Shear Strength of Cohesive and Cohesionless Soils

Outcomes

After completing this unit the learner should be able to define the governing state variables, calculate representative quantities with correct units, explain when the idealized relation breaks down, interpret the corresponding laboratory models, and defend a field investigation or verification decision.

Core lessons

Lesson 5.1 — Effective-stress failure and Mohr–Coulomb

Central question. In frictional soils the limiting shear resistance commonly increases with effective normal stress.

Governing relationships and definitions. The idealized Mohr–Coulomb envelope τf=c′+σ′n tanφ′ is a fit over a chosen stress range, not a perfect law at all confinements.

Worked calculation / observation. For c′=8 kPa, φ′=30° and σ′n=120 kPa, τf=8+120×0.57735=77.3 kPa. Raising pore pressure by 20 kPa, at fixed total normal stress, reduces predicted strength by 11.5 kPa.

Why interpretation matters. Effective strength parameters must be associated with drainage and stress path. A saturated fine soil may behave undrained in a rapid short-term loading event and drained after enough time, even though the grains and initial moisture remain the same. It is not correct to select a test based solely on the soil name “clay.”

Field case: A temporary excavation in saturated silt. In a temporary excavation in saturated silt, the engineer should first sketch the geometry, state the loading or groundwater history, and identify which of the measured quantities are direct observations. Next identify the controlling stress, volume, hydraulic or strength variable from this lesson. A prediction is only as defensible as the link between the laboratory variable and its field counterpart. For example, the stress relevant to settlement may be an effective stress after excess water pressure dissipates, whereas the load relevant to a temporary wall brace may be a short-term total-stress action. Mixing those questions can produce a plausible-looking but physically inconsistent answer.

From observation to a defensible model. A practical investigation has three stages. Observation: identify the features of a temporary excavation in saturated silt that a field crew can actually record, such as layer thickness, water-table history, moisture state or deformation. Interpretation: select the idealized law described above, explain the chosen signs and units, and identify which soil parameters need independent tests. Prediction: evaluate a baseline and at least two credible alternatives. A single baseline is rarely an adequate basis for choosing a foundation treatment, drainage provision or construction method.

Verification practice. A useful checking sequence is dimensional analysis, limiting behavior and independent arithmetic. Insert the stated units before using a formula; a conductivity given in cm/s cannot be entered as m/s unchanged. Ask whether a computed stress or discharge tends toward zero when its driving load is removed. Then compare with a second hand calculation, a plotted numerical response or a bounding case. A graph from an interactive model provides feedback, not evidence that the assumed governing equation is correct for the site.

Reasoning exercise. Draw the physical control volume or stress path for a temporary excavation in saturated silt. Label the measured variables, the inferred variables and the quantities that could change after construction. Which variable would be the cheapest to measure more accurately? Describe how two alternative assumptions could change the outcome in opposite directions. Finally explain which decision would be postponed until a more representative specimen, groundwater observation or construction-stage measurement becomes available.

Model boundary. Separate peak, critical-state and residual strengths. A single φ′ from one stress level can conceal nonlinearity or particle crushing. Present the result with an explicit number of significant digits justified by the inputs; do not confuse that numerical precision with geological certainty. Professional design requires the governing standard, site-specific investigation and independent review.

Lesson 5.2 — Direct shear and drained triaxial testing

Central question. Drainage conditions and loading rate control pore pressure response and therefore which strength parameters can be inferred.

Governing relationships and definitions. Direct shear imposes a failure plane; triaxial testing permits controlled cell pressure and drainage measurements. For triaxial compression, q=σ′1−σ′3 and p′=(σ′1+2σ′3)/3.

Worked calculation / observation. At effective principal stresses σ′1=300 kPa and σ′3=100 kPa, q=200 kPa and p′≈166.7 kPa. The point is a stress state; failure is not determined without a calibrated criterion.

Why interpretation matters. Direct shear plots can be easy to interpret because the shear plane is prescribed, but that convenience obscures some stress uniformity and drainage assumptions. Triaxial testing gives richer pore-pressure and volume-change measurements, although specimen preparation and end effects also introduce interpretation choices.

