Shallow Foundations: Bearing Capacity and Settlement Criteria

Bearing capacity, contact pressure and settlement

Footings are deceptively simple objects: a concrete pad and a patch of soil. Yet a footing is an interface where structurally determined forces meet soils with path-dependent, nonlinear, spatially variable response. The two main families of performance questions are resistance to ground failure and tolerable movement under repeated service loads. These are related but not interchangeable. The bearing mechanism develops through a volume of soil, while differential settlement reflects stiffness variations across the foundation group and within the depth of influence.

Contact pressure is not necessarily uniform. A load with moment shifts the resultant and changes the contact region. If the calculated linear elastic distribution contains tension, the assumption of full contact is wrong. Gravity cannot pull soil into the footing, so compression-only behavior must be respected. The foundation can also be subject to uplift, sliding or rotation, particularly when structures experience wind, earthquake or transient equipment actions. A complete design therefore uses a consistent suite of actions and ground conditions for each limit state.

A common error is to treat a modulus estimate as equivalent to a measured soil constant. Real soil stiffness varies with confinement, strain, drainage and sampling disturbance. A nominal elastic calculation can still teach important scale relationships: larger pressure raises movement, greater stiffness reduces it, and greater footing width mobilizes a deeper influence zone. The model becomes useful when the designer acknowledges that it is one part of a settlement assessment rather than a surrogate for investigation.

When comparing footing options, the engineering rationale should include excavation stability, temporary groundwater control, footing interference, nearby property foundations, construction tolerance and inspection. Soil that passes a laboratory strength test may still create a field risk if its continuity is poorly known. This is where the investigation, loading, bearing and settlement lessons meet.

Case integration: loading beneath a machine foundation. A small production hall contains a machine whose vertical weight is modest relative to the building but which imposes cyclic force and can be relocated. A preliminary footing has adequate nominal bearing resistance under static gravity loads. That conclusion does not establish vibration serviceability, nor does it account for eccentric column loads and the behavior of soil under repeated small strains. The foundation engineer needs the machine's dynamic loading frequency and expected response, building frame stiffness, groundwater condition and support layout. Construct two analyses: one based on the simplified static bearing-pressure model and one examining sensitivity of immediate movement and frequency-dependent stiffness to soil properties. Consider a soft seam beneath one corner and its effect on tilt. Do not claim that a static modulus or isolated footing calculator supplies a complete dynamic soil–structure interaction result. Before changing the footing dimensions, consider whether more testing, a different machine support arrangement or an isolated foundation is needed. This example illustrates why a foundation model should be selected from the performance objective rather than from whatever calculation package is available.

Four lessons and their applications

Lesson 2.1 — Footings as load-transfer systems

Why this mechanism matters

A spread footing transfers column or wall actions through contact pressure into deformable ground. Its behavior depends on foundation geometry, installation depth, soil strength, water regime, load inclination, and time. Engineers must distinguish ultimate bearing resistance from the service deformation limit. A footing can have ample apparent resistance yet experience differential settlement that damages the supported frame. Local shear, punching and general shear require different conceptual models.

Evidence, scales and the engineering model

The choice of model changes the interpretation of footings as load-transfer systems: a single observation establishes a condition at a place and time, while an engineering prediction requires a link from that observation to a repeatable physical mechanism. Establish the relevant load or environmental scenario, then identify whether the response is governed by strength, deformation, pore pressure, geometry, or construction sequence. The question to ask before entering numerical values is not simply which equation applies? but which field evidence could make the equation inapplicable? The answer often determines whether the apparent precision of an output is justified.

Resistance mobilization. Maximum resistance depends on compatibility of movement and mechanism. Different components of an embedded foundation may require different displacements to mobilize. For this lesson, make the assumed boundary conditions visible in a section drawing and distinguish measured inputs from correlations. A parameter that is merely convenient should not be presented as if a project investigation had confirmed it.

A useful field-to-model workflow is to identify an observed quantity, the geological or structural mechanism, the chosen constitutive assumption, the calculation, and a practical verification measurement. If the observation cannot distinguish two mechanisms, carry both possibilities into the sensitivity analysis. The reported result should preserve units, stress convention, drainage condition, stage and uncertainty bounds. It must be possible to trace a final design recommendation back to the underlying evidence.

