# Energy Evaluation of Triggering Soil Liquefaction Based on the Response Surface Method
## 2 Approaches Based on Laboratory Test Results  
By inspecting and monitoring the number of site responses to earthquakes' time history accelerations in the West of the United States and by introducing normalized maximum energy, which is the area under the stress-strain of earthquake motion at depths, [PERSON] [39] derived a relationship between maximum energy, \(D_{r}\), \(a_{max}\), and initial effective confining stress. Additionally, [PERSON] [13] derived the equations by performing 74 torsional shear tests on Reid Bedford sand, Lower San Fernando Dam (LSFD) silty sand, and Lapis Luster dried sand (LSI-30):  
Reid Bedford:  
\[\log_{10}(\delta W)=1.982+0.00477\sigma_{c}^{\prime}+0.0116D_{r}+0.0376\Gamma \ \ \ R^{2}=0.942 \tag{1}\]  
\[\log_{10}(\delta W)=2.002+0.00477\sigma_{c}^{\prime}+0.0116D_{r}\ \ \ R^{2}=0.937 \tag{2}\]  
Reid Bedford sand:  
\[\log_{10}(\delta W)=1.2062+0.0039\sigma_{c}^{\prime}+0.0124D_{r}+0.0376\Gamma \ \ \ R^{2}=0.995 \tag{3}\]  
LSFD sand:  
\[\log_{10}(\delta W)=2.484+0.00477\sigma_{c}^{\prime}+0.00052D_{r}\ \ \ R^{2}=0.995 \tag{4}\]  
\[\log_{10}(\delta W)=2.529+0.00477\sigma_{c}^{\prime}\ \ \ R^{2}=0.994 \tag{5}\]  
Figure 1: Flowchart of the process to derive the response surface method (RSM) equations in this study; BB: Box-Behnken; CC: central composite; HCC: half central composite. LSI-30 sand:  
\[\log_{10}(\delta W)=2.0554+0.004824\sigma_{c}^{\prime}+0.01267D_{r}\ \ \ \mathrm{R}^{2}=0.888 \tag{6}\]  
where \(\delta W\) is the cumulative unit energy (J/m\({}^{3}\)), \(\Gamma\) is the shear strain amplitude, and R\({}^{2}\) is the coefficient of determination. Furthermore, by conducting 27 strain-controlled torsional triaxial tests on Reid Bedford sand, [PERSON] [23] derived the following two regression equations:  
\[\log_{10}(\delta W) =1.8281+0.0051\sigma_{c}^{\prime}+0.0147D_{r}-0.1448f\ \ \ \mathrm{R}^{2}=0.93873 \tag{7}\] \[\log_{10}(\delta W) =1.8198+0.0051\sigma_{c}^{\prime}+0.0147D_{r}-0.1448f\ \ \ \mathrm{R}^{2}=0.93822 \tag{8}\]  
where \(f\) is the cyclic rate (Hz). For the first time, [PERSON] et al. [24] and [PERSON] [25] inspected the influence of particle size distribution on the potential of liquefaction according to the strain energy-based procedures. [PERSON] performed some cyclic torsional shear tests on the sand samples from Nevada, incorporating the \(C_{u}\) and the \(C_{c}\), and the author presented the equations expressed as:  
\[\log_{10}(\delta W)=3.6746+0.004877\sigma_{c}^{\prime}+0.01039D_{r}+0.21802C_{ u}-2.1444C_{c}\ \ \ \mathrm{R}^{2}=0.8195 \tag{9}\]  
\[\log_{10}(\delta W)=2.002+0.00477\sigma_{c}^{\prime}+0.0116D_{r}\ \ \ R^{2}=0.937 \tag{2}\]0  
\[\log_{10}(\delta W)=2.002+0.00477\sigma_{c}^{\prime}+0.0116D_{r}\ \ \ R^{2}=0.937 \tag{2}\]1  
where \(D_{i}\) is the particle diameter, which is given by a grain-size distribution for a given percent finer, denoted by the subscript \(i\). Through a statistical method, using the test results data of [PERSON] [40] and [PERSON] [25], [PERSON] [41] presented mathematical equations for Reid Bedford sand, LSFD silty sand, and Nevada sand. [PERSON] et al. [26] collected a large database containing 284 cyclic, triaxial, torsional shear, and simple shear test results, and they developed two ANN models. Thereafter, by comparing the ANN models' results with data from 18 centrifuge tests, they evaluated their model. Their first model included six input parameters--\(\sigma_{c}^{\prime}\), \(D_{r}\%\), \(FC\%\), \(C_{u}\), \(D_{50}\) (in mm), and \(C_{c}\)--and the output (the target) was \(W\) (J/m\({}^{3}\)). In the second model, they eliminated \(C_{c}\) and developed the model with the five extra input parameters. With the same database as [PERSON] et al. [26] and using genetic programming (GP), linear genetic programming (LGP), and multi-expression programming (MEP), [PERSON] et al. [28] developed three equations to evaluate \(W\). [PERSON] et al. [32] utilized an ANFIS on the same database as [PERSON] et al. [26] and illustrated the effect of input parameters by graphical representation.