A simplified method to predict failure of sands under general cyclic simple shear loading Open Access

a, b, c

regression coefficients

C_u

coefficient of uniformity of the grain size distribution of the test sand

D₅₀

mean particle size of the test sand

D_r

relative density, also known as density index (I _D), of the test sample

E_f

cumulative specific energy dissipated by the sample from initial conditions until failure

e_max

maximum void ratio of the test sand

e_min

minimum void ratio of the test sand

f

frequency of the cyclic load

G_s

specific gravity of the test sand

H

height of the sample

γ

shear strain

ΔW_i

dissipated energy for the ith shear stress load cycle

σ_atm

atmospheric pressure at sea level (1 atm or 101.325 kPa)

$σv′$

effective vertical stress acting on the sample at any time during a cyclic simple shear test

$σvo′$

effective vertical stress applied to the sample at the start of a cyclic simple shear test

τ

shear stress applied to the sample at any time during a cyclic simple shear test

τ_max

maximum shear stress applied to a sample in each cycle

$τmax/σvo′$

cyclic stress ratio for a cyclic simple shear test subjected to harmonic cyclic shear loading

Accurately predicting the cyclic behaviour of sands subjected to either drained or undrained loading conditions remains a challenge in geotechnical engineering. The main challenge is predicting the accumulation of deformations and strains during cyclic loading and the occurrence of failure. This paper presents a simplified approach to predicting the failure of dry sands under constant-volume (CV) cyclic simple shear (CSS) independent of the characteristics of the shear stress–time history (e.g. the method can be applied also to non-uniform or irregular cyclic loading). The method is based on using a multivariable regression obtained using a relatively small number of experimental results performed on the sand of interest prepared at different sample initial states (i.e. D_r and $σvo′$⁠) selected based on the project range of interest and subjected simply to only one level of sinusoidal simple shear loading until failure is achieved. The regression is performed on only one test per initial sample state, and it requires only the use of a readily accessible, conventional CSS device. It is important to point out that the proposed methodology will yield a sand-specific regression to capture the intrinsic material properties that are unique to each test sand and are independent of the state of the sand. Examples of intrinsic material properties that make the regression specific to each test sand include grain shape, grain size distribution, mineralogy, mineral-to-mineral friction angle and specific gravity. To highlight this, the proposed methodology was applied to two different test sands yielding different regressions to capture the unique intrinsic characteristics of each soil. The influence of the intrinsic properties in the shear behaviour and strength of sands has been reported extensively in the literature (e.g. Been et al., 1991; Cho et al., 2006). Therefore, the regression obtained using the proposed simplified methodology is applicable only to the sand used in the testing programme. The proposed simplified method is based on the cumulative energy hypothesis reported in the literature that states that the dissipated energy per sample volume required to reach failure depends only on the initial state of the sample and is independent of the characteristics of the cyclic loading applied to the sample. Thus, this cumulative energy hypothesis allows using a multivariable regression of the specific energy to failure measured using a small number of data based on easy standardised CSS tests that apply uniform sinusoidal cyclic shear loading. Note that since the cumulative energy hypothesis states that only the initial sample state is what determines the required specific energy to failure, only one level of amplitude and one frequency level for the sinusoidal CSS test need to be performed for each sample state. This study shows that a multivariable regression based on a relatively small and simple experimental programme consisting of uniform cyclic simple tests on the different initial sample conditions of interest can predict with reasonable accuracy the failure of the same test sand under more general cyclic shear loading time histories, including complex irregular cyclic shear stress–time histories that are similar to the ones generated by an earthquake. It is important to point out that in the literature, researchers have proposed regressions for their full experimental data set to help identify the main variables that influence the results of dissipated energy to failure. However, the use of a multivariable regression that is based on using the smallest possible number of test results obtained using only one simple shear cyclic test per sample initial state subjected to uniform sinusoidal load cycles with only one frequency and amplitude level is proposed in this paper.

