Shear Strength and Constitutive Behavior

Introduction

Shear strength is the capacity of soil to resist sliding, distortion, and failure along a plane. It governs the stability of slopes, retaining walls, foundations, embankments, excavations, and earth dams. Unlike many construction materials, soil is a particulate material whose response depends on effective stress, drainage, density, fabric, stress history, and loading rate.

A reliable geotechnical assessment therefore requires more than assigning a single “soil strength.” The engineer must identify the likely failure mechanism, determine whether loading is drained or undrained, select representative strength parameters, and relate laboratory test results to field conditions. This chapter develops the Mohr–Coulomb model, stress paths, drained and undrained behavior, and the interpretation of direct shear, triaxial, and unconfined compression tests.

Learning Objectives

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

1. Explain shear stress, normal stress, effective stress, and shear strength.
2. Apply the Mohr–Coulomb failure criterion.
3. Distinguish total-stress and effective-stress strength parameters.
4. Interpret drained, undrained, and partially drained behavior.
5. Explain stress paths and their use in soil testing and design.
6. Interpret direct shear, triaxial, and unconfined compression tests.
7. Select appropriate parameters for sands, clays, and different construction stages.
8. Recognize the limitations of idealized constitutive models.

1. Fundamental Concepts

1.1 Stress and effective stress

A soil mass carries applied load through a combination of the soil skeleton and pore water. Terzaghi’s effective-stress principle is:

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 equation applies to saturated soils when pore pressure is defined relative to the same reference pressure used for total stress. In unsaturated soils, suction and the degree of saturation also affect strength, so the simple saturated form is incomplete.

Effective stress controls frictional resistance because it controls the contact forces between soil particles. An increase in pore pressure reduces effective stress; dissipation of pore pressure increases effective stress.

1.2 Shear strength

Shear stress acts tangentially to a plane, whereas normal stress acts perpendicular to it. Soil failure occurs when the applied shear stress reaches the available shear resistance. In granular soils, resistance arises mainly from friction, particle interlocking, and dilation. In clays, it may arise from friction, bonding, cementation, structure, and—in undrained loading—the development of pore pressure.

The peak strength is the maximum resistance reached during shearing. Critical-state or ultimate strength is the approximately constant resistance reached after substantial distortion, when the soil continues to shear without significant change in volume. Peak strength can be important for dense sands and overconsolidated clays, but relying on it requires control of deformation and an understanding of progressive failure.

2. Mohr–Coulomb Strength Theory

The Mohr–Coulomb criterion represents shear strength as a linear function of effective normal stress:

where τ_f is shear stress at failure, c′ is effective cohesion, σ′_n is effective normal stress on the failure plane, and φ′ is the effective angle of shearing resistance. The equation is applicable as an idealized envelope over the stress range represented by the tests. Units of c′, σ′_n, and τ_f are kPa; φ′ is an angle.

The total-stress form is:

where c_u and φ_u are total-stress parameters. For the idealized saturated clay model under short-term undrained loading, φ_u is often taken as approximately zero, giving:

This simplification is not a universal soil property. It is a modeling choice valid only when the test and field loading conditions are comparable.

The friction angle can be interpreted through the slope of a strength envelope. A larger φ′ indicates a stronger increase in resistance with effective normal stress. The intercept c′ should not automatically be interpreted as true particle cohesion. Apparent cohesion can result from suction, cementation, bonding, curvature of the failure envelope, or fitting a straight line over a limited stress range.

2.1 Mohr-circle interpretation

For principal stresses σ′_1 and σ′_3, the Mohr circle has:

where σ′_m is the centre and R is the radius. A soil reaches failure when the Mohr circle becomes tangent to the Mohr–Coulomb envelope. For a cohesionless soil, c′ = 0, and the envelope passes through the origin.

For triaxial compression, the major principal stress is vertical and the minor principal stress is the cell pressure. For a linear effective-stress envelope, the failure relation is:

This relation assumes axisymmetric triaxial compression, a Mohr–Coulomb envelope, and effective principal stresses at failure.

