## 2 Methods  
### Derivation of Completely Similar Macro Conditions for Semi-Rigid Connections  
This chapter analyzes the unique characteristics of semi-rigid connections and derives the completely similar conditions for semi-rigid steel nodes [11]. This is the theoretical basis for further similar error analysis. The'so-called' macro similarity means that the target physical quantity in the derivation process is a macro quantity. For example, the dimension parameter L refers to all geometric properties and physical quantities of the node, including length, width, plate thickness, and aperture. At the same time, the stress includes the stress in a series of components, such as steel beams, steel columns, and joint core areas. First, the main physical quantities involved in the physical process according to the analysis elements are listed in Table 1. All relevant quantities in the analysis should be included. During the static test of semi-rigid beam-column joints, the physical quantities involved are summarized as follows:  
Sixteen physical quantities were selected (shown above). This research chose F L T, which are commonly used in mechanical systems, as the basic dimensional system. The a symbol introduced by [PERSON] has been used to represent the dimension of any quantity 'a'. Table 2 shows all the relevant system parameters used in the case study. \begin{table}
\begin{tabular}{c c} \hline \hline
**Physical Quantity 1** & **Physical Quantity 2** \\ \hline \(L\): geometric dimension. & \(A\): sectional area. \\ \(g\): acceleration of gravity. & \(\gamma\): Poisson ratio. \\ \(F\): static load. & \(\rho\): density. \\ \(M\): moment & \(\sigma\): stress. \\ E: elastic modulus. & \(\varepsilon\): strain. \\ \(G\): shear modulus. & \(s\): displacement. \\ \(K\): initial stiffness of semi-rigid joint. & \(\theta\): angle. \\ \(I\): moment of inertia & \(T\): time. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Table of the physical quantities. According to the second law of similitude theory (Buckingham \(\pi\) theorem), if there is a physically meaningful equation involving a certain number (\(n\)) of physical variables, then the original equation can be rewritten in terms of a set of \(p=n-k\) dimensionless parameters constructed from the original variables. \[f(\pi_{1},\pi_{2},\cdots,\pi_{n-k})=0 \tag{1}\]  
The number of basic physical quantities is 3. Then \(n=16\), \(k=3\), and \(n-k=13\), and the above formula can be rewritten as:  
\[f(M,E,G,K,I,A,\gamma,\rho,\sigma,\varepsilon,s,\theta,T)=0 \tag{2}\]  
According to the principle of dimensional homogeneity, the dimensions of both sides of the equations should be the same. Assuming the power exponent as \(x_{1}\), \(x_{2}\), \(x_{3,}\) and \(x_{4}\), the equation representing \(M\) (moment) in the form of basic parameters is expressed as followed:  
\[L^{x_{1}}\cdot(LT^{-2})^{x_{2}}\cdot F^{x_{3}}=(F\cdot L)^{x_{4}} \tag{3}\]  
Since the equation is harmonious around the dimension, sorting out the relationship gives:  
\[L^{x_{1}+x_{2}}\cdot T^{-2x_{2}}\cdot F^{x_{3}}=F^{x_{4}}\cdot L^{x_{4}} \tag{4}\]  
Make the power exponent of both sides equal,  
\[\left\{ \begin{array}{l}x_{3}=x_{4}\\ x_{1}+x_{2}=x_{4}\\ -2\cdot x_{2}=0\end{array} \right. \tag{5}\]  
Solve the equations, assuming \(x_{1}=1\), thus,  
\[x_{2}=0 \tag{6}\]  
\[x_{1}=x_{3}=x_{4} \tag{7}\]  
\[L\cdot F=M \tag{8}\]  
From this, the first dimensionless number can be calculated:  
\[\pi_{1}=\frac{M}{L\cdot F} \tag{9}\]