Rock as a material; Elasticity and strength of rock; Linear elasticity and laboratory testing of rocks.
A term paper on
CVEN834: Rock Mechanics
Onah Mary Omada
Engr. Prof K.J. Osinubi
Department of Civil Engineering
Faculty of Engineering
Ahmadu Bello University, Zaria.
Some knowledge of rock mechanics is very important in the study of civil engineering as it is the branch of geotechnical engineering concerned with the properties of rocks and the special methodology required for design of work related components of engineering schemes. Geotechnology is a term used to describe both the science and engineering of soil deposits, rock masses and the fluids they contain. In general, civil engineering deals with two types of earth materials: soils and rocks.
Rock as a material is a natural geological substance, a solid aggregate of one or more minerals. It is discontinuous, anisotropic, inhomogeneous and inelastic which contains numerous randomly oriented zones of initiation of potential failure, like initials joints, defects, cavities or other natural flaws. Rocks has been used by humankind throughout history and the minerals and metals in rocks have been essential to human civilization. The major classes of rocks according to the process that result into their formation are: igneous, sedimentary and metamorphic rocks. These three classes, in turn are subdivided into numerous groups and types on the bases of various factors, the most I important of which are chemical, mineralogical and textural attributes.
Consequently, the ability to predict the strength behavior of rock materials is very important for the design engineer because it allows him better appreciation of the problem in hand, clarifies the influence of different variables and provides an estimate of the design parameters.
A combination of laboratory testing of small samples of rocks, empirical analysis, and field observation should be employed to determine the requisite engineering properties. Common engineering properties typically obtained from laboratory test include specific gravity, point load strength, compressive strength, tensile strength, shear strength, modulus and durability.
2.0ROCK AS A MATERIAL
Rock material refers to the intact rock within the frame work of discontinuity. In other words, this is the smallest element of rock block not cut by any fracture. Rock or stone is a natural substance, a solid aggregate of one or more minerals. For example, granite, a common rock is a combination of the minerals quartz, feldspar and biotite. Rocks has been used by humankind throughout history and can be used in themselves as raw sources for construction materials (aggregates, construction stones, decorative stones, etc..). The use of rocks has had a huge impact on the cultural and technological development. Also, many engineering activities involve rocks either as construction or foundation n material. These include:
Design of foundations for buildings, bridges, dams, towers, etc…
Design of rock slopes and surface excavations for canals, highways, railways, spillways, pipelines, penstocks, dam abutments, open pit mines, quarries, etc.
Design of underground excavations such as tunnels, mines and other underground chambers,
Design of structures associated with energy development such as underground nuclear plants, repositories for storage of nuclear and chemical wastes, LNG and oil.
rock is a very complex material that can be:
Discontinuous with micro-discontinuities (pores, micro cracks) and macro-discontinuities (joints, shears, faults)
Anisotropic if its properties vary with directions as for sedimentary rocks, foliated metamorphic rocks and regularly jointed rocks;
Heterogeneous if its properties vary from point to point as in multilayered rock masses.
Note that rocks and especially rock masses can rarely be described as isotropic, homogeneous continua.
2.0CLASSIFICATION AND INDEX PROPERTIES OF ROCKS
2.1Classification of rocks
Rocks are classified according to characteristics such as mineral and chemical composition, permeability, texture of the constituent particles and particle size. These physical properties are the result of the processes that forms the rock. Over the course of time, rocks can be transform from one type into another as described by a geological model called the rock cycle. These transformation produces three general classes of rocks:
Igneous rock: This group of rock is formed through the cooling and solidification of molten magma or lava. The magma can be derived from partial melts of existing rocks in either a planet’s mantle or crust. Typically, the melting is caused by one or more of three processes:
An increase in temperature,
A decrease in pressure, or
A change in composition.
Solidification into rock occurs either below the surface as intrusive rocks or on the surface as extrusive rocks. Igneous rock may form with crystallization to form granular, crystalline rock or without crystallization to form natural glasses. Examples of this Examples of this rocks basalt, granite, rhyolite, obsidian, pumice.
sedimentary rock: Are the types of rock that are formed by the deposition and subsequent cementation of that material at the Earth’s surface and within bodies of water. The particles that form a sedimentary rock by accumulating are called sediment. Before being deposited, the sediments was formed by weathering and erosion from the source area, and then transported to the place of deposition by water, wind, ice, mass movement, glaciers which are called agents of denudation. Examples of this rocks include: conglomerate, breccia, sandstone, shale.
Metamorphic rock: This rock arises from the transformation of existing rock types, in a process called metamorphism, which means change in form. The original rock (photolith) is subjected to heat (temperatures greater than 150 to 200 o C) and pressure (100 mega Pascal (1,000 bar) or more), causing profound physical or chemical change. The photolith may be a sedimentary, igneous or existing metamorphic rock. Examples of this rocks include: quartzite, marble, slate, phyllite, schist, gneiss.
