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  • INTRODUCTION OF SOIL MECHANICS
    • GEOTECHNICAL LESSONS FROM FAILURES
    • BASIC GEOLOGY
    • INTRODUCTION OF SOILS INVESTIGATION
    • PHASE RELATIONSHIPS
    • Importance of soil compaction
    • HEAD AND PRESSURE VARIATION IN A FLUID AT REST
    • GEOLOGICAL CHARACTERISTICS AND PARTICLE SIZES OF SOILS
    • Composition of the Earth’s Crust
    • PHASES OF A SOILS INVESTIGATION
    • PHYSICAL STATES AND INDEX PROPERTIES OF FINE-GRAINED SOILS
    • INTERPRETATION OF PROCTOR TEST RESULTS
    • DARCY’S LAW
    • COMPOSITION OF SOILS
    • SOILS EXPLORATION PROGRAM
    • DETERMINATION OF THE LIQUID, PLASTIC, AND SHRINKAGE LIMITS
    • SOIL CLASSIFICATION SCHEMES
    • FIELD COMPACTION
    • FLOW PARALLEL TO SOIL LAYERS
    • Surface Forces and Adsorbed Water
    • Soil Identifi cation in the Field
    • DETERMINATION OF THE HYDRAULIC CONDUCTIVITY
    • DETERMINATION OF PARTICLE SIZE OF SOILS
    • Soil Sampling
    • Falling-Head Test
    • Particle Size of Fine-Grained Soils
    • Groundwater Conditions
    • Pumping Test to Determine the Hydraulic Conductivity
    • COMPARISON OF COARSE-GRAINED AND FINE-GRAINED SOILS
    • Types of In Situ or Field Tests
    • GROUNDWATER LOWERING BY WELLPOINTS

  • SOIL WATER AND WATER FLOW
    • STRESSES AND STRAINS
    • STRESS AND STRAIN INVARIANTS
    • IDEALIZED STRESS–STRAIN RESPONSE AND YIELDING
    • Hooke’s Law Using Stress and Strain Invariants
    • PLANE STRAIN AND AXIAL SYMMETRIC CONDITIONS
    • STRESS PATHS
    • Axisymmetric Condition
    • Plotting Stress Paths Using Two-Dimensional Stress Parameters
    • ANISOTROPIC, ELASTIC STATES
    • Mohr’s Circle for Stress States
    • Mohr’s Circle for Strain States
    • The Principle of Effective Stress
    • Effective Stresses Due to Geostatic Stress Fields
    • Effects of Capillarity
    • Effects of Seepage
    • LATERAL EARTH PRESSURE AT REST
    • STRESSES IN SOIL FROM SURFACE LOADS
    • Strip Load
    • Uniformly Loaded Rectangular Area
    • Vertical Stress Below Arbitrarily Shaped Areas

  • STRESS DISTRIBUTIONCOMPRESSIBILITY AND SETTLEMENT
    • BASIC CONCEPTS
    • TYPICAL RESPONSE OF SOILS TO SHEARING FORCES
    • BASIC CONCEPTS
    • Consolidation Under a Constant Load—Primary Consolidation
    • Effects of Increasing the Normal Effective Stress
    • Soil Yielding
    • Void Ratio and Settlement Changes Under a Constant Load
    • Effects of Soil Tension
    • Primary Consolidation Parameters
    • Coulomb’s Failure Criterion
    • CALCULATION OF PRIMARY CONSOLIDATION SETTLEMENT
    • Taylor’s Failure Criterion
    • Procedure to Calculate Primary Consolidation Settlement
    • Mohr–Coulomb Failure Criterion
    • ONE-DIMENSIONAL CONSOLIDATION THEORY
    • PRACTICAL IMPLICATIONS OF THE FAILURE CRITERIA
    • Solution of Governing Consolidation Equation Using Fourier Series
    • INTERPRETATION OF THE SHEAR STRENGTH OF SOILS
    • Finite Difference Solution of the Governing Consolidation Equation
    • LABORATORY TESTS TO DETERMINE SHEAR STRENGTH PARAMETERS
    • SECONDARY COMPRESSION SETTLEMENT
    • Conventional Triaxial Apparatus
    • Oedometer Test
    • Unconfi ned Compression (UC) Test
    • Determination of the Coeffi cient of Consolidation
    • Consolidated Undrained (CU) Compression Test
    • Determination of the Past Maximum Vertical Effective Stress
    • POREWATER PRESSURE UNDER AXISYMMETRIC UNDRAINED LOADING
    • PRECONSOLIDATION OF SOILS USING WICK DRAINS
    • OTHER LABORATORY DEVICES TO MEASURE SHEAR STRENGTH
    • Hollow-Cylinder Apparatus
    • FIELD TESTS

