## 4 Detrimental Effects of SSI
### Evaluation of System Frequency  
Involving SSI in the seismic designs could result in neutral/negligible, beneficial or detrimental effects on structure safety. Due to the uncertainties of seismic motions, there is little necessity for the seismic analysis on SSI having beneficial effects, while it is necessary to develop a method that determines the detrimental effects to ensure the seismic safety of structures. However, seismic hazard assessments on SSI are costly, even for linear problems. For a seismic engineering analysis, an efficient method that is applicable in coded standard procedures is needed. For an engineering seismic analysis, it needs efficient method that is applicable in coded standard procedures. Since the structure period (frequency) is the most important parameter in the seismic designs, it makes sense to predict on the shift of structure period (frequency) in engineering practice. In the following, we describe a method, using Figure 5 to predict the system frequency of the Millikan Library. The parameters of the structure are \(a=11.65\) m, \(b=12.55\) m, \(c=4.3\) m, \(H_{e}=29.3\) m, \(M_{0}=1.43\)\(\times\)\(10^{6}\) kg, \(M_{\text{b}}=1.07\)\(\times\)\(10^{7}\) kg, so we have \(b/a=1.08\), \(c/a=0.37\), \(H_{e}/a=2.52\), \(M_{\text{b}}/M_{0}=7.48\). The fifth layer (\(\beta_{5}=609.6\) m/s) is assumed as bedrock, and we simplify the multi-layered half-space to a uniform half-space by the equivalent shear wave velocity of the site  
\[\beta^{\text{eq}}=\frac{5.49+4.26+3.66+6.71}{298.7}=\frac{5.49+4.26+3.66+6.71}{ 387.1}=\frac{3.66+6.71}{454.2}=\frac{6.71}{887.7}=393.0\text{ m/s} \tag{3}\]  
so we have the dimensionless frequency in the NS direction in the equivalent uniform half-space as  
\[\eta_{\text{b}}=\frac{(2.16\text{Hz}\times 2\pi)\times 11.65\text{m}}{393.0 \text{m/s}\times\pi}=0.128 \tag{4}\]  
Figure 7: Ratio of dynamic responses between the SSI system and the fixed-base oscillator (shown by “FB”) (red circle: Hollywood Storage Building, gray star: Millikan Library, blue triangle: Sherman Oaks Building, green rectangle: Walnut Creek Building). We find the curve in parameters \(b/a=1\), \(H_{e}/a=2\), \(M_{\rm b}/M_{0}=10\) in uniform half-space at the \(x\)-axis of \(\eta_{\rm b}=0.128\), then we measure the value of \(y\)-axis of \(\widetilde{\eta}_{\rm b}/\eta_{\rm b}\ =\ 0.85\), to evaluate the system frequency \(\widetilde{\omega}_{\rm b}/2\pi\ =\ 1.836\) Hz. The evaluation of the system frequency using Figure 5 is very close to the true value 1.84 Hz. While in an EW direction, it gives \(\eta_{\rm b}\ =\ 0.075\), and we measure \(\widetilde{\eta}_{\rm b}/\eta_{\rm b}\ =\ 0.94\) from \(y\)-axis, to evaluate the system frequency \(\widetilde{\omega}_{\rm b}/2\pi\ =\ 1.18\) Hz, which is also very close to the true value 1.20 Hz. The parameters of the other three buildings are a little far from the parameter combinations described in Figure 5, so it is difficult to use Figure 5 to predict their system frequency. It is necessary to find a general method capable of covering a variety of structure and site parameters. However, the SSI system is multi-parametric, and dozens of target parameters are necessary to obtain the global function, which maps subdomain parameters to the system frequency. At last, we describe a possible way, based on an artificial neural network, to realize the evaluation of system frequency. The artificial neural network is an algorithm that fits a complex mapping of multiple inputs and outputs. Through appropriate machine learning, this algorithm could theoretically converge to any function. The system frequency of SSI depends on the properties of the structure itself as well as the underlying soil. The soil that can determine the system frequency is the surface layer across dozens or hundreds of meters, the constructions and the parameters of which are statistically consistent in adjacent areas with the same geological environment. Moreover, the structural parameters follow general principles in structural designs and engineering practices.