Worked case 5 — method and interpretation

Given. A wall footing is 2.0 m wide and embedded 1.0 m in uniform drained sand with γ=18 kN/m³; let Nq=18.4 and Nγ=22 for an explicitly assumed factor set.

Solution. The illustrative bearing sum is γDNq+0.5γBNγ = 18×1×18.4 + 0.5×18×2×22 = 727.2 kPa.

Interpretation. This is a nominal ultimate idealization, not allowable stress; groundwater, inclination and local bearing failure were omitted.

The numerical answer above uses deliberately narrow assumptions to make the dependency transparent. If any assumption is relaxed, the expression may need a different correction or an altogether different mechanical model. Before applying the result to another footprint, soil layer or construction state, ask whether the test procedure and the selected parameters represent the new condition. Check dimensional consistency and confirm that the direction of a predicted change agrees with the underlying physics.

Deeper technical application

Diagnostic study — choosing a bearing failure model. Classical bearing equations represent an idealized mechanism around a foundation, often with assumptions of uniform soil and manageable geometry. The friction-angle factors rise rapidly, which makes a small parameter change consequential. If a value of φ comes from a loose sand layer but the same model assumes dense-sand general shear, the internal physical story is inconsistent. A wide footing may mobilize a deeper stress bulb than a narrow pad and therefore encounter a different layer. Groundwater lowers effective unit weight in relevant parts of the mechanism. Design should compare plausible depth and width combinations rather than treating a single equation output as definitive.

Decision exercise. Use the nominal strip-footing expression once with φ=30° and once with φ=28° while holding unit weight, width and depth constant. Calculate the bearing factors consistently from the chosen convention and quantify the percentage loss in resistance. Then introduce a soft layer within the depth of influence: the homogeneous assumption is no longer guaranteed, and an alternative layered analysis may govern even if the first nominal pressure looks large. Explain why comparing q_ult/3 with service contact pressure is only an illustrative teaching calculation, not necessarily the governing code format. Prepare a table separating measured field conditions, correlated parameters and corrections that have not yet been considered. Where construction excavation loosens the founding horizon, the assumed φ may be invalid after placement.

Counterfactual test

In this lesson, begin by writing the input assumption in a calculation sheet and introduce one perturbation at a time. Consider a 20% increase in the most uncertain load, a plausible change in groundwater, and a different depth or stiffness of a critical layer. Do not automatically multiply every output by the same factor: soil response is often nonlinear and the mechanism itself can change. Record which sensitivity is purely numerical and which requires more field data. If the decision to use shallow versus deep foundations changes, that is evidence that additional investigation may have a higher value than further decimal places.

Field practice and failure-first review

Practical action. Sketch the complete load path and governing load cases, including moments and uplift. Prioritize direct observations whenever a correlation could conceal a critical stratum or groundwater path. Document who reviews anomalies, whether repeat tests are required, and which construction stage is exposed. Failure trap: Bearing failure and unacceptable settlement are separate limit states.

Make a brief alternative hypothesis for the same observations. Ask whether a weak pocket, high water level, unrecognized load eccentricity, or installation defect might explain an unexpectedly large deformation. The right response is not always more conservative material strength; sometimes it is a different foundation system, a modified construction sequence, or a targeted test that resolves the uncertainty. Protect neighboring structures and utilities when the mechanism can propagate outside the footprint.

Study questions

1. Explain the difference between a measured site observation and an inferred parameter in the context of footings as load-transfer systems. Which information would you insist on before approving a preliminary calculation?
2. Rework the example after increasing the most influential input by 20%. Show units and explain whether a linear response is physically justified.
3. Sketch resistance and displacement, separating load-transfer components.
4. Give one mode of failure or serviceability loss that the worked equation cannot detect, and propose a corresponding observation or laboratory test.
5. With the two unit laboratories, Strip Footing Bearing Capacity and Footing Pressure Under Eccentric Load, describe one experiment that would reveal the limitations of an oversimplified model.

Lesson 2.2 — Bearing resistance and drainage

Why this mechanism matters

Classical bearing equations superpose cohesion, surcharge and unit-weight terms for idealized soil, geometry and failure mode. Terzaghi strip-footing form is q_ult=cNc+qNq+0.5 gamma B Ngamma, where q=gamma D in a simplified dry uniform profile. In clays the short-term undrained condition may use undrained strength; sands require an effective-stress and drainage interpretation. Shape, depth, load inclination, groundwater and layered soils demand corrections or different methods.

