## 2 Risers' Geometries and Materials and FE Modeling
### Materials for the Risers  
High grade steel X80 was selected as the material for Riser 1, while AS4/PEEK composites with PEEK liner were chosen for Risers 2 and 3. The properties of these materials are listed in Table 2. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Riser**} & \(I.D.\) & \(O.D.\) & \(Length\) & \multirow{2}{*}{**Lay-Ups**} & \(t\_{l}\_{l}\) & \(t\_{l}\theta^{\circ}\) & \(t\_{\pm}\theta^{\circ}\) & \(t\_{90}^{\circ}\) & \(\pm\theta\) \\  & **(m)** & **(m)** & **(m)** & **(m)** & **(mm)** & **(mm)** & **(mm)** & **(°)** \\ \hline Riser 1 & 0.25 & 0.3 & 12.5 & Monolithic steel & & & & & \\ Riser 2 & 0.25 & 0.326 & 12.5 & [liner/90/(/90)\({}_{10}\)] & 6 & 1.165 & & 1.85 & \\ Riser 3 & 0.25 & 0.31 & 12.5 & [liner/0\({}_{3}\)/(+52,\(-\)52)\({}_{5}\)/90\({}_{4}\)] & 6 & 1.48 & 1.30 & 1.64 & 52 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometries of risers. Figure 1: Cross-section for: (**a**) steel riser joint; and (**b**) composite riser joint. The material properties listed in the table were utilized for steel riser directly, as well as composite risers with layered-structure method (LSM). In terms of the composite risers with effective modulus method (EMM), a MATLAB code (Version 9.1 R2016b, MathWorks, Inc., Natick, MA, USA, 2016) of 3D laminate property theory [34,35] was created for calculating composite tube's 3D effective properties using Equations (1)-(6). Table 3 shows the effective properties of FRP composite risers for effective modulus method. \[\left[S_{ij}^{\prime}\right]^{(k)}=\left[ \begin{array}{cccccccc}1/E_{1}&-v_{ 12}/E_{1}&-v_{13}/E_{1}&0&0&0\\ -v_{21}/E_{2}&1/E_{2}&-v_{23}/E_{2}&0&0&0\\ -v_{13}/E_{3}&-v_{32}/E_{3}&1/E_{3}&0&0&0\\ 0&0&0&1/G_{23}&0&0\\ 0&0&0&0&1/G_{13}&0\\ 0&0&0&0&0&1/G_{12}\end{array} \right] \tag{1}\]  
\[\left[C_{ij}^{\prime}\right]^{(k)}=\left[S_{ij}^{\prime}\right]^{-1(k)} \tag{2}\]  
\[\left[C_{ij}\right]=\left[ \begin{array}{cccccccc}c^{2}&s^{2}&0&0&0&-cs\\ s^{2}&c^{2}&0&0&0&-2 cs\\ 0&0&1&0&0&0\\ 0&0&0&c&-s&0\\ 0&0&0&s&c&0\\ -cs&cs&0&0&0&c^{2}-s^{2}\end{array} \right]\left[C_{ij}^{\prime}\right]=\left[ \begin{array}{cccccccc}c^{2}&s^{2}&0&0&0&cs\\ s^{2}&c^{2}&0&0&0&-cs\\ 0&0&1&0&0&0\\ 0&0&0&c&-s&0\\ 0&0&0&s&c&0\\ -2 cs&2 cs&0&0&0&c^{2}-s^{2}\end{array} \right] \tag{3}\]  
\[\left[\overline{C}\right]=\left[ \begin{array}{cccccccc}\overline{C}_{11}& \overline{C}_{12}&\overline{C}_{13}&0&0&\overline{C}_{16}\\ \overline{C}_{12}&\overline{C}_{22}&\overline{C}_{23}&0&0&\overline{C}_{26} \\ \overline{C}_{13}&\overline{C}_{23}&\overline{C}_{33}&0&0&\overline{C}_{36} \\ 0&0&0&\overline{C}_{44}&\overline{C}_{45}&0\\ 0&0&0&\overline{C}_{45}&\overline{C}_{55}&0\\ \overline{C}_{16}&\overline{C}_{26}&\overline{C}_{36}&0&0&\overline{C}_{66} \end{array} \right] \tag{4}\]  
\[\left[\overline{S}\right]=\left[\overline{C}\right]^{-1} \tag{5}\]  
\[\overline{E}_{x}=\frac{1}{\overline{S}_{11}};\overline{E}_{y}=\frac{1}{ \overline{S}_{22}};\overline{E}_{z}=\frac{1}{\overline{S}_{33}};\overline{v} _{yz}=-\frac{\overline{S}_{23}}{\overline{S}_{22}};\overline{v}_{xz}=-\frac{ \overline{S}_{31}}{\overline{S}_{11}}; \tag{6}\]  
\[\overline{v}_{xy}=-\frac{\overline{S}_{21}}{\overline{S}_{11}};\overline{C}_{ yz}=-\frac{1}{\overline{S}_{44}};\overline{C}_{xz}=-\frac{1}{\overline{S}_{55}}; \overline{C}_{xy}=-\frac{1}{\overline{S}_{66}} \tag{7}\]  
where \(\left[C_{ij}\right]\) is the lamina elasticity matrix, \(\left[C_{ij}^{\prime}\right]^{(k)}\) is the principle elasticity matrix of each composite lamina, \(\left[\overline{S}\right]\) is the effective elastic compliance matrix of the composite laminate, \(\left[\overline{C}\right]\) is the laminate stiffness, and \(\left[S_{ij}^{\prime}\right]^{(k)}\) is the principal compliance matrix for transversely isotropic composite lamina.