Modeling of Stone Column-Supported Embankment Under Axi-Symmetric Condition

Abstract

In the present work, a simplified model has been developed to study the behavior of stone column-supported embankment under axi-symmetric loading condition. The rate of consolidation of stone column-reinforced soft ground under axi-symmetric condition has also been presented in the paper. Mechanical model elements such as Pasternak shear layer, spring–dashpot system are used to model the different components such as granular layer, soft soil, stone columns etc. The governing differential equations are solved by finite difference technique. Parametric study has also been carried out to show the effect of different model variables on the settlement, stress concentration ratio of the foundation system. It is observed that for lower diameter ratio, at a particular time, the degree of consolidation predicted by the present method for axi-symmetric loading condition is almost same or lower than the degree of consolidation obtained by unit cell approach, but as the diameter ratio increases present analysis predicts higher degree of consolidation as compared to the unit cell approach. The maximum settlement decreases as the modular ratio increases and beyond the modular ratio value 30, the reduction rate of settlement decreases.

Appendix: Rate of Consolidation

Figure 20 shows ith equivalent stone ring and radial flow from the influence zone of the stone ring. Following Indraratna et al. (2008), the rate of flow in the radial direction from the inner impermeable boundary to the hollow cylindrical column wall is expressed by Darcy’s law

$$\frac{\partial Q}{\partial t} = \frac{{k_{h} }}{{\gamma_{w} }}\frac{\partial u}{\partial r}A$$

(31)

where Q is the flow in soil, u is the excess pore water pressure, γ _w is the unit weight of water, k _h is the coefficient of horizontal permeability of soft soil, t is the time and A is the cross-sectional area of the flow at a distance r and can be expressed as 2πr(dz), r is radial distance from center of embankment.

Fig. 20

Single stone ring and its influenced zone

The rate of volume in the vertical direction of the soil can be written as

$$\frac{\partial V}{\partial t} = \frac{\partial \varepsilon }{\partial t}\pi \left\{ {\left( {r_{i} - S_{e} /2} \right)^{2} - r^{2} } \right\}dz$$

(32)

where V is the volume change of the soil, ε is the vertical strain, r _i is the radius of ith stone ring, S _e is spacing between equivalent stone rings. Assuming radial flow rate is equal to the rate of volume change of the soil in the vertical direction, the excess pore pressure gradient for the idealized foundation can be expressed as:

$$\frac{\partial u}{\partial r} = \frac{1}{2}\frac{{\gamma_{w} }}{{k_{h} }}\frac{\partial \varepsilon }{\partial t}\frac{{\left( {r_{i} - {{S_{e} } \mathord{\left/ {\vphantom {{S_{e} } 2}} \right. \kern-0pt} 2}} \right)^{2} - r^{2} }}{r}\quad {\text{for}}\;r_{i} - {{S_{e} } \mathord{\left/ {\vphantom {{S_{e} } 2}} \right. \kern-0pt} 2} \le r \le r_{i} - {{t_{c} } \mathord{\left/ {\vphantom {{t_{c} } 2}} \right. \kern-0pt} 2}$$

(33)

where t _c is the thickness of equivalent stone ring.

Integrating Eq. (33), one can have expression of excess pore pressure for inner and outer zone (as shown in Fig. 20) as:

$$u_{in} = \frac{{\gamma_{w} }}{{k_{h} }}\frac{\partial \varepsilon }{\partial t}\left[ {\left( {r_{i} - S_{e} /2} \right)^{2} \ln \left( {\frac{r}{{r_{i} - t_{c} /2}}} \right) - \frac{1}{2}\left\{ {r^{2} - \left( {r_{i} - t_{c} /2} \right)^{2} } \right\}} \right]\quad {\text{for}}\;r_{i} - {{S_{e} } \mathord{\left/ {\vphantom {{S_{e} } 2}} \right. \kern-0pt} 2} \le r \le r_{i} - {{t_{c} } \mathord{\left/ {\vphantom {{t_{c} } 2}} \right. \kern-0pt} 2}$$

