IN
DEGREE PROJECT THE BUILT ENVIRONMENT, SECOND CYCLE, 30 CREDITS
STOCKHOLM SWEDEN 2019,
Evaluation of methods for rock mass characterization and design of rock slopes in crystalline rock
JOHANNA GOTTLANDER
KTH ROYAL INSTITUTE OF TECHNOLOGY
SCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENT
TRITA TRITA-ABE-MBT-1929
www.kth.se
Abstract
Construction of rock slopes is needed in many civil work projects. It is for example very common in road and railway cuts, but other applications include excavation for tunneling or building founda- tions, where perhaps sensitive constructions are present in the immediate vicinity. In Sweden the majority of the rock is hard crystalline bedrock of relatively good quality, and the fracture orien- tation have a large effect of the stability of the rock slope.
If the geology is not properly considered for when the design of the slope is carried out, it can result in slope failure, with severe consequences. This applies especially if the rock slope is high, but unwanted effects like increased excavation and construction costs, could occur also in smaller slopes if the risks are not identified and managed. However, it is difficult to standardize design of rock slopes in fractured hard crystalline rock because of the uncertainties and variations in the geological conditions during the design phase.
Rock mass characterization systems like Rock mass rating, RMR (Bieniawski 1989) and the Q- system (Barton, Lien och Lunde 1974) are commonly used to describe the general rock mass quality.
Whilst a good rock mass quality is generally easier to construct in, stability problems do occur due to structural geology in rock slopes even in good rock mass quality. The application of these systems in rock slopes can be problematic as they do not describe the geometry of the slope and how the fractures daylight in the slope face. Instead, stereonets can be used to visualize this, but fracture parameters of large importance for stability (persistence, roughness and alteration) are traditionally not presented in a stereonet analysis. Additionally, these parameters and the structu- ral geological conditions can be difficult to predict and can vary significantly over short distances, why it can be difficult to forecast and predict failure in the design face.
Slope instability due to large sliding and wedge failures have been observed in a large number of slopes in crystalline rock, and a standard method for design of rock slopes is lacking. This has given rise to the research question of how best to describe rock mass conditions, how to design slopes in crystalline rock and how to manage these risks during construction.
To investigate this, three rock slopes where large failure had occurred were selected for the case study and were mapped, characterized and analyzed. The results from the case study showed that all rock slopes investigated had in common that failure had occurred along fractures of large per- sistence. This highlights the importance of evaluating the geology in the correct scale and suggest that lineament studies and other elevation based interpretation are perhaps more useful than pre- viously thought. It also came to light that the planning and design process is inflexible which in many cases lead to large risks being overlooked or left unnoticed.
A flow chart for risk classification was produced and suggested for aid in decision making aiming at simplifying project management of rock slopes, as well as aiding in visualization of the risk that rock slopes can be associated to.
Keywords
Geology, rock mechanics, rock slope stability in fractured crystalline rock, rock mass characteriza- tion and classification, rock slope design, slope failure.
Sammanfattning
Bergslänter är nödvändiga konstruktioner i många infrastrukturprojekt. Det är till exempel mycket vanligt vid väg- och järnvägsskärningar, men även vid förskärning för tunnlar eller bergschakter eller för anläggning av husfundament, där känsliga konstruktioner ofta finns i omedelbar närhet.
I Sverige är den största delen av berggrunden hårdkristallin och av relativt god kvalitet, varför sprickorna och sprickornas orientering har en stor effekt på stabiliteten i bergsslänter.
Om den rådande geologin inte tas hänsyn till när släntens lutning och utformning beslutas kan oönskade brott ske i slänten, vilket i sin tur kan få allvarliga konsekvenser. Detta gäller särskilt om bergslänten är hög, men oönskade konsekvenser såsom utökad utgrävningsvolym och/eller ökade byggkostnader kan också uppstå även i mindre slänter om riskerna inte identifieras och hanteras på förhand.
Det är svårt att standardisera designen av bergsslänter i sprickigt hårt kristallint berg på grund av variationen i de geologiska förhållandena och osäkerheter under designfasen. Det finns ett an- tal vanliga system för karaktärisering av berg. Rock mass-rating, RMR (Bieniawski 1989) och Q-systemet (Barton, Lien och Lunde 1974) är metoder som används för att beskriva den allmänna bergmassans kvalitet. Dessa system beskriver inte specifikt geometrin i slänten och hur sprickor skär bergskärningen, utan för att beskriva detta kan istället ett stereonet användas. Sprickparametrar med stor betydelse för stabiliteten (uthållighet, råhet och vidd/fyllning) tydliggörs emellertid inte i en stereonetanalys. Dessutom kan dessa parametrar samt de strukturella geologiska förhållandena vara svåra att identifiera och förutsäga.
Släntinstabilitet på grund av stora glid- och kilbrott har observerats i ett antal slänter i uppsprucket hårt kristallint berg. Detta har givit upphov till frågan om hur man bör beskriva dessa bergför- hållanden och hur man bör hantera de risker som dessa ger upphov till, vilket den här uppsatsen siktar på att ge svar på.
För att undersöka detta genomfördes en enkätstudie och en fallstudie. Enkätstudien visade att hur bergslänter hanteras av ingenjörer i Sverige idag skiljer sig och bekräftade frågeställnigen för uppsatsen. En skrivbordsstudie utfördes för att identifiera vanliga brottsmekanismer i bergslänter i dessa bergförhållanden och för att välja ut ett antal fall att ytterligare studera under fallstudien.
Tre bergsslänter där stora brott skett valdes ut. Dessa karterades och karaktäriserades, och var senare föremål för disskution under intervjuer med sju deltagare från enkätstudien.
Resultaten visade att vad alla tre bergsslänter i fallstudien hade gemensamt var att utfall hade skett längs med stora utsträckta sprickor i bergmassan. Detta belyser vikten av att utvärdera geo- login i rätt skala och indikerar att lineamentsstudier och annan höjdkurvsbaserad tolkning kan vara mer användbar än tidigare trott. Det framkom också att planerings- och designprocessen är oflexibel vilket i många fall leder till stora risker som förbises eller inte identifieras.
Ett flödesschema för riskklassificering togs fram inom ramarna för uppsatsen och föreslås för använ- ding som stöd i beslutsfattande och för att förenkla projekthanteringen när det gäller bergslänter.
Flödesschemat bidrar också till att visualisera vilka risker som kan kopplas till bergslänter.
Nyckelord
Geologi, bergmekanik, släntstabilitet i uppsprucket kristallint berg, bergkaraktärisering och klas- sificering, dimensionering av bergslänt, stora bergutfall/brott.
Preface
This thesis has been written as part of the Master’s Programme Environmental engineering and sustainable infrastructure in KTH Royal Institute of Technology, Sweden. It was written during the period June 2018 to January 2019 and was based on a thesis proposal by Robert Swindell.