In this paper, the multivariable regression is based on data from CV CSS tests that are shown schematically in Figure 1(a). The dissipated energy per unit volume of a sample subjected to a load cycle of a CV CSS test is equal to the area of the hysteresis loop in the plot of shear stress against shear strain, as shown in Figure 1(b) as the shaded area for the ith stress-controlled sinusoidal shear stress load cycle of a CSS test. These tests, when run under CV conditions and under stress-controlled cycles, show reduced energy dissipation for the initial stress–strain cycles and increased energy dissipation as the number of load cycles increases. The cumulative specific dissipated energy to failure (E_f) is computed by adding the different hysteresis loop areas (ΔW_i) of the successive load cycles until sample failure (as defined in the study that is being conducted) is reached. A detailed description of the procedure used to compute the cumulative specific dissipated energy to failure (E_f) for CV CSS tests, under uniform or irregular shear stress–time histories, can be found in the paper by Zavala et al. (2022).

Several researchers have evaluated the cumulative energy dissipation of sands for different types of cyclic tests (e.g. Azeiteiro et al., 2017; Fardad Amini and Noorzad, 2018; Figueroa et al., 1994; Green and Terri, 2005; Kokusho, 2017; Kokusho and Kaneko, 2018; Kokusho and Mimori, 2015; Lasley, 2015; Lasley et al., 2016; Liang et al., 1995; Polito et al., 2013). A summary of key findings relevant to this paper from these studies is presented below.

Figueroa et al. (1994) and Liang et al. (1995) performed a series of cyclic undrained hollow cylinder torsional shear tests on Reid Bedford sand samples that had been prepared at three relative densities and three initial effective confining stresses. A total of 36 tests (uniform sinusoidal and random loadings) were run, and the dissipated energy to failure was calculated in each case. These studies reported that the cumulative dissipated energy per unit volume required to reach failure was dependent only on the initial sample state defined by the initial relative density and the initial effective confining pressure and was reported to be independent on the dynamic loading form used in their experiments. This finding supports the cumulative energy hypothesis mentioned earlier that states that the dissipated energy per sample volume required to reach failure depends only on the initial state of the sample and is independent of the characteristics of the cyclic loading applied to the sample. Furthermore, these two studies developed logarithmic regressions based on all their data, using tests with both uniform and random loadings, to estimate the dissipated energy to failure as a function of the initial state parameters. These studies recommended use of dissipated energy to evaluate liquefaction potential of sands under general earthquake loads.

Polito et al. (2013) performed 28 cyclic triaxial tests using five different shapes of periodic deviatoric load time histories and a range of amplitudes defined in terms of cyclic stress ratios (CSRs) to identical specimens of Ottawa 20/30 sand, prepared at an initial sample state with a 22% relative density and an initial confining stress of 100 kPa. The authors reported that the dissipated energy required to reach initial liquefaction was independent of the load shape used. Therefore, this is another study that supports the cumulative energy hypothesis mentioned earlier.

Researchers at Virginia Tech (Green and Terri, 2005; Lasley, 2015; Lasley et al., 2016) have also promoted the use of energy-based approaches to investigate the behaviour of sands under cyclic loading. For example, Green and Terri (2005) presented an energy-based alternative to the Palmgren–Miner fatigue hypothesis to account more appropriately for the non-linear behaviour of soils. Lasley (2015) and Lasley et al. (2016) presented correlations to find the number of equivalent uniform cycles to represent general earthquake loading using an energy-based approach. A comprehensive experimental programme that included 49 uniform loading and 24 earthquake loading CV CSS tests performed on Monterrey 0/30 sand samples was reported by Lasley (2015). The initial states for the test samples of this study were compacted to relative densities between 30 and 80% and subjected to three different levels of the initial effective vertical stress. The results from this study also support the validity of the cumulative energy hypothesis. Lasley (2015) presents an exponential regression developed using all the test results that correlates the dissipated energy to failure, normalised to the initial effective stress level, with the initial relative density of the sample. This study is described in more detail later in this paper, as its results were used to evaluate the simplified methodology proposed herein. The regression reported by Lasley (2015) is based on using the full data set, in contrast to the approach proposed in this study, which uses a multivariable regression developed using only a small data set of dissipated energy to failure measured from only one simple uniform CSS test per sample initial state.

Kokusho and Kaneko (2018) reported results from torsional simple shear tests on Futsu beach sand with 31 tests using uniform harmonic loading and six tests using different irregular cyclic loading types. This study reported that cumulative dissipated energy predicted cyclic liquefaction failure reasonably well for a specified initial state and was independent of the loading stress–time history type used.