(34)

$$u_{out} = \frac{{\gamma_{w} }}{{k_{h} }}\frac{\partial \varepsilon }{\partial t}\left[ {\left( {r_{i} + S_{e} /2} \right)^{2} \ln \left( {\frac{r}{{r_{i} + t_{c} /2}}} \right) - \frac{1}{2}\left\{ {r^{2} - \left( {r_{i} + t_{c} /2} \right)^{2} } \right\}} \right]\quad {\text{for}}\;r_{i} + {{t_{c} } \mathord{\left/ {\vphantom {{t_{c} } 2}} \right. \kern-0pt} 2} \le r \le r_{i} + {{S_{e} } \mathord{\left/ {\vphantom {{S_{e} } 2}} \right. \kern-0pt} 2}$$

(35)

The average excess pore pressure under axi-symmetric condition ($\bar{u}_{axi}$) can be written as:

$$\bar{u}_{axi} \pi \left\{ {\left( {r_{i} + S_{e} /2} \right)^{2} - (r_{i} + t_{c} /2){}^{2} + (r_{i} - t_{c} /2){}^{2} - \left( {r_{i} - S_{e} /2} \right)^{2} } \right\} = \int\limits_{{r_{i} - S_{e} /2}}^{{r_{i} - t_{c} /2}} {2\pi u_{in} rdr + } \int\limits_{{r_{i} + t_{c} /2}}^{{r_{i} + S_{e} /2}} {2\pi u_{out} rdr}$$

(36)

After substituting Eqs. (34) and (35) and integrating Eq. (36) one can get

$$\overline{u}_{axi} = \frac{{\gamma_{w} }}{{k_{h} }}\frac{\partial \varepsilon }{\partial t}\frac{1}{8}d_{e}^{2} \mu_{axi}$$

(37)

where d _e is the diameter of unit cell, and

$$\begin{aligned} \mu_{axi} & = \frac{1}{{r_{i} \left( {S_{e} - t_{c} } \right)}}\left[ {2\left( {r_{i} - 0.5S_{e} } \right)^{4} \ln \frac{{r_{i} - 0.5t_{c} }}{{r_{i} - 0.5S_{e} }} + \frac{1}{4}\left( {2r_{i} S_{e} - 0.5S_{e}^{2} - 2r_{i} t_{c} + 0.5t_{c}^{2} } \right)\left( { - 2r_{i}^{2} + 3r_{i} S_{e} - 0.75S_{e}^{2} - r_{i} t_{c} + 0.25t_{c}^{2} } \right)} \right. \\ & \quad \left. { + 2\left( {r_{i} + 0.5S_{e} } \right)^{4} \ln \frac{{r_{i} + 0.5S_{e} }}{{r_{i} + 0.5t_{c} }} - \frac{1}{4}\left( {2r_{i} S_{e} + 0.5S_{e}^{2} - 2r_{i} t_{c} - 0.5t_{c}^{2} } \right)\left( {2r_{i}^{2} + 3r_{i} S_{e} + 0.75S_{e}^{2} - r_{i} t_{c} - 0.25t_{c}^{2} } \right)} \right] \\ \end{aligned}$$

(38)

Substituting r _i  = iS _e , S _e  = ψd _e , t _c  = β (r ² _c /S _e ) and r _c /S _e  = n _c in Eq. (38), one can get

$$\begin{aligned} \mu_{axi} & = \frac{{\psi^{2} }}{{i\left( {1 - \beta n_{c}^{2} } \right)}}\left[ {2\left( {i - 0.5} \right)^{4} \ln \frac{{i - 0.5\beta n_{c}^{2} }}{i - 0.5} + \frac{1}{4}\left( {2i - 0.5 - 2i\beta n_{c}^{2} + 0.5\beta^{2} n_{c}^{4} } \right)\left( { - 2i^{2} + 3i - 0.75 - i\beta n_{c}^{2} + 0.25\beta^{2} n_{c}^{4} } \right)} \right. \\ & \quad \left. { + 2\left( {i + 0.5} \right)^{4} \ln \frac{i + 0.5}{{i + 0.5\beta n_{c}^{2} }} - \frac{1}{4}\left( {2i + 0.5 - 2i\beta n_{c}^{2} - 0.5\beta^{2} n_{c}^{4} } \right)\left( {2i^{2} + 3i + 0.75 - i\beta n_{c}^{2} - 0.25\beta^{2} n_{c}^{4} } \right)} \right] \\ \end{aligned}$$

(39)

where r _c is the radius of the stone column (d _c /2), Ψ is equal to 0.865 and 1.0 for triangular and square arrangement of stone columns, respectively and β is equal to 3 and 4 for triangular and square arrangement of stone columns, respectively. One can relate the excess pore pressure ($\bar{u}_{axi}$) with diameter ratio (N = d _e /d _c ) by substituting n _c  = 1/(2ΨN) in Eq. (38).

