Shear Strength and Soil Behaviour Models

Introduction

Shear strength is the resistance of soil to distortion and failure under applied loading. It governs the stability of slopes, foundations, retaining structures, embankments, excavations and earth dams. Unlike many structural materials, soil strength is not solely an intrinsic material property. It depends on effective stress, drainage conditions, stress history, density, fabric, saturation, strain level and loading path.

A reliable geotechnical analysis therefore requires more than selecting a cohesion and friction angle from a handbook. The engineer must identify whether loading is drained or undrained, establish the relevant stress path, distinguish peak from critical-state strength, select appropriate laboratory tests and use a constitutive model consistent with the problem. This chapter develops these concepts and explains their application in engineering analysis.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Explain total stress, effective stress and the Mohr–Coulomb failure criterion.

2. Distinguish drained, undrained and partially drained soil behaviour.

3. Construct and interpret stress paths and critical-state relationships.

4. Select and interpret appropriate shear-strength laboratory tests.

5. Evaluate peak, critical-state, residual and undrained strengths.

6. Choose constitutive models for practical numerical and analytical studies.

7. Recognise uncertainty, scale effects, anisotropy and limitations in strength assessment.

1. Fundamental Concepts of Soil Shear Strength

1.1 Stress carried by soil

For a saturated soil, the total normal stress is divided between the soil skeleton and pore water:

\[

\sigma' = \sigma - u

\]

where \(\sigma'\) is effective normal stress, \(\sigma\) is total normal stress and \(u\) is pore-water pressure. The effective stress controls interparticle contact forces and is therefore central to deformation and shear strength. The equation is applicable to saturated soils under conditions where pore pressure is represented by a single scalar and the soil skeleton carries the remaining stress.

For unsaturated soils, suction contributes to stress state and a single effective-stress expression may be inadequate. Extended formulations use net stress and suction stress, but their parameters are strongly dependent on the soil-water characteristic curve and degree of saturation.

1.2 Mohr–Coulomb criterion

The Mohr–Coulomb effective-stress criterion is

\[

\tau_f = c' + \sigma'_n \tan\phi'

\]

where \(\tau_f\) is shear stress at failure, \(c'\) is effective cohesion intercept, \(\sigma'_n\) is effective normal stress on the failure plane and \(\phi'\) is effective friction angle. It is an idealised linear envelope. Parameters are meaningful only over the stress range, drainage condition and stress history represented by the tests used to obtain them.

For a cohesionless soil, \(c'\) is commonly taken as approximately zero, although an apparent intercept can arise from testing range, particle interlock, suction, cementation or data-fitting effects. A nonzero fitted intercept should not automatically be interpreted as true bonding.

In principal stresses, the Mohr–Coulomb condition can be expressed, for triaxial compression, as

\[

\sigma'_1 = \sigma'_3\frac{1+\sin\phi'}{1-\sin\phi'} + \frac{2c'\cos\phi'}{1-\sin\phi'}

\]

where \(\sigma'_1\) and \(\sigma'_3\) are major and minor effective principal stresses at failure. This form assumes conventional triaxial compression, homogeneous stress conditions and a linear frictional envelope.

The corresponding stress ratio is

\[

M = \frac{\sigma'_1}{\sigma'_3}

= \frac{1+\sin\phi'}{1-\sin\phi'}

\]

for a cohesionless triaxial-compression material. In critical-state soil mechanics, the critical-state stress ratio is often written as

\[

M_{tc}=\frac{6\sin\phi'_{cs}}{3-\sin\phi'_{cs}}

\]

where \(\phi'_{cs}\) is the critical-state friction angle in triaxial compression and \(q=\sigma'_1-\sigma'_3\), \(p'=(\sigma'_1+2\sigma'_3)/3\). The two expressions are consistent when \(M=q/p'\) is used rather than \(\sigma'_1/\sigma'_3\).

1.3 Components of strength

Soil strength may include:

• friction due to sliding and particle contact;
• dilation-related resistance caused by interlocking;
• apparent cohesion caused by suction or fabric;
• true bonding caused by cementation or chemical agents; and
• structural resistance associated with intact natural fabric.

