## 3 Application
### Failure Probability Analysis Using MCS
#### 3.2.2 Limit State Function  
From a design point of view, a cofferdam can be regarded as a main dam with a low design flood frequency to determine the size of the cofferdam. Therefore, the limit state function for a cofferdam is utilized as that for a main dam. The failure mode is the process of failure occurrence and refers to the root cause and failure effect in risk analysis. [PERSON] et al. proposed the failure modes of a main dam with concrete gravity type and their limit state functions based on numerous reviews in the domestic and international literature on dam failure cases and design standards [9]. The limit state function for overturning (\(f_{o}\)) is defined as  
\[f_{o}=\sum M_{r}-\sum M_{o} \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{2}{c}{**Design Flood**} & \multicolumn{1}{c}{**No. of**} & \multicolumn{1}{c}{**Upstream Water Level**} & \multicolumn{1}{c}{**Freeboard (m)**} & \multicolumn{1}{c}{**Dam Crest**} & \multicolumn{1}{c}{**Cofferdam**} \\ \multicolumn{2}{c}{**Years**} & \multicolumn{1}{c}{**Discharge**} & \multicolumn{1}{c}{**Diversion**} & \multicolumn{1}{c}{**(EL m)**} & \multicolumn{1}{c}{**[b]**} & \multicolumn{1}{c}{**[a + b]**} & \multicolumn{1}{c}{**(m)**} \\ \hline \(1\) & \(1761\) & \(2\) & \(502.0\) & \(1.0\) & \(503.0\) & \(31.5\) \\ \hline \(2\) & \(4187\) & \(4\) & \(507.6\) & \(1.0\) & \(508.6\) & \(37.1\) \\ \hline \(5\) & \(6499\) & \(4\) & \(517.9\) & \(1.0\) & \(518.9\) & \(47.4\) \\ \hline \(10\) & \(8084\) & \(4\) & \(521.6\) & \(1.0\) & \(522.6\) & \(51.1\) \\ \hline \end{tabular}
\end{table} 
Table 1: Flood routing analysis. Figure 3: Diversion structure. where \(M_{r}\) is the resistance moment and \(M_{o}\) is the overturning moment. The limit state function for sliding (\(f_{s}\)) is expressed as  
\[f_{s}=\sum S-\sum H=\sum S-\sum(N\tan\varnothing+cL) \tag{2}\]  
where \(S\) is the resistance force, \(H\) is the horizontal force, \(\varnothing\) is the angle of internal friction, \(c\) is the cohesion intercept, \(N\) is the resultant of forces normal to the assumed sliding plane, and \(L\) is the length of the base in compression for a unit strip of the dam. The limit state function for base pressure (\(f_{b}\)) is expressed as  
\[f_{b}=\sigma_{a}-\sigma_{max} \tag{3}\]  
where \(\sigma_{a}\) is the allowable bearing capacity on the foundation and \(\sigma_{max}\) is the maximum applied basal pressure. To analyze the limit state function, the related loads have to be calculated in advance. The related loads for the cofferdam are (1) the dead load of the dam weight, (2) the headwater pressure exerted by hydrostatic water pressure, (3) the uplift acting on the bottom of the dam, (4) the silt pressure activated by sediment deposition behind the dam, and (5) the seismic force developed by earthquake loading [10]. Among these loads, the headwater pressure and the uplift acting on the bottom should be modified because the cofferdam has a low design flood frequency; it means that a flood exceeding the design flood frequency can overflow a cofferdam during main dam construction. The loads are described in detail as follows:  
1. Dead load. The unit weight of concrete generally should be assumed to be 23 kN per cubic meter (kN/m\({}^{3}\)) in the design stage. 2. Headwater pressure. External water pressure loads are to be applied to the structure using the upstream water level. When the overflow has occurred, the headwater pressure is higher than the cofferdam height as shown in Figure 4.  
The hydrostatic pressure to the dam is a function of the water depth and the unit weight of water. \[p_{w}=\gamma_{w}\times h\]  
where  
\(p_{w}\): water pressure at depth \"\(h\)\" (kN/m\({}^{2}\));  
\(\gamma_{w}\): unit weight of water (10 kN/m\({}^{3}\));  
\(h\): water depth (m). Figure 4: Headwater pressure when overflowing. (3)  
Uplift pressure. The uplift pressure resulting from headwater and tailwater has to be considered for stability analysis which exists through cross sections, at the bottom of the cofferdam.