A recent study by the authors (Zavala et al., 2022) also investigated experimentally the cumulative dissipative energy hypothesis. The authors performed a comprehensive experimental programme that tracked dissipated energy from start to failure using 258 CV CSS tests on Ottawa 20/30 sand. The authors considered nine sample initial states and a wide variety of cyclic loading types that included uniform and non-uniform shearing loading. The results of this study supported the validity of the hypothesis that the cumulative specific dissipated energy required to reach failure of a uniform dry sand sample subjected to CV stress-controlled CSS testing is reasonably constant and independent of the type of stress–time history used in the testing.

The literature summarised above that involves detailed experimental studies on uniform sands under different types of cyclic loading (uniform or complex or irregular) supports the general validity of the hypothesis of a cumulative specific dissipated energy that is dependent on only the initial state of the sand sample and independent of the characteristics of the cyclic loading applied to the sample. However, it is important to note that most experimental studies show some scatter in the measured dissipated energies to failure, which is common in these kinds of tests in granular materials. Regardless of the inherent variability of experimental E_f values, the literature reports the hypothesis as valid.

This hypothesis is the foundation of the proposed simplified methodology for predicting failure of uniform sands under general cyclic loading based on a small data set that can involve readily available and easy-to-perform uniform, sinusoidal, CSS tests. The proposed methodology is described in the next section and then evaluated and validated using the data sets reported by the authors (Zavala et al., 2022) and by an independent study published by Lasley (2015).

A simplified method that offers a convenient way to estimate the cumulative dissipated energy to failure for a specific granular material subjected to general cyclic loading is proposed herein based on the constant cumulative dissipated energy hypothesis that is described in the previous section. The proposed method involves making predictions using a multivariable regression developed using a small data set of CSS tests performed using the simplest form of cyclic loading of uniform sinusoidal cyclic loading. The method is based on the validity of the constant cumulative dissipated energy hypothesis, and as such, only one test per initial sample state (i.e. sample relative density and initial effective vertical stress) is needed. The size of the proposed reduced data set will depend on the range of sample initial states involved in the project or application of interest.

A flow chart showing the main steps of the proposed simplified method is shown in Figure 2. Steps 1 and 2 involve selecting the initial sample states (D_r and $σvo′$⁠) that will be tested using the simplified CSS testing. The third step in the flow chart involves performing one CV CSS test per sample initial state using uniform harmonic shear loading at only one frequency and amplitude level (or CSR level). Step 4 in the simplified method involves computing the cumulative specific dissipated energy to failure measured in each of the CV CSS tests of step 3 performed for the different selected initial sample states. The final step involves developing a multivariable regression that relates the cumulative specific dissipated energy to failure to the initial state variables D_r and $σvo′$⁠.

The multivariable regression involved in step 5 of Figure 2 can be of any form selected by the user. For example, as mentioned before, Figueroa et al. (1994) and Liang et al. (1995) used logarithmic multivariable regressions to relate the cumulative dissipated specific energy to failure (E_f) to the different sample initial states using the full experimental data set. In contrast, Lasley (2015) used an exponential type of regression to estimate E_f based on the sample initial states and also used the full experimental data set. In this study, several forms of regression equations were investigated, but a logarithmic equation was found to work well and to be convenient. Furthermore, it is recommended to use a dimensionless regression by normalising the cumulative specific dissipated energy to failure (E_f in kJ/m³, which is equivalent to kPa) and the initial effective vertical stress (⁠$σvo′$ in kPa) with respect to the atmospheric pressure (σ_atm = 101.325 kPa), as follows:

where a, b and c are the regression coefficients and the other variables are as defined before.

The regression coefficients in Equation 1 are obtained using the reduced data set involving one CV CSS test with uniform sinusoidal loading per initial sample state. The obtained multivariable regression can now be used to predict the cumulative specific dissipated energy to failure for any sample initial state within the range of interest and for general cyclic loading conditions including different periodic loading functions, non-periodic loading or general irregular cyclic loading such as those generated by earthquakes.

This section shows the results of applying the proposed simplified method to data from two comprehensive experimental programmes that involved CV CSS tests on uniform dry sands subjected to a wide range of cyclic loading types (e.g. uniform, periodic, non-periodic and complex irregular earthquake loading). The two data sets are from the study by Zavala et al. (2022), performed by the authors, and from an independent test programme by Lasley (2015). The main features of both experimental programmes are summarised in Table 1.

