Appendix A
Rigidity matrix of laminates:
$$\begin{aligned} \left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{N_s}}\\ {{N_r}}\\ {{N_{sr}}}\\ {{N_{rs}}} \end{array}}\\ {\begin{array}{*{20}{c}} {{M_s}}\\ {{M_r}}\\ {{M_{sr}}}\\ {{M_{rs}}} \end{array}} \end{array}}\\ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {N_s^*}\\ {N_r^*}\\ {N_{sr}^*}\\ {N_{rs}^*} \end{array}}\\ {\begin{array}{*{20}{c}} {M_s^*}\\ {M_r^*}\\ {M_{sr}^*} \end{array}} \end{array}}\\ {M_{rs}^*} \end{array}} \right\} = \left[ {\begin{array}{*{20}{c}} {\overline{A_{11}}}&{}{A}_{12}&{}{\overline{A_{16}}}&{}A_{16}&{}{\overline{B_{11}}}&{}B_{12}&{}{\overline{B_{16}}}&{}B_{16}&{}{\overline{C_{11}}}&{}C_{12}&{}C_{16}&{}{\overline{C_{16}}}&{}{\overline{D_{11}}}&{}D_{12}&{}{\overline{D_{16}}}&{}D_{16}\\ A_{12}&{}{\widetilde{A_{22}}}&{}A_{26}&{}{\widetilde{A_{26}}}&{}{B}_{12}&{}{\widetilde{B_{22}}}&{}B_{26}&{}{\widetilde{B_{26}}}&{}{C}_{12}&{}{\widetilde{C_{22}}}&{}{C}_{26}&{}{\widetilde{C_{26}}}&{}D_{12}&{}{\widetilde{D_{22}}}&{}D_{26}&{}{\widetilde{D_{26}}}\\ {\overline{A_{16}}}&{}A_{{26}}&{}{\overline{A_{66}}}&{}A_{66}&{}{\overline{B_{16}}}&{}{B}_{26}&{}{\overline{B_{66}}}&{}{B}_{66}&{}{\overline{C_{16}}}&{}{C}_{26}&{}{\overline{C_{66}}}&{}{C}_{66}&{}{\overline{D_{16}}}&{}{D}_{26}&{}{\overline{D_{66}}}&{}{D}_{66}\\ A_{16}&{}{\widetilde{A_{26}}}&{}A_{66}&{}{\widetilde{A_{66}}}&{}{B}_{16}&{}{\widetilde{B_{26}}}&{}{B}_{66}&{}{\widetilde{B_{66}}}&{}{C}_{16}&{}{\widetilde{C_{26}}}&{}{C}_{66}&{}{\widetilde{C_{66}}}&{}{D}_{16}&{}{\widetilde{D_{26}}}&{}{D}_{66}&{}{\widetilde{D_{66}}}\\ {\overline{B_{11}}}&{}{B}_{12}&{}{\overline{B_{16}}}&{}{B}_{16}&{}{\overline{C_{11}}}&{}{C}_{12}&{}C_{16}&{}{\overline{C_{16}}}&{}{\overline{D_{11}}}&{}{D}_{12}&{}{\overline{D_{16}}}&{}{D}_{16}&{}{\overline{E_{11}}}&{}{E}_{22}&{}{\overline{E_{16}}}&{}{E}_{16}\\ {\overline{B_{12}}}&{}{\widetilde{B_{22}}}&{}{B}_{26}&{}{\widetilde{B_{26}}}&{}{C}_{12}&{}{\widetilde{C_{22}}}&{}{C}_{26}&{}{\widetilde{C_{26}}}&{}{D}_{12}&{}{\widetilde{D_{22}}}&{}{D}_{26}&{}{\widetilde{D_{26}}}&{}{E}_{12}&{}{\widetilde{E_{22}}}&{}{E}_{26}&{}{\widetilde{E_{26}}}\\ {\overline{B_{16}}}&{}{B}_{26}&{}{\overline{B_{66}}}&{}B_{66}&{}{\overline{C_{16}}}&{}{C}_{26}&{}{\overline{C_{66}}}&{}{C}_{66}&{}{\overline{D_{16}}}&{}{D}_{26}&{}{\overline{D_{66}}}&{}D_{66}&{}{\overline{E_{16}}}&{}E_{26}&{}{\overline{E_{66}}}&{}E_{66}\\ {B}_{16}&{}{\widetilde{B_{26}}}&{}{B}_{66}&{}{\widetilde{B_{66}}}&{}{C}_{16}&{}{\widetilde{C_{26}}}&{}{C}_{66}&{}{\widetilde{C_{66}}}&{}D_{16}&{}{\widetilde{D_{26}}}&{}{D}_{66}&{}{\widetilde{D_{66}}}&{}E_{16}&{}{\widetilde{E_{26}}}&{}E_{66}&{}{\widetilde{E_{66}}}\\ {\overline{C_{11}}}&{}{C}_{12}&{}{\overline{C_{16}}}&{}{C}_{16}&{}{\overline{D_{11}}}&{}{D}_{12}&{}{\overline{D_{16}}}&{}D_{16}&{}{\overline{E_{11}}}&{}{E}_{22}&{}{\overline{E_{16}}}&{}E_{16}&{}{\overline{F_{11}}}&{}F_{12}&{}{\overline{F_{16}}}&{}{F}_{16}\\ {C}_{12}&{}{\widetilde{C_{22}}}&{}{C}_{26}&{}{\widetilde{C_{26}}}&{}D_{12}&{}{\widetilde{D_{22}}}&{}D_{26}&{}E_{12}&{}{\widetilde{E_{22}}}&{}E_{26}&{}{\widetilde{E_{26}}}&{}F_{12}&{}{\overline{F_{12}}}&{}{\widetilde{F_{22}}}&{}F_{26}&{}{\overline{F_{26}}}\\ {\overline{C_{16}}}&{}{C}_{26}&{}{\overline{C_{66}}}&{}C_{66}&{}{\overline{D_{16}}}&{}D_{26}&{}{\overline{D_{66}}}&{}D_{66}&{}{\overline{E_{16}}}&{}{\overline{E_{26}}}&{}{\overline{E_{66}}}&{}{E}_{66}&{}{\overline{F_{16}}}&{}{F}_{26}&{}{\overline{F_{66}}}&{}F_{66}\\ {C}_{16}&{}{\widetilde{C_{26}}}&{}C_{66}&{}{\overline{C_{66}}}&{}D_{16}&{}{\widetilde{D_{26}}}&{}D_{66}&{}{\widetilde{D_{66}}}&{}E_{16}&{}{\widetilde{E_{26}}}&{}E_{66}&{}{\widetilde{E_{66}}}&{}F_{16}&{}{\widetilde{F_{26}}}&{}F_{66}&{}{\widetilde{F_{66}}}\\ {\overline{D_{11}}}&{}{D}_{12}&{}{\overline{D_{16}}}&{}D_{16}&{}{\overline{E_{11}}}&{}E_{22}&{}{\overline{E_{16}}}&{}E_{16}&{}{\overline{F_{11}}}&{}F_{12}&{}{\overline{F_{16}}}&{}F_{16}&{}{\overline{G_{11}}}&{}G_{12}&{}{\overline{G_{16}}}&{}G_{16}\\ D_{12}&{}{\widetilde{D_{22}}}&{}D_{26}&{}{\widetilde{D_{26}}}&{}E_{12}&{}{\widetilde{E_{22}}}&{}E_{26}&{}{\widetilde{E_{26}}}&{}F_{12}&{}{\widetilde{F_{22}}}&{}F_{26}&{}{\widetilde{F_{26}}}&{}G_{12}&{}{\widetilde{G_{22}}}&{}G_{26}&{}{\widetilde{G_{26}}}\\ {\overline{D_{16}}}&{}{D}_{26}&{}{\overline{D_{66}}}&{}D_{66}&{}{\overline{E_{16}}}&{}E_{26}&{}{\overline{E_{66}}}&{}E_{66}&{}{\overline{F_{16}}}&{}F_{26}&{}{\overline{F_{66}}}&{}F_{66}&{}{\overline{G_{16}}}&{}G_{26}&{}{\overline{G_{66}}}&{}G_{66}\\ D_{16}&{}{\widetilde{D_{26}}}&{}D_{66}&{}{\overline{D_{66}}}&{}E_{16}&{}{\widetilde{E_{26}}}&{}E_{66}&{}{\widetilde{E_{66}}}&{}F_{16}&{}{\widetilde{F_{26}}}&{}F_{66}&{}{\widetilde{F_{66}}}&{}{G}_{16}&{}{\widetilde{G_{66}}}&{}G_{66}&{}{\widetilde{G_{66}}}\\ \end{array}} \right] \left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{\varepsilon _{so}}}\\ {{\varepsilon _{ro}}} \end{array}}\\ {{\varepsilon _{sro}}}\\ {{\varepsilon _{rso}}} \end{array}}\\ {{\kappa _s}} \end{array}}\\ {{\kappa _r}} \end{array}}\\ {{\kappa _{sr}}}\\ {{\kappa _{rs}}} \end{array}}\\ {\varepsilon _{so}^*}\\ {\varepsilon _{ro}^*} \end{array}}\\ {\begin{array}{*{20}{c}} {\varepsilon _{sro}^*}\\ {\varepsilon _{rso}^*}\\ {\kappa _s^*}\\ {\kappa _r^*} \end{array}}\\ {\kappa _{sr}^*}\\ {\kappa _{rs}^*} \end{array}} \right\} \end{aligned}$$
$$\begin{aligned} \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} Q_{s}\\ Q_{r}\\ \end{array} }\\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} S_{s}\\ S_{r}\\ \end{array} }\\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} Q_{s}^{*}\\ Q_{r}^{*}\\ \end{array} }\\ {\begin{array}{*{20}c} S_{s}^{*}\\ S_{r}^{*}\\ \end{array} }\\ \end{array} }\\ \end{array} }\\ \end{array} } \right\} =\left[ {\begin{array}{*{20}c} \overline{A_{44}} &{}\quad A_{45} &{}\quad \overline{B_{44}} &{}\quad B_{45} &{}\quad \overline{C_{44}} &{}\quad C_{45} &{}\quad \overline{D_{44}} &{}\quad D_{45}\\ A_{45} &{}\quad \widetilde{A_{55}} &{}\quad B_{45} &{}\quad \widetilde{B_{55}} &{}\quad C_{45} &{}\quad \widetilde{C_{55}} &{}\quad D_{45} &{}\quad \widetilde{D_{55}}\\ \overline{B_{44}} &{}\quad B_{45} &{}\quad \overline{C_{44}} &{}\quad C_{45} &{}\quad \overline{D_{44}} &{}\quad D_{45} &{}\quad \overline{E_{44}} &{}\quad E_{45}\\ B_{45} &{}\quad \widetilde{B_{55}} &{}\quad C_{45} &{}\quad \widetilde{C_{55}} &{}\quad D_{45} &{}\quad \widetilde{D_{55}} &{}\quad E_{45} &{}\quad \widetilde{E_{55}}\\ \overline{C_{44}} &{}\quad C_{45} &{}\quad \overline{D_{44}} &{}\quad D_{45} &{}\quad \overline{E_{44}} &{}\quad E_{45} &{}\quad \overline{F_{44}} &{}\quad F_{45}\\ C_{45} &{}\quad \widetilde{C_{55}} &{}\quad D_{45} &{}\quad \widetilde{D_{55}} &{}\quad E_{45} &{}\quad \widetilde{E_{55}} &{}\quad F_{45} &{}\quad \widetilde{F_{55}}\\ \overline{D_{44}} &{}\quad D_{45} &{}\quad \overline{E_{44}} &{}\quad E_{45} &{}\quad \overline{F_{44}} &{}\quad F_{45} &{}\quad \overline{G_{44}} &{}\quad G_{45}\\ D_{45} &{}\quad \widetilde{D_{55}} &{}\quad E_{45} &{}\quad \widetilde{E_{55}} &{}\quad F_{45} &{}\quad \widetilde{F_{55}} &{}\quad G_{45} &{}\quad \widetilde{G_{55}}\\ \end{array} } \right] \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} \epsilon _{szo}\\ \epsilon _{rzo}\\ \end{array} }\\ \kappa _{sz}\\ {\begin{array}{*{20}c} \kappa _{rz}\\ \epsilon _{szo}^{*}\\ {\begin{array}{*{20}c} \epsilon _{rzo}^{*}\\ {\begin{array}{*{20}c} \kappa _{sz}^{*}\\ \kappa _{rz}^{*}\\ \end{array} }\\ \end{array} }\\ \end{array} }\\ \end{array} } \right\} \end{aligned}$$
