## 5 Applications
### Application 2: Draft Tube  
Figure 12 shows the geometry of the draft tube RC structure with its rock foundation and embedded reinforcement. A Mohr-Coulomb contact condition was used between concrete and rock and a fixed condition was used for the bottom face of the rock. Plane strain condition was imposed to all vertical rock surfaces and to the upstream concrete face of draft tube. The existing reinforcement was modeled and embedded in the model as shown. It was originally designed using linear elastic finite element analysis. Thermal effects were disregarded in the design stage and considered later for verification purposes using ANFEA [3]. Figure 11: Results for spiral case example: (**a**) stresses in reinforcement before and after strength reduction; (**b**) \(\beta\) field before and after strength reduction; (**c**) yielding state of reinforcement at the end of strength reduction. In this example, thermal effects are considered. Due to these effects, and as explained before, S&T and linear elastic finite element methods are not applicable in this example. Only ANFEA can be used. SRAM is applied within the global resistance factor reliability approach. The geometry and material input parameters are taken similar to [3], for comparison purposes with ANFEA results. The load case considered is the empty load case which governed the design of main reinforcement. Hydrostatic pressure loads are applied on external faces of the draft tube. The load combination used at service level is 1.0 \(GL+1.0\)\(HP+1.0\)\(\Delta T\). Loads are applied successively in the order: \(GL\), \(\Delta T\) and \(HP\). The factored ultimate load considered is \(\lambda_{GL}GL+\lambda_{HP}HP+\lambda_{\Delta T}\Delta T\) where \(\lambda_{GL}=1.25\), \(\lambda_{HP}=1.25\) and \(\lambda_{\Delta T}=1.25\) are the load factors respectively for gravity, hydrostatic pressure and thermal loads. Hence, all loads are simultaneously increased starting from the service level and up to failure; following the incremental load direction (0.25; 0.25; 0.25) at ultimate which was considered critical for the roof of draft tube. Similar to the approach in [3], the mean values for concrete compressive strength and steel yield strength are: \(f^{\prime}_{c}=30\) MPa at \(f_{y}=400\) MPa. The coefficients of variation (C.o.vs) for concrete and steel are 0.2 and 0.09, respectively. Figure 10b shows the evolution of kinetic over internal energy ratio during loading, reduction and reloading phases for the analysis with mean input parameters (\(R_{m}\)). Failure is observed during the reloading phase. When compared to the previous application example (Figure 10a), an important peak can be noticed during the loading phase corresponding to thermal effects. Considering the model with mean material properties, Figure 13 presents different results regarding the evolution of analysis and the final failure mechanism. Even if it is difficult to talk about a real damage pattern in strength reduction analysis, the maximum principal strain \(\varepsilon_{max}\) field, can still be used to represent the amount of elements that undergone softening. Based on authors experience, pattern of this deformation field is quite similar to the cracking damage pattern that can be observed with ANFEA and the smeared crack approach. However, 'damage' appears clearly _more smeared_ in the case of strength reduction analysis. Figure 13a presents the evolution of \(\varepsilon_{max}\) field during analysis. The roof failure can be clearly seen at the end of the analysis with a damage pattern consistent with the punching shear failure reported in [3]. Figure 13b,c compare \(\sigma_{min}\) and \(\beta\) fields at the ultimate load level and at failure. The final 3D strut pattern can be clearly seen just before failure. Figure 14a shows [PERSON]'s analyses results (\(2^{2}\) analyses plus an additional analysis using mean properties). The load in Figure 14 corresponds to the vertical load transferred by the roof, normalized by its service level value.