Evidence, scales and the engineering model

The choice of model changes the interpretation of bearing resistance and drainage: a single observation establishes a condition at a place and time, while an engineering prediction requires a link from that observation to a repeatable physical mechanism. Establish the relevant load or environmental scenario, then identify whether the response is governed by strength, deformation, pore pressure, geometry, or construction sequence. The question to ask before entering numerical values is not simply which equation applies? but which field evidence could make the equation inapplicable? The answer often determines whether the apparent precision of an output is justified.

Testing and response. Define how field observations will validate model assumptions and what a failing test will trigger. Inspection without decision thresholds is documentation, not control. For this lesson, make the assumed boundary conditions visible in a section drawing and distinguish measured inputs from correlations. A parameter that is merely convenient should not be presented as if a project investigation had confirmed it.

A useful field-to-model workflow is to identify an observed quantity, the geological or structural mechanism, the chosen constitutive assumption, the calculation, and a practical verification measurement. If the observation cannot distinguish two mechanisms, carry both possibilities into the sensitivity analysis. The reported result should preserve units, stress convention, drainage condition, stage and uncertainty bounds. It must be possible to trace a final design recommendation back to the underlying evidence.

Worked case 6 — method and interpretation

Given. A 3.0 m by 4.0 m footing carries 1,200 kN with eccentricity e = 0.25 m along the 4 m side.

Solution. Average pressure = 1200/12 = 100 kPa; B/6=0.667 m, so full contact is possible; qmax=100(1+6×0.25/4)=137.5 kPa and qmin=62.5 kPa.

Interpretation. The relevant dimension is the direction of eccentricity; check that lateral force and structural rigidity assumptions are compatible.

The numerical answer above uses deliberately narrow assumptions to make the dependency transparent. If any assumption is relaxed, the expression may need a different correction or an altogether different mechanical model. Before applying the result to another footprint, soil layer or construction state, ask whether the test procedure and the selected parameters represent the new condition. Check dimensional consistency and confirm that the direction of a predicted change agrees with the underlying physics.

Deeper technical application

Diagnostic study — loss of contact under moment. In the simplified rectangular footing diagram, the resultant shifts from the center as overturning moment grows. When its eccentricity is within the middle third, a linear pressure distribution can remain wholly compressive under the chosen rigid-footing idealization. As the resultant moves outside the middle third, the linear full-contact prediction contains tensile soil pressure and becomes physically inadmissible. The compression block then changes geometry; a triangular diagram is one plausible first approximation for a rigid base. None of these sketches proves that soil beneath the toe remains stable under extreme local pressure. Connection stiffness, uplift anchors, cyclic reversal and two-axis moment create further mechanisms.

Decision exercise. Begin with the 3 m × 4 m base, 1,200 kN force and 0.25 m eccentricity. Now increase e to 0.85 m along the 4 m dimension. Compute the compressed contact length a=3(B/2−e)=3.45 m and qmax≈2×1200/(3×3.45)=232 kPa, while qmin becomes zero outside contact. Notice that the peak pressure has increased sharply despite unchanged vertical force. Compare this with the original linear contact pressure, then ask whether a structural footing really behaves as the assumed rigid body. The correct next step is not to extend a linear equation beyond its domain but to select a contact-consistent model and check all associated sliding, rotation and resistance modes.

Counterfactual test

In this lesson, begin by writing the input assumption in a calculation sheet and introduce one perturbation at a time. Consider a 20% increase in the most uncertain load, a plausible change in groundwater, and a different depth or stiffness of a critical layer. Do not automatically multiply every output by the same factor: soil response is often nonlinear and the mechanism itself can change. Record which sensitivity is purely numerical and which requires more field data. If the decision to use shallow versus deep foundations changes, that is evidence that additional investigation may have a higher value than further decimal places.

Field practice and failure-first review

Practical action. Check the compatibility of strength parameters, stress basis and chosen bearing equation. Prioritize direct observations whenever a correlation could conceal a critical stratum or groundwater path. Document who reviews anomalies, whether repeat tests are required, and which construction stage is exposed. Failure trap: Never combine total-stress cohesion with effective-stress friction angle casually.

