Shallow Foundations and Rafts

Introduction

Shallow foundations transfer structural actions to soil at depths that are small relative to their plan dimensions. They include isolated footings, strip footings, combined footings and raft foundations. Their design is governed not merely by ultimate bearing capacity but by a coordinated assessment of bearing resistance, sliding, overturning, uplift, total and differential settlement, groundwater, structural strength and soil–structure interaction.

A technically safe foundation may still be unserviceable if differential settlement damages partitions, façades, utilities or sensitive equipment. Conversely, excessive conservatism can increase excavation, concrete consumption and embodied carbon without improving performance. Modern practice therefore uses limit-state design supported by a suitable ground model and explicit consideration of uncertainty.

Learning Objectives

After studying this chapter, the reader should be able to:

• select an appropriate shallow-foundation system;
• calculate bearing resistance under vertical, eccentric and inclined loading;
• determine contact pressures and identify partial contact;
• evaluate groundwater effects on resistance, uplift and sliding;
• estimate immediate and consolidation settlement;
• distinguish ultimate and serviceability limit states;
• apply a clearly defined limit-state design approach;
• assess raft behaviour using rigid, flexible and numerical models; and
• interpret geotechnical calculations within professional and regulatory practice.

1. Foundation Types and Selection

1.1 Principal forms

An isolated footing supports one column or pier. A strip footing supports a wall or closely spaced columns. A combined footing supports two or more columns when isolated footings would overlap or when a property boundary prevents concentric loading. A raft is a large slab supporting many columns or walls over most or all of the building footprint.

A raft is often appropriate when:

• individual footings would occupy a substantial proportion of the plan area;
• allowable settlement, rather than bearing resistance, governs;
• columns are closely spaced or heavily loaded;
• variable ground conditions require structural load redistribution;
• a basement slab can also function as the foundation; or
• a compensated foundation can reduce net ground pressure.

Rafts do not eliminate differential settlement. Their stiffness redistributes contact pressure and may reduce distortion, but performance still depends on soil variability, loading pattern, construction sequence and raft rigidity.

1.2 Ground model

The design ground model should identify strata, groundwater regimes, characteristic soil parameters, weak layers, compressible zones and credible spatial variability. Parameters for bearing resistance may differ from those used for settlement. For example, drained strength parameters may govern long-term bearing capacity, whereas operational stiffness at service strain levels governs immediate settlement.

Characteristic values should be cautious estimates of the values governing the relevant limit state, not automatically statistical means or minimum laboratory results.

2. Limit-State Framework

2.1 Ultimate limit states

Relevant ultimate limit states include:

• GEO: failure or excessive deformation of the ground;
• STR: structural failure of the footing or raft;
• EQU: loss of equilibrium, such as overturning;
• UPL: loss of stability due to uplift;
• HYD: hydraulic heave, piping or internal erosion.

Bearing failure, sliding and overall stability are principally GEO checks. Punching shear, one-way shear and flexure are STR checks. Deep excavations and basement rafts may also require explicit UPL and HYD verification.

2.2 Serviceability limit states

Serviceability verification considers:

• total settlement;
• differential settlement and angular distortion;
• rotation or tilt;
• cracking and redistribution in the superstructure;
• loss of drainage gradients or utility alignment; and
• vibration or machine-foundation requirements where applicable.

Permissible movement is structure-specific. A single universal allowable settlement is technically inappropriate.

2.3 Eurocode 7 illustration without mixing formats

A design must follow the Design Approach and factors specified in the applicable National Annex. For example, the first-generation EN 1997-1 framework commonly expresses Design Approach 1 as:

• Combination 1: A1 + M1 + R1;
• Combination 2: A2 + M2 + R1.

Using the recommended Annex A values as an illustration, A1 applies larger action factors, while M1 generally leaves soil strength unfactored. A2 uses a different action-factor set, while M2 reduces soil strength; for example,

tan φ′d = tan φ′k/γMφ.

