A Family of Constitutive Models Implemented in PLAXIS to Simulate Cemented Mine Backfill
M.G. Sottile, N.A. Labanda, I. Garcia Mendive, O. Ledesma, A. O. Sfriso
AA F amily of Constitutive
Models
Implemented in P L A X IS to Simulate
Cemented
Mine
Bac k fill Mauro SOTTILE a, b , Nicolás LABANDA a, b ,1 , Iña k i GARCÍA MENDIVE a , Osvaldo LEDESMA a and Alejo O. SFRISO a, b a SRK Consulting. b University of Buenos Aires.
Abstract. A family of constitutive models for mine cemented backfill is presented. Four formulas for the density - and pressure - dependency of elastic moduli, five formulas for the density - and pressure - dependency of friction angle and four formulas for the age - dependency of the elastic moduli and effective cohesion are incorporated into an isotropic hypoelasticity w ith Mohr - Coulomb perfect plasticity frame w ork and implemented in PL AX IS as a user - defined material model. This family includes the standard
Mohr - Coulomb, B olton, Leps, B arton and Hoek -B ro w n models as trivial cases w hen both nonlinear elasticity and age - dependency are s w itched off. In this paper, the formulation of the models is introduced, the basis of the numerical implementation is outlined, and a case history of the application to the cemented backfill of a sublevel stoping mine is presented as an example. Keywords.
Constitutive modelling, cemented backfill, underground mining. . I n troduction Cemented roc k fill and cemented paste have been widely used in the mining industry for bac k filling underground openings to provide wall stability while mining adjacent stopes. Modelling mining processes incorporating cemented bac k fill requires a reliable estimate of the changes in the strength and stiffness of the material with basis of the numerical implementation is outlined , and a case history of the application to the cemented bac k fill of a sublevel stoping mine is presented as an example. For brevity , formulas are introduced with little explanation; the derivations and support behind several of these formulas can be found in the relevant references. . M odel formulation State variables
State variables are age (cid:2) and void ratio (cid:3) . Their update formulas are Nicolás A . Labanda, SR K Consulting A rgentina, Chile B uenos A ires, A rgentina; E - mail : nlabanda @ fi.uba.ar, nlabanda @ srk.com.ar. Geotechnical Engineering in the XXI Century: Lessons learned and future challengesN.P. López-Acosta et al. (Eds.)© 2019 The authors and IOS Press.This article is published online with Open Access by IOS Press and distributed under the termsof the Creative Commons Attribution Non-Commercial License 4.0 (CC BY-NC 4.0).doi:10.3233/STAL190087 (cid:2)(cid:3)(cid:4) (cid:4) (cid:2) (cid:5) (cid:5) (cid:6) (cid:2) (1a) (cid:3) (cid:2)(cid:3)(cid:4) (cid:4) (cid:3) (cid:5) (cid:5) (cid:4) (cid:7) (cid:5) (cid:3) (cid:5) (cid:5) (cid:6) (cid:6) (cid:6) (1b) where (cid:2) (cid:5) and (cid:3) (cid:5) are the age and void ratio at the beginning of the step , (cid:6) (cid:2) is the time increment and (cid:6) (cid:6) (cid:6) is the increment of volumetric strain. Elasticity