These three classes are subdivided into many groups. These are, however, no hard and fast boundaries between allied rocks. By increase or decrease in the proportions of their minerals, they pass through gradations from one to the other; the distinct structures of the kind of rock may then be traced gradually merging into those of another. In general, the name of a rock is not sufficient to narrow down its engineering properties. It is not because we have a limestone or a granite that we have a strong rock. Information supplied by geologists to engineers may appear at a first glance to be of limited value. This is not so in geological engineering where geological data are of prime importance. The cooperative work between geologists, engineering geologists and engineers can help engineers in deriving useful engineering conclusions in the early stage of a project. Long term problems and costly remedial actions can be avoided.
The lithological name of a rock gives a range within which the engineering properties of the intact rock should fall. This can be useful in the preliminary design stage where test data are not yet available and preliminary decisions need to be made. Most textbooks in rock mechanics and engineering geology have tables of compressive strength, Young’s modulus, etc.. for various rock types (Goodman, 1989; U.S. Bureau of Reclamation, 1953).
The lithological name, the age, the texture and the fabric of a rock (i.e the uniformity of the texture within the rock) can provide qualitative information on its engineering properties. For instance, rocks with a crystalline texture consists of highly interlocked crystals of silicates, carbonates or sulfates. They are usually strong, elastic and brittle when unweathered (fresh). Carbonates and sulfates may show a ductile behavior rather than a brittle behavior at medium to high temperatures and confining levels. The engineering properties of rocks with clastic textures will depend on the relative proportion of particles and cement and durability of the cement. For instance, a poorly cemented sandstone will certainly show a lower strength, higher deformability and weather ability than a highly cemented one. Foliation and bedding planes result in highly directional (anisotropic) rock strength and deformability. Finer rock grain size leads to higher fracture strength. Older rocks that have been buried at larger depths will show more compaction and smaller porosities. Rocks with fine textures and rich in silica can be expected to be more abrasive and wear drilling bits and machine cutters.
The petrographic description of the rock may give some information about minerals that could create some engineering problems such as gypsum, montmorillonite, chert, feldspars, asbestos, etc. The lithological name may be associated with specific features that could cause engineering problems such as karsts in limestone formations or columnar jointing in basalt. Clay bearing rocks are expected to be very sensitive to water and weathering and to be susceptible to slaking and swelling. The combination of more than one rock can give properties much worse than each rock alone; a good example would be a sedimentary rock mass consisting of multiple layers.
Finally, field observations can give information about the degree of rock mass fracturing and weathering.
2.2INDEX PROPERTIES OF ROCKS
As a result of the vast range in properties of rocks, which reflects varieties of structures, fabrics and components, we rely on a number of basic measurements to describe rocks quantitatively. Certain properties that are relatively easy to measure are valuable in this regards and may be designated index properties for rock specimens.
Porosity: The shape, size and nature of packing of the grains of a rock give rise to the property of porosity or development of pore spaces within the rock. Numerically, it is expressed as the ratio between the total volume of the rock sample. And represented as:
Where, n=Porosity, Vp is the volume of pores in total volume Vt.
Porosity is commonly given in percentage terms. Presence of interlocking crystals, angular grains of various sizes and abundant cementing materials are responsible for low porosity of stones.
Porosity is an important engineering property of rock. It accounts for the fluid adsorption value of the stones in most cases and also that a higher porosity signifies a lesser density which generally means a lesser compressive strength.
Absorption value: It defines the capacity of a stone to absorb moisture when immersed in water for 72 hours or till it gets full saturation. It is generally expressed in percentage terms of original dry weight of the mass. It may be obtained from the relationship
QUOTE x 100
Where Ws = weight at saturation; Wo = dry weight of the sample used.
Hydraulic permeability and conductivity: Measurement of the permeability of a rock sample may have direct bearing on a practical problem, for example, pumping water, oil, or gas into or out of a porous formation, disposing of brine waste in porous formation, storing fluid in mined caverns for energy conversion, assessing the water tightness of a reservoir, dewatering a deep chamber, or predicting water inflows into a tunnel.in many instances, the system of discontinuities will radically modify the permeability values of the rock in the field as compared to that in the lab. Permeability is the capacity of the rock to transmit water. Sand stones and lime stones may show high values for adsorption or 10% or even more. Selection of such high porous verities of these stones for use in building construction, especially in most situations, would be greatly objectionable. The presence of water within the pores not only decreases the strength of the rock but also makes the stones very vulnerable to frost action, in cold and humid climatic conditions. Also, the degree to which the permeability changes by changing the permeant from air to water expresses interaction between the water and the minerals or binders of the rock and can be detect subtle but fundamental flaws in the integrity of the rock; this promising aspect of permeability as an index has not been fully researched.
Most rocks obey Darcy’s law. For many applications in civil engineering practice, which may involve water at about 20oC, it is common to write Darcy’s law in the form
Where, qx is the flow rate (L3T-1) in the x direction. h, is the hydraulic head with dimension
L. A is the cross sectional area normal to x (dimension L2).
The coefficient k is termed the hydraulic conductivity. Permeability can be determined in the laboratory by measuring the time for a calibrated volume of fluid to pass through the specimen when a constant air pressure acts over the surface of the fluid.
Density: it is defined as weight per unit volume of a substance, but in the case of rock, it is not only the solid mineral matter which wholly accounts for the total volume of a given specimen. A part of the rock may comprise of pores or open spaces, which may be empty, partly filled or wholly filled with water. Accordingly, three types of density may be distinguished in rocks. They are:
Dry density: it is the weight per unit volume of an absolutely dried rock specimen, it includes the volume of the pore spaces present in the rock.
