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300 Solved Problems Soil / Rock Mechanics and Foundations Engineering by Luis A. Prieto-Portar PhD, PE

300 Solved Problems Soil / Rock Mechanics and Foundations Engineering These notes are provided to you by Professor Prieto-Portar, and in exchange, he will be grateful for your comments on improvements.

All problems are graded according to difficulty as follows: * Easy; defines general principles; typical of the PE examination; ** Slightly more difficult; typical of Master’s level problems; ***

Professional level (“real-life”) problems. by Luis A. Prieto-Portar PhD, PE Professor of Civil and Environmental Engineering Florida International University, Miami, Florida Former Professor, United States Military Academy (West Point)

Table of Contents

Table of Contents………………………………………………………………………………………………….. i Chapter 1 Soil Exploration…………………………………………………………………………………….. 1 Symbols for Soil Exploration ……………………………………………………………………………… 1 *Exploration–01. Find the required number of borings and their depth. ………………….. 2 *Exploration–02. The sample’s disturbance due to the boring diameter. …………………. 3 *Exploration–03. Correcting the SPT for depth and sampling method. ……………………. 4 *Exploration–04. Three methods used for SPT depth corrections……………………………. 6 *Exploration–05. SPT corrections under a mat foundation. ……………………………………. 7 *Exploration–06. The Shear Vane Test determines the in-situ cohesion…………………… 9 *Exploration–07. Reading a soil boring log……………………………………………………….. 10 *Exploration–08: Using a boring log to predict soil engineering parameters…………… 11 **Exploration–09. Find the shear strength of a soil from the CPT Report. …………….. 14 Chapter 2 Phase Relations of Soil…………………………………………………………………………. 16 Symbols for Phase Relations of soils …………………………………………………………………. 16 Basic Concepts and Formulas for the Phases of Soils…………………………………………… 17 *Phases of soils-01: Convert from metric units to SI and US units. ……………………….. 21 *Phases of soils–02: Compaction checked via the voids ratio. …………………………….. 22 *Phases of soils–03: Value of the moisture when fully saturated. ………………………….. 23 *Phases of soils–04: Finding the wrong data. ……………………………………………………… 24 *Phases of soils–05: Increasing the saturation of a soil. ……………………………………….. 25 *Phases of soils–06: Find γd, n, S and Ww. ………………………………………………………… 26 *Phases of soils–07: Use the block diagram to find the degree of saturation. ………….. 27 *Phases of soils–08: Same as Prob-07 but setting the total volume V=1 m3. …………… 28 *Phases of soils–09: Same as Problem #5 with a block diagram……………………………. 29 *Phases of soils–10: Block diagram for a saturated soil. ……………………………………… 30 *Phases of soils–11: Find the weight of water needed for saturation. …………………….. 31 *Phases of soils–12: Identify the wrong piece of data. …………………………………………. 32 *Phases of soils–13: The apparent cheapest soil is not!………………………………………… 33 *Phases of soils–14: Number of truck loads. ………………………………………………………. 34 *Phases of soils–15: How many truck loads are needed for a project?……………………. 35 *Phases of soils–16: Choose the cheapest fill supplier. ………………………………………… 36 ii *Phases of soils–17: Use a matrix to the find the missing data………………………………. 38 **Phases of soils–18: Find the voids ratio of“muck” (a highly organic soil). ………….. 40 Chapter 3 Classification of Soils and Rocks…………………………………………………………… 41 Symbols for Classification of soils…………………………………………………………………….. 41 *Classify–01: Percentage of each of the four grain sizes (G, S, M & C)…………………. 42 *Classify–02: Coefficients of uniformity and curvature of granular soils. ………………. 43 *Classify-03: Classify two soils using the USCS…………………………………………………. 44 *Classify-04: Manufacturing a “new” soil. …………………………………………………………. 45 Classify – 05 …………………………………………………………………………………………………… 47 Classify – 06 …………………………………………………………………………………………………… 48 Classify – 07 …………………………………………………………………………………………………… 49 Classify – 08 …………………………………………………………………………………………………… 50 Classify – 09 …………………………………………………………………………………………………… 51 Classify – 10 …………………………………………………………………………………………………… 52 Classify – 11 …………………………………………………………………………………………………… 53 Chapter 4 Compaction and Soil Improvement………………………………………………………… 54 Symbols for Compaction ………………………………………………………………………………….. 