Design of Shallow Foundations

Design of isolated, combined, strap footings, and mat/raft foundations.

Overview

The design of shallow foundations is a process that involves selecting the appropriate foundation type (isolated, combined, strap, or mat) and determining its dimensions to satisfy two primary criteria: 1) The applied bearing pressure must not exceed the allowable bearing capacity of the soil, and 2) the anticipated settlement (both total and differential) must be within acceptable limits for the supported structure.

Types of Shallow Foundations

The choice of shallow foundation depends primarily on the column loads, soil conditions, and spatial constraints (like property lines).

Procedure

  • Isolated (Spread) Footings: These are individual pads (usually square or rectangular) supporting single columns. They are the most common and economical type when columns are widely spaced and soil bearing capacity is adequate. The required plan area (AreqA_{req}) is simply the total column load (PP) plus the weight of the footing itself divided by the allowable soil pressure (qallowq_{allow}).
  • Combined Footings: Used when two or more columns are very close together (so their individual footings would overlap) or when an exterior column is located directly on a property line, precluding a centered isolated footing. They are designed so the centroid of the footing area aligns with the resultant of the column loads to ensure uniform soil pressure.
  • Strap (Cantilever) Footings: A variation of a combined footing used when an exterior column is on a property line but the interior column is relatively far away. A rigid beam (strap) connects the exterior footing to an interior footing, counteracting the overturning moment on the eccentric exterior footing.
  • Mat (Raft) Foundations: A single, large, continuous reinforced concrete slab that supports all columns and walls of a structure.

Rigid vs. Flexible Foundation Design Approach

Rigid vs. Flexible Assumptions

The structural design of the foundation slab depends on its relative stiffness compared to the soil.
  • Rigid Approach: Assumes the foundation is infinitely stiff compared to the soil. As a result, the settlement is uniform, but the soil contact pressure distribution is non-uniform (higher at edges in cohesive soils, higher in the center for granular soils). For simplicity, most structural design assumes a uniform or linear pressure distribution under rigid footings to calculate bending moments.
  • Flexible Approach: Assumes the foundation deforms with the soil. A uniformly loaded flexible footing (like a steel tank bottom) will exert uniform contact pressure, but will experience non-uniform settlement (dishing in the center).
Concrete footings generally fall somewhere in between, but are typically designed using the rigid assumption for simplicity unless they are very large mats.

Mat/Raft Foundations Details

Mats are typically employed under specific challenging conditions:

When to Use Mat Foundations

  • The soil bearing capacity is very low, requiring individual footings to cover more than 50-70% of the building footprint.
  • Column loads are exceptionally high (e.g., in high-rise buildings).
  • Differential settlement across the site must be strictly minimized, as the rigid mat bridges over weak soil pockets.
  • The structure has a basement located below the groundwater table, where the mat acts as a water barrier and resists hydrostatic uplift forces.

Compensated (Floating) Foundation

A compensated foundation is a specific design strategy for mat foundations on highly compressible soils (like deep soft clays). The mat is placed at a depth (DfD_f) such that the weight of the excavated soil roughly equals the total weight of the new structure. This results in nearly zero net increase in pressure (qnetq_{net}) on the soil below, dramatically minimizing anticipated consolidation settlement.
qnet=QbldgA−γDf≈0 q_{net} = \frac{Q_{bldg}}{A} - \gamma D_f \approx 0

Coefficient of Subgrade Reaction (Winkler Model)

Winkler Foundation Model

The Winkler model idealizes the soil beneath a mat foundation as a bed of closely spaced, independent linear elastic springs. The stiffness of these imaginary springs is defined by the Coefficient of Subgrade Reaction (kk).
This approach allows structural engineers to analyze the mat foundation as a structural slab supported on elastic supports, calculating internal bending moments and shears more accurately than assuming uniform soil pressure.
q=k⋅δ q = k \cdot \delta
Where qq is the soil pressure, kk is the coefficient of subgrade reaction (e.g., in kN/m3\text{kN/m}^3), and δ\delta is the settlement or deflection at that specific point.

Limitations of the Winkler Model

Because the springs are independent, the model does not account for the continuous nature of soil. A localized heavy load will only compress the spring directly beneath it, without causing adjacent "springs" to deform, which contradicts real soil behavior. To address this, more advanced models (like the coupled spring model or pseudo-coupled models) are often employed in complex finite element analyses.