These components evolve with strain. Dense sand may display a peak strength followed by softening toward a critical-state value. Normally consolidated clay commonly exhibits contractive behaviour and approaches critical state without a pronounced peak. Heavily overconsolidated clay may show significant peak strength and post-peak reduction.

2. Drained, Undrained and Partially Drained Behaviour

2.1 Drainage and loading time

A drained test permits sufficient water movement that excess pore pressure dissipates during shearing. The measured strength is generally interpreted using effective stresses. An undrained test prevents significant volume change during shearing. In saturated soil, the tendency to contract or dilate is expressed primarily through pore-pressure change, and total-stress strength is often reported using an undrained shear strength \(s_u\).

Drainage is not determined by soil type alone. It depends on permeability, drainage-path length, loading rate, layer geometry and the time scale of construction or loading. A sand foundation can behave undrained during a very rapid earthquake loading, while a clay layer can behave drained during slow construction or long-term service.

2.2 Undrained strength

For saturated clay under a particular stress history and testing procedure, the undrained criterion is often approximated by

\[

\tau_f = s_u

\]

in total-stress analysis. This representation is convenient but not universal. Undrained strength varies with water content, anisotropy, strain rate, confining stress, stress path, sample disturbance and orientation of shearing.

The relationship between undrained strength and effective stress parameters can be illustrated for saturated triaxial compression, but no universal conversion exists. For a normally consolidated soil, the ratio \(s_u/\sigma'_{vc}\) may be approximately constant over a limited stress range, where \(\sigma'_{vc}\) is the consolidation vertical effective stress. Its value depends on compression characteristics, friction angle and stress path. It must therefore be measured or calibrated rather than assumed.

2.3 Pore-pressure response

The sign and magnitude of excess pore pressure are linked to the soil’s tendency to change volume. Contractive soil generates positive excess pore pressure in undrained loading, reducing effective stress. Dilative soil generates negative excess pore pressure, increasing effective stress. Consequently, two specimens with the same initial density can exhibit different responses if their confining stress or stress history differs.

Skempton’s pore-pressure relationship is often written for triaxial loading as

\[

\Delta u = B[\Delta\sigma_3 + A(\Delta\sigma_1-\Delta\sigma_3)]

\]

where \(A\) and \(B\) are pore-pressure coefficients and \(\Delta u\), \(\Delta\sigma_1\), and \(\Delta\sigma_3\) are pore-pressure and principal-stress increments. The equation is an empirical incremental relationship, not a universal constitutive law. It is most useful for interpreting saturated laboratory tests and requires attention to sign conventions and stress history.

3. Stress Paths and State Variables

A stress path describes how the stress state changes during loading. Common representations include the \(p'-q\) plane, the \(\sigma'_1-\sigma'_3\) plane and the \(q-\varepsilon\) plane. Stress paths clarify why identical initial stresses can result in different strengths under different boundary conditions.

For axisymmetric triaxial conditions:

\[

p'=\frac{\sigma'_1+2\sigma'_3}{3}, \qquad q=\sigma'_1-\sigma'_3

\]

Under isotropic consolidation followed by undrained triaxial compression, total mean stress may change while the effective mean stress is altered by pore-pressure generation. Under drained compression, pore pressure is approximately constant and the effective stress path follows the applied stress increments.

Stress-path analysis is particularly important for embankments, excavations and staged construction. An embankment may produce an initial undrained loading path, followed by consolidation and a later drained loading path. An excavation may reduce total stress rapidly, causing negative pore pressures initially and subsequent strength changes as pressures equalise.

A stress path should be defined with:

1. the initial effective stress and stress history;

2. the stress increment or boundary condition;

3. the drainage condition and loading rate;

4. the direction of principal-stress rotation, if relevant; and

5. the failure or critical-state condition.

Ignoring principal-stress rotation can be serious in retaining walls, offshore foundations and cyclic loading, where the direction of major principal stress changes during loading.

4. Critical-State Soil Mechanics

4.1 Critical state

At critical state, soil shears continuously at constant effective stress ratio and approximately constant specific volume. The soil may undergo large shear strains without further systematic change in volume. Critical state is distinct from peak failure: peak strength may include temporary interlocking or structure that is lost with continued shearing.

