## 2 Why Strength Reduction for Large RC Structures?
### Strength Reduction  
The strength reduction is a method originally developed in the geotechnical engineering field for quantifying slope stability. It was introduced by [PERSON] et al. (1975) [15] and is based on the gradual reduction of the resistive strength of materials along potential planes of weakness. The _tensile strength reduction_ analysis was developed by the authors as an alternative design solution for complex reinforced concrete structures in 2D configurations [16]. Recently, this approach was generalized to the case of 3D state of stresses [17], referred to as SRDM-3D for Strength Reduction Design Method. The concept involves applying the load on the structure using a simple linear constitutive model, followed by a progressive reduction of the tensile strength to allow the redistribution of stresses into purely compressive stresses in concrete (struts) and purely tensile stresses in reinforcement (ties). Figure 2a schematically illustrates the analysis procedure, where P represents the external load applied smoothly and gradually during an initial loading phase. This load is maintained constant for the second strength reduction phase, during which the tensile strength of concrete gradually decreases to zero. If the initial tensile strength is chosen reasonably high (\(f_{t}^{\prime}\) at t = 0), the loading phase remains purely linear elastic. The redistribution of stresses between concrete and reinforcement in the reduction phase, is highly non-linear. Therefore, an explicit dynamics solver is employed using an incremental numerical scheme with a small stable increment time. The problem is solved using dynamic equilibrium equations, where conventional nodal forces are converted into inertia forces by assigning lumped masses to nodal degrees of freedom. The loading P is applied _slowly enough_ to minimize the kinetic over internal energy ratio of the system \(E_{c}/E_{i}\). Figure 2: Strength reduction analysis procedure [17]: (**a**) Application of loads and reduction of strength; (**b**) Explicit analysis: Kinetic energy and internal energy ratio. As depicted in Figure 2, intermediate non-linear events may occur during the strength reduction phase, such as crack initiation and propagation at \(\mathfrak{t}_{2}\). Two different situations may occur: (i) failure before the end of analysis (at \(\mathfrak{t}_{3}\)), detectable through an irreversible increase of the kinetic over internal energy ratio; (ii) no failure before the end of the strength reduction phase at time \(\mathfrak{t}_{4}\), where the tensile strength reaches zero. The advantage of this approach is that it uses a relatively simple constitutive modelling of concrete (compared to those used in ANFEA), requiring mainly the compressive strength of concrete as input parameter. In the typical _design mode_ of SRDM-3D, the constitutive model of the reinforcement is chosen linear elastic and the required quantity of reinforcement can therefore be assessed at the end of the analysis, in case of no failure (t = \(\mathfrak{t}_{4}\)). In fact, failure in this case can only occur in concrete, either through the crushing of the struts in pure compression or by incompatibility failure in nodal areas near the intersections of tensile ties in reinforcement and compressive struts in concrete. This failure can be monitored by the state variable \(\beta\), defined as the ratio between the principal compressive stress \(\sigma_{min}\) and the effective compressive strength \(f_{c}\):  
\[\beta=\frac{\sigma_{min}}{f_{c}} \tag{1}\]  
As defined in [17], the effective compressive strength \(f_{c}\) is a resistance value less than the uni-axial compressive strength of tested concrete \(f_{c}^{\prime}\) due to the presence of incompatibility effects between concrete tensile strains (originating from strain compatibility with embedded reinforcement) and concrete compressive strains, as initially suggested in the modified compression field theory or MCFT [18].