Shear Strength and Stress-Path Behaviour

Introduction and Context

Shear strength is the resistance of soil to distortion and failure when subjected to shearing stresses. It governs the stability of slopes, foundations, retaining structures, embankments, excavations, tunnels and earth dams. In advanced soil mechanics, shear strength cannot be treated as a single constant property. It depends on effective stress, drainage conditions, stress history, density, fabric, loading direction, strain level, particle crushing and the path followed by stresses in stress space.

A soil element may experience the same final stresses through different loading paths. The resulting pore pressure, deformation and failure state may nevertheless be different. This is the central significance of stress-path analysis. For example, rapid loading of saturated clay may occur under nearly undrained conditions, whereas long-term loading may permit drainage and consolidation. A dense sand may dilate during shearing, while a loose sand may contract. A structured clay may exhibit a high peak strength but lose much of that strength after remoulding or large displacement.

This chapter develops the effective-stress basis of shear strength, distinguishes drained and undrained behaviour, explains contractancy and dilation, examines peak and residual strength, and introduces anisotropy, sensitivity and stress-path methods for engineering interpretation.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Explain total-stress and effective-stress descriptions of soil strength.

2. Evaluate drained and undrained shear strength using appropriate parameters.

3. Interpret pore-pressure response during shearing.

4. Distinguish contractive, dilative, peak, critical-state and residual behaviour.

5. Construct and interpret stress paths in p′–q and principal-stress space.

6. Assess the effects of stress history, anisotropy and sensitivity.

7. Select suitable laboratory tests and parameters for engineering design.

8. Apply shear-strength concepts to a practical slope or foundation problem.

1. Effective Stress and the Mohr–Coulomb Framework

For saturated soil, Terzaghi’s effective-stress principle is

σ′ = σ − u

where σ′ is effective normal stress, σ is total normal stress and u is pore-water pressure. All stresses must be expressed in consistent units, commonly kPa. The principle assumes that the stress carried by the soil skeleton is the difference between total stress and pore-fluid pressure. For unsaturated soils, suction and the degree of saturation must also be considered; the simple saturated expression is then incomplete.

The Mohr–Coulomb effective-stress strength criterion is

τf = c′ + σ′n tan φ′

where τf is shear stress at failure, c′ is effective cohesion intercept, σ′n is effective normal stress on the failure plane and φ′ is the effective angle of shearing resistance. This equation is an idealised linear envelope over a specified stress range. It should not be extrapolated beyond the stress range used to determine the parameters, particularly for structured, crushable or heavily overconsolidated soils.

For many clean sands, c′ is approximately zero and strength is mainly frictional. An apparent cohesion intercept may arise from cementation, suction, bonding or fitting a straight line to a curved envelope. It should not automatically be interpreted as a true material cohesion.

In total-stress analysis, particularly for short-term undrained clay problems, strength is commonly expressed as

τf = cu

where cu is the undrained shear strength. More generally, a total-stress Mohr–Coulomb form may be written as τf = cu + σn tan φu. For saturated normally consolidated clays tested under conventional undrained conditions, φu is often approximated as zero, but this is not a universal law. The value of cu depends on water content, stress history, sampling, strain rate, anisotropy and the test procedure.

2. Stress Invariants and Stress Paths

Three-dimensional stress states are often represented using mean effective stress and deviator stress. In triaxial compression, with axial effective stress σ′1 and radial effective stress σ′3,

p′ = (σ′1 + 2σ′3)/3

q = σ′1 − σ′3

Here p′ is the mean effective stress and q is the deviator stress. These definitions are specific to axisymmetric triaxial conditions. In general three-dimensional stress states, p′ is one-third of the first effective-stress invariant and q is commonly defined from the second deviatoric invariant.

A stress path is the locus of stress states followed by a soil element during loading. Common paths include:

• Isotropic compression: p′ increases while q remains zero.
• Triaxial compression with constant σ′3: q increases and p′ increases at a related rate.
• Undrained shearing: total mean stress may remain constant, while p′ changes because pore pressure changes.
• Constant-p′ shearing: q changes while p′ is held approximately constant.
• Extension: the major and minor principal stress roles differ from triaxial compression.

