v117n4a6 Modelling of caving and deformation …Modelling of caving and deformation mechanisms of...

The Malmberget mine is owned and operatedby the Luossavaara-Kiirunavaara AB (LKAB)mining company. Iron ore is extracted bylarge-scale sublevel caving. The minecomprises a total of 20 orebodies, about 10 of which are in production today (Figure 1).The mine is located in the municipality ofGällivare, about 100 km north of the ArcticCircle (Figure 2).

Successful sublevel caving requiresprogressive caving of the orebody after initialblasting. This results in caving of thehangingwall, sometimes all the way to surface,resulting in surface deformation. As miningprogresses to deeper levels a larger area isaffected by deformation. Over the years,extraction from the Malmberget orebodies hasled to deformation of the ground surface,affecting residential areas and existinginfrastructure.

Forecasting the caving process is notstraightforward as it can be rapid or slow,depending on the rock mass conditions, stressconditions, the failure mechanisms at work,etc. Moreover, several of the orebodies inMalmberget are non-daylighting, which makesreliable caving analysis even more difficult,primarily since there is less availableexperience from such cases. The present studywas designed to improve the understanding ofthe caving and deformation mechanisms at

Malmberget, and in particularly for thePrintzsköld orebody.

Mining of the non-daylighting orebodies atMalmberget, including Printzsköld, will leave acap rock, or crown pillar, between theextracted areas and the ground surface. Asmining proceeds deeper in the Printzsköldorebody, caving will gradually progressupward in the crown pillar and hangingwall.The caving propagation can lead to groundsurface deformation, depending on the extentand rate of caving, and whether breakthroughdevelops or not. It is therefore essential toinvestigate, describe, and quantify thisphenomenon and its governing factors.

Umar, Sjöberg, and Nordlund (2013)described the rock mass in the Printzsköldorebody as highly jointed in many places. Thisrock mass is characterized by fractures, faults,and weak zones. The ore is quite competentbut it is affected by the weak schistosity ofbiotite schists in some places on the boundarybetween magnetite (ore) and the grey leptitehorizon.

The caving is believed to be controlled by,among other factors, the stress conditions inthe rock mass. Umar, Sjöberg, and Nordlund(2013) presented an investigation of the stressredistribution in the hangingwall of thePrintzsköld orebody in which a two-dimensional conceptual model was constructedusing the continuum code Phase2 (RocscienceInc., 2002). A sensitivity analysis of strengthparameters was performed in which cohesion(c); tensile strength ( tm), and the angle ofinternal friction ( ) of the rock mass werevaried. This study showed that the stability of

Modelling of caving and deformationmechanisms of the hangingwall of thePrintzsköld oreboby at Malmbergetmineby S.B. Umar*, J. Sjöberg†, and T. Savilahti‡

Sublevel caving in Malmberget mine results in mining-induced surfacedeformation. One of the currently mined orebodies is the Printzsköldorebody. As mining deepens there is need to assess the behaviour of thecave formed in the subsurface above this orebody. Numerical analysis wasused to assess the effects of extraction to deeper levels and performstrength parametric studies. Stress redistribution was studied, and theresults showed high stress buildups in the hangingwall and the crownpillar. Two failure mechanisms have been identified – shear and tensile.Reducing cohesion by 50% increased the area of the yielded zone by morethan 100% in the hangingwall.

numerical modelling, sublevel caving, caving mechanism, strengthparametric study, stress redistribution, hangingwall stability.

* Department of Civil, Environmental and NaturalResources Engineering, Luleå University ofTechnology, Luleå, Sweden.

† Itasca Consultants AB, Luleå, Sweden.‡ Loussavaara-Kiirunavaara AB (LKAB)

Malmberget Mine, Gällivare, Swede .© The Southern African Institute of Mining and

Metallurgy, 2017. ISSN 2225-6253. Paper receivedAug 2015; revised paper received Nov. 2015.