where the values of above matrix coefficients are given by
$$\begin{aligned} \left( {{A_{ij}},~~{B_{ij}},~{C_{ij}},~{D_{ij}},~{E_{ij}},~{F_{ij}},~{G_{ij}},~{H_{ij}}~} \right) = \mathop \sum \limits _{k = 1}^n {\overline{Q}} _{ij}^{\left( k \right) }\mathop \smallint \nolimits _{{z_k}}^{{z_{k + 1}}} \left( {1,~z,{z^2},{z^3},{z^4},{z^5},{z^6},{z^7}} \right) ~d \end{aligned}$$
and
$$\begin{aligned} \overline{{A_{ij} }}= & {} A_{ij} -C_0 B_{ij} ,\quad \widetilde{{A_{ij} }}=A_{ij} +C_0 B_{ij} \\ \overline{{B_{ij} }}= & {} B_{ij} -C_0 C_{ij} ,\quad \widetilde{{ B_{ij} }}=B_{ij} +C_0 C_{ij} \\ \overline{{C_{ij} }}= & {} C_{ij} -C_0 D_{ij} ,\quad \widetilde{{{C_{ij} }}} =C_{ij} +C_0 D_{ij} \\ \overline{{D_{ij} }}= & {} D_{ij} -C_0 E_{ij} ,\quad \widetilde{{ D_{ij} }}=D_{ij} +C_0 E_{ij} \\ \overline{{E_{ij} }}= & {} E_{ij} -C_0 F_{ij} , \quad \widetilde{{{E_{ij} }}} =E_{ij} +C_0 F_{ij} \\ \overline{{F_{ij} }}= & {} F_{ij} -C_0 G_{ij} , \quad \widetilde{{{{F_{ij} }}}} =F_{ij} +C_0 G_{ij} \\ \overline{{G_{ij} }}= & {} G_{ij} -C_0 H_{ij} ,\quad \widetilde{{ G_{ij} }}=G_{ij} +C_0 H_{ij} \end{aligned}$$
where \(i, j = 1, 2, 4, 5, 6\)
and \(C_0 =\left( {\frac{1}{R_s }-\frac{1}{R_r }} \right) \)
Appendix B
The non-zero terms of strain–displacement matrix [B] are as follows:
$$\begin{aligned} B_{11}= & {} B_{32} =B_{54} =B_{75} =B_{96} =B_{11,7} =B_{13,8} =B_{15,9} =B_{17,3} =\frac{\partial N_i }{\partial s} \\ B_{13}= & {} -B_{17,1} =\frac{N_i }{R_s } \\ B_{22}= & {} B_{41} =B_{65} =B_{84} =B_{10,7} =B_{12,6} =B_{14,9} =B_{16,8} =B_{18,3} =\frac{\partial N_i }{\partial r} \\ B_{23}= & {} -B_{18,2} =\frac{N_i }{R_r } \\ B_{17,4}= & {} B_{18,5} =N_i \\ B_{19,6}= & {} B_{20,7} =2N_i \\ B_{21,6}= & {} \frac{N_i }{R_s } \\ B_{22,7}= & {} \frac{N_i }{R_r } \\ B_{21,8}= & {} B_{22,9} =3N_i \\ B_{23,8}= & {} \frac{2N_i }{R_s } \\ B_{24,9}= & {} \frac{2N_i }{R_r } \end{aligned}$$
Appendix C
The first derivatives of \({\bar{Q}}_{ij} \) with respect to \(\theta \) are as follows:
$$\begin{aligned} \frac{\partial {\bar{Q}}_{11} }{\partial \theta }= & {} 4\left( {Q_{12} -Q_{11} +2Q_{66} } \right) \sin \theta \cos ^{3}\theta -4(Q_{12} -Q_{22} +2Q_{66} )\sin ^{3}\theta \cos \theta \\ \frac{\partial {\bar{Q}}_{12} }{\partial \theta }= & {} 2\left( {Q_{11} +Q_{22} -2Q_{12} -4Q_{66} } \right) \sin \theta \cos ^{3}\theta -2(Q_{11} +Q_{22} -2Q_{12} -4Q_{66} )\sin ^{3}\theta \cos \theta \\ \frac{\partial {\bar{Q}}_{22} }{\partial \theta }= & {} 4\left( {Q_{12} -Q_{22} +2Q_{66} } \right) \sin \theta \cos ^{3}\theta -4(Q_{12} -Q_{11} +2Q_{66} )\sin ^{3}\theta \cos \theta \\ \frac{\partial {\bar{Q}}_{16} }{\partial \theta }= & {} 3\left( {2Q_{12} -Q_{11} -Q_{22} +4Q_{66} } \right) \sin ^{2}\theta \cos ^{2}\theta \\&-(Q_{12} -Q_{22} +2Q_{66} )\sin ^{4}\theta -\left( {Q_{12} -Q_{11} +2Q_{66} } \right) \cos ^{4}\theta \\ \frac{\partial {\bar{Q}}_{26} }{\partial \theta }= & {} -3\left( {2Q_{12} -Q_{11} -Q_{22} +4Q_{66} } \right) \sin ^{2}\theta \cos ^{2}\theta \\&+(Q_{12} -Q_{11} +2Q_{66} )\sin ^{4}\theta +\left( {Q_{12} -Q_{22} +2Q_{66} } \right) \cos ^{4}\theta \\ \frac{\partial {\bar{Q}}_{66} }{\partial \theta }= & {} 2\left( {Q_{11} -2Q_{12} +Q_{22} -4Q_{66} } \right) \sin \theta \cos ^{3}\theta -2(Q_{11} -2Q_{12} +Q_{22} -4Q_{66} )\sin ^{3}\theta \cos \theta \\ \frac{\partial {\bar{Q}}_{44} }{\partial \theta }= & {} 2\left( {Q_{55} -Q_{44} } \right) \sin \theta \cos \theta \\ \frac{\partial {\bar{Q}}_{45} }{\partial \theta }= & {} \left( {Q_{44} -Q_{55} } \right) (\sin ^{2}\theta -\cos ^{2}\theta ) \\ \frac{\partial {\bar{Q}}_{55} }{\partial \theta }= & {} 2\left( {Q_{44} -Q_{55} } \right) \sin \theta \cos \theta \end{aligned}$$
Appendix D
The first derivatives of shape function \(N_{i}\) with respect to natural coordinates \(\xi \) and \(\eta \) are as follows:
$$\begin{aligned} \frac{\partial N_1 }{\partial {\xi }}= & {} \frac{\left( {\eta +1} \right) \left( {{\xi }-\eta +1} \right) }{4} +\frac{\left( {{\xi }-1} \right) \left( {\eta +1} \right) }{4} \\ \frac{\partial N_2 }{\partial {\xi }}= & {} -{\xi }\left( {\eta +1} \right) \\ \frac{\partial N_3 }{\partial {\xi }}= & {} \frac{\left( {\eta +1} \right) \left( {{\xi }+\eta -1} \right) }{4}+ \frac{\left( {{\xi }+1} \right) \left( {\eta +1} \right) }{4} \\ \frac{\partial N_4 }{\partial {\xi }}= & {} \frac{1}{2}-\frac{\eta ^{2}}{2} \\ \frac{\partial N_5 }{\partial {\xi }}= & {} \frac{\left( {\eta - 1} \right) \left( {\eta -{\xi }+1} \right) }{4} -\frac{\left( {{\xi }+1} \right) \left( {\eta -1} \right) }{4} \\ \frac{\partial N_6 }{\partial {\xi }}= & {} {\xi }\left( {\eta -1} \right) \\ \frac{\partial N_7 }{\partial {\xi }}= & {} -\frac{\left( {\eta -1} \right) \left( {{\xi }+\eta +1} \right) }{4} -\frac{\left( {{\xi }-1} \right) \left( {\eta -1} \right) }{4} \\ \frac{\partial N_8 }{\partial {\xi }}= & {} \frac{\eta ^{2}}{2}-\frac{1}{2} \end{aligned}$$
