Foundations and Earth-Retaining Elements

Learning purpose. Develop an explicit force, geometry, materials and evidence model before any final code compliance calculation. Numerical examples in this unit are educational and not an authorization to construct.

Guided investigation: foundation forces do not stop at the concrete surface

A footing works with soil, water and nearby foundations. Start from an applied column reaction and approximate the contact pressure for a centered load on a nominally rigid footing. The familiar q=P/A relation is a statement about average pressure, not a complete soil-structure model. The distribution may become nonuniform if the load is eccentric, the soil is compressible, the footing rotates or the support includes a significant horizontal moment. A designer needs geotechnical recommendations on bearing resistance and settlement that are appropriate to the actual load level.

A square isolated footing can be drawn in plan and section. Show the column, its load, the footing boundaries, the upward soil pressure and a trial effective depth. To develop one-way bending demand, isolate a cantilever strip from the column face toward the footing edge. To develop a punching free body, cut around a control perimeter and sum the soil reaction outside or inside the cut consistently. The punching control perimeter is code-defined and changes at edges, corners, openings and nonrectangular supports. Do not rely on a fixed simple perimeter as a general design.

Run a parametric study with plan widths increasing while load remains fixed. Average soil pressure falls with B², but the punching cut also moves relative to the footing boundary. A deeper footing changes the perimeter and the available shear transfer dimension. This illustrates why 'just increase the footing width' is not always a complete response to a punching concern. In real design, footing self-weight increases contact pressure, groundwater may introduce buoyancy, and site setbacks or adjacent structures limit available dimensions.

For retaining walls, add the geotechnical lateral pressure. Under a particular dry level-backfill and active Rankine idealization, pressure increases with depth; the resultant is triangular and acts above the base. But traffic surcharge, compaction, hydrostatic pressure and seismic earth action require separate treatment. The reinforced concrete stem, heel, toe and base key may face different bending signs and soil-pressure patterns. Global sliding, bearing, overturning and settlement must be reviewed alongside member capacity.

Produce a coordination sheet with separate signatures or review responsibilities for structural actions, soil investigation, water levels, excavation stability and reinforcement drawings. The concrete designer cannot derive a safe settlement criterion from a flexural moment formula. Use your foundation lab results to ask sharper questions of the soil model, not to bypass it.

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Lesson 6.1: Footing soil reaction and flexure

Physical meaning and governing ideas

A reinforced concrete footing transfers column load to soil while protecting against flexural, one-way shear and punching failure. Uniform contact pressure is a convenient centered-load approximation; real soil pressure depends on eccentricity, foundation stiffness and soil-structure response. Geotechnical bearing and settlement assessments are separate from concrete resistance checks.

Analysis and design reasoning

Combine service or factored loads with the corresponding geotechnical or strength requirement without mixing levels. Determine bearing pressure, cantilever projection from column face, moment per metre, shear sections and punching perimeter. Include footing self-weight, groundwater and uplift where important. Reinforcement continuity, cover against soil and bar development deserve the same attention as capacity.

Worked numerical or geometric illustration

A centered P=1200 kN load on a B=2.8 m square footing gives q=1200/2.8²≈153.1 kPa. For a centered 0.45 m square column, the cantilever length is (2.8−0.45)/2=1.175 m and the idealized moment is q×1.175²/2≈105.7 kN·m per metre.

Interpretation and failure checks

The load level used for a service bearing-pressure check may be different from that for structural flexure and punching checks.

Study task 6.1. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Lesson 6.2: Punching and one-way foundation shear

Physical meaning and governing ideas

Punching critical sections surround columns, walls or concentrated supports at a specified distance dependent on the governing method. One-way shear develops across a strip spanning the footing projection. Soil reaction inside a punching perimeter reduces the net upward shear crossing that perimeter in a concentric uniform-contact idealization.

Analysis and design reasoning

Draw the free body inside and outside the prospective shear perimeter. For column moments and eccentric load, soil pressure varies and punching stress may be nonuniform. Edge footings, combined footings and raft foundations require modified geometry. Check the actual governing code for allowable resistance and shear reinforcement provisions.

Worked numerical or geometric illustration

For the square footing example with d=0.5 m, the idealized concentric punching perimeter side is c+d=0.95 m, length b0=3.8 m. Net shear outside perimeter V=q(2.8²−0.95²)=about 1062 kN; average punching stress is V/(b0 d)=559 kPa. This remains demand only, not verified resistance.