351VOLUME 117 �

http://dx.doi.org/10.17159/2411-9717/2017/v117n4a6

Modelling of caving and deformation mechanisms of the hangingwall

the hangingwall rock mass was more sensitive to variationsin cohesion. However, the two-dimensional model wasjudged to be rather simplistic, and a three-dimensional modelstudy is thus warranted. No three-dimensional stressmodelling has previously been conducted for the Printzsköldorebody with the aim of studying failure mechanisms in orderto understand the caving process.

The Printzsköld orebody was selected for this study becauseof (i) its importance as one of the main areas for futureproduction at Malmberget, and (ii) the importance of betterunderstanding future caving behaviour and possible grounddeformation in the central area of Malmberget. The paperpresents a conceptual model for studying stress redistributionand yielding in the hangingwall of this orebody as a functionof mining at depth and the caving activities. However, thecaving process itself was not explicitly simulated. Rather, thestudy was focused on the specific objectives outlined below:

i. To provide an understanding of the rock mass

behaviour in the crown pillar of a non-daylightingorebody as mining progresses deeper

ii. To assess the sensitivity of rock mass strengthparameters with respect to possible caving anddeformation mechanisms

iii. To identify the failure mechanism likely to affect thehangingwall and the crown pillar of the Printzsköldorebody as mining deepens.

Three-dimensional numerical modelling was employed usingthe Three Dimensional Distinct Element Code (3DEC) (ItascaConsulting Group Inc., 2013). The rock mass was assumed tobe an equivalent continuum (fractures and joints thus notexplicitly simulated) and both elastic and elasto-plasticanalyses were conducted. In the elastic analysis, stresspatterns were evaluated for effects of mining at certaindepths. It was necessary to determine where stress buildupsdeveloped in relation to the crown pillar and the hangingwall.

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352 VOLUME 117

UPPLANDSLAVEN

TINGVALL SKULL

HEMATITE

GROPEN - THE PIT

KAPTEN

FABIAN

1000 METERLEVEL

PARTA

600 METER LEVEL

1050 METER LEVEL

DENNIWITZ

815 METER LEVEL

VITAFORS SHAFT

VITAFORS RIDDERSTOLPE

ALLIANS SHAFT

BELT CONVEYORINCLINED SHAFT

MALMBERGET

JOSEFINAJOHANNES 600 METER LEVEL

PRITZSCHOLD

VITAFORSINDUSTRIAL SITE

ALLIANSSHAFT

In the elasto-plastic analysis, the extent of yielding as afunction of mining was investigated. Moreover, a parametricstudy was conducted in which strength parameter valueswere changed and the effect on stress distribution, as well asthe extent and type of yielding, was studied. The simulationswith the plastic material model were all done using the Mohr-Coulomb yield criterion and a non-associated flow rule.

The Printzsköld orebody has a complex geometry. Extractionfrom this orebody requires a controlled approach in whichstresses are redistributed uniformly to avoid high stressbuildups in certain mining areas as well as in thehangingwall. Mining of this orebody by sublevel cavingstarted at the 780 m level. As mining progressed deeper, anupward cave developed, which as of 2012 had propagatedupward to approximately 300 m below the ground surface (atabout the 500 m level in the local mine coordinate system).The orebody has a plunge hangingwall which is defined bythe flat-lying cave back (see Figure 3). Umar, Sjöberg, andNordlund (2013) only examined a vertical cross-section ofthe Printzsköld orebody in a two-dimensional model, thusleaving out much of the effects of the plunge hangingwall. Atwo-dimensional model of a longitudinal section of theorebody was also modelled, but not reported. This work wasincluded in the present study to compare with the three-dimensional model.

Another simplification that has been made concerns themining sequence of the Printzsköld orebody in the currentmodel, as shown in Figure 3. The draw schedule within eachsublevel, or at different sublevels simultaneously, has notbeen simulated; instead, each sublevel has been excavated infull for each mining step in the model.