And
$$\begin{aligned} \frac{\partial N_1 }{\partial \eta }= & {} \frac{\left( {{\xi }-1} \right) \left( {{\xi }-\eta +1} \right) }{4}-\frac{\left( {{\xi }-1} \right) \left( {\eta +1} \right) }{4} \\ \frac{\partial N_2 }{\partial \eta }= & {} \frac{1}{2}-\frac{{\xi }^{2}}{2} \\ \frac{\partial N_3 }{\partial \eta }= & {} \frac{\left( {{\xi }+1} \right) \left( {{\xi }+\eta -1} \right) }{4}+ \frac{\left( {{\xi }+1} \right) \left( {\eta +1} \right) }{4} \\ \frac{\partial N_4 }{\partial \eta }= & {} -\eta \left( {{\xi }+1} \right) \\ \frac{\partial N_5 }{\partial \eta }= & {} \frac{\left( {{\xi }+1} \right) \left( {\eta -{\xi }+1} \right) }{4}+\frac{\left( {{\xi }+1} \right) \left( {\eta -1} \right) }{4} \\ \frac{\partial N_6 }{\partial \eta }= & {} \frac{{\xi }^{2}}{2}-\frac{1}{2} \\ \frac{\partial N_7 }{\partial \eta }= & {} -\frac{\left( {{\xi }-1} \right) \left( {{\xi }+\eta +1} \right) }{4} -\frac{\left( {{\xi }-1} \right) \left( {\eta -1} \right) }{4} \\ \frac{\partial N_8 }{\partial \eta }= & {} \eta \left( {{\xi }-1} \right) \end{aligned}$$
Appendix E
The non-zero terms of the first derivatives the elements of [B] matrix with respect to \(R_{s}\):
$$\begin{aligned} \frac{\partial B_{11} }{Rs}= & {} \frac{\partial B_{32} }{Rs}=\frac{\partial B_{54} }{Rs}=\frac{\partial B_{75} }{Rs}\\= & {} \frac{\partial B_{96} }{Rs}=\frac{\partial B_{11,7} }{Rs}=\frac{\partial B_{13,8} }{Rs}=\frac{\partial B_{15,9} }{Rs}=\frac{\partial B_{17,3} }{Rs} \\= & {} - \,\frac{{\partial {N_i}}}{{\partial \xi }} \times \frac{{\partial z}}{{\partial \xi }} \times \sum \limits _{i = 1}^8 {\left( {\frac{{\partial {N_i}}}{{\partial \xi }}\frac{{\partial {Z_i}}}{{\partial {R_s}}}} \right) } /{\left\{ {{{\left( {\frac{{\partial x}}{{\partial \xi }}} \right) }^2} + {{\left( {\frac{{\partial y}}{{\partial \xi }}} \right) }^2} + {{\left( {\frac{{\partial z}}{{\partial \xi }}} \right) }^2}} \right\} ^{3/2}}\\ \frac{\partial B_{13} }{Rs}= & {} \frac{-\partial B_{17,1} }{Rs}=\frac{\partial B_{21,6} }{Rs}=-\frac{N_i }{R_s^2 } \\ \frac{\partial B_{22} }{Rs}= & {} \frac{\partial B_{41} }{Rs}=\frac{B_{65} }{Rs}=\frac{\partial B_{84} }{Rs} =\frac{\partial