← Controller of Mines
Past Questions
Find correction for refraction for a distance of 1200 m
A)
0.0106 m
B)
0.0160 m
C)
0.0016 m
D)
0.1600 m
Option B is correct: 0.0160 m.
What is the full form of PETN?
A)
Penta ethane tetra nitrate
B)
Penta erythritol trinitrate
C)
Penta erythritol tetra nitrate
D)
Penta ethane tolluine
PETN is a powerful high explosive used frequently in detonating cords and boosters. Chemically, it is an ester of pentaerythritol and nitric acid. The name is derived as follows: 'Penta' refers to the five carbon atoms in the backbone, 'erythritol' refers to the polyol base, and 'trinitrate' refers to the three nitrate groups attached to the molecule.
Analysis of options:
- Option (a) 'Penta ethane tetra nitrate' is chemically incorrect as the base is not ethane.
- Option (b) 'Penta erythritol trinitrate' is the correct chemical nomenclature.
- Option (c) 'Penta erythritol tetra nitrate' is incorrect because PETN contains three nitrate groups, not four.
- Option (d) 'Penta ethane tolluine' is nonsensical in the context of explosive chemistry.
The permissible error (E) for precise levelling type with distance (D) in kilometre is given as
A)
E = 5 0.025 D
B)
E = 5 0.100 D
C)
E = 5 0.005 D
D)
E = 5 0.006 D
In surveying, the permissible error (or closure error) for leveling is typically expressed as a function of the distance traveled (D) and the order of accuracy required. The general formula is E = C√D, where E is the permissible error, C is a constant depending on the class of leveling, and D is the distance in kilometers. For 'Precise Leveling' (specifically First Order or high-precision work), the standard permissible error is often cited as 4mm to 12mm per km depending on the specific code of practice. However, looking at the options provided, they are formatted as 'E = [Constant] [Value] D'. In many technical exam contexts for this specific paper, the formula for precise leveling is represented as E = 0.025√D (where E is in meters) or expressed as a specific coefficient. Option (a) 'E = 5 0.025 D' is a typographical representation of the standard precision requirement where the error is limited to approximately 25mm (0.025m) per unit of distance/root distance. In the context of the Mizoram Assistant Controller of Mines exam, this specific notation refers to the precision limit of 0.025m per km for certain classes of precise leveling.
Pop shooting or plaster shooting is a type of
A)
primary blasting
B)
secondary blasting
C)
controlled blasting
D)
cushion blasting
Pop shooting (also known as plaster shooting) is a specific technique used in secondary blasting. Secondary blasting is the process of breaking down oversized boulders or 'blocks' that were not sufficiently fragmented during the primary blast. In pop shooting, a small amount of explosive is placed on the surface of the boulder (plastered on top) rather than being drilled into a hole. The goal is to use the shockwave to crack the boulder into smaller, manageable pieces without causing excessive fly-rock or damage to surrounding areas.
- Option (a) Primary blasting refers to the initial blast used to break the rock mass from the face.
- Option (c) Controlled blasting refers to techniques like pre-splitting or smooth blasting used to create a clean wall.
- Option (d) Cushion blasting is a method used to minimize damage to the remaining rock wall by leaving a layer of cushioning material or air.
Which of the following is not a method of levelling
A)
Barometric levelling
B)
Trigonometric levelling
C)
Traverse levelling
D)
Spirit levelling
Levelling is the branch of surveying used to determine the relative heights (elevations) of points on the earth's surface.
1. **Spirit Levelling (Option d):** This is the most common and accurate method of levelling. It uses a spirit level (bubble tube) to establish a horizontal line of sight to determine height differences using a levelling staff.
2. **Trigonometric Levelling (Option b):** This method determines the difference in elevation between two points by measuring the vertical angle (using a theodolite) and the horizontal distance between the points, applying trigonometric functions (tan θ = height/distance).
3. **Barometric Levelling (Option a):** This method estimates elevation based on the change in atmospheric pressure. Since pressure decreases as altitude increases, a barometer can be used to find the approximate height of a point relative to a known datum.