Field case: A retaining-wall backfill subject to drainage changes. In a retaining-wall backfill subject to drainage changes, the engineer should first sketch the geometry, state the loading or groundwater history, and identify which of the measured quantities are direct observations. Next identify the controlling stress, volume, hydraulic or strength variable from this lesson. A prediction is only as defensible as the link between the laboratory variable and its field counterpart. For example, the stress relevant to settlement may be an effective stress after excess water pressure dissipates, whereas the load relevant to a temporary wall brace may be a short-term total-stress action. Mixing those questions can produce a plausible-looking but physically inconsistent answer.

From observation to a defensible model. A practical investigation has three stages. Observation: identify the features of a retaining-wall backfill subject to drainage changes that a field crew can actually record, such as layer thickness, water-table history, moisture state or deformation. Interpretation: select the idealized law described above, explain the chosen signs and units, and identify which soil parameters need independent tests. Prediction: evaluate a baseline and at least two credible alternatives. A single baseline is rarely an adequate basis for choosing a foundation treatment, drainage provision or construction method.

Verification practice. A useful checking sequence is dimensional analysis, limiting behavior and independent arithmetic. Insert the stated units before using a formula; a conductivity given in cm/s cannot be entered as m/s unchanged. Ask whether a computed stress or discharge tends toward zero when its driving load is removed. Then compare with a second hand calculation, a plotted numerical response or a bounding case. A graph from an interactive model provides feedback, not evidence that the assumed governing equation is correct for the site.

Reasoning exercise. Draw the physical control volume or stress path for a retaining-wall backfill subject to drainage changes. Label the measured variables, the inferred variables and the quantities that could change after construction. Which variable would be the cheapest to measure more accurately? Describe how two alternative assumptions could change the outcome in opposite directions. Finally explain which decision would be postponed until a more representative specimen, groundwater observation or construction-stage measurement becomes available.

Model boundary. Record whether the test is CD, CU with pore-pressure measurement, or UU. Total-stress parameters must not be substituted into an effective-stress analysis. Present the result with an explicit number of significant digits justified by the inputs; do not confuse that numerical precision with geological certainty. Professional design requires the governing standard, site-specific investigation and independent review.

Lesson 5.3 — Undrained response and rapid loading

Central question. Fine soils may respond approximately undrained when loading is faster than drainage, generating excess pore pressure.

Governing relationships and definitions. For a saturated clay idealized with φu≈0 in total stresses, undrained shear strength su is often represented as half the deviator stress at failure in an unconfined compression test: su=qu/2.

Worked calculation / observation. If unconfined compressive strength qu=90 kPa under the assumptions of this test, su≈45 kPa. This says nothing about drained friction angle without additional information.

Why interpretation matters. An effective-stress strength line is usually fitted to several tests at different normal or confining stresses. A single peak measurement cannot uniquely identify both cohesion intercept and friction angle. Include a selected stress range and whether the reported line describes peak, critical or residual response.

Field case: A road embankment on soft marine clay. In a road embankment on soft marine clay, the engineer should first sketch the geometry, state the loading or groundwater history, and identify which of the measured quantities are direct observations. Next identify the controlling stress, volume, hydraulic or strength variable from this lesson. A prediction is only as defensible as the link between the laboratory variable and its field counterpart. For example, the stress relevant to settlement may be an effective stress after excess water pressure dissipates, whereas the load relevant to a temporary wall brace may be a short-term total-stress action. Mixing those questions can produce a plausible-looking but physically inconsistent answer.

From observation to a defensible model. A practical investigation has three stages. Observation: identify the features of a road embankment on soft marine clay that a field crew can actually record, such as layer thickness, water-table history, moisture state or deformation. Interpretation: select the idealized law described above, explain the chosen signs and units, and identify which soil parameters need independent tests. Prediction: evaluate a baseline and at least two credible alternatives. A single baseline is rarely an adequate basis for choosing a foundation treatment, drainage provision or construction method.

Verification practice. A useful checking sequence is dimensional analysis, limiting behavior and independent arithmetic. Insert the stated units before using a formula; a conductivity given in cm/s cannot be entered as m/s unchanged. Ask whether a computed stress or discharge tends toward zero when its driving load is removed. Then compare with a second hand calculation, a plotted numerical response or a bounding case. A graph from an interactive model provides feedback, not evidence that the assumed governing equation is correct for the site.

Reasoning exercise. Draw the physical control volume or stress path for a road embankment on soft marine clay. Label the measured variables, the inferred variables and the quantities that could change after construction. Which variable would be the cheapest to measure more accurately? Describe how two alternative assumptions could change the outcome in opposite directions. Finally explain which decision would be postponed until a more representative specimen, groundwater observation or construction-stage measurement becomes available.

