# Energy Evaluation of Triggering Soil Liquefaction Based on the Response Surface Method
## 1 Introduction
In addition, [PERSON] [35] investigated the complicated influence of \(FC\) and illustrated that liquefaction resistance, in terms of the unit energy, starts to increase with an increase in \(FC\) above 28%. They also indicated that the liquefaction resistance becomes less dependent on relative density when \(FC\) is less than 28%. [PERSON] et al. [36] investigated 7000 dataset case histories from the 2010-2011 Canterbury Earthquakes and indicated that the evaluation of liquefaction is less accurate when soils have a high \(FC\) value. Although these studies have indicated an altered influence of a high \(FC\) value on liquefaction assessment, it has not yet been taken into account to propose a model. In this study, a larger dataset is first collected. Then, to evaluate the influence of \(FC\) on \(W\), two ANN models are constructed that include the following six input parameters: \(c_{c}^{\prime}\), \(D_{r}\)?o, \(FC\)?o, \(C_{u}\), \(D_{50}\), and \(C_{c}\). To analyze the complicated influence of \(FC\), the first ANN model, without any constraints in the range of input parameters, is derived similarly to other studies that have been performed and explained herein, and the second ANN model is derived through a database with \(FC\) values less than 28%, as in Tao [35]. To increase the accuracy and capability of the ANN models, the dataset is divided into three groups by considering the statistical aspects of parameters with similar mean as well as mean coefficient of variation (COV) values, instead of random division. The first group is for the training phase, the second is for testing, and the third is for the validating phase, to prevent overtraining in the training phase. In the second step, six visualized equations are captured by using the response surface method (RSM), which was demonstrated to be a capable method for evaluating liquefaction in sandy soil by [PERSON] et al. [37]. To the best of our knowledge, no other studies have been conducted on liquefaction using RSM. In Section 5, the dataset and two derived ANN models and their characteristics are described. According to any ANN model, three different design of experiments (DOEs) are performed. Therefore, three equations are obtained to illustrate and calculate correlation between six defined parameters and \(W\) as a target. During this step, the meaningfulness of all terms of the equations are analyzed through hypothesis testing, and to obtain more capable and reliable equations, some equation terms that do not provide a meaningful correlation with the target are eliminated, instead of performing an overall elimination of parameters such as \(C_{c}\). The final equations thus contain all six parameters and are presented in Section 6. Furthermore, by applying three different DOEs, their influence and capability are studied to determine the best DOE that can be applied for similar issues. Finally, to demonstrate the accuracy and capability of the presented equations, their predicted results are compared with four existing, well-known, and highly rated models that are currently used. Section 7 presents this comparison, using 20 samples from Dief [38] that are not included in the database to develop the ANN and RSM models for this study. Figure 1 illustrates the flowchart of the process applied in this study to develop the RSM equations.