Settlement of Shallow Foundations

Bearing capacity alone does not guarantee a safe design. A foundation must also satisfy settlement criteria to prevent structural damage or functional impairment. Total settlement consists of three components:
Stotal=Si+Sc+Ss S_{total} = S_i + S_c + S_s
Where:
  • SiS_i = Immediate (Elastic) Settlement: Occurs rapidly during or immediately after construction as load is applied. Dominant in granular soils (sands, gravels). Calculated using elastic theory.
  • ScS_c = Primary Consolidation Settlement: A slow, time-dependent process occurring over months or years as pore water is squeezed out of saturated, fine-grained soils (clays, silts) under sustained load.
  • SsS_s = Secondary Compression (Creep): A very slow, continuous deformation of soil fabric at constant effective stress after primary consolidation is complete. Significant primarily in highly organic soils (peats).

Stress Distribution in Soil

Before any settlement (immediate or consolidation) can be calculated, engineers must determine how the applied foundation pressure (q0q_0) dissipates with depth (zz).

Methods for Stress Increase (Δσz\Delta \sigma_z)

  • Boussinesq's Point Load Theory: The foundational theory for calculating the vertical stress increase Δσz\Delta \sigma_z at any depth zz and radial distance rr from a concentrated point load PP on an elastic half-space. It forms the basis for integrating stress beneath continuous foundations.
    Δσz=3P2πz2[11+(r/z)2]5/2\Delta \sigma_z = \frac{3P}{2\pi z^2} \left[ \frac{1}{1 + (r/z)^2} \right]^{5/2}
  • 2:1 Method (Approximate Method): A rapid, empirical method assuming the stress zone spreads outward at a slope of 2 vertical to 1 horizontal. For a rectangular footing of dimensions B×LB \times L carrying a total load QQ, the average stress increase at depth zz is:
    Δσz≈Q(B+z)(L+z)\Delta \sigma_z \approx \frac{Q}{(B + z)(L + z)}

Calculating Immediate Settlement (SiS_i)

Elastic Theory Method

Immediate settlement in granular soils or unsaturated cohesive soils is estimated using elastic theory:
Si=q0B(1−μ2)EsIs S_i = \frac{q_0 B (1 - \mu^2)}{E_s} I_s
Where q0q_0 is the applied pressure, BB is the foundation width, μ\mu is Poisson's ratio, EsE_s is the modulus of elasticity of the soil, and IsI_s is a shape/influence factor depending on the foundation geometry and rigidity.

Angular Distortion and Differential Settlement Limits

Differential Settlement

While total settlement is important, differential settlement (the difference in settlement between two adjacent columns, ΔS\Delta S) is usually the governing design criterion. Excessive differential settlement induces severe shear stresses in the superstructure, leading to cracked walls, jammed doors, and potential structural failure.
The severity is measured by Angular Distortion (θ\theta):
θ=ΔSL \theta = \frac{\Delta S}{L}
Where LL is the span distance between the two columns. Typical angular distortion limits (ΔS/L\Delta S/L) for buildings range from 1/300 (structural damage begins) to 1/500 (safe limit for buildings in which cracking is not permissible).
Key Takeaways
  • The choice between isolated, combined, strap, or mat foundations depends on load magnitude, column spacing, soil bearing capacity, and property line constraints.
  • Mat foundations are ideal for low-bearing soils, minimizing differential settlement, and providing a water barrier in basements.
  • Compensated (floating) foundations excavate soil equal in weight to the building to produce zero net pressure increase, drastically reducing consolidation settlement in deep soft clays.
  • Stress distribution below a foundation can be accurately calculated via integrated Boussinesq theory or rapidly approximated using the 2:1 method.
  • Total settlement comprises immediate (elastic), primary consolidation, and secondary compression components. Differential settlement (angular distortion) between adjacent structural elements is usually the most critical factor governing foundation design.
  • Combined footings must be dimensioned such that the centroid of their bearing area coincides precisely with the resultant of the applied column loads to maintain uniform soil pressure.

Shallow Foundation Sizing & Pressure

Required Square Width (BB)
0.0 m
Maximum Soil Pressure (qmaxq_{max})
0.0 kPa
Eccentricity (e=M/Pe = M/P)
0.00 mLimit B/6B/6: 0.00 m
Pq_maxq_minB

The simulator automatically increases the footing width (BB) until the maximum soil pressure (qmaxq_{max}) is below the allowable bearing capacity. When a moment is applied, the pressure becomes trapezoidal. If eccentricity (ee) exceeds B/6B/6, tension develops at the heel (shown as qmin=0q_{min} = 0).

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