Dissipated energy in undrained cyclic triaxial tests

Energy-based methods are an emerging tool for the evaluation of liquefaction potential. These methods relate excess pore water pressure build-up to seismic energy dissipated per unit volume. Further development of these methods require their validation through laboratory testing. In this paper, a comprehensive study of energy dissipated during cyclic triaxial tests is undertaken. Results of undrained cyclic triaxial tests performed on air-pluviated samples of Hostun sand prepared with different initial densities and subjected to several confining pressures and loading amplitudes are presented. The energy dissipated per unit volume is estimated from the experimental results and correlated to the generated excess pore water pressure. The correlation between those quantities appear to be independent of the initial relative density of the sample, isotropic consolidation pressure and cyclic stress ratio used in the tests. Moreover, the relationship between observed doubleamplitude axial strain and the energy dissipated per unit volume is examined. It is found that this relationship is greatly dependent on the relative density of the sample. Introduction Earthquake-induced soil liquefaction involves significant loss of the soil’s strength and stiffness due to pore water pressure generation, which takes place concurrently with the dissipation of energy, mainly due to the rearrangement of soil’s particles (Nemat-Nasser & Shokooh, 1979). The cumulative energy dissipated per unit volume of soil has been satisfactorily correlated to the pore water pressure build-up prior to the onset of liquefaction (e.g. Nemat-Nasser & Shokooh, 1979; Simcock et al., 1983; Okada & Nemat-Nasser, 1994; Polito et al., 2013) instigating the development of several energy-based approaches to evaluate site liquefaction potential (e.g. Davis & Berrill, 1982; Berrill & Davis, 1985; Figueroa et al., 1994; Kokusho, 2013). According to Liang et al. (1995), when compared with stress-based or strain-based alternative procedures, approaches for the assessment of liquefaction potential based on energy offer some potential advantages, namely: (a) it is not necessary to decompose the erratic time-history of earthquake-induced stresses (or strains) to find an equivalent uniform loading; (b) the dissipated energy per unit volume depends on both stress and strain; (c) energy is a scalar quantity, which has the potential to simplify the liquefaction evaluation. In this paper, a comprehensive study of energy dissipated during cyclic triaxial tests is undertaken. Results of undrained cyclic triaxial tests performed on air-pluviated samples of Hostun sand prepared with different initial densities and subjected to different confining pressures are 1PhD student, Dep. of Civil Engineering, University of Coimbra, Coimbra, Portugal, ricardoazeiteiro@uc.pt 2Assistant Professor, Dep. of Civil Engineering, University of Coimbra, Coimbra, Portugal, pac@dec.uc.pt 3Lecturer, Dep. of Civil and Environmental Engineering, Imperial College London, UK d.taborda@imperial.ac.uk 4Assistant Professor, Dep. of Civil Engineering, University of Coimbra, Coimbra, Portugal, graza@dec.uc.pt presented. The energy dissipated per unit volume is determined from the experimental results and its correlation to excess pore water pressure generation and accumulated strain amplitude is thoroughly examined. Specifically, the effects of the confining pressure, void ratio and cyclic stress ratio (CSR) on those relationships are carefully investigated. The outcome of this experimental study intends to contribute to the further development of energy-based approaches for liquefaction assessement, in order to enhance its use in current practice. Laboratory testing programme Material and equipment used All tests were performed on Hostun RF sand, which is a fine-grained, sub-angular to angular, silica sand (Flavigny et al., 1990). In order to avoid segregation, the sand is uniformly graded between no. 20 (0.850 mm) and no. 200 (0.075 mm) sieves of ASTM series. The mean particle diameter, D50, and the uniformity coefficient, Cu, are close to 0.33 mm and 1.4, respectively. The density of soil particles, Gs, is 2.64 and the minimum and maximum void ratios, emin and emax, are approximately 0.66 and 1.00, respectively (according to ASTM D4253-00 (2006) and ASTM D4254-00 (2006), respectively). Over the last decades, this sand has been used as a reference material for liquefaction studies (e.g. Konrad, 1993; Marques et al., 2014). A fully computer-controlled hydraulic triaxial apparatus of the Bishop & Wesley (1975) type, designed for 38 mm diameter specimens, was used to perform all tests. In order to apply extension loading, a flexible sleeve connecting a top cap to a reaction head was used. The instrumentation consisted of cell and pore water pressure transducers, a submersible load cell, an externally mounted LVDT and a volume gauge. The data provided by these instruments were