A robust empirical model to estimate earthquake-induced excess pore water pressure in saturated and non-saturated soils

Abstract

In engineering practice, the liquefaction potential of a sandy soil is usually evaluated with a semi-empirical, stress-based approach computing a factor of safety in free field conditions, defined as the ratio between the liquefaction resistance (capacity) and the seismic demand. By so doing, an estimate of liquefaction potential is obtained, but nothing is known on the pore pressure increments (often expressed in the form of normalized pore pressure ratio r_u) generated by the seismic action when the safety factor is higher than 1. Even though r_u can be estimated using complex numerical analyses, it would be extremely useful to have a simplified procedure to estimate them consistent with the stress-based approach adopted to check the safety conditions. This paper proposes such a procedure with reference to both saturated and unsaturated soils, considering the latter as soils for which partial saturation has been artificially generated with some ground improvement technology to increase cyclic strength and thus tackle liquefaction risk. A simple relationship between the liquefaction free field safety factor FS, and r_u(S_r) is introduced, that generalizes a previous expression proposed by Chiaradonna and Flora (Geotech Lett, 2020. https://doi.org/10.1680/jgele.19.00032) for saturated soils. The new procedure has been successfully verified against some experimental data, coming from laboratory constant amplitude cyclic tests and from centrifuge tests with irregular acceleration time histories for soils having different gradings and densities.

Appendices

Appendix 1: Saturated and non-saturated laboratory tests

In this section, the results of saturated (Mele et al. 2018; Lirer and Mele 2019) and non-saturated (Mele et al. 2018) tests have been summarized in Tables 5 and 6, respectively. Some of such tests have been processed in this paper to extend the charts FS-r_u of Chiaradonna and Flora (2020) for saturated soils to non-saturated ones (Sect. 3). The charts FS-r_u have been also validated on some of such laboratory tests (Sect. 4).

Table 5 Results of cyclic saturated tests

Table 6 Results of all cyclic non-saturated triaxial tests (Mele et al. 2018)

Appendix 2: Calculation of the damage parameter versus time for an irregular shear stress history

For any irregular shear loading history, normalized to the initial effective stress state as:

$$\tau^{ * } (t) = \frac{{\left| {\tau (t)} \right|}}{{\sigma^{\prime}_{0} }}$$

(25)

the damage parameter is calculated as:

$$\kappa \left( t \right) = \kappa_{0} + d\kappa$$

(26)

whereκ₀ is the damage cumulated at the last reversal point of the function $\left( {\tau^{*} - CSR_{t} } \right)$ reached at the time instant t. The parameter κ₀ can be defined as follows:

$$\kappa_{0} = \left\{ {\begin{array}{*{20}c} {\kappa \left( {t - dt} \right)} & {{\text{if}}\quad \dot{\tau }^{*} \left( t \right) = 0} & {{\text{or}}\quad \tau^{*} \left( t \right) = CSR_{t} } \\ {\kappa_{0} \left( {t - dt} \right)} & {{\text{if}}\quad \dot{\tau }^{*} \left( t \right) \ne 0} & {{\text{or}}\quad \tau^{*} \left( t \right) \ne CSR_{t} } \\ \end{array} } \right.$$

(26)

i.e., κ₀ is a stepwise function assuming the value of the damage parameter gained at the time step (t—dt) every time the stress ratio reaches a local maximum value, or when $\tau^{*} = CSR_{t}$ (see Chiaradonna et al. 2018 for details).

The increment of the damage parameter, dκ, in the time interval dt is given by:

$$d\kappa = \left\{ {\begin{array}{*{20}c} 0 & {{\text{if }}\tau^{*} \left( t \right) < CSR_{t} } \\ {\left[ {\tau_{0}^{*} \left( t \right) - \tau \left( t \right)} \right]^{\alpha } } & {{\text{if }}\tau^{*} \left( t \right) \ge CSR_{t} } \\ \end{array} } \right.$$

(27)

where $\tau_{0}^{ * } = \tau_{\max }^{ * }$ if ${\tau}^{\cdot}*$ (t) < 0 and $\tau_{0}^{ * } = CSR_{t}$ otherwise.

It is important to point out that the damage parameter reaches the maximum value κ_L (Eq. 18) when liquefaction is attained, and this maximum value cannot be overcome. Consequently, when liquefaction triggers κ/κ_L = 1 and FS = 1.

Appendix 3: Detailed description of centrifuge tests simulations

The centrifuge tests described in Sect. 5 were simulated through a 1D numerical analysis in effective stress conditions using a non-linear computer code in the time domain (Tropeano et al. 2019).

In the simulations, the time history of horizontal acceleration measured at the base of the model (Fig. 

Fig. 15

Profiles of V_S at prototype scale (a), acceleration time history of the reference input motion for saturated (b) and unsaturated (c) centrifuge test

15b, c for saturated and unsaturated model, respectively) was applied at the base of the profile (12.5 m and 13 m depth at prototype scale for saturated and unsaturated model), assumed as rigid bedrock. The profile of V_S as a functions of z (Fig. 15a) is characterized by a mean value of approximately 130 m/s, which was calculated as V_S = L∕T, where L is the distance between the two furthest accelerometers and T is the travel time, as proposed by Ghosh and Madabhushi (2002).

The non-linear and dissipative properties of Pieve di Cento sand (Fig. 

Fig. 16

a Grain size distribution, b normalized shear modulus and damping ratio vs shear strain, c cyclic resistance curve, d pore pressure relationships adopted for Pieve di Cento and Ticino sand in the numerical simulations

16b) were defined based on the experimental data obtained from cyclic laboratory tests, performed on undisturbed samples retrieved on a site 20 km far from Pieve di Cento with the same geological background and a grain size distribution (Chiaradonna et al. 2019). Conversely, the mean curve proposed by Seed and Idriss (1971) for sand has been adopted for Ticino sand, since its grain size distribution is without fine (Fig. 16a).The pore pressure relationship of the model proposed by Chiaradonna et al. (2018) was based on results of cyclic simple shear and cyclic triaxial tests carried out on reconstituted specimens of Pieve di Cento and Ticino sand with a relative density of about 40% (Mele et al. 2019). The pore pressure relationship was calibrated on the experimental data (Fig. 16d) assuming that the influence of relative density on the shape of this curve can be neglected. Finally, a permeability coefficient equal to 1 × 10^–5 m/s was estimated and used for Pieve di Cento sand, and 1 × 10^–4 m/s for Ticino sand. Once all the input data were known, the cyclic resistance curve adopted for both sands in the analysis has been modified, starting from the available laboratory data, until the best fitting of the experimental data has been obtained, as discussed in Sect. 5.2. The obtained curves are plotted in Fig. 16c.