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An analytical solution for the settlement of stone columns beneath rigid footings


This paper presents a new approximate solution to study the settlement of rigid footings resting on a soft soil improved with groups of stone columns. The solution development is fully analytical, but finite element analyses are used to verify the validity of some assumptions, such as a simplified geometric model, load distribution with depth and boundary conditions. Groups of stone columns are converted to equivalent single columns with the same cross-sectional area. So, the problem becomes axially symmetric. Soft soil is assumed as linear elastic, but plastic strains are considered in the column using the Mohr–Coulomb yield criterion and a non-associated flow rule, with a constant dilatancy angle. Soil profile is divided into independent horizontal slices, and equilibrium of stresses and compatibility of deformations are imposed in the vertical and horizontal directions. The solution is presented in a closed form and may be easily implemented in a spreadsheet. Comparisons of the proposed solution with numerical analyses show a good agreement for the whole range of common values, which confirms the validity of the solution and its hypotheses. The solution also compares well with a small-scale laboratory test available in the literature.

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a r :

Area replacement ratio: \( a_{\text{r}} = {{A_{\text{c}} } \mathord{\left/ {\vphantom {{A_{\text{c}} } {A_{\text{l}} }}} \right. \kern-0pt} {A_{\text{l}} }} \)

c :


d c :

Column diameter

K a :

Coefficient of active earth pressure

K 0 :

Coefficient of lateral pressure at rest

p a :

Uniform applied vertical pressure

r l, r c :

Radius of the loaded area and of the column

s :

Centre-to-centre column spacing

s r :

Radial displacement at the soil/column interface

s z :


s z0 :

Settlement without columns

Δt :

Slice thickness

u :


x, y, z :

Cartesian coordinates

A :

Cross-sectional area

B :

Footing width

D :

Footing diameter

E :

Young’s modulus

E m :

Oedometric (constrained) modulus: \( E_{\text{m}} = {{\left[ {E\left( {1 - \nu } \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {E\left( {1 - \nu } \right)} \right]} {\left[ {(1 + \nu )\left( {1 - 2\nu } \right)} \right]}}} \right. \kern-0pt} {\left[ {(1 + \nu )\left( {1 - 2\nu } \right)} \right]}} \)

G :

Shear modulus: \( G = {E \mathord{\left/ {\vphantom {E {\left[ {2\left( {1 + \nu } \right)} \right]}}} \right. \kern-0pt} {\left[ {2\left( {1 + \nu } \right)} \right]}} \)

H :

Soft soil layer thickness

L :

Column length

N :

Number of columns in the group

β :

Settlement reduction factor: \( \beta = {{s_{z} } \mathord{\left/ {\vphantom {{s_{z} } {s_{z0} }}} \right. \kern-0pt} {s_{z0} }} \)


Effective unit weight

ε :


λ :

Lamé’s constant: \( \lambda = {{2G\nu } \mathord{\left/ {\vphantom {{2G\nu } {\left( {1 - 2\nu } \right)}}} \right. \kern-0pt} {\left( {1 - 2\nu } \right)}} = E_{m} - 2G \)

ν :

Poisson’s ratio

σ :


ϕ :

Friction angle

ψ :

Dilatancy angle

c, s, l:

Column, soil, loaded area

e, p:

Elastic, plastic


Equivalent central column

i, y, f:

Initial (prior to loading), at yielding, final

r, z, θ :

Cylindrical coordinates


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Correspondence to Jorge Castro.

Additional information

Compressive stresses and strains are assumed as positive throughout the paper. Effective stresses are used throughout the paper, which are equal to total stresses but for hydrostatic pore pressures because drained conditions are assumed. For the sake of brevity, dash notation is not used.