Interpretation and failure checks

A footing that passes bearing may still fail punching. Enlarging plan dimensions does not necessarily reduce shear demand in the same proportion.

Study task 6.2. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Lesson 6.3: Retaining walls and lateral earth actions

Physical meaning and governing ideas

Retaining structures resist earth and water pressures, surcharge, sliding and overturning and must provide a workable drainage path. A reinforced concrete stem is usually a cantilever or propped plate; heel and toe slabs act with soil contact and base reactions. Soil parameters, construction sequence and groundwater often dominate the accuracy of a structural calculation.

Analysis and design reasoning

Start from a geotechnical lateral pressure envelope rather than assuming an arbitrary line load. For simple dry level backfill with active Rankine conditions, the lateral pressure increases linearly with depth, giving a triangular resultant. Evaluate reinforcement on the appropriate tension faces and check sliding, bearing and overall stability separately.

Worked numerical or geometric illustration

For Ka=1/3, unit weight γ=18 kN/m³ and retained height H=3 m, the base horizontal soil pressure is KaγH=18 kPa, lateral force per metre wall is KaγH²/2=27 kN/m, and base moment is KaγH³/6=27 kN·m/m, before water or surcharge.

Interpretation and failure checks

If drainage is blocked, hydrostatic pressure can change the design substantially. Never treat weep holes as a substitute for a drainage investigation.

Study task 6.3. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Lesson 6.4: Rafts, combined footings and tie beams

Physical meaning and governing ideas

Rafts and combined footings redistribute reactions among multiple supports. The soil behaves as a deformable foundation, so uniform pressure under each column is unlikely when loads and stiffness differ. Differential settlement can induce moments that a rigid isolated-footing analysis misses.

Analysis and design reasoning

Use compatible soil and structural models: a rigid resultant check for an initial sanity estimate, then a more appropriate raft-on-springs, continuum or advanced geotechnical model where project demands it. Account for construction joints, punching at each column, edge strips and the need to anchor tie beams into the supported foundations.

Worked numerical or geometric illustration

Two columns carrying 800 and 1400 kN at 4 m spacing have resultant 2200 kN acting 1400×4/2200≈2.545 m from the smaller column, ignoring footing weight. A rectangular combined foundation should place its contact-area centroid consistently with this resultant for a centered-pressure idealization.

Interpretation and failure checks

Check foundation serviceability with site-specific settlement predictions; larger concrete thickness cannot repair an unrealistic soil model.

Study task 6.4. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Numerical practice — solved and interpreted

The eight problems below vary geometry and actions within a consistent instructional model. They develop reproducible calculation habits, but do not substitute for independent code checks.

Practice scenario 6.1

Problem. A 2.15 m square footing supports 750 kN concentric load from a 0.36 m square column with effective depth 0.375 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=162.25 kPa. Projection from column face a=(B−c)/2=0.895 m, and bending demand is q a²/2=64.98 kN·m per metre width. Perimeter b0=4(c+d)=2.940 m; net outside-perimeter shear Vu=662.35 kN and mean stress v=Vu/(b0 d)=600.77 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 6.2

Problem. A 2.30 m square footing supports 850 kN concentric load from a 0.37 m square column with effective depth 0.400 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=160.68 kPa. Projection from column face a=(B−c)/2=0.965 m, and bending demand is q a²/2=74.81 kN·m per metre width. Perimeter b0=4(c+d)=3.080 m; net outside-perimeter shear Vu=754.73 kN and mean stress v=Vu/(b0 d)=612.61 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 6.3

Problem. A 2.45 m square footing supports 950 kN concentric load from a 0.38 m square column with effective depth 0.425 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=158.27 kPa. Projection from column face a=(B−c)/2=1.035 m, and bending demand is q a²/2=84.77 kN·m per metre width. Perimeter b0=4(c+d)=3.220 m; net outside-perimeter shear Vu=847.44 kN and mean stress v=Vu/(b0 d)=619.25 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 6.4

Problem. A 2.60 m square footing supports 1050 kN concentric load from a 0.39 m square column with effective depth 0.450 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=155.33 kPa. Projection from column face a=(B−c)/2=1.105 m, and bending demand is q a²/2=94.83 kN·m per metre width. Perimeter b0=4(c+d)=3.360 m; net outside-perimeter shear Vu=940.40 kN and mean stress v=Vu/(b0 d)=621.96 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 6.5