Model boundary. Structure, strain rate, anisotropy and disturbance can make a single su misleading. Unconfined compression is not applicable to all soils. Present the result with an explicit number of significant digits justified by the inputs; do not confuse that numerical precision with geological certainty. Professional design requires the governing standard, site-specific investigation and independent review.

Lesson 5.4 — Stress path and interpretation

Central question. The journey through stress space matters because soil response depends on stress history, drainage, and how boundaries impose deformation.

Governing relationships and definitions. Compare total and effective stress paths with the same applied total stress increment. Use q and p′ to track changes while pore pressure changes; interpretation requires a defined sign convention.

Worked calculation / observation. A 60 kPa increase of mean total stress accompanied by 45 kPa excess pore pressure increases p′ by only 15 kPa. Interpreting the full 60 kPa as increased effective confinement would overestimate frictional resistance.

Why interpretation matters. In a stress-path plot, the same point may be reached through different loading histories that produce different fabric and pore pressure. A complete explanation includes the initial state, whether volume change was permitted and how quickly loading occurred. The model becomes much less informative if only a final friction angle is shown.

Field case: A bridge foundation where sampling has disturbed the soil. In a bridge foundation where sampling has disturbed the soil, the engineer should first sketch the geometry, state the loading or groundwater history, and identify which of the measured quantities are direct observations. Next identify the controlling stress, volume, hydraulic or strength variable from this lesson. A prediction is only as defensible as the link between the laboratory variable and its field counterpart. For example, the stress relevant to settlement may be an effective stress after excess water pressure dissipates, whereas the load relevant to a temporary wall brace may be a short-term total-stress action. Mixing those questions can produce a plausible-looking but physically inconsistent answer.

From observation to a defensible model. A practical investigation has three stages. Observation: identify the features of a bridge foundation where sampling has disturbed the soil that a field crew can actually record, such as layer thickness, water-table history, moisture state or deformation. Interpretation: select the idealized law described above, explain the chosen signs and units, and identify which soil parameters need independent tests. Prediction: evaluate a baseline and at least two credible alternatives. A single baseline is rarely an adequate basis for choosing a foundation treatment, drainage provision or construction method.

Verification practice. A useful checking sequence is dimensional analysis, limiting behavior and independent arithmetic. Insert the stated units before using a formula; a conductivity given in cm/s cannot be entered as m/s unchanged. Ask whether a computed stress or discharge tends toward zero when its driving load is removed. Then compare with a second hand calculation, a plotted numerical response or a bounding case. A graph from an interactive model provides feedback, not evidence that the assumed governing equation is correct for the site.

Reasoning exercise. Draw the physical control volume or stress path for a bridge foundation where sampling has disturbed the soil. Label the measured variables, the inferred variables and the quantities that could change after construction. Which variable would be the cheapest to measure more accurately? Describe how two alternative assumptions could change the outcome in opposite directions. Finally explain which decision would be postponed until a more representative specimen, groundwater observation or construction-stage measurement becomes available.

Model boundary. Constitutive models used in numerical analysis require calibration to a test path near the expected field path; do not transfer a curve blindly. Present the result with an explicit number of significant digits justified by the inputs; do not confuse that numerical precision with geological certainty. Professional design requires the governing standard, site-specific investigation and independent review.

Laboratory activities

Laboratory 9: Mohr–Coulomb Direct Shear. Peak strength envelope and pore-pressure sensitivity. First predict the direction of change for the output when one control changes. Hold all other controls fixed; record a baseline, two variations, and a short discussion of why the governing assumptions are or are not credible. Navigate to ../labs/lab-09.html. Important: Linear c′–φ′ fit at a selected effective normal stress.

Laboratory 10: Triaxial Stress Path. Stress invariants and pore-pressure effect. First predict the direction of change for the output when one control changes. Hold all other controls fixed; record a baseline, two variations, and a short discussion of why the governing assumptions are or are not credible. Navigate to ../labs/lab-10.html. Important: Axisymmetric triaxial compression state; no automatic failure criterion.

Integrated design conversation

Imagine that the site data are incomplete and construction will begin in phases. Construct two defensible interpretations of the subsurface conditions. For each, state the best estimate of the controlling material parameter and a credible low and high value. Explain how a field observation or laboratory test could reject one interpretation. The deliverable is not only a number but a documented chain from field evidence to calculation, checking, and the resulting construction choice.

