## 2 C.F.R.P. Material Analysis
### Failure Criteria  
#### 2.1.1. Tsai-Wu Failure Criterion  
One of the most frequently used failure criteria is the quadratic interaction criterion or the Tsai-Wu failure criterion [1]. That criterion, being based on the total strain energy failure theory of Beltrami, possesses the additional advantage of considering the stress  
Figure 1: Examples of different chassis layouts (**a**) space frame and (**b**) hybrid.  
Figure 2: Ranking of alternative chassis designs.  
interactions between \(\sigma_{1}\) and \(\sigma_{2}\). Also, its mathematical representation is materialized via a summation of terms, which are exhibited in Formulas (1) and (2) for three-dimensional and plane stress cases [1; 18]. The constants (\(c_{l}\)) used in the corresponding formula for the three-dimensional stress state are used to express the coupling coefficients, whose value is \(-1\). Finally, for plane stress, the lamina's strength parameters H\({}_{1}\), H\({}_{2}\), H\({}_{6}\), H\({}_{11}\), H\({}_{22}\), and H\({}_{66}\) are calculated experimentally by utilizing the data derived from the tensile, compressive, and shear loading of the lamina.  
\[\text{Tsai Wu failure index}:I_{f}=\left[-\frac{B}{2A}+\sqrt{\left(\frac{B}{ 2A}\right)^{2}+\frac{1}{A}}\right]^{-1} \tag{1}\]  
\[H_{1}\sigma_{1}+H_{2}\sigma_{2}+H_{6}\tau_{12}+H_{11}\sigma_{1}^{2}+H_{22} \sigma_{2}^{2}+H_{66}\tau_{12}^{2}+2H_{12}\sigma_{1}\sigma_{2}<1 \tag{2}\]  
where  
\[\begin{array}{c}H_{1}=\frac{1}{\left(\sigma_{1}^{T}\right)_{alt}}-\frac{1 }{\left(\sigma_{1}^{T}\right)_{alt}}\;\;H_{2}=\frac{1}{\left(\sigma_{2}^{T} \right)_{alt}}-\frac{1}{\left(\sigma_{2}^{T}\right)_{alt}}\;\;H_{6}=0\\ H_{11}=\frac{1}{\left(\sigma_{1}^{T}\right)_{alt}\left(\sigma_{1}^{T}\right)_ {alt}}\;\;H_{22}=\frac{1}{\left(\sigma_{2}^{T}\right)_{alt}\left(\sigma_{2}^ {T}\right)_{alt}}\;\;H_{66}=\frac{1}{\left(\tau_{12}\right)_{alt}^{T}}\\ A=\frac{\sigma_{1}^{2}}{F_{1}^{T}F_{1}^{+}}+\frac{\sigma_{2}^{2}}{F_{2}^{T}F_{2 }^{+}}+\frac{\sigma_{1}^{2}}{F_{3}^{T}F_{3}^{+}}+\frac{\sigma_{2}^{2}}{F_{4}^ {2}}+\frac{\sigma_{3}^{2}}{F_{5}^{2}}+\frac{\sigma_{6}^{2}}{F_{6}^{2}}+\frac{ \sigma_{6}^{2}}{F_{6}^{2}}+\frac{\sigma_{6}^{2}}{\sqrt{\tau_{1}\tau_{1}\tau_{1 }\tau_{1}\tau_{2}\tau_{3}\tau_{3}}}+c_{6}\frac{\sigma_{1}\sigma_{2}}{\sqrt{F_{ 1}\tau_{1}\tau_{1}\tau_{2}\tau_{2}\tau_{3}}}+c_{6}\frac{\sigma_{1}\sigma_{2}} {\sqrt{F_{1}\tau_{1}\tau_{1}\tau_{1}\tau_{2}\tau_{2}\tau_{3}}}\\ \end{array} \tag{3}\]  
\[B=\left(F_{1t}^{-1}-F_{1c}^{-1}\right)\sigma_{1}+\left(F_{2t}^{-1}-F_{2}^{-1} \right)\sigma_{2}+\left(F_{3t}^{-1}-F_{3c}^{-1}\right)\sigma_{3} \tag{4}\]  
Additionally, as far as the evaluation of parameter H\({}_{12}\) is concerned, its values are also determined experimentally, in a similar manner to the rest of the constants, with the yielded values exhibiting a significant dependence on the applied load combination. Due to that reason, empirical formulae are derived from the literature in order to calculate H\({}_{12}\)[1].  
Among the implemented criteria, the Tsai-Wu method was chosen due to its ability to model multiaxial stress interactions in the composite, as it incorporates interaction terms between normal and shear stresses, especially the \(\sigma_{1}\)-\(\sigma_{2}\) coupling, which is an important factor in the case of this research, capturing complex loads in C.F.R.P. structures. Unlike simpler criteria, Tsai-Wu accounts for C.F.R.P.'s anisotropy, making it ideal for finite element analysis where local stresses vary [1]. Moreover, despite concerns such as its inability to explicitly identify specific failure modes, or the fact that its accuracy relies on experimental strength parameters, like the empirically estimated \(H_{12}\) coefficient, it remains a practical, computationally efficient tool for assessing C.F.R.P. monocoque failure.  
\[H_{12}=-1/\left(2{\left(\sigma_{1}^{T}\right)_{alt}}^{2}\right),\qquad perTsai-Hill\;failure\;theory \tag{5}\]