Make a brief alternative hypothesis for the same observations. Ask whether a weak pocket, high water level, unrecognized load eccentricity, or installation defect might explain an unexpectedly large deformation. The right response is not always more conservative material strength; sometimes it is a different foundation system, a modified construction sequence, or a targeted test that resolves the uncertainty. Protect neighboring structures and utilities when the mechanism can propagate outside the footprint.

Study questions

1. Explain the difference between a measured site observation and an inferred parameter in the context of bearing resistance and drainage. Which information would you insist on before approving a preliminary calculation?
2. Rework the example after increasing the most influential input by 20%. Show units and explain whether a linear response is physically justified.
3. Write a hold point and a concrete action for a field result outside the assumed range.
4. Give one mode of failure or serviceability loss that the worked equation cannot detect, and propose a corresponding observation or laboratory test.
5. With the two unit laboratories, Strip Footing Bearing Capacity and Footing Pressure Under Eccentric Load, describe one experiment that would reveal the limitations of an oversimplified model.

Lesson 2.3 — Eccentric loads and contact

Why this mechanism matters

A footing under moment has a nonuniform contact pressure. In a simplified elastic fully contacting rectangular base, the pressure can be represented by N/A times one plus or minus six e/B about the relevant axis. When e exceeds B/6, that linear full-contact expression predicts tension, which soil contact cannot sustain; a compression-only distribution is needed. Two-axis eccentricity, overturning and lateral actions demand additional checks.

Evidence, scales and the engineering model

The choice of model changes the interpretation of eccentric loads and contact: a single observation establishes a condition at a place and time, while an engineering prediction requires a link from that observation to a repeatable physical mechanism. Establish the relevant load or environmental scenario, then identify whether the response is governed by strength, deformation, pore pressure, geometry, or construction sequence. The question to ask before entering numerical values is not simply which equation applies? but which field evidence could make the equation inapplicable? The answer often determines whether the apparent precision of an output is justified.

Mechanism of improvement. Treat water, soil fabric and load paths as separate levers. A technique that accelerates drainage may not reduce ultimate compression. For this lesson, make the assumed boundary conditions visible in a section drawing and distinguish measured inputs from correlations. A parameter that is merely convenient should not be presented as if a project investigation had confirmed it.

A useful field-to-model workflow is to identify an observed quantity, the geological or structural mechanism, the chosen constitutive assumption, the calculation, and a practical verification measurement. If the observation cannot distinguish two mechanisms, carry both possibilities into the sensitivity analysis. The reported result should preserve units, stress convention, drainage condition, stage and uncertainty bounds. It must be possible to trace a final design recommendation back to the underlying evidence.

Worked case 7 — method and interpretation

Given. A 2.5 m wide footing receives net q=150 kPa on soil with E=25 MPa, ν=0.3 and influence factor 1.12.

Solution. s=qB(1−ν²)I/E =150×2.5×0.91×1.12/25000=0.0153 m, approximately 15.3 mm.

Interpretation. Elastic settlement cannot represent long-term clay consolidation without an additional component.

The numerical answer above uses deliberately narrow assumptions to make the dependency transparent. If any assumption is relaxed, the expression may need a different correction or an altogether different mechanical model. Before applying the result to another footprint, soil layer or construction state, ask whether the test procedure and the selected parameters represent the new condition. Check dimensional consistency and confirm that the direction of a predicted change agrees with the underlying physics.

Deeper technical application

Diagnostic study — modulus is not a universal soil constant. The immediate-settlement expression can be derived from linear-elastic strain scaling, where q/E is dimensionless and multiplication by footing width gives a displacement. It is therefore a useful model for examining units and sensitivity. Its weakness is that a real soil modulus depends on confinement, strain amplitude, loading history and drainage. A plate-load test, CPT correlation and laboratory triaxial result may sample different conditions and provide different moduli without any of them being simply erroneous. A foundation spanning two layers cannot be assumed to respond as if one homogeneous half-space extended indefinitely.