Additional details regarding these two experimental studies, such as the sample preparation procedure used, detailed test results (e.g. stress–strain load cycles), complex cyclic shear loading such as earthquake-type loading and other information can be found in the papers by Zavala et al. (2022) and Lasley (2015).

The following subsections present the multivariable regressions obtained for both data sets using the simplified procedure and the steps in the flow chart of Figure 2. The simplified procedure was applied using one CV CSS test with uniform sinusoidal loading per initial sample state within the range of interest for each study. It should be noted that due to the non-linear nature of the multivariable regression, a minimum of nine initial states is recommended to capture reasonably the curvature of the regression surface.

Zavala et al. (2022) considered a total of nine initial sample states corresponding to initial vertical effective stresses (⁠$σvo′$⁠) of 100, 200 and 400 kPa and relative density levels of loose, dense and very dense (see Table 1 for actual D_r values). Therefore, a total of nine simple CV CSS tests performed using uniform harmonic shear stress cycles at a frequency of 0.1 Hz at only one level of CSR were used to obtain the multivariable regression of the proposed simplified approach. Obviously, a multivariable regression could have been developed using more tests (or even the full data set), to include, for example, more CSR values, frequencies or test repetitions that are commonly performed to capture the inherent variability of experimental results related to variations of the sample D_r and fabric of the sand samples. However, the purpose of this paper is to assess the accuracy and usefulness of using a regression obtained based on the smallest amount of data without sacrificing accuracy when predicting failure of the same test sand under other cyclic loading including more complex earthquake-like cyclic loading. The subset of data used to develop the multivariable regression for the experimental study by Zavala et al. (2022) is summarised in Table 2.

Using the regression form presented in Equation 1 and the reduced set of data in Table 2, the following multivariable regression was obtained for the test sand tested by Zavala et al. (2022):

The above multivariable regression has a coefficient of multiple determination (or R²) of 0.946. A three-dimensional (3D) plot representation of this multiple regression is shown in Figure 3. In this figure, the two horizontal axes correspond to the independent variables (i.e. the sample initial state defined by the normalised initial effective stress and the sample initial relative density) and the vertical axis corresponds to the dependent variable – that is, the normalised cumulative specific dissipated energy. The regression defined by Equation 2 is shown as a 3D mesh surface in the figure. Besides the 3D surface that corresponds to the regression, Figure 3 also shows the nine experimental data points listed in Table 2. A blue line projecting the points across the 3D surface indicates that the point is above the surface, whereas a red dotted line indicates that the point is below the 3D surface. This figure qualitatively shows a good correlation between the nine data points and the regression in Equation 2.

The regression obtained using the simplified approach and the results from the selected nine uniform CV CSS tests in Table 2 are evaluated in this section. This was done by comparing the predicted values of cumulative specific dissipated energy to failure using the regression in Equation 2 with the energy to failure values measured in experiments from the full data set reported by Zavala et al. (2022) that were not used to develop the regression.

The full data set reported by Zavala et al. (2022) involved a wide range of CV CSS tests prepared at the nine initial sample states mentioned earlier and listed in Table 2. Zavala et al. (2022) reported results from a large number of CV CSS tests using the following five types of load patterns: (a) uniform harmonic cyclic loading (six levels of CSR and three frequencies), (b) alternating sine waves of variable CSR values at a constant frequency (considered combinations of four CSR levels and three frequencies), (c) alternating sine waves of different frequency but constant CSR level (combined 0.1 and 0.5 Hz and considered five CSR levels), (d) alternating sine waves with different frequency and CSR levels (considered combinations of four CSR levels and combined 0.1 and 0.5 Hz) and (e) spike loading consisting of a set of low-CSR uniform sinusoidal waves followed by a sudden large CSR spike, and this sequence was repeated until failure. A detailed description of these experiments and results is outside the scope of the paper, but additional information of this study can be found in the paper of Zavala et al. (2022).

As mentioned earlier, the results of the Zavala et al. (2022) study supported the general validity of the constant cumulative dissipated energy to failure hypothesis. However, as in other studies, experimental results have the usual inherent level of variability. A summary of the experimental values of cumulative specific dissipated energy to failure measured in this study for the nine initial sample states is presented in Table 3, and statistical descriptors such as mean, standard deviation and other parameters are reported based on all the E_f values measured for each sample initial state considering all tests and all types of cyclic load time histories reported in this study.