By substituting ∂ε/∂t = −(m _vs m _vc A _s )/(m _vc A _s  + m _vs A _c ) (∂$\bar{u}$/∂t) for stone column-improved ground (Han and Ye 2001) in Eq. (37), one can get

$$\overline{u}_{axi} = - \frac{{\gamma_{w} }}{{k_{h} }}\frac{{m_{vs} m_{vc} A_{s} }}{{m_{vc} A_{s} + m_{vs} A_{c} }}\frac{{\partial \overline{u} }}{\partial t}\frac{1}{8}d_{e}^{2} \mu_{axi} = - \frac{{\gamma_{w} }}{{k_{h} }}\frac{{m_{vs} }}{\lambda }\frac{{\partial \overline{u} }}{\partial t}\frac{1}{8}d_{e}^{2} \mu_{axi}$$

(40)

where

$$\lambda = 1 + n_{s} \frac{1}{{\left( {\frac{{4\psi^{2} }}{\beta }} \right)N^{2} - 1}}$$

(41)

$$n_{s} = \xi \frac{{E_{c} }}{{E_{s} }}$$

(42)

$$\xi = \frac{{\left( {1 + \nu_{s} } \right)\left( {1 - 2\nu_{s} } \right)\left( {1 - \nu_{c} } \right)}}{{\left( {1 + \nu_{c} } \right)\left( {1 - 2\nu_{c} } \right)\left( {1 - \nu_{s} } \right)}}$$

(43)

$$\frac{{A_{c} }}{A} = a_{s} = \beta n_{c}^{2}$$

(44)

$$A = A_{c} + A_{s}$$

(45)

and $\bar{u}$ is the average excess pore water pressure, ξ is the Poisson ratio factor, A _c and A _s are cross-sectional areas of the stone ring and the surrounding soft soil, respectively, A is the total area, a _s is the area replacement ratio, m _vc and m _vs are the coefficient of volume compressibility of stone column and soft soil, respectively, E _c and E _s are elastic modulus of stone column and soft soil, respectively, ν _c and ν _s are the Poisson ratios of stone column and soft soil, respectively. The n _s is steady-stress concentration ratio can also be written as:

$$n_{s} = \frac{{q_{c} }}{{q_{s} }}$$

(46)

After rearranging Eq. (40), one can write

$$\frac{1}{{\overline{u}_{axi} }}\partial \overline{u} = - \frac{8}{{\mu_{axi} }}\frac{{c_{r}^{\prime} }}{{d_{e}^{2} }}\partial t$$

(47)

where $c_{r}^{\prime}$ is the modified coefficient of radial consolidation and can be expressed as:

$$c_{r}^{\prime} = c_{r} \lambda = \frac{{\lambda k_{h} }}{{m_{vs} \gamma_{w} }}$$

(48)

Integrating Eq. (47) with initial boundary condition $\overline{u}$ = σ _i at t = 0, one can get

$$\frac{{\overline{u}_{axi} }}{{\sigma_{i} }} = \exp \left( { - \frac{{8T_{r}^{\prime} }}{{\mu_{axi} }}} \right)$$

(49)

where σ _i is the initial vertical stress, $T_{r}^{\prime}$ is the modified time factor that can be written as:

$$T_{r}^{\prime} = {{c_{r}^{\prime} t} \mathord{\left/ {\vphantom {{c_{r}^{\prime} t} {d_{e}^{2} }}} \right. \kern-0pt} {d_{e}^{2} }}$$

(50)

Therefore, the degree of consolidation (U _axi ) of stone column improved soft ground under axi-symmetric condition at any time t can be written as:

$$U_{axi} = 1 - \exp \left( { - \frac{{8T_{r}^{\prime} }}{{\mu_{axi} }}} \right)$$

(51)