# Incomplete Similarity Error Analysis Method of Steel Frame Structure Based on Correlation
## 2 Methods
### Derivation of Completely Similar Conditions for Semi-rigid Steel Frames
\begin{array}{c}\pi_{2}=\frac{L^{2}\cdot E}{F},\pi_{3}=\gamma,\pi_{4}=\frac{g \cdot l^{2}}{L^{5}\cdot F^{2}},\pi_{5}=\frac{K_{r}}{L\cdot F},\\ \pi_{6}=\frac{L^{3}\cdot g\cdot p}{F},\pi_{7}=\frac{g}{L\cdot f^{2}},\pi_{8}= \frac{g}{a},\pi_{9}=\frac{g}{a_{r}},\\ \pi_{10}=\frac{F}{\alpha\cdot L^{2}},\pi_{11}=\frac{X}{L}\end{array}  \tag{9}\]  
If the scaled down model is completely similar to the prototype, then all the parameters should satisfy the equations below,  
\[ \begin{array}{c}K_{m}=S_{K}\cdot K_{p},E_{m}=S_{E}\cdot E_{p},\\ \gamma_{m}=S_{\gamma}\cdot\gamma_{p},I_{m}=S_{\gamma}\cdot I_{p},K_{r_{m}}=S_ {K_{r}}\cdot K_{r_{p}},\\ \rho_{m}=S_{p}\cdot\rho_{p},f_{m}=S_{f}\cdot f_{p},a_{m}=S_{a}\cdot a_{p},\\ a_{\gamma m}=S_{a_{r}}\cdot a_{\gamma p},a_{m}=S_{\sigma}\cdot\sigma_{p},X_{m }=S_{X}\cdot X_{p}\end{array}  \tag{10}\]  
In the equations, \(S_{\text{K}}\), \(S_{\text{E}}\), \(S_{\text{Y}}\), \(S_{\text{Y}}\), \(S_{\text{Kr}}\), \(S_{\text{P}}\), \(S_{\text{f}}\), \(S_{\text{a}}\), \(S_{\text{ar}}\), \(S_{\text{\sigma}}\), \(S_{\text{X}}\), are the similarity ratios between the model and the prototype. \(K_{\text{m}}\), \(E_{\text{m}}\), \(\gamma_{\text{m}}\), \(\gamma_{\text{m}}\), \(K_{\text{m}}\), \(\rho_{\text{m}}\), \(f_{\text{m}}\), \(a_{\text{m}}\), \(a_{\text{m}}\), \(\sigma_{\text{m}}\), \(X_{\text{m}}\), are the parameters of the model, \(K_{\text{P}}\), \(E_{\text{P}}\), \(\gamma_{\text{p}}\), \(\gamma_{\text{p}}\), \(K_{\text{rp}}\), \(\rho_{\text{p}}\), \(f_{\text{p}}\), \(a_{\text{rp}}\), \(\sigma_{\text{p}}\), \(X_{\text{p}}\), are the parameters of the prototype. According to the first law of similitude theory, the dimensionless parameters are equal in all the models of one prototype,11 similarity ratio equations can be derived, thus,  
\[\begin{array}{c}\frac{S_{K}}{S_{\text{L}}S_{\text{F}}}=1,\frac{S_{\text{L}} \cdot S_{E}}{S_{\text{F}}}=1,S_{\text{Y}}=1,\\ \frac{S_{\text{g}}\cdot S_{\text{I}}}{S_{\text{L}}S_{\text{S}}\cdot S_{\text{F }}}=1,\frac{S_{\text{K}}}{S_{\text{L}}\cdot S_{\text{F}}}=1,\frac{S_{\text{ \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Factor** & **Similarity Ratio** & **Factor** & **Similarity Ratio** \\ \hline \(S_{\text{L}}\) & 1:4 & \(S_{\text{F}}\) & 2:1 \\ \(S_{\text{F}}\) & 1:16 & \(S_{\text{X}}\) & 1:4 \\ \(S_{\text{K}}\) & 1:64 & \(S_{\text{ar}}\) & 1:1 \\ \(S_{\text{E}}\) & 1:1 & \(S_{\text{\sigma}}\) & 1:1 \\ \(S_{\text{g}}\) & 1:1 & \(S_{\text{Kr}}\) & 1:64 \\ \(S_{\text{\gamma}}\) & 1:1 & \(S_{\text{a}}\) & 1:1 \\ \(S_{\text{\rho}}\) & 4:1 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Similarity ratio of each factor. Figure 1: Semi-rigid frame prototype schematic. (B\({}_{\text{M}}\): Middle beam; B\({}_{\text{S}}\): Side beam; J\({}_{\text{S}}\): Strong axis; J\({}_{W}\): Weak axis). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Prototype** & \
\begin{tabular}{c} **Component Modulus** \\ **(mm)** \\ \end{tabular} & **Model** &
\begin{tabular}{c} **Component Modulus of** \\ **Completely Similar (mm)** \\ \end{tabular} \\ \hline \(\text{B}_{\text{S}}\) & \(\text{HN300}\times 150\times 6.5\times 9\) & \(\text{B}_{\text{S}}\) & \(75\times 37.5\times 1.625\times 2.25\) \\ \(\text{B}_{\text{M}}\) & \(\text{HN400}\times 150\times 8\times 13\) & \(\text{B}_{\text{M}}\) & \(100\times 37.5\times 2\times 3.25\) \\ \(\text{C}_{\text{C}}\) & \(\text{HN500}\times 200\times 10\times 16\) & \(\text{C}_{\text{C}}\) & \(125\times 50\times 2.5\times 4\) \\ \(\text{C}_{\text{S}}\) & \(\text{HN400}\times 200\times 5\times 13\) & \(\text{C}_{\text{S}}\) & \(100\times 50\times 1.25\times 3.25\) \\ \(\text{C}_{\text{M}}\) & \(\text{HN450}\times 200\times 9\times 14\) & \(\text{C}_{\text{M}}\) & \(112.5\times 50\times 2.25\times 3.5\) \\ \hline \hline \end{tabular} Note: BS: Side beam; BM: Middle beam; C\({}_{\text{S}}\): Side column; C\({}_{\text{C}}\): Corner column; C\({}_{\text{M}}\): Middle column.  
\end{table} 
Table 3: Component modulus of completely similar.