# Water Basin Effect of Cofferdam Foundation Pit
## 3 Deformation Evolution Mechanism of Cofferdam Foundation Pit
### Deformation Characteristics of the Bottom Sealing  
To analyze the stress-deformation behavior of the concrete bottom sealing, the material is idealized as a continuous, uniform, isotropic medium undergoing small deformations [36]. A unit-width lining segment is selected for stress analysis, which can be simplified as a beam \(AB\) with length \(l\) and thickness \(t\), as depicted in Figure 4.  
The base structure is primarily subjected to three vertical actions: self-weight \(G=\gamma tl\), adhesive force \(f_{1}=\sigma l\) between the base and underlying stratum, and water buoyancy \(f_{2}=\gamma_{\rm w}h_{\rm w}l\). Here, \(\gamma\) denotes the specific weight of base concrete, \(\sigma\) the adhesive strength per unit area between the base and stratum, \(\gamma_{\rm w}\) the specific weight of water, and \(h_{w}\) the difference in water levels inside and outside the cofferdam. The base is thus modeled as a doubly fixed-end beam \(AB\) under an equivalent uniform distributed load \(q=\gamma_{\rm w}h_{\rm w}-\gamma t-\sigma\). According to the theory of material mechanics, the vertical load \(F_{A}=F_{B}=\frac{ql}{2}\) and end bending moments \(M_{A}=M_{B}\) of this statically indeterminate beam  
Figure 4: Calculation model for cofferdam bottom sealing.  
are analyzed by selecting a simply supported beam as the basic static system. The geometric compatibility equation for the rotational deformation at support \(A\) is expressed as follows:  
\[\theta_{A}=\theta_{Aq}+\theta_{AM_{A}}+\theta_{AM_{B}}=0 \tag{24}\]  
where, \(\theta_{A}\) is the total rotation angle at support \(A\), \(\theta_{Aq}\), \(\theta_{AM_{A}}\), and \(\theta_{AM_{B}}\) denote the rotational angles at \(A\) induced by the uniform load \(q\), support bending moment \(M_{A}\), and support \(B\) bending moment \(M_{B}\), respectively.  
Substituting the force-rotation angle relationships Equations (1)-(3) for 123 working conditions into Equation (24), the fixed-end bending moments \(M_{A}=-\frac{ql^{2}}{12}\) is derived. Taking support \(A\) as the coordinate origin and the beam axis as the \(x\)-axis, the bending moment at any position \(x\) along the fixed-end beam is as follows:  
\[M(x)=-\frac{qx^{2}}{2}+\frac{qlx}{2}-\frac{ql^{2}}{12} \tag{25}\]  
Using the differential relationship between bending moment and deflection curve (\(EI\omega^{\prime\prime}(x)=-M(x)\)) and applying the fixed-end boundary conditions (\(x=0\), \(\omega(x)=0\), \(x=l\), \(\omega(x)=0\)), the deflection curve equation is obtained as follows:  
\[\omega(x)=\frac{1}{EI}\bigg{(}-\frac{qx^{4}}{24}+\frac{qlx^{3}}{12}-\frac{ql^{ 2}x^{2}}{24}\bigg{)} \tag{26}\]  
The maximum midspan deflection under vertical loading is \(\omega_{0}=\frac{ql^{4}}{384 EI}\). For the vertical deformation of the bottom sealing, in addition to the effective vertical load \(q\), the additional bending moment \(M_{0B}\) and horizontal load \(F_{0B}\) at support \(B\), resulting from the differential water and soil pressure at the ends of the steel pipe piles must be considered. As the foundation pit water level is gradually lowered, these additional loads increase incrementally. Thus, only the deformations induced during the fourth-stage precipitation to the casting stage are analyzed. The midspan deflection due to the end additional bending moment \(M_{0B}\) at casting completion is \(\Delta_{M_{0B}}=\frac{M_{0B}l^{2}}{16 EI}\), and the shrinkage deformation from the end additional forces \(F_{0B}\) is \(\Delta_{F_{0B}}=\frac{F_{0B}l^{2}}{EA}\). Using the triangular relationship for deformed pressure-bending members, the vertical midspan deformation \(\Delta_{F_{0}}=\sqrt{\frac{|\Delta_{F_{0B}}}{2}-\frac{\Delta_{F_{0B}}}{4}}\) caused by horizontal compression of the bottom sealing is derived. The total midspan deformation during casting is therefore the superposition of these components:  
\[\Delta l=\omega_{0}-2\Delta_{M_{0}}+\Delta_{F_{0}} \tag{27}\]