3.2 shows the ntal UDEC block geometry, consstng of a number of concentrc crcular blocks, and the zonng wthn the blocks. The UDEC data fle to calculate the two excavaton stages and montor energy components s lsted n Example 3.1. Note that adaptve global dampng (DAMPING auto) s used for ths calculaton. As dscussed n Note 12 n Secton 3.9 n the User s Gude, ths dampng s more computatonally effcent than local dampng for an elastc analyss. Smlar results for the energy components wll also be calculated f local dampng s used. JOB TITLE : ENERGY CALCULATION FOR CIRCULAR TUNNEL (*10^1) UDEC (Verson 5.00) LEGEND Dec :27:48 cycle tme 1.442E-02 sec zones n fdef blocks block plot Itasca Consultng Group, Inc. Mnneapols, Mnnesota USA (*10^1) Fgure 3.2 UDEC ntal geometry

24 3-24 Theory and Background Example 3.1 Energy calculatons for excavaton of a crcular hole new ;fle energy.dat ttle ENERGY CALCULATION FOR CIRCULAR TUNNEL round 0.01 ;use auto dampng and turn off mass scalng damp auto mscale off ;set geometry (before excavaton) block crcular crack crack tunnel tunnel tunnel tunnel ;create zonng (4 dfferent szes) gen quad=0.4 range ann gen quad=0.8 range ann gen quad=1.6 range ann gen quad=3 range ann gen edge.4 ;set stresses bound stress nstu stress szz -40 ;materal propertes prop m=1 d=.002 k=38.9e3 g=29.17e3 prop jm=1 jfrc=40.0 jcoh=10e8 jtens=10e7 jkn=6e8 jks=6e8 ;hstores (dsplacements and stresses) at rad=5,10,20 hst solve type 1 hst unbal hst n=20 xds 1 0 xds 2 0 hst n=20 sxx 1 0 sxx 2 0 sxx 3 0 hst n=20 syy 1 0 syy 2 0 syy 3 0 solve rat 1e-5 save energy0.sav set energy on hst energy be gen be mat 1 be fx be stff ; excavaton step 1

25 ENERGY CALCULATION 3-25 del range ann 0,0 0 1 solve rat 1e-5 set log on prnt ener set log off save energy1.sav ; excavaton step 2 del range ann 0,0 1,2 solve rat 1e-5 set log on prnt ener set log off save energy2.sav Fgure 3.3 gves an example of a hstory plot for the damped and knetc energy components for the frst excavaton stage. In ths fgure, the knetc energy term s not summed over tme, but decays to zero as the model comes to equlbrum. At the same tme, damped energy, whch s summed, approaches a constant value. JOB TITLE : ENERGY CALCULATION FOR CIRCULAR TUNNEL UDEC (Verson 5.00) 1.80 LEGEND 30-Dec :27:48 cycle tme 1.442E-02 sec hstory plot Y-axs: 11 - current knetc ener 21 - mass dampng work X-axs: Number of cycles Itasca Consultng Group, Inc. Mnneapols, Mnnesota USA (e+004) Fgure 3.3 Plot of the hstory of the damped, W k (hst 21), and ncremental knetc energy, U k (hst 11), components. (The knetc energy drops to zero as the model comes to equlbrum, whereas the damped (summed knetc) energy approaches a constant value.)

26 3-26 Theory and Background Table 3.1 shows a typcal result of the PRINT energy command after the frst stage of excavaton. All of the current energy components and ther rates of change are prnted. Table 3.1 Totals for energy stored and dsspated n system current knetc energy (Uk) = 2.588E 06 total block stran energy (Ucb) = 2.051E 01 total fll stran energy (Ucf) = 0.000E+00 total jont stran energy (Ucj) = 1.099E 04 total materal stran energy (Uc=Ucb+Ucf+Ucj) = 2.050E 01 total block energy excavated (Umb) = 3.210E 01 total jont energy excavated (Umj) = 8.700E 05 total stran energy excavated (Um=Umb+Umj) = 3.211E 01 total block volume excavated (Vm) = 3.121E+00 total change n potental energy (Ub) = 0.000E+00 total mass dampng work (Wk) = 5.216E 01 total vscous boundary work (Wv) = 0.000E+00 total frcton work (Wj) = 0.000E+00 total plastc stran work (Wp) = 0.000E+00 total boundary loadng work (W) = 1.048E+00 total energy released (Wr=W-Uc-Ub-Wj-Wp) = 8.431E 01 total energy released (Wr=Uk+Wk+Wv+Um) = 8.427E 01 breakdown of energy stored n jonts (Ucj) total energy stored n tenson (Ujt) = 0.000E+00 total energy stored n compresson (Ujc) = 1.100E 04 total energy stored n shear (Ujs) = 6.468E 08

27 ENERGY CALCULATION Comparson to Salamon Soluton Table 3.2 summarzes the results from the analytcal soluton and UDEC. The comparson s good, generally wthn 3%. Table 3.2 Summary of results from the analytcal soluton and UDEC ENERGY Excavaton Stage I Excavaton Stage II COMPONENT analytc udec error (%) analytc udec error (%) U c U m V m W k W W R W R NOTES 1. UDEC energy components at stage II are obtaned be subtractng components at stage I from the total components reported wth PRINT energy at stage II. 2. U c n UDEC s dfferent from that gven by Salamon (1984). The relaton s U (UDEC ) c = U (Salamon) c U m 3. W R1 = W U (UDEC ) c 4. W R2 = W k + U m

28 3-28 Theory and Background 3.6 Reference Jaeger, J. C., and N. G. W. Cook. Fundamentals of Rock Mechancs, 3rd Ed. London: Chapman and Hall (1979). Salamon, M. D. G. Energy Consderatons n Rock Mechancs: Fundamental Results, J. S. Afr. Inst. Mn. Metall., 84(8), (1984).