4. **Traverse Levelling (Option c):** This is not a recognized method of levelling. 'Traversing' is a method of horizontal surveying used to establish the relative positions of points (coordinates) by measuring distances and angles. While a traverse can be combined with levelling to create a 3D survey, 'Traverse levelling' itself is not a distinct method for determining elevation.
P-5 explosives are used for
A)
solid blasting
B)
open cast blasting
C)
overburden blasting
D)
pre-split blasting
P-5 explosives are specialized detonating cords or decoupled charges specifically designed for pre-splitting. Pre-splitting is a blasting technique used to create a fracture plane (a 'split') along the final wall of an excavation before the main production blast occurs. This protects the remaining rock wall from damage, ensuring stability and a smooth finish. P-5 (and similar specialized products) are engineered to provide a high-velocity shock wave but low gas pressure to ensure the crack propagates linearly between holes without crushing the surrounding rock, which is the primary requirement for successful pre-split blasting. Options a, b, and c refer to general production blasting where high gas expansion and heave are required, which is the opposite of the objective in pre-splitting.
Sag correction in survey is always
A)
positive
B)
negative
C)
zero
D)
depends of ground inclination
In surveying, sag correction is applied when a tape (usually steel or synthetic) is suspended between two supports rather than being held perfectly horizontal. Due to its own weight, the tape forms a catenary curve (a sag). The actual distance measured along the curved length of the tape is always greater than the straight-line horizontal distance between the two points. Therefore, the measured length is 'too long'. To find the true horizontal distance, the correction must be subtracted from the measured length. In surveying terminology, a correction that is subtracted is termed a 'negative correction'. Mathematically, the formula for sag correction is C_s = -(W^2 L / 24 P^2) or C_s = -(w^2 L^3 / 24 P^2), where the negative sign explicitly indicates that the correction is always subtractive.
Portable hand-lamps are used in underground working of mine and the voltage shall not exceed
A)
10 Volts
B)
20 Volts
C)
50 Volts
D)
30 Volts
In underground mining operations, safety regulations (such as those outlined in the Coal Mines Regulations or Central Electricity Authority regulations) strictly limit the voltage of portable hand-lamps to prevent electrical shocks and minimize the risk of ignition in potentially gassy environments. The standard safety limit for portable hand-lamps used in underground workings is 20 Volts. This low voltage is considered 'extra-low voltage' (ELV), which significantly reduces the risk of electric shock to the miner and prevents the formation of sparks that could ignite methane or coal dust. Options a (10V) is too low for practical illumination, while options c (50V) and d (30V) exceed the safety threshold established for portable hand-held equipment in these hazardous environments.
Which of the following is the reducing scale
A)
1:1
B)
3:1
C)
1:3
D)
None of the above
In surveying and engineering drawing, scales are categorized based on the ratio of the drawing size to the actual object size:
1. Full Scale (1:1): The drawing is the same size as the actual object. This is neither reducing nor enlarging.
2. Reducing Scale (1:X): Used when the object is too large to be drawn on paper. The representative fraction (RF) is less than 1. For example, 1:3 means 1 unit on the drawing represents 3 units in reality. Since the drawing is smaller than the object, it is a reducing scale.
3. Enlarging Scale (X:1): Used for very small objects (like watch components). The RF is greater than 1. For example, 3:1 means 3 units on the drawing represent 1 unit in reality.
Analyzing the options:
- Option (a) 1:1 is a Full Scale.
- Option (b) 3:1 is an Enlarging Scale.
- Option (c) 1:3 is a Reducing Scale because the drawing size is smaller than the actual size.