Applied dossier: conflicting shear test interpretations

A saturated fine-grained sample is tested under different drainage conditions. A rapid unconfined compression test indicates a peak axial stress of 100 kPa; under the ideal assumption of zero confining pressure and an undrained φu≈0 response, su≈50 kPa. A separate consolidated-drained program returns an effective-stress envelope with c′≈5 kPa and φ′≈29°. These results are not competing measurements of one identical strength. They refer to different variables and potentially different loading times.

At effective normal stress 100 kPa, the drained Mohr–Coulomb fit predicts τf≈5+100 tan29°≈60.4 kPa. If pore pressure rises from 20 to 50 kPa while total normal stress stays 140 kPa, the effective normal stress falls from 120 to 90 kPa and the linear-fit strength drops by approximately 30 tan29°≈16.6 kPa. In a slow drained test, the internal pore pressure can dissipate as drainage proceeds; in a rapid undrained field event it may rise sharply. Therefore the relevant strength envelope cannot be chosen without a loading and drainage narrative.

Review the triaxial stresses in an axisymmetric state. Suppose the cell total principal stress is 160 kPa, the major total principal stress is 340 kPa and pore pressure is 35 kPa. Then σ′3=125 kPa, σ′1=305 kPa, q=180 kPa and p′=(305+2×125)/3=185 kPa. An increase in pore pressure by 20 kPa at fixed total stresses shifts p′ left by 20 kPa while leaving q unchanged. This is an important stress-path fact that can be visualized without invoking a particular failure surface.

The case should conclude by reviewing specimen disturbance, test orientation, rate and confining pressure range. A peak result at one stress state may not represent critical-state resistance, while residual strength may govern a reactivated sliding surface. Good reporting labels every strength parameter with its derivation and relevant stress path rather than presenting a bare friction angle as a universal soil property.

Technical methods notebook: interpreting strength in stress space

Strength in effective normal stress. For the linear Mohr–Coulomb approximation τf=c′+σ′n tanφ′, the input normal stress must be effective. Suppose a direct-shear box is subjected to 160 kPa total normal stress and develops 40 kPa pore pressure on the assumed failure plane. The effective normal stress is 120 kPa. With c′=10 kPa and φ′=28°, the shear resistance predicted by the fit is about 73.8 kPa. Replacing 120 kPa by the total 160 kPa would overstate the friction component by 40 tan28°≈21.3 kPa.

Stress changes without changed deviator stress. In axisymmetric triaxial compression, q=σ′1−σ′3 and p′=(σ′1+2σ′3)/3. Subtracting the same pore pressure u from both total principal stresses leaves q unchanged but decreases p′ by u. This explains why an increase in excess pore pressure can move an effective stress path toward failure even when the total deviator stress is approximately constant. The plot does not imply that q must increase when u increases.

An instrumented test. A consolidated-undrained triaxial compression test may measure pore pressure continuously. Its total-stress path and effective-stress path then provide different information. A simple unconsolidated-undrained test without pore-pressure measurement cannot be interpreted as a definitive drained φ′ value. A consolidated-drained test seeks to permit adequate drainage, often requiring slow shearing in low-permeability soils. State drainage conditions and any saturation checks when presenting the result.

Peak, critical and residual values. Dense sands may dilate and develop peak strength above their critical-state frictional resistance, while fissured or overconsolidated clays may soften after a peak. A reactivated landslide may involve much smaller residual friction. A laboratory fit must be identified by mobilized displacement or strain range; a peak parameter may not be appropriate for a long-term progressive failure scenario. A single line fits only the stress range in which it was calibrated.

Reconcile two test types. If an unconfined compression test yields qu=80 kPa, the simple saturated φu=0 total-stress approximation suggests su≈40 kPa. Independently, a drained test gives φ′=31° and c′≈0. The first result is not necessarily inconsistent with the second: undrained resistance depends on initial state and stress history and cannot be computed from φ′ alone without further constitutive assumptions. A credible note should explain the loading rate and drainage path that make each measurement relevant.

Interpretation exercise. On a single p′–q chart, show initial stress state (σ′1=200 kPa, σ′3=100 kPa), a hypothetical 30 kPa increase in both total principal stresses and 20 kPa generated pore pressure. Find the new q and p′. Then draw a second path with the same total principal stress changes but zero pore-pressure increase. Explain why the two stress paths can support different strength interpretations despite having identical total-stress increments.

