Lateral Earth Pressure and Slope Stability Analysis

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 7.1 — Infinite-slope stability

Central question. A shallow planar failure parallel to the ground surface can be approximated using a slice of uniform thickness on a long slope.

Governing relationships and definitions. For a dry cohesionless slope, FS=tanφ′/tanβ. With cohesion and pore pressure, FS=[c′+(γz cos²β−u)tanφ′]/[γz sinβ cosβ].

Worked calculation / observation. For β=25°, φ′=32°, dry cohesionless FS≈tan32°/tan25°≈1.34. Saturation increases pore pressure and reduces the frictional resistance for a fixed total stress.

Why interpretation matters. A factor of safety compares a simplified available resistance with mobilized demand. It does not directly quantify failure probability or predict displacement. A result near unity motivates closer investigation; a high result does not prove a design reliable if a relevant failure mode or groundwater condition was omitted.

Field case: A road cut with shallow slip surfaces after heavy rain. In a road cut with shallow slip surfaces after heavy rain, 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 cut with shallow slip surfaces after heavy rain 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 cut with shallow slip surfaces after heavy rain. 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. This model is not suitable for deep circular failures, seismic loads, nonuniform stratigraphy, or progressive failure. An FS number is not a probability of failure. 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 7.2 — Retaining-system actions and drainage

Central question. Wall pressure is controlled by soil strength, movement, surcharge, water pressure and construction sequence.

Governing relationships and definitions. For a simple active Rankine soil component Pa=0.5KaγH²; a uniform surcharge q adds Pa,q=KaqH. Hydrostatic water adds Pw=0.5γw hw² and must not be included twice.

Worked calculation / observation. For H=4 m, Ka=0.33, γ=18 and q=15 kPa, soil thrust≈47.5 kN/m, surcharge thrust≈19.8 kN/m, excluding water. Their application heights differ.

Why interpretation matters. The infinite-slope model is a deliberately narrow view of a long shallow failure plane. Real slopes can fail along curved or compound surfaces, and layered interfaces may govern. The simplicity is useful for controlled sensitivity studies, particularly for understanding friction angle, slope angle, depth and pore pressure.

Field case: A retaining wall with uncertain drainage condition. In a retaining wall with uncertain drainage condition, 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 with uncertain drainage condition 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 with uncertain drainage condition. 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. Rigid walls at rest may have larger pressures than walls that move enough to mobilize active conditions. Drainage details and outlet maintenance are critical. 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 7.3 — Bearing-capacity concepts

Central question. A shallow foundation may reach a shear failure mechanism if contact pressure becomes too large, but serviceability may govern first.

Governing relationships and definitions. For a strip footing on level soil under centered vertical loading and a simplified Terzaghi general shear model, qult=cNc+qNq+0.5γBNγ, where q=γDf. Bearing factors depend on φ and chosen convention.

Worked calculation / observation. For φ=0° in the classic idealized model, Nc≈5.7, Nq=1 and Nγ=0, giving qult≈5.7c+q. If c=40 kPa and q=20 kPa, qult≈248 kPa.

Why interpretation matters. Retaining wall earth-pressure diagrams are actions on a structure, not a substitute for evaluating overall stability, overturning, sliding, basal bearing or internal reinforcement. Students should track each resultant magnitude and application height before performing moment equilibrium.

Field case: A strip footing near the edge of a slope. In a strip footing near the edge of a slope, 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 strip footing near the edge of a slope 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 strip footing near the edge of a slope. 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. Do not turn this classroom formula directly into an allowable footing pressure: groundwater, shape, inclination, embedment, layered soils, settlement and code-required resistance factors matter. 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 7.4 — Stability, sensitivity and engineering decisions

Central question. Because soil is spatially variable, a single factor-of-safety computation needs a record of inputs and plausible ranges.

Governing relationships and definitions. Perform one-at-a-time sensitivity, separate correlated inputs, and identify which measurement has most influence on the safety conclusion.