Mining was simulated to progress downwards on levelsseparated by 25–30 m (corresponding to actual sublevelheight). This mining simulation started from the currentactive production level at 920 m and progressed to the future1225 m level with a total extraction of approximately 70 Mtplanned by November 2030. It should be noted that thepossible continued caving as mining proceeds deeper was notexplicitly simulated in the model. Rather, the cave back thathad developed up until 2012 was kept at the same positionand the possible behaviour of the rock around the caveanalyzed as mining progressed downward.

The model size was set to 2800 m × 3629 m × 2050 m toreduce possible model boundary interference (see Figure 4).A graded discretization was used with fine meshing close tothe orebody and a gradually coarser mesh toward the modelboundaries. The boundary conditions were that the modelwas fixed at the bottom, and with roller boundaries (zerohorizontal velocity) on all vertical boundaries. The topsurface was modelled as a free surface.

It was further assumed that neighbouring orebodies didnot affect the Printzsköld orebody, hence only the Printzsköldorebody was included in the model. This assumption needs tobe verified in future mine-scale models, but initial stressmodels by Perman, Sjöberg, and Dahnér (2011) indicatedthat interaction effects may be limited.

Initial stresses were obtained from Sjöberg (2008), in whichthe vertical stresses were set equal to the pressure fromoverburden material and the horizontal stresses were foundto be (through stress calibration):

[1]

[2]

in which z is the vertical depth in metres. Here, H is themaximum horizontal stress and h is the minimum horizontalstress, with H having an orientation of 130.6° from the localnorth of the mine. It should be noted that the modelboundaries were oriented parallel and perpendicular, respec-tively, to the initial major horizontal stress, to avoid havingto apply shear stresses to the model boundaries. The strengthparameters used for elastic-perfectly plastic models areshown in Table I. These values were taken from Umar,Sjöberg, and Nordlund, (2013), and are based on a study bySjöberg and Jacobsson (2007). These values are consideredequivalent strengths for the rock mass, when modelled as acontinuum.

For plastic analysis, three scenarios were analysed toassess the effects of varying cohesion and tensile strength ofthe hangingwall rock. The three scenarios were base case,and low and high values for cohesion and tensile strength.The friction angle was kept at a value of 50.7° for thesensitivity analysis (see Table II). The cohesion value wasvaried by ± 3 MPa for the low,and high,cohesion cases. Forthe tensile strength, zero tensile strength was chosen for thelow-strength case, and an approximately twice as high atensile strength (1.5 vs. 0.7 MPa) for the high-strength case.


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Major Principal StressSxx direction

2800 m

3629

m

2050

m


Comparisons with the 2D models were made for vertical andlongitudinal cross-sections of the orebody. The mining stagescorresponding to levels 970 and 1225 m were selected forinterpretation and evaluation. The 970 m level was selectedas this area will be mined in the near future. The 1225 mlevel was selected as it represented a significant difference inthe amount of overall extraction compared to that of thecurrent mining state.

Three query lines were set in the vertical cross-sectionalong the hangingwall to study parameter effects on themodel in comparison to the two-dimensional model analysispresented in Umar, Sjöberg, and Nordlund (2013). Theselines represented three evaluation planes that wereestablished in the vertical cross-section view. Threeevaluation planes were also established in the longitudinalcross-section of the Printzsköld orebody. These planes wereused to monitor and evaluate the stress redistribution asmining progressed. Three cross-sectional view planes – C1,C2, and C3 – were chosen to evaluate how far into thehangingwall the stress redistribution reached. Plane C1,placed about 100 m from the cave boundary, was set as astarting point and established for comparison with the two-dimensional Phase2 model stress evaluations reported inUmar, Sjöberg, and Nordlund (2013) (Figures 5 and 6).Evaluation plane C2 was 50 m away from plane C1 and C3was placed 50 m away from C2.