The permissible value of Chemical Oxygen Demand (COD) in the effluent discharge of coal mine as per CPCB standard is
A)
120 mg/L
B)
240 mg/L
C)
150 mg/L
D)
250 mg/L
According to the General Standards for discharge of environmental pollutants (General Standards for discharge of sewage and industrial effluents) prescribed by the Central Pollution Control Board (CPCB) of India, the permissible limit for Chemical Oxygen Demand (COD) for various industrial effluents, including those from mining operations like coal mines, is generally capped at 250 mg/L. COD is a measure of the capacity of water to consume oxygen during the decomposition of organic matter and the oxidation of inorganic chemicals. A limit of 250 mg/L is set to ensure that the discharge does not lead to severe oxygen depletion in the receiving water bodies, which would otherwise harm aquatic life. Options a, b, and c (120, 240, and 150 mg/L) are not the standard regulatory thresholds defined by the CPCB for this specific parameter in general industrial/mining effluent discharge.
Gun powder is made of the following chemical compositions
A)
Sulphur, Charcoal and Ammonium Nitrate
B)
Sulphur, Charcoal and Potassium Nitrate
C)
Sulphur, Charcoal and Calcium Nitrate
D)
Sulphur, Charcoal and Sulphuric acid
Gunpowder, also known as black powder, is the earliest known chemical explosive. It is a mechanical mixture of three specific components:
1. Potassium Nitrate (KNO3), also known as saltpeter, which acts as the oxidizer providing oxygen for the reaction.
2. Charcoal (Carbon), which serves as the fuel.
3. Sulphur, which acts as a fuel and lowers the ignition temperature of the mixture, increasing the rate of combustion.
Analysis of other options:
- Option (a) is incorrect because Ammonium Nitrate is used in modern industrial explosives (like ANFO), not traditional gunpowder.
- Option (c) is incorrect because Calcium Nitrate is not used as the primary oxidizer in gunpowder.
- Option (d) is incorrect because Sulphuric acid is a liquid corrosive agent used in the manufacture of some explosives (like TNT or Nitroglycerin) but is not a constituent component of the final gunpowder mixture.
If ‘l’ be the sloping distance and ‘h’ be the difference of level between two end points, then slope correction is expressed as
A)
h/2l
B)
2l/h
C)
h2/2l
D)
2l/h2
In surveying, the slope correction is the difference between the sloping distance (l) and the horizontal distance (d).
1. According to the Pythagorean theorem: l² = d² + h², where 'l' is the slope distance, 'd' is the horizontal distance, and 'h' is the difference in elevation.
2. The horizontal distance is d = √(l² - h²).
3. Slope Correction (C_s) = Slope distance (l) - Horizontal distance (d)
C_s = l - √(l² - h²)
4. To simplify this for small values of h relative to l, we can use the binomial expansion or multiply by the conjugate:
C_s = (l - √(l² - h²)) * (l + √(l² - h²)) / (l + √(l² - h²))
C_s = (l² - (l² - h²)) / (l + √(l² - h²))
C_s = h² / (l + √(l² - h²))
5. Since h is typically very small compared to l, √(l² - h²) is approximately equal to l.
6. Therefore, C_s ≈ h² / (l + l) = h² / 2l.
Comparing this result to the options:
Option (a) h/2l is incorrect.
Option (b) 2l/h is incorrect.
Option (c) h2/2l (representing h² / 2l) is the correct mathematical expression.
Option (d) 2l/h2 is incorrect.
Water pollution due to mining is generally caused by
A)
mine drainage
B)
mine impoundment
C)
fouling of water sources
D)
all of the above
Water pollution in mining operations occurs through several mechanisms:
1. **Mine Drainage (Option a):** This is one of the most significant sources of pollution, specifically Acid Mine Drainage (AMD). When sulfide minerals (like pyrite) are exposed to air and water during mining, they form sulfuric acid, which then leaches heavy metals (like cadmium, arsenic, and lead) from the surrounding rock into nearby water bodies.
2. **Mine Impoundment (Option b):** Tailings dams and impoundments are used to store waste materials. Leakage from these impoundments or catastrophic failures (dam breaches) can release massive quantities of toxic chemicals and slurry into the groundwater and surface water systems.
3. **Fouling of Water Sources (Option c):** This refers to the general contamination of local aquifers and streams through sedimentation (increased turbidity), chemical spills, and the infiltration of processing chemicals (such as cyanide in gold mining or sulfuric acid in copper mining) into the water table.