Worked calculation / observation. Suppose β=28° and measured φ′ spans 29°–35°: dry cohesionless FS ranges from tan29°/tan28°≈1.04 to tan35°/tan28°≈1.32. This variation is large relative to the nominal margin.

Why interpretation matters. Ground bearing and settlement are separate limit states. A footing can be well below a classroom ultimate shear capacity and still undergo unacceptable differential settlement. Conversely a small nominal bearing ratio does not establish serviceability. Real foundation decisions need stratigraphy, loading, groundwater and code-specific factors.

Field case: A buried culvert embankment with uncertain fill properties. In a buried culvert embankment with uncertain fill properties, 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 buried culvert embankment with uncertain fill properties 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 buried culvert embankment with uncertain fill properties. 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. An uncertain test parameter should not be converted into spurious precision by displaying five decimal places. Identify a field test that narrows the decision interval. 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 13: Infinite-Slope Stability. Shear versus effective normal resistance in a shallow planar slope. 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-13.html. Important: Uniform infinite slope with failure plane parallel to ground surface.

Laboratory 14: Simplified Strip Bearing Capacity. Classical φ=0 strip-footing benchmark. 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-14.html. Important: Undrained homogeneous φ=0 classroom idealization; no allowable pressure.

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: shallow slope after a storm

A long, approximately planar colluvial slope has β=26° and an estimated drained friction angle φ′=33°. For a cohesionless dry infinite-slope interpretation, FS=tan33°/tan26°≈1.33. This result is a ratio of modeled resistance to modeled sliding demand for a very specific shallow mechanism. It does not imply a one-in-three chance of failure, nor does it describe the margin for deeper rotational instability.

Suppose a 2 m-deep slip plane is adopted, with γ=19 kN/m³ and c′=4 kPa. Using the chosen vertical-depth form of the infinite-slope equation, the driving shear is γz sinβ cosβ≈14.97 kPa and total normal stress is γz cos²β≈30.70 kPa. If pore pressure at the plane is 0 kPa, the friction component is about 19.94 kPa and FS≈(4+19.94)/14.97≈1.60. If pore pressure rises to 18 kPa, that term drops to approximately 8.25 kPa and FS≈0.82. The effect comes from pore pressure reducing effective normal stress, not from reducing soil mass.

These values are purposefully sharp for a classroom sensitivity analysis, but the field pore pressure may vary along the slip surface and through time. A shallow rainfall front, perched water, cracks and heterogeneous weathering can create local pressure variations. The assumed constant values must be linked to monitoring or justified conservatively; applying a uniform pressure is an idealization.

For the associated road cutting, the next investigation might combine slope geometry survey, inclinometer interpretation and short-response piezometers. If observations suggest a deeper sliding mass or a curved scarp, the infinite-slope approximation should be replaced by a more appropriate limit-equilibrium or numerical model. The critical lesson is to recognize when the model's own geometry fails to match the observed mechanism.

Technical methods notebook: stability measures and their limits

Infinite-slope force equilibrium. Consider a unit-width slice resting on a plane parallel to a long slope of inclination β. With vertical thickness z, unit weight γ, effective cohesion c′, and pore pressure u at the plane, the commonly used vertical-depth expression gives shear demand γz sinβ cosβ and total normal stress γz cos²β. The effective normal stress is the latter minus u. The simplified factor of safety is [c′+(γz cos²β−u)tanφ′]/[γz sinβ cosβ] where effective normal stress remains physically meaningful. Every quantity in numerator and denominator is stress, so the ratio is dimensionless.

Dry cohesionless limit. If c′=0 and u=0, the common γz cosβ terms cancel, leaving FS=tanφ′/tanβ. The result is independent of z and γ for that special case, making it an excellent unit test. For β=20° and φ′=30°, FS≈1.59. For β near φ′, FS approaches 1. If β exceeds φ′, the idealized ratio falls below one. A model that responds to γ or z under those exact dry cohesionless assumptions is likely implementing the algebra incorrectly.

