## 2 Similar Model Test Design
### Similarity Constant  
The model size is scaled down from the actual dimensions of the prototype structure to an appropriate proportion. In the initial phase, to ensure genuine and reliable test outcomes, efforts are made to facilitate the ease of conducting similar model tests. The deformation and stress of the similar model are meticulously monitored using a variety of specialized equipment. The results derived from the similar model tests are notably reliable and readily applicable. However, to theoretically guarantee the similarity ratio between the model and the actual engineering scenario, the fundamental principles of elastic mechanics are employed. This ensures that the similar model and the actual engineering project adhere to specific similarity indices. Such adherence allows the test results to accurately mirror the mechanical phenomena occurring in real engineering situations right from the source. The theory of similarity encompasses the first, second, and third theorems of similarity, which collectively underpin the validity and applicability of the model's findings to real-world engineering contexts. In this study, a derivation method for similarity criteria based on dimensional analysis was employed to ascertain the test similarity constant. Within a typical engineering system, three fundamental dimensions generally prevail: length [\(L\)], time [\(T\)], and either force [\(F\)]--constituting the _FLT_ system--or mass [\(M\)]--forming the _MLT_ system. Specifically, this study adopts mass, length, and time as the foundational physical quantities, utilizing kilograms, meters, and seconds as their respective base units. Through a meticulous analysis of the dimensions of physical quantities, a corresponding dimensional matrix was constructed. Adhering to the principle of dimensional equilibrium, the similarity criterion for the test is systematically deduced. This approach ensures that the relationships between the model and the prototype are accurately represented, thereby facilitating a reliable extrapolation of the test results to real-world engineering applications. The basic physical quantities selected are mass (\(M\)), length (\(L\)), and time (\(T\)). Their dimensions are as follows:  
\[\left\{ \begin{array}{ll}[M]&=M\\ [L]&=L\\ [T]&=T\end{array} \right. \tag{1}\]  
The physical quantities discussed below include elastic modulus (\(E\)), density (\(\rho\)), stress (\(\sigma\)), strain (\(\epsilon\)), displacement (\(\Delta\)), velocity (\(v\)), acceleration (\(a\)), time (\(t\)), frequency (\(f\)), etc. The following are dimensional expressions of the relevant physical quantities:  
\[a_{5}=2a_{2}-a_{6}-a_{7} \tag{7}\]  
\[a_{9}=2a_{2}-a_{6}-2a_{7}+a_{8} \tag{8}\]  
Assuming that there are 6 known quantities, the number of similarity criteria is 6. \(a_{2},\ a_{3},\ a_{4},\ a_{6},\ a_{7},\ a_{8}\) is set to 6 sets of parameter values respectively, that is,  
when \(a_{2}\)=1, \(a_{3}=a_{4}=a_{6}=a_{7}=a_{8}=0\), then \(a_{1}=-1\), \(a_{5}=2\), \(a_{9}=2\);  
when \(a_{3}\)=1, \(a_{2}=a_{4}=a_{6}=a_{7}=a_{8}=0\), then \(a_{1}=-1\), \(a_{5}\)= 0, \(a_{9}=0\);  
when \(a_{4}\)=1, \(a_{2}=a_{3}=a_{6}=a_{7}=a_{8}=0\), then \(a_{1}=0\), \(a_{5}=0\), \(a_{9}=0\);  
when \(a_{6}\)=1, \(a_{2}=a_{3}=a_{4}=a_{7}=a_{8}=0\), then \(a_{1}=0\), \(a_{5}=-1\), \(a_{9}=-1\);  
when \(a_{7}\)=1, \(a_{2}=a_{3}=a_{4}=a_{6}=a_{8}=0\), then \(a_{1}=0\), \(a_{5}=-1\), \(a_{9}=-2\);  
when \(a_{8}\)=1, \(a_{2}=a_{3}=a_{4}=a_{6}=a_{7}=0\), then \(a_{1}=0\), \(a_{5}=0\), \(a_{9}=1\). Through the above analysis, the following \(\pi\) matrix shown in Table 2 can be obtained. Each row represents a dimensionless factor, and each column represents a different physical quantity and its corresponding power exponent. The values in the table represent the power of each physical quantity in a dimensionless factor. According to the Buckingham \(\pi\) theorem [24], the number of dimensionless factors is \(n-m=6\), where \(n=9\) is the total number of physical quantities, and \(m=3\) is the number of fundamental quantities.