The subject of research was inspired by the need to evaluate the investigation and design process of rock slopes, specifically in fractured hard crystalline rock, as a large improvement potential of how to design and plan for rock slopes in these kind of conditions had been identified. This also became evident during the study, especially during the interviews.
Hopefully, the work presented in this thesis can lay base for and inspire discussion between different stake holders such as owners, consultants and contractors in Sweden and perhaps lead to a better, more iterative design and planning process in the future.
Stockholm, Januari 2019 Johanna Gottlander
Acknowledgements
I would like to thank my supervisor Robert Swindell for giving me the opportunity to write my thesis on this interesting topic, for valuable discussions and feedback and for cheering me on du- ring the project. Also, I would like to thank my supervisor Fredrik Johansson for valuable feedback and support during this time. Both supervisors have my gratitude for sharing their expertise and knowledge with me.
My gratitude also goes out to all who took time to participate in the survey and a special thanks to those how offered to meet with me during interview sessions. I also want to thank SWECO for the opportunity to write my thesis in a pleasant work space and for giving me access to construction sites, as well as supplying me with tools necessary to investigate this subject.
Finally, thanks to friends and fellow students at KTH who helped with review of this thesis and support in other ways.
Nomenclature
Abbreviations
F S Factor of safety
F Sp Factor of safety, planar failure F Sw Factor of safety, wedge failure GSI Geological strength index Ja Joint surface alteration Jn Joint set number Jr Joint roughness number J RC Joint roughness coefficient
J CS Joint surface compressive strength Jw Joint water inflow
Jw,ice Joint water inflow, adapted to rock slope engineering L10 Length of core piece longer than 10 cm
Ltot Total core length RMR Rock mass rating
RQD Rock quality designation number
S Spacing
Sapp Apparent spacing SRF Stress reduction factor
SRFslope Stress reduction factor, adapted to rock slope engineering Q Rock mass quality number
Qslope Rock mass quality number, adapted to rock slope engineering
Symbols
A Area [m2]
αd dip direction of fracture plane [ °]
αf dip direction of face [ °]
αs direction of sliding [ °]
αt direction of toppling [ °]
αi dip direction, line of intersection [ °]
b Distance from tension crack to slope crest [m]
β Steepest allowed angle [ °]
β Angle between midpoint and side of the wedge [ °]
c Cohesion [kPa]
cA Cohesion of plane A [kPa]
cB Cohesion of plane B [kPa]
δ Shear displacement [mm]
∆x Width of toppling slab [cm]
γ Unit weight [kN/m3]
γr Unit weight of rock mass [kN/m3]
γw Unit weight of water [kN/m3]
H Total height of slope [m]
Hw Total height of wedge [m]
i Angle of asperities [ °]
K Wedge factor [-]
mb Hoek-Brown constant [-]
φ Friction angle [ °]
φA Friction angle of plane A [ °]
φB Friction angle of plane B [ °]
φb Basic friction angle [ °]
φbp Friction angle of base plane, toppling analysis [ °]
φd Respective friction angle for slabs, toppling analysis [ °]
φr Residual friction angle [ °]
φp Peak friction angle [ °]
ψ5 Line of intersection between plane A and B [ °]
ψd Dip of planes that make up sides of slabs, toppling analysis [ °]
ψf Dip of slope face [ °]
ψi Plunge of intersecting line between planes [ °]
ψs Angle of slope above crest [ °]
ψp Dip angle of sliding plane [ °]
RA Normal reaction for plane A [-]
RB Normal reaction for plane B [-]
σn Normal stress [kPa]
σn0 Effective normal stress [kPa]
σ10 Effective major principal stress [kPa]
σ30 Effective minor principal stress [kPa]
σci Uni-axial compressive strength of intact rock [kPa]
τ Shear strength [kPa]
θ Angle between face of outcrop and fracture dip direction [ °]
U Uplifting water pressure [kPa]
V Vertical water pressure [kPa]
W Weight [kg]
ξ Angle between planes forming wedge [ °]
y Height of slab that can topple [cm]
z Height of tension crack [m]
zw Water depth in tension crack [m]
Contents
1 Introduction 1
1.1 Background . . . 1
1.2 Aim . . . 1
1.3 Disposition . . . 1
1.4 Limitations . . . 2
2 Literature study 3 2.1 Rock mass investigation methods . . . 3
2.1.1 Geological mapping . . . 3
2.1.2 Diamond drilling . . . 4
2.1.3 Geophysical investigations . . . 4
2.1.4 Photogrammetry . . . 5
2.2 Fracture characteristics . . . 5
2.3 Rock mass characterization and classification systems . . . 6
2.3.1 RQD . . . 6
2.3.2 Q-base and Q-slope . . . 6
2.3.3 RMR-base and RMR-slope . . . 7
2.3.4 GSI . . . 7
2.3.5 Stereonet analysis . . . 9
2.4 Rock mass strength . . . 10
2.4.1 Intact rock . . . 10
2.4.2 Shear strength of fractures . . . 11
2.5 Rock slope design methods . . . 13
2.5.1 Q-slope . . . 13
2.5.2 Plane failure analysis . . . 13
2.5.3 Wedge failure analysis . . . 14
2.5.4 Limit equilibrium to calculate stability against toppling . . . 17
2.5.5 Circular failure and numerical analysis . . . 18
2.5.6 Engineering judgment . . . 19
2.6 Excavation methods . . . 19
2.7 Rock and soil excavations . . . 20
2.8 Requirements and standards for rock slope stability design . . . 21
2.8.1 Eurocode 7 . . . 21
2.8.2 AMA 17 . . . 22
2.8.3 TK Geo . . . 23
2.8.4 Guidelines and other recommendations . . . 24
2.9 Project management . . . 25
2.10 Summary of literature study . . . 25
3 Method 27 3.1 Survey . . . 27
3.2 Case study . . . 27
3.2.1 Desktop study . . . 28
3.2.2 Geological mapping . . . 29
3.2.3 Interviews . . . 29
4 Result 31 4.1 Survey . . . 31
4.1.1 Praxis and methods . . . 31
4.1.2 Planning phase and design . . . 31
4.1.3 Construction phase . . . 34
4.2 Case study . . . 35
4.2.1 Desktop study . . . 35
4.2.2 Interchange ’Tureberg’, Kista . . . 35
4.2.3 Norvik, Nynäshamn . . . 38
4.2.4 Nynäsvägen, Trångsund . . . 42
4.2.5 Interview summary . . . 44
5 Discussion 47
5.1 Risk management . . . 47
5.2 Pre-investigation and geological analysis . . . 48
5.3 Design . . . 51
5.4 Construction phase . . . 53
6 Conclusion and suggestions for future research 55 6.1 Rock mass characterization and critical parameters . . . 55
6.2 Project and risk management . . . 55
6.3 Future research suggestions . . . 56 A Desktop study
B Mapping protocols of case study rock slopes
B.1 Norvik . . . . B.2 Trångsund . . . . B.3 Kista . . . . B.4 GSI assessments . . . . C Case study interviews
1 Introduction
1.1 Background
Construction in rock slopes is needed in many infrastructure projects. It is for example very com- mon in road and railway cuts, but other applications include excavation to prepare for tunnel construction and excavation for the foundation of buildings, where sensitive constructions can be present in the immediate vicinity. If the geology is not accounted for when the design of the slope is carried out, it can result in slope failure, with severe consequences. This applies especially if the rock slope is high, but unwanted effects like increased excavation and construction cost, also occur in smaller slopes if the risks are not identified and managed. The rock mass can be highly heterogeneous, why there are risks associated to rock constructions. The strength of rock mass varies around the world, depending both on its origin (magmatic, sedimentary or metamorphic) and the local tectonic regime. The intact crystalline rock is generally very strong, however, if subjected to tectonic forces it will fracture, thus reducing the strength of the total rock mass.