54 *Compaction–01: Find the optimum moisture content (OMC). …………………………….. 55 *Compaction–02: Find maximum dry unit weight in SI units. ………………………………. 57 *Compaction-03: What is the saturation S at the OMC? ………………………………………. 59 *Compaction-04: Number of truck loads required……………………………………………….. 61 *Compaction-05: What is the saturation S at the OMC? ………………………………………. 62 *Compaction-06: Definition of the relative compaction (RC)……………………………….. 63 *Compaction-07: The relative compaction (RC) of a soil. ……………………………………. 64 *Compaction-08: Converting volumes from borrow pits and truck loads. ………………. 65 **Compaction-09: Ranges of water and fill required for a road…………………………….. 66 **Compaction-10: Find the family of saturation curves for compaction…………………. 68 **Compaction-11: Water needed to reach maximum density in the field. ………………. 71 **Compaction-12: Fill volumes and truck load requirements for a levee. ………………. 73 **Compaction-13: Multiple choice compaction problem of a levee. ……………………… 75 Chapter 5 Permeability of Soils ……………………………………………………………………………. 78 Symbols for Permeability …………………………………………………………………………………. 78 *Permeability–01: Types of permeability tests and common units…………………………. 79 iii *Permeability-02: Use of Hazen’s formula to estimate the k of an aquifer. …………….. 80 *Permeability-03: Flow in a sand layer from a canal to a river. …………………………….. 81 *Permeability-04: Find the equivalent horizontal permeability of two layers. …………. 82 *Permeability-05: Equivalent vertical and horizontal permeabilities. …………………….. 83 *Permeability-06: Ratio of horizontal to vertical permeabilities. …………………………… 84 *Permeability–07: Do not confuse a horizontal with a vertical permeability. ………….. 85 *Permeability-08: Permeability as a function of the voids ratio e. …………………………. 86 *Permeability–09: Uplift pressures from vertical flows………………………………………… 87 *Permeability-10: Capillary rise in tubes of differing diameters. …………………………… 88 *Permeability-11: Rise of the water table due to capillarity saturation. ………………….. 90 *Permeability-12: Find the capillary rise hc in a silt stratum using Hazen. ……………… 91 *Permeability-13: Back-hoe trench test to estimate the field permeability………………. 92 **Permeability-14: Seepage loss from an impounding pond…………………………………. 93 Chapter 6 Seepage and Flow-nets…………………………………………………………………………. 97 Symbols for Seepage and Flow-nets ………………………………………………………………….. 97 *Flownets-01: Correcting flawed flow-nets. ……………………………………………………….. 98 *Flow-nets-02: A flow-net beneath a dam with a partial cutoff wall………………………. 99 *Flow-nets-03: The velocity of the flow at any point under a dam. ……………………… 100 *Flow-nets-04: Flow through an earth levee……………………………………………………… 101 *Flow-nets-05: Finding the total, static and dynamic heads in a dam. ………………….. 102 **Flow nets-06: Hydraulic gradient profile within an earth levee………………………… 103 **Flow-net-07: Flow into a cofferdam and pump size………………………………………… 105 *Flow-nets-08: Drainage of deep excavations for buildings………………………………… 108 *Flow-nets-09: Dewatering a construction site………………………………………………….. 110 *Flow-net-10: Dewatering in layered strata. ……………………………………………………… 111 **Flownets-11: Flow through the clay core of an earth dam. ………………………………. 113 Chapter 7 Effective Stresses and Pore Water Pressure…………………………………………… 117 Symbols for Effective Stresses and Pore Water Pressure…………………………………….. 117 *Effective Stress–01: The concept of buoyancy. ……………………………………………….. 118 *Effective Stress–02: The concept of effective stress. ………………………………………… 119 *Effective Stress–03: The concept of effective stress with multiple strata…………….. 120 Effective Stress-03B………………………………………………………………………………………. 121 Chapter 8 Dams and Levees ………………………………………………………………………………. 122 iv Symbols for Dams and Levees ………………………………………………………………………… 122 *Dams-01: Find the uplift pressure under a small concrete levee………………………… 123 *Dams-02: Determine the uplift forces acting upon a concrete dam. ……………………. 124 Chapter 9 Stresses in Soil Masses……………………………………………………………………….. 