Stress paths may be plotted in p′–q space. The critical-state line is often represented by

q = M p′

where M is the critical-state stress ratio. For a triaxial compression convention,

M = 6 sin φ′cs / (3 − sin φ′cs)

where φ′cs is the critical-state friction angle in triaxial compression. This relationship assumes the conventional critical-state idealisation and axisymmetric conditions. Different values arise in extension or under alternative definitions of q.

The p′–q diagram is useful because it separates volumetric stress from distortional stress. A stress path can show whether loading causes increasing effective confinement, decreasing confinement, or movement toward a state boundary. It also clarifies why two tests with the same final deviator stress may produce different pore pressures and strains.

3. Drained and Undrained Shear Behaviour

3.1 Drained loading

Under drained conditions, water is permitted to escape or enter sufficiently rapidly that excess pore pressure is negligible during the relevant shearing stage. The soil volume may change, and the effective stress path is close to the total-stress path. Drained parameters c′ and φ′ are therefore directly relevant.

Drained strength is appropriate for long-term stability of many slopes, retaining structures and foundations in permeable soils, provided the field loading rate permits drainage. In low-permeability soils, a process may be practically undrained during construction but drained over the service life.

3.2 Undrained loading

Under undrained loading, no net water can leave a saturated soil element during the shearing stage. Volume change of the whole specimen is approximately zero, but the soil skeleton may tend to contract or dilate. This tendency is accommodated by pore-pressure change.

For saturated soil,

Δσ′ = Δσ − Δu

where Δ denotes a change. If total stress is prescribed, a positive excess pore pressure Δu reduces effective stress; a negative excess pore pressure increases it. Consequently, contractive soils commonly lose effective stress during undrained shearing, while dilative soils may gain effective stress.

Undrained strength is not a unique intrinsic constant. It depends on the initial effective stress, overconsolidation ratio, stress path, direction of shearing, strain rate and sample quality. The notation su is often used for undrained shear strength, but the test conditions should always be stated, such as triaxial compression, simple shear or direct simple shear.

3.3 Pore-pressure parameterisation

For triaxial testing, Skempton’s pore-pressure parameters are commonly written as

Δu = B[Δσ3 + A(Δσ1 − Δσ3)]

where A and B are pore-pressure coefficients, Δσ1 and Δσ3 are changes in major and minor principal total stresses, and B reflects saturation and compressibility effects. For a nearly saturated specimen, B is close to unity. The value of A varies with stress history and shearing stage; it is not a universal constant.

During undrained triaxial compression with constant cell pressure, Δσ3 = 0, so

Δu = BAΔq

because Δq = Δσ1 − Δσ3. A positive A indicates pore-pressure generation during the relevant stage, while a negative value may indicate dilation. The equation is an interpretation tool, not a substitute for direct measurement, and its use requires consistent sign conventions.

4. Contractancy, Dilation and Critical State

Contractancy is the tendency of a soil skeleton to reduce volume during shear. Loose sands and normally consolidated clays are generally contractive over important strain ranges. Under undrained conditions, contractancy tends to generate positive pore pressure and reduce p′.

Dilation is the tendency of the soil skeleton to increase volume during shear. Dense sands and heavily overconsolidated clays commonly dilate after sufficient shearing. In undrained conditions, dilation tends to generate negative excess pore pressure and increase p′.

The observed response depends on density relative to the critical state, not merely on absolute density. The critical state is the condition at large shear strain in which soil continues to shear at approximately constant stress ratio, effective mean stress and volume. The critical-state friction angle is generally lower than the peak friction angle of a dense or heavily overconsolidated soil because the latter includes structural and dilative contributions.

A useful conceptual distinction is:

• Peak state: maximum measured shear resistance, often associated with dilation, bonding or particle interlocking.
• Critical state: continuing shear at constant stress and volume, after the influence of initial fabric and density has largely disappeared.
• Residual state: large-displacement strength, often involving particle alignment and polished shear surfaces, especially in clay-rich soils.