In rock slope engineering, the degree of fracturing and quality of rock is important for the overall stability of the construction, but the most important factor for stability of a rock slope is how fractures intersect the rock slope face, i.e. the fracture orientation. This determines what blocks and wedges that can form in the slope and if they can potentially fall out, leading to so called slope failure or block falls. Rock slope design is identification of potential failure modes and adjustment of excavation geometry to avoid these failures or to secure against them by design reinforcement.
It is difficult to standardize design of rock slopes in crystalline rock because the structural prop- erties of the rock mass has a large influence on the rock slope stability, and because crystalline rock can be associated to large geological uncertainties. For example, ancient crystalline rock commonly have complex structural geology due to the many tectonic events it has experienced, and large local variation is common. This makes it difficult to forecast the geological conditions.
Additionally, commonly used rock classification systems such as RMR and Q do not reveal the structural geological conditions which are often the controlling factor for rock slope stability in crystalline rock. It is also common that site investigations are limited, but without knowledge of the geological conditions a good rock slope design cannot be performed.
Slope instability due to sliding, wedge and toppling failure occur in many, almost all, slopes that are constructed in fractured hard crystalline rock. Failure in describing and communicating the geological conditions and uncertainties before, during and after the construction of a rock slope is related to large risks that can even result in fatal accidents. This has given rise to the question of how to best describe these rock conditions and how to design, reinforce and manage risks in rock slopes in crystalline rock.
1.2 Aim
The aim of this thesis is to:
1. Investigate and evaluate commonly used methods for characterization of both rock mass and fracture properties in rock slope application in a range of different geological and production related settings in crystalline bedrock.
2. Investigate which rock engineering parameters that are critical for the design and stability of rock slopes, and how they can be identified and investigated in the planning phase.
3. Investigate how the design and management of a project can better take the geological conditions and related risks into account regarding rock slope stability.
1.3 Disposition
This paper is divided in 5 main sections: introduction, literature study, method, result and dis- cussion. The introduction states the study aims and limitations as well as the background to
the study, whilst the literature study chapter goes into theory of the subject. The method de- scribes how the survey and case study was performed, and the results describes what was found in these two studies. In the discussion section, these results are evaluated and a general approach to characterization and design of rock slopes is suggested.
1.4 Limitations
This thesis was written towards rock slope engineering in civil works in Sweden, focusing on the geological conditions in this region, however the results in this thesis can be applied in other geographical areas where the rock mass conditions are similar. The aim in this work was not to evaluate the different limit equilibrium methods to calculate stability against plane, wedge or toppling failure, as the main focus is how large failures of rock slopes can be identified, described and avoided. The safety factor for design is also not discussed. Instead the geological uncertainties are handled by discussion of how to plan for differing geological conditions and how to identify the critical parameters for stability. The results in this thesis cannot be applied in weak rock conditions such as sedimentary rocks and weakly metamorphosed rocks. As this thesis is partly a qualitative analysis, exploring trends and reasons behind these, it is more sensitive to subjectivity and biases than a quantitative research would have been.
2 Literature study
The general design concept of rock slopes varies around the world, because the geological conditions differ, but also because of tradition. In countries with hard crystalline rock the rock is generally considered a good building material, because rock excavations are to a larger extent stable if the over-all rock mass is of good quality. Traditionally this h to extensive construction in rock where risks could be overlooked if the fracture orientation in the rock is favorable with regards to the orientation and shape of the excavation. In rock slope engineering with these geological condi- tions, rock reinforcement might not be needed even in very steep slopes. Alternatively, in some instances, for example in mountainous or rural areas, slopes can be allowed to fail. In these cases it is common to install mitigation measures such as nets, ditches or barriers, to catch rock falls or rock avalanches that could otherwise harm people and/or infrastructure.
Generally, rock slopes in fractured hard crystalline rock are stable if the fracture orientation is favorable in regards to the slope face. A favorable orientation means that fractures are oriented in such a way that they do not intersect the face and one another in such a way that a block can form and slide, fall or topple out of the slope face. The common failure modes that can occur in a slope are thus sliding, wedge and toppling failure, as shown in Figure 1.
(a) Sliding failure. (b) Wedge failure. (c) Toppling failure.
Figure 1: Common failure modes in fractured hard crystalline rock (Wyllie and Mah 2014).
2.1 Rock mass investigation methods
2.1.1 Geological mapping
Geological mapping in the context of rock slope engineering is carried out on outcrops and slope faces. The practice includes collecting data on fracture orientations, parameters regarding shear strength properties and rock mass assessment as described above. Geological mapping of outcrops is a widely used method for identifying geological structures and to gain knowledge of the overall rock mass characteristics. It can however be difficult to determine the small scale properties of fractures, for example weathering of the surface can lead to overestimation of the fracture aperture, as fractures that look open on the surface can be tight within the rock mass, especially at depth.
Other parameters that can be difficult to interpret correctly is the true orientation of fractures. The orientation of the outcrop face in relation to the fracture orientation can result in misinterpretation of orientation measurements. Spacing can also be measured on an outcrop, but it should be noted that unless the fractures are oriented in such a way that they are perfectly perpendicular to the outcropping face it is only the apparent spacing of discontinuities that can be measured. The true spacing can be found by the so called Terzaghi correction, equation 1:
S = Sappsin θ (1)
where S is the true spacing, Sapp is the apparent spacing and θ is the angle between the face of the outcrop and the dip direction of the discontinuity (Wyllie and Mah 2014).