Unit 5: Shear strength

Laboratory tests, drainage and failure envelopes.

Outcomes

After completing this unit the learner should be able to define the governing state variables, calculate representative quantities with correct units, explain when the idealized relation breaks down, interpret the corresponding laboratory models, and defend a field investigation or verification decision.

Core lessons

Lesson 5.1 — Effective-stress failure and Mohr–Coulomb

Central question. In frictional soils the limiting shear resistance commonly increases with effective normal stress.

Governing relationships and definitions. The idealized Mohr–Coulomb envelope τf=c′+σ′n tanφ′ is a fit over a chosen stress range, not a perfect law at all confinements.

Worked calculation / observation. For c′=8 kPa, φ′=30° and σ′n=120 kPa, τf=8+120×0.57735=77.3 kPa. Raising pore pressure by 20 kPa, at fixed total normal stress, reduces predicted strength by 11.5 kPa.

Why interpretation matters. Effective strength parameters must be associated with drainage and stress path. A saturated fine soil may behave undrained in a rapid short-term loading event and drained after enough time, even though the grains and initial moisture remain the same. It is not correct to select a test based solely on the soil name “clay.”

Field case: A temporary excavation in saturated silt. In a temporary excavation in saturated silt, the engineer should first sketch the geometry, state the loading or groundwater history, and identify which of the measured quantities are direct observations. Next identify the controlling stress, volume, hydraulic or strength variable from this lesson. A prediction is only as defensible as the link between the laboratory variable and its field counterpart. For example, the stress relevant to settlement may be an effective stress after excess water pressure dissipates, whereas the load relevant to a temporary wall brace may be a short-term total-stress action. Mixing those questions can produce a plausible-looking but physically inconsistent answer.

From observation to a defensible model. A practical investigation has three stages. Observation: identify the features of a temporary excavation in saturated silt that a field crew can actually record, such as layer thickness, water-table history, moisture state or deformation. Interpretation: select the idealized law described above, explain the chosen signs and units, and identify which soil parameters need independent tests. Prediction: evaluate a baseline and at least two credible alternatives. A single baseline is rarely an adequate basis for choosing a foundation treatment, drainage provision or construction method.

Verification practice. A useful checking sequence is dimensional analysis, limiting behavior and independent arithmetic. Insert the stated units before using a formula; a conductivity given in cm/s cannot be entered as m/s unchanged. Ask whether a computed stress or discharge tends toward zero when its driving load is removed. Then compare with a second hand calculation, a plotted numerical response or a bounding case. A graph from an interactive model provides feedback, not evidence that the assumed governing equation is correct for the site.

Reasoning exercise. Draw the physical control volume or stress path for a temporary excavation in saturated silt. Label the measured variables, the inferred variables and the quantities that could change after construction. Which variable would be the cheapest to measure more accurately? Describe how two alternative assumptions could change the outcome in opposite directions. Finally explain which decision would be postponed until a more representative specimen, groundwater observation or construction-stage measurement becomes available.

Model boundary. Separate peak, critical-state and residual strengths. A single φ′ from one stress level can conceal nonlinearity or particle crushing. Present the result with an explicit number of significant digits justified by the inputs; do not confuse that numerical precision with geological certainty. Professional design requires the governing standard, site-specific investigation and independent review.

Lesson 5.2 — Direct shear and drained triaxial testing

Central question. Drainage conditions and loading rate control pore pressure response and therefore which strength parameters can be inferred.

Governing relationships and definitions. Direct shear imposes a failure plane; triaxial testing permits controlled cell pressure and drainage measurements. For triaxial compression, q=σ′1−σ′3 and p′=(σ′1+2σ′3)/3.

Worked calculation / observation. At effective principal stresses σ′1=300 kPa and σ′3=100 kPa, q=200 kPa and p′≈166.7 kPa. The point is a stress state; failure is not determined without a calibrated criterion.

Why interpretation matters. Direct shear plots can be easy to interpret because the shear plane is prescribed, but that convenience obscures some stress uniformity and drainage assumptions. Triaxial testing gives richer pore-pressure and volume-change measurements, although specimen preparation and end effects also introduce interpretation choices.