The evaluation planes in Figure 5 were situated in thetrue hangingwall while the longitudinal evaluation planes inFigure 6 cut across the orebody. Planes L1 and L3 in Figure 6were set up to evaluate the behaviour in the cap rock andplunge hangingwall, respectively. On each plane, principalstresses and yielding were evaluated. L2 was used to giveinfill stress distribution information in the plungehangingwall to ascertain the trend of failure as stresses buildup in the cap rock.

The results are presented for both elastic and elasto-plastic analyses. These results are compared with thoseobtained in the two-dimensional analysis using Phase2(Umar, Sjöberg, and Nordlund 2013). The 1109 m level wasused as a control, to observe the trends and tendencies ofrock mass behaviour. It was found that the rock massbehaviour when mining level 1109 m did not depart muchfrom that when mining the 1225 m level.

The sign convention for stress results in 3DEC is negativefor compressive stresses and positive for tensile. However,for the Phase2 results the convention is the opposite, i.e.,positive for compressive stresses and negative for tensilestress.

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Table I

Hangingwall c (MPa) 5.18(°) 50.7

tm (MPa) 0.71

Orebody c (MPa) 4.81(°) 50.7

tm (MPa) 0.48

Footwall c (MPa) 6.67(°) 52.9

tm (MPa) 1.3

Table II

Cohesion 5.18 5.18 5.18 8.18 2.18c (MPa)

Tensile 0.71 1.5 0 0.71 0.71strengtht (MPa)

c = cohesion; = internal friction angle; tm = tensile strength

C3 C2 C1

Cave

Footwall

970 m level

1225 m level

Hangingwall

L3 L2 L1

Results for the 970 m and 1225 m level are shown in thevertical cross-section and the longitudinal section in Figures7 through 9. In the longitudinal section, a buildup of themajor stresses in the cap rock and the plunge hangingwallcan be observed. In Figures 7 and 8, (a) represents resultsevaluated on a plane that cuts through the orebody, while(b), (c), and (d) represent the major principal stress invarious parts of the model on the evaluation planes (C1, C2,C3) for the 970 m and 1225 m levels. A high concentration ofcompressive stress was seen to develop in the cap rock whenmining reached the 1225 m level.

In the cross-sectional direction (shown in Figure 6) threemonitoring planes (L1, L2, and L3) were used, which enabledassessment of the stress redistribution in the cap rock, truehangingwall, and plunge hangingwall. The major principalstresses on these evaluation planes are shown in Figures 9through 11.

The evaluation plane L1 in Figure 9 cuts through the roofof the cave and the Printzsköld true hangingwall. Acompressive stress regime is built up in the roof of the cave.A destressed region can be observed on the boundary of thecave going outwards into the hangingwall to about 100 m.Evaluation plane L2 (Figure 10) also shows a high-stressregion in the upper part of the plunge hangingwall.

Figure 11 shows the monitoring plane L3 whichaddresses the plunge hangingwall on the fringes of the

orebody. The major principal stress was also analysed alonga line traced through evaluation plane L1 (from Figure 6) andresults monitored from evaluation planes C1, C2, and C3.This was to determine how the major principal stresschanged for the various distances of the planes from the caveboundary.

Figure 12 shows the major principal stresses plottedagainst distance for the three evaluation planes. The bottomof the cave is represented at the origin in the graph while thesurface is at 1100 m (upper part of the graph). Figure 12 alsoshows zones of high stressing and destressing along eachevaluation plane.

A stress variation evaluation was undertaken along a linetraced along the evaluation plane C1 in Figure 5. The resultswere compared to the stress variations observed along thecorresponding evaluation (or query) line analysis from thetwo-dimensional model analysed in Phase2 (Umar, Sjöberg,and Nordlund, 2013), shown in Figure 13.