With the commonly recommended M2 value γMφ = 1.25, a characteristic friction angle must therefore be transformed through its tangent, not divided directly by 1.25. R1 commonly has γRv = 1.0 for bearing resistance, subject to the National Annex.

The two combinations must be checked independently. One must not combine factored actions from one combination, characteristic strength from another and an arbitrary global resistance factor. Other jurisdictions may prescribe DA2, DA3, load-and-resistance-factor design or allowable-stress procedures; the adopted standard and edition must be stated in the design basis.

Serviceability calculations normally use representative service actions and stiffness appropriate to the expected strain range, again subject to the governing standard.

3. Bearing Resistance

3.1 General equation

A widely used drained bearing-resistance expression for a shallow foundation is

qR = c′Nc sc dc ic + q′Nq sq dq iq + 0.5γ′B′Nγ sγ dγ iγ,

where:

• qR is gross bearing resistance per unit effective area;
• c′ and φ′ are effective-stress strength parameters;
• q′ is effective overburden pressure at foundation level;
• γ′ is the relevant effective unit weight below the base;
• B′ is effective footing width;
• Nc, Nq and Nγ are bearing-capacity factors;
• s, d and i are shape, depth and load-inclination factors.

The equation assumes a compatible bearing-capacity model, adequate founding depth, an approximately level base and ground surface, and soil conditions represented by the selected parameters. Factors from different published formulations must not be mixed because definitions of Nγ and correction factors vary.

For drained loading,

Nq = exp(π tan φ′) tan²(45° + φ′/2),

Nc = (Nq − 1)cot φ′.

Nγ depends on the chosen bearing-capacity formulation and must be taken consistently from that formulation.

For undrained total-stress analysis of saturated clay under short-term loading, φu = 0 and a model based on undrained shear strength cu may be used. The drainage condition must reflect loading rate, permeability and stress history.

3.2 Groundwater correction: worked example

Consider a long strip footing with width B = 2.0 m and depth D = 1.0 m in sand with φ′ = 30°, c′ = 0 and moist unit weight 18 kN/m³ above the base. The water table is at foundation level, and the submerged unit weight below the base is 10 kN/m³. Ignore correction factors and use Nq = 18.4 and Nγ = 22.4 from one consistent formulation.

Effective surcharge:

q′ = 18 × 1.0 = 18 kPa.

Gross bearing resistance:

qR = 18(18.4) + 0.5(10)(2)(22.4)

= 331.2 + 224.0

= 555.2 kPa.

Using the moist unit weight below the water table would incorrectly overestimate the third term. If the water table lies within the bearing influence zone, an appropriate average effective unit weight or a layered analysis is required.

This value is not automatically a design resistance. It must be converted according to the selected limit-state format and checked against the correspondingly factored action effect.

3.3 Layering and weak strata

The classical equation is unreliable where a strong crust overlies a weak layer within the failure mechanism, where thin compressible seams exist, or where the footing is close to a slope or excavation. Layered bearing-capacity analysis, numerical limit analysis, finite-element analysis or punching-through models may then be required.

4. Eccentric Loading and Contact Pressure

4.1 Linear full-contact distribution

For a rigid rectangular base of width B in the x-direction and length L in the y-direction, carrying vertical resultant V through eccentricities ex and ey measured from the base centre, elastic full-contact pressure is

q(x,y) = V/(BL)[1 + 12exx/B² + 12eyy/L²].

Here ex = My/V and ey = Mx/V, with signs determined by the adopted axes. This expression follows from

q = V/A + Myx/Iy + Mxy/Ix,

where Iy = LB³/12 and Ix = BL³/12.

For uniaxial eccentricity ey along length L,

qmax,min = V/(BL)(1 ± 6ey/L)

= V/(BL)(1 ± 6Mx/(VL)).

Full compression requires |ex| ≤ B/6 and |ey| ≤ L/6 when eccentricity acts independently. Under biaxial loading, all four corner pressures must be checked; positive corner pressures are required for the assumed full-contact linear distribution.