3. Constitutive Behavior of Soil

A constitutive model describes the relationship between stress, strain, and often pore pressure or volume change. The simplest model is elastic-perfectly plastic Mohr–Coulomb behavior: soil deforms elastically before reaching a yield or failure surface and then shears at constant strength. Real soils are more complex.

Important features include:

• Nonlinearity: stiffness changes with stress level and strain.
• Anisotropy: behavior depends on the direction of loading and deposition.
• Stress history: overconsolidated soils commonly behave differently from normally consolidated soils.
• Dilatancy: dense granular soils and heavily overconsolidated clays may increase in volume during shear.
• Contractancy: loose sands and normally consolidated clays tend to decrease in volume.
• Strain softening: peak strength may decline toward a lower residual strength.
• Creep and rate effects: time-dependent deformation and strength changes can be important in clay.

Critical-state soil mechanics explains that, after sufficient shearing, soil tends toward a condition in which stress ratio, density, and effective stress evolve consistently while deformation continues. In practical design, critical-state or residual parameters are often preferable where large displacement or progressive failure is expected.

4. Drainage and Loading Conditions

4.1 Drained loading

In drained loading, water is able to escape sufficiently during shearing, so excess pore pressure is negligible or dissipates as loading proceeds. The soil volume may change. Drained tests provide effective-stress parameters c′ and φ′ and are commonly relevant to long-term stability in permeable soils and to slow loading in clay.

4.2 Undrained loading

In undrained loading, water cannot escape during the relevant loading period. The total volume of a saturated soil specimen remains approximately constant, but pore pressure changes. Strength may be described using total-stress undrained strength c_u, or using effective stresses if pore pressure is measured.

Undrained conditions are common during rapid construction on saturated low-permeability clay, rapid excavation, and short-term loading. They are not determined solely by soil type; they depend on permeability, drainage distance, loading rate, and the duration of construction.

4.3 Partially drained loading

Many field events are neither perfectly drained nor perfectly undrained. Staged embankment construction, cyclic traffic, rainfall infiltration, and excavation may produce partial drainage. Consolidation analyses and coupled stress–pore-pressure calculations may then be required.

A practical selection rule is to compare the loading time with the time required for pore-pressure dissipation. The consolidation time is influenced by drainage path length, compressibility, permeability, and the coefficient of consolidation. Short drainage paths and high permeability promote drained behavior; long drainage paths and low permeability promote undrained behavior.

5. Stress Paths

A stress path is the route followed by a soil element through stress space. It is more informative than the initial and final stresses alone because different loading histories can produce different pore pressures, strains, and strengths.

Common stress variables include mean effective stress:

and deviator stress in triaxial compression:

For axisymmetric triaxial testing, σ′_2 = σ′_3. A stress path may be plotted in q–p′ space. Drained shearing generally changes both q and p′ because volume change alters effective stress. Undrained shearing can produce a nearly constant total mean stress path while p′ changes as pore pressure develops.

A normally consolidated clay commonly contracts during undrained shear, generating positive excess pore pressure and reducing p′. A dense sand may dilate, generate negative excess pore pressure, and temporarily increase p′. These trends explain why the same total stress change can produce different undrained strengths in different soils.

6. Laboratory Strength Tests

6.1 Direct shear test

In a direct shear test, a specimen is placed in a split box. A prescribed normal load is applied, and one half of the box is displaced horizontally while shear force and displacement are recorded.

Advantages include simple equipment, rapid testing, and direct control of normal stress. The test is useful for sands, interfaces, and remoulded materials. Its limitations are that the failure plane is predetermined, stress distribution is nonuniform, and drainage control is less precise than in a high-quality triaxial test. For each normal stress, the peak or large-displacement shear stress is obtained. A plot of τ_f against σ′_n gives an approximate strength envelope; its intercept estimates c′ and its slope estimates tan φ′.

For granular soil, the peak friction angle may exceed the critical-state angle because particle interlocking causes dilation. The large-displacement angle is often more appropriate for movements involving substantial shear displacement.