127 Symbols for Stresses in Soil Masses ………………………………………………………………… 127 *Mohr-01: Simple transformation from principal to general stress state……………….. 129 *Mohr – 02: Find the principal stresses and their orientation. ……………………………… 130 *Mohr – 03: Find the principal stresses and their orientation. ……………………………… 131 *Mohr – 04: ………………………………………………………………………………………………….. 132 *Mohr – 05: Normal and shear stress at a chosen plane. …………………………………….. 133 **Mohr – 07: Back figure the failure angle ………………………………………………………. 134 *Mohr – 08: find the Principle pressure using Mohr ………………………………………….. 135 *Mohr – 09: Relation between θ and φ. ……………………………………………………………. 136 *Mohr – 10: ………………………………………………………………………………………………….. 137 *Mohr–11: ……………………………………………………………………………………………………. 138 *Mohr – 12: ………………………………………………………………………………………………….. 139 *Mohr – 13: Data from Mohr-Coulomb failure envelope……………………………………. 140 **Mohr – 14: ………………………………………………………………………………………………… 141 *Mohr – 15: Derive the general formula for horizontal stress. …………………………….. 142 *Newmark–01: Stress beneath a tank at different depths…………………………………….. 143 *Newmark-02: The stress below the center of the edge of a footing. ……………………. 144 *Newmark-03: Stress at a point distant from the loaded footing. …………………………. 145 *Newmark-04: Stresses coming from complex shaped foundations……………………… 146 *Newmark-05: Stress beneath a circular oil tank………………………………………………. 147 **Newmark-06: Use Newmark with a settlement problem. ………………………………… 148 *Stress–01: Stress increase at a point from several surface point loads…………………. 150 *Stress-02: Find the stress under a rectangular footing……………………………………….. 151 *Stress-03: The effect of the WT on the stress below a rectangular footing…………… 152 *Stress–04: Finding the stress outside the footing area……………………………………….. 153 *Stress-05: Stress below a footing at different points. ……………………………………….. 154 *Stress-06: Stress increase from a surcharge load of limited width………………………. 155 *Stress-07: Finding a stress increase from a surface load of limited width. …………… 156 **Stress-08: Stress increase as a function of depth…………………………………………….. 157 v Chapter 10 Elastic Settlements …………………………………………………………………………… 158 Symbols for Elastic Settlements ………………………………………………………………………. 158 *Elastic Settlement-01: Settlement (rutting) of a truck tire………………………………….. 159 *Elastic Settlement-02: Schmertmann method used for granular soils………………….. 160 *Elastic Settlement-03: Schmertmann method used for a deeper footings. ……………. 161 *Elastic Settlement-04: The 2:1 method to calculate settlement…………………………… 163 *Elastic Settlement-05: Differential settlement between two columns………………….. 165 *Elastic Settlement-06: Compare the settlements predicted by the Boussinesq, Westergaard, and the 2:1 methods……………………………………………………………………. 166 *Elastic Settlement-07: Schmertmann versus the strain methods. ………………………… 169 *Elastic Settlement-08: The Schmertmann method in multiple strata. ………………….. 170 **Elastic Settlement-09: Settlement of a mat foundation. …………………………………… 172 Chapter 11 Plastic Settlements……………………………………………………………………………. 174 Symbols for Plastic Settlements ………………………………………………………………………. 174 *Plastic Settlement–01: Porewater pressure in a compressible soil. ……………………… 175 *Plastic Settlement-02: Total settlement of a single layer. ………………………………….. 177 *Plastic Settlement-03: Boussinesq to reduce the stress with depth. …………………….. 178 *Plastic Settlement -04: Surface loads with different units………………………………….. 180 *Plastic Settlement-05: Pre-consolidation pressure pc and index Cc……………………… 181 *Plastic Settlement-06: Final voids ratio after consolidation……………………………….. 183 *Plastic Settlement-07: Settlement due to a lowered WT. …………………………………… 184 *Plastic Settlement-08: The over-consolidation ratio (OCR)……………………………….. 185 **Plastic Settlement-09: Coefficient of consolidation Cv. …………………………………… 186 *Plastic Settlement -10: Secondary rate of consolidation. …………………………………… 188 *Plastic Settlement-11: Using the Time factor Tv. ……………………………………………… 189 *Plastic Settlement-12: The time rate of consolidation……………………………………….. 190 *Plastic Settlement-13: Time of consolidation t…………………………………………………. 