4.2 Partial contact

Soil cannot normally sustain foundation tension. If a uniaxial resultant lies outside the middle third but remains within the base, contact becomes triangular. For eccentricity e along dimension L, the compression length is

a = 3(L/2 − e),

for L/6 < e < L/2. The maximum pressure is

qmax = 2V/(Ba).

This is a special uniaxial result and should not be applied directly to biaxial eccentricity.

4.3 Effective-area method

For bearing-resistance assessment under eccentric loading, a common simplified method uses

B′ = B − 2|ex|, L′ = L − 2|ey|,

and A′ = B′L′. The resultant is treated as acting centrally on the effective area. Applicability and correction factors must follow the chosen standard. The method is not identical to calculating elastic corner pressures.

5. Inclined Loading, Sliding and Uplift

Horizontal loading reduces bearing resistance and creates a sliding limit state. Bearing-capacity formulations account for inclination through factors ic, iq and iγ, whose equations are model-specific. They should not be transferred between incompatible formulations.

For a drained granular interface, a simplified sliding check may be written as

Hd ≤ V′d tan δd + Aad,

where Hd is the design horizontal action, V′d is the design effective compressive normal force, δd is the design interface friction angle, A is contact area and ad is any permissible design adhesion. Adhesion is commonly neglected for conservative design, and passive resistance should be included only when it can be maintained throughout the design life.

Crucially,

V′d = downward design actions − design uplift force.

Uplift and pore-water pressures must be evaluated separately; they are not automatically accounted for merely because bearing resistance was calculated using effective stresses. Water may also exert horizontal pressure and reduce passive resistance. Drainage systems cannot be assumed effective unless their inspection, redundancy and maintenance are assured.

6. Settlement Analysis

6.1 Components

Foundation settlement may contain:

• immediate distortion in granular or unsaturated soil;
• primary consolidation caused by dissipation of excess pore pressure;
• secondary compression or creep;
• collapse, swelling or wetting-induced movement in susceptible soils;
• construction-induced movement and stress relief.

6.2 Immediate settlement

For a homogeneous elastic half-space, a simplified expression is

si = qnet B(1 − ν²)Is/Es,

where qnet is net service pressure, Es is representative soil modulus, ν is Poisson's ratio and Is is an influence factor for footing shape, rigidity and embedment. Because soil stiffness is nonlinear and strain-dependent, Es should correspond to the stress path and operational strain level.

A layer-based alternative is

s = Σ(Δσ′z,i Hi/Mi),

where Δσ′z,i is representative vertical effective stress increase in layer i, Hi is layer thickness and Mi is an appropriate constrained or operational modulus.

6.3 Primary consolidation

For normally consolidated clay under one-dimensional conditions,

Sc = H Cc/(1 + e0) log10[(σ′v0 + Δσ′v)/σ′v0],

where H is layer thickness, Cc is compression index, e0 is initial void ratio, σ′v0 is initial effective vertical stress and Δσ′v is the stress increase. Overconsolidated clay requires separate recompression and virgin-compression segments where the final stress crosses the preconsolidation pressure.

Consolidation time depends on drainage path and coefficient of consolidation. A settlement magnitude calculation alone does not establish when movement will occur.

6.4 Differential settlement

If adjacent points settle by s1 and s2 over distance l, angular distortion is approximately

β = |s1 − s2|/l.

Acceptability depends on structural form, façade brittleness, partition detailing and operational requirements. Soil and structural engineers should therefore coordinate deformation criteria rather than rely on a generic allowable value.

7. Raft Foundations and Soil–Structure Interaction

7.1 Rigid and flexible behaviour

A raft behaves as relatively rigid when its flexural stiffness is high compared with the deformability of the supporting ground. A rigid model enforces approximately planar deformation and determines pressures from equilibrium. A flexible raft develops deformation compatible with local loading and soil response.

The actual behaviour lies between these idealizations. Column stiffness, core walls, construction joints and superstructure framing may materially affect pressure redistribution.