Field case: A retaining-wall backfill subject to drainage changes. In a retaining-wall backfill subject to drainage changes, the engineer should first sketch the geometry, state the loading or groundwater history, and identify which of the measured quantities are direct observations. Next identify the controlling stress, volume, hydraulic or strength variable from this lesson. A prediction is only as defensible as the link between the laboratory variable and its field counterpart. For example, the stress relevant to settlement may be an effective stress after excess water pressure dissipates, whereas the load relevant to a temporary wall brace may be a short-term total-stress action. Mixing those questions can produce a plausible-looking but physically inconsistent answer.

From observation to a defensible model. A practical investigation has three stages. Observation: identify the features of a retaining-wall backfill subject to drainage changes that a field crew can actually record, such as layer thickness, water-table history, moisture state or deformation. Interpretation: select the idealized law described above, explain the chosen signs and units, and identify which soil parameters need independent tests. Prediction: evaluate a baseline and at least two credible alternatives. A single baseline is rarely an adequate basis for choosing a foundation treatment, drainage provision or construction method.

Verification practice. A useful checking sequence is dimensional analysis, limiting behavior and independent arithmetic. Insert the stated units before using a formula; a conductivity given in cm/s cannot be entered as m/s unchanged. Ask whether a computed stress or discharge tends toward zero when its driving load is removed. Then compare with a second hand calculation, a plotted numerical response or a bounding case. A graph from an interactive model provides feedback, not evidence that the assumed governing equation is correct for the site.

Reasoning exercise. Draw the physical control volume or stress path for a retaining-wall backfill subject to drainage changes. Label the measured variables, the inferred variables and the quantities that could change after construction. Which variable would be the cheapest to measure more accurately? Describe how two alternative assumptions could change the outcome in opposite directions. Finally explain which decision would be postponed until a more representative specimen, groundwater observation or construction-stage measurement becomes available.

Model boundary. Record whether the test is CD, CU with pore-pressure measurement, or UU. Total-stress parameters must not be substituted into an effective-stress analysis. Present the result with an explicit number of significant digits justified by the inputs; do not confuse that numerical precision with geological certainty. Professional design requires the governing standard, site-specific investigation and independent review.

Lesson 5.3 — Undrained response and rapid loading

Central question. Fine soils may respond approximately undrained when loading is faster than drainage, generating excess pore pressure.

Governing relationships and definitions. For a saturated clay idealized with φu≈0 in total stresses, undrained shear strength su is often represented as half the deviator stress at failure in an unconfined compression test: su=qu/2.

Worked calculation / observation. If unconfined compressive strength qu=90 kPa under the assumptions of this test, su≈45 kPa. This says nothing about drained friction angle without additional information.

Why interpretation matters. An effective-stress strength line is usually fitted to several tests at different normal or confining stresses. A single peak measurement cannot uniquely identify both cohesion intercept and friction angle. Include a selected stress range and whether the reported line describes peak, critical or residual response.

Field case: A road embankment on soft marine clay. In a road embankment on soft marine clay, the engineer should first sketch the geometry, state the loading or groundwater history, and identify which of the measured quantities are direct observations. Next identify the controlling stress, volume, hydraulic or strength variable from this lesson. A prediction is only as defensible as the link between the laboratory variable and its field counterpart. For example, the stress relevant to settlement may be an effective stress after excess water pressure dissipates, whereas the load relevant to a temporary wall brace may be a short-term total-stress action. Mixing those questions can produce a plausible-looking but physically inconsistent answer.

From observation to a defensible model. A practical investigation has three stages. Observation: identify the features of a road embankment on soft marine clay that a field crew can actually record, such as layer thickness, water-table history, moisture state or deformation. Interpretation: select the idealized law described above, explain the chosen signs and units, and identify which soil parameters need independent tests. Prediction: evaluate a baseline and at least two credible alternatives. A single baseline is rarely an adequate basis for choosing a foundation treatment, drainage provision or construction method.

Verification practice. A useful checking sequence is dimensional analysis, limiting behavior and independent arithmetic. Insert the stated units before using a formula; a conductivity given in cm/s cannot be entered as m/s unchanged. Ask whether a computed stress or discharge tends toward zero when its driving load is removed. Then compare with a second hand calculation, a plotted numerical response or a bounding case. A graph from an interactive model provides feedback, not evidence that the assumed governing equation is correct for the site.

Reasoning exercise. Draw the physical control volume or stress path for a road embankment on soft marine clay. Label the measured variables, the inferred variables and the quantities that could change after construction. Which variable would be the cheapest to measure more accurately? Describe how two alternative assumptions could change the outcome in opposite directions. Finally explain which decision would be postponed until a more representative specimen, groundwater observation or construction-stage measurement becomes available.