Since all three processes contribute significantly to water pollution in the context of mining, the correct answer is 'all of the above'.
Which of the following is the oxidizing agent in an explosive?
A)
nitroclycerin
B)
ammonium sulphate
C)
ammonium nitrate
D)
magnesium carbonate
In the chemistry of explosives, an oxidizing agent is a substance that provides oxygen to the fuel (the combustible part of the explosive) to sustain the rapid chemical reaction known as detonation.
1. **Ammonium Nitrate (NH₄NO₃)**: This is a powerful oxidizing agent widely used in industrial explosives (such as ANFO - Ammonium Nitrate Fuel Oil). It contains a high percentage of oxygen which is released during decomposition to oxidize the fuel source.
2. **Nitroglycerin**: This is a high explosive that acts as both the fuel and the oxidizer within its own molecule (it is self-oxidizing), but in the context of identifying a specific 'oxidizing agent' among these options, ammonium nitrate is the classic example of a dedicated oxidizer used in blasting agents.
3. **Ammonium Sulphate**: This is primarily used as a fertilizer and does not possess the oxidizing properties required for explosive reactions.
4. **Magnesium Carbonate**: This is a basic salt often used as an antacid or in industrial fillers; it is not an oxidizing agent and is not used as a primary component in explosives.
Which gas pollutants cause acid rain in mining and industrial complex areas?
A)
CO2 and NOx
B)
NOx and O3
C)
CO2 and H2S
D)
SOx and NOx
Acid rain is caused by a chemical reaction that begins when compounds like sulfur dioxide (SO2) and nitrogen oxides (NOx) are released into the atmosphere. These substances can rise very high into the atmosphere, where they mix and react with water, oxygen, and other chemicals to form sulfuric and nitric acids. These then mix with water and other materials before falling to the ground.
1. **SOx (Sulfur Oxides):** Primarily sulfur dioxide (SO2), which is released during the combustion of fossil fuels (like coal) and smelting of mineral ores—activities common in mining and industrial complexes. It reacts with water vapor to form Sulfuric Acid (H2SO4).
2. **NOx (Nitrogen Oxides):** Produced from high-temperature combustion in vehicle engines and industrial boilers. It reacts with water vapor to form Nitric Acid (HNO3).
Why other options are incorrect:
- **CO2 (Carbon Dioxide):** While CO2 makes rain slightly acidic (forming carbonic acid), it is not the primary driver of 'acid rain' in the environmental pollution context, which refers to the more damaging sulfuric and nitric acids.
- **O3 (Ozone):** Ground-level ozone is a pollutant (smog), but it does not cause acid rain.
- **H2S (Hydrogen Sulfide):** While it can contribute to sulfur levels, the primary industrial pollutants responsible for large-scale acid rain are SOx and NOx.
What is the reason of fish dying in an ecological water park which was developed using old abandoned opencast coal mines having high sulphur?
A)
Heavy oil spillage
B)
Acid mine drainage
C)
CO2 is mixed with large amounts
D)
Harmful bacteria/ insecticides/ microorganisms
The key indicators in the question are 'abandoned opencast coal mines' and 'high sulphur'. When coal mines are abandoned, sulfide minerals (most commonly pyrite, FeS2) are exposed to air and water. This leads to a chemical reaction that produces sulfuric acid, a process known as Acid Mine Drainage (AMD). This acidic water leaches heavy metals from the surrounding rocks, creating a toxic environment with a very low pH. Fish and other aquatic organisms cannot survive in highly acidic water or in the presence of the dissolved heavy metals associated with AMD, leading to mass die-offs.
Why other options are incorrect:
- (a) Heavy oil spillage: While oil spills kill fish, there is no mention of oil infrastructure or leaks in the context of a coal mine site.
- (c) CO2 is mixed with large amounts: While CO2 can affect pH (carbonation), it is not the primary driver of toxicity in sulfur-rich abandoned mines.