Problem. A 2.75 m square footing supports 1150 kN concentric load from a 0.40 m square column with effective depth 0.475 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=152.07 kPa. Projection from column face a=(B−c)/2=1.175 m, and bending demand is q a²/2=104.97 kN·m per metre width. Perimeter b0=4(c+d)=3.500 m; net outside-perimeter shear Vu=1033.57 kN and mean stress v=Vu/(b0 d)=621.70 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 6.6

Problem. A 2.90 m square footing supports 1250 kN concentric load from a 0.41 m square column with effective depth 0.500 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=148.63 kPa. Projection from column face a=(B−c)/2=1.245 m, and bending demand is q a²/2=115.19 kN·m per metre width. Perimeter b0=4(c+d)=3.640 m; net outside-perimeter shear Vu=1126.92 kN and mean stress v=Vu/(b0 d)=619.19 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 6.7

Problem. A 3.05 m square footing supports 1350 kN concentric load from a 0.42 m square column with effective depth 0.525 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=145.12 kPa. Projection from column face a=(B−c)/2=1.315 m, and bending demand is q a²/2=125.47 kN·m per metre width. Perimeter b0=4(c+d)=3.780 m; net outside-perimeter shear Vu=1220.40 kN and mean stress v=Vu/(b0 d)=614.97 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 6.8

Problem. A 3.20 m square footing supports 1450 kN concentric load from a 0.43 m square column with effective depth 0.550 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=141.60 kPa. Projection from column face a=(B−c)/2=1.385 m, and bending demand is q a²/2=135.81 kN·m per metre width. Perimeter b0=4(c+d)=3.920 m; net outside-perimeter shear Vu=1314.01 kN and mean stress v=Vu/(b0 d)=609.46 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Knowledge check with explanatory answers

Question 6.1

What is the distinction between soil bearing demand and concrete footing strength?

Discussion answer. Soil bearing checks examine contact pressures, geotechnical resistance and settlement at relevant load levels. Concrete footing checks examine flexural, one-way shear, punching, development, crack and durability behavior. Passing a soil pressure limit does not prove that the concrete section can transfer the column action, and passing punching does not prove acceptable settlement.

Question 6.2

Why can footing punching demand change with plan width?

Discussion answer. A wider footing decreases average pressure under the same centered load, but the punching free body excludes or includes a soil reaction area defined by the control perimeter. The net force crossing that perimeter and its perimeter length depend on width, column size and effective depth. The change is not captured by pressure alone and must be compared with proper code resistance.

Question 6.3

When does a constant soil pressure assumption fail?

Discussion answer. Eccentric loads, overturning, unequal soil stiffness, loss of contact and significant raft flexibility can cause nonuniform contact stress. The soil may be nonlinear or layered, and groundwater can influence effective stress. A rigid centered-footing approximation must be replaced or supplemented as site conditions demand.

Question 6.4

What is the implication of a blocked retaining-wall drain?

Discussion answer. Water pressure may build behind the wall in addition to earth pressure, increasing lateral force, overturning, sliding demand and possibly uplift. The stability and concrete bending checks must include a credible groundwater scenario. A drainage detail requires maintenance and constructability; it is not inherently reliable without evidence.

Question 6.5

Why should two unequal column reactions be combined by moments?

Discussion answer. The resultant load location follows equilibrium and depends on force-weighted positions, not merely the midpoint between supports. A combined footing that ignores this eccentric resultant may develop nonuniform soil pressure even under otherwise uniform conditions. For columns with force P1 at x1 and P2 at x2, xR=(P1x1+P2x2)/(P1+P2).

Unit project and experiment

A 1200 kN column reaction acts on a preliminary square isolated footing. For candidate widths of 2.4, 2.8 and 3.2 m, calculate uniform bearing pressures under a centered-load assumption. For each width, estimate cantilever moments and the net punching demand around a 450 mm square column with stated effective depth. Plot pressure and punching stress against width and explain why enlarging the plan is not a substitute for depth or reinforcement verification. Add footing and soil self-weight to a revised model, and create a separate eccentric-load sketch. Propose what the geotechnical consultant must provide: allowable or design bearing resistance, settlement, groundwater, excavation and construction constraints.

Deliverable: A transparent engineering memo with assumptions, diagrams, units, hand-checks, source references and a list of issues that need a governing-code and professional review.

Related interactive simulations

• [Footing bearing and punching tradeoff](../labs/lab-footprint.html) — adjust inputs, inspect curves, export data and explain limits of the model.

Unit 6: Foundations and earth-retaining elements

Learning purpose. Develop an explicit force, geometry, materials and evidence model before any final code compliance calculation. Numerical examples in this unit are educational and not an authorization to construct.