191 *Plastic Settlement-14: Laboratory versus field time rates of settlement. ……………… 192 *Plastic Settlement-15: Different degrees of consolidation. ………………………………… 193 **Plastic Settlement-16: Excavate to reduce the settlement. ……………………………….. 194 **Plastic Settlement-17: Lead time required for consolidation of surcharge. ………… 196 **Plastic Settlement-18: Settlement of a canal levee………………………………………….. 198 **Plastic Settlement-19: Differential settlements under a levee. ………………………….. 200 ***Plastic Settlement-20: Estimate of the coefficient of consolidation cv……………… 202 vi **Plastic Settlement-21: The apparent optimum moisture content……………………….. 204 **Plastic Settlement-22: The differential settlement between two buildings. ………… 205 **Plastic Settlement-23: Settlement of a bridge pier. …………………………………………. 210 Chapter 12 Shear Strength of Soils……………………………………………………………………… 212 Symbols for Shear Strength of Soils…………………………………………………………………. 212 *Shear strength–01: Maximum shear on the failure plane. ………………………………….. 213 *Shear strength–02: Why is the maximum shear not the failure shear? ………………… 214 *Shear strength–03: Find the maximum principal stress σ1. ………………………………… 215 *Shear strength–04: Find the effective principal stress……………………………………….. 216 *Shear strength–05: Using the p-q diagram. ……………………………………………………… 217 **Shear strength–06: Consolidated-drained triaxial test……………………………………… 218 **Shear strength–07: Triaxial un-drained tests. …………………………………………………. 220 **Shear strength-08: Consolidated and drained triaxial test. ……………………………….. 222 ***Shear strength-09: Plots of the progressive failure in a shear-box. ………………….. 224 **Shear strength-10: Shear strength along a potential failure plane. …………………….. 227 ***Shear strength-11: Use of the Mohr-Coulomb failure envelope. …………………….. 228 ***Shear strength-11b: Use of the Mohr-Coulomb failure envelope. …………………… 230 **Shear strength-12: Triaxial un-drained tests…………………………………………………… 232 **Shear strength-12b: Triaxial un-drained tests…………………………………………………. 233 **Shear strength-13: Determine the principal stresses of a sample. ……………………… 234 **Shear strength-13b: Determine the principal stresses of a sample. ……………………. 237 **Shear strength-14: Formula to find the maximum principal stress. …………………… 240 Chapter 13 Slope Stability …………………………………………………………………………………. 242 Symbols for Slope Stability…………………………………………………………………………….. 242 *Slope-01: Factor of Safety of a straight line slope failure………………………………….. 243 *Slope-02: Same as Slope-01 but with a raising WT………………………………………….. 244 *Slope-03: Is a river embankment safe with a large crane? …………………………………. 245 *Slope-04: Simple method of slices to find the FS. ……………………………………………. 246 **Slope-05: Method of slices to find the factor of safety of a slope with a WT……… 247 **Slope-06: Swedish slip circle solution of a slope stability. ………………………………. 249 Chapter 14 Statistical Analysis of Soils……………………………………………………………….. 252 Symbols for the Statistical Analysis of Soils……………………………………………………… 252 Chapter 15 Lateral Pressures from Soils………………………………………………………………. 253 vii Symbols for Lateral Pressures from Soils …………………………………………………………. 253 *Lateral-01: A simple wall subjected to an active pressure condition. ………………….. 257 *Lateral–02: Compare the Rankine and Coulomb lateral coefficients………………….. 258 *Lateral-03: Passive pressures using the Rankine theory. …………………………………… 259 *Lateral-04: The “at-rest” pressure upon an unyielding wall……………………………….. 260 *Lateral-05: The contribution of cohesion to reduce the force on the wall. …………… 261 **Lateral-06: The effect of a rising WT upon a wall’s stability. ………………………….. 262 *Lateral-07: The effects of soil-wall friction upon the lateral pressure. ………………… 264 *Lateral-08: What happens when the lower stratum is stronger? …………………………. 265 *Lateral-09: Strata with different parameters…………………………………………………….. 266 *Lateral-10: The effects of a clay stratum at the surface. …………………………………… 268 **Lateral-11: Anchoring to help support a wall…………………………………………………. 270 **Lateral-12: The effect of five strata have upon a wall…………………………………….. 