Pore pressure sensitivity. Take β=25°, φ′=32°, γ=19 kN/m³, z=2 m, c′=5 kPa and pore pressure u=10 kPa. The shear demand is γz sin25° cos25°≈14.56 kPa, and effective normal stress is 19×2×cos²25°−10≈21.21 kPa. Resistance is 5+21.21 tan32°≈18.25 kPa; FS≈1.25. Raising u by a further 8 kPa reduces frictional resistance by 8 tan32°≈5.00 kPa while leaving the simplified driving stress unchanged. The predicted FS therefore drops substantially. These numbers are classroom screening quantities, not a model of rainfall infiltration.

Geometry mismatch. Infinite-slope analysis cannot describe the circular arc commonly assessed beneath a high embankment, a compound slip surface that follows a weak seam, or the effects of a retaining structure with finite width. It also does not model seismic inertia or progressive strain softening. A field sketch should show observed scarps, seepage exits, tension cracks, bedding planes and soil boundaries, then select a failure model that matches those features. A large FS for the wrong mechanism is not evidence of safety.

Bearing alongside slope stability. The simple Terzaghi φ=0 strip footing relation qult≈5.7su+q is a benchmark for homogeneous undrained general shear. It is not a general shallow foundation method covering finite rectangles, sloping ground, groundwater, inclination, punching failure or code factors. A footing on a slope may experience both inadequate local bearing and global slope failure; these are separate checks. A capacity ratio from one ideal expression should not be described as a certified overall factor of safety.

Quality of a decision. Show the model's applicable conditions, report the parameter sources, compare at least two credible pore-pressure or strength scenarios, and identify the observation most likely to reverse the conclusion. Stability calculations are only as defensible as the chosen failure surface and site characterization.

Unit 7: Soil stability

Slopes, retaining earth systems, foundations and limit equilibrium.

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 7.1 — Infinite-slope stability

Central question. A shallow planar failure parallel to the ground surface can be approximated using a slice of uniform thickness on a long slope.

Governing relationships and definitions. For a dry cohesionless slope, FS=tanφ′/tanβ. With cohesion and pore pressure, FS=[c′+(γz cos²β−u)tanφ′]/[γz sinβ cosβ].

Worked calculation / observation. For β=25°, φ′=32°, dry cohesionless FS≈tan32°/tan25°≈1.34. Saturation increases pore pressure and reduces the frictional resistance for a fixed total stress.

Why interpretation matters. A factor of safety compares a simplified available resistance with mobilized demand. It does not directly quantify failure probability or predict displacement. A result near unity motivates closer investigation; a high result does not prove a design reliable if a relevant failure mode or groundwater condition was omitted.

Field case: A road cut with shallow slip surfaces after heavy rain. In a road cut with shallow slip surfaces after heavy rain, 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 cut with shallow slip surfaces after heavy rain 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 cut with shallow slip surfaces after heavy rain. 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. This model is not suitable for deep circular failures, seismic loads, nonuniform stratigraphy, or progressive failure. An FS number is not a probability of failure. 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 7.2 — Retaining-system actions and drainage

Central question. Wall pressure is controlled by soil strength, movement, surcharge, water pressure and construction sequence.

Governing relationships and definitions. For a simple active Rankine soil component Pa=0.5KaγH²; a uniform surcharge q adds Pa,q=KaqH. Hydrostatic water adds Pw=0.5γw hw² and must not be included twice.

Worked calculation / observation. For H=4 m, Ka=0.33, γ=18 and q=15 kPa, soil thrust≈47.5 kN/m, surcharge thrust≈19.8 kN/m, excluding water. Their application heights differ.

