Geostatic Stress and Induced Stress Distributions

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 6.1 — Elastic point-load stress distribution

Central question. A surface point load induces a depth- and radius-dependent vertical stress increment in an ideal elastic half-space.

Governing relationships and definitions. Boussinesq solution Δσz=(3Q/(2πz²)) [1+(r/z)²]^(−5/2) for a vertical surface point load acting on a homogeneous isotropic linear elastic half-space.

Worked calculation / observation. For Q=100 kN, z=2 m and r=0, Δσz≈11.94 kPa; at r=z=2 m the factor is 2^(−5/2), giving about 2.11 kPa.

Why interpretation matters. The Boussinesq point-load solution is a kernel for more complex loadings, not a realistic description of a footing contact patch by itself. Dividing a finite surface into small areas and integrating point effects is one way to approximate a stress bulb; numerical quadrature and edge corrections must be checked for convergence.

Field case: A shallow footing close to a buried service. In a shallow footing close to a buried service, 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 shallow footing close to a buried service 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 shallow footing close to a buried service. 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. The point-load singularity at z→0 is nonphysical. Real footings require integrating over loaded area and considering soil layering. 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 6.2 — Footing stress and influence concepts

Central question. A foundation distributes load over a finite plan area; stress bulbs intersect compressible layers at different depths.

Governing relationships and definitions. For a uniform rectangular foundation, approximate contact pressure q=Q/A if resultant is central and contact is fully compressive. A rough 2:1 distribution uses Δσz≈Q/[(B+z)(L+z)], a heuristic rather than an elastic solution.

Worked calculation / observation. For Q=1200 kN, B=2 m, L=3 m, z=2 m, 2:1 estimate Δσz=1200/(4×5)=60 kPa. Compare with a proper elastic integration before final design.

Why interpretation matters. Two foundations with the same contact pressure may create very different stress changes at depth because their widths differ. The foundation width sets an important length scale for how far load spreads, while compressible-layer depth and initial effective stress govern the settlement consequences.

Field case: A rectangular spread footing above a deep compressible seam. In a rectangular spread footing above a deep compressible seam, 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 rectangular spread footing above a deep compressible seam 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 rectangular spread footing above a deep compressible seam. 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. A footing near an excavation or with eccentric loading violates the assumed uniform and symmetric stress field. 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 6.3 — Rankine lateral earth pressure

Central question. A simple retaining-wall model relates effective vertical stress to active and passive horizontal stresses under ideal conditions.

Governing relationships and definitions. For cohesionless horizontal backfill and a vertical smooth wall with sufficient movement, Ka=tan²(45°−φ′/2), Kp=tan²(45°+φ′/2). Active thrust from soil alone Pa=0.5 Ka γ H² per unit wall length.

Worked calculation / observation. At φ′=30°, Ka≈1/3; for γ=18 kN/m³ and H=5 m, Pa=0.5×(1/3)×18×25=75 kN/m. Resultant acts at H/3 above the base for the triangular component.

Why interpretation matters. Rankine active pressure is a failure-state model. A stiff basement wall that cannot move enough to mobilize the active condition should not automatically use the active coefficient. A separate at-rest model may be more appropriate, and construction compaction can add loads not represented by the simple triangular pressure diagram.

Field case: A constrained retaining wall adjoining a property line. In a constrained retaining wall adjoining a property line, 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 constrained retaining wall adjoining a property line 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 constrained retaining wall adjoining a property line. 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. Wall friction, sloping backfill, surcharge, groundwater, compaction pressure and movement constraints require other models. Passive resistance especially needs large movement. 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 6.4 — Stress paths under loading and excavation

Central question. Loading and unloading alter total and effective stress and can rotate principal stress directions around a structure.

Governing relationships and definitions. For a saturated surface loading event, calculate Δσ′=Δσ−Δu. Excavation usually reduces total vertical stress and may cause rebound before drainage equilibrates.

Worked calculation / observation. A 50 kPa surcharge with an initial 40 kPa excess pore pressure increases effective stress by about 10 kPa immediately; after complete dissipation, the additional effective stress approaches 50 kPa if other conditions stay fixed.

Why interpretation matters. Water behind a wall changes both load and effective stress. The hydrostatic water load on the wall cannot be replaced by adding water density to the entire soil unit weight without checking what part of the skeleton stress should be evaluated with effective properties. Decomposing soil and water thrusts avoids double counting.

Field case: A staged excavation near an operating building. In a staged excavation near an operating building, 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 staged excavation near an operating building 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 staged excavation near an operating building. 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. A laboratory triaxial path may not represent a retaining-wall excavation path. Use staged construction and ground-water history in any defensible field interpretation. 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 11: Boussinesq Stress Bulb. Vertical stress under a concentrated surface load. 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-11.html. Important: Homogeneous isotropic linear elastic half-space; z strictly positive.

