A large-scale excavation in soft Holocene deposit and its elasto-viscoplastic analysis

Abstract

In the present study, the large-scale excavation in the construction is numerically back-analyzed using a soil–water-coupled finite element method with an elasto-viscoplastic model which considers the strain-induced degradation. The measurements of the deformation have been performed during the construction of a new railway station in Osaka, Japan, in which a large and deep excavation has been successfully carried out using a special deep mixing type of soil improvement method with earth retaining walls through the thick Holocene Osaka Umeda clay deposit. A comparison between the numerical results and the measurements of the excavation at Osaka shows that the simulation method can reproduce the overall deformation of the soft ground and the earth retaining walls including the time-dependent behaviour during the excavation and a deep mixing soil improvement method as an additional technique for stability are effective.

Appendices

Appendix 1: Elasto-viscoplastic constitutive model for clay considering structural changes

For the constitutive model for clay, an elasto-viscoplastic model which considers structural changes is used. The model is an over-stress type of viscoplasticity theory considering the material instability brought about by microstructural changes in the geomaterials [10].

The deviatoric and volumetric viscoplastic strain rates are given by

$$\dot{e}_{ij}^{vp} = C_{01} \exp \left\{ {m^{\prime } \left( {\bar{\eta }_{{}}^{*} + \tilde{M}^{*} \ln \frac{{\sigma_{m}^{\prime } }}{{\sigma_{\text{mb}}^{\prime } }}} \right)} \right\}\frac{{\eta_{ij}^{*} - \eta_{ij(0)}^{*} }}{{\bar{\eta }^{*} }}$$

(5)

$$\dot{\varepsilon }_{\text{kk}}^{vp} = C_{02} \exp \left\{ {m^{\prime } \left( {\bar{\eta }_{{}}^{*} + \tilde{M}^{*} \ln \frac{{\sigma_{m}^{\prime } }}{{\sigma_{\text{mb}}^{\prime } }}} \right)} \right\}\left\{ {\tilde{M}^{*} - \frac{{\eta_{mn}^{*} \left( {\eta_{mn}^{*} - \eta_{mn(0)}^{*} } \right)}}{{\bar{\eta }^{*} }}} \right\}$$

(6)

$$\bar{\eta }^{*} = \left\{ {\left( {\eta_{ij}^{*} - \eta_{ij(0)}^{*} } \right)\left( {\eta_{ij}^{*} - \eta_{ij(0)}^{*} } \right)} \right\}^{1/2} ,\quad \eta_{ij}^{*} = S_{ij} /\sigma_{m}^{\prime }$$

(7)

$$C_{ijkl} = a\delta_{ij} \delta_{kl} + b(\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk} ),\quad C_{01} = 2b,\quad C_{02} = 3a + 2b$$

where $\dot{e}_{ij}^{vp}$ is the deviatoric viscoplastic strain rate tensor, $\dot{\varepsilon }_{\text{kk}}^{vp}$ is the viscoplastic volumetric strain rate, $\sigma_{m}^{\prime }$ is the mean effective stress, C ₀₁, C ₀₂, and m′ are viscoplastic parameters, $\bar{\eta }^{*}$ is the relative stress ratio, $\eta_{ij}^{*}$ is the stress ratio tensor, subscript (0) indicates the initial state, Sij is the deviatoric stress tensor, $\tilde{M}^{*}$ is the dilatancy coefficient, fp is the plastic potential function, $\sigma_{ij}^{\prime }$ is the effective stress tensor, and $\sigma_{\text{mb}}^{\prime }$ is a hardening–softening parameter given as follows:

$$\sigma_{\text{mb}}^{\prime } = \sigma_{\text{ma}}^{\prime } \exp \left( {\frac{1 + e}{\lambda - \kappa }\varepsilon_{kk}^{vp} } \right)$$

(8)

$$\sigma_{\text{ma}}^{\prime } = \sigma_{\text{maf}}^{\prime } + (\sigma_{\text{mai}}^{\prime } - \sigma_{\text{maf}}^{\prime } )\exp \,( - \beta z)$$

(9)

$$z = \int_{0}^{t} {\dot{z}} {\text{d}}t,\quad \dot{z} = (\dot{\varepsilon }_{ij}^{vp} \dot{\varepsilon }_{ij}^{vp} )^{1/2}$$

(10)

where $\sigma_{\text{maf}}^{\prime }$ and $\sigma_{\text{mai}}^{\prime }$ are the final and the initial values of structural parameter $\sigma_{\text{ma}}^{\prime }$, respectively, β is a parameter denoting the rate of degradation of the structure, e is the void ratio, λ is the compression index, κ is the swelling index, and. z is an accumulation of the second invariant of viscoplastic strain rate $\dot{\varepsilon }_{ij}^{vp}$.

