# Reinforced Concrete Slab Optimization with Simulated Annealing
## 2 Structural Model
### Internal Work  
Consider a rectangular plate, whose dimensions are \(L_{x}\) and \(L_{y}\), with both top and bottom reinforcements arranged in the direction of the axes of the reference system \(x\), \(y\). \({A_{x}}^{+}\) represents the area, per unit length, of the reinforcement located in the bottom part of the section parallel to the \(x\)-axis, while \({A_{x}}^{-}\) denotes the area, per unit length, of the top reinforcements in the \(x\)-direction. Similarly, in the \(y\) direction, \({A_{y}}^{+}\) and \({A_{y}}^{+}\) represent the area, per unit length, of the bottom and top reinforcements, respectively.  
It is necessary to introduce the following coefficients of \(k\), characterizing the orthotropy, and \(g\), the anisotropy of the slab:  
\[k=\frac{A_{y}^{+}}{A_{x}^{+}}=\frac{A_{y}^{-}}{A_{x}^{-}}, \tag{2}\]  
\[g=\frac{A_{x}^{-}}{A_{x}^{+}}=\frac{A_{y}^{-}}{A_{y}^{+}}, \tag{3}\]  
The ultimate limit bending moment of the cross section can be expressed as a function of its geometrical and mechanical characteristics. In a conservative approach, the role of the compressive reinforcements is neglected, thus, considering the bottom reinforcements along the \(x\)-axis area, \(A_{x}^{+}\), is characterized by a tensile strength, \(f_{yd}\), it is possible to define the ultimate bending moment \(M_{x}^{+}\):  
\[M_{x}^{+}=A_{x}^{+}\times z\times f_{yd} \tag{4}\]  
where \(z\) represents the distance between the tensile force and the compressive force.  
Considering that RC slabs have relatively low levels of reinforcement compared to beams and the values of \(z\) are almost constant, the ultimate bending moments can be considered proportional to the reinforcement areas. This approximation makes it possible to express the other bending moments as a function of the \(M_{x}^{+}\) as defined in Equation (4):  
\[M_{y}^{+}=k\times M_{x}^{+} \tag{5}\]  
\[M_{x}^{-}=g\times M_{x}^{+} \tag{6}\]  
\[M_{y}^{-}=k\times g\times M_{x}^{+} \tag{7}\]  
where \(k\) is the orthotropic coefficient and \(g\) denotes the anisotropic coefficient. Given the \(k\)-th yield line characterized by length \(\Delta l\) parallel to the \(x\)-axis, the internal work \(D_{i}\) can be expressed by Equation (8).  
\[D_{k}=M_{n}\times\Delta l\times|\theta| \tag{8}\]  
where \(M_{n}\) is the ultimate bending moment per unit length corresponding to the yield line while \(\theta\) is the relative rotation between the rigid parts separated by the given yield line. Considering a generic yield line, characterized by an inclination angle \(\varphi\) with respect to the \(x\)-axis, the ultimate bending moments corresponding to the top \(M_{p}^{-}\) and to the bottom \(M_{p}^{+}\) reinforcements can be expressed following [PERSON]'s criterion [PERSON] (1967) (see Figure 1a).  
\[M_{p}^{+}=M_{x}^{+}\times\left(\sin^{2}\varphi+k\cos^{2}\varphi\right) \tag{9}\]  
\[M_{p}^{-}=M_{x}^{+}\times g\times\left(\sin^{2}\varphi+k\cos^{2}\varphi\right) \tag{10}\]In this case, the principle of virtual work is defined by Equation (11).  
\[W_{i}+W_{\varphi}=W_{e} \tag{11}\]  
where \(W_{i}\) is the internal work corresponding to the yield lines; \(W_{\varphi}\) is the fan internal work; and \(W_{e}\) is the external work.