A comparison of the regression obtained using the simplified method and the reduced set of data (i.e. Equation 2) with the full data set of 258 CV CSS test results reported by Zavala et al. (2022) is shown in Figure 4. Like Figure 3, in this figure, blue lines projecting from the data points across the 3D surface indicate that the points are above the surface, whereas red dotted lines indicate that the points are below the 3D surface. This figure qualitatively shows that even though the dissipated energy to failure in tests with larger relative density and larger initial effective stress has more scatter, the simplified method estimation is close to the average dissipated energy values.

As an alternative way to assess the regression presented in Equation 2 to predict the cumulative specific dissipated energy to failure from any CV CSS test, even general cyclic loading, a comparison was made between the predicted (i.e. using Equation 2) and the measured cumulative specific dissipated energy to failure as shown in Figure 5. This plot also shows the 1:1 equality line that would correspond to a perfect prediction and two lines corresponding to predictions that are ±25% with respect to the measured values; 54% of the predictions fall within this ±25% range. The percentage of predictions that fell within the larger range of ±40%, with respect to the 1:1 line, was found to be 77%.

An additional evaluation of the simplified method was made by comparing the accuracy of Equation 2 with the level of accuracy of predictions of E_f values using a multivariable regression developed using the whole data set reported by Zavala et al. (2022). As mentioned earlier, different regression equation types were considered, but for the sake of brevity, only the results for the logarithmic-type regression are reported. Using the whole data set reported by Zavala et al. (2022), the following multivariable regression was obtained:

where the variables are as defined before.

The value of the coefficient of multiple determination (or R²) for the regression presented in Equation 3, which used all 258 CV CSS test results from this study, was 0.802. In general, the higher the R², the better the model fits the data. The computed relatively high value of R² suggests that the proposed model fits the experimental data reasonably well.

Figure 6 shows a comparison plot of the predicted (i.e. using Equation 3) and the measured cumulative specific dissipated energy to failure. It also shows the 1:1 equality line that would correspond to a perfect prediction and two lines corresponding to predictions that are ±25% with respect to the measured values; 57% of the predictions fell within this ±25% range. The percentage of predictions that fell within the larger range of ±40%, with respect to the 1:1 line, was found to be 83%.

An additional comparison of the two regressions – that is, Equation 2 (simplified method using nine tests) and Equation 3 (using the whole data set, n = 258) – can be made by comparing the root mean square error (RMSE) and the mean absolute error (MAE) of the measured against the predicted data. The RMSE is the standard deviation of the residuals (prediction errors) and is a standard way to measure the error of a model in predicting quantitative data. The MAE is the average of the errors between predicted and measured data and is another way of measuring the accuracy of prediction models. The value of the RMSE between the measured dissipated energy and the predicted dissipated energy (using the simplified method, i.e. Equation 2) is 0.414 kJ/m³, and its value between measured and estimated dissipated energy using the whole-data-set regression (i.e. Equation 3) is 0.389 kJ/m³. Likewise, the value of the MAE between the measured and predicted dissipated energy using Equation 2 is 0.314 kJ/m³, and its value between measured and estimated dissipated energy using Equation 3 is 0.289 kJ/m³. These close RMSE and MAE values suggest that the correlations between measured and estimated data using both equations have a similar level of accuracy. A graphical comparison of the predictions made using Equation 2 (proposed simplified method) and Equation 3 (whole-data-set regression) is shown in Figure 7. This plot shows that the two correlations have similar predictions, as they are very close to the 1:1 equality line shown in this figure.

Besides to the 258 tests reported by Zavala et al. (2022), the authors also performed several unpublished CV CSS tests subjected to complex shear stress–time history like the one shown in Figure 8. For the sake of brevity, only one test with general irregular shear loading is presented and discussed. The shear stress–time history applied to the sample was obtained using one-dimensional seismic site analyses with the program Shake91 (Idriss and Sun, 1992) to replicate the cyclic stresses that a sand sample would experience during an earthquake. The time history shown in Figure 8 has a duration of 44 s, a predominant frequency of 0.8 Hz and a peak normalised shear stress (NSS) of 0.156.