The critical-state line in the \(q-p'\) plane is commonly written

\[

q=Mp'

\]

where \(M\) depends on the critical-state friction angle and the stress path. In a specific-volume versus logarithm of mean effective stress plane, the critical-state line is often represented as

\[

v=\Gamma-\lambda\ln p'

\]

where \(v=1+e\) is specific volume, \(e\) is void ratio, \(\Gamma\) is an intercept dependent on the chosen stress units and reference convention, and \(\lambda\) is the critical-state compression slope. The logarithm requires consistent pressure units and a clearly defined reference.

4.2 State parameter

The state parameter is commonly defined as

\[

\psi=e-e_{cs}(p')

\]

where \(e\) is the current void ratio and \(e_{cs}(p')\) is the void ratio on the critical-state line at the same effective mean stress. A positive \(\psi\) indicates a looser, contractive state; a negative value indicates a denser, potentially dilative state. State-parameter concepts are valuable for evaluating liquefaction susceptibility, sand strength and flow behaviour.

4.3 Normal consolidation and overconsolidation

The overconsolidation ratio is

\[

OCR=\frac{\sigma'_{vc}}{\sigma'_{v0}}

\]

where \(\sigma'_{vc}\) is the maximum past vertical effective stress and \(\sigma'_{v0}\) is the current vertical effective stress. Normally consolidated clay has \(OCR\approx1\); overconsolidated clay has \(OCR>1\). OCR influences compressibility, pore-pressure response, peak strength and dilation.

Critical-state theory provides a unified interpretation: loading normally consolidated soil tends to move toward the critical-state surface through contraction, whereas overconsolidated soil may initially dilate and mobilise a peak strength before reaching the critical state.

5. Laboratory Evaluation of Strength

5.1 Direct shear test

In a direct shear test, a specimen is sheared along a prescribed plane under a controlled normal load. It is relatively simple and useful for sands, interfaces and large particles. Drained tests can provide peak and critical or residual strength if sufficient displacement is applied.

Limitations include nonuniform stress and strain, a forced failure plane, changing shear area and difficulty measuring pore pressure. The test is less suitable for obtaining fundamental undrained parameters in saturated clay unless carefully adapted.

5.2 Triaxial tests

The triaxial apparatus allows control of confining stress, drainage, consolidation and pore-pressure measurement. Principal tests include:

• unconsolidated–undrained (UU), in which specimens are not consolidated and are sheared without drainage;
• consolidated–undrained (CU), in which specimens are consolidated and then sheared undrained, preferably with pore-pressure measurement; and
• consolidated–drained (CD), in which consolidation and shearing occur with drainage.

A CU test with measured pore pressure can determine effective stress at failure. A CU test without pore-pressure measurement is primarily a total-stress test. A CD test requires a sufficiently slow rate to prevent significant excess pore pressure.

Specimen quality is crucial. Tube sampling disturbance, fissures, trimming, membrane penetration, leakage and nonuniform saturation can substantially affect results. Saturation is commonly checked using the Skempton B-value; a value close to unity indicates good saturation for many laboratory applications, but the acceptable criterion depends on the test standard and material.

5.3 Ring shear and residual strength

For highly plastic clays and reactivated landslides, residual strength may control long-term stability. Ring-shear testing permits very large displacement and avoids the displacement limitation of conventional shear boxes. Residual strength is usually lower than peak strength and may be associated with realignment of platy clay minerals along a polished shear surface.

5.4 In situ tests and field correlation

Field vane tests are useful for soft saturated clay and provide an estimate of undrained strength, subject to correction for anisotropy, strain rate and progressive failure. Cone penetration testing provides continuous profiles and can support estimates of undrained strength and state, but correlations are empirical and site-specific. Pressuremeter, dilatometer and seismic methods provide complementary information about stiffness, stress history and anisotropy rather than direct replacements for laboratory strength testing.

A robust investigation combines high-quality samples, in situ testing, groundwater observations and geological interpretation. No single test should be treated as universally definitive.

6. Constitutive Models for Geotechnical Analysis

A constitutive model relates stress increments to strain increments and may also represent plasticity, hardening, critical state, anisotropy, destructuration and rate effects. Model selection must reflect the analysis objective and available data.