B_{10,7 } }{Rs}=\frac{\partial B_{12,6} }{Rs}=\frac{\partial B_{14,9} }{Rs}=\frac{\partial B_{16,8} }{Rs}=\frac{\partial B_{18,3} }{Rs} \\= & {} -\, \frac{{\partial {N_i}}}{{\partial \xi }} \times \frac{{\partial z}}{{\partial \xi }} \times \sum \limits _{i = 1}^8 {\left( {\frac{{\partial {N_i}}}{{\partial \xi }}\frac{{\partial {Z_i}}}{{\partial {R_s}}}} \right) } /{\left\{ {{{\left( {\frac{{\partial x}}{{\partial \xi }}} \right) }^2} + {{\left( {\frac{{\partial y}}{{\partial \xi }}} \right) }^2} + {{\left( {\frac{{\partial z}}{{\partial \xi }}} \right) }^2}} \right\} ^{3/2}} \\ \frac{\partial B_{23,8} }{Rs}= & {} -\frac{2N_i }{R_s^2 } \end{aligned}$$
The non-zero terms of the first derivatives the elements of [B] matrix with respect to \(R_{r}\):
$$\begin{aligned} \frac{\partial B_{11} }{Rr}= & {} \frac{\partial B_{32} }{Rr}=\frac{\partial B_{54} }{Rr}=\frac{\partial B_{75} }{Rr}=\frac{\partial B_{96} }{Rr}=\frac{\partial B_{11,7} }{Rr}=\frac{\partial B_{13,8} }{Rr}=\frac{\partial B_{15,9} }{Rr}=\frac{\partial B_{17,3} }{Rr} \\= & {} -\, \frac{{\partial {N_i}}}{{\partial \xi }} \times \frac{{\partial z}}{{\partial \xi }} \times \sum \limits _{i = 1}^8 {\left( {\frac{{\partial {N_i}}}{{\partial \xi }}\frac{{\partial {Z_i}}}{{\partial {R_s}}}} \right) } /{\left\{ {{{\left( {\frac{{\partial x}}{{\partial \xi }}} \right) }^2} + {{\left( {\frac{{\partial y}}{{\partial \xi }}} \right) }^2} + {{\left( {\frac{{\partial z}}{{\partial \xi }}} \right) }^2}} \right\} ^{3/2}}\\ \frac{\partial B_{22} }{Rr}= & {} \frac{\partial B_{41} }{Rr}=\frac{B_{65} }{Rr}=\frac{\partial B_{84} }{Rr} =\frac{\partial B_{10,7 } }{Rr}=\frac{\partial B_{12,6} }{Rr}=\frac{\partial B_{14,9} }{Rr}=\frac{\partial B_{16,8} }{Rr}=\frac{\partial B_{18,3} }{Rr} \\= & {} -\, \frac{{\partial {N_i}}}{{\partial \xi }} \times \frac{{\partial z}}{{\partial \xi }} \times \sum \limits _{i = 1}^8 {\left( {\frac{{\partial {N_i}}}{{\partial \xi }}\frac{{\partial {Z_i}}}{{\partial {R_s}}}} \right) } /{\left\{ {{{\left( {\frac{{\partial x}}{{\partial \xi }}} \right) }^2} + {{\left( {\frac{{\partial y}}{{\partial \xi }}} \right) }^2} + {{\left( {\frac{{\partial z}}{{\partial \xi }}} \right) }^2}} \right\} ^{3/2}}\\ \frac{\partial B_{23} }{Rr}= & {} \frac{-\partial B_{18,2} }{Rr}=\frac{\partial B_{22,7} }{Rr}=-\frac{N_i }{R_r^2 } \\ \frac{\partial B_{24,9} }{Rr}= & {} -\frac{2N_i }{R_r^2 } \end{aligned}$$
