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
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*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
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*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
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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
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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
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**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
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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
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**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
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Symbols for Mechanically Stabilized Earth Walls……………………………………………… 373
**MSE Walls-01: Design the length L of geotextiles for a 16 ft wall. ………………….. 374