continuously acquired by a data logger and transferred to a computer. Experimental procedures Air-pluviation of dry sand was used to prepare all samples with a height/diameter ratio close to 2. Different relative densities were attained by varying the rate of pouring. For moderately loose samples, a miniature container was used, with the rate of pouring depending on the number and size of the openings. In the case of dense samples, the multiple sieving pluviation technique was used. Mass and volume measurements after preparation were performed to determine the density of the produced samples. In order to minimise undesirable non-measured variations of the void ratio, a small suction of about 5 kPa was used to sustain the sample after dismounting the mould used during sand pluviation. Subsequently, during the saturation stage, the flow of de-aired water was induced by a small differential pressure (in general, lower than 5 kPa). A Skempton’s B-value above 0.98 was measured in all tests before proceeding to the consolidation stage. All samples were isotropically consolidated to an effective pressure of 80 or 135 kPa (Table 1), before shearing. Two different stages of undrained cyclic shearing were established by varying the axial stress only. In the first stage, the test was stress-controlled using a frequency of 1.5 cycles/h. As samples approached initial liquefaction and a hysteresis-type stress-strain behaviour became apparent, strain-controlled loading with reversal governed by a chosen deviatoric stress amplitude was used, allowing a better control of the test. A constant axial strain rate of ±1 %/h was initially specified, being increased by two, four and eight times once the axial strain exceeded 0.5, 1 and 2 %, respectively. All samples were tested until large strains and excess pore water pressures were measured. Table 1 summarises the initial conditions of all conducted tests. Table 1. Summary of the initial conditions of the cyclic triaxial tests performed Test ID § e0 ‡ s'0 (kPa) ‡ Dq (kPa) CSR = |Dq| / (2 s’0) ICUCT 0.803/80/0.225 0.803 80 ± 36 0.225 ICUCT 0.832/80/0.250 0.832 80 ± 40 0.250 ICUCT 0.804/80/0.300 0.804 80 ± 48 0.300 ICUCT 0.793/135/0.250 0.793 135 ± 67.5 0.250 ICUCT 0.579/80/0.250 0.579 80 ± 40 0.250 § The designation identifies: 1) the type of consolidation – IC for isotropic consolidation; 2) the type of drainage –U for undrained test; 3) the type of loading –CT for cyclic triaxial loading; 4) the void ratio immediately after consolidation; 5) the isotropic consolidation stress; 6) the cyclic stress ratio (CSR). ‡ Post-consolidation values. Energy dissipated under undrained cyclic triaxial loading The energy dissipated per unit volume of soil per loading cycle, , can be determined using the area of the stress-strain hysteresis loop corresponding to that cycle, . Although conceptually simple, difficulties can arise from the fact that the measured stress-strain loops are usually not closed (Figure 1a). In order to overcome that problem, the following methodology was used in this study: 1) isolate each half-cycle stress-strain loop; 2) create a symmetric image of each half-cycle stress-strain loop about its centre; 3) centre the half stress-strain loop and its corresponding mirror image at the origin; an entire stress-strain loop is then defined by the original and the reflected half-loop (Figure 1b); 4) estimate the area enclosed by each closed loop using a trapezoidal approximation (Equation (1)), which represents the energy dissipated per unit volume per loading cycle, : = ≈ 1 2 ( ) + ( ) ( ) − ( ) (1) where q is the deviatoric stress, ea is the axial strain and k is the total number of points in which the stress-strain loop is discretised. Figure 1. Methodology used for the estimation of the energy dissipated per unit of volume per loading cycle (ICUCT 0.832/80/0.250 test) Experimental results and analysis Residual pore water pressure build-up as a function of dissipated energy Figure 2a) presents the excess pore water pressure ratio generation, = ∆ ′ ⁄ , as a function of the number of loading cycles, , for the ICUCT 0.832/80/0.250 test. As expected, initially increases when subjected to compression loading, while decreasing under extension loading. Once the Phase Transformation line (Ishihara et al., 1975) is crossed and the behaviour of sand changes from plastic contraction to plastic dilation and vice-versa, a double-frequency variation of with occurs (in the case of ICUCT 0.832/80/0.250 test, this can be observed from the 5th cycle of loading). With the purpose of relating the excess pore water pressure generation to the dissipated energy, the values of corresponding to a null deviatoric stress, i.e. the residual values, ( ) , were computed (circle points shown in Figure 2a). These values were then plotted against the accumulation of dissipated energy per unit volume, ∆ , normalised by the initial confining pressure, ′ , as presented in Figure 2b). Due to the previously mentioned double-frequency varia