Bulk density: it is the weight per unit volume of rock sample with natural moisture content where pores are only partially filled with water.
Saturated density: it is the density of the saturated rock or weight per unit volume of a rock in which all the pores are completely filled with water.
Strength: The value of having an index to rock strength is self evident. The problem is that strength determinations or rock usually require careful test setup and specimen preparation and the result are highly sensitive to the method and style of loading. An index is useful only if the properties are reproducible from one laboratory to another and can be measured inexpensively. Such a strength index is now available using the point load test described by Broch and Frankine (1972). In this test, a rock is loaded between hardened steel cones causing failure by the development of tensile cracks parallel to the axis of loading. The test is an outgrowth of experiment with compression of irregular pieces of rock in which it was found that the shape and the size effects were relatively small and could be accounted for and in which the failure was usually by induced tension. In the Broch and Franklin apparatus, which is commercially available, the point load strength is
Where, P is the load at rupture and d is the distance between the point loads.
Slaking and durability
Durability of rocks is fundamentally important for all applications. Changes in the properties of rocks are produced by exfoliation, hydration, decrepitation (slaking), solution, oxidation, abrasion and other processes. In some shales and some volcanic rocks, radical deterioration in rock quality occurs rapidly after a new surface is uncovered. Fortunately, such changes usually act imperceptible through the body of the rock and only the immediate surface is degraded in terms of years. At any rate, some index to the degree of alterability of rock is required. Since the paths to rock destruction dense by nature are many and varied, no test can reproduce expectable service conditions for more than a few special situations. This, an index to alteration is useful mainly in offering a relative ranking of rock durability.
Other physical properties: Other physical properties of rocks such as Abrasive resistance, frost and fire resistance are important to specific engineering tasks in rocks.
3.0ELASTICITY AND STRENGTH OF ROCKS
3.1Elasticity of rocks
The theory of elasticity is used widely in rock mechanics to predict how rock masses respondto loads and excavation (surface and underground). The assumptions inherent to the theory ofelasticity are:
the material is elastic (linear or non-linear) which implies an immediate response duringloading and a fully reversible response upon unloading,
the material behaves as a continuum.
If time is involved (time-deferred response), the theory of viscoelasticity should be used instead.Elasticity is the property of reversibility of deformation when subjected to a load. Hook’s law describes the behavior of elastic materials and states that for a small deformation, the resulting strain is proportional to the applied stress.
Stress is the force applied per unit area.
Strain is the fractional distortion that results because of the acting force.
The modulus of elasticity is the ratio of stress to strain
Many fresh, hard rocks are elastic when considered as laboratory specimens. But on the field scale rocks can be expected to contain fractures, fissures, bedding planes, contacts, zones of altered rock and clays with plastic properties. Therefore, most rocks do not exhibit perfect elasticity. The extent of irrecoverability of strain in response to load cycles may be important for the design and can be determined by the slope of the load/deformation curve. Most problems in rock mechanics are three-dimensional. Under certain assumptions of plane strain or plane stress, a two-dimensional approximation can be made. In plane stress problems, the geometry of the rock mass is essentially that of a plate with one dimension much smaller than the others. Plane stress is used for instance to model surface problems. In plane strain problems, the geometry of the body is essentially that of a prismatic cylinder with one dimension much larger than the others. These notes are limited to two-dimensional elasto-static problems.
Rocks and rocks masses do not always behave elastically or as continua. Nevertheless, for a wide range of engineering problems, useful solutions may be obtained by treating the rock as a homogeneous, linearly elastic continuum. If necessary, anisotropy and nonlinearity may be taken into account. Despite its limitation, the elastic analysis can be used to evaluate a number of factors of importance in rock engineering.
Examination of solutions to elastic stress distribution problems can provide useful qualitative guidelines for the design of engineering structures built in or on rock. For instance, elastic stress analysis provides an assessment of the extent of compression and tension zones, the zones of influence of excavations, the stress intensity and the extent of highly stressed zones. A failure criterion can be superposed on the elastic solution to assess the extent of overstressed rock around the excavations of interest. The basic assumption behind this approach is that the existence of the plastic zone does not disturb the stress distribution. A failure criterion can also be included in the solution (elasto-plastic analysis). Elastic analysis can also help in determining the rock deformation associated with the stresses. Deformation, and not stress, may sometimes become the critical factor in rock engineering design.
3.2CLOSED FORM SOLUTIONS VS NUMERICAL METHOD
Elasticity problems can be solved using closed-form solutions or numerical methods depending on the complexity of the material of interest and the geometry of the problem being addressed. Closed-form solutions are usually used when the material is homogeneous, isotropic (or anisotropic), and the boundary of the problem is of simple shape.
Closed form solutions: are available in various texts such as Obert and Duvall (1967), Timoshenko and Goodier (1970), Lekhnitskii (1977), Jaeger and Cook (1979), Poulos and Davis (1974), etc. Closed-form solutions can be derived, for instance, to study the response of a rock mass to surface loads, and surface or underground excavation. Various closed-form solutions are also available for the analysis of field and laboratory rock mechanics tests.