VOLUME 117 355 �


Figure 13 from Phase2 compares well with Figure 9b.Both figures have been taken showing the stress after miningthe 1225 m level. However, larger stresses were found fromthe Phase2 model compared to the 3DEC model. Stressvariation graphs from 3DEC and from Phase2 (Figure 14)show a similar trend in the behaviour of the stresses alongplane C1 in the hangingwall for all mining levels. Themaximum stress value in the 3DEC model in the cap rock is20 MPa while the Phase2 model shows a maximum stressjust above 100 MPa at the same location (both for the 1225m mining step). One reason for this seemingly largedifference is that the two-dimensional model does not includethe abutment effect from the third dimension, which wouldlead to stresses being overestimated compared to a three-dimensional model It should also be noted that zone sizesare larger in the 3DEC model, which further reduces thestress concentrations compared to those in the Phase2models.

For both the 3DEC and the Phase2 models, the 970 mlevel stress behaviour shows a high stress buildup at the 300m distance mark because this is the toe of the cave adjacentto the position of the line in the hangingwall. For the 1225 mlevel this point has become destressed and so the highstresses have been redistributed to lower levels. Destressedregions were observed in the hangingwall on all the threeevaluation planes in the longintudinal and cross-sectional

directions. In the cap rock, as seen from Figures 9 and 14,there does not seem to be much change in the major principalstress when mining from the 970 to the 1225 m level. Thiscould be because the thickness of the crown pillar remainsthe same when mining to the deeper levels.

Further stress evaluation was conducted in thehangingwall and the cap rock of the Printzsköld orebody.Two evaluation lines were established, one starting from thecave back going vertically upwards to the ground surfacewith length of about 300 m (A) and the other from the caveboundary generated after mining the 1225 m level to about1000 m into the hangingwall oriented in the northeasterndirection and inclined at a dip of approximately 30° (B) (seeFigure 15). The graphs in Figure 15 also show the plots ofmajor and minor principal stresses against distance along theevaluation plane into the hangingwall. These plots illustratethe variations between the two principal stresses along thelines A and B.

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356 VOLUME 117

Major and minor principal stresses were also plottedalong the two lines, with the results shown in Figures 16 and17. In the cave back along line A, both minor and majorprincipal stresses show a steady decrease when movingupward from the boundary of the cave. The stress buildup inthe immediate cave back is accompanied by highconfinement. In the hangingwall (along line B), the minorprincipal stress is about 5 MPa less than the major principalstress. The graph for line B shows a lack of confinement atthe cave boundary in the hangingwall.

Plots of the major and minor principal stresses were usedto check for confinement in the crown pillar and thehangingwall. It was evident that there was a high stressconfinement at the cave back along line A after mining wassimulated for the 1225 m level; the hangingwall showed anincrease in confinment along line B from about 300 m to 800m from the cave boundary. Thereafter there was a reductionin confinment at about 1200 m into the hangingwall.

The results for the base case scenario are shown in Figure 18for mining of levels 970 m and 1225 m. The different coloursin the legend indicate the current and past yield states for anelement, with shear-p and tension-p denoting shear andtensile yielding in the past (previous time steps), respectively,and shear-n and tension-n denoting shear and tensileyielding ‘now’, i.e. at the particular time step being evaluated.The blue ‘background’ colour indicates that the material isstill elastic, i.e. that no yielding has occurred. The resultsindicate that tensile failure was dominant in the hangingwall,whereas shear failure developed below the active mininglevel. This simulation also shows that as mining progressesdeeper, the area of yielding increases. Also, mixed shear andtensile yielding is observed to be developing in the lower partof the hangingwall, and shear failure can also be noted in thefootwall. The area of hangingwall yielding in this cross-section for mining the 970 m level is 35 600 m2, compared to54 055 m2 for mining of the 1225 m level.

Figure 19 shows a simulation of yielding patterns in thetrue and plunge hangingwall sections after mining level 1225 m. Both shear and tensile failure have developed, withtensile yelding predominating on the C1 evaluation plane.The plunge hangingwall at this level fails primarily in shear.Similar tensile failure patterns were observed on theevaluation planes C2 and C3.