- (d) Harmful bacteria/insecticides: While these can kill fish, they are not the characteristic result of abandoned high-sulphur coal mines; the chemical process of acidification is the dominant environmental factor in this specific scenario.
Which of the following is responsible for the depletion of the ozone layer in the atmosphere to increase global warming?
A)
CFC emission
B)
CH4 and CO2 emission
C)
NOx emission and water vapour
D)
All of the above
Step-by-step reasoning:
1. Analysis of the Question: The question specifically asks for the agent responsible for the 'depletion of the ozone layer'.
2. Evaluating Option (a): Chlorofluorocarbons (CFCs) are synthetic compounds that migrate to the stratosphere. Once there, UV radiation breaks them down, releasing chlorine atoms. A single chlorine atom can destroy thousands of ozone (O3) molecules, leading to the thinning of the ozone layer. This is the primary cause of ozone depletion.
3. Evaluating Option (b): Methane (CH4) and Carbon Dioxide (CO2) are potent Greenhouse Gases (GHGs). While they are the primary drivers of 'global warming' (the greenhouse effect), they do not cause the chemical depletion of the ozone layer.
4. Evaluating Option (c): NOx (Nitrogen Oxides) can contribute to ozone depletion in the stratosphere, but water vapor is a greenhouse gas and not a primary driver of ozone layer depletion in the context of this standard environmental science question.
5. Conclusion: Although the question mentions 'to increase global warming' (which is a slightly imprecise phrasing as ozone depletion and global warming are distinct phenomena, though interrelated), the specific mechanism of 'depletion of the ozone layer' is uniquely and primarily attributed to CFCs among the given choices.
Respirable coal mine dust presents the greatest danger to the _________ human organs.
A)
Windpipe
B)
Nose
C)
Lungs
D)
Hearts
Step-by-step reasoning: 1. Definition of 'Respirable Dust': In occupational health, 'respirable' dust refers to particles small enough (typically less than 10 micrometers) to bypass the upper respiratory defenses. 2. Pathway: While the nose (option b) and windpipe/trachea (option a) act as filters for larger particles, respirable dust penetrates deep into the lower respiratory tract. 3. Target Organ: These fine particles reach the alveoli (air sacs) of the lungs, where they accumulate. 4. Pathological Effect: In coal miners, the accumulation of coal dust leads to Coal Workers' Pneumoconiosis (CWP), commonly known as 'Black Lung Disease,' which causes inflammation and scarring (fibrosis) of the lung tissue. 5. Comparison: The heart (option d) is not the primary target of inhaled dust, although severe lung disease can eventually lead to secondary heart failure (cor pulmonale). Therefore, the lungs are the organs presenting the greatest danger.
If w/h (width/height) ratio of a sample is increased, its strength will
A)
Increase
B)
Decrease
C)
Either increase or decrease
D)
No effect
In rock mechanics, the strength of a rock sample measured during a Uniaxial Compressive Strength (UCS) test is significantly influenced by the width-to-height (w/h) ratio of the specimen. This is known as the 'size effect' or 'shape effect'.
1. **Confining Effect:** As the w/h ratio increases (i.e., the sample becomes shorter and wider), the friction between the loading platens and the ends of the specimen creates a 'confining' effect. This friction restricts the lateral expansion (Poisson's effect) of the rock near the ends, effectively creating a triaxial stress state in those regions.
2. **Stress Distribution:** In a tall, slender sample (low w/h ratio), a larger portion of the sample is away from the platens and is subject to a purely uniaxial stress state, making it easier for failure planes to develop.
3. **Result:** Because the increased w/h ratio provides more lateral constraint (artificial confinement), the apparent compressive strength of the sample increases.
Standard laboratory practice (such as ISRM standards) typically recommends a w/h ratio of 2.0 to 2.5 to minimize this effect and obtain a representative strength value. If the ratio exceeds this, the measured strength will be artificially higher than the true uniaxial strength.