Numerical methods: are used instead when the material has a complex constitutive behavior (non-linear behavior, elasto-plastic, etc.) and/or the problem geometry is complicated. The most frequently applied numerical procedures in rock engineering and rock mechanics are the finite element method (FEM), boundary element methods (BEM), and the discrete element methods(DEM). Both FEM and BEM are used when the rock mass is modelled as a continuum with several discrete planes of weakness. DEM is used when the rock mass contains a large number of blocks, and its deformation is controlled mostly by the opening of, closing of, and sliding along the discontinuities and to a lesser extent by the block deformation. Hybrid methods are also used in order to preserve the advantages of each method and eliminate their disadvantages.
Finite Element Method
Efficient numerical solution procedure.
Complex constitutive behavior can be modelled.
Requires discretization of the complete problem domain.
Arbitrary external boundaries are needed.
The size of the numerical problem to be solved is related to the volume of the problem domain.
Boundary Element Methods
No requirement to define arbitrary external boundaries from the problem area.
Size of the numerical problem increases with the size of surface area of excavation.
Volume of the problem is considered explicitly in the analysis.
Discretization of problem boundaries only.
Low demand on computer storage.
Simplicity of data input.
Calculate stresses, strains and displacements at points of interest.
Limited to rock masses with linear constitutive behavior.
Discrete Element Methods
Analysis of large block movement in geologically complex regions having many joint blocks.
Can be performed by a microcomputer and displayed interactively.
Precise location of joints needs to be input.
BEM, FEM, DEM Coupling
BEM for far-field rock mass (continuum) and DEM near-field rock mass (discontinuum).
Coupling BEM and FEM eliminates boundaries and uncertainties associated with outer boundary conditions.
Far-field rock mass is a homogeneous continuum modeled with BEM and near-field rock mass is modeled as a continuum with zones (usually small and localized) of complex constitutive behavior.
The word “failure’ ‘connotes an almost total loss of integrity in a sample of rock; in an engineering context, it usually implies loss of ability to perform the intended function. Obviously, the phenomena that constitute failure will depend on the function varying from loss of a commodity in storage to structural collapse, property damage and death.
Rock failure modes, however are complex and difficult to quantify or predict. A compressive study on rock failure modes at laboratory scale, is therefore potentially important as it helps recognize the adequacy of the support designed on the basis of the nature of an engineering work. The varieties of load configuration in practice are such that no single mode of rock failure predominates. Intact, flexure, shear, tension and compression can each prove most critical in particular instances.
Flexural: Refers to failure by bending, with development and propagation of tensile crack.
Shear failure: Refers to formation of a surface of rupture where the shear stress has become critical, followed by release of the shear stress as the rock suffers a displacement along the rupture surface. This is common in slopes cut in weak, soil-like rocks such as weathered clay shales and crushed rock of fault zones.
Direct tension: is occasionally set up in rock layers resting on convex upward slope surfaces and in sedimentary rocks o0n the flank of an anticline. The base of the slope has layers inclined more steeply than friction will allow and the balance of support for the weight of the layers is the tensile pull from the stable part.
Crushing or comprression failure: occurs in intensely shortened volumes or rock penetrated by a stiff punch. Examination of processes of crushing shows i to be a highly complex mode, including formation of tensile cracks and their growth and interaction through flexure and shear.
3.3.1 Parameters affecting rock strength
Rock strength depends on many parameters (Paterson, 1978) including:
Rock type and composition
Rock grain size
Rock density and porosity
Rate of loading
confining stresses QUOTE 2 and QUOTE 3
Geometry, size and shape of the test specimens
Water pore pressure and saturation
Testing apparatus (end effects, stiffness)
3.3.2 Theory of strength
Several theories of strength have been applied to rocks. They include;
Mohr (1900) proposed that when shear failure takes place across a plane, the shear and normal
stress components QUOTE and QUOTE n acting across that plane are uniquely related as follows
QUOTE = QUOTE n) (1)
Several assumptions are inherent to this criterion:
Failure takes place when the Mohr circle with radius ( QUOTE 1- QUOTE 3)/2 is tangent to the Mohr envelope representing equation (1). Failure is assumed to depend only on the major and minor applied principal stresses QUOTE 1 and QUOTE 3 and is independent on the value of the intermediate principal stress QUOTE 2.
Two failure planes develop in a direction parallel to QUOTE 2. The angle between the planes of failure and QUOTE 1 increases with ( QUOTE 1+ QUOTE 3)/2; this trend has been observed in laboratory tests.
Tensile and compressive strengths are not equal.
The simplest form of the Mohr envelope is a straight line with equation
Where So is the so-called inherent shear strength of the material and QUOTE is the so-called coefficient of internal friction. The latter is equal to tan QUOTE where QUOTE is the so-called friction angle by analogy with surface sliding. Values of So and QUOTE can be found in Goodman (1989).
According to the Mohr-Coulomb criterion, failure takes place along two conjugate planes inclined at QUOTE f = QUOTE /4- QUOTE 2 with respect to the major principal stress, QUOTE 1, and parallel to the intermediate stress QUOTE 2. Also, the orientation of the failure planes does not depend on the stress level.