In geological mapping the scale of the investigation is very important. Normally mapping is car- ried out in a scale of 1:200. There is a risk that large structures in an area are overlooked when the mapping is carried out in a small perspective (at larger scale). It can therefore be useful to carry out a lineament analysis, especially in countries where strong erosive forces have been acting (Wyllie and Mah 2014). In Sweden, the glacial period led to an extensive erosion of weaker rock.
This resulted in the characteristic landscape that is found in many parts of Sweden today: out- cropping rock in the elevated points in the landscape and clay filled valleys in depressions where the general rock mass quality is lower and more easily eroded, due to deformation as an effect of tectonic movement. A lineament analysis is carried out by studying how the elevation changes in the landscape, and thereby identifying and analyzing the geological properties of the bedrock.
There are several guidelines on geological mapping and how it should be carried as a means to standardize the process and the collected results, for example as stated by Lindfors et al. (2014), which is often complied in Sweden. However, it has been showed that geological mapping is subjective and that the range in results between different geologists can be large (Ewan, West, and Temporal 1983), which may need to be considered when geological data is collected and analyzed.
2.1.2 Diamond drilling
Diamond drilling is the drilling of a core in a rock mass. The core is then extracted, as much as possible in its entirety. The amount of fracturing in the rock and the overall quality is assessed by a geologist. If the rock core is oriented, the fracture orientation can be measured. Diamond drilling is a good way to map specific fracture properties (see chapter 2.2), like spacing, infill, roughness, aperture, and an overall rock mass interpretation. However it can be difficult to find important parameters for stability assessment of a rock slope, because the core recovers only a small piece of information of the entire rock mass. It can also be difficult to identify the critical fractures, for example those with large persistence and low roughness, as persistence and roughness (over a larger distance) in rock fractures cannot be measured in a drilled rock core. Core mapping can be extensive, collecting parameters such as mineral composition and properties and fracture orientations, or general, assessing the degree of fracturing in the rock core and classification of rock mass according to rock mass quality systems RMR and Q (chapter 2.3).
2.1.3 Geophysical investigations
Seismic refraction can be used to identify rock masses with different properties. It is a geophys- ical measuring method that works by induction of a seismic wave. The seismic wave travels the shortest possible way though the media, and is refracted back to the surface. By measuring the seismic wave velocity in the different materials, one can distinguish between different geological formations. In the application on fractured hard crystalline rock, a seismic survey is most useful to detect the surface of the bedrock, as the seismic wave velocity is much larger in the bedrock than in the soil. It is difficult to localize fractures using this method (Wyllie and Mah 2014).
Ground penetrating radar works similarly to seismology but instead uses high frequency electro- magnetic waves, which are sent into the ground in pulses. The waves are reflected on materials of different density in the ground, which are then interpreted by a geologist. Electric resistivity surveys are measuring the induced electricity in materials. All materials have a different resistivity to electric current. By measuring and mapping the resistivities one can interpret the geological units (Milsom 2003).
There are also other geophysical surveying methods that can be suitable rock mass investigation methods (Lindfors et al. 2014); for example VLF (very low frequency) which is a electromagnetic measurement, as well as pure magnetic measuring. These methods both measure magnetism, however while magnetic measuring measures the magnetism in the material, VLF measures induced magnetism in materials, which is transmitted by VLF transmitters such as radio antennas (Milsom 2003). Generally when it comes to geophysical measuring, it is important to know the limitations.
As an example, when using seismic refraction, if there is a denser layer overlaying a less dense layer the waves will always travel faster in the dense layer than in the less dense layer, why the less dense geological unit will not be visible in the results. Because it is as important to know what you can see as knowing what you cannot see, the interpretation of the data is more important and more difficult than geological mapping of outcrops and cores.
2.1.4 Photogrammetry
A way to collect information about the rock mass, or more specifically the fractures and fracture orientations, is photogrammetry. This technology is a remote sensing method that interprets objects based on photos, and can be applied to geotechnical engineering by identifying fractures and their orientation. It has been demonstrated that interpreted measurements of fracture orientations correspond well to measurements carried out during traditional geological mapping (Osterman 2017). The information obtained through photogrammetry can then be processed in a stereonet analysis as well as in a general geological judgment.
2.2 Fracture characteristics
It is of large importance to investigate the geology whenever constructing in rock, because the stability of the construction is depending on it. However it is especially important in rock slopes, where the specific geological parameters can have a large effect on the overall stability. For ex- ample, in hard rock conditions, the orientation and shear properties of fractures can determine weather the slope is stable at a certain angle or not. In tunnel construction, the orientation is usually less important as the stresses exerted by the surrounding rock help stabilizing the con- struction (bridging effect). It is often considered common practice to collect information about at least the below mentioned geological properties regarding fractures as well as performing a general rock mass assessment (Wyllie and Mah 2014).
Orientationis how the fracture plane is orientated in space, i.e. what direction and angle it dips towards. The dip direction is measured with a compass, and can range from 0-360°and the dip angle is measured with an inclinometer and can range from 0-90°. The orientation of fractures relative to the rock slope surface helps determine what kind of failure can be expected in the slope.
Persistence is the length of fractures, i.e. how far they extend in space. It is an important parameter to determine as it is directly related to the size of the blocks that potentially form in a slope, but it can be difficult to measure, especially in high/very high persistence fractures as it can be difficult to access the full extent of these fractures.
Spacingis the distance between fractures of the same set, a fracture set being defined as fractures with similar geological properties, i.e. approximately the same orientation and shear strength.
Together with fracture persistence, the spacing determines the size of the blocks that can form in the slope.
Infillis a measure of alteration of rock fractures. The infill material in a fracture can be of loose character like clay and silt, or hard, like mineral growth on the fracture surfaces. The infill material can have a large influence on the shear strength of the fracture.
Roughnessof the fracture surface is an important parameter for the shear strength in a fracture, at least in cases where there is still interlocking of the fracture surfaces. The roughness should be measured at a distance of at least 2 meters and can be described in small scale: stepped, undulating or planar structure, and in large scale: rough, smooth or slickensided (very smooth) surface.
Apertureis the measure of how open a fracture is, and is often linked to the hydraulic conductiv- ity of the rock mass. It is important to identify water-bearing fractures, as they can be problematic in rock slope engineering.
Rock mass assessmentnormally includes definition of the rock type, a description of the char- acteristics of the rock mass such as the fracturing system, and a rock mass quality assessment by classification systems, which is further described in chapter 2.3.
As there is large variation both in geological settings and in rock slope design in different projects and different countries, it is difficult to specify one type of investigation program that is always suitable for the purpose of rock mass investigation in rock slopes. Generally regarding geological investigations however, it is important to know and understand the purpose of the investigation
to be able to collect the relevant data. For this reason it is also good if the same person that is responsible for the design carries out or at least takes part in the geological investigation.