272 **Lateral-13: The stability of a reinforced concrete wall. ………………………………….. 274 ***Lateral-14: Derive a formula that provides K and σH as a function of σv. ………… 277 **Lateral-15: The magnitude and location of a seismic load upon a retaining wall… 280 **Lateral-16: Seismic loading upon a retaining wall………………………………………….. 282 Chapter 16 Braced Cuts for Excavations ……………………………………………………………… 283 Symbols for Braced Cuts for Excavations…………………………………………………………. 283 *Braced-cuts-01: Forces and moments in the struts of a shored trench. ………………… 284 **Braced cuts-02: A 5 m deep excavation with two struts for support………………….. 289 *Braced cuts-03: Four-struts bracing a 12 m excavation in a soft clay…………………. 293 Chapter 17 Bearing Capacity of Soils………………………………………………………………….. 296 Symbols for the Bearing Capacity of Soils ……………………………………………………….. 296 *Bearing–01: Terzaghi’s bearing capacity formula for a square footing……………….. 299 *Bearing–02: Meyerhof’s bearing capacity formula for a square footing. …………….. 300 *Bearing–03: Hansen’s bearing capacity formula for a square footing. ………………… 301 *Bearing–04: Same as #01 but requiring conversion from metric units. ……………….. 302 *Bearing–05: General versus local bearing capacity failures. ……………………………… 303 *Bearing–06: Comparing the Hansen and Meyerhof bearing capacities. ………………. 304 *Bearing–07: Increase a footing’s width if the WT is expected to rise. ……………….. 305 **Bearing–08: The effect of the WT upon the bearing capacity. …………………………. 307 *Bearing–09: Finding the gross load capacity. …………………………………………………. 309 viii **Bearing–10: The effect of an eccentric load upon bearing capacity. …………………. 311 **Bearing–11: The effect of an inclined load upon the bearing capacity. ……………… 312 **Bearing-12: Interpretation of borings to estimate a bearing capacity. ……………….. 314 Chapter 18 Shallow Foundations………………………………………………………………………… 316 Symbols for Shallow Foundations……………………………………………………………………. 316 *Footings–01: Analyze a simple square footing. ……………………………………………….. 318 *Footings–02: Add a moment to the load on a footing. ………………………………………. 322 *Footings–03: Find the thickness T and the As of the previous problem……………….. 324 *Footings–04: Find the dimensions B x L of a rectangular footing………………………. 329 *Footings–05: Design the steel for the previous problem……………………………………. 331 *Footings–06: Design a continuous footing for a pre-cast warehouse wall…………… 335 **Footings–07: Design the footings of a large billboard sign………………………………. 340 Chapter 19 Combined Footings ………………………………………………………………………….. 344 Symbols for Combined Footings……………………………………………………………………… 344 Chapter 20 Mat Foundations………………………………………………………………………………. 345 Symbols for Mat Foundations …………………………………………………………………………. 345 *Mat Foundations–01: Ultimate bearing capacity in a pure cohesive soil……………… 346 Chapter 21 Deep Foundations – Single Piles ………………………………………………………… 347 Symbols for Single Piles of Deep Foundations………………………………………………….. 347 *Single-Pile–01: Pile capacity in a cohesive soil. ………………………………………………. 348 Chapter 22 Deep Foundations – Pile Groups and Caps…………………………………………… 349 Symbols for Pile Groups and Caps of Deep Foundations ……………………………………. 349 **Pile-caps–01: Design a pile cap for a 9-pile cluster. ……………………………………….. 350 Chapter 23 Deep Foundations: Lateral Loads ………………………………………………………. 353 Symbols for Lateral Loads on Deep Foundations ………………………………………………. 353 **Lateral loads on piles-01: Find the lateral load capacity of a steel pile………………. 354 Chapter 24 Reinforced Concrete Retaining Walls and Bridge Abutments………………… 358 Symbols for Reinforced Concrete Retaining Walls ……………………………………………. 358 **RC Retaining Walls–01: Design a RC wall for a sloped backfill. …………………….. 359 Chapter 25 Steel Sheet Pile Retaining Walls………………………………………………………… 367 Symbols for Steel Sheet Pile Retaining Walls……………………………………………………. 367 **Sheet-pile Wall-01: Free-Earth for cantilevered walls in granular soils. ……………. 368 Chapter 26 MSE (Mechanically Stabilized Earth) Walls ……………………………………….. 373 ix Symbols for Mechanically Stabilized Earth Walls……………………………………………… 373 **MSE Walls-01: Design the length L of geotextiles for a 16 ft wall. ………………….. 374