The CV CSS test had an Ottawa sand sample with an initial state defined by a very dense sand state (D_r = 91.3%) and an initial effective vertical stress of 200 kPa. The peak shear stress of the applied signal was 31.1 kPa, which occurred at a signal time of 18.5 s. The sample was tested using the same device and procedure described in the paper by Zavala et al. (2022). The results for this CV CSS test under earthquake-like loading are summarised in Figure 9. The summary of results is presented using four plots with matching axes. For this specific test, the figure shows that the shear strains are very low when the applied NSS is low and below 0.05, which occurs during the first 17 s of the excitation signal. For this test, NSS values are the value of the applied shear stress normalised with respect to 200 kPa. Larger shear strains start to develop in the sample when the NSS values reach about 0.10, but are still not significant in magnitude. As shown in Figure 9, the shear strains increase dramatically when the peak NSS of 0.156 is reached at a time of about 18.5 s. The strains continue to accumulate past the load cycle with the peak shear stress until failure is reached. As described in the paper by Zavala et al. (2022), failure was defined when the sample reached a double amplitude strain of 7.5%. For this test, failure occurred at a time of 20.6 s.

The cumulative specific dissipated energy to failure measured in this test was 1.326 kJ/m³. Using the simplified methodology, based on the regression in Equation 2, the estimated dissipated energy to failure for a sample with an initial state of D_r = 91.3% and an initial effective vertical stress of 200 kPa is 1.460 kJ/m³. This difference between the predicted and measured E_f values corresponds to an error of 10.1%, which is a reasonably good prediction for this type of parameter.

The proposed simplified method was further evaluated using the independent experimental data set published by Lasley (2015). This author conducted 49 CV CSS tests on Monterey 0/30 sand samples using uniform cyclic loading and 24 tests using earthquake-like loading. The main features of this experimental study are summarised in Table 1.

According to the methodology explained above, nine representative tests were selected, which include the three initial vertical stresses considered (60, 100 and 250 kPa) and relative densities between 25.9 and 70.6%. The data points used for this regression were selected to cover the range of initial conditions corresponding to most of the full data set and are shown in Table 4.

The obtained regression formula obtained using this simplified data set is as follows:

Figure 10 shows the 3D mesh plot of the regression corresponding to Equation 4 and the nine points that represent the selected initial conditions (horizontal axes) and the measured normalised cumulative specific dissipated energy to failure (vertical axis) for the reduced data set. The 3D mesh shows a relatively good correlation with the selected data points, except for the point corresponding to the highest relative density and initial effective stress. Even though this point was one of the nine data points chosen to develop the regression, the inherent variability of this type of test may cause values of dissipated energy to not match the prediction very well. The value of the coefficient of multiple determination (or R²) of the regression presented in Equation 4 is 0.958.

The regression obtained with the simplified set of data (Equation 4) is plotted against the full data set in Figure 11. This figure shows a 3D mesh plot of that regression and the measured normalised cumulative specific dissipated energy to failure for the full data set. The graph qualitatively shows a reasonably good correlation, except for the data point corresponding to the highest relative density and initial effective stress, which was also mentioned above.

To assess further the ability of the regression presented in Equation 4 to predict the dissipated energy to failure in a CSS test, a comparison was made between the predicted and measured normalised cumulative specific dissipated energy to failure. The former was calculated using the regression obtained through the simplified set of data, and the latter was measured for each of the tests in the full data set from Lasley (2015). This comparison is shown in Figure 12, which indicates that in most cases the dissipated energy prediction is close to the measured dissipated energy for the same initial testing conditions.

As mentioned before, the RMSE and the MAE are standard ways to measure the error of a model in predicting quantitative data. The value of the RMSE between the measured and predicted dissipated energy using Equation 4 for all of the data set is 1.235 kJ/m³, and the MAE value for the same is 0.632 kJ/m³. To evaluate further the predictions obtained using the simplified method and compare them with those made with a full-data-set regression, such regression was made using the Lasley (2015) whole data set, although it is not presented for the sake of brevity. The value of the RMSE between the measured and predicted dissipated energy using such regression for the all the data set is 1.422 kJ/m³, and the MAE value for the same is 0.442 kJ/m³.

Table 5 summarises the RMSE and the MAE obtained with the predictions using the simplified method and whole-data-set regression, for both data sets.

Table 5 shows that for both data sets, the values of RMSE and MAE for the dissipated energy predictions are similar when using the simplified method as compared with those when using the whole-data-set regression and supports the idea of using a reduced data set to create a regression to predict the cumulative specific dissipated energy to failure for sands subjected to cyclic shear loading.