6.1 Perfectly plastic Mohr–Coulomb model

The Mohr–Coulomb model uses friction angle, cohesion, dilation angle, elastic stiffness and Poisson’s ratio. It is appropriate for preliminary analyses, limit-equilibrium support and problems where failure load is more important than detailed deformation. Its main limitations are constant strength parameters, simplified yielding, abrupt failure and inadequate representation of stress-dependent stiffness.

The dilation angle should not automatically equal the friction angle. Dense granular soils may dilate at peak strength, whereas loose soils may contract. In many practical models, dilation reduces toward zero as critical state is approached.

6.2 Tresca and undrained models

The Tresca model represents undrained clay using a shear-strength parameter, often \(s_u\), with no explicit effective-stress pore-pressure calculation in a total-stress formulation. It is useful for short-term stability when undrained conditions and spatial variation of \(s_u\) are well established. It cannot independently predict consolidation, pore-pressure dissipation or changes in effective stress.

6.3 Modified Cam Clay

Modified Cam Clay is an isotropic critical-state elastoplastic model for normally consolidated and overconsolidated soils. Its yield surface in \(p'-q\) space is commonly written

\[

f=q^2+M^2p'(p'-p'_c)=0

\]

where \(p'_c\) is the preconsolidation pressure and \(M\) is the critical-state stress ratio. The model uses elastic and plastic compressibility parameters and assumes isotropic hardening. It can reproduce consolidation, contractive response and critical-state behaviour, but may not represent anisotropy, bonding, destructuration or sharp peak strength adequately.

6.4 Advanced models

Bounding-surface, anisotropic critical-state, hypoplastic, bounding-plasticity and structured-soil models can represent stress-dependent stiffness, cyclic response, fabric, destructuration and memory effects more realistically. Their use requires more parameters and stronger calibration evidence. Model complexity is justified only when the additional mechanisms influence the engineering decision.

For cyclic loading and liquefaction, models must reproduce pore-pressure accumulation, cyclic mobility, stiffness degradation and density effects. A static Mohr–Coulomb model is not sufficient for such predictions.

7. Worked Example: Drained Triaxial Strength

A drained triaxial compression test on a clean sand gives \(\sigma'_3=100\) kPa and \(\phi'=34^\circ\), with negligible effective cohesion. Determine the major effective principal stress and deviator stress at failure.

For a cohesionless soil:

\[

\sigma'_1=\sigma'_3\frac{1+\sin\phi'}{1-\sin\phi'}

\]

Since \(\sin34^\circ\approx0.559\),

\[

\sigma'_1=100\left(\frac{1.559}{0.441}\right)\approx353.5\text{ kPa}

\]

Therefore,

\[

q_f=\sigma'_1-\sigma'_3=353.5-100=253.5\text{ kPa}

\]

The result assumes triaxial compression, drained conditions, a linear Mohr–Coulomb envelope and negligible cohesion. It should not be transferred directly to plane strain or a different stress range without considering the change in mobilised friction and stress path.

8. Applied Case: Staged Embankment on Soft Clay

Background

Consider an illustrative case in which a 6 m-high road embankment is planned over a 12 m layer of normally consolidated soft clay. The clay has low permeability, and construction is expected to occur in two stages.

Problem

A rapid first-stage construction may generate positive excess pore pressures. If the embankment is raised too quickly, effective stress and undrained strength may be insufficient, causing a foundation failure or lateral spreading.

Analysis

An investigation should combine CPTu soundings, high-quality sampling, oedometer tests and CU triaxial tests with pore-pressure measurement. The oedometer tests establish compressibility, preconsolidation stress and consolidation rate. CU tests provide undrained strength and effective stress parameters under representative consolidation paths. Piezometers monitor excess pore pressure during construction.

The first-stage analysis is generally undrained or partially drained, using an appropriate undrained-strength profile and stability method. A consolidation analysis then predicts pore-pressure dissipation and strength gain. As effective stress increases, the clay’s available undrained strength generally increases, but the relationship must be calibrated from testing rather than assumed linear over all stress levels.

Approach and outcome

A staged construction programme is selected: the first lift is placed, piezometers confirm that pore pressures remain below the trigger level, and construction pauses until sufficient dissipation occurs. The second lift is then placed. Stability is checked for short-term undrained and long-term drained conditions, including potential circular failure surfaces and foundation bearing failure.

