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Fully coupled solution for the consolidation of poroelastic soil around elastoplastic stone column

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Abstract

The paper presents a new fully coupled elastoplastic solution for the response of a poroelastic thick-walled soil cylinder around an elastoplastic stone column using Biot’s (J Appl Phys 12:155–164, 1941) consolidation theory. A unit cell concept is adopted for the soil–stone column analysis, and the problem is formulated in cylindrical coordinates. Expressions for excess pore pressure, stresses and displacements in the Laplace domain are derived analytically taking into account elastic or plastic behavior of the column. The inverse of the Laplace transform is evaluated numerically using an efficient scheme to obtain the final elastoplastic solution in time domain. The validity of the new solution has been checked against finite element solution and compared with some previously developed analytical methods for the stone column analysis. The results showing settlements, change in excess pore pressures and stresses with time are presented in terms of time factor. The proposed solution can be used to calculate transient state of settlements, distribution of deformations, stresses and excess pore pressures in soil and column under instantaneous or time-dependent monotonically increasing rigid vertical load.

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Abbreviations

\( A_{j} ,A_{j}^{i} \) :

Coefficients

A r :

Area replacement ratio

B :

Skempton parameter

B 1 :

Unit cell dimensionless coefficient

B 2 :

Unit cell coefficient (m)

c :

Diffusivitiy coefficient (m2/s)

C R :

Radial coefficient of consolidation (m2/s)

C 0 :

Unit cell coefficient (m4 Pa)

C 1 :

Unit cell coefficient (m3 Pa2)

C 2 :

Unit cell coefficient (m4 Pa2)

E, E c :

Soil and stone column Young modulus (Pa)

E v, E h :

Vertical/horizontal Young modulus (Pa)

D c :

Material constant—stone column (Pa)

D 0 :

Unit cell constant (m4 Pa)

D 1 :

Unit cell constant (m3 Pa)

D 2 :

Unit cell constant (m2 Pa)

G :

Shear modulus (Pa)

I 0 :

Modified Bessel function of the first kind

k :

Hydraulic conductivity (m/s)

K :

Soil bulk modulus (Pa)

K ini :

Initial lateral pressure coefficient

K 0 :

Modified Bessel function of the second kind

K w :

Water bulk modulus (Pa)

K ϕ :

Passive earth pressure coefficient

K Ψ :

Dilation constant

\( K^{\prime }\) :

Stress ratio

L :

Column length (m)

M :

Biot modulus (Pa)

N :

Replacement ratio

n :

Porosity

p :

Excess pore pressure (Pa)

\( \bar{p} \) :

Average excess pore pressure (Pa)

\( {\bar{p}}_{0} \) :

Initial average pore pressure (Pa)

r :

Radial distance (m)

R a :

Stone column radius (m)

R b :

Unit cell radius (m)

s :

Laplace transform parameter

S :

Storage coefficient (1/Pa)

SCF:

Stress concentration factor

t :

Time (s)

T R :

Time factor

u r :

Radial displacement (m)

u z :

Vertical displacement (m)

U z :

Surface settlement (m)

U p :

Average degree of pore pressure dissipation

z :

Depth (m)

q, q 0 :

Transient and final load (Pa)

α :

Biot’s effective stress coefficient

γ :

Unit weight (N/m3)

δ :

Load proportionality factor

ε :

Volumetric strain

ε rr :

Principal radial strain

ε θθ :

Principal tangential strain

ε zz :

Principal vertical strain

κ :

Permeability (m2)

ζ :

Volumetric variation of pore fluid content

η :

Poroelastic stress coefficient

λ :

Lamé parameter

μ :

Pore fluid viscosity (Pa s)

ν :

Poisson’s ratio

ν u :

Poisson’s ratio for undrained conditions

ξ :

Dimensionless radial coordinate

σ rc :

Radial stress–stone column (Pa)

σ rr :

Principal radial stress (Pa)

σ θθ :

Principal tangential stress (Pa)

σ zc :

Radial stress–stone column (Pa)

σ zz :

Principal vertical stress (Pa)

ϕ :

Friction angle (°)

ψ :

Dilatancy angle (°)

a, b:

Inner and outer radius of the thick-walled cylinder

c, w:

Column, water

ini:

Initial value

e, p:

Elastic, plastic

\( \bar{} \) (upper bar):

Average value along the radius

\( {\prime } \) (apostrophe):

Effective value

\( \tilde{} \) (tilde overbar):

Laplace transform

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  1. Faculty of Civil and Geodetic Engineering, University of Ljubljana, Jamova 2, 1000, Ljubljana, Slovenia

    Boštjan Pulko & Janko Logar

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  1. Boštjan Pulko
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  2. Janko Logar
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Correspondence to Boštjan Pulko.

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Pulko, B., Logar, J. Fully coupled solution for the consolidation of poroelastic soil around elastoplastic stone column. Acta Geotech. 12, 869–882 (2017). https://doi.org/10.1007/s11440-016-0484-2

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