For a dense granular soil, the stress–strain curve may rise to a peak and then soften toward the critical-state strength. For a loose soil, the curve may increase monotonically toward the critical state. In a clay, destructuration or pore-pressure generation may produce substantial strain softening.

The state parameter is often used in critical-state soil mechanics:

ψ = e − ecs(p′)

where e is the current void ratio and ecs(p′) is the critical-state void ratio at the same p′. A positive ψ generally indicates a looser-than-critical state and a tendency toward contractancy; a negative ψ indicates a denser-than-critical state and a tendency toward dilation. The interpretation depends on the adopted critical-state line and the stress range.

5. Peak, Critical and Residual Strength

5.1 Peak strength

Peak strength is the maximum shear resistance mobilised during a test. It is important in short-displacement failures and in dense or structured soils. Peak strength may include:

• friction between particles;
• interlocking and dilation;
• cementation or bonding;
• suction in partially saturated soil; and
• fabric-related anisotropy.

Because dilation requires work against confinement, peak strength generally increases with density and decreases as effective normal stress becomes sufficiently large. At high stresses, particle crushing can suppress dilation and reduce the peak friction angle.

5.2 Residual strength

Residual strength is the resistance mobilised after very large displacement, when a well-developed shear surface has formed. In highly plastic clay, platy particles may become aligned parallel to the shear surface, producing a low residual friction angle. Residual strength is relevant to reactivated landslides, fissured clay slopes, spoil heaps and old shear zones.

Residual strength should not be estimated simply by assigning an arbitrary reduction to peak strength. It should be measured using ring-shear or carefully designed repeated-shear tests when residual conditions govern. The effective-stress envelope may be curved, and the residual parameter can depend on mineralogy, normal stress and shear displacement.

6. Stress History, Overconsolidation and Sensitivity

The overconsolidation ratio is

OCR = σ′p / σ′v0

where σ′p is the maximum past vertical effective stress and σ′v0 is the current vertical effective stress. Normally consolidated soils have OCR close to one; overconsolidated soils have OCR greater than one.

Normally consolidated clays tend to contract during undrained shearing, generate positive pore pressure and follow an effective stress path toward lower p′. Overconsolidated clays may initially behave more rigidly and dilatively, generate lower or negative pore pressure, and exhibit a pronounced peak followed by softening. As OCR decreases toward unity, the response becomes more contractive and peak strength becomes less prominent.

Sensitivity is the ratio of the undrained strength of an intact cohesive soil to that of the same soil after remoulding at approximately the same water content:

St = su,intact / su,remoulded

The ratio is dimensionless. High sensitivity indicates that fabric, bonding or cementation contributes strongly to intact strength. Sensitive or quick clays may lose strength rapidly when disturbed, creating major hazards for excavations, embankments and natural slopes. Sensitivity is not the same as thixotropy, although thixotropic strength recovery may occur after remoulding in some clays.

7. Anisotropy and Directional Strength

Soil fabric is commonly directional because of deposition, consolidation, particle orientation and geological history. Strength measured in vertical compression may therefore differ from strength measured in horizontal or inclined shearing.

Important forms include:

• inherent anisotropy: caused by depositional fabric;
• induced anisotropy: caused by previous loading or shearing;
• strength anisotropy: variation of failure resistance with loading direction; and
• stiffness anisotropy: directional variation of stress–strain response.

For natural clays, active failure in triaxial compression, passive failure in extension and direct simple shear may produce different undrained strengths. A design based only on vertical compression tests can therefore be unconservative for an embankment or slope if the governing field failure mode involves different principal-stress orientations.

Anisotropy is addressed through sampling and testing in relevant directions, empirical correction only when justified by high-quality data, and constitutive models that incorporate rotational hardening or fabric evolution. Applying a single isotropic strength envelope to all loading paths is a simplification that requires engineering judgement.

8. Laboratory Evaluation of Strength

8.1 Direct shear test

The direct shear test is relatively simple and permits control of normal stress and measurement of shear displacement. It is useful for sands, interfaces and residual-strength studies. However, the failure plane is imposed, stress distribution is non-uniform, and drainage conditions may be difficult to define precisely.