Learning points

1. Drainage condition is controlled by time and permeability, not by soil classification alone.

2. Monitoring can convert an uncertain design into a controlled observational procedure.

3. Strength parameters must correspond to the stress path and construction stage.

4. Long-term stability may be governed by effective-stress strength even when short-term risk is governed by undrained strength.

5. Spatial variability and progressive failure require factors of safety and trigger levels appropriate to the consequence of failure.

9. Parameter Selection, Uncertainty and Professional Practice

Parameter selection should begin with the failure mechanism and design situation. For temporary excavation stability, an undrained analysis may be appropriate if the clay response is rapid. For a long-term slope, effective-stress peak, critical or residual strength may control. For a dense sand foundation under rapid cyclic loading, state-dependent and cyclic parameters are necessary.

Best practice includes:

• matching laboratory stress paths to field loading;
• separating peak, critical and residual strength;
• reporting specimen quality, saturation, strain rate and drainage condition;
• using effective stresses for drained and pore-pressure-informed analyses;
• checking anisotropy and principal-stress rotation where relevant;
• performing sensitivity analyses for strength, groundwater and geometry;
• calibrating numerical models against laboratory and field observations; and
• documenting assumptions, parameter sources and model limitations.

Uncertainty should be represented explicitly through characteristic values, statistical interpretation, spatial variability and scenario analysis. A mean laboratory value is not automatically a safe design value. Geological uncertainty, sampling bias and model uncertainty may exceed the numerical uncertainty of the calculation itself.

Digital tools and AI can assist with CPT interpretation, parameter databases, automated quality checks and probabilistic back-analysis. They do not replace engineering judgement: an AI-generated correlation may be invalid outside its calibration population, and all automated results require geological and mechanical verification.

Challenges and Limitations

The Mohr–Coulomb model is useful but does not capture stress-dependent friction, progressive failure, anisotropy, cyclic degradation or evolving fabric. Undrained strength is not a universal constant and can vary with strain rate, stress path and sample quality. Critical-state parameters may be obscured by bonding or destructuration. Laboratory specimens may not represent fissured, layered or naturally structured field soil. Finally, two-dimensional analyses can misrepresent three-dimensional failure mechanisms and spatial variability.

Chapter Summary

Soil shear strength depends on effective stress, drainage, stress history, density, fabric, strain and loading path. The Mohr–Coulomb criterion provides a practical linear failure envelope, while critical-state concepts distinguish transient peak strength from the large-strain state reached during continued shearing. Drained tests evaluate effective-stress behaviour; undrained tests assess short-term response and pore-pressure generation. Stress paths provide the link between boundary conditions and observed strength.

Direct shear, triaxial, ring-shear and in situ tests each have specific strengths and limitations. Parameter selection must match the design mechanism and construction time scale. Perfectly plastic, Tresca, Modified Cam Clay and advanced constitutive models offer progressively richer representations, but complexity must be supported by data. Reliable practice combines testing, field monitoring, uncertainty analysis and transparent engineering judgement.

Key Terms

Effective stress: Stress carried by the soil skeleton after subtracting pore-water pressure from total stress.

Total stress: The externally applied stress before separation into skeleton and pore-fluid components.

Mohr–Coulomb criterion: A linear relationship between shear strength, effective normal stress, cohesion and friction angle.

Effective cohesion: Cohesion intercept obtained in an effective-stress strength representation.

Friction angle: Parameter describing the frictional component of soil shear strength.

Undrained strength: Shear strength mobilised when significant drainage is prevented during loading.

Drained strength: Strength measured when excess pore pressure is allowed to dissipate.

Stress path: Trajectory followed by a soil stress state during loading.

Critical state: Condition of continuous shearing at constant stress ratio and approximately constant volume.

Critical-state line: Locus of critical states in stress–volume space.

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

Overconsolidation ratio: Maximum past effective stress divided by current effective stress.

Peak strength: Maximum strength mobilised before post-peak softening or stabilisation.

Residual strength: Large-displacement strength after substantial shearing and fabric realignment.

Pore-pressure coefficient: Parameter relating stress increments to excess pore-pressure change.

Constitutive model: Mathematical relationship describing stress, strain and material evolution.

Modified Cam Clay: An isotropic critical-state elastoplastic model for soil behaviour.