Guided investigation: foundation forces do not stop at the concrete surface

A footing works with soil, water and nearby foundations. Start from an applied column reaction and approximate the contact pressure for a centered load on a nominally rigid footing. The familiar q=P/A relation is a statement about average pressure, not a complete soil-structure model. The distribution may become nonuniform if the load is eccentric, the soil is compressible, the footing rotates or the support includes a significant horizontal moment. A designer needs geotechnical recommendations on bearing resistance and settlement that are appropriate to the actual load level.

A square isolated footing can be drawn in plan and section. Show the column, its load, the footing boundaries, the upward soil pressure and a trial effective depth. To develop one-way bending demand, isolate a cantilever strip from the column face toward the footing edge. To develop a punching free body, cut around a control perimeter and sum the soil reaction outside or inside the cut consistently. The punching control perimeter is code-defined and changes at edges, corners, openings and nonrectangular supports. Do not rely on a fixed simple perimeter as a general design.

Run a parametric study with plan widths increasing while load remains fixed. Average soil pressure falls with B², but the punching cut also moves relative to the footing boundary. A deeper footing changes the perimeter and the available shear transfer dimension. This illustrates why 'just increase the footing width' is not always a complete response to a punching concern. In real design, footing self-weight increases contact pressure, groundwater may introduce buoyancy, and site setbacks or adjacent structures limit available dimensions.

For retaining walls, add the geotechnical lateral pressure. Under a particular dry level-backfill and active Rankine idealization, pressure increases with depth; the resultant is triangular and acts above the base. But traffic surcharge, compaction, hydrostatic pressure and seismic earth action require separate treatment. The reinforced concrete stem, heel, toe and base key may face different bending signs and soil-pressure patterns. Global sliding, bearing, overturning and settlement must be reviewed alongside member capacity.

Produce a coordination sheet with separate signatures or review responsibilities for structural actions, soil investigation, water levels, excavation stability and reinforcement drawings. The concrete designer cannot derive a safe settlement criterion from a flexural moment formula. Use your foundation lab results to ask sharper questions of the soil model, not to bypass it.

---

Lesson 6.1: Footing soil reaction and flexure

Physical meaning and governing ideas

A reinforced concrete footing transfers column load to soil while protecting against flexural, one-way shear and punching failure. Uniform contact pressure is a convenient centered-load approximation; real soil pressure depends on eccentricity, foundation stiffness and soil-structure response. Geotechnical bearing and settlement assessments are separate from concrete resistance checks.

Analysis and design reasoning

Combine service or factored loads with the corresponding geotechnical or strength requirement without mixing levels. Determine bearing pressure, cantilever projection from column face, moment per metre, shear sections and punching perimeter. Include footing self-weight, groundwater and uplift where important. Reinforcement continuity, cover against soil and bar development deserve the same attention as capacity.

Worked numerical or geometric illustration

A centered P=1200 kN load on a B=2.8 m square footing gives q=1200/2.8²≈153.1 kPa. For a centered 0.45 m square column, the cantilever length is (2.8−0.45)/2=1.175 m and the idealized moment is q×1.175²/2≈105.7 kN·m per metre.

Interpretation and failure checks

The load level used for a service bearing-pressure check may be different from that for structural flexure and punching checks.

Study task 6.1. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Lesson 6.2: Punching and one-way foundation shear

Physical meaning and governing ideas

Punching critical sections surround columns, walls or concentrated supports at a specified distance dependent on the governing method. One-way shear develops across a strip spanning the footing projection. Soil reaction inside a punching perimeter reduces the net upward shear crossing that perimeter in a concentric uniform-contact idealization.

Analysis and design reasoning

Draw the free body inside and outside the prospective shear perimeter. For column moments and eccentric load, soil pressure varies and punching stress may be nonuniform. Edge footings, combined footings and raft foundations require modified geometry. Check the actual governing code for allowable resistance and shear reinforcement provisions.

Worked numerical or geometric illustration

For the square footing example with d=0.5 m, the idealized concentric punching perimeter side is c+d=0.95 m, length b0=3.8 m. Net shear outside perimeter V=q(2.8²−0.95²)=about 1062 kN; average punching stress is V/(b0 d)=559 kPa. This remains demand only, not verified resistance.

Interpretation and failure checks

A footing that passes bearing may still fail punching. Enlarging plan dimensions does not necessarily reduce shear demand in the same proportion.