7.2 Analysis models

Uniform or linear pressure models are suitable for preliminary equilibrium calculations but cannot reproduce local column effects.

Winkler models represent soil using independent springs, p = ks, where ks is the coefficient of subgrade reaction. The coefficient is not a fundamental soil constant: it depends on loaded area, shape, stiffness, depth and strain level. Independent springs also neglect shear interaction within the soil.

Continuum and finite-element models represent soil interaction more realistically. Advanced models may include stress-dependent stiffness, plasticity, consolidation, staged construction and interface behaviour. Their apparent precision is meaningful only when supported by adequate site characterization and sensitivity studies.

7.3 Structural design

A raft must be checked for flexure, punching shear around columns, one-way shear, local bearing, crack control and robustness. Geotechnical pressure distributions and structural deformation should be iterated because slab stiffness affects pressure, while pressure affects slab actions.

8. Illustrative Integrated Case: Office-Building Raft

Background

This fictional case concerns a 20 m × 20 m raft supporting an office building on medium-dense granular soils. The net service vertical resultant is V = 18,400 kN. A service moment Mx = 12,000 kN·m acts about the transverse axis, causing pressure variation along the 20 m length. The resultant acts centrally across the width.

Problem

Determine the preliminary edge pressures and estimate service settlement. The available operational constrained moduli are:

• 0–5 m: M = 40 MPa;
• 5–15 m: M = 70 MPa;
• 15–30 m: M = 120 MPa.

For preliminary stress distribution, use the 2:1 approximation

Δσz = qnet BL/[(B + z)(L + z)],

where z is depth below the raft and qnet is average net service pressure. The approximation is used only for screening; final design would require a calibrated interaction model.

Contact-pressure analysis

Average pressure:

qavg = 18,400/(20 × 20) = 46.0 kPa.

Eccentricity:

ey = Mx/V = 12,000/18,400 = 0.652 m.

Since ey < L/6 = 3.333 m, full compression is predicted.

Edge pressures:

qmax,min = 46[1 ± 6(12,000)/(18,400 × 20)].

The moment multiplier is

6M/(VL) = 72,000/368,000 = 0.19565.

Therefore,

qmax = 46(1.19565) = 55.0 kPa,

qmin = 46(0.80435) = 37.0 kPa.

Both pressures are compressive.

Settlement analysis

Use the midpoint depth of each layer.

For the 0–5 m layer, z = 2.5 m:

Δσ1 = 46(400)/(22.5²) = 36.35 kPa,

s1 = 36.35(5)/40,000 = 0.00454 m = 4.54 mm.

For the 5–15 m layer, z = 10 m:

Δσ2 = 46(400)/(30²) = 20.44 kPa,

s2 = 20.44(10)/70,000 = 0.00292 m = 2.92 mm.

For the 15–30 m layer, z = 22.5 m:

Δσ3 = 46(400)/(42.5²) = 10.19 kPa,

s3 = 10.19(15)/120,000 = 0.00127 m = 1.27 mm.

Estimated average settlement:

s = 4.54 + 2.92 + 1.27 = 8.73 mm.

As a preliminary upper-level screening assumption, settlement may be scaled with edge pressure. This gives approximately

shigh = 8.73(55/46) = 10.44 mm,

slow = 8.73(37/46) = 7.02 mm.

Estimated differential settlement is 3.42 mm over 20 m, corresponding to an angular distortion of approximately

β = 3.42/20,000 = 1/5,850.

Approach and outcome

The raft remains in full compression, with service pressures from approximately 37 to 55 kPa. The preliminary settlement estimate is modest. Nevertheless, final design must examine column-level punching shear, local pressure concentrations, stiffness variability, construction sequence and groundwater uplift. The edge-scaling estimate is not a substitute for raft–soil interaction analysis.

Learning points

• Moment pressure depends on the dimension over which pressure varies.
• The correct multiplier is 6M/(VL), not 6M/(VL²).
• Settlement should be checked using service actions and operational stiffness.
• Average settlement alone does not demonstrate acceptable differential performance.
• Structural and geotechnical analyses must use compatible pressure–deformation assumptions.