Appendix 1: Solution for the soil hollow cylinder

The soil surrounding the column is a hollow cylinder, subjected to a vertical uniform pressure \( \sigma_{{z{\text{s}}}} \), a radial internal pressure \( \sigma_{{r{\text{s}}}} \) at the soil/column interface (r = r c) and a null radial strain (ε r  = 0) at the external boundary (r = r l) (Fig. 4). The solution is obtained by superposition of a state A of confined vertical compression and a state B in plane strain conditions (e.g. [2]). In state A, there is only vertical deformation, ε z , and in state B, there are only horizontal deformations, ε r and ε θ . The solution for state A is

$$ \left[ {\begin{array}{*{20}c} {\sigma_{{z{\text{s}},A}} } \\ {\sigma_{{r{\text{s}},A}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\lambda_{\text{s}} + 2G_{\text{s}} } & 0 \\ {\lambda_{\text{s}} } & 0 \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {\varepsilon_{z} } \\ {2\varepsilon_{{\theta {\text{s}}}} } \\ \end{array} } \right] $$

where ε θs is the hoop strain at the soil/column interface (r = r c) and is related to the radial displacement at the soil/column interface, ε θs = s r/r c.

The solution for state B starts with the displacement field for a hollow cylinder in plane strain conditions and axial symmetry (u θ  = 0):

$$ u_{r} = Ar + \frac{B}{r} $$

where A and B are constants that satisfy the boundary conditions. From the displacement field (Eq. 26), the strains may be calculated

$$ \begin{gathered} \varepsilon_{r} = \frac{{\partial u_{r} }}{\partial r} = A - \frac{B}{{r^{2} }} \hfill \\ \varepsilon_{\theta } = \frac{{u_{r} }}{r} = A + \frac{B}{{r^{2} }} \hfill \\ \end{gathered} $$

The stresses are

$$ \begin{aligned} \sigma_{r} & = 2\left( {\lambda_{\text{s}} + G_{\text{s}} } \right)A - 2G\frac{B}{{r^{2} }} \\ \sigma_{\theta } & = 2\left( {\lambda_{\text{s}} + G_{\text{s}} } \right)A + 2G\frac{B}{{r^{2} }} \\ \sigma_{z} & = \nu \left( {\sigma_{r} + \sigma_{\theta } } \right) = 2\lambda_{s} A \\ \end{aligned} $$

and the boundary conditions are

$$ \begin{aligned} & \varepsilon_{r} \left( {r = r_{\text{l}} } \right) = 0 \\ & \sigma_{r} \left( {r = r_{\text{c}} } \right) = \sigma_{{r{\text{s}}}} \\ \end{aligned} $$

Applying the boundary conditions (Eq. 29) to Eqs. (27) and (28), the constants A and B and the solution for state B are calculated.

$$ \left[ {\begin{array}{*{20}c} {\sigma_{{z{\text{s}},B}} } \\ {\sigma_{{r{\text{s}},B}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 & {\frac{{a_{\text{r}} \lambda_{\text{s}} }}{{1 + a_{\text{r}} }}} \\ 0 & {\frac{{a_{\text{r}} \left( {\lambda_{\text{s}} + G_{\text{s}} } \right) - G_{\text{s}} }}{{1 + a_{\text{r}} }}} \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {\varepsilon_{z} } \\ {2\varepsilon_{{\theta {\text{s}}}} } \\ \end{array} } \right] $$

The final solution is obtained by superposition of states A (Eq. 25) and B (Eq. 26).

$$ \left[ {\begin{array}{*{20}c} {\sigma_{{z{\text{s}}}} } \\ {\sigma_{{r{\text{s}}}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\lambda_{\text{s}} + 2G_{\text{s}} } & {\frac{{a_{\text{r}} \lambda_{\text{s}} }}{{1 + a_{\text{r}} }}} \\ {\lambda_{\text{s}} } & {\frac{{a_{\text{r}} \left( {\lambda_{\text{s}} + G_{\text{s}} } \right) - G_{\text{s}} }}{{1 + a_{\text{r}} }}} \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {\varepsilon_{z} } \\ {2\varepsilon_{{\theta {\text{s}}}} } \\ \end{array} } \right] $$

Appendix 2: Code for the analytical solution

The analytical solution may be implemented in a spreadsheet or any programming language. This appendix shows an example of code in MATLAB to get the footing settlement. The data of the calculation example in Sect. 4 are used here.

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Castro, J. An analytical solution for the settlement of stone columns beneath rigid footings. Acta Geotech. 11, 309–324 (2016).

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  • Analytical solution
  • Design
  • Footings
  • Ground improvement
  • Settlement
  • Stone columns