# Incomplete Similarity Error Analysis Method of Steel Frame Structure Based on Correlation
## 2 Methods  
### Derivation of Completely Similar Conditions for Semi-rigid Steel Frames  
In the study of scaled down model tests, little research has been conducted to investigate the influence of the semi-rigid characteristic of the joint [7]. In this section the Buckingham \(\pi\) theorem and dimensional analysis are used to develop the similarity relations in the semi-rigid frame [8]. A few dimensionless numbers are obtained through the derivation, which are the necessary conditions of complete similarity in the scaled down model frame [9]. The relevant system parameters are listed in the first place to suit the requirement of the derivation of similarity criteria [10; 11; 12; 13]. The number of parameters should be exact to ensure the accuracy of the similarity relation. Dynamic parameters are also included for future study of dynamic similitude law [14; 15; 16]. The relevant parameters are listed as follows: \begin{tabular}{c c} \hline \hline _L_: geometric dimension. & \(\gamma\): poisson ratio. \\ _g_: acceleration of gravity. & \(\rho\): density. \\ _F_: static load. & _I_: moment of inertia. \\ _a_: prompting acceleration. & \(\sigma\): stress. \\ _K_: joint stiffness of semi-rigid frame. & \(x\): displacement. \\ _K_: joint stiffness of semi-rigid column base. & _f_: frequency. \\ _E_: elastic modulus. & \(a_{r}\): response acceleration. \\ \hline \hline \end{tabular}   
The primary dimensions selected were (_F_, \(L\), _T_). Representing the force (_F_), length (_L_) and time (_T_). The number of listed parameters is 14 in total; the number of restrictions is equal to the number of primary dimensions. Table 1 shows all the relevant system parameters used in the case study. 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. \[f(\pi_{1},\pi_{2},\cdots,\pi_{n-k})=0 \tag{1}\]  
Geometric dimension \(L\), acceleration of gravity \(g\), static load \(F\), these three parameters are chosen as the basic parameters with independent dimension. So the 14 relevant system parameters can be transformed into certain function of 11 dimensionless parameters. The function can be derived as followed:  
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 \(K\) (joint stiffness of semi-rigid frame) in the form of basic parameters are expressed as followed:  
\[L^{x_{1}}\cdot\left(LT^{-2}\right)^{x_{2}}\cdot F^{x_{3}}=\left(F\cdot L\right) ^{x_{4}} \tag{2}\]  
Classify the dimensions in the form of basic parameters,  
\[L^{x_{1}+x_{2}}\cdot T^{-2x_{2}}\cdot F^{x_{3}}=F^{x_{4}}\cdot L^{x_{4}} \tag{3}\]  
Make the power exponent of both sides equal,  
\[\left\{ \begin{array}{l}x_{3}=x_{4}\\ x_{1}+x_{2}=x_{4}\\ -2\cdot x_{2}=0\end{array} \right. \tag{4}\]  
Solve the equations,  
\[x_{2}=0 \tag{5}\]  
\[x_{1}=x_{3}=x_{4} \tag{6}\]  
Assuming \(x_{1}=1\), thus,  
\[L\cdot F=K \tag{7}\]  
Finally, the first dimensionless parameter is obtained:  
\[\pi_{1}=\frac{K}{L\cdot F} \tag{8}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Basic Dimension** & \(L\) & \(g\) & \(F\) & \(K\) & \(E\) & \(\gamma\) & \(I\) \\ \hline \(F\) & 0 & 0 & 1 & 1 & 1 & 0 & 1 \\ \(L\) & 1 & 1 & 0 & 1 & \(-2\) & 0 & 2 \\ \(T\) & 0 & \(-2\) & 0 & 0 & 0 & 0 & 1 \\ \hline
**Basic Dimension** & \(K\tau\) & \(\rho\) & \(f\) & \(a\) & \(a\tau\) & \(\sigma\) & \(X\) \\ \hline \(F\) & 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ \(L\) & 1 & \(-4\) & 0 & 1 & 1 & \(-2\) & 1 \\ \(T\) & 0 & 2 & \(-1\) & \(-2\) & \(-2\) & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Basic dimension of related parameters. In the same way, the left 10 dimensionless parameters can be derived and shown as followed:  
\[