6.2 Triaxial test

A cylindrical specimen is enclosed in a membrane and subjected to cell pressure. Axial load produces the major principal stress. The main forms are:

• Consolidated drained (CD): specimen consolidates and is sheared slowly with drainage open; volume change is measured.
• Consolidated undrained (CU): specimen consolidates, then is sheared without drainage; pore pressure may be measured.
• Unconsolidated undrained (UU): specimen is sheared without consolidation or drainage; results are total-stress strengths.

Triaxial testing allows control of drainage, measurement of pore pressure, and observation of volume change. CU tests with pore-pressure measurement are especially valuable because effective stresses at failure can be calculated. Specimen preparation, saturation, membrane correction, end restraint, strain rate, and sample disturbance must be controlled.

The pore-pressure parameter B is commonly used to assess saturation during back-pressure saturation. A value close to 1 indicates high saturation, although the acceptable value depends on the test standard and purpose. The parameter A may be used to describe pore-pressure response during shear, but its value depends strongly on stress history and strain level.

6.3 Unconfined compression test

The unconfined compression test applies axial compression to a cohesive specimen without lateral confinement. The unconfined compressive strength is:

where P is axial load at failure and A_c is the corrected cross-sectional area. If the specimen is assumed incompressible during undrained axial strain, a common area correction is:

where A_0 is the initial area and ε_a is axial strain expressed as a decimal. The undrained shear strength is commonly estimated as:

This relationship assumes saturated cohesive soil, undrained φ_u = 0, and a valid axial failure test. The test is unsuitable for clean sand, fissured specimens that cannot stand unsupported, or soils whose behavior is strongly affected by confinement.

7. Worked Examples

Example 1: Effective-stress shear strength

A drained test indicates c′ = 8 kPa and φ′ = 30°. The effective normal stress on a potential failure plane is 120 kPa. Determine the shear strength.

Using:

Since tan 30° ≈ 0.577:

The estimated shear strength is 77 kPa. This result is valid only if the drained effective-stress parameters represent the soil and field stress range.

Example 2: Unconfined compression

A clay specimen has diameter 38 mm and fails under an axial load of 1.20 kN. The axial strain at failure is 10%. Determine q_u and c_u using the area correction.

Initial area:

Corrected area:

Unconfined compressive strength:

Therefore:

The estimated undrained strength is 476 kPa. Such a high value should prompt checks for specimen dimensions, units, sample disturbance, test validity, and whether the load was actually recorded in kN rather than N.

8. Selecting Parameters for Design and Construction

Parameter selection must reproduce the field stress history and drainage condition rather than simply use the largest laboratory value.

For sands, φ′ is usually the dominant parameter and c′ should generally be taken as zero unless bonding or cementation is demonstrated. For intact or structured clays, effective cohesion may be meaningful over a defined stress range, but it may reduce after remoulding or large displacement. For fissured, slickensided, or highly overconsolidated clays, residual strength may control reactivated landslides.

Construction sequencing is central. An embankment placed rapidly on soft clay may initially be undrained, whereas the same embankment may experience increased long-term strength as consolidation dissipates pore pressure. Instrumentation—such as piezometers, settlement plates, and inclinometers—can verify whether the assumed behavior is occurring.

9. Illustrative Case: Staged Embankment on Soft Clay

Background

A highway embankment is planned over 8 m of soft, saturated clay. The embankment will be built in two lifts, with a planned waiting period between lifts. Site investigation indicates low permeability and an undrained strength that increases with depth.

Problem

A rapid placement schedule could generate excess pore pressure, reduce effective stress, and cause lateral spreading or foundation failure. A long-term analysis based only on initial undrained strength would also fail to recognize strength gain from consolidation.

Analysis

Short-term stability is assessed using undrained strength and total-stress methods. Piezometers are installed beneath and beside the embankment. Consolidation analysis estimates pore-pressure dissipation during the waiting period. Long-term stability is assessed using effective-stress parameters and the anticipated post-construction stress state.