Study task 6.2. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Lesson 6.3: Retaining walls and lateral earth actions

Physical meaning and governing ideas

Retaining structures resist earth and water pressures, surcharge, sliding and overturning and must provide a workable drainage path. A reinforced concrete stem is usually a cantilever or propped plate; heel and toe slabs act with soil contact and base reactions. Soil parameters, construction sequence and groundwater often dominate the accuracy of a structural calculation.

Analysis and design reasoning

Start from a geotechnical lateral pressure envelope rather than assuming an arbitrary line load. For simple dry level backfill with active Rankine conditions, the lateral pressure increases linearly with depth, giving a triangular resultant. Evaluate reinforcement on the appropriate tension faces and check sliding, bearing and overall stability separately.

Worked numerical or geometric illustration

For Ka=1/3, unit weight γ=18 kN/m³ and retained height H=3 m, the base horizontal soil pressure is KaγH=18 kPa, lateral force per metre wall is KaγH²/2=27 kN/m, and base moment is KaγH³/6=27 kN·m/m, before water or surcharge.

Interpretation and failure checks

If drainage is blocked, hydrostatic pressure can change the design substantially. Never treat weep holes as a substitute for a drainage investigation.

Study task 6.3. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Lesson 6.4: Rafts, combined footings and tie beams

Physical meaning and governing ideas

Rafts and combined footings redistribute reactions among multiple supports. The soil behaves as a deformable foundation, so uniform pressure under each column is unlikely when loads and stiffness differ. Differential settlement can induce moments that a rigid isolated-footing analysis misses.

Analysis and design reasoning

Use compatible soil and structural models: a rigid resultant check for an initial sanity estimate, then a more appropriate raft-on-springs, continuum or advanced geotechnical model where project demands it. Account for construction joints, punching at each column, edge strips and the need to anchor tie beams into the supported foundations.

Worked numerical or geometric illustration

Two columns carrying 800 and 1400 kN at 4 m spacing have resultant 2200 kN acting 1400×4/2200≈2.545 m from the smaller column, ignoring footing weight. A rectangular combined foundation should place its contact-area centroid consistently with this resultant for a centered-pressure idealization.

Interpretation and failure checks

Check foundation serviceability with site-specific settlement predictions; larger concrete thickness cannot repair an unrealistic soil model.

Study task 6.4. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

---

Numerical practice — solved and interpreted

The eight problems below vary geometry and actions within a consistent instructional model. They develop reproducible calculation habits, but do not substitute for independent code checks.

Practice scenario 6.1

Problem. A 2.15 m square footing supports 750 kN concentric load from a 0.36 m square column with effective depth 0.375 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=162.25 kPa. Projection from column face a=(B−c)/2=0.895 m, and bending demand is q a²/2=64.98 kN·m per metre width. Perimeter b0=4(c+d)=2.940 m; net outside-perimeter shear Vu=662.35 kN and mean stress v=Vu/(b0 d)=600.77 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 6.2

Problem. A 2.30 m square footing supports 850 kN concentric load from a 0.37 m square column with effective depth 0.400 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=160.68 kPa. Projection from column face a=(B−c)/2=0.965 m, and bending demand is q a²/2=74.81 kN·m per metre width. Perimeter b0=4(c+d)=3.080 m; net outside-perimeter shear Vu=754.73 kN and mean stress v=Vu/(b0 d)=612.61 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 6.3

Problem. A 2.45 m square footing supports 950 kN concentric load from a 0.38 m square column with effective depth 0.425 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=158.27 kPa. Projection from column face a=(B−c)/2=1.035 m, and bending demand is q a²/2=84.77 kN·m per metre width. Perimeter b0=4(c+d)=3.220 m; net outside-perimeter shear Vu=847.44 kN and mean stress v=Vu/(b0 d)=619.25 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 6.4

Problem. A 2.60 m square footing supports 1050 kN concentric load from a 0.39 m square column with effective depth 0.450 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=155.33 kPa. Projection from column face a=(B−c)/2=1.105 m, and bending demand is q a²/2=94.83 kN·m per metre width. Perimeter b0=4(c+d)=3.360 m; net outside-perimeter shear Vu=940.40 kN and mean stress v=Vu/(b0 d)=621.96 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 6.5

Problem. A 2.75 m square footing supports 1150 kN concentric load from a 0.40 m square column with effective depth 0.475 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=152.07 kPa. Projection from column face a=(B−c)/2=1.175 m, and bending demand is q a²/2=104.97 kN·m per metre width. Perimeter b0=4(c+d)=3.500 m; net outside-perimeter shear Vu=1033.57 kN and mean stress v=Vu/(b0 d)=621.70 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 6.6