2.3 Rock mass characterization and classification systems
Rock mass characterization and classification systems have been developed as a way to assess and describe rock with regards to its suitability for construction purposes. They help standardize the method of rock mass definition, and thus contribute to a more uniform and systematic process. The difference between rock mass characterization and rock mass classification is that characterization only considers the strength and deformation properties of the rock itself, whereas classification also considers these properties in relation to the planned construction and divide the rock mass quality into predefined classes. This means that rock mass classification also depends on the construction, and includes for example how fractures are oriented in relation to the construction.
These systems were originally developed for underground excavations, but have been adapted to rock slope engineering as described below.
2.3.1 RQD
Rock quality designation number, RQD, is a way to describe the degree of fracturing in a drilled rock core, and is calculated by equation 2:
RQD =P L10
Ltot × 100% (2)
where L10 is the length of all pieces longer than 10 cm and Ltot is the total core length. The associated rock mass quality assessment is then found in Table 1. The RQD-value is used as a parameter in the rock mass classification systems RMR and Q, and can also be interpreted during geological mapping. It is uncommon that it is used in itself to describe rock mass quality.
Table 1: Rock mass quality assessment based on the RQD-value.
Very poor
rock Poor
rock Fair rock Good
rock Excellent
0-25% 25-50% 50-75% 75-90% 90-100%
2.3.2 Q-base and Q-slope
The Q-system (Barton, Lien, and Lunde 1974) was first developed for use in tunnel construction and suggests if the rock can be considered of good or poor quality.
Q =RQD Jn ×Jr
Ja × Jw
SRF (3)
where Jn is a number that relates to the number of joint sets present in the rock mass, Jr is the joint roughness and Ja is the alteration of the joint surface [joint is synonymous with fracture in this report]. The obtained Q-value in equation 3 is a representative value and describes the overall quality of the rock, as clarified in Table 2. For above ground applications of the Q-system, the parameters for stress reduction, SRF , and for water in-leakage, Jw, cannot be applied directly. For overground conditions, the system can therefore be simplified according to Eq 4 (Lindfors et al.
2014).
Qbase= RQD Jn ×Jr
Ja (4)
Table 2: Rock mass quality assessment based on the Q-system.
Exceptionally
poor Extremely
poor Very
poor Poor Fair Good Very
good Extremely
good Exceptionally 0.001-0.01 0.01-0.1 0.1-1 1-4 4-10 10-40 40-100 100-400 good400-1000
The Q-slope characterization system (Bar and Barton 2017) is an adaptation of the Q-method.
The term in the equation that accounts for friction, JJar is adapted so that the intersection of the
fractures and the slope face is considered. The parameters for water inleakage, Jw, and the strength reduction factor, SRF , are adjusted to a rock slope situation rather than those in a tunnel. This means that the effects of weathering, such as the effect of expansion of water filled fractures by freezing, are included in the rock slope quality assessment. The Qslopeis calculated by equation 5.
Qslope= RQD J n × Jr
Ja
O
× Jw,ice SRFslope
(5) Where Jw,ice is the environmental and geological condition number and SRFslope is the strength reduction factor for the slope, including physical condition, stress and major discontinuity. The index O is the discontinuity orientation factor and allows for orientation adjustments for fractures day-lighting in rock slopes. The value obtained can then be used to find the steepest allowed slope angle without the need of rock mass reinforcement (Barton and Bar 2015), see chapter 2.5.1.
2.3.3 RMR-base and RMR-slope
Rock mass rating, RMR (Bieniawski 1989), is a method for rock mass characterization and clas- sification which is calculated based on rating of 6 parameters. The sum of these 6 parameters then make up the RMR-score. In Sweden, a base version of the RMR system is used in charac- terization of the rock mass quality (Lindfors et al. 2014), where only the parameters regarding strength, fracturing and alteration make up the RMR-rating. These ratings are here described as (ri): uni-axial compressive strength of the rock (r1), RQD (r2), spacing (r3), condition and orientation of discontinuities (r4), and ground water conditions (r5). Each of these parameters are rated depending on the parameters effect on the overall rock mass quality, according to equation 6. Table 3 presents the associated rock mass quality assessment.
RM Rbase= r1+ r2+ r3+ r4+ r5 (6)
Table 3: Rock mass quality assessment based on the RMR-system.
Very poor
rock Poor
rock Fair
rock Good
rock Very good rock
<21 21-40 41-60 61-80 81-100
Similarly to the Q-system, an adaptation for rock slopes can be applied in the RMR-system, equation 7 (Bieniawski 1989). It is done by rating how favourable the fractures are oriented to the slope (r6), and will result in a score ranging from 0 to -60, thus lowering the total rating of the rock mass. The rock mass quality according to RMR-slope can guide the engineer in terms of overall stability of the rock slope.
RM Rslope= r1+ r2+ r3+ r4+ r5+ r6 (7) 2.3.4 GSI
Geological strength index, GSI (Hoek 2000), is an index that is used to estimate the rock mass strength using the Hoek-Brown failure criterion, described in Chapter 2.4.1. It can also be used in itself as a way to characterize the rock mass according to its’ properties, see Figures 2 and 3.
Unlike RMR- and Q-systems, GSI has no direct application towards rock slopes.
Figure 2: Geological strength index for fractured rock produced by Hoek (2000) and Hoek and Karzulovic (2000).
Figure 3: Geological strength index for schistose metamorphic rocks (Hoek and Karzulovic 2000).
2.3.5 Stereonet analysis
Stereonet analysis is not a traditional rock mass characterization system, like the above mentioned, and does not result in a classification in terms of rock mass quality. Instead it is a way to present the amount of fractures and their orientations in the rock mass. The stereonet is a projection of planes onto a hemisphere, which can aid in interpretation of how planes intersect each other. Both the planes that are represented by the fractures in the area and the plane represented by the rock slope can be drawn onto the stereonet. This is a good way to visualize what types of failures that can be expected in the rock slope: wedge, sliding and/or toppling failure, see Figure 4. Stereonet projections are also a good way to visualize the natural variation in fracture orientations, and can aid in grouping fractures into fracture sets by plotting the poles of the measured fractures, which can then be grouped according to the the concentration they occur in. More importantly, it is a good way to describe and visualize the variation in fracture orientation in a fracture set.
(a) Sliding failure. (b) Wedge failure. (c) Toppling failure.
Figure 4: Common failure modes in fractured hard crystalline rock (Wyllie and Mah 2014).
Figure 4 can be compared to Figure 1 to illustrate how stereonets aid in visualization of the geological situation in the rock slope. Stereonets are also used to determine the direction of sliding in plane and wedge failure. The use of stereonets to identify potential failure modes in rock slope and how blocks that form can slide out of the slope is called a kinematic analysis. There are many software applications developed to help visualize fracture systems and performing kinematic analysis, such as Dips and S-wedge which are developed by RocScience.