9. Professional Practice, Technology and Governance

Geotechnical design is inherently uncertain. Good practice includes sensitivity analysis for groundwater, strength, stiffness and layer thickness; independent review for consequential structures; clear documentation of characteristic values and design approaches; and verification during construction.

Instrumentation such as settlement markers, piezometers and load cells can support observational design. Building-information models and finite-element tools improve coordination, but automated outputs require engineering validation. Machine-learning correlations may assist parameter screening, yet they should not replace site-specific investigation or conceal uncertainty.

Ethical practice requires reporting adverse ground conditions, avoiding unsupported precision and considering whole-life consequences. Drainage assumptions must include maintenance responsibility. Lower-carbon foundation design should be pursued through better characterization and efficient sizing, not by compromising reliability.

Contemporary practice increasingly uses nonlinear three-dimensional interaction models, probabilistic reliability assessment, digital monitoring and performance-based deformation criteria. Designers must nevertheless apply the standard legally adopted in the project jurisdiction and manage transitions between standard editions explicitly.

10. Design Workflow and Best Practices

A defensible workflow is:

1. Establish structural actions and performance requirements.

2. Develop a ground and groundwater model.

3. Select candidate foundation systems.

4. Define the governing standard, National Annex and Design Approach.

5. Check bearing, sliding, overturning, uplift and hydraulic limit states.

6. Calculate settlement using service actions and appropriate stiffness.

7. Analyse eccentricity, partial contact and load inclination.

8. Complete structural design for flexure, shear and punching.

9. Iterate soil–structure interaction where stiffness affects pressure materially.

10. Assess parameter sensitivity and construction stages.

11. Specify inspection, drainage, monitoring and acceptance requirements.

Chapter Summary

Shallow-foundation design requires integrated verification of ultimate and serviceability limit states. Bearing resistance depends on soil strength, effective stress, foundation geometry, groundwater and load inclination. Eccentric loads produce nonuniform pressure and may cause partial contact; the pressure equation must be consistent with coordinates measured from the base centre. Groundwater affects effective overburden, unit weight, uplift, sliding and hydraulic stability and must be treated explicitly.

Settlement analysis requires service loads and stiffness appropriate to operational strains. Rafts redistribute actions through soil–structure interaction, so rigid, spring and continuum models have different capabilities and limitations. Code-compliant design must apply one specified Design Approach consistently rather than mix partial factors from incompatible formats.

Key Terms

Angular distortion: Differential settlement divided by the distance between two points.

Bearing resistance: Maximum foundation pressure sustainable under a specified failure model.

Characteristic value: Cautious estimate of a parameter governing a limit state.

Combined footing: One footing supporting two or more columns.

Contact pressure: Normal stress transmitted between a foundation and the ground.

Differential settlement: Difference in vertical movement between foundation locations.

Effective area: Reduced foundation plan area used for eccentric bearing assessment.

Effective stress: Stress carried by the soil skeleton after subtracting pore-water pressure.

Eccentricity: Offset between a load resultant and the foundation centroid.

Ground model: Engineering representation of strata, properties and groundwater conditions.

Immediate settlement: Settlement occurring without long-term pore-pressure dissipation.

Limit state: Condition beyond which specified performance requirements are not satisfied.

Net pressure: Foundation pressure after deducting the relevant pre-existing overburden effect.

Partial contact: Condition in which only part of the foundation base remains compressive.

Primary consolidation: Time-dependent compression caused by dissipation of excess pore pressure.

Raft foundation: Large slab supporting multiple structural elements over a substantial plan area.

Soil–structure interaction: Coupled response of soil deformation and structural stiffness.

Uplift: Upward action caused principally by water pressure beneath a foundation.