  • SHEAR STRENGTH
    • ALLOWABLE STRESS AND LOAD AND RESISTANCE FACTOR DESIGN
    • COLLAPSE LOAD USING THE LIMIT EQUILIBRIUM METHOD
    • Prediction of the Behavior of Coarse-Grained Soils Using CSM
    • BEARING CAPACITY EQUATIONS
    • ELEMENTS OF THE CRITICAL STATE MODEL
    • MAT FOUNDATIONS
    • FAILURE STRESSES FROM THE CRITICAL STATE MODEL
    • BEARING CAPACITY OF LAYERED SOILS
    • Undrained Triaxial Test
    • SETTLEMENT CALCULATIONS
    • MODIFICATIONS OF CSM AND THEIR PRACTICAL IMPLICATIONS
    • Primary Consolidation Settlement
    • RELATIONSHIPS FROM CSM THAT ARE OF PRACTICAL SIGNIFICANCE
    • DETERMINATION OF BEARING CAPACITY AND SETTLEMENT OF COARSE-GRAINED SOILS
    • Relationships Among the Tension Cutoff, Mean Effective Stress, and Preconsolidation Stress
    • Cone Penetration Test (CPT)
    • Relationships Among Undrained Shear Strength, Critical State Friction Angle, and Preconsolidation Ratio
    • Plate Load Test (PLT)
    • Relationship Between the Normalized Undrained Shear Strength of One-Dimensionally Consolidated or Ko-Consolidated and Isotropically
    • SHALLOW FOUNDATION ANALYSIS USING CSM
    • Relationship Between the Normalized Undrained Shear Strength at Initial Yield and at Critical State for Overconsolidated Fine-Grained Soils Under Triaxial Test Condition
    • Dense, Coarse-Grained Soils
    • Relationship Between Direct Simple Shear Tests and Triaxial Tests
    • Relationship for the Application of Drained and Undrained
    • Relationship Among Excess Porewater Pressure, Preconsolidation Ratio, and Critical State Friction Angle
    • Undrained Shear Strength, Liquidity Index, and Sensitivity
    • SOIL STIFFNESS
    • STRAINS FROM THE CRITICAL STATE MODEL
    • Shear Strains
    • CALCULATED STRESS–STRAIN RESPONSE
    • APPLICATION OF CSM TO CEMENTED SOILS