Standard isotropic hypoelasticity is employed. The elastic operator (cid:8) is of the form (cid:8) (cid:4) (cid:9) (cid:8) (cid:9) (cid:7) (cid:5) (cid:10) (cid:7) (cid:10) (cid:9) (cid:10) (cid:7) ⊗ (cid:7) (cid:5) (cid:11) (cid:7) (cid:11) (2) where (cid:8) is the shear modulus , (cid:10) is Poisson’s ratio , (cid:7) is the 2 nd order unit tensor , ⊗ the tensor product operator , and (cid:11) (cid:7) is the 4 th order deviator projector tensor. Four expressions for (cid:8) were implemented: (cid:8) (cid:4) (cid:8) (cid:8)(cid:9)(cid:10) (cid:12)(cid:13)(cid:14) (cid:11) (cid:10) (cid:15)(cid:16) (cid:12) (cid:15) (cid:8)(cid:9)(cid:10) (cid:17) (cid:12) (cid:13)(cid:7)(cid:14) (3a) (cid:8) (cid:4) (cid:8) (cid:8)(cid:9)(cid:10) (cid:4)(cid:14) (cid:9) (cid:10) (cid:3)(cid:5) (cid:13) (cid:12) (cid:4) (cid:7) (cid:5) (cid:3)(cid:5) (cid:12)(cid:13)(cid:14) (cid:11) (cid:10) (cid:15)(cid:16) (cid:12) (cid:15) (cid:8)(cid:9)(cid:10) (cid:17) (cid:12) (cid:13)(cid:9)(cid:14) (3b) (cid:8) (cid:4) (cid:8) (cid:8)(cid:9)(cid:10) (cid:12) (cid:4)(cid:14) (cid:9) (cid:5) (cid:4) (cid:7) (cid:10) (cid:14) (cid:9) (cid:5) (cid:3) (cid:13) (cid:5) (cid:12)(cid:13)(cid:14) (cid:11) (cid:10) (cid:15)(cid:16) (cid:12) (cid:15) (cid:8)(cid:9)(cid:10) (cid:17) (cid:12) (cid:13)(cid:15)(cid:14) (3c) (cid:8) (cid:4) (cid:8) (cid:8)(cid:9)(cid:10) (cid:12) (cid:3) (cid:14) (cid:2) (cid:12)(cid:13)(cid:14) (cid:11) (cid:10) (cid:15)(cid:16) (cid:12) (cid:15) (cid:8)(cid:9)(cid:10) (cid:17) (cid:12) (cid:13)(cid:16)(cid:14) (3d) where (cid:8) (cid:8)(cid:9)(cid:10) (cid:18)(cid:2)(cid:19) and (cid:14) (cid:11) (cid:18)(cid:2)(cid:19) are age-dependent parameters , (cid:14) (cid:9) and (cid:20) are parameters , (cid:15) is mean pressure (tension positive) , (cid:3) is void ratio and (cid:15) (cid:8)(cid:9)(cid:10) is a reference pressure. Shear modulus in tension is equal to (cid:8)(cid:21) (cid:11)(cid:15)(cid:16) . The reader is referred to Ref. [1-4] for further discussion around these formulas. Yield function and plastic potential
A standard Mohr-Coulomb yield surface with tension cut-off is employed. A curved envelope is obtained by employing density- and pressure-dependent formulas for the friction angle (cid:22) , following the procedure described in [5]. Five expressions were implemented , namely constant friction angle , Leps/Barton [6 , , Bolton [8] , Sfriso [9] , and Hoe k -Brown [10]. (cid:22) (cid:4) (cid:22) (cid:16) (4a) (cid:22) (cid:4) (cid:22) (cid:16) (cid:10) (cid:23)(cid:22) (cid:24)(cid:25)(cid:26) (cid:4)(cid:16) (cid:18)(cid:27) (cid:5) (cid:19) (cid:17) (cid:22) (cid:14)(cid:6) (cid:13)(cid:18)(cid:19) (cid:20)(cid:14) (4b) (cid:22) (cid:4) (cid:22) (cid:14)(cid:6) (cid:5) (cid:23)(cid:22) (cid:13)(cid:28) (cid:8) (cid:13)(cid:29) (cid:10) (cid:24)(cid:30)(cid:31) (cid:10)(cid:7)(cid:21)(cid:21) (cid:15) (cid:12) (cid:15) (cid:8)(cid:9)(cid:10) !