From the above equations, it is seen that the decrease in $\sigma_{\text{mb}}^{\prime }$ in Eq. (8) can lead to softening behaviour and that Eq. (6) can describe the positive dilatancy depending on the stress ratio.

In the above formulation, no pure elastic region is taken into consideration although the over-stress type elasto-viscoplasticity can, in general, include the elastic region. However, in the present analysis, the viscoplastic parameters C ₀₁ and C ₀₂ are small as 10⁻¹¹/s. Hence the effect of the elastic region is negligible even for the unloading process.

Appendix 2: Soil–water-coupled finite element method

For the numerical simulation, we use a finite element method for two-phase mixtures based on the finite deformation theory. In general the strain level in the analysis is not high in the whole domain but locally the strain may be not so small since the ground deformation may includes strain localization. In addition, the strain level induced in the construction activity cannot be known before analysis, and the finite deformation analysis is sufficient in general, but the small strain analysis is not. Based on the above consideration, we have used a finite deformation theory in the present analysis.

Biot’s type of two-phase mixture theory is adopted to give the governing equations of the soil–water coupling problem (e.g., [13, 15]. An updated Lagrangian method with the objective Jaumann rate of Cauchy stress is used for the weak form of the rate type of equilibrium equations for the soil–water whole mixture which is given by the following equation:

$$\int_{V} {{\text{div}}\,\dot{\varvec{S}}_{\varvec{t}} \cdot \delta \varvec{v}} = 0$$

(11)

$$\dot{\varvec{S}}_{\varvec{t}} \equiv \dot{\varvec{T}} + \varvec{T}{\kern 1pt} {\text{tr}}{\kern 1pt} \varvec{L} - \varvec{TL}^{\text{T}}$$

(12)

in which $\delta \varvec{v}$ is the virtual velocity vector, $\dot{\varvec{S}}_{\varvec{t}}$ is the nominal stress rate tensor, T denotes the Cauchy stress tensor, L denotes the velocity gradient tensor, and the superimposed dots indicate the time differentiation.

For the fluid phase, the weak form of the continuity equation is employed. The types of elements used in the three-dimensional analysis are eight-node isoparametric elements with a reduced Gaussian (2 × 2) integration for the soil skeleton and four-node isoparametric elements with a full (2 × 2) integration for the pore fluid. Detailed formulations of the equilibrium equation and the continuity equation can be found in the references, e.g., Oka et al. [13] and Higo et al. [6, 7].

Appendix 3

The results of the parametric study are illustrated in the Figs. 18, 19, 20, 21, 22 and 23. The results show that the present value for E is reasonable considering the displacement of the retaining wall and the deformation of soil. Hence, considering the recommendation by RTRI (16) and the parametric study on E, we have adopted a Young’s modulus which is 1.5 times larger for the Dg1 layer and one which is 2.0 times larger for the Dg2 layer, which is consistent with the fact that the Pleistocene gravel layers are stiff and the expected strains are small.

Fig. 18

Displacement of the retaining wall (E = 2500 N)

Fig. 19

Vertical settlement–time profile at a depth of GL-2.5 m behind the retaining wall (E = 2500 N)

Fig. 20

Vertical settlement–time profile at a depth of GL-20.7 m in the excavation area (E = 2500 N)

Fig. 21

Displacement of the retaining wall (E = 3 × 2500 N)

Fig. 22

Vertical settlement–time profile at a depth of GL-2.5 m behind the retaining wall (E = 3 × 2500 N)

Fig. 23

Vertical settlement–time profile at a depth of GL-20.7 m in the excavation area (E = 3 × 2500 N)