2.4 Rock mass strength
The concept of rock mass strength is divided into strength of the intact rock, depending on com- position, texture and minerals, and into the strength of the rock mass, based on amount and characteristics of fractures/discontinuities within the rock mass. The effect of fractures on the rock mass strength is depending both on fracture properties such as orientation, spacing and per- sistence which determine the blockiness of the rock mass, but also on properties like roughness and infill, which determine the shear strength of the fractures.
2.4.1 Intact rock
The Hoek-Brown failure criterion (Hoek, Carranza-Torres, and Corkum 2002) can be used to determine at what effective stress the rock mass will fail. In rock slope engineering, this criterion should not be used to describe a geology and setting where the properties of the fractures in the rock mass has a large effect on the stability, but instead when the rock mass can be considered to be of an isotropic character. That is, when the rock mass is intact or when there are plenty of fracture sets, thus more crushed character. For rock mass with many fracture orientations/crushed rock, equation 8 is used:
σ10 = σ30 + σci
mb
σ30 σci
+ s
a
(8) where σ01and σ03are the effective principal stresses, σciis the uni-axial compressive strength of the intact rock, mbis the Hoek-Brown constant for the rock mass and s and a are constants depending on the rock mass characteristics. For application in intact rock, Equation 9 is used:
σ10 = σ30 + σci
mi
σ30 σci
+ 1
0.5
(9) where mi is a constant. In excess to performing tri-axial tests to identify the parameters of stress and strength, σci, it is also necessary to determine the GSI-value, which is assessed according to Figure 2 and 3.
For uniaxial compressional strength of rock, σ30 is equal to 0, thereby simplifying equation 8 to equation 10.
σc= σcisa (10)
For applications in rock slopes, when the rock tend to fail in the rock mass rather than along a discrete fracture, the concept of a "global rock mass strength" can be useful. The rock mass strength, σcm0 , is calculated using the Hoek Brown failure criterion in equation 11 (Hoek and Brown 1997).
σ0cm= σci(mb+ 4s − a(mb− 8s))(mb/4 + s)a−1
2(1 + a)(2 + a) (11)
using the Mohr-Coulomb relaionship, equation 12.
σ0cm= 2c0cos φ0
1 − sin φ0 (12)
where c0 is the cohesive strength and φ0 is the angle of friction.
2.4.2 Shear strength of fractures
In the cases where the strength of a rock slope is depending on the structural geological conditions and a failure can be expected along the fracture surface, the friction angle and cohesion must be determined for the fracture in order to perform stability analyses. In rock, these parameters are depending on infill material in the fracture, but in a fresh fracture, the cohesion will be zero and the friction angle will be related to the grain size and shape of grains in the rock, ranging from approximately 20°-40°. The shear strength, τ, can be expressed by equation 13 according to the Mohr Coulomb failure criterion. However, the failure envelope of a fracture is curved while the Mohr Coulomb criterion is a linear approximation of this curve. Thus, depending on the stress range, an apparent cohesion could sometimes be used for fractures.
τ = c + σntan φp (13)
Where φpis the peak friction angle and c is the cohesion. At some shear stress, the fracture surface will fail, and the shear stress required for displacement will become constant, see Figure 5a. This state is called the residual shear strength, and when plotting the residual shear strength values at a given normal stress in a Mohr diagram (Figure 5b) it is evident that also the cohesion is lost.
This is expressed by equation 14:
τ = σntan φr (14)
where φr is the residual friction angle. Equation 13 and 14 are valid assuming that the fracture surface is parallel to the shear stress direction.
(a) Plot showing shear displacement over shear stress.
(b) Mohr Coulomb plot of peak and residual shear strength.
Figure 5: Plots showing the concept of shearing on fracture surfaces (Wyllie and Mah 2014).
Notice also that the residual friction angle is less than the peak friction angle. This is explained by shearing of the fracture surface, thus crushing parts of the irregularities. These irregularities
are called asperities and increase the shear strength of fractures (Patton 1966). Under low normal stresses, when sliding over the asperities occur, the shear strength could be expressed as described in equation 15:
τ = σntan(φr+ i) (15)
where i is the angle of the asperities in relation to the direction in which the shear force is act- ing. It is important to consider the effect of the scale in which the asperities are measured, as the roughness of a surface can be described both in large structure, such as how the surface undulates, and in small structures on the surface, such as bumps and ripples, which will have a larger i-value. This is defined as first order and second order asperities, respectively (Patton 1966).
The peak shear strength, τf of a rock fracture was also described by Barton (1973) and Barton and Choubey (1977) and is expressed according to equation 16:
τf = σn0 tan
φb+ J RC log10 JCS σ0n
(16) where φb is the basic friction angle (measured from sawed sample). However, when the joint surface is altered the residual friction angle should instead be used. JRC is the joint roughness coefficient and JCS is the compressive strength of the joint surface [joint is synonymous to fracture in this report], both which can be estimated in the field (Barton and Choubey 1977). Equation 16 illustrate that asperities are sheared off at high stress levels relative to the strength of the intact rock, i.e. when the quotient of JCS and σn is equal to 1, the whole equation will be zero, and at low stress levels the quotient of JRC and σn will approach infinity and the roughness angle will thus be large. This is illustrated in Figure 6, where the roughness profiles (under A, B and C) are intended as approximations of roughness given by the JRC values in each chart.
Figure 6: Empirical law of friction (Barton and Choubey 1977). Each curve is numbered with the appropriate JCS value (MN/m2).
Barton and Choubey (1977) performed an extensive experiment with sliding block tests, using only the self weight of the block to determine the total friction angle in order to estimate JRC. By doing so, it was proved that with larger fracture persistence, the parameters JRC and JCS were reduced, thus introducing scale as an effect on shear strength of fractures.
An empirical formula for scale correction of JRC- and JCS-parameters (JRCn and JCSn) based on data from this study was then proposed by Barton and Bandis (1982), equation 17 and 18:
J RCn= J RC0 Ln L0
−0.02JRC0
(17)
J CSn= J CS0
Ln L0
−0.03JRC0
(18) where the laboratory measured joint parameters, JRC0 and JCS0, are scaled depending on the length of the blocks, Ln, along which the joint is sheared and the length of the sample L0.
2.5 Rock slope design methods
The practice of rock slope design includes determining the geometry of the slope and the necessity of possible support measures. In order to design a safe rock slope it is important to understand how it can fail. As mentioned the most common failure modes that can be expected are plane, wedge and toppling failures, but this can occur both as block fall and a larger failure of the rock mass, which generally concerns larger blocks sliding out of the slope. Circular failures can occur in large slopes or in slopes with very low strength. In large and/or complex slopes where this type of failure can occur, numerical analysis could be performed, in order to understand in what situation a failure would occur. For slopes in fractured crystalline rock, those cases are however very uncommon, why circular failure and numerical analysis are not evaluated in depth in this thesis.