Approach and solution

The first lift is placed conservatively. Construction is paused when measured pore pressure approaches the trigger value established in the observational method. Settlement and lateral movement are monitored. During the waiting period, pore pressure dissipates and effective stress increases. The second lift is then placed after confirming acceptable stability indicators.

Outcome

The staged approach reduces the risk of undrained failure and allows part of the soil’s future strength to develop before final loading. The design is more reliable than applying either a purely drained or purely undrained model to the entire construction period.

Learning points

1. Drainage condition changes with time and construction sequence.
2. Monitoring can test and refine the geotechnical model.
3. Strength parameters must correspond to the analysis stage.
4. A laboratory value is not automatically a field design value.

10. Limitations, Safety, and Professional Practice

Mohr–Coulomb theory is valuable because it is simple and transparent, but it does not fully represent nonlinear envelopes, anisotropy, cyclic loading, strain softening, creep, or complex pore-pressure response. Advanced numerical models may use hardening plasticity, critical-state formulations, or coupled consolidation analyses; however, greater model complexity does not compensate for poor soil data.

Engineers should document sample quality, stress history, saturation, drainage conditions, strain rate, failure definition, and parameter selection. Laboratory testing should follow the applicable national or international standard, and quality assurance should include calibration, specimen measurements, repeat tests, and independent review of anomalous results.

Safety decisions must recognize uncertainty. Conservative values should be selected where consequences of failure are high, but excessive conservatism can also cause unnecessary construction cost. The observational method, staged construction, instrumentation, and trigger-action response plans provide ways to manage uncertainty transparently.

Digital and AI tools can assist with data screening, plotting stress paths, identifying anomalous test curves, and performing sensitivity studies. They should not replace engineering judgment, traceable calculations, laboratory quality control, or responsibility by a qualified engineer. Any data-driven prediction must be checked against soil mechanics, site geology, and applicable codes.

Chapter Summary

Shear strength is the resistance of soil to sliding and distortion. The Mohr–Coulomb model expresses strength as τ_f = c′ + σ′_n tan φ′ in effective-stress form, or through total-stress parameters when undrained conditions justify that approach. Effective stress, not total stress alone, controls frictional contact between soil particles.

Drained loading permits pore-pressure dissipation and volume change; undrained loading generates pore-pressure changes and often requires total-stress undrained strength or measured effective stresses. Stress paths reveal how loading history, contractancy, dilation, and pore pressure influence behavior. Direct shear, triaxial, and unconfined compression tests each have distinct strengths and limitations.

Design parameters must be selected for the soil type, stress history, loading rate, drainage condition, construction stage, and expected displacement. Peak, critical-state, and residual strengths are not interchangeable. Sound practice combines appropriate laboratory testing, field monitoring, conservative interpretation, and professional judgment.

Key Terms

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

Total stress: The externally applied stress, including the contribution balanced by pore water.

Shear strength: Maximum resistance of soil to shearing under specified conditions.

Mohr–Coulomb criterion: Linear relationship between shear strength and normal effective stress.

Cohesion: Strength-intercept parameter in a soil strength envelope; may be true or apparent.

Friction angle: Angle defining the stress dependence of frictional resistance.

Drained loading: Loading during which excess pore pressure can dissipate.

Undrained loading: Loading during which water cannot escape during the relevant period.

Peak strength: Maximum shear resistance reached during a test.

Critical-state strength: Strength associated with continued shearing at approximately constant stress ratio and volume condition.

Residual strength: Large-displacement strength after particle alignment or structural breakdown.

Dilatancy: Volume increase during shearing.

Contractancy: Volume decrease during shearing.

Stress path: Route followed by a soil element in stress space.

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

Direct shear test: Test using a predetermined horizontal failure plane.

Triaxial test: Test applying controlled cell pressure and axial stress to a cylindrical specimen.

Unconfined compression test: Axial compression test on cohesive soil without lateral confinement.