(cid:16) (cid:10) (cid:7) (cid:16) (cid:22) (cid:22) (cid:12)(cid:17)(cid:18) (cid:22) (cid:22) (cid:12)(cid:17)(cid:18) (cid:13)(cid:23)(cid:14) (4c) M. Sottile et al. / A Family of Constitutive Models Implemented in PLAXIS (cid:4) (cid:22) (cid:14)(cid:6) (cid:5) (cid:23)(cid:22) (cid:13)(cid:28) (cid:8) (cid:24)(cid:30)(cid:31)(cid:4) (cid:10) (cid:15) (cid:3) (cid:13) (cid:2) (cid:19) (cid:5) (cid:12) (cid:13)(cid:15) (cid:8) (cid:15) (cid:8)(cid:9)(cid:10) (cid:16)! (cid:10) (cid:7) (cid:16) (cid:22) (cid:22) (cid:12)(cid:17)(cid:18) (cid:13)(cid:24)(cid:14) (4d) (cid:22) (cid:4) " (cid:20)(cid:4) $ (cid:15) (cid:20) %(cid:27) (cid:14) (cid:12) (cid:12)%(cid:20) (cid:13) (cid:27) (cid:14) (cid:5) (cid:15)(cid:18) " (cid:27) (cid:14) (cid:10) (cid:15)(cid:18) (cid:20) (cid:15) (cid:5) (cid:9) (cid:20)%(cid:27) (cid:14) (cid:17)& (cid:17) (cid:17) (cid:22) (cid:14)(cid:6) (cid:13)(cid:25)(cid:14) (4e) where (cid:22) (cid:16) , (cid:23)(cid:22) , (cid:29) , (cid:15) (cid:8) , (cid:20) , " are parameters , (cid:27) (cid:14) (cid:18)(cid:2)(cid:19) is an age-dependent unconfined compression strength , (cid:22) (cid:14)(cid:6) is the constant volume friction angle , (cid:22) (cid:12)(cid:17)(cid:18) is an upper limit of the friction angle , (cid:28) (cid:8) is relative density and (cid:27) (cid:5) is the normal pressure in the sliding plane , computed in turn as a function of (cid:15) . Age-dependent cohesion (cid:14)(cid:18)(cid:2)(cid:19) is employed in the standard Mohr-Coulomb model (Eq. (4a)). Models derived from Eqs. (4b) to (4e) employ (cid:27) (cid:14) (cid:18)(cid:2)(cid:19) instead. The reader is referred to [5] for the derivation of Eq. (4) and details on the implementation of Hoe k -Brown model as a Mohr-Coulomb-type model. A standard Vermeer-deBorst plastic potential is employed. Except for the standard Mohr-Coulomb model which employs a constant value , the dilatancy angle is made dependent on confining pressure and density through the expression [8]. ' (cid:4) (cid:21)(cid:26)(cid:23) (cid:4)(cid:22) (cid:10) (cid:22) (cid:14)(cid:6) (cid:5) (cid:27) ' (cid:12)(cid:17)(cid:18) (5) where ' (cid:12)(cid:17)(cid:18) is a limiting value. Strain-softening can occur if Eqs. (4c) or (4d) are employed in combination with Eq. (5). While strain-softening is realistic and does occur in dilating cemented materials , it produces a range of numerical issues including poor convergence and mesh-dependency in standard FEM models. Age-dependency
Hardening of cement paste produces an increment in strength and stiffness that can be modelled by incorporating age-dependency in the input parameters (see for instance [11-14]). In this family of models , four evolution laws were implemented. ( (cid:4) ( (cid:21) (6a) ( (cid:4) ( (cid:21) (cid:5) (cid:13)( (cid:10) (cid:10) ( (cid:21) (cid:16) (cid:4)(cid:2) (cid:10) (cid:2) (cid:21) (cid:5) (cid:12) (cid:13)(cid:2) (cid:10) (cid:10) (cid:2) (cid:21) (cid:16) (6b) ( (cid:4) (cid:13)( (cid:21) (cid:10) ( (cid:10) (cid:16) (cid:3)((cid:15)(cid:31) (cid:10)(cid:15) (cid:4)(cid:2) (cid:10) (cid:2) (cid:21) (cid:5) (cid:12) (cid:13)(cid:2) (cid:10) (cid:10) (cid:2) (cid:21) (cid:16)! (cid:5) ( (cid:10) (6c) ( (cid:4) ( (cid:21) (cid:5) ( (cid:10) (cid:10) ( (cid:21) (cid:9) ) (cid:7) (cid:5) (cid:2)*(cid:30) (cid:28) +(cid:24)(cid:25)(cid:26)(cid:18) (cid:7)(cid:24) (cid:19) (cid:2) (cid:10) (cid:13)(cid:2) (cid:10) (cid:5) (cid:2) (cid:21) (cid:16) (cid:12)(cid:9) (cid:2) (cid:10) (cid:10) (cid:2) (cid:21) ,- (6d) where (cid:2) stands for the relevant age-dependent parameter , and (cid:2) (cid:5) | (cid:2) (cid:6) are the initial and final values at ages (cid:3) (cid:5) | (cid:3) (cid:6) . The user is able to switch on/off any combination of (cid:4) (cid:7)(cid:8)(cid:6) , (cid:14) (cid:11) , (cid:5) , and/or (cid:6) (cid:9) as age-dependent parameters. M. Sottile et al. / A Family of Constitutive Models Implemented in PLAXIS . N umerical implementation Inte g ration in ti m e The me an v a l u e (cid:2) o f the r elev an t v ar i a ble withi n the time step (cid:3)(cid:4), (cid:4) (cid:6) Δ(cid:4)(cid:8) is co mp u te d by the ex ac t ( ana lyti ca l) i n te gra ti on o f the r elev an t Eq . (6 a ) t o (6 d ) . (cid:3) (cid:4) (cid:5) (cid:3)(cid:6)(cid:7)(cid:8) (cid:2)(cid:3)(cid:4)(cid:2)(cid:2) (cid:9)(cid:7)/Δ(cid:7) (7) This i n te gra ti on is i nd epe nd e n t o f the i n te gra ti on i n st ra i n and is pe r f or me d bef or e the l a tte r , p roduc i ng mi n im a l imp ac t i n the effi c ie nc y o f the nu me r i ca l a l gor ithm . . Inte g ration in strain A con ve n ti ona l, f u lly impli c it i n te gra ti on s c heme i n st ra i n is empl o ye d. The t r i a l st r ess (cid:12)(cid:13) is co mp u te d a ss u mi ng ( non li n e ar ) el a sti c beh a vi or by impli c it i n te gra ti on o f the r elev an t Eqs . (3 a ) t o (3 d ) . O nc e (cid:12)(cid:13) is kno w n , the el a sti c o pe ra t or D (Eq . ( )), she ar m odu l u s (Eq . (3)), f r i c ti on ang le (cid:15) (Eq . (4)), d il a t anc y ang le (cid:16) (Eq . (5)) and the yiel d f unc ti on a t the t r i a l st r ess (cid:17)(cid:6)(cid:12)(cid:13)(cid:8) ar e co mp u te d. If (cid:17)(cid:6)(cid:12)(cid:13)(cid:8) (cid:18) 0 , the st ra i n step is ( non li n e ar ) el a sti c ; the u p da te d st r ess is r e ad ily co mp u te d a s (cid:12) (cid:5)(cid:3)(cid:6) (cid:4) (cid:12)(cid:13) and r ep or te d b ack t o the m a i n p rogra m . If (cid:9)(cid:10)(cid:11)(cid:12)(cid:13) (cid:14) 0 the st ra i n step is el a st o pl a sti c. I n this ca se, the u p da te d st r ess (cid:11) (cid:3)(cid:4)(cid:5) is i n iti a lize d (cid:11) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:16) (cid:11)(cid:12) , whe r e (cid:17)0(cid:18) st and s f or the st ar ti ng v a l u e o f the ite ra t or. A n este d u p da te a l gor ithm is the n empl o ye d a s ou tli n e d i n Fi gur e 1 . Fi r st, the f unc ti on s o f st a te v ar i a bles ar e f ro ze n a t the g ive n st a te . The n , a M o h r -C ou l o mb yiel d s ur f ac e with pe r fe c t pl a sti c ity is empl o ye d f or the st r ess u p da te [15] . O nc e (cid:11) (cid:3)(cid:4)(cid:5) is u p da te d , the r elev an t f unc ti on s o f st a te v ar i a bles ar e u p da te d , and the l oo p is r epe a te d un til con ve rg e nc e, whi c h is u s ua lly ac hieve d i n tw o t o five ite ra ti on s . The r e ad e r is r efe rr e d t o Ref . [5] f or f ur the r d et a ils o f this nu me r i ca l impleme n t a ti on. Figure 1. Fl o w c h a r t o f t he a lg o r it h m f o r t he st re ss upd at e. M. Sottile et al. / A Family of Constitutive Models Implemented in PLAXIS . A pplication to a suble v el stoping mining case Description of the problem