Laboratory 12: Rankine Earth Pressure. Active/passive coefficients and soil/surcharge thrust. 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-12.html. Important: Cohesionless level backfill, smooth vertical wall, ideal movement.

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: stress beneath a footing beside a wall

A 100 kN vertical point load acts at ground surface. At a depth of 3 m immediately below the load, the elastic half-space expression gives Δσz=3×100/(2π×3²)≈5.31 kPa. At a horizontal radius of 3 m at the same depth, multiply by [1+(3/3)²]^(−5/2)=2^(−5/2)≈0.177, yielding roughly 0.94 kPa. This rapid decay with offset is useful for intuition; it does not claim that real foundations deliver forces through a mathematical point.

A rectangular footing carrying 1,500 kN over 2.5 m×3 m has average central contact pressure 200 kPa under the uniform-contact assumption. A 2:1 approximation two metres below the base would give 1,500/[(2.5+2)(3+2)]≈66.7 kPa. Compare this heuristic with a true integration of an elastic solution before using it for detailed settlement prediction. A second footing with the same pressure but greater dimensions can influence deeper soil even when the pressure at the contact is unchanged.

Beside the foundation stands a 4 m retaining wall. For dry level cohesionless backfill with φ′=30°, Ka≈1/3. With γ=18 kN/m³, the ideal active soil thrust is 0.5×(1/3)×18×4²≈48 kN per metre of wall. A uniform 12 kPa surcharge adds (1/3)×12×4=16 kN/m. The soil triangular component acts H/3 above the base; the surcharge rectangular component acts H/2 above the base. They cannot be combined into one thrust magnitude and given an arbitrary lever arm.

A basement wall prevented from moving may not reach active conditions, and a high water table can create an additional hydrostatic action. A credible concept sketch separates water pressure, soil effective pressure, structural surcharge and construction compaction. This prevents an overly favorable result produced by choosing a coefficient that requires wall movements the structure cannot actually undergo.

Technical methods notebook: checking pressure diagrams

A point load is a mathematical idealization. For an isotropic linear elastic half-space, the Boussinesq vertical stress increment at radius r and depth z beneath a vertical surface point load Q is Δσz=3Q/(2πz²)[1+(r/z)²]⁻⁵ᐟ². Its dimensions are force divided by area, provided Q is in kN and z in metres, so the result is kPa. The r/z ratio is dimensionless. A common coding mistake is to use radial distance squared inside the power without first dividing by z². This error may be hidden at r=0, where both formulas agree.

Check the shape. The maximum Δσz at fixed z occurs at r=0; the pressure decays as lateral offset increases. At r=z, the stress equals 2⁻⁵ᐟ²≈0.177 times its value on the axis. At the same radial ratio, doubling the depth reduces stress under an unchanged point load by a factor of four. This symmetry provides two quick analytical tests for the plotted model. As z approaches zero, the stress approaches infinity at the point. The singularity announces the inadequacy of a point-load model close to a real footing contact patch.

Finite foundation areas. An actual footing distributes load over area. The stress increment below the footing can be represented as an integral of point-load contributions, but the contact pressure distribution may not be uniform when loading is eccentric or the footing is flexible. A rough 2:1 distribution method is a separate engineering approximation and should not be labeled an exact Boussinesq integral. The two approaches can diverge meaningfully near footing edges and in layered ground.

Rankine pressure from stress ratio. For cohesionless backfill, vertical smooth wall, horizontal surface and ideal active deformation, Ka=tan²(45°−φ′/2). With uniform γ, the horizontal effective stress is Kaγz at depth z, so integration from zero to wall height H produces Pa=0.5KaγH² per unit length. The triangular action acts H/3 above the base. A uniformly distributed surcharge q contributes Ka q at every depth, a rectangular diagram with action Ka q H at H/2. This separation is required when computing overturning moments.

Water pressure is a separate curve. Hydrostatic water behind a wall adds a triangular distribution γw z measured from the water surface, with its own lever arm. The soil pressure used with effective strength parameters should be based on effective stress. Adding the water diagram and an earth-pressure calculation based on saturated total stress without proper decomposition can double-count pore effects.

Alternative states. A basement wall held rigid may stay close to at-rest stress. Passive resistance requires more wall movement than active mobilization in many contexts. A design that silently chooses active on one side and fully mobilized passive on the other may be unconservative. The laboratory model is useful for learning thrust integration, but support movement, wall friction, groundwater and construction sequence must be judged separately.