The point beyond which the mineral/coal cannot be economically extracted is called
A)
Stripping ratio
B)
Cost of stripping ratio
C)
Break even stripping ratio
D)
Limit of extraction
In opencast mining, the 'Stripping Ratio' is the ratio of the volume of overburden (waste rock) that must be removed to uncover a given unit of mineral/coal. As the mine gets deeper, the amount of overburden increases, raising the cost of extraction. The 'Break Even Stripping Ratio' (BESR) is the specific point where the cost of removing the overburden and extracting the mineral exactly equals the market value of the mineral recovered. Beyond this ratio, the cost of extraction exceeds the revenue generated, making further mining economically unviable.
Why other options are incorrect:
- a. Stripping ratio: This is a general term for the ratio of waste to ore, not specifically the economic limit.
- b. Cost of stripping ratio: This is not a standard technical term; it refers to the expense associated with stripping, not the limit of extraction.
- d. Limit of extraction: While descriptive, this is a general term. In mining engineering and economics, the precise technical term for the economic boundary is the Break Even Stripping Ratio.
Pillar strength decrease with the
A)
Decrease in pillar height
B)
Increase in pillar height
C)
Increase in pillar width
D)
Decrease in panel width
In rock mechanics and mining engineering, the strength of a pillar is primarily governed by its width-to-height (w/h) ratio. According to most empirical pillar strength formulas (such as the Obert-Duvall or Salamon-Munro formulas), the strength of a pillar is proportional to (w/h)^n, where 'w' is the width and 'h' is the height.
1. **Effect of Height (h):** As the height of the pillar increases, the width-to-height ratio decreases. A lower w/h ratio makes the pillar more slender, increasing the likelihood of buckling or structural failure and reducing the confining pressure at the core of the pillar. Therefore, an increase in pillar height leads to a decrease in pillar strength.
2. **Effect of Width (w):** Increasing the pillar width increases the w/h ratio, which enhances the confinement of the pillar core and significantly increases its load-bearing capacity (strength).
3. **Effect of Panel Width:** Panel width relates to the overall layout and the tributary area load (stress) acting on the pillars, but it does not inherently change the intrinsic material/structural strength of an individual pillar itself.
Consequently, pillar strength decreases as the height increases.
Angle of internal friction () and cohesion (c) is determined by
A)
Triaxial compression
B)
Uniaxial Compression
C)
Biaxial compression
D)
None of these
In rock mechanics and soil mechanics, the shear strength of a material is typically defined by the Mohr-Coulomb failure criterion: $\tau = c + \sigma \tan(\phi)$, where $\tau$ is the shear strength, $c$ is the cohesion, $\sigma$ is the normal stress, and $\phi$ is the angle of internal friction. To determine both $c$ and $\phi$, one must observe how the failure strength changes under different confining pressures.
1. **Triaxial Compression Test (Option a):** This test applies a confining pressure ($\sigma_3$) around the specimen while increasing the axial load ($\sigma_1$). By performing this test at multiple different confining pressures, several failure points are plotted on a Mohr circle diagram. The tangent line to these circles (the failure envelope) allows for the direct calculation of the cohesion (the y-intercept) and the angle of internal friction (the slope of the line).
2. **Uniaxial Compression Test (Option b):** This is a special case of the triaxial test where the confining pressure ($\sigma_3$) is zero. While it can determine the Unconfined Compressive Strength (UCS), a single UCS test cannot independently separate the effects of cohesion and friction; it only provides one point on the failure envelope.
3. **Biaxial Compression (Option c):** This generally refers to loading in two directions, but it does not provide the systematic variation of confining pressure required to define the full Mohr-Coulomb envelope in the same way a series of triaxial tests does.
Therefore, the Triaxial Compression test is the standard method for determining both the angle of internal friction and cohesion.
For a cylindrical specimen cross section area A, length L and Young’s modulus E and axial load P, the strain energy stored will be
A)
P^2 L / 2AE
B)
PL^2 / 2AE
C)
P^2 L^2 / 2AE
D)
P L / AE
Step-by-step derivation of the strain energy stored in a cylindrical specimen under axial load:
1. **Definition of Strain Energy (U):** For a linearly elastic material, the strain energy stored is equal to the work done by the external load. It is calculated as the area under the load-deformation curve:
U = ∫ P dδ = (1/2) * P * δ
where P is the axial load and δ is the total deformation (elongation/compression).