Decision exercise. For q=150 kPa, B=2.5 m and E=25 MPa, the example gives roughly 15 mm immediate displacement. Hold pressure and geometry fixed and compare moduli 15, 25 and 40 MPa; the ideal elastic formula predicts approximately 25.5, 15.3 and 9.6 mm. Report that variation as a sensitivity range, not as a statistical confidence interval. Add a consolidating clay layer below the assumed elastic horizon and discuss how long-term movement would be assembled separately. Sketch settlement at two neighboring columns whose soil profiles differ, because relative displacement rather than average settlement can control damage. State what tests would improve both short-term stiffness and long-term compression estimates.

Counterfactual test

In this lesson, begin by writing the input assumption in a calculation sheet and introduce one perturbation at a time. Consider a 20% increase in the most uncertain load, a plausible change in groundwater, and a different depth or stiffness of a critical layer. Do not automatically multiply every output by the same factor: soil response is often nonlinear and the mechanism itself can change. Record which sensitivity is purely numerical and which requires more field data. If the decision to use shallow versus deep foundations changes, that is evidence that additional investigation may have a higher value than further decimal places.

Field practice and failure-first review

Practical action. Draw both the assumed distribution and the true admissible compression region. Prioritize direct observations whenever a correlation could conceal a critical stratum or groundwater path. Document who reviews anomalies, whether repeat tests are required, and which construction stage is exposed. Failure trap: The middle-third rule is a contact condition, not proof of overall stability.

Make a brief alternative hypothesis for the same observations. Ask whether a weak pocket, high water level, unrecognized load eccentricity, or installation defect might explain an unexpectedly large deformation. The right response is not always more conservative material strength; sometimes it is a different foundation system, a modified construction sequence, or a targeted test that resolves the uncertainty. Protect neighboring structures and utilities when the mechanism can propagate outside the footprint.

Study questions

1. Explain the difference between a measured site observation and an inferred parameter in the context of eccentric loads and contact. Which information would you insist on before approving a preliminary calculation?
2. Rework the example after increasing the most influential input by 20%. Show units and explain whether a linear response is physically justified.
3. Show which input to the foundation model changes after treatment and which does not.
4. Give one mode of failure or serviceability loss that the worked equation cannot detect, and propose a corresponding observation or laboratory test.
5. With the two unit laboratories, Strip Footing Bearing Capacity and Footing Pressure Under Eccentric Load, describe one experiment that would reveal the limitations of an oversimplified model.

Lesson 2.4 — Immediate and time-dependent settlement

Why this mechanism matters

Settlement can contain immediate distortion, consolidation and longer-term creep. Simplified elastic solutions use soil modulus and Poisson ratio with a footprint influence factor, but a soil modulus is dependent on strain level, drainage, stress path and test method. Layered integration often improves transparency. Differential settlement matters to buildings, while service time and construction sequence alter the response of fine-grained ground.

Evidence, scales and the engineering model

The choice of model changes the interpretation of immediate and time-dependent settlement: a single observation establishes a condition at a place and time, while an engineering prediction requires a link from that observation to a repeatable physical mechanism. Establish the relevant load or environmental scenario, then identify whether the response is governed by strength, deformation, pore pressure, geometry, or construction sequence. The question to ask before entering numerical values is not simply which equation applies? but which field evidence could make the equation inapplicable? The answer often determines whether the apparent precision of an output is justified.

Limits and communication. Use explicit design basis, sensitivity study and independent check. Charts are decision aids whose input assumptions should be inspectable. For this lesson, make the assumed boundary conditions visible in a section drawing and distinguish measured inputs from correlations. A parameter that is merely convenient should not be presented as if a project investigation had confirmed it.

A useful field-to-model workflow is to identify an observed quantity, the geological or structural mechanism, the chosen constitutive assumption, the calculation, and a practical verification measurement. If the observation cannot distinguish two mechanisms, carry both possibilities into the sensitivity analysis. The reported result should preserve units, stress convention, drainage condition, stage and uncertainty bounds. It must be possible to trace a final design recommendation back to the underlying evidence.

Worked case 8 — method and interpretation

Given. Four equal columns each carry 800 kN. The available raft area is 20 m×12 m.

Solution. Total column force=3,200 kN; the area-average applied reaction would be 3200/240=13.33 kPa, ignoring raft self-weight.

Interpretation. The local pressure distribution is not uniform; check uplift, stiffness, column concentrations and groundwater buoyancy.