8.2 Triaxial compression test

The triaxial test permits control of consolidation and drainage history. Common procedures include:

• consolidated drained (CD): drainage allowed during consolidation and shearing;
• consolidated undrained (CU): consolidation drained, shearing undrained, with pore pressure measured; and
• unconsolidated undrained (UU): neither consolidation nor shearing drainage permitted.

A CU test with reliable pore-pressure measurement can provide effective-stress strength parameters, whereas a UU test generally provides a total-stress response for its particular initial condition. Specimen saturation, membrane compliance, end restraint, strain rate and sample disturbance must be assessed.

8.3 Simple shear and ring shear

Simple shear is valuable for embankment and slope problems because it approximates some field stress rotations and plane-strain conditions. Ring shear permits very large displacement and is particularly suitable for residual strength. No single test reproduces every field stress path; test selection must follow the anticipated mechanism.

9. Worked Example: Drained and Undrained Interpretation

A saturated clay element is subjected to an increase in total vertical stress of 120 kPa while lateral total stress remains unchanged. During rapid loading, the measured excess pore pressure is 80 kPa. If the initial vertical effective stress is 180 kPa and the initial lateral effective stress is 100 kPa, determine the final principal effective stresses and comment on the response.

Assume the vertical direction is principal direction 1 and the lateral direction is principal direction 3. The total stress increments are Δσ1 = 120 kPa and Δσ3 = 0. Since the excess pore pressure is 80 kPa,

Δσ′1 = Δσ1 − Δu = 120 − 80 = 40 kPa

Δσ′3 = Δσ3 − Δu = 0 − 80 = −80 kPa

Therefore,

σ′1,final = 180 + 40 = 220 kPa

σ′3,final = 100 − 80 = 20 kPa

The final deviator stress is q = 220 − 20 = 200 kPa, and the final mean effective stress is

p′ = (220 + 2×20)/3 = 260/3 = 86.7 kPa.

The large positive pore pressure substantially reduced lateral effective confinement. The response is consistent with a contractive undrained tendency. Although the vertical effective stress increased, the mean effective stress decreased from (180 + 2×100)/3 = 126.7 kPa to 86.7 kPa. This example demonstrates why total-stress loading alone cannot predict effective stress or strength mobilisation.

10. Applied Case: Illustrative Embankment on Sensitive Clay

Background

Consider an illustrative case involving a 7 m-high road embankment constructed rapidly over a soft, saturated, sensitive clay deposit. Site investigation shows a natural water content near the liquid limit, low permeability, and an intact undrained strength of 35 kPa near the centre of the deposit. Remoulded strength is 7 kPa, giving a sensitivity of 5.

Problem

The embankment is to be built in one stage. A conventional analysis using only intact peak strength predicts an acceptable factor of safety. However, field observations from nearby construction indicate lateral heave and cracking near the toe.

Analysis

Rapid construction produces an approximately undrained response. The clay tends to contract, generating positive pore pressure and reducing effective stress. The likely failure mechanism involves a combination of direct simple shear beneath the embankment and passive-type shearing near the toe. Neither may be represented adequately by a single triaxial compression value.

Sensitivity indicates that local disturbance or progressive failure could cause substantial strength loss. If a shear band develops, peak strength may not be available throughout the potential failure surface. Consolidation after construction will increase effective stress and may improve long-term strength, but intermediate construction stages may be critical.

Approach and solution

A suitable investigation would include high-quality sampling, piezometers, vane-shear testing with disturbance checks, CU triaxial tests with pore-pressure measurement, and simple-shear or DSS tests at representative stress states. Construction should be staged, with waiting periods based on measured pore-pressure dissipation. Stability should be checked using undrained strength appropriate to the field stress path, and instrumentation should include settlement plates, inclinometers and piezometers.

If monitoring shows pore pressures approaching design limits, construction should pause or the embankment geometry should be modified using berms or staged widening. Long-term analysis should use effective-stress parameters and account for possible progressive or residual strength where a pre-existing shear surface exists.