The general design approach in geotechnical projects is often the so called "observational method".
This means that the design include a set of options for differing geotechnical conditions in order to take geological variation into account, see chapter 2.8.1. Thus the design is reviewed as more knowledge is obtained in the building and maintenance phase. This approach is an effect of the inevitable variation and uncertainty in soil and rock constructions, but few projects are documented were this method is applied as defined in Eurocode 7 (Spross 2014). Further, according to current requirements, safety margins for the stability of the construction are not mentioned when the observational method is applied, which is a limitation of the standard (Spross 2014).
2.5.1 Q-slope
The Q-slope method was developed as a simple way to determine a suitable slope angle, without the need of reinforcement, based on the rock mass quality calculated using equation 5. The steepest allowed angle, β, is then calculated according to equation 19 (Barton and Bar 2015).
β = 20 log10Qslope+ 65° (19)
It should be noted that the Q-slope method is not intended for stability assessment in large slopes with several bench levels, such as in large open pit mines, but can be applied for example in road and railway cuts and individual benches. The application of Q-slope on hard rock has been evaluated by back analysis of slope failures, both in the case of sliding failure and wedge failure, where the Q-slope method proved to provide slope angles sufficient for slope stability (Bar and Barton 2016).
2.5.2 Plane failure analysis
Plane failure (Figure 1a) can occur when the fracture plane is oriented parallel or close to parallel to the slope face and the dip angle of the fracture plane is less than that of the slope, but larger than the friction angle of the fracture plane. If a potential failure plane has been identified in a rock slope, a limit equilibrium analysis can be carried out to determine if the slope will fail or
not. The limit equilibrium analysis is carried out by weighing the resisting forces to the driving forces. If the quotient is larger than 1, the resisting forces are larger than the driving forces, and the slope is stable. But because there are inevitable uncertainties in geological parameters used for calculation, a factor of safety, FS, against sliding is needed. The factor of safety should be at least 1.2-1.4, but is often larger, and is calculated as shown in equation 20 (Wyllie and Mah 2014).
FS = cA + (W cos ψp− U − V sin ψp) tan φ
W sin ψp+ V cos ψp (20)
where c is the cohesion, A is the area of the sliding plane, W is the weight of the sliding block and ψp is the dip angle of the fracture plane that is subject to sliding. As previously mentioned φ is the friction angle of the rock. If the slope is drained the terms regarding uplifting water pressure, U, and the vertical water pressure, V , can be neglected, otherwise they are calculated according to equation 21 and 22:
U = 1
2γwzw(H + b tan ψs− z) 1
sin ψp (21)
V = 1
2γwzw2 (22)
where γwis the water density, z is the height of the tension crack and zwis the water depth in the tension crack. The total height of the slope, H, the distance from the slope crest to the tension crack, b and the dip of the slope above the crest, ψs, are used to calculate the geometry of the fracture plane on which the uplifting forces are acting. The concept of tension cracks (Figure 7) is important in plane failure analysis, because if the fracture that the plane failure would occur along does not daylight above the slope crest, they will be the reason for failure. They form because of small shear movements in the rock, and an important conclusion is that if there are tension cracks, shear failure has occurred in the rock. In this way tension cracks are an indication of a possible instability (Barton 1971).
(a) Slope with tension crack below slope crest (b) Slope with tension crack above slope crest Figure 7: Slope failure with water filled tension crack (Wyllie and Mah 2014).
For drained conditions, equation 20 can be developed into equation 23, where the normal stress acting on the fracture plane as well as the surface roughness are considered in the resisting forces, defined by the shear strength acting on the fracture plane, τA, as shown in Chapter 2.4.2.
FS = σntan(φ + J RC log10(J CS/σn))A
W sin ψp (23)
2.5.3 Wedge failure analysis
Wedge failures are more common than plane failure, and occur when at least two fracture planes intersect and form a wedge, see Figure 1b. Wedge failure occurs when the plunge of the line of intersection between the planes is flatter than the dip of the face and steeper than the average friction angle of the two slide planes. Using a stereonet, a kinematic analysis can be performed to determine the direction of sliding and the shape of the wedge. The line that is made up of
the intersecting point of the planes and the center point in the stereonet illustrates the direction of sliding, and the shape of the wedge is evident on the stereonet. The friction angle of the rock material can also be drawn onto the stereonet to help determine if a failure is possible. Similarly to in plane failure analysis, limit equilibrium analysis is used to calculate the factor of safety of wedge failure, as shown in equation 24 (Wyllie and Mah 2014).
FS =(RA+ RB) tan φ
W sin ψi (24)
where ψi is the plunge of intersecting line between the planes, and RA and RB are the normal reactions for plane A and B, see Figure 8a, and are solved by dividing them into normal and parallel components to the direction of the intersecting line, as in equation 25 and 26:
RAsin(β −1
2ξ) = RAsin(β +1
2ξ) (25)
RAcos(β −1
2ξ) + RAcos(β +1
2ξ) = W cos ψi (26)
where ξ is the angle between the planes forming the wedge and β is the angle between the midpoint of the wedge (i.e. 12ξ) and the slope face as shown in Figure 8b.
(a) Cross section of a wedge, seen from slope face, showing the reactions RAand RB.
(b) Wedge illustrated on a stereonet, showing the angles ξ and β.
Figure 8: Definition of RA, RB, β and ξ (Wyllie and Mah 2014).
The reactions RA and RB are found from equation 25 and 26 by solving as described in equation 27:
RA+ RB= W cos ψisin β
sin(ξ/2) (27)
Which means:
FS = sin β sin(ξ/2)
tan φ
tan ψi (28)
or,
FSw= KFSp (29)
where FSwis the factor of safety against wedge failure and FSpis the factor of safety against plane failure with the same dip as the intersection line, and the same friction angle, as the corresponding wedge failure. The constant K is the so called wedge factor. The above described methods of determining the factor of safety does not include cohesion in the fracture planes. If assumed that the sliding of the wedge occurs along the intersection line of the two fracture planes, cohesion can be included in the wedge failure analysis (Hoek, Bray, and Boyd 1973), according to equation 30:
FS = 3 γrHw
cAX + cBY
+
A − γw
2γr
X
tan φA+
B − γw 2γr
Y
tan φB (30) where cA and cB is the cohesion of the fracture planes, and φA and φB are the friction angles of the fracture planes. γr and γw are the unit weights of rock and water and Hw is the total height of the wedge. The parameters X, Y , A, and B are given by equations 31, 32, 33 and 34:
X = sin θ24
sin θ45cos θ2.na (31)
Y = sin θ13
sin θ35cos θ1.nb (32)
A = cos ψa− cos ψbcos θna.nb
sin ψ5sin θna.nb (33)
B = cos ψb− cos ψacos θna.nb
sin ψ5sin θna.nb (34)
where ψaand ψbcorrespond to the dip angles of the planes A and B, and ψ5is the line of intersection between these. The θ-angles are defined by the geometry of the wedge, as illustrated in Figure 9.