The numerical answer above uses deliberately narrow assumptions to make the dependency transparent. If any assumption is relaxed, the expression may need a different correction or an altogether different mechanical model. Before applying the result to another footprint, soil layer or construction state, ask whether the test procedure and the selected parameters represent the new condition. Check dimensional consistency and confirm that the direction of a predicted change agrees with the underlying physics.

Deeper technical application

Diagnostic study — load cases precede footing area. A foundation plan contains several force paths: permanent structural weight, occupancy or equipment loads, lateral forces and potentially uplift. The area-average base pressure is a convenient diagnostic but not a complete contact analysis. Column reactions can change with frame stiffness and loading sequence; the self-weight of the footing and backfill also influences net versus gross pressure. Adjacent pads can have overlapping stress zones, and excavating one foundation can disturb the soil supporting another. A large nominal area is therefore neither proof of low movement nor proof of acceptable resistance.

Decision exercise. For four 800 kN columns and a 20 m × 12 m raft, calculate the 13.3 kPa simple average and add a fictional uniform raft weight of 8 kPa. The new gross average is 21.3 kPa, but net pressure relative to excavated overburden may differ. Determine the resultant position for an asymmetric column pattern and show why an average cannot satisfy moment equilibrium. Next compare a stand-alone footing under each column with an integrated raft: identify potential differences in differential settlement, excavation logistics, punching reinforcement, and groundwater control. Prepare a decision table for a low-water case and a high-water case. A solution that appears economical in concrete alone may be unbuildable if dewatering causes adjacent structures to settle.

Counterfactual test

In this lesson, begin by writing the input assumption in a calculation sheet and introduce one perturbation at a time. Consider a 20% increase in the most uncertain load, a plausible change in groundwater, and a different depth or stiffness of a critical layer. Do not automatically multiply every output by the same factor: soil response is often nonlinear and the mechanism itself can change. Record which sensitivity is purely numerical and which requires more field data. If the decision to use shallow versus deep foundations changes, that is evidence that additional investigation may have a higher value than further decimal places.

Field practice and failure-first review

Practical action. Calculate a range rather than a false single-value prediction; identify the sensitivity to modulus and water level. Prioritize direct observations whenever a correlation could conceal a critical stratum or groundwater path. Document who reviews anomalies, whether repeat tests are required, and which construction stage is exposed. Failure trap: A small computed settlement is only as reliable as the selected deformation parameters.

Make a brief alternative hypothesis for the same observations. Ask whether a weak pocket, high water level, unrecognized load eccentricity, or installation defect might explain an unexpectedly large deformation. The right response is not always more conservative material strength; sometimes it is a different foundation system, a modified construction sequence, or a targeted test that resolves the uncertainty. Protect neighboring structures and utilities when the mechanism can propagate outside the footprint.

Study questions

1. Explain the difference between a measured site observation and an inferred parameter in the context of immediate and time-dependent settlement. Which information would you insist on before approving a preliminary calculation?
2. Rework the example after increasing the most influential input by 20%. Show units and explain whether a linear response is physically justified.
3. State the observation that would make the current recommendation unacceptable.
4. Give one mode of failure or serviceability loss that the worked equation cannot detect, and propose a corresponding observation or laboratory test.
5. With the two unit laboratories, Strip Footing Bearing Capacity and Footing Pressure Under Eccentric Load, describe one experiment that would reveal the limitations of an oversimplified model.

Unit 2 synthesis

This unit connects its four lessons through one overarching design decision: the right model follows from the mechanism and evidence, not from a convenient formula. Prepare a one-page design-basis table listing assumptions, parameter origin, uncertainty, sensitivity, and the condition requiring professional escalation. Pair that table with a labeled cross-section. For the investigation or structure in your own scenario, justify whether the controlling concern is ultimate resistance, unacceptable movement, unexpected groundwater, temporary construction loading or interaction with neighbors. Do not confuse a numerical check with an approval; identify the external standard and reviewer required for a real project.

Paired simulations: [Lab 03](../labs/lab-03.html) and [Lab 04](../labs/lab-04.html).

Assessment: [Unit 02 questions](../assessments/unit-02.html).

Project: Footing alternatives under eccentric loading — Compare two rectangular spread footings supporting synthetic column loads 800 kN and 1,400 kN. Include area, resultant eccentricity, assumed contact distribution, simplified bearing margins and elastic settlement sensitivity. Identify the data required before either preliminary option becomes defensible.