Model boundary. Structure, strain rate, anisotropy and disturbance can make a single su misleading. Unconfined compression is not applicable to all soils. Present the result with an explicit number of significant digits justified by the inputs; do not confuse that numerical precision with geological certainty. Professional design requires the governing standard, site-specific investigation and independent review.

Lesson 5.4 — Stress path and interpretation

Central question. The journey through stress space matters because soil response depends on stress history, drainage, and how boundaries impose deformation.

Governing relationships and definitions. Compare total and effective stress paths with the same applied total stress increment. Use q and p′ to track changes while pore pressure changes; interpretation requires a defined sign convention.

Worked calculation / observation. A 60 kPa increase of mean total stress accompanied by 45 kPa excess pore pressure increases p′ by only 15 kPa. Interpreting the full 60 kPa as increased effective confinement would overestimate frictional resistance.

Why interpretation matters. In a stress-path plot, the same point may be reached through different loading histories that produce different fabric and pore pressure. A complete explanation includes the initial state, whether volume change was permitted and how quickly loading occurred. The model becomes much less informative if only a final friction angle is shown.

Field case: A bridge foundation where sampling has disturbed the soil. In a bridge foundation where sampling has disturbed the soil, the engineer should first sketch the geometry, state the loading or groundwater history, and identify which of the measured quantities are direct observations. Next identify the controlling stress, volume, hydraulic or strength variable from this lesson. A prediction is only as defensible as the link between the laboratory variable and its field counterpart. For example, the stress relevant to settlement may be an effective stress after excess water pressure dissipates, whereas the load relevant to a temporary wall brace may be a short-term total-stress action. Mixing those questions can produce a plausible-looking but physically inconsistent answer.

From observation to a defensible model. A practical investigation has three stages. Observation: identify the features of a bridge foundation where sampling has disturbed the soil that a field crew can actually record, such as layer thickness, water-table history, moisture state or deformation. Interpretation: select the idealized law described above, explain the chosen signs and units, and identify which soil parameters need independent tests. Prediction: evaluate a baseline and at least two credible alternatives. A single baseline is rarely an adequate basis for choosing a foundation treatment, drainage provision or construction method.

Verification practice. A useful checking sequence is dimensional analysis, limiting behavior and independent arithmetic. Insert the stated units before using a formula; a conductivity given in cm/s cannot be entered as m/s unchanged. Ask whether a computed stress or discharge tends toward zero when its driving load is removed. Then compare with a second hand calculation, a plotted numerical response or a bounding case. A graph from an interactive model provides feedback, not evidence that the assumed governing equation is correct for the site.

Reasoning exercise. Draw the physical control volume or stress path for a bridge foundation where sampling has disturbed the soil. Label the measured variables, the inferred variables and the quantities that could change after construction. Which variable would be the cheapest to measure more accurately? Describe how two alternative assumptions could change the outcome in opposite directions. Finally explain which decision would be postponed until a more representative specimen, groundwater observation or construction-stage measurement becomes available.

Model boundary. Constitutive models used in numerical analysis require calibration to a test path near the expected field path; do not transfer a curve blindly. Present the result with an explicit number of significant digits justified by the inputs; do not confuse that numerical precision with geological certainty. Professional design requires the governing standard, site-specific investigation and independent review.

Laboratory activities

Laboratory 9: Mohr–Coulomb Direct Shear. Peak strength envelope and pore-pressure sensitivity. First predict the direction of change for the output when one control changes. Hold all other controls fixed; record a baseline, two variations, and a short discussion of why the governing assumptions are or are not credible. Navigate to ../labs/lab-09.html. Important: Linear c′–φ′ fit at a selected effective normal stress.

Laboratory 10: Triaxial Stress Path. Stress invariants and pore-pressure effect. First predict the direction of change for the output when one control changes. Hold all other controls fixed; record a baseline, two variations, and a short discussion of why the governing assumptions are or are not credible. Navigate to ../labs/lab-10.html. Important: Axisymmetric triaxial compression state; no automatic failure criterion.

Integrated design conversation

Imagine that the site data are incomplete and construction will begin in phases. Construct two defensible interpretations of the subsurface conditions. For each, state the best estimate of the controlling material parameter and a credible low and high value. Explain how a field observation or laboratory test could reject one interpretation. The deliverable is not only a number but a documented chain from field evidence to calculation, checking, and the resulting construction choice.