Why interpretation matters. The infinite-slope model is a deliberately narrow view of a long shallow failure plane. Real slopes can fail along curved or compound surfaces, and layered interfaces may govern. The simplicity is useful for controlled sensitivity studies, particularly for understanding friction angle, slope angle, depth and pore pressure.

Field case: A retaining wall with uncertain drainage condition. In a retaining wall with uncertain drainage condition, 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 with uncertain drainage condition 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 with uncertain drainage condition. 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. Rigid walls at rest may have larger pressures than walls that move enough to mobilize active conditions. Drainage details and outlet maintenance are critical. 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 7.3 — Bearing-capacity concepts

Central question. A shallow foundation may reach a shear failure mechanism if contact pressure becomes too large, but serviceability may govern first.

Governing relationships and definitions. For a strip footing on level soil under centered vertical loading and a simplified Terzaghi general shear model, qult=cNc+qNq+0.5γBNγ, where q=γDf. Bearing factors depend on φ and chosen convention.

Worked calculation / observation. For φ=0° in the classic idealized model, Nc≈5.7, Nq=1 and Nγ=0, giving qult≈5.7c+q. If c=40 kPa and q=20 kPa, qult≈248 kPa.

Why interpretation matters. Retaining wall earth-pressure diagrams are actions on a structure, not a substitute for evaluating overall stability, overturning, sliding, basal bearing or internal reinforcement. Students should track each resultant magnitude and application height before performing moment equilibrium.

Field case: A strip footing near the edge of a slope. In a strip footing near the edge of a slope, 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 strip footing near the edge of a slope 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 strip footing near the edge of a slope. 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. Do not turn this classroom formula directly into an allowable footing pressure: groundwater, shape, inclination, embedment, layered soils, settlement and code-required resistance factors matter. 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 7.4 — Stability, sensitivity and engineering decisions

Central question. Because soil is spatially variable, a single factor-of-safety computation needs a record of inputs and plausible ranges.

Governing relationships and definitions. Perform one-at-a-time sensitivity, separate correlated inputs, and identify which measurement has most influence on the safety conclusion.

Worked calculation / observation. Suppose β=28° and measured φ′ spans 29°–35°: dry cohesionless FS ranges from tan29°/tan28°≈1.04 to tan35°/tan28°≈1.32. This variation is large relative to the nominal margin.

Why interpretation matters. Ground bearing and settlement are separate limit states. A footing can be well below a classroom ultimate shear capacity and still undergo unacceptable differential settlement. Conversely a small nominal bearing ratio does not establish serviceability. Real foundation decisions need stratigraphy, loading, groundwater and code-specific factors.

Field case: A buried culvert embankment with uncertain fill properties. In a buried culvert embankment with uncertain fill properties, 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 buried culvert embankment with uncertain fill properties 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 buried culvert embankment with uncertain fill properties. 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. An uncertain test parameter should not be converted into spurious precision by displaying five decimal places. Identify a field test that narrows the decision interval. 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 13: Infinite-Slope Stability. Shear versus effective normal resistance in a shallow planar slope. 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-13.html. Important: Uniform infinite slope with failure plane parallel to ground surface.

Laboratory 14: Simplified Strip Bearing Capacity. Classical φ=0 strip-footing benchmark. 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-14.html. Important: Undrained homogeneous φ=0 classroom idealization; no allowable pressure.

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: shallow slope after a storm

A long, approximately planar colluvial slope has β=26° and an estimated drained friction angle φ′=33°. For a cohesionless dry infinite-slope interpretation, FS=tan33°/tan26°≈1.33. This result is a ratio of modeled resistance to modeled sliding demand for a very specific shallow mechanism. It does not imply a one-in-three chance of failure, nor does it describe the margin for deeper rotational instability.