Outcome and learning points

The illustrative case demonstrates that a satisfactory long-term drained analysis does not guarantee short-term construction stability. The relevant strength is governed by loading rate, stress path, anisotropy, sensitivity and the possibility of progressive failure. Instrumentation and observational control are therefore part of the engineering design, not merely post-construction quality checks.

11. Design Interpretation, Limitations and Best Practice

Strength parameters must be selected to represent the anticipated failure mechanism and design situation. The following principles are particularly important:

1. Define whether the analysis is total-stress or effective-stress based.

2. Match drainage assumptions to construction and service times.

3. Identify the likely stress path, including principal-stress rotation where relevant.

4. Use peak, critical-state or residual strength according to the expected displacement and failure history.

5. Consider anisotropy and avoid applying compression-test data indiscriminately.

6. Check sample quality, saturation, strain rate and test reproducibility.

7. Use pore-pressure measurements in undrained tests whenever effective parameters are required.

8. Examine progressive failure in sensitive, fissured or strain-softening soils.

9. Apply anisotropic or advanced constitutive models only when data quality justifies their complexity.

10. Report test conditions, stress history and parameter derivation transparently.

Common limitations include non-uniform deformation in direct shear tests, imperfect saturation, scale effects, uncertainty in field drainage, and the difficulty of reproducing natural fabric. Numerical modelling can incorporate advanced constitutive laws such as critical-state, bounding-surface or anisotropic hardening models, but model sophistication cannot compensate for poor site characterisation.

Digital monitoring and automated interpretation are increasingly useful. Piezometers, distributed fibre-optic systems, inclinometers and remotely sensed deformation can support stress-path interpretation through time. Artificial intelligence may assist anomaly detection and parameter back-analysis, but it should not replace physical understanding, uncertainty assessment or professional responsibility. Training data may not represent rare failure mechanisms, and all automated predictions require independent engineering review.

Chapter Summary

Shear strength is a stress- and history-dependent response rather than a single universal soil constant. Effective stress controls the resistance of the soil skeleton, while drainage conditions determine whether loading generates pore pressure and changes effective stress. The Mohr–Coulomb criterion provides a useful engineering approximation, but its parameters must be linked to a specified stress range, stress path and failure condition.

Stress paths in p′–q space help interpret contractancy, dilation, critical state and undrained pore-pressure response. Loose and normally consolidated soils tend to contract and generate positive pore pressure; dense and overconsolidated soils may dilate and generate negative pore pressure. Peak strength may include interlocking, dilation and bonding, whereas critical and residual strengths represent progressively larger-deformation states.

Overconsolidation, anisotropy, sensitivity, mineralogy and sample disturbance can significantly alter measured strength. Test selection must reproduce, as far as practical, the field drainage condition, stress history and loading direction. Reliable design integrates laboratory testing, field monitoring, appropriate strength selection and explicit consideration of uncertainty and progressive failure.

Key Terms

Effective stress: Stress carried by the soil skeleton after pore pressure is deducted from total stress.

Total stress: The externally applied stress before separating pore-fluid pressure.

Drained condition: Loading condition in which drainage occurs sufficiently to prevent significant excess pore pressure.

Undrained condition: Loading condition in which no net drainage occurs during the relevant stage.

Pore pressure: Pressure of water or another pore fluid within soil voids.

Contractancy: Tendency of the soil skeleton to decrease volume during shear.

Dilation: Tendency of the soil skeleton to increase volume during shear.

Stress path: Locus of stress states followed during loading.

Mean effective stress: Average effective normal stress, denoted p′ in triaxial notation.

Deviator stress: Difference between major and minor principal stresses, denoted q in triaxial compression.

Critical state: Large-strain state of continuing shear at approximately constant stress and volume.

Peak strength: Maximum shear resistance mobilised during a test.

Residual strength: Large-displacement strength after a persistent shear surface has formed.

Overconsolidation ratio: Ratio of past maximum effective stress to current effective stress.

Sensitivity: Ratio of intact to remoulded undrained strength.

Anisotropy: Directional variation in soil strength or stiffness.