Consider the failure plane shown in Figure 13a and inclined at an angle QUOTE f with respect to QUOTE 1. Using coordinate transformation rules, the normal and shear stresses acting on that plane are equal to
QUOTE n = (( QUOTE 1 + QUOTE 3)/2 – ( QUOTE 1 – QUOTE 3)/2)Sin QUOTE (3)
QUOTE = (( QUOTE 1 – QUOTE 3)/2)Cos QUOTE
Substituting the expression of QUOTE n and QUOTE into equation (2), the Mohr-Coulomb criterion can also be expressed in terms of principal stresses as follows;
QUOTE 1 = Co + QUOTE 3(Co/TIo)(4)
CO = 2SO tan( QUOTE /4- QUOTE 2) (5)
Note that TIo is not the uniaxial tensile strength of the material since equation (2) implies that, at the time of failure, the normal stress QUOTE n is positive. It can be shown that this condition implies that QUOTE 3 must be larger than -TIo/2 and QUOTE 1 must be larger than Co/2. Thus, the two-parameter (So, QUOTE ) Mohr-Coulomb criterion can be expressed as
QUOTE 1 = Co + QUOTE 3(Co/TIo) when QUOTE 1> Co/2 (6)
QUOTE 3 = -To = TIo/2 when QUOTE 1< Co/2
According to the two-parameter criterion, the tensile strength is still related to So and QUOTE . The horizontal line QUOTE 3 = -To is called the tension cutoff.
A three-parameter Mohr-Coulomb criterion has been proposed. The three parameters include So and QUOTE as before and the tensile strength To which is now assumed to be independent of the other two parameters. The criterion is expressed as follows
QUOTE 1 = Co + QUOTE 3(Co/TIo) when QUOTE 1> QUOTE 1c (7)
QUOTE 3 = -To when QUOTE 1< QUOTE 1c
with QUOTE 1c=Co(1-To/TIo). Note that To must be smaller than TIo /2 and therefore F1c must be larger than Co/2 in order to keep the normal stress across the fracture planes always positive.
Table 3 gives the value of the strength ratio Co/To when N=30° for the three cases considered above, i.e. no tension cutoff, two-parameter criterion with tension cutoff, and three-parameter with tension cutoff.
Hoek and Brown Criterion
This is an empirical criterion that does not depend on the intermediate principal stress. It was first proposed by Hoek and Brown (1980b) to model the strength of intact rock and rock masses. In terms of principal stresses, it is expressed as follows
QUOTE 1 = QUOTE 3 + QUOTE c QUOTE 3 + S QUOTE 2c) (8)
QUOTE 1 and QUOTE 3 are the major and minor principal stresses at failure,
QUOTE c=Co is the uniaxial compressive strength of the intact part of the rock mass, and,
m, s are constants that depend on the extent to which the rock mass has been broken before being subjected to the principal stresses. s=1 for intact rock and m and s decrease as the rock mass becomes more fractured.
The three parameters m, s and QUOTE c can be determined by regression analysis. The intact rock tensile strength, To= QUOTE t, predicted by the Hoek and Brown criterion is obtained by setting QUOTE 1=0 and QUOTE 3=-To in equation (8). This gives,
QUOTE t = To = QUOTE c/2 + QUOTE )-m) (9)
According to equation (8) the strength ratio QUOTE c / QUOTE t (or Co/To) depends on the value of m only. For limestone m=5.4 and Co/To=5.6; for granite m=29.2 and Co/To=29.2; for sandstone m=14.3 and Co/To=14.3.
The Hoek and Brown criterion for intact rock can also be expressed in terms of normal and shear stresses as follows
FORMULA QUOTE n = A( QUOTE n – QUOTE on)B(10)
QUOTE n = QUOTE / QUOTE c ; QUOTE n= QUOTE / QUOTE c ; QUOTE on = QUOTE o/ QUOTE c
QUOTE = QUOTE 3 + ( QUOTE 2 m/ QUOTE m + m QUOTE c/8)(11)
? = ( QUOTE – QUOTE 3) QUOTE /4 QUOTE m)) ; QUOTE m = ( QUOTE 1 – QUOTE 3)/2
In equation (9), A and B are found by regression analysis. Equation (8) is the equation of the Mohr envelope in the normal and shear stress space. According to the Hoek and Brown criterion, the failure surface is inclined at an angle QUOTE to the QUOTE 1 axis with sin 2 QUOTE = QUOTE m.
Both the m and s coefficients appearing in equation (7) have been related to the RMR and Q ratings. As discussed by Bieniawski (1993), for intact rock m=mi and s=1. For fractured rock masses, m and s are related to the basic (unadjusted) RMR as follows
m = mi exp(RMR – 100)/28
s = exp(RMR – 100)/9
for smooth-blasted or machine-bored excavations in rock and
m = mi exp(RMR – 100)/14
s = exp(RMR – 100)/6
Note that the Mohr-Coulomb friction and cohesion can be determined from the Hoek and Brown failure criterion (Hoek, 1990).