Review Questions

Short Answer Questions

1. Distinguish between GEO and STR limit states for a raft.

2. Why must the tangent of friction angle be factored rather than the angle itself in an M2-type format?

3. Define the middle-third condition for uniaxial eccentricity.

4. Why is the coefficient of subgrade reaction not a fundamental soil constant?

5. Explain why groundwater uplift requires a separate sliding assessment.

Long Answer Questions

1. Explain the factors controlling selection between isolated footings, combined footings and rafts.

2. Derive the full-contact pressure equation for a rectangular base under biaxial eccentric loading.

3. Discuss immediate, consolidation and differential settlement, including parameter selection.

4. Compare rigid, Winkler and continuum methods of raft analysis.

5. Explain how a Eurocode 7 Design Approach should be applied without mixing action, material and resistance factors.

Analytical / Application Questions

1. A 3 m × 4 m footing carries 3,600 kN and a moment of 720 kN·m causing variation along the 4 m dimension. Calculate the two edge pressures and assess full contact.

2. Evaluate how a rise in groundwater from well below a footing to foundation level affects bearing resistance, sliding and uplift.

3. Develop a verification plan for a basement raft founded on layered soil, including ULS, SLS, groundwater and monitoring requirements.

Multiple Choice Questions with Answers

Question 1

For a rectangular footing with uniaxial eccentricity e along length L, which expression gives the full-contact edge pressures?

A. V/(BL)(1 ± 6e/L)

B. V/(BL)(1 ± 6e/L²)

C. V/(BL)(1 ± 12e/L)

D. V/(BL)(1 ± e/6L)

Correct Answer: A. V/(BL)(1 ± 6e/L)

Question 2

A 20 m × 20 m raft carries V = 18,400 kN and M = 12,000 kN·m, producing pressure variation along the 20 m dimension. What are the approximate edge pressures?

A. 46 and 46 kPa

B. 55 and 37 kPa

C. 64 and 28 kPa

D. 92 and 0 kPa

Correct Answer: B. 55 and 37 kPa

Question 3

Under uniaxial eccentricity, full compression is maintained when which condition is satisfied?

A. |e| ≤ L/2

B. |e| ≤ L/4

C. |e| ≤ L/6

D. |e| ≤ L/12

Correct Answer: C. |e| ≤ L/6

Question 4

Which action must be deducted when determining effective normal force for a sliding check?

A. Soil self-weight below the footing

B. Water uplift beneath the foundation

C. Passive resistance in front of the footing

D. Bearing-capacity surcharge

Correct Answer: B. Water uplift beneath the foundation

Question 5

In a commonly used Eurocode 7 M2 treatment, how is the design friction angle obtained?

A. φ′d = φ′k/γMφ

B. tan φ′d = tan φ′k/γMφ

C. φ′d = γMφφ′k

D. tan φ′d = γMφ tan φ′k

Correct Answer: B. tan φ′d = tan φ′k/γMφ

Question 6

What is the main limitation of the Winkler soil model?

A. It cannot represent vertical settlement

B. It assumes zero foundation stiffness

C. Its independent springs neglect soil shear interaction

D. It can be used only for circular foundations

Correct Answer: C. Its independent springs neglect soil shear interaction

Question 7

A raft has settlements of 12 mm and 8 mm at points 20 m apart. What is the approximate angular distortion?

A. 1/500

B. 1/1,000

C. 1/5,000

D. 1/20,000

Correct Answer: C. 1/5,000

Question 8

Which soil property should normally be used below a foundation base when soil is submerged?

A. Dry unit weight

B. Saturated unit weight without correction

C. Effective or submerged unit weight

D. Concrete unit weight

Correct Answer: C. Effective or submerged unit weight

Question 9

Which limit state principally addresses hydraulic heave or piping?

A. STR

B. EQU

C. HYD

D. GEO settlement

Correct Answer: C. HYD

Question 10

Why should factors from different bearing-capacity formulations not be mixed?

A. All formulations use identical Nγ values

B. Their bearing factors and correction factors may have incompatible definitions

C. Shape factors are dimensionless

D. Only one formulation permits groundwater correction

Correct Answer: B. Their bearing factors and correction factors may have incompatible definitions