The sensitivity of the model to changing strength parameterswas studied. This rock mass contains several weak zones and


VOLUME 117 357 �


biotite schist zones, which affect the rock mass strength. Inthis initial study, the strength was changed for the entirehangingwall rock mass, to increase the understanding ofwhat strength parameters are important to quantify in thecontinued work. Figures 20 and 21 show the results from thecases of high and low cohesion (cf. Table II), respectively,which may be compared to the base case shown above. Theresults show a marked difference in the volumes of yield inthe hangingwall, with a larger yielding obtained for the caseof the lower cohesion value.

Tensile strength was increased to twice that of the base casevalues for the high-value case and reduced to zero for thelow-value case (cf. Table II). Figures 22 and 23 show theresults for the two cases. It can be seen that the overall areaof yielding in the hangingwall and cap rock does not signifi-cantly change as the tensile strength is changed. However,the failure mechanism changes from shear to tensile failure,as could be expected. Moreover, for the low-strength case,some yielding near the ground surface can also be observed.

The changes observed for the variations of cohesion arehigher in terms of area of yielding compared to the cases withvaried tensile strength. However, changing the tensile

strength affected the yielding mechanism. It was clearlyinferred from model simulations that cohesion played animportant role in the amount of yielding of the hangingwallrock mass. The type of yielding for low cohesion was more of shear, while that for high cohesion was predominantlytensile. Evaluation plane L1 was used to estimate the arealchanges on this plane, which represented a volume variationalong the evaluation plane. The variations, shown in Figure24, show the differences in the area of the hangingwallaffected by the cohesion changes (in terms of tensile or shearyielding) on the L1 evaluation plane.

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358 VOLUME 117

Ground surface

Ground surface Ground surface

Ground surface

Ground surface

Comparisons were made between the plastic analysisresults in 3DEC and the numerical modeling results presentedin Umar, Sjöberg, and Nordlund (2013) in which Phase2code was used. It was found that the results from the Phase2model showed more conservative values for the hangingwallyielding than those shown in 3DEC, (compare Figure 25 andFigure 18). A larger area of yielding was noted in the Phase2model compared to that in 3DEC. As stated earlier, the plane-strain assumption used in Phase2 will generally result inexaggerated yielding compared to a 3D model.

The results from the three-dimensional modelling presentedin this paper have been compared with results from two-dimensional modelling presented in Umar, Sjöberg, andNordlund (2013) using Phase2. Elastic modelling showed agood match with the Phase2 results, especially regarding thestress redistribution pattern. However, there is a difference inthe values of the stress buildup, especially in the cave back.The two-dimensional Phase2 model gave higher stressescompared to those found from the 3DEC model. The primaryreason for this is that the two-dimensional analysis assumesinfinite cross-sections and thus it eliminates the effect ofabutment from the walls perpendicular to the model plane.Hence it can be reasonably assumed that the 3DEC modelyields more realistic stress values for a complicated orebodygeometry such as that for the Printzsköld orebody.

Numerical simulations showed that tensile yielding in thehangingwall was more predominant compared to shearyielding, with the latter being observed primarily in the

crown pillar and at the toe of the cave. Varying strengthparameter values led to changes in the behaviour of themodel in the hangingwall. It was found that loweringcohesion in the hangingwall caused an increase in the overallvolume of yielding in either tension or shear. Since there areweak zones in this rock mass, this finding can be used toestimate failure extents within the hangingwall rock masswith low cohesion, such as the biotite schist zones foundnear the Printzsköld orebody.

The destressed zone in the true hangingwall can bethought of as a beam ‘fixed’ to the cave back and the toe ofthe hangingwall-cave boundary (see Figure 26). Bending ofthis ‘beam’ will induce tensile failure in the hangingwall. Thecave back, on the other hand, is likely to fail in shear due tohigh compressive stresses, and thus advance upwards. Thiswould lead to an extension of the exposed hangingwall‘beam’, which in turn will promote additional tensile failure,thus resulting in progressive cave growth.