4.0 LINEAR ELASTICITY AND LABORATORY TESTING OF ROCKS
4.1Linear elasticity of rocks
Linear Elasticity is a model of elasticity used to describe the states of stress of a rock. A rock is linearly elastic if it undergoes strain linearly proportional to the magnitude of the applied stress. Or to put it another way, plots as a straight line on a stress strain curve. Furthermore, at the elastic limit if the load is removed, the specimen will still return back to its original shape and form. The uniaxial relation between stress QUOTE and strain QUOTE for a perfectly elastic solid is
QUOTE = E QUOTE
Where E is the young’s modulus and is an intrinsic property of the material. The mechanical model for a perfectly elastic material is a simple spring. This equation is known as Hooke’s law and is often written in terms of force F and displacement x (as in simple harmonic motion):
F = – k x
Where k, is called the spring constant, is the elastic proportionality constant between applied stress and strain.
4.2Fundamental of elasticity
When an internal or external force, is applied to a continuum every point of this continuum is in influenced by this force. It is common to denote internal forces as body forces and external forces as contact forces. The most common body force results from the acceleration due to gravity. Body forces are proportional to the volume of the medium and to its density and have the unit force per volume. Contact forces depend on the surface they are acting on and have the unit force per area.
Imagine external forces acting on a continuum. In general, these forces will lead to a deformation of the medium resulting in changes of size and shape. Internal forces acting within the medium try to resist this deformation. As a consequence, the medium will return to its initial shape and volume when the external forces are removed. If this recovery of the original shape is perfect, the medium is called elastic.
The constitutive law relating the applied force to the resulting deformation is Hooke’s Law. It is defined in terms of stress and strain. The exact form of the stress, the state of stress, at an arbitrary point P of the continuum depends on the orientation of the force acting on P and the orientation of the reference plane with respect to a reference coordinate system. To quantify the state of stress at a point P resulting from the force F, P is imagined as an infinitesimal small cube. The stress acting on each of the six sides of the cube can be resolved into components normal to the face and within it. This situation is illustrated in Figure A for the plane normal to the 2 axes. In the following, a plane oriented normal to an axes i is called the i-plane
Figure A: Stress components acting on the 2-plane.
A stress QUOTE ij is defined as acting on the i-plane and being oriented in the j direction. Components of the stress tensor with repeating indices, e.g. QUOTE 11, are denoted as normal stress while a stress component with different indices is called a shear stress. Consequently, this gives six shear and three normal stress components acting on the cube. If the medium is in static equilibrium the sum of all stress components acting in the 1,2, and 3 directions and the total moment is zero. This means:
QUOTE ij = QUOTE ji:
Thus, the stress tensor QUOTE ij completely describes the state of stress at any point P of
Normal stresses with positive values directed outward from faces are called tensional stress, and negative values correspond to compressional stress. The SI unit for stress is Pa. In geoscientic practice, stresses are usually given in mega pascal (1 MPa = 106P). 1 A special state of stress is found when all normal stresses are equal, i.e., QUOTE 11 = QUOTE 22 = QUOTE 33 and all shear stresses are zero. Then, the stress tensor is independent of the reference coordinate system, and the stress can be understood as a scalar, thus, as a pressure. This pressure is given as P = QUOTE ii. Such a stress state is often denoted as hydrostatic, because it is similar to the pressure in a fluid, which is always equal in all directions. However, this state of stress in a solid material depends on the orientation and magnitude of the externally or internally applied forces. In a fluid, it results from the general property of fluids that they cannot resist shear stress. Therefore, this state of stress in a solid material should be correctly denotes as isostatic and hydrostatic should refer to pressure in a fluid.
As mentioned above, when an elastic body is subjected to stress, changes in size and shape occur and these deformations are called strain. Per definition, strain is the relative (fractional) change of a dimension of a body. In the three dimensional case the strain at point P is determined by the strain tensor QUOTE ij , assuming the deformations to be sufficiently small:
The elements of the strain tensor with repeating indices are denoted as normal strain,
all others as shear strain. Just as the stress tensor the strain tensor has six independent
QUOTE ij = QUOTE ji:
The volume change of a body is given by the diagonal elements QUOTE ii of the strain tensor only. Normalizing this volume change to a unit volume defines the dilatation QUOTE of a body:
QUOTE = QUOTE ii;
where summation over repeated indices is assumed
5.0LABORATORY TESTING OF ROCKS
Laboratory rock testing is performed to determine the strength and elastic properties of intact specimens and the potential for degradation and disintegration of the rock material. The derived parameters are used in part for the design of rock fills, cut slopes, shallow and deep foundations, tunnels, and the assessment of shore protection materials (rip-rap).
Common laboratory tests for intact rocks include measurements of strength (point load index, compressive strength, Brazilian test, direct shear), stiffness (ultrasonics, elastic modulus), and durability (slaking, abrasion)
The laboratory determination of intact rock strength is accomplished by the following tests: point load index, unconfined compression, triaxial compression, Brazilian test, and direct shear.
5.1.1Point Load Index Test
The point load test is used as an index strength for strength classification materials. The following procedures are used in carrying out the test:
The diametral test is conducted on rock core sample. Minimum of 10 test specimens are required to find out the average value of point load strength index.
