## 2 Material and Methods
### Semirigid Connection Frame Theory with Lateral Displacement
#### 2.2.2 Consider the Node Semirigid Crossbar Stiffness Coefficient Correction  
As shown in Figure 5, the two ends are semirigidly restrained with lateral shift of the crossbar, \(L_{b}\) is the length of the crossbar, \(\theta_{1}\) and \(\theta_{2}\) are the turning angles of the two ends of the crossbar, and \(R_{k1}\) and \(R_{k2}\) are the turning stiffness of the crossbar nodes. Under the action of end moments \(M_{1}\) and \(M_{2}\), the relative turning angles of the left and right ends of the crossbar are \(\theta_{1}=\frac{M_{1}}{R_{k1}}\) and \(\theta_{2}=\frac{M_{2}}{R_{k2}}\), respectively. Angle-displacement equation:  
\[M_{1}=\frac{EI_{b}}{L_{b}R^{*}}\bigg{[}\bigg{(}4+\frac{12 EI_{b}}{L_{b}R_{k2}} \bigg{)}\theta_{1}+2\theta_{2}\bigg{]} \tag{5}\]  
Figure 5: Semirigid restraint bars at both ends. \[M_{2}=\frac{EI_{b}}{L_{b}R^{*}}\bigg{[}\bigg{(}4+\frac{12 EI_{b}}{L_{b}R_{k}}\bigg{)} \theta_{2}+2\theta_{1}\bigg{]} \tag{6}\]  
In the formula:  
\[R^{*}=\bigg{(}1+\frac{4 EI_{b}}{L_{b}R_{k1}}\bigg{)}\bigg{(}1+\frac{4 EI_{b}}{L_{b }R_{k2}}\bigg{)}-\bigg{(}\frac{EI_{b}}{L_{b}}\bigg{)}^{2}\frac{4}{R_{k1}R_{k2}} \tag{7}\]  
For the formwork bracket with lateral shift, the crossbar produces the same direction of turning angle, so the crossbar end moment \(M_{1}\) is:  
\[M_{1}=\alpha_{u}\frac{6 EI_{b}}{L_{b}}\theta_{1} \tag{8}\]  
In the formula:  
\[\alpha_{u}=\bigg{(}1+\frac{2 EI_{b}}{L_{b}R_{k2}}\bigg{)}/R^{*} \tag{9}\]  
\(\alpha_{u}\) indicates the stiffness correction coefficient of the crossbar of the formwork bracket with a side-shift semirigid connection. 2.3 Calculation of Stable Bearing Capacity of Vertical Bar Based on Theory of Semirigid Connection Frame with Lateral Displacement  
Through the theoretical study of the semirigid connection frame with lateral displacement, it is found that the effective length coefficient of the vertical bar of the fastener-type steel pipe formwork support is obtained according to the constraint coefficient of the upper and lower ends of the vertical bar. Therefore, the semirigid characteristics of the formwork support joint are considered by modifying the end stiffness of the transverse bar in the formwork support. In the derivation of the effective length coefficient of the vertical rod of the fastener-type steel pipe formwork support, there are the following assumptions: (1) Simultaneous buckling of the neutral rod on the same layer; (2) The rotation angles at both ends of the horizontal bar are equal; that is, the rotational stiffness at both ends of the horizontal bar is \(R_{k1}=R_{k2}\), and the rotational stiffness at both ends of the horizontal bar is the rotational stiffness of the right angle fastener \(R_{k}\); (3) The axial force of the bar is very small and can be ignored. From Formulas (7) and (9), the calculation formula (10) of the stiffness correction coefficient \(\alpha_{u}\) of the bar can be obtained. \[\alpha_{u}=\frac{1+\frac{2 EI}{L_{b}R_{k}}}{\left(1+\frac{4 EI}{L_{b}R_{k}} \right)^{2}-\left(\frac{EI}{L_{b}}\right)^{2}\frac{4}{R_{k}R^{2}}} \tag{10}\]  
\(EI\)--Bending stiffness of steel tube  
\(R_{k}\)--Rotational stiffness of right angle fastener  
\(L_{b}\)--Lag spacing  
Q235 steel pipe is used in the vertical and horizontal rods of the fastener-type steel pipe formwork support; that is, the bending stiffness \(EI\) of the vertical rod and the horizontal rod is the same. The Formula (2) and (3) are simplified to obtain the constraint coefficient \(K\) formula (11). \[K=\frac{\alpha_{u}L_{c}}{L_{b}} \tag{11}\]  
\(L_{c}\)--Extension  
\(L_{b}\)--Lag spacing  
The effective length coefficient \(\mu\) of the pole is calculated by the simplified Formula (12):  
\[\mu=\sqrt{\frac{7.5K^{2}+8K+1.52}{7.5K^{2}+2K}} \tag{12}\]The stability of the pole under different node stiffness values is calculated.