At an underground gold mine in South America the tabular orebody lies approximately 250 m below ground surface. The ore is recovered via sublevel stoping , stopes being 15 m long and 20 m high , and progressively bac k -filled with cemented roc k fill (CRF). A 2D plane strain numerical model was set up in Plaxis to evaluate the stress distribution due to the extraction sequence and to assess the adequacy of the CRF strength and stiffness to control deformations of the country roc k . Geometry and mesh
The model is 500 m wide and 170 m deep , has 6552 15-noded triangular elements with an average size of 3.34 m (Figure 2). The weight of the overlying 200 m of roc k is modelled by a layer of equivalent weight. Figure 2. finite element mesh. Materials
The orebody was modelled using the Hoe k -Brown model available in Plaxis. Both the cemented and uncemented bac k fills were modelled using Eq. (3a) , Eq. (4a) and Eq. (6c) , simplest choice for CRF. Age-independent parameters are presented in Table 1 . See [16] for the description of the parameters of the Hoe k -Brown criterion in Plaxis. Table 1. A ge - independent material parameters. (cid:10) (cid:11) (cid:12) (cid:13) (cid:14) (cid:15) (cid:11) (cid:2) (cid:16) (cid:17) (cid:18) (cid:3) [kN/m ] [ - ] [ - ] [ o ] [ o ] [GPa] [ - ] [ - ] [ - ] [MPa] O rebody - - CRF - - - - - B ackfill - - - - -
The age-dependent parameters of the CRF were fitted against unconfined compression test results , see Figure 3. For non-cemented bac k fill , the same initial parameters were employed , with no age dependency. Parameters are presented in Table 2. M. Sottile et al. / A Family of Constitutive Models Implemented in PLAXIS able 2.
Age-depe n de nt mat er ia l p a r am e t er s . (cid:3) (cid:2)(cid:3)(cid:4)(cid:3) (cid:3) (cid:2)(cid:5)(cid:3)(cid:4) (cid:4) (cid:6)(cid:7)(cid:5)(cid:3)(cid:4)(cid:3) (cid:4) (cid:6)(cid:7)(cid:5)(cid:5)(cid:3)(cid:4) (cid:3) (cid:3)(cid:4)(cid:3) (cid:3) (cid:5)(cid:3)(cid:4) (cid:5) (cid:3)(cid:4)(cid:3) (cid:5) (cid:5)(cid:3)(cid:4) [kP a ] [kP a ] [MP a ] [MP a ] [kP a ] [kP a ] [d a y] [d a y] CRF
35 7
00 10
56 B ac kf i ll - 35 - - - - Figure 3. T im e ev o lu tion o f u ncon f in ed com pre ssi ve st re n g t h. E x per im e nta l re s ul ts an d f its . M odellin g of t h e m inin g se q uence The m a i n m od elli ng st ag es can be s u mm ar ize d a s f o ll o ws: i) i n iti a l st r ess co mp u te d u si ng the (cid:20) (cid:7) p roc e dur e; ii) seq u e n ti a l ex ca v a ti on and filli ng o f the m a i n st o pes with CRF; iii) seq u e n ti a l ex ca v a ti on and filli ng o f the se condar y st o pes with unc eme n te d rock fill . F or the st ag es r el a te d t o the m a i n st o pes (ii), ex ca v a ti on is pe r f or me d by bl a sti ng and is simply m od elle d by d e ac tiv a ti ng the r elev an t c l u ste r ; filli ng is m od elle d by c h ang i ng the m a te r i a l t o c eme n te d b ack fill and r e- ac tiv a ti ng the c l u ste r , a t a ra te o f 0 .