Undrained shear strength: Shear strength measured or interpreted for undrained loading.

Progressive failure: Failure that develops through sequential local yielding and strength reduction.

Review Questions

Short Answer Questions

1. Define effective stress and explain its significance in soil strength.
2. Distinguish drained from undrained loading.
3. What is the physical meaning of the friction angle φ′?
4. State two advantages and two limitations of the direct shear test.
5. Why may peak strength be inappropriate for a reactivated landslide?

Long Answer Questions

1. Explain the Mohr–Coulomb failure criterion and its effective-stress and total-stress forms.
2. Describe the procedure, interpretation, and limitations of a consolidated undrained triaxial test.
3. Explain how contractancy and dilatancy influence undrained pore pressure and effective stress.
4. Compare direct shear, triaxial, and unconfined compression tests for clay and sand.
5. Discuss how construction sequence affects the selection of strength parameters for an embankment on soft clay.

Analytical / Application Questions

1. A soil has c′ = 12 kPa, φ′ = 28°, and effective normal stress of 150 kPa. Calculate its Mohr–Coulomb shear strength and state the assumptions behind the calculation.
2. An embankment is placed rapidly on saturated clay, but construction is paused for six months. Explain which strength parameters should be used immediately, during the pause, and for long-term stability.
3. A dense sand produces a high peak friction angle in a direct shear test but a lower large-displacement angle. Recommend parameters for a permanent slope and justify your choice.

Multiple Choice Questions with Answers

Question 1

Which equation expresses Terzaghi’s effective-stress principle?

A. σ′ = σ + u

B. σ′ = σ − u

C. σ′ = u − σ

D. σ′ = σu

Correct Answer: B. σ′ = σ − u

Question 2

A drained Mohr–Coulomb strength calculation uses which parameters?

A. c′ and φ′

B. c_u and φ_u only

C. q_u and E

D. Permeability and liquid limit only

Correct Answer: A. c′ and φ′

Question 3

For saturated clay modeled with undrained φ_u = 0, the total-stress shear strength is:

A. τ_f = σ_n tan φ′

B. τ_f = c_u + σ_n

C. τ_f = c_u

D. τ_f = c′ + u

Correct Answer: C. τ_f = c_u

Question 4

Which test permits direct measurement of pore pressure during undrained shearing?

A. Standard sieve analysis

B. Consolidated undrained triaxial test with instrumentation

C. Unconfined compression without load measurement

D. Atterberg limits test

Correct Answer: B. Consolidated undrained triaxial test with instrumentation

Question 5

A dense sand commonly tends to do what during drained shearing?

A. Dilate

B. Remain perfectly incompressible

C. Lose all friction

D. Generate no shear stress

Correct Answer: A. Dilate

Question 6

If c′ = 10 kPa, σ′_n = 100 kPa, and φ′ = 30°, what is τ_f approximately?

A. 58 kPa

B. 68 kPa

C. 110 kPa

D. 173 kPa

Correct Answer: B. 68 kPa

Question 7

The unconfined compression test is most appropriate for:

A. Clean, loose sand

B. Unsupported cohesive soil specimens

C. Rock joints only

D. Dry gravel under high confinement

Correct Answer: B. Unsupported cohesive soil specimens

Question 8

In a triaxial compression test, the cell pressure commonly represents:

A. The minor principal stress

B. The deviator stress only

C. The pore-water pressure only

D. The shear strain

Correct Answer: A. The minor principal stress

Question 9

Rapid construction on saturated low-permeability clay is commonly analyzed initially using:

A. Undrained strength

B. Residual sand friction only

C. Drained strength with zero pore pressure

D. Void ratio alone

Correct Answer: A. Undrained strength

Question 10

Why can residual strength be important in a reactivated landslide?

A. Large displacement may reduce strength through particle alignment and structural breakdown

B. Residual strength is always greater than peak strength

C. Pore pressure is always zero

D. The failure plane cannot develop

Correct Answer: A. Large displacement may reduce strength through particle alignment and structural breakdown