This test can be conducted on the core specimens which are completely dry or after soaking it for 7days.
Measure the total length(l) and diameter (d) of the core specimen. Specimen of QUOTE = 1.5, are considered to be suitable for this test.
Place the specimen horizontally two platens in such a way that the distance between the contact point and the nearest free end (L) is at least 0.75times the diameter of the core (d).
Measure the distance between two platen contact points (D) with the help of the scale attached with the loading frame. (Note-in case of diametral test, the diameter of the core 9d) and the distance (D) will be same)
Apply load to the core specimen such that failure occur within 10-60 sec. record the failure load ‘P’
Point load strength index (Is) = (P*1000)/D2 Mpa
Where, P is breaking load in KN, D is the distance between platens in mm.
Corresponding point load strength index for the standard core size of 50mm (Is50) diameter is given by the following equation.
Is50= (P*1000)/(D1.5? QUOTE ) MPa
Uniaxial compressive strength of rock may be predicted from the following equation
qc =22*Is50 Mpa
5.1.2Uniaxial Compression Test
This test is use to determine the uniaxial compressive strength of rock (qu = Fu = FC)
Procedure: In this test, cylindrical rock specimens are tested in compression without lateral confinement. The test procedure is similar to the unconfined compression test for soils and concrete. The test specimen should be a rock cylinder of length-to-width ratio (H/D) in the range of 2 to 2.5 with flat, smooth, and parallel ends cut perpendicular to the cylinder axis.
The uniaxial compression test is most direct means of determining rock strength. The results are influenced by the moisture content of the specimens, and thus should be noted. The rate of loading and the condition of the two ends of the rock will also affect the final results. Ends should be planar and parallel per ASTM D 4543. The rate of loading should be constant as per the ASTM test procedure. Inclined fissures, intrusions, and other anomalies will often cause premature failures on those planes. These should be noted so that, where appropriate, other tests such as triaxial or direct shear tests can be required.
5.1.3Direct Shear Strength
This test is use to determine the shear strength characteristics of rock along a plane of weakness
Procedure: The laboratory test equipment is shown below in Figure 1. The specimen is placed in the lower half of the shear box and encapsulated in either synthetic resin or mortar. The specimen must be positioned so that the line of action of the shear force lies in the plane of the discontinuity to be investigated, and the normal force acts perpendicular to this surface. Once the encapsulating material has hardened, the specimen is mounted in the upper half of the shear box in the same manner. A strip approximately 5 mm wide above and below the shear surface must be kept free of encapsulating material. The test is then carried out by applying a horizontal shear force T under a constant normal load, N.
FIGURE 1: General Set-up for Direct Shear Strength Testing of Rock (Wittke, 1990)
Determination of shear strength of rock specimens is an important aspect in the design of structures such as rock slopes, foundations and other purposes. Pervasive discontinuities (joints, bedding planes, shear zones, fault zones, schistosity) in a rock mass, and genesis, crystallography, texture, fabric, and other factors can cause the rock mass to behave as an anisotropic and heterogeneous discontinuum. Therefore, the precise prediction of rock mass behavior is difficult. For nonplanar joints or discontinuities, shear strength is derived from a combination base material friction and overriding of asperities (dilatancy), shearing or breaking of the asperities, rotations at or wedging of the asperities (Patton, 1966). Sliding on and shearing of the asperities can occur simultaneously. When the normal force is not sufficient to restrain dilation, the shear mechanism consists of the overriding of the asperities. When the normal load is large enough to completely restrain dilation, the shear mechanism consists of the shearing off of the asperities. Using this test method to determine the shear strength of intact rock may generate overturning moments that induce premature tensile breaking. Thus, the specimen would fail in tension first rather than in shear. Rock shear strength is influenced by the overburden stresses; therefore, the larger the overburden stress, the larger the shear strength. In some cases, it may be desirable to conduct tests in-situ rather than in the laboratory to more accurately determine a representative shear strength of the rock mass, particularly when design is controlled by discontinuities filled with very weak material.
5.2Splitting Tensile (Brazilian) Test
This test is carried out to evaluate the (indirect) tensile shear of intact rock core, FT.
Procedure: Core specimens with length-to-diameter ratios (L/D) of between 2 to 2.5 are placed in a compression loading machine with the load platens situated diametrically across the specimen. The maximum load (P) to fracture the specimen is recorded and used to calculate the split tensile strength.
The Brazilian or split-tensile strength (FT) is significantly more convenient and practicable for routine measurements than the direct tensile strength test (T0). The test gives very similar results to those from direct tension (Jaeger ; Cook, 1976). It is a more fundamental strength measurement of the rock material, as this corresponds to a more likely failure mode in many situations than compression. Also, note that the point load index is actually a type of Brazilian tensile strength, that is correlated back to compressive strength.
5.3 DURABILITY TESTS
The evaluation of rock durability becomes an issue when the materials are to be subjected to the natural elements, seasonal weather, and repeated cycles of temperature (e.g., flowing water, wetting and drying, wave action, freeze and thaw, etc.) in its proposed use. Tests to measure durability depend on the type of rock, on its use in construction, and on the elements to which the rock will be subjected. The basis for durability tests are empirical and the results produced are an indication of the rock’s resistance to natural processes; the rock’s behavior in actual use may vary greatly from the test results. These tests, however, provide reasonably reliable tools for quality control. The suitability of various types of rock for different uses should, in addition to these test results, depend on their performance in previous applications. An example of the use of rock durability tests is in the evaluation of shale in rock fill embankments.