300 Solved Problems Soil / Rock Mechanics and Foundations Engineering Former Professor, United States Military Academy (West Point) Telephone 305-348-2825 / Fax 305-348-2802 Table of Contents

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• Appendix: Solutions to Exercises

Principles of rock mechanics

Home Courses Principles of rock mechanics Subjects Appendix: Solutions to Exercises

Throughout the sixteen chapters several exercises are given in order to make the theory more understandable. In this appendix, the solutions to these exercises are given.

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• 300 Solved Problems Soil Rock Mechanics and Foundations Engineering

Table Of Contents

• Soil Exploration
• Symbols For Soil Exploration
• Find The Required Number Of Borings And Their Depth
• The Sample’s Disturbance Due To The Boring Diameter
• Correcting The SPT For Depth And Sampling Method
• Three Methods Used For SPT Depth Corrections
• SPT Corrections Under A Mat Foundation
• The Shear Vane Test Determines The In-Situ Cohesion
• Reading A Soil Boring Log

Using A Boring Log To Predict Soil Engineering Parameters

• Find The Shear Strength Of A Soil From The CPT Report
• Phase Relations Of Soil
• Symbols For Phase Relations Of Soils
• Basic Concepts And Formulas For The Phases Of Soils
• Convert From Metric Units To SI And US Units
• Compaction Checked Via The Voids Ratio
• Value Of The Moisture When Fully Saturated
• Finding The Wrong Data
• Increasing The Saturation Of A Soil
• Find Γd, N, S And Ww
• 300 Solved Problems Soil
• Use The Block Diagram To Find The Degree Of Saturation
• Same As Prob- But Setting The Total Volume V= M
• Same As Problem # With A Block Diagram
• Block Diagram For A Saturated Soil
• Find The Weight Of Water Needed For Saturation
• Identify The Wrong Piece Of Data
• The Apparent Cheapest Soil Is Not!
• Number Of Truck Loads

How Many Truck Loads Are Needed For A Project?

• Choose The Cheapest Fill Supplier
• Use A Matrix To The Find The Missing Data
• Find The Voids Ratio
• Classification Of Soils And Rocks
• Symbols For Classification Of Soils
• Percentage Of Each Of The Four Grain Sizes (G, S, M & C)
• Coefficients Of Uniformity And Curvature Of Granular Soils
• Classify Two Soils Using The USCS
• Manufacturing A “New” Soil
• Compaction And Soil Improvement
• Symbols For Compaction
• Compaction–: Find The Optimum Moisture Content (OMC)
• Compaction–: Find Maximum Dry Unit Weight In SI Units
• What Is The Saturation S At The OMC?
• Number Of Truck Loads Required
• Definition Of The Relative Compaction (RC)
• The Relative Compaction (RC) Of A Soil

Converting Volumes From Borrow Pits And Truck Loads

• Ranges Of Water And Fill Required For A Road
• Find The Family Of Saturation Curves For Compaction
• Water Needed To Reach Maximum Density In The Field
• Fill Volumes And Truck Load Requirements For A Levee
• Multiple Choice Compaction Problem Of A Levee
• Permeability Of Soils
• Symbols For Permeability
• Types Of Permeability Tests And Common Units

300 Solved Problems Soil Rock Mechanics

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Rock deformation problems

• Published: December 2004
• Volume 40 , pages 1311–1322, ( 2004 )

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• A. F. Bulat 1  

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Studies into the stress-strain state of rock mass are reviewed. The solutions of a wide range of applied problems are analyzed and methods used to solve them are generalized. Important areas of further research in rock mechanics are pointed out.

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Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 3–17, December 2004.

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Bulat, A.F. Rock deformation problems. Int Appl Mech 40 , 1311–1322 (2004). https://doi.org/10.1007/s10778-005-0039-y

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