Unit 6: Stress distributions

Elastic stresses, foundations, lateral pressures and stress paths.

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 6.1 — Elastic point-load stress distribution

Central question. A surface point load induces a depth- and radius-dependent vertical stress increment in an ideal elastic half-space.

Governing relationships and definitions. Boussinesq solution Δσz=(3Q/(2πz²)) [1+(r/z)²]^(−5/2) for a vertical surface point load acting on a homogeneous isotropic linear elastic half-space.

Worked calculation / observation. For Q=100 kN, z=2 m and r=0, Δσz≈11.94 kPa; at r=z=2 m the factor is 2^(−5/2), giving about 2.11 kPa.

Why interpretation matters. The Boussinesq point-load solution is a kernel for more complex loadings, not a realistic description of a footing contact patch by itself. Dividing a finite surface into small areas and integrating point effects is one way to approximate a stress bulb; numerical quadrature and edge corrections must be checked for convergence.

Field case: A shallow footing close to a buried service. In a shallow footing close to a buried service, 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 shallow footing close to a buried service 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 shallow footing close to a buried service. 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. The point-load singularity at z→0 is nonphysical. Real footings require integrating over loaded area and considering soil layering. 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 6.2 — Footing stress and influence concepts

Central question. A foundation distributes load over a finite plan area; stress bulbs intersect compressible layers at different depths.

Governing relationships and definitions. For a uniform rectangular foundation, approximate contact pressure q=Q/A if resultant is central and contact is fully compressive. A rough 2:1 distribution uses Δσz≈Q/[(B+z)(L+z)], a heuristic rather than an elastic solution.

Worked calculation / observation. For Q=1200 kN, B=2 m, L=3 m, z=2 m, 2:1 estimate Δσz=1200/(4×5)=60 kPa. Compare with a proper elastic integration before final design.

Why interpretation matters. Two foundations with the same contact pressure may create very different stress changes at depth because their widths differ. The foundation width sets an important length scale for how far load spreads, while compressible-layer depth and initial effective stress govern the settlement consequences.

Field case: A rectangular spread footing above a deep compressible seam. In a rectangular spread footing above a deep compressible seam, 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 rectangular spread footing above a deep compressible seam 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 rectangular spread footing above a deep compressible seam. 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. A footing near an excavation or with eccentric loading violates the assumed uniform and symmetric stress field. 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 6.3 — Rankine lateral earth pressure

Central question. A simple retaining-wall model relates effective vertical stress to active and passive horizontal stresses under ideal conditions.

Governing relationships and definitions. For cohesionless horizontal backfill and a vertical smooth wall with sufficient movement, Ka=tan²(45°−φ′/2), Kp=tan²(45°+φ′/2). Active thrust from soil alone Pa=0.5 Ka γ H² per unit wall length.

Worked calculation / observation. At φ′=30°, Ka≈1/3; for γ=18 kN/m³ and H=5 m, Pa=0.5×(1/3)×18×25=75 kN/m. Resultant acts at H/3 above the base for the triangular component.

Why interpretation matters. Rankine active pressure is a failure-state model. A stiff basement wall that cannot move enough to mobilize the active condition should not automatically use the active coefficient. A separate at-rest model may be more appropriate, and construction compaction can add loads not represented by the simple triangular pressure diagram.

Field case: A constrained retaining wall adjoining a property line. In a constrained retaining wall adjoining a property line, 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 constrained retaining wall adjoining a property line 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 constrained retaining wall adjoining a property line. 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. Wall friction, sloping backfill, surcharge, groundwater, compaction pressure and movement constraints require other models. Passive resistance especially needs large movement. 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 6.4 — Stress paths under loading and excavation

Central question. Loading and unloading alter total and effective stress and can rotate principal stress directions around a structure.

Governing relationships and definitions. For a saturated surface loading event, calculate Δσ′=Δσ−Δu. Excavation usually reduces total vertical stress and may cause rebound before drainage equilibrates.

Worked calculation / observation. A 50 kPa surcharge with an initial 40 kPa excess pore pressure increases effective stress by about 10 kPa immediately; after complete dissipation, the additional effective stress approaches 50 kPa if other conditions stay fixed.

Why interpretation matters. Water behind a wall changes both load and effective stress. The hydrostatic water load on the wall cannot be replaced by adding water density to the entire soil unit weight without checking what part of the skeleton stress should be evaluated with effective properties. Decomposing soil and water thrusts avoids double counting.

Field case: A staged excavation near an operating building. In a staged excavation near an operating building, 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 staged excavation near an operating building 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 staged excavation near an operating building. 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. A laboratory triaxial path may not represent a retaining-wall excavation path. Use staged construction and ground-water history in any defensible field interpretation. 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 11: Boussinesq Stress Bulb. Vertical stress under a concentrated surface load. 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-11.html. Important: Homogeneous isotropic linear elastic half-space; z strictly positive.

