# Water Basin Effect of Cofferdam Foundation Pit
## 3 Deformation Evolution Mechanism of Cofferdam Foundation Pit
### Deformation Development Characteristics of Steel Pipe Pile
#### 3.3.2 Stage 2: Installation of the Second Support  
During the installation of the second support, the steel pipe pile is simplified as a single-span cantilever beam model, \(BE\), as illustrated in Figure 3b. The deflection equation at this stage results from the superposition of triangular loads \(S_{B}\) and \(W_{D}\) defined in conditions 8 and conditions 11, combined with the moment load \(M_{B}\) in Case 11. The geometric compatibility condition, \(\theta_{BE}=0\), for the rotation angle at end \(B\) is satisfied. For the cantilever beam \(BE\), the following relationships hold:  
\[F_{E}+F_{B}=\frac{1}{2}S_{B}h_{3}+\frac{1}{2}W_{D}h_{1} \tag{15a}\] \[F_{E}H_{w}=M_{B}+\frac{1}{2}W_{D}h_{1}\bigg{(}H_{w}-\frac{2}{3}h_{1}\bigg{)}+ \frac{1}{6}S_{B}h_{3}{}^{2}\] (15b) \[\frac{M_{B}H_{w}}{3 EI}+\frac{S_{B}h_{3}{}^{2}}{360 EIH_{w}}\bigg{(}15h_{3}H_{w}-20 H_{w}{}^{2}-3h_{3}{}^{2}\bigg{)}-\frac{W_{D}h_{1}{}^{2}}{90 EIH_{w}}\bigg{(}3h_{1}{}^{2}-5 H_{w}{}^{2}\bigg{)}=0 \tag{15c}\]  
Based on the expression group (15), \(M_{B}\), \(F_{B}\) and \(F_{E}\) can be obtained:  
\[M_{B}=-\frac{S_{B}h_{3}{}^{2}}{120H_{w}{}^{2}}\bigg{(}15h_{3}H_{w}-20H_{w}{}^ {2}-3h_{3}{}^{2}\bigg{)}+\frac{W_{D}h_{1}{}^{2}}{30H_{w}{}^{2}}\bigg{(}3h_{1}{ }^{2}-5H_{w}{}^{2}\bigg{)} \tag{16a}\] \[F_{E}=-\frac{S_{B}h_{3}{}^{2}}{120H_{w}{}^{3}}\bigg{(}15h_{3}H_{w}-20H_{w}{}^{ 2}-3h_{3}{}^{2}\bigg{)}+\frac{W_{D}h_{1}{}^{2}}{30H_{w}{}^{3}}\bigg{(}3h_{1}{} ^{2}-5H_{w}{}^{2}\bigg{)}+\frac{1}{2}\frac{W_{D}}{H_{w}}h_{1}\bigg{(}H_{w}- \frac{2}{3}h_{1}\bigg{)}+\frac{1}{6H_{w}}S_{B}h_{3}{}^{2}\] (16b) \[F_{B}=\frac{1}{2}S_{B}h_{3}+\frac{1}{2}W_{D}h_{1}+\frac{S_{B}h_{3}{}^{2}}{120 H_{w}{}^{3}}\bigg{(}15h_{3}H_{w}-20H_{w}{}^{2}-3h_{3}{}^{2}\bigg{)}-\frac{W_{D}h_{ 1}{}^{2}}{30H_{w}{}^{3}}\bigg{(}3h_{1}{}^{2}-5H_{w}{}^{2}\bigg{)}-\frac{1}{2} \frac{W_{D}}{H_{w}}h_{1}\big{(}H_{w}-\frac{2}{3}h_{1}\big{)}\] (16c) \[-\frac{1}{6H_{w}}S_{B}h_{3}{}^{2}\]  
By solving the above equations and establishing a coordinate system with support \(B\) as the origin (\(x\)-axis along the beam axis), the deflection deformation expression of the steel pipe pile at this construction stage is derived through the superposition of simply supported beam conditions 11, 8 and 11.  
\[\omega(x)=-\frac{S_{B}}{360 EIH_{w}{}_{2}}\bigg{[}3H_{w}x^{5}-15h_{3}H_{w}x^{4} +10h_{3}{}^{2}(3H_{w}-h_{3})x^{3}+15h_{3}{}^{4}H_{w}x-20h_{3}{}^{3}H_{w}{}^{2}x -3h_{3}{}^{5}x\bigg{]} \tag{17a}\] \[+\frac{M_{B}x}{6 EIH_{w}}\bigg{(}2H_{w}{}^{2}-3H_{w}x+x^{2}\bigg{)}-\frac{W_{ D}h_{1}{}^{2}}{90 EIH_{w}}\bigg{(}5x^{3}-5H_{w}{}^{2}x+3h_{1}{}^{2}x\bigg{)},\ (0\leq x\leq h_{3})\] \[\omega(x)=-\frac{S_{B}h_{3}^{2}}{360 EIH_{w}}\bigg{(}-10x^{3}+30H_{w }x^{2}-20H_{w}{}^{2}x-3h_{3}{}^{2}x+3h_{3}{}^{2}H_{w}\bigg{)}+\frac{M_{B}x}{6 EIH _{w}}\bigg{(}2H_{w}{}^{2}-3H_{w}x+x^{2}\bigg{)}\] (17b) \[-\frac{W_{D}h_{1}{}^{2}}{90 EIH_{w}}\bigg{(}5x^{3}-5H_{w}{}^{2}x+3h_{1}{}^{2}x \bigg{)},\ (h_{3}\leq x\leq h_{3}+h_{2})\] \[\omega(x)=-\frac{S_{B}h_{3}{}^{2}}{360 EIH_{w}}\bigg{(}-10x^{3}+30H_{ w}x^{2}-20H_{w}{}^{2}x-3h_{3}{}^{2}x+3h_{3}{}^{2}H_{w}\bigg{)}+\frac{M_{B}x}{6 EIH _{w}}\bigg{(}2H_{w}{}^{2}-3H_{w}x+x^{2}\bigg{)}\] \[-\frac{W_{D}}{360 EIH_{w}{}_{2}}\bigg{[}20h_{1}{}^{3}x^{3}-15h_{1}H_{ w}(x-H_{w}+h_{1})^{4}+3H_{w}(x-H_{w}+h_{1})^{5}-20h_{1}{}^{3}H_{w}{}^{2}x+12h_{1}{}^{5}x \bigg{]},\] (17c) \[(h_{3}+h_{2}\leq x\leq H_{w})\]