The hangingwall of the Printzsköld orebody was analysednumerically using elastic and plastic material models. Thefollowing conclusions can be drawn from this study.

� As mining deepens there are high stress buildups inthe cave back. Lowering the cohesion has an overalleffect of increased yielding in the hangingwall.Simulations with high cohesion values produces verylittle yielding in the hangingwall, as expected

� The tensile strength variation had a limited effect onyielding in the hangingwall compared to that fromvariations in cohesion


VOLUME 117 359 �


� The failure pattern observed in the hangingwall was amixture of tensile failure and shear failure. Shearfailure was observed mostly near the crown pillar andalso at the toe of the hangingwall.

The following recommendations are made with respect tocontinued studies.

� This simulation has been undertaken withoutconsidering the geological structures that may bepresent in the rock mass. It is therefore envisaged thatfurther analysis of this orebody should incorporatepotential large-scale geological features to address theirpossible effects on the stability of the hangingwall

� The potential for extension cracking, possibly resultingfrom caving, should be investigated further. This maybe conducted using continuum modelling. However, itis also recommended that for future work more effortsshould be made in explicitly simulating caving andcave advance, and the resulting cave geometry in thePrintzsköld orebody. For this, a ’caving’ model isrequired, and different approaches need to beinvestigated

� For this study, historical observations of caving wereused qualitatively to understand the caving zone. Forfuture work, it is envisaged that e.g., seismicmonitoring data may be used as more quantitativedata-set, thus allowing an improved calibration of themodels.

This paper was written as part of an ongoing research projectfor Luossavaara Kirunavaara AB (LKAB) Malmberget mine,

Sweden. The project is funded through the HjalmarLundbohm Research Centre (HLRC). The support from LKABand the Centre for Applied Mining and Metallurgy isacknowledged. The authors would also like to thank thefollowing members of the research team for their contri-butions in making this research a smooth run: Jimmy Töyrä,Linda Jonsson, Joel Andersson, and Fredrik Ersholm, all fromLKAB. Also, thanks go to the assistant supervisor of the firstauthor, Catrin Edelbro, as well as to Professor ErlingNordlund, both at Luleå University of Technology, for theirguidance and support during this research.

Itasca Consulting Group Inc. 2013. 3DEC (3-Dimensional Distinct ElementCode), Version 5.0. Minneapolis.

LKAB, 2014. Malmberget Mine Level. .https://www.lkab.com/en/Future/Urban-Transformations/Why/New-Mine-Level/Malmberget-Mine-Level/

PERMAN, F., SJÖBERG, J., and DAHNÉR, C. 2011. Detailed three-dimensional stressanalysis of complex orebody geometry–model setup and results for theMalmberget Mine. Proceedings of the 2nd International FLAC/DEMSymposium, Melbourne, Australia, 14-16 February 2011. ItascaInternational Inc., Minneapolis. Paper no. 02–04.

ROCSCIENCE INC. 2002. Phase2. Version 6.028 - Finite Element Analysis andSupport Design for Excavations. Toronto, Ontario, Canada.http://www.rocscience.com

SJÖBERG, J. 2008. Three-dimensional unit stress tensor modeling of complexorebody geometry. Proceedings of the 42nd US Rock MechanicsSymposium (ARMA), 29 June–2 July 2008. American Rock MechanicsAssociation (ARMA), San Francisco, CA.

SJÖBERG, J. and JACOBSSON, L. 2007. Bergmekaniska analyser för ny huvudnivå iMUJ (M1250). LKAB Utredning 07-701. (in Swedish).

UMAR, S.B., SJÖBERG, J., and NORDLUND, E. 2013. Rock mass characterization andconceptual modeling of the Printzsköld orebody of the Malmberget mine,Sweden. Journal of Earth Sciences and Geotechnical Engineering, vol. 3,no. 4. pp. 147–173. �

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