This test is conducted to determine the durability of shale or other weak or soft rocks subjected to cycles of wetting and drying.
Procedure: In this test dried fragments of rock of known weight are placed in a drum fabricated with 2.0 mm square mesh wire cloth. Figure 2 shows a schematic of the test apparatus. The drum is rotated in a horizontal position along its longitudinal axis while partially submerged in distilled water to promote wetting of the sample. The specimens and the drum are dried at the end of the rotation cycle (10 minutes at 20 rpm) and weighed. After two cycles of rotating and drying the weight loss and the shape and size of the remaining rock fragments are recorded and the Slake Durability Index (SDI) is calculated. Both the SDI and the description of the shape and size of the remaining particles are used to determine the durability of soft rocks.
FIGURE 2: Rotating Drum Assembly and Setup of Slake Durability Equipment. (ASTM D 4644, 1995)
This test is typically performed on shales and other weak rocks that may be subject to degradation in the service environment. When some shales are newly exposed to atmospheric conditions, they can degrade rapidly and affect the stability of a rock fill or cut, the subgrade on which a foundation is to be placed, or the base and side walls of drilled shafts prior to placement of concrete.
5.4Soundness of Riprap
This is test is conducted in order to determine the soundness of rock subjected to erosion.
Procedure: The procedure is known as the Rock Slab Soundness Test. Two representative, sawed, rock slab specimens are immersed in a solution of sodium or magnesium sulfate and dried and weighed for five cycles. The percent weight loss as a result of these tests is expressed as percent soundness.
Comment: One of the most effective means to control erosion along riverbanks and coastal beaches is by covering exposed soil with rip-rap, or a combination of geosynthetics and rip-rap. Rock or stone used in this mode is subject to degradation from weathering effects due to repeated cycles of wetting ; drying, as well as repeated exposure to salts used in deicing of roadways. This test is used to estimate this type of degradation.
5.5Durability Under Freezing and Thawing
This test is carried out to determine the resistance of rock used for erosion control to repeated cycles of freezing and thawing.
Procedure: Slabs of representative rock specimens are subjected to freezing and thawing cycles in the laboratory. The loss of dry weight at the end of five successive cycles of freezing, thawing, and drying is expressed as percent loss due to freeze/thaw.
This test is useful in assessing the durability of rock due to weathering effects, in particularly for stone and gravel aggregates used in northern climates where seasonal winters will degrade their use in highway construction. It can also be used to assess the durability of armor stones placed for shore protection or rip-rap placed for shoreline protection or dam embankment protection.
As discussed above, none of these tests provide results which can be used independent of each other or independent of other tests and experience. Often the behavior of rip-rap stone in actual use will vary widely from the laboratory behavior.
5.6 STIFFNESS TESTS
The stiffness of rocks is represented by an equivalent elastic modulus at small- to intermediate-strains.
This test is used to determine the deformation characteristics of intact rock at intermediate strains and permit comparison with other intact rock types.
Procedure: This test is performed by placing an intact rock specimen in a loading device and recording the deformation of the specimen under axial stress. The Young’s modulus, either average, secant, or tangent moduli, can be determined by plotting axial stress versus axial strain curves.
FIGURE 3: Definitions for Determining Elastic (Young’s) Modulus from Axial Stress-Strain Measurements During Compression Loading, including (a) Tangent, (b) Average, and (c) Secant Values. (ASTM D 3148)
Comment: The results of these tests cannot always be replicated because of localized variations in each unique rock specimen. They provide reasonably reliable data for engineering applications involving rock classification type, but must be adjusted to take into account rock mass characteristics such as jointing, fissuring, and weathering.
Omissions or errors introduced during laboratory testing, if undetected, will be carried through the process of design and construction, possibly resulting in costly or unsafe facilities.
It is a well-known fact that rocks play a vital role in constructing the structures which are destined to be strong, appealing and economical. Different classes of rocks which have been considered gives a clear guideline for an engineer to choose the right type of naturally occurring rocks or stones to be used to build such structures.
Engineering properties of rocks are very essential properties to be determined in every project of civil engineering, construction engineering and structural engineering. This rock property can be divided into two categories: intact rock properties and rock mass properties. By choosing all the properties judiciously in conjunction with one another, it is possible to adhere to the safety regulations prescribed in building standards
The complicated structure of the rock mass with its defects and inhomogeneity’s and the wide range of its applications cause challenges and problems in rock engineering and construction which often involve considerations that are of relatively little or no concern in most other branches of engineering.
A combination of laboratory testing of small samples, empirical analysis and field observations should be employed to determine the requisite engineering properties.
ASTM D 2936-84 Standard test method for direct tensile strength of intact rock core specimen.
ASTM D 2938-86 Standard test method for unconfined compressive strength of intact rock core specimens.
ASTM D 3967-92 Standard test method for splitting tensile strength of intact rock core specimens
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