Anisotropy: Dependence of soil response on direction of loading or fabric orientation.

Progressive failure: Development of failure through successive mobilisation of different soil zones.

Review Questions

Short Answer Questions

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

2. Distinguish drained from undrained shear conditions.

3. What is meant by critical state?

4. State two limitations of the direct shear test.

5. Define overconsolidation ratio.

Long Answer Questions

1. Explain the Mohr–Coulomb failure criterion and discuss the meaning and limitations of its parameters.

2. Describe stress paths for drained and undrained triaxial compression and explain their engineering significance.

3. Explain peak, critical-state and residual strengths, including the soil conditions associated with each.

4. Compare UU, CU and CD triaxial tests and identify suitable applications for each.

5. Compare the Mohr–Coulomb, Tresca and Modified Cam Clay models in terms of assumptions, parameters and applications.

Analytical / Application Questions

1. A saturated clay foundation is loaded rapidly by an embankment. Identify the likely drainage condition, required investigations and appropriate short-term and long-term analyses.

2. A dense sand shows a peak friction angle of 40° and a critical-state friction angle of 32°. Explain why the two values differ and identify which value is more appropriate for a long-term large-displacement slope analysis.

3. Design a testing and monitoring programme for staged construction of an embankment on soft clay, explaining how test results and pore-pressure observations would control construction decisions.

Multiple Choice Questions with Answers

Question 1

Which stress most directly controls interparticle contact forces in saturated soil?

A. Total stress

B. Effective stress

C. Atmospheric pressure

D. Hydrostatic head only

Correct Answer: B. Effective stress

Question 2

For a cohesionless soil in triaxial compression, which relationship gives the major effective principal stress at failure?

A. σ′1 = σ′3 tan φ′

B. σ′1 = σ′3(1 − sin φ′)/(1 + sin φ′)

C. σ′1 = σ′3(1 + sin φ′)/(1 − sin φ′)

D. σ′1 = c′ + σ′3 tan φ′

Correct Answer: C. σ′1 = σ′3(1 + sin φ′)/(1 − sin φ′)

Question 3

What is the principal characteristic of critical state?

A. Continuous shearing at approximately constant stress ratio and volume

B. Zero effective stress at failure

C. Maximum strength at the first increment of strain

D. No particle rearrangement

Correct Answer: A. Continuous shearing at approximately constant stress ratio and volume

Question 4

A CU triaxial test with pore-pressure measurement can be used primarily to determine:

A. Only total unit weight

B. Effective-stress strength parameters

C. Only permeability

D. Only residual strength

Correct Answer: B. Effective-stress strength parameters

Question 5

A positive state parameter generally indicates that a sand is:

A. Dense and strongly dilative

B. Loose relative to its critical state and potentially contractive

C. Fully cemented

D. At zero effective stress

Correct Answer: B. Loose relative to its critical state and potentially contractive

Question 6

If σ′3 = 100 kPa and φ′ = 30° for a cohesionless soil in triaxial compression, σ′1 at failure is closest to:

A. 173 kPa

B. 200 kPa

C. 300 kPa

D. 433 kPa

Correct Answer: C. 300 kPa

Question 7

Which model is specifically formulated as an isotropic critical-state elastoplastic model?

A. Tresca

B. Modified Cam Clay

C. Linear elastic model

D. Rigid-perfectly rough interface model

Correct Answer: B. Modified Cam Clay

Question 8

Residual strength is most relevant to:

A. Initial loading of intact dense sand only

B. Long-term stability of reactivated shear surfaces

C. Calculating unit weight

D. Determining groundwater elevation alone

Correct Answer: B. Long-term stability of reactivated shear surfaces

Question 9

During undrained loading of a contractive saturated soil, the usual pore-pressure response is:

A. Positive excess pore pressure

B. Negative excess pore pressure only

C. No pore-pressure change

D. Immediate atmospheric pressure

Correct Answer: A. Positive excess pore pressure

Question 10

Why should peak friction angle not automatically be used for a long-term large-displacement slope?

A. Peak strength may include temporary interlocking that is lost with displacement

B. Peak strength is always lower than residual strength

C. Peak strength applies only to water

D. Peak strength is independent of stress history

Correct Answer: A. Peak strength may include temporary interlocking that is lost with displacement