## 2 Proposed Optimization Problem
### Structural Analysis  
The placement of buttresses, usually at the back of the wall, means that the slab can be modelled as a continuous slab on supports. The placement of the buttresses on the inner part allows the rigidity of the joining system against flexural bending stresses to be the necessary one. In this way, the upper part works as a T-shaped section. The wall works as a section with a variable edge in a bracket where the edge is maximum in the area where the stem joins the foundation. The structure becomes hyperstatic because of the constraints suffered by both the stem and the foundation due to the existence of the buttresses. The buttress was calculated in the same way as a cantilevered T-beam and the sections were checked at different depths. Horizontal deflection is similar to a continuous slab supported on pillars because of the reduced dimensions of the cross-section of the stem of the buttresses. In the structural analysis, the buttress was considered to work as the web of a T-beam with a depth that varied in relation to the depth of the wall. The structural checks carried out were those indicated in the Spanish regulations and recommendations [45; 46]. The ultimate limit states of shear and bending and the serviceability limit state of cracking were verified. The hyperstatic structure testing method used was that by [PERSON] [47]. All limit states were calculated taking into account a uniform overload on the slope surface [41]. The wall earth pressures were calculated from the characteristics of the backfill and the surface loads. The forces used for the calculation of the stem were: horizontal ground thrust, elevation weight, heel weight, toe weight, surface load, and passive resistance exerted by the ground on the toe. The stresses on the buttresses were calculated according to the horizontal pressure of the ground in the stem and the spacing between buttresses. The effects of bending moment and shear in the stem are reduced by the effect of the buttresses placed at a distance (\(d_{b}\)). The top of the stem works as a cantilever, while the lower part is coerced by the embedding in the foundation at the base of the buttresses. Equations (4) and (5) give the value of the bending moments that appear in the middle section between buttresses:  
\[M_{1}=-0.03p_{1}db(H-ft), \tag{4}\]  
\[M_{2}=-0.0075p_{1}db(H-ft), \tag{5}\]  
In Equations (4) and (5), \(p_{1}\) represents the pressure in the contact zone of the stem with the footing, \(M_{1}\) is the bending moment in this zone, and \(M_{2}\) is the maximum bending moment produced in the stem slab. Equation (6) shows the value of the shear resistance at the connection of the stem to the footing if the value of the distance between buttresses is less than half the height:  
\[s=0.4p_{1}db \tag{6}\]  
The distribution of pressures due to flexural stresses between the different stem spans can be considered trapezoidal [47]. The maximum value of these stresses is at the top of the foundation. The bending moment is one of the constraints that conditions the cross-sections and can therefore be used to define the thickness of those sections. Vertical bending moment is neglected in [PERSON]'s calculation method [47]. For the bending check of the T-cross-sections, the effective width was obtained, as indicated in the Model Code [48]. The expressions to evaluate the mechanical strength of the sections were obtained from [PERSON] [41]. In these expressions, the restrictions imposed by the Spanish Structural Concrete Code [46] were considered. In addition to the resistance checks of the sections, the overturning and sliding of the wall were also checked. On the one hand, to satisfy the overturning condition, the favorable moments must be greater than the unfavorable ones, considering an overturning safety factor of 1.8 for frequent events.