1. Background
Soil liquefaction is a complex phenomenon associated with pore water pressure development and the corresponding reduction in volume during undrained loading that may result in several additional failure mechanisms, such as lateral spreading, slope failure, excessive settlement, and bridge and building foundation failure, as described by several researchers [
1,
2,
3,
4,
5,
6,
7].
Historically, liquefaction effects have been observed and documented in many earthquakes all over the world. In particular, liquefaction-induced effects observed during field reconnaissance from the 1964 Niigata, 1990 Luzon, 2010–2011 Canterbury, and 2011 Tohoku earthquakes have been shown to depend on many input parameters such as shaking intensity, duration, and frequency [
8,
9,
10] as well as material and geometric factors such as relative density, fines content, soil thickness, building weight and width. Many relationships between liquefaction resistance and soil parameters have been proposed [
11,
12,
13,
14,
15,
16,
17,
18,
19,
20]. Among them, Ref. [
13] applied SPT results to a second-moment statistical analysis as a development of the empirical procedure introduced by Robertson and Wride [
18].
In addition, these approaches assess liquefaction-induced settlement with semi-empirical procedures that predict post-liquefaction, one-dimensional, consolidation settlement for free-field conditions [
8,
9,
10]. Other contributions (i.e., [
21,
22]) investigated post-liquefaction free-field settlements by considering the cyclic stress ratio and soil relative density to estimate a volumetric strain. In particular, Ref. [
21] considered that this procedure is affected by a potential error ranging from 25–50% and stated that liquefaction conditions are more complex than accounted for in their prediction.
In addition, Ref. [
23] proposed an expression for the maximum earthquake-induced shear stress that considers both peak ground acceleration (PGA) and peak ground velocity (PGV), demonstrating that the conventional method based on PGA neglects the effects of the frequency content of the input motion.
Other researchers focused on experimental tests to assess the soil properties [
24,
25,
26] that drive liquefaction potential. In particular, Refs. [
27,
28] performed centrifuge tests while shaking table tests were used in [
29,
30,
31,
32,
33,
34]. Centrifuge modelling has been and continues to be a valuable resource in the advancement of liquefaction understanding. The centrifuge community has had two large multi-facility round-robin validation exercises, with VELACS (Verification of Liquefaction Analysis by Centrifuge Studies) [
35] in the 1990s and, more recently, LEAP (Liquefaction Experiments and Analysis Projects) [
36]. In the last few years, LIQUEFACT [
37] used centrifuge testing to calibrate constitutive models that capture both free-field response as well as soil-structure interaction. Moreover, Ref. [
38] considered the effect of acceleration and frequency of base shaking on the liquefaction potential by performing 1-g free-field liquefaction studies on sand through uniaxial shaking table tests. The assessment of the parameters that may drive liquefaction has been investigated by many researchers. Among the other contributions, Ref. [
39] investigated the role of magnitude, mean grain size, vertical effective stress, peak ground acceleration, and cone resistance by applying the artificial neural network method. In addition, Ref. [
40] developed a logistic regress model for evaluating soil liquefaction probability via the normalized resistance from cone penetration tests (CPT) and soil behaviour type indices. Moreover, Ref. [
41] considered the effective stress, soil type, shear wave velocity, peak horizontal acceleration, and earthquake magnitude with the vector machine method. Another contribution [
42] analysed the effects of magnitude, standard penetration number, mean diameter, and groundwater table by considering the fuzzy comprehensive evaluation method. In addition, Ref. [
43] considered the magnitude, peak ground acceleration, vertical effective stress, and standard penetration number as the main parameters by adopting a genetic algorithm. Likewise, Refs. [
44,
45] identified 16 significant parameters of soil liquefaction by considering the Analytic Hierarchy Process (AHP) and the entropy method, and [
46] separately used AHP and rough set theory to calculate the weights of factors for soil liquefaction.
In this context, liquefaction has been studied primarily through two different approaches: (1) laboratory tests, and (2) site observations of past earthquakes. Only a few contributions [
47,
48,
49,
50,
51,
52,
53,
54,
55] proposed advanced numerical simulations that assess liquefaction potential. This paper targets this gap through focused numerical simulations that study the factors that affect liquefaction, specifically concentrating on those connected with the input motion characteristics. Several parametric studies are proposed to evaluate the influence of various factors (duration, amplitude, and frequency) that define the input motions. The soil layers were taken from the profile previously analyzed in [
50], and free-field conditions are applied in order to focus on the effects of liquefaction on the soil. The main novelties of the proposed methodology consist in (1) performing parametric studies with advanced numerical simulations as an alternative to the more common approaches (i.e., experimental investigations and site observations) generally adopted in literature, (2) concentrating on the loading factors themselves while most other research endeavours focused on the effects of soil parameters on the liquefaction potential, and thus the liquefaction-induced damage scenarios, and (3) that the outcomes may be considered for revisions or proposals of new code provisions that may account for loading factors in the assessments of liquefaction potential.
The paper is divided into four sections.
Section 2 describes the case study by focusing on (1) the performed numerical model, (2) the constitutive materials used to model the soil layers, and (3) the non-linear analyses that have been performed.
Section 3 describes the three parametric studies that have been performed to assess the role of (1) the number of cycles, (2) the frequency, and (3) the amplitude. Finally, the results are compared with the existing literature.
4. Conclusions
The paper investigates the effects of loading factors on the development of pore pressure and consequently on the liquefaction-induced effects on a soil deposit. Three parametric studies have been performed that consider the number of cycles, frequency, and duration of the input motions, and several numerical models have been compared, with specific findings as summarized below:
The number of cycles has a profound impact on the pore water pressure generated, with longer duration sinusoidal input motions resulting in higher pressures. Further, a larger number of cycles increases the deformation of the soil and thus the shear strain accumulation.
The role of frequency may be significant in the development of pore pressure; however, this effect is non-linear and depends on the displacement of the input motion as well as the soil characteristics. In the parametric study performed here, there was a threshold frequency where the rate of pore pressure generation increased with frequency (i.e., from 1 to 2 Hz) as well as an observed trend in which the magnitude of the maximum ru decreased with frequency.
The soil response is substantially affected by variations in the acceleration amplitude. For all the acceleration amplitudes less than 0.14 g, there were increases in ru without reaching the point at which liquefaction is considered to occur (maximum ru values: 0.86 and 0.75, respectively, for 0.08 g and 0.06 g), since their curves do not reach ru = 1. For an amplitude of 0.14 g, the buildup of pore pressure is slower than that for 0.20 g and ru reaches a maximum value of 1.35 for PS3-014, compared with 1.61 for PS3-020. It is important to note that the time to reach liquefaction (ru = 1) decreases as the amplitude increases.
The results have been discussed by considering the existing literature, and the effects of the loading factors have been analysed on a free-field model of a realistic soil deposit. In particular, the novelty of the work consists in proposing numerical models to represent the most common experimental studies. These two approaches (experiments and numerical simulations) may be used together to support and advance modelling as whole. The advanced non-linear analyses described in this paper have been performed with the state-of-the-art 3D finite elements OpenSeesPL that may consider the mechanisms of cycle mobility inside the soil that has been previously shown in the laboratory tests. The numerical models have been described in order to detail the various assumptions that have been considered to allow the study of the various performances in terms of pore pressure development inside the soil deposit. The most significant results have been discussed in terms of shear strain relationships. Overall, the paper demonstrates that soil liquefaction may significantly depend on the loading factors, and they need to be accounted for design purposes. Although the findings are limited to the performed conditions, they may potentially be useful to propose code provisions. To this end, further parametric numerical studies on the response of other soil deposits will be performed.