Applied dossier: conflicting shear test interpretations

A saturated fine-grained sample is tested under different drainage conditions. A rapid unconfined compression test indicates a peak axial stress of 100 kPa; under the ideal assumption of zero confining pressure and an undrained φu≈0 response, su≈50 kPa. A separate consolidated-drained program returns an effective-stress envelope with c′≈5 kPa and φ′≈29°. These results are not competing measurements of one identical strength. They refer to different variables and potentially different loading times.

At effective normal stress 100 kPa, the drained Mohr–Coulomb fit predicts τf≈5+100 tan29°≈60.4 kPa. If pore pressure rises from 20 to 50 kPa while total normal stress stays 140 kPa, the effective normal stress falls from 120 to 90 kPa and the linear-fit strength drops by approximately 30 tan29°≈16.6 kPa. In a slow drained test, the internal pore pressure can dissipate as drainage proceeds; in a rapid undrained field event it may rise sharply. Therefore the relevant strength envelope cannot be chosen without a loading and drainage narrative.

Review the triaxial stresses in an axisymmetric state. Suppose the cell total principal stress is 160 kPa, the major total principal stress is 340 kPa and pore pressure is 35 kPa. Then σ′3=125 kPa, σ′1=305 kPa, q=180 kPa and p′=(305+2×125)/3=185 kPa. An increase in pore pressure by 20 kPa at fixed total stresses shifts p′ left by 20 kPa while leaving q unchanged. This is an important stress-path fact that can be visualized without invoking a particular failure surface.

The case should conclude by reviewing specimen disturbance, test orientation, rate and confining pressure range. A peak result at one stress state may not represent critical-state resistance, while residual strength may govern a reactivated sliding surface. Good reporting labels every strength parameter with its derivation and relevant stress path rather than presenting a bare friction angle as a universal soil property.

Technical methods notebook: interpreting strength in stress space

Strength in effective normal stress. For the linear Mohr–Coulomb approximation τf=c′+σ′n tanφ′, the input normal stress must be effective. Suppose a direct-shear box is subjected to 160 kPa total normal stress and develops 40 kPa pore pressure on the assumed failure plane. The effective normal stress is 120 kPa. With c′=10 kPa and φ′=28°, the shear resistance predicted by the fit is about 73.8 kPa. Replacing 120 kPa by the total 160 kPa would overstate the friction component by 40 tan28°≈21.3 kPa.

Stress changes without changed deviator stress. In axisymmetric triaxial compression, q=σ′1−σ′3 and p′=(σ′1+2σ′3)/3. Subtracting the same pore pressure u from both total principal stresses leaves q unchanged but decreases p′ by u. This explains why an increase in excess pore pressure can move an effective stress path toward failure even when the total deviator stress is approximately constant. The plot does not imply that q must increase when u increases.

An instrumented test. A consolidated-undrained triaxial compression test may measure pore pressure continuously. Its total-stress path and effective-stress path then provide different information. A simple unconsolidated-undrained test without pore-pressure measurement cannot be interpreted as a definitive drained φ′ value. A consolidated-drained test seeks to permit adequate drainage, often requiring slow shearing in low-permeability soils. State drainage conditions and any saturation checks when presenting the result.

Peak, critical and residual values. Dense sands may dilate and develop peak strength above their critical-state frictional resistance, while fissured or overconsolidated clays may soften after a peak. A reactivated landslide may involve much smaller residual friction. A laboratory fit must be identified by mobilized displacement or strain range; a peak parameter may not be appropriate for a long-term progressive failure scenario. A single line fits only the stress range in which it was calibrated.

Reconcile two test types. If an unconfined compression test yields qu=80 kPa, the simple saturated φu=0 total-stress approximation suggests su≈40 kPa. Independently, a drained test gives φ′=31° and c′≈0. The first result is not necessarily inconsistent with the second: undrained resistance depends on initial state and stress history and cannot be computed from φ′ alone without further constitutive assumptions. A credible note should explain the loading rate and drainage path that make each measurement relevant.

Interpretation exercise. On a single p′–q chart, show initial stress state (σ′1=200 kPa, σ′3=100 kPa), a hypothetical 30 kPa increase in both total principal stresses and 20 kPa generated pore pressure. Find the new q and p′. Then draw a second path with the same total principal stress changes but zero pore-pressure increase. Explain why the two stress paths can support different strength interpretations despite having identical total-stress increments.

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