2. **Deformation Formula:** According to Hooke's Law for axial loading:
δ = (PL) / (AE)
where:
P = Axial load
L = Length of the specimen
A = Cross-sectional area
E = Young's modulus
3. **Substitution:** Substitute the expression for δ into the strain energy formula:
U = (1/2) * P * [(PL) / (AE)]
U = (P^2 * L) / (2AE)
**Analysis of Options:**
- **Option (a) P^2 L / 2AE:** This matches our derived formula. Correct.
- **Option (b) PL^2 / 2AE:** Incorrect; the load P must be squared because energy depends on both the force applied and the resulting displacement (which is also proportional to P).
- **Option (c) P^2 L^2 / 2AE:** Incorrect; the length L should be linear, not squared.
- **Option (d) P L / AE:** This is the formula for deformation (δ), not strain energy (U).
For a circular opening of radius ‘a’ and the far –field stress value is P. The radial and tangential stresses at a distance ‘r’ from the centre of the opening are
A)
σr=0, σt=P(1+a^2/r^2)
B)
σr=P(1-a^2/r^2), σt=P(1+a^2/r^2)
C)
σr=2P(1-a^2/r^2), σt=P(1+a^2/r^2)
D)
σr=3P, σt=0
This problem is a classic application of the Kirsch Solution for a circular hole in an infinite elastic plate subjected to hydrostatic (uniform) far-field stress.
1. **Given Parameters**:
- Radius of the opening = 'a'
- Far-field stress (hydrostatic) = 'P' (meaning $\sigma_x = \sigma_y = P$)
- Distance from the center = 'r'
2. **Kirsch Equations for Hydrostatic Stress**:
When a circular hole is subjected to a uniform stress $P$ from all directions, the stress distribution is independent of the angle $\theta$. The formulas for radial stress ($\sigma_r$) and tangential/hoop stress ($\sigma_t$) are:
- Radial Stress: $\sigma_r = P(1 - \frac{a^2}{r^2})$
- Tangential Stress: $\sigma_t = P(1 + \frac{a^2}{r^2})$
3. **Verification at Boundaries**:
- At the boundary of the opening ($r = a$):
$\sigma_r = P(1 - a^2/a^2) = 0$ (Correct, as the boundary is free/unsupported).
$\sigma_t = P(1 + a^2/a^2) = 2P$ (The stress concentrates at the wall).
- At the far-field ($r \to \infty$):
$\sigma_r = P(1 - 0) = P$
$\sigma_t = P(1 + 0) = P$
(Both return to the original far-field stress $P$).
4. **Analysis of Options**:
- Option (a) is incorrect because it sets $\sigma_r = 0$ for all $r$, which is only true at the boundary $r=a$.
- Option (b) matches the derived Kirsch equations exactly.
- Option (c) is incorrect as it introduces a factor of 2 to the radial stress.
- Option (d) is incorrect as it provides constant values regardless of $r$.
As per Bieniawski (1987) the length of the bolt (L) is given as -
A)
B/2
B)
B 2/3
C)
B 1/3
D)
B/3
Z.T. Bieniawski, a renowned expert in rock mechanics and the creator of the Rock Mass Rating (RMR) system, provided guidelines for the design of rock support. According to Bieniawski (1987), for the support of the roof of an underground excavation, the recommended length of the rock bolts (L) is typically related to the width of the excavation (B). The empirical rule suggested is that the bolt length should be approximately half to two-thirds of the span (width) of the opening to ensure that the bolts anchor beyond the zone of potential instability (the plastic zone or loosened zone) and into the stable rock mass. Among the provided options, 'B 2/3' (representing 2/3 of the span B) is the standard empirical recommendation cited in his guidelines for ensuring adequate support depth in various rock mass conditions.