Laboratory 12: Rankine Earth Pressure. Active/passive coefficients and soil/surcharge thrust. 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-12.html. Important: Cohesionless level backfill, smooth vertical wall, ideal movement.

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: stress beneath a footing beside a wall

A 100 kN vertical point load acts at ground surface. At a depth of 3 m immediately below the load, the elastic half-space expression gives Δσz=3×100/(2π×3²)≈5.31 kPa. At a horizontal radius of 3 m at the same depth, multiply by [1+(3/3)²]^(−5/2)=2^(−5/2)≈0.177, yielding roughly 0.94 kPa. This rapid decay with offset is useful for intuition; it does not claim that real foundations deliver forces through a mathematical point.

A rectangular footing carrying 1,500 kN over 2.5 m×3 m has average central contact pressure 200 kPa under the uniform-contact assumption. A 2:1 approximation two metres below the base would give 1,500/[(2.5+2)(3+2)]≈66.7 kPa. Compare this heuristic with a true integration of an elastic solution before using it for detailed settlement prediction. A second footing with the same pressure but greater dimensions can influence deeper soil even when the pressure at the contact is unchanged.

Beside the foundation stands a 4 m retaining wall. For dry level cohesionless backfill with φ′=30°, Ka≈1/3. With γ=18 kN/m³, the ideal active soil thrust is 0.5×(1/3)×18×4²≈48 kN per metre of wall. A uniform 12 kPa surcharge adds (1/3)×12×4=16 kN/m. The soil triangular component acts H/3 above the base; the surcharge rectangular component acts H/2 above the base. They cannot be combined into one thrust magnitude and given an arbitrary lever arm.

A basement wall prevented from moving may not reach active conditions, and a high water table can create an additional hydrostatic action. A credible concept sketch separates water pressure, soil effective pressure, structural surcharge and construction compaction. This prevents an overly favorable result produced by choosing a coefficient that requires wall movements the structure cannot actually undergo.

Technical methods notebook: checking pressure diagrams

A point load is a mathematical idealization. For an isotropic linear elastic half-space, the Boussinesq vertical stress increment at radius r and depth z beneath a vertical surface point load Q is Δσz=3Q/(2πz²)[1+(r/z)²]⁻⁵ᐟ². Its dimensions are force divided by area, provided Q is in kN and z in metres, so the result is kPa. The r/z ratio is dimensionless. A common coding mistake is to use radial distance squared inside the power without first dividing by z². This error may be hidden at r=0, where both formulas agree.

Check the shape. The maximum Δσz at fixed z occurs at r=0; the pressure decays as lateral offset increases. At r=z, the stress equals 2⁻⁵ᐟ²≈0.177 times its value on the axis. At the same radial ratio, doubling the depth reduces stress under an unchanged point load by a factor of four. This symmetry provides two quick analytical tests for the plotted model. As z approaches zero, the stress approaches infinity at the point. The singularity announces the inadequacy of a point-load model close to a real footing contact patch.

Finite foundation areas. An actual footing distributes load over area. The stress increment below the footing can be represented as an integral of point-load contributions, but the contact pressure distribution may not be uniform when loading is eccentric or the footing is flexible. A rough 2:1 distribution method is a separate engineering approximation and should not be labeled an exact Boussinesq integral. The two approaches can diverge meaningfully near footing edges and in layered ground.

Rankine pressure from stress ratio. For cohesionless backfill, vertical smooth wall, horizontal surface and ideal active deformation, Ka=tan²(45°−φ′/2). With uniform γ, the horizontal effective stress is Kaγz at depth z, so integration from zero to wall height H produces Pa=0.5KaγH² per unit length. The triangular action acts H/3 above the base. A uniformly distributed surcharge q contributes Ka q at every depth, a rectangular diagram with action Ka q H at H/2. This separation is required when computing overturning moments.

Water pressure is a separate curve. Hydrostatic water behind a wall adds a triangular distribution γw z measured from the water surface, with its own lever arm. The soil pressure used with effective strength parameters should be based on effective stress. Adding the water diagram and an earth-pressure calculation based on saturated total stress without proper decomposition can double-count pore effects.

Alternative states. A basement wall held rigid may stay close to at-rest stress. Passive resistance requires more wall movement than active mobilization in many contexts. A design that silently chooses active on one side and fully mobilized passive on the other may be unconservative. The laboratory model is useful for learning thrust integration, but support movement, wall friction, groundwater and construction sequence must be judged separately.

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