Problem. A 2.90 m square footing supports 1250 kN concentric load from a 0.41 m square column with effective depth 0.500 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=148.63 kPa. Projection from column face a=(B−c)/2=1.245 m, and bending demand is q a²/2=115.19 kN·m per metre width. Perimeter b0=4(c+d)=3.640 m; net outside-perimeter shear Vu=1126.92 kN and mean stress v=Vu/(b0 d)=619.19 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 6.7

Problem. A 3.05 m square footing supports 1350 kN concentric load from a 0.42 m square column with effective depth 0.525 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=145.12 kPa. Projection from column face a=(B−c)/2=1.315 m, and bending demand is q a²/2=125.47 kN·m per metre width. Perimeter b0=4(c+d)=3.780 m; net outside-perimeter shear Vu=1220.40 kN and mean stress v=Vu/(b0 d)=614.97 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 6.8

Problem. A 3.20 m square footing supports 1450 kN concentric load from a 0.43 m square column with effective depth 0.550 m. Find average pressure, cantilever bending demand and idealized mean punching stress.

Worked resolution. Average pressure is P/B²=141.60 kPa. Projection from column face a=(B−c)/2=1.385 m, and bending demand is q a²/2=135.81 kN·m per metre width. Perimeter b0=4(c+d)=3.920 m; net outside-perimeter shear Vu=1314.01 kN and mean stress v=Vu/(b0 d)=609.46 kPa.

Engineering interpretation. Do not interpret v as an allowable stress. Subsoil resistance, settlement, moment-transfer eccentricity and the governing concrete punching resistance require independent checks.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Knowledge check with explanatory answers

Question 6.1

What is the distinction between soil bearing demand and concrete footing strength?

Discussion answer. Soil bearing checks examine contact pressures, geotechnical resistance and settlement at relevant load levels. Concrete footing checks examine flexural, one-way shear, punching, development, crack and durability behavior. Passing a soil pressure limit does not prove that the concrete section can transfer the column action, and passing punching does not prove acceptable settlement.

Question 6.2

Why can footing punching demand change with plan width?

Discussion answer. A wider footing decreases average pressure under the same centered load, but the punching free body excludes or includes a soil reaction area defined by the control perimeter. The net force crossing that perimeter and its perimeter length depend on width, column size and effective depth. The change is not captured by pressure alone and must be compared with proper code resistance.

Question 6.3

When does a constant soil pressure assumption fail?

Discussion answer. Eccentric loads, overturning, unequal soil stiffness, loss of contact and significant raft flexibility can cause nonuniform contact stress. The soil may be nonlinear or layered, and groundwater can influence effective stress. A rigid centered-footing approximation must be replaced or supplemented as site conditions demand.

Question 6.4

What is the implication of a blocked retaining-wall drain?

Discussion answer. Water pressure may build behind the wall in addition to earth pressure, increasing lateral force, overturning, sliding demand and possibly uplift. The stability and concrete bending checks must include a credible groundwater scenario. A drainage detail requires maintenance and constructability; it is not inherently reliable without evidence.

Question 6.5

Why should two unequal column reactions be combined by moments?

Discussion answer. The resultant load location follows equilibrium and depends on force-weighted positions, not merely the midpoint between supports. A combined footing that ignores this eccentric resultant may develop nonuniform soil pressure even under otherwise uniform conditions. For columns with force P1 at x1 and P2 at x2, xR=(P1x1+P2x2)/(P1+P2).

Unit project and experiment

A 1200 kN column reaction acts on a preliminary square isolated footing. For candidate widths of 2.4, 2.8 and 3.2 m, calculate uniform bearing pressures under a centered-load assumption. For each width, estimate cantilever moments and the net punching demand around a 450 mm square column with stated effective depth. Plot pressure and punching stress against width and explain why enlarging the plan is not a substitute for depth or reinforcement verification. Add footing and soil self-weight to a revised model, and create a separate eccentric-load sketch. Propose what the geotechnical consultant must provide: allowable or design bearing resistance, settlement, groundwater, excavation and construction constraints.

Deliverable: A transparent engineering memo with assumptions, diagrams, units, hand-checks, source references and a list of issues that need a governing-code and professional review.

Related interactive simulations

• [Footing bearing and punching tradeoff](labs/lab-footprint.html) — adjust inputs, inspect curves, export data and explain limits of the model.