# Energy Evaluation of Triggering Soil Liquefaction Based on the Response Surface Method
## 1 Introduction  
Liquefaction is one of the most destructive effects of earthquakes. This phenomenon, which has occurred several times during recent earthquakes, is caused by seismic shear waves that propagate upward to the surface layers, increasing pore water pressure in saturated, relatively loose or loose sandy deposits. Liquefaction occurs when rapid earthquake motion prevents drainage, thereby increasing excess pore water pressure to as much as initial overburden stress. Liquefaction causes severe property damage and fatalities. After liquefaction, the strength and stiffness of the liquefied soil are considerably decreased, often resulting in a range of structural failures. Evaluating the liquefaction potential guides the decision of which method is to be applied to prevent the disaster. A number of approaches and models have been presented to assess the liquefaction potential of soils. Some of these approaches follow a stress-based procedure, on the basis of the equation presented by [PERSON] and [PERSON] [1]. During this process, the shear stress induced by an earthquake is first determined according to the peak ground surface acceleration (\(a_{max}\)). Then, the cyclic stress ratio (CSR) and the cyclic shear strength (CRR) are determined, and by comparing them the potential of liquefaction is analyzed [2; 3; 4; 5; 6; 7; 8; 9; 10; 11]. Strain-based methods have also been conducted by supposing that pore water pressure grows by control of the cyclic shear strain during dynamic loads [12; 13]. In contrast, other researchers have assessed the potential of liquefaction through an energy-based method, which considers the energy dissipated into the soil by earthquake motions. This methodcan be divided into three groups based on case histories during earthquakes that occurred in the past [14; 15; 16], the Arias intensity (\(I_{h}\)) [17; 18], and laboratory test results [19; 20]. [PERSON] et al. [21] performed some uniaxial shaking tests and demonstrated that the difference between strain energy generated in the soil caused by biaxial and uniaxial shaking tests is negligible. [PERSON] et al. [22] studied the effect of non-proportionality and the phase angle of the induced shear stresses on rising pore water pressure. Moreover, researchers investigated the influence of five parameters, including the initial effective mean confining pressure (\(c_{c}^{\prime}\)), initial relative density (\(D_{r}\))?o, fine content (\(FC\))?o, coefficient of uniformity (\(C_{u}\)), and mean grain size (\(D_{50}\)), on the capacity energy of soils (\(W\)) [13; 23; 24; 25; 26; 27; 28; 29; 30; 31] by considering laboratory test results. [PERSON] et al. [26] collected a large number of datasets with a wide range of test results, including six parameters, and they divided them randomly into a testing and training phase in order to present artificial neural networks (ANNs). Furthermore, they eliminated the coefficient of curvature (\(C_{c}\)) due to no increase in the model's accuracy, and they presented new artificial neural network (ANN) and multi-linear regression (MLR) models, including five parameters containing \(c_{c}^{\prime}\), \(D_{r}\)?o, \(FC\)?o, \(C_{u}\), and (\(D_{50}\)) in mm. With the same dataset collected by [PERSON] et al. [26], [PERSON] et al. [28] developed three new models. By adding new data to the dataset of [PERSON] et al. [26] and applying a neuro-fuzzy interface system (ANFIS), [PERSON] et al. [32] developed a model that included six parameters containing \(C_{c}\) and demonstrated its influence. However, data division was conducted randomly in all studies, without considering the statistical aspects of parameters. Furthermore, data was divided into two groups, testing and training, without performing a validating phase to prevent overtraining of the ANN model. A validating phase is applied to minimize overfitting of the trained model [33; 34].