Suppose a 2 m-deep slip plane is adopted, with γ=19 kN/m³ and c′=4 kPa. Using the chosen vertical-depth form of the infinite-slope equation, the driving shear is γz sinβ cosβ≈14.97 kPa and total normal stress is γz cos²β≈30.70 kPa. If pore pressure at the plane is 0 kPa, the friction component is about 19.94 kPa and FS≈(4+19.94)/14.97≈1.60. If pore pressure rises to 18 kPa, that term drops to approximately 8.25 kPa and FS≈0.82. The effect comes from pore pressure reducing effective normal stress, not from reducing soil mass.

These values are purposefully sharp for a classroom sensitivity analysis, but the field pore pressure may vary along the slip surface and through time. A shallow rainfall front, perched water, cracks and heterogeneous weathering can create local pressure variations. The assumed constant values must be linked to monitoring or justified conservatively; applying a uniform pressure is an idealization.

For the associated road cutting, the next investigation might combine slope geometry survey, inclinometer interpretation and short-response piezometers. If observations suggest a deeper sliding mass or a curved scarp, the infinite-slope approximation should be replaced by a more appropriate limit-equilibrium or numerical model. The critical lesson is to recognize when the model's own geometry fails to match the observed mechanism.

Technical methods notebook: stability measures and their limits

Infinite-slope force equilibrium. Consider a unit-width slice resting on a plane parallel to a long slope of inclination β. With vertical thickness z, unit weight γ, effective cohesion c′, and pore pressure u at the plane, the commonly used vertical-depth expression gives shear demand γz sinβ cosβ and total normal stress γz cos²β. The effective normal stress is the latter minus u. The simplified factor of safety is [c′+(γz cos²β−u)tanφ′]/[γz sinβ cosβ] where effective normal stress remains physically meaningful. Every quantity in numerator and denominator is stress, so the ratio is dimensionless.

Dry cohesionless limit. If c′=0 and u=0, the common γz cosβ terms cancel, leaving FS=tanφ′/tanβ. The result is independent of z and γ for that special case, making it an excellent unit test. For β=20° and φ′=30°, FS≈1.59. For β near φ′, FS approaches 1. If β exceeds φ′, the idealized ratio falls below one. A model that responds to γ or z under those exact dry cohesionless assumptions is likely implementing the algebra incorrectly.

Pore pressure sensitivity. Take β=25°, φ′=32°, γ=19 kN/m³, z=2 m, c′=5 kPa and pore pressure u=10 kPa. The shear demand is γz sin25° cos25°≈14.56 kPa, and effective normal stress is 19×2×cos²25°−10≈21.21 kPa. Resistance is 5+21.21 tan32°≈18.25 kPa; FS≈1.25. Raising u by a further 8 kPa reduces frictional resistance by 8 tan32°≈5.00 kPa while leaving the simplified driving stress unchanged. The predicted FS therefore drops substantially. These numbers are classroom screening quantities, not a model of rainfall infiltration.

Geometry mismatch. Infinite-slope analysis cannot describe the circular arc commonly assessed beneath a high embankment, a compound slip surface that follows a weak seam, or the effects of a retaining structure with finite width. It also does not model seismic inertia or progressive strain softening. A field sketch should show observed scarps, seepage exits, tension cracks, bedding planes and soil boundaries, then select a failure model that matches those features. A large FS for the wrong mechanism is not evidence of safety.

Bearing alongside slope stability. The simple Terzaghi φ=0 strip footing relation qult≈5.7su+q is a benchmark for homogeneous undrained general shear. It is not a general shallow foundation method covering finite rectangles, sloping ground, groundwater, inclination, punching failure or code factors. A footing on a slope may experience both inadequate local bearing and global slope failure; these are separate checks. A capacity ratio from one ideal expression should not be described as a certified overall factor of safety.

Quality of a decision. Show the model's applicable conditions, report the parameter sources, compare at least two credible pore-pressure or strength scenarios, and identify the observation most likely to reverse the conclusion. Stability calculations are only as defensible as the chosen failure surface and site characterization.

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