  • SLOPE STABILITY
    • TYPES OF PILES AND INSTALLATION
    • TWO-DIMENSIONAL FLOW OF WATER THROUGH POROUS MEDIA
    • BASIC CONCEPTS OF LATERAL EARTH PRESSURES
    • SOME CAUSES OF SLOPE FAILURE
    • Pile Installation
    • FLOWNET SKETCHING
    • COULOMB’S EARTH PRESSURE THEORY
    • Construction Activities
    • LOAD CAPACITY OF SINGLE PILES
    • INTERPRETATION OF FLOWNET
    • RANKINE’S LATERAL EARTH PRESSURE FOR A SLOPING BACKFILL AND A SLOPING WALL FACE
    • INFINITE SLOPES
    • PILE LOAD TEST (ASTM D 1143)
    • FLOW THROUGH EARTH DAMS
    • LATERAL EARTH PRESSURES FOR A TOTAL STRESS ANALYSIS
    • ROTATIONAL SLOPE FAILURES
    • METHODS USING STATICS FOR DRIVEN PILES
    • SOIL FILTRATION
    • APPLICATION OF LATERAL EARTH PRESSURES TO RETAINING WALLS
    • METHOD OF SLICES
    • PILE LOAD CAPACITY OF DRIVEN PILES BASED ON SPT AND CPT RESULTS
    • TYPES OF RETAINING WALLS AND MODES OF FAILURE
    • APPLICATION OF THE METHOD OF SLICES
    • LOAD CAPACITY OF DRILLED SHAFTS
    • STABILITY OF RIGID RETAINING WALLS
    • PROCEDURE FOR THE METHOD OF SLICES
    • PILE GROUPS
    • STABILITY OF FLEXIBLE RETAINING WALLS
    • STABILITY OF SLOPES WITH SIMPLE GEOMETRY
    • ELASTIC SETTLEMENT OF PILES
    • Analysis of Sheet Pile Walls in Mixed Soils
    • CONSOLIDATION SETTLEMENT UNDER A PILE GROUP
    • BRACED EXCAVATION
    • SETTLEMENT OF DRILLED SHAFTS
    • MECHANICAL STABILIZED EARTH WALLS
    • PILE-DRIVING FORMULAS AND WAVE EQUATION
    • OTHER TYPES OF RETAINING WALLS
    • LATERALLY LOADED PILES
    • MICROPILES

Branch : Civil Engineering
Subject : Soil Mechanics
Unit : SOIL WATER AND WATER FLOW

Mohr’s Circle for Stress States


STRESS AND STRAIN STATES:
Access www.wiley.com/college/budhu, and click Chapter 7 and then Mohrcircle.zip to download an application to plot, interpret, and explore a variety of stress states interactively.

Stresses on a two-dimensional element and Mohr’s circle.

 

Mohr’s Circle for Stress States:
Suppose a cuboidal sample of soil is subjected to the stresses shown in Figure . We would like to know what the stresses are at a point, say, A, within the sample due to the applied stresses. One approach to fi nd the stresses at A, called the stress state at A, is to use Mohr’s circle. The stress state at a point is the set of stress vectors corresponding to all planes passing through that point. For simplicity, we will consider a two-dimensional element with stresses, as shown in Figure 7.12a. Let us draw Mohr’s circle. First, we have to choose a sign convention. We have already decided that compressive stresses are positive for soils. We will assume counterclockwise shear is positive and sz . sx. The two coordinates of the circle are (sz, tzx) and (sx, txz). Recall from your strength of materials course that, for equilibrium, txz 5 2tzx; these are called complementary shear stresses and are orthogonal to each other. Plot these two coordinates on a graph of shear stress (ordinate) and normal stress (abscissa), as shown by A and B in Figure 7.12b. Draw a circle with AB as the diameter. The circle crosses the normal stress axis at 1 and 3.


The stresses at these points are the major principal stress, s1, and the minor principal stress, s3. The principal stresses are related to the stress components sz, sx, tzx by

The angle between the major principal stress plane and the horizontal plane (c) is

The stresses on a plane oriented at an angle u from the major principal stress plane are

 

The stresses on a plane oriented at an angle u from the horizontal plane are

In the above equations, u is positive for counterclockwise orientation. The maximum (principal) shear stress is at the top of the circle with magnitude

 

For the stresses shown in Figure 7.9 we would get three circles, but we have simplifi ed the problem by plotting one circle for stresses on all planes perpendicular to one principal direction. The stress sz acts on the horizontal plane and the stress sx acts on the vertical plane for our case. If we draw these planes in Mohr’s circle, they intersect at a point, P. Point P is called the pole of the stress circle. It is a special point because any line passing through the pole will intersect Mohr’s circle at a point that represents the stresses on a plane parallel to the line. Let us see how this works. Suppose we want to fi nd the stresses on a plane inclined at an angle u from the horizontal plane, as depicted by MN in Figure . Once we locate the pole, P, we can draw a line parallel to MN through P as shown by M9N9 in Figure. The line M9N9 intersects the circle at N9 and the coordinates of N9, (su, tu), represent the normal and shear stresses on MN.

Questions of this topic


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