This paper describes a simplified approach proposed to predict the failure of dry sands subjected to general shear stress–time histories such as the ones experienced during earthquakes. The proposed simplified method is based on the cumulative energy hypothesis reported in the literature that states that the dissipated energy per sample volume required to reach failure depends only on the initial state of the sample and is independent of the characteristics of the cyclic loading applied to the sample. Therefore, the proposed method allows the prediction of failure of sands under general cyclic loading, without the need for advanced and costly cyclic testing and can be based on a multivariable regression developed using results from a relatively small number of standard and readily accessible CSS tests involving uniform sinusoidal loading. The proposed simplified methodology is sand specific, as the failure of granular soils is controlled by their intrinsic properties (e.g. particle shape, grain size distribution, mineralogy); therefore, the obtained regressions are valid for each specific sand used to develop them. The small data set required for the regression involves only one test per sample initial state (e.g. D_r and $σvo′$⁠) that should be selected based on the project range of interest. The test required for each sample initial state involves a routine CV CSS test with only one level of sinusoidal simple shear loading tested until failure. The proposed simplified method was evaluated using two comprehensive experimental studies involving two different types of sands. The first data set is from an experimental programme by the authors reported in the paper by Zavala et al. (2022) that involved 20/30 Ottawa sand subjected to different cyclic loading types that included simple sinusoidal cyclic shear and also complex cyclic shear time histories. This paper also compares the simplified approach using an unpublished CV CSS performed on the same Ottawa sand but under general earthquake-type loading. Additionally, the proposed approach was also evaluated using an independently published data set by Lasley (2015) that used Monterey 0/30 sand. The predictions using the simplified approach, which is based on a small data set of simple CV CSS tests, were found to yield predictions with a similar level of accuracy to the predictions using regressions based on the full data set. Therefore, for the different sources of experimental data considered herein, the simplified approach was found to yield reasonably good predictions of the failure of the two test sands when subjected to complex and irregular shear stress loading.

The proposed simplified approach can help predict the failure of sands under general cyclic loading by tracking the dissipated energy of the sand sample during the test. This might be useful because it predicts the failure of sands under general cyclic loading without the need for specialised testing equipment usually required to apply this kind of loading scheme to a sand sample. This approach also has the potential to be extended to the prediction of failure of sands in the field when the expected applied energy to the soil can be estimated.

The main conclusions and recommendations drawn from this study are the following.

• A reduced data set of simple CSS tests, based on one CV CSS test per sample initial state within a range of interest, can be used to produce a regression equation that can predict reasonably well the dissipated energy to failure in the same test sand under general or complex cyclic loading. In the analysis made with the two data sets considered, the values of the RMSE and MAE between predicted and measured dissipated energy, using the simplified method, were found to be very similar to the ones obtained using a whole-data-set regression equation. The close values of RMSE and MAE suggest that the proposed simplified methodology has a similar level of accuracy to the more costly and time-consuming full-data-set regression.

• The simplified method was assessed and validated. Thus, it is possible to predict reasonably well the failure of a specific sand when subjected to general or complex simple shear loading by tracking the dissipated energy during the test and using the proposed simplified approach that is based on the cumulative energy hypothesis reported in the literature.

• The comparison of the predicted cumulative energy to failure against the measured experimental values in general not only supports the cumulative energy hypothesis reported in the literature but also shows that the dissipated energy values required to reach failure for a given sample initial state have important scatter that could be related to the inherent scatter of experimental test results in sands due to small differences in sample density and fabric.

• The regression obtained using the simplified set of CSS tests predicts fairly well the dissipated energy to failure of a large set of CSS tests under a wide range of loading conditions, including those subjected to an earthquake-type CSS test.

• It is recommended that the simplified method uses a set of CV CSS tests based on at least nine sample initial states (defined by the relative density and initial vertical stress) based on the range of interest of the project. Specifically, the recommendation is that the nine points should be based on the 3 × 3 matrix formed by the minimum, average and maximum values of the relative density and the initial effective stress (⁠$σvo′$⁠) of interest.

• This simplified method is particularly useful when access to advanced CSS testing devices is not available.

Recommendations for further studies in this topic include extending the assessment to other test sands and different types of lab tests (e.g. cyclic triaxial and cyclic torsional tests) and eventually to field conditions.