66 m pe r da y . C ur i ng st ag es we r e add e d betwee n the fill o f a m a i n st o pe and the ex ca v a ti on o f the n e ar by se condar y st o pes t o accoun t f or m a te r i a l h ard e n i ng. F or the st ag es r el a te d t o the se condar y st o pes (iii), the bl a sti ng and filli ng seq u e nc es ar e m od elle d i n the s a me w a y . While the unc eme n te d rock fill h a s no ag e- d epe nd e nc y, r e a listi c time i n te r v a ls we r e an yw a y u se d t o a ll o w the CRF t o con ti nu e its h ard e n i ng p roc ess whe n r elev an t . Fi gur e 4 sh o ws the mi n i ng seq u e nc e and the ac tiv a ti on time o f e ac h c l u ste r. E ac h time st a mp is u se d by the m od el t o co mp u te ag es con t ro lli ng the ev o l u ti on o f stiff n ess and st r e ng th o f e ac h i nd ivi dua l c l u ste r with time . Figure 4. C onst ru ction s eque nc e , dep ict ed by t he acti v ation tim e o f t he c lu st er s, in d a y s . Gr a y c lu st er s repre s e nt c e m e nt ed b ac kf i ll; wh it e c lu st er s repre s e nt u nc e m e nt ed r oc kf i ll. M. Sottile et al. / A Family of Constitutive Models Implemented in PLAXIS . . Results A n ex a mple o f the ev o l u ti on o f the f unc ti on s o f st a te v ar i a bles with ag e is sh o w n i n Fi gur e 5, whe r e the she ar stiff n ess is pl o tte d a t da y 196, j u st a fte r the sixth st o pe is b ack fille d. N on - un if or mity o f (cid:21) f or a g ive n c l u ste r is expl a i n e d be cau se (cid:21) is a f unc ti on o f b o th ag e and con fi n i ng p r ess ur e, whi c h is no t un if or m withi n the c l u ste r. Fi gur e 6 sh o ws the ve r ti ca l d ispl ac eme n t (cid:22) (cid:8) a t the e nd o f the mi n i ng seq u e nc e, pl o tte d a t a h or iz on t a l li n e l oca te d j u st a b o ve the u ppe r ro w o f st o pes . It is sh o w n th a t the c e n t ra l CRF pill ar und e rgo es settleme n ts th a t ar e sli g htly l arg e r th an th o se o f the l a te ra l CRF pill ar s ( co l or e d gra y), and no ti c e a bly sm a lle r th an th o se o f the RF pill ar s a t e ac h si d e . This is an expe c te d and r e a listi c r es u lt, si nc e the c e n t ra l pill ar is the fi r st on e t o be co mplete d , and the r ef or e the on e carr yi ng the hi g hest l oad. Fi gur e 7 sh o ws the d ist r ib u ti on o f ve r ti ca l st r ess a t da y 460, i . e . j u st bef or e the ex ca v a ti on and b ack filli ng o f the u ppe r m o st exte rna l st o pes (see Fi gur e 4) . The effe c t o f the mi n i ng seq u e nc e and o f the d iffe r e n t stiff n ess and st r e ng th o f the th r ee CRF pill ar s is e a sily no ti c e a ble . The c e n t ra l CRF pill ar carr ies the hi g hest l oad , the r i g ht pill ar carr ies m or e l oad th an the left pill ar , and the unc eme n te d rock fill is l arg ely st r ess-f r ee . It m u st be emph a size d th a t this r es u lt w a s ac hieve d with ou t c h ang i ng an y m a te r i a l p ara mete r th roug h ou t the m od elli ng p roc ess . Figure 5.
She a r mo dulu s o f t he v a r io u s CRF c lu st er s at d a y , e n d o f b ac kf i ll in g o f sto pe st er s in l i gh t blue a re not ye t min ed. N on -u ni f o r mit y o f G due to b ot h a ge an d pre ss ure depe n de nc y. Figure 6. T ota l ver tica l d is pl ac e m e nts at to p o f o reb o dy. M ax . v a lue 8 mm . N ot e t he effe ct o f t he minin g s eque nc e on rel ati ve d is pl ac e m e nts o f t he v a r io u s CRF p i ll a r s . M. Sottile et al. / A Family of Constitutive Models Implemented in PLAXIS
Figure 7. V ertical stress at day just before the excavation and backfilling of the last t w o stopes. Note the arching effect and the higher load carried by the central CRF pillar, produced by its early construction. Conclusions
A family of perfect plasticity constitutive models with age- , density- and pressure-dependency of strength and stiffness has been presented. Models incorporate four of the most-employed expressions for the shear modulus , five for the friction angle , and four for age-dependency reproducing cement hardening , thus producing a set of 80 options to model the behavior of cemented bac k fill. The family of models was implemented in Plaxis as a user-defined constitutive model; a short introduction of the algorithm was described here. Finally , a case history of the mining sequence in a South American mine was presented to prove that the model is capable of capturing the evolution of strength and stiffness of CRF pillars with age and confinement. Moreover , it is shown that a realistic simulation can be obtained while k eeping material parameters of both cemented and uncemented bac k fills the same except for the few that account for the effect of age. A c k no w ledgements The authors wish to ac k nowledge the support of the team at the University of Buenos Aires and SRK Consulting during the development , calibration and beta-testing of the models presented in this paper. R eferences [1] N. Janbu,
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