Figure 9: Definitions of the angles used as input data in equations 31-34 (Wyllie and Mah 2014).
Numbers 1-5 represent the intersecting lines of the fracture planes and the face (1 and 2), intersec- tion of the fracture planes and the ground surface above the slope crest (3 and 4), and intersection of the fracture planes themselves (5). If there is no cohesion and the slope is drained, equation 30 can be simplified to equation 35:
FS = A tan φA+ BφB (35)
The limitation of the analysis described in these equations are however that it does not consider a tension crack, nor external forces, like bolts for reinforcement. For theses cases, a more extensive analysis based on equation 24 may be needed (Wyllie and Mah 2014).
2.5.4 Limit equilibrium to calculate stability against toppling
Toppling failure occurs when the fracture plane dips towards the slope face, see Figure 1c. Top- pling failure of blocks involves rotation around a fixed base, and the failure of the slope is when the blocks are falling out of the slope because the center of gravity of the blocks are located outside the base of the block. Another kind of toppling failure is flexural failure which occurs in rock with schistose/slaty characteristics, where slabs of rock dipping steeply towards the slope break by flexural tension. Failure can also occur as the result of a combination of block toppling and flexural toppling. A toppling failure in a slope will often consist of a system of blocks interacting with one another, where some will be stable, some will slide and some will topple as shown in Figure 10. A limit equilibrium analysis can be preformed based on this figure to find the factor of safety of the whole slope (Goodman 1976).
Figure 10: Toppling failure in slope, equilibrium analysis model (Wyllie and Mah 2014).
The block shape test is a quick way to determine if a block at risk of toppling will fail, using simple mechanics. The block is stable against sliding when the dip of the base plane, ψp, is less than the friction angle between the block and the base plane, φbp, as the condition in equation 36 implies.
The block is unstable against toppling when the quotient of the block width, δx, over the block height, y, is less than tan ψp as the condition in 37 shows.
ψp< φbp (36)
∆x
y < tan ψp (37)
The inter-layer slip test determines if sliding is possible along slabs, and the conditions shown in equation 38 must be satisfied for a toppling failure to occur:
ψd ≤ 90 − ψf+ φd (38)
where ψd is the dip of the planes that make up the sides of the slabs, φd is the respective friction angle and ψf is the dip of the slope face. Another condition for failure is that the fracture planes subject to toppling failure are oriented parallel or close to parallel to the face, therefore equation 39 must be fulfilled:
|αf− αd| < 10° (39)
where αf is the dip direction of the slope face and αd is the dip direction of the fracture plane.
There are many softwares developed to preform limit equilimbrium analysis, for example RocFall RocPlane RocTopple, Slide (2 and 3) and RS (2 and 3), which are all developed by RocSience.
These programs are using different assumptions for calcultation and it is important that these assumptions are verified in every case.
2.5.5 Circular failure and numerical analysis
Circular failures are typically dealt with in the practice of soil mechanics, as they normally occur in weaker materials like clay or sand. In heavily weathered rock circular failure can however occur, but it is an uncommon mode of failure, especially in the application of hard crystalline rock.
Circular failure analysis is carried out by dividing the potentially unstable part of the slope over an interpreted circular sliding surface into vertical slices, see Figure 11a. For each slice the resisting and the driving moments are calculated as illustrated in Figure 11b. By adding these together, the factor of safety for the slope can be calculated (Wyllie and Mah 2014).
(a) Cross section of a potentially un- stable circular slope surface
(b) Slice, limit equilibrium analysis is carried out on each slice
Figure 11: Circular slope analysis by slice calculation (Wyllie and Mah 2014).
Charts to determine the factor of safety for circular failures have been developed. These charts were developed as a way to quickly estimate the factor of safety and they are adapted for different ground water conditions, ranging from fully drained to fully saturated. An example showing a circular failure chart that can be used in slopes with fully drained conditions is showed in Figure 12. The chart is used by calculating the ratio c/(γH tan φ) and finding this value on the outer circular scale, then following the radial line to the intersection of the curve of the corresponding slope angle valid for the slope. Then read the x- or y-axis value, depending on which is most convenient, and solve for the factor of safety, FS.
In the subject of rock mechanics, numerical analysis is the use of mathematics to define and calcu- late the mechanical properties with regards to stability. Systems of equations are built to determine the stability problem. These can be very extensive as no approximations or simplifying assump- tions are needed, and can therefore be very difficult or even impossible to solve in other ways than by the aid of a computer (Wyllie and Mah 2014). This can be applied to complex failures where for example more than one failure mechanism is included, or rotational failures, which cannot be analyzed with limit equilibrium methods. There are limitations to numerical analysis however.
When using the continuum approach, the assumption is that the rock mass is and behaves like a homogeneous unit. This is a simplification that is rarely fulfilled in reality. If the discontinuous approach is instead chosen, the rock mass need to be divided into discrete blocks intersected by fractures to represent the reality, which will also yield a simplified model and includes uncertainties related to the analysis of the site conditions. Additionally, it is time-consuming to set up, why numerical analysis focus mainly on large open pit mines and to study land slides (Wyllie and Mah 2014). An example of a discrete numerical tool that can be used to set up a numerical analysis is 3DEC.
Figure 12: Circular slope analysis by circular failure chart, here in fully drained conditions (Wyllie and Mah 2014).
2.5.6 Engineering judgment
Limit equilibrium analysis, which is the most commonly used method for calculation of failure in a slope, is a relatively quick way to determine the rock slope stability. Often, however, it is not necessary at all to calculate the stability against failure, because no critical failure is expected. In the case of small plane, wedge or toppling failures, it is instead common that wedges or blocks are reinforced based on engineering judgment. Spot bolting is an example and means that rock bolts are installed where a possible failure in the rock can occur, for example in a wedge formation, and how many bolts that may be needed to secure the block is then determined based on experience.
2.6 Excavation methods
The excavation method is very important in rock slope engineering as excavation can induce dam- age and fracturing in the rock, thus reducing its’ strength. To retain he strength of the over-all rock mass and limit the degree of fracturing is especially important in steep slopes where no or very limited rock fall can be tolerated. In civil projects, like foundations for buildings, it is common to design steep slopes and in these cases the rock should be excavated carefully, to maintain as much strength as possible. In hard crystalline rock, it is common to excavate by drill and blast, but wire sawing and hydraulic fracturing are other methods that are used. The advantage of the latter two is that the damage on the rock is much smaller.