State parameter: Difference between current void ratio and critical-state void ratio at the same mean effective stress.

Strain softening: Reduction in strength with increasing deformation after a peak.

Review Questions

Short Answer Questions

1. Define effective stress and explain its significance in saturated soil.

2. Distinguish drained and undrained loading.

3. What is the difference between peak strength and residual strength?

4. Explain why contractive soil generates positive pore pressure under undrained shear.

5. Define overconsolidation ratio and sensitivity.

Long Answer Questions

1. Explain the construction and engineering interpretation of a p′–q stress-path diagram.

2. Discuss the effects of density and stress history on contractancy, dilation and pore-pressure response.

3. Compare direct shear, triaxial and simple-shear tests for determining soil strength.

4. Explain how anisotropy and principal-stress direction influence undrained strength.

5. Discuss the selection of peak, critical-state and residual parameters in slope design.

Analytical / Application Questions

1. A saturated soil experiences Δσ1 = 150 kPa, Δσ3 = 20 kPa and Δu = 70 kPa. Calculate Δσ′1 and Δσ′3, and determine whether the pore pressure reduces or increases effective confinement.

2. An intact clay has an undrained strength of 48 kPa and a remoulded strength of 12 kPa. Calculate its sensitivity and discuss the implications for rapid embankment construction.

3. Design a laboratory and field-monitoring programme for a rapidly constructed embankment on soft, anisotropic clay. Justify the tests, drainage assumptions and strength parameters selected.

Multiple Choice Questions with Answers

Question 1

Which expression defines effective stress in saturated soil?

A. σ′ = σ + u

B. σ′ = σ − u

C. σ′ = u − σ

D. σ′ = σu

Correct Answer: B. σ′ = σ − u

Question 2

Under undrained loading, a contractive saturated soil generally produces:

A. Positive excess pore pressure

B. Zero total stress

C. Negative excess pore pressure

D. Immediate full drainage

Correct Answer: A. Positive excess pore pressure

Question 3

For triaxial compression, which definition of q is used in this chapter?

A. q = σ′1 + σ′3

B. q = σ′1/σ′3

C. q = σ′1 − σ′3

D. q = σ′3 − σ′1

Correct Answer: C. q = σ′1 − σ′3

Question 4

Residual strength is most directly associated with:

A. Initial elastic deformation

B. Very large displacement along a developed shear surface

C. Zero effective stress in every soil

D. Undrained isotropic compression

Correct Answer: B. Very large displacement along a developed shear surface

Question 5

A soil has intact undrained strength 40 kPa and remoulded strength 8 kPa. Its sensitivity is:

A. 0.2

B. 5

C. 32

D. 48

Correct Answer: B. 5

Question 6

During undrained shearing, negative excess pore pressure causes effective stress to:

A. Increase

B. Decrease to zero necessarily

C. Remain unchanged

D. Become equal to total stress plus pressure

Correct Answer: A. Increase

Question 7

For a saturated clay, a CU triaxial test with reliable pore-pressure measurement can be used to obtain:

A. Only particle-size distribution

B. Effective-stress strength parameters

C. Only residual strength

D. Permeability without consolidation data

Correct Answer: B. Effective-stress strength parameters

Question 8

The overconsolidation ratio is defined as:

A. Current effective stress divided by past maximum stress

B. Past maximum effective stress divided by current effective stress

C. Total stress divided by pore pressure

D. Peak strength divided by residual strength

Correct Answer: B. Past maximum effective stress divided by current effective stress

Question 9

A positive state parameter commonly indicates a soil state that is:

A. Looser than the critical state and contractive

B. Denser than the critical state and always elastic

C. At zero stress

D. Fully remoulded in every case

Correct Answer: A. Looser than the critical state and contractive

Question 10

Why may one triaxial compression strength value be inadequate for a natural slope analysis?

A. Soil strength never depends on direction

B. The slope may involve different stress paths and principal-stress orientations

C. Triaxial tests cannot measure stress

D. Slope failures occur only under drained conditions

Correct Answer: B. The slope may involve different stress paths and principal-stress orientations