Appendix A
Assumption 1
$$\begin{aligned} &\frac{1}{\overline{r}_{i}}I_{N} = \left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \frac{1}{\overline{r}_{1}} & & & 0 \\ & \frac{1}{\overline{r}_{2}} & & \\ & & \ddots & \\ 0 & & & \frac{1}{\overline{r}_{N}} \end{array} \right ],\\ & \frac{1}{\overline{r}_{i}}I_{N - 1} = \left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \frac{1}{\overline{r}_{1}} & & & 0 \\ & \frac{1}{\overline{r}_{2}} & & \\ & & \ddots & \\ 0 & & & \frac{1}{\overline{r}_{N - 1}} \end{array} \right ],\\ &\frac{1}{\overline{r}_{i}}I_{N - 2} = \left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \frac{1}{\overline{r}_{2}} & & & 0 \\ & \frac{1}{\overline{r}_{3}} & & \\ & & \ddots & \\ 0 & & & \frac{1}{\overline{r}_{N - 1}} \end{array} \right ] \end{aligned}$$
Assumption 2
If f=f
ij
(i=n,…,n′,j=m,…,m′):
$$\begin{aligned} &(1) \ \frac{1}{\overline{r}_{i}}f = \left [ \begin{array}{@{}cccc@{}} \frac{1}{\overline{r}_{n}}f_{n,m} & \frac{1}{\overline{r}_{n}}f_{n,m + 1} & \cdots & \\ \frac{1}{\overline{r}_{n + 1}}f_{n + 1,m} & \frac{1}{\overline{r}_{n + 1}}f_{n + 1,m + 1} & \cdots & \\ \vdots & \vdots & \ddots & \\ & & & \frac{1}{\overline{r}_{n'}}f_{n',m'}\end{array} \right ]\\ &(2) \ \frac{1}{\overline{r}_{j}}f\\ &\quad = \left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \frac{1}{\overline{r}_{m}}f_{n,m} & \frac{1}{\overline{r}_{m + 1}}f_{n,m + 1} & \cdots & \\ \frac{1}{\overline{r}_{m}}f_{n + 1,m} & \frac{1}{\overline{r}_{m + 1}}f_{n + 1,m + 1} & \cdots & \\ \vdots & \vdots & \ddots & \\ & & & \frac{1}{\overline{r}_{m'}}f_{n',m'} \end{array} \right ] \end{aligned}$$
(c–c):
$$M_{b} = \left [ \begin{array}{@{}cccccccc@{}} - \lambda I_{N - 2} & 0 & 0 & - \frac{h^{2}\overline{C}_{55}}{a^{2}}f_{cc} & - \frac{h}{a} ( g_{cc}^{ ( 1 )} + \frac{1}{\overline{r}_{i}}I_{N - 2} ) & - \frac{mh}{a}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 & - \frac{h^{2}\overline{e}_{5}}{a^{2}}f_{cc} \\ 0 & 0 & 0 & - \frac{h}{a}g_{cc}^{ ( 1 )} & \frac{1}{\overline{C}_{55}}I_{N - 2} & 0 & 0 & - \frac{h\overline{e}_{5}}{a\overline{C}_{55}}g_{cc}^{ ( 1 )} \\ 0 & 0 & 0 & \frac{mh}{a}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 & \frac{1}{\overline{C}_{44}}I_{N - 2} & 0 & \frac{mh\overline{e}_{4}}{a\overline{C}_{44}}\frac{1}{\overline{r}_{i}}I_{N - 2} \\ s_{1}I_{N - 2} & s_{2}g_{cc}^{ ( 1 )} + s_{3}\frac{1}{\overline{r}_{i}}I_{N - 2} & s_{4}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 & 0 & 0 & s_{5}I_{N - 2} & 0 \\ s_{6}g_{cc}^{ ( 1 )} + s_{7}\frac{1}{\overline{r}_{i}}I_{N - 2} & M_{52} & M_{53} & 0 & - \lambda I_{N - 2} & 0 & s_{19}g_{cc}^{ ( 1 )} + s_{20}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 \\ s_{21}\frac{1}{\overline{r}_{i}}I_{N - 2} & M_{62} & M_{63} & 0 & 0 & - \lambda I_{N - 2} & s_{30}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 \\ 0 & 0 & 0 & - \frac{h^{2}\overline{e}_{5}}{a^{2}}f_{cc} & s_{31} ( g_{cc}^{ ( 1 )} + \frac{1}{\overline{r}_{i}}I_{N - 2} ) & s_{32}\frac{1}{\overline{r}_{i}}I_{N - 2} & - \lambda I_{N - 2} & M_{78} \\ s_{35}I_{N - 2} & s_{36}g_{cc}^{ ( 1 )} + s_{37}\frac{1}{\overline{r}_{i}}I_{N - 2} & s_{38}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 & 0 & 0 & s_{39}I_{N - 2} & 0 \end{array} \right ] $$
$$\begin{aligned} &\begin{aligned}M_{52} &= s_{8}g_{cc}^{ ( 2 )} + s_{9}f_{cc} + s_{11}\frac{1}{\overline{r}_{j}}g_{cc}^{ ( 1 )} + s_{12}\frac{1}{\overline{r}_{i}}g_{cc}^{ ( 1 )}\\ &\quad + s_{14}\frac{1}{\overline{r}_{i}^{2}}I_{N - 2},\end{aligned}\\ &M_{53} = s_{16}\frac{1}{\overline{r}_{j}}g_{cc}^{ ( 1 )} + s_{17}\frac{1}{\overline{r}_{i}}g_{cc}^{ ( 1 )} + s_{18}\frac{1}{\overline{r}_{i}^{2}}I_{N - 2} \\ &M_{62} = s_{22}\frac{1}{\overline{r}_{j}}g_{cc}^{ ( 1 )} + s_{23}\frac{1}{\overline{r}_{i}}g_{cc}^{ ( 1 )} + s_{24}\frac{1}{\overline{r}_{i}^{2}}I_{N - 2},\\ &M_{63} = s_{25}g_{cc}^{ ( 2 )} + s_{26} \frac{1}{\overline{r}_{j}}g_{cc}^{ ( 1 )} + s_{27} \frac{1}{\overline{r}_{i}}g_{cc}^{ ( 1 )} + s_{28} \frac{1}{\overline{r}_{i}^{2}}I_{N - 2} \\ &\begin{aligned}M_{78} &= s_{32} \biggl( g_{cc}^{ ( 2 )} + \frac{1}{\overline{r}_{i}}g_{cc}^{ ( 1 )} \biggr) + s_{33}f_{cc}\\ &\quad + s_{34}\frac{1}{\overline{r}_{i}^{2}}I_{N - 2}\end{aligned} \\ &g_{cc}^{ ( 1 )} = g_{ij}^{ ( 1 )}\ ( i,j = 2, \ldots,N - 1 ),\\ &g_{cc}^{ ( 2 )} = g_{ij}^{ ( 2 )}\ ( i,j = 2,\ldots,N - 1 ),\\ & f_{cc} = g_{i1}^{ ( 1 )}g_{1j}^{ ( 1 )} + g_{iN}^{ ( 1 )}g_{Nj}^{ ( 1 )}\ ( i,j = 2, \ldots,N - 1 ) \end{aligned}$$
(s–c):
$$M_{b} = \left [ \begin{array}{@{}cccccccc@{}} - \lambda I_{N - 2} & 0 & 0 & - \frac{h^{2}\overline{C}_{55}}{a^{2}}f_{3sc} & - \frac{h}{a} ( g_{4sc}^{ ( 1 )} + \frac{1}{\overline{r}_{i}}I_{sc} ) & - \frac{mh}{a}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 & - \frac{h^{2}\overline{e}_{5}}{a^{2}}f_{3sc} \\ 0 & 0 & 0 & - \frac{h}{a}g_{3sc}^{ ( 1 )} & \frac{1}{\overline{C}_{55}}I_{N - 1} & 0 & 0 & - \frac{h\overline{e}_{5}}{a\overline{C}_{55}}g_{3sc}^{ ( 1 )} \\ 0 & 0 & 0 & \frac{mh}{a}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 & \frac{1}{\overline{C}_{44}}I_{N - 2} & 0 & \frac{mh\overline{e}_{4}}{a\overline{C}_{44}}\frac{1}{\overline{r}_{i}}I_{N - 2} \\ s_{1}I_{N - 2} & s_{2}g_{4sc}^{ ( 1 )} + s_{3}\frac{1}{\overline{r}_{i}}I_{sc} & s_{4}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 & 0 & 0 & s_{5}I_{N - 2} & 0 \\ s_{6}g_{3sc}^{ ( 1 )} + s_{7}\frac{1}{\overline{r}_{i}}I_{sc}^{T} & M_{52} & M_{53} & 0 & - \lambda I_{N - 1} & 0 & s_{19}g_{3sc}^{ ( 1 )} + s_{20}\frac{1}{\overline{r}_{i}}I_{sc}^{T} & 0 \\ s_{21}\frac{1}{\overline{r}_{i}}I_{N - 2} & M_{62} & M_{63} & 0 & 0 & - \lambda I_{N - 2} & s_{30}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 \\ 0 & 0 & 0 & - \frac{h^{2}\overline{e}_{5}}{a^{2}}f_{3sc} & s_{31} ( g_{4sc}^{ ( 1 )} + \frac{1}{\overline{r}_{i}}I_{sc} ) & s_{32}\frac{1}{\overline{r}_{i}}I_{N - 2} & - \lambda I_{N - 2} & M_{78} \\ s_{35}I_{N - 2} & s_{36}g_{4sc}^{ ( 1 )} + s_{37}\frac{1}{\overline{r}_{i}}I_{sc} & s_{38}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 & 0 & 0 & s_{39}I_{N - 2} & 0 \end{array} \right ] $$
$$\begin{aligned} &\begin{aligned} M_{52} &= s_{8}g_{1sc}^{ ( 2 )} + s_{9}f_{1sc} + s_{10}f_{2sc} \\ &\quad + s_{11} \biggl[ [ 0 ]_{N - 1 \times 1} \biggl[ \frac{1}{\overline{r}_{j}}g_{3sc}^{ ( 1 )} \biggr] \biggr] + \left [ \begin{array}{l} {[ s_{13}\frac{1}{\overline{r}_{1}}g_{1j}^{ ( 1 )} ]} \\ {[ s_{12}\frac{1}{\overline{r}_{i}}g_{4sc}^{ ( 1 )} ]} \end{array} \right ] \\ &\quad + \left [ \begin{array}{l} {[ s_{15}\frac{1}{\overline{r}_{1}^{2}} ]} [ 0 ]_{1 \times N - 2} \\ {[ s_{14}\frac{1}{\overline{r}_{i}^{2}}I_{sc} ]} \end{array} \right ] \end{aligned}\\ &M_{53} = s_{16}\frac{1}{\overline{r}_{j}}g_{3sc}^{ ( 1 )} + s_{17}\frac{1}{\overline{r}_{i}}g_{3sc}^{ ( 1 )} + s_{18}\frac{1}{\overline{r}_{i}^{2}}I_{sc}^{T},\\ &M_{62} = s_{22}\frac{1}{\overline{r}_{j}}g_{4sc}^{ ( 1 )} + s_{23}\frac{1}{\overline{r}_{i}}g_{4sc}^{ ( 1 )} + s_{24}\frac{1}{\overline{r}_{i}^{2}}I_{sc} \\ &M_{63} = s_{25}g_{2sc}^{ ( 2 )} \,{+}\, s_{26}\frac{1}{\overline{r}_{j}}g_{2sc}^{ ( 1 )} + s_{27}\frac{1}{\overline{r}_{i}}g_{2sc}^{ ( 1 )} + s_{28}\frac{1}{\overline{r}_{i}^{2}}I_{N - 2},\\ & M_{78} = s_{32} \biggl( g_{2sc}^{ ( 2 )} + \frac{1}{\overline{r}_{i}}g_{2sc}^{ ( 1 )} \biggr) + s_{33}f_{3sc} + s_{34}\frac{1}{\overline{r}_{i}^{2}}I_{N - 2} \\ &g_{2sc}^{ ( 1 )} = g_{ij}^{ ( 1 )} \ ( i,j = 2, \ldots,N - 1 ),\\ &g_{3sc}^{ ( 1 )} = g_{ij}^{ ( 1 )} \ ( i = 1,\ldots,N - 1,j = 2,\ldots,N - 1 ) \\ &g_{4sc}^{ ( 1 )} = g_{ij}^{ ( 1 )} \ ( i = 2, \ldots,N - 1,j = 1,\ldots,N - 1 ),\\ &g_{1sc}^{ ( 2 )} = g_{ij}^{ ( 2 )}\ ( i,j = 1,\ldots,N - 1 ) \\ &g_{2sc}^{ ( 2 )} = g_{ij}^{ ( 2 )} \ ( i,j = 2, \ldots,N - 1 ),\\ &f_{1sc} = g_{i1}^{ ( 1 )}g_{1j}^{ ( 1 )} + g_{iN}^{ ( 1 )}g_{Nj}^{ ( 1 )} \ ( i,j = 1, \ldots,N - 1 ) \\ &f_{2sc} = g_{i1}^{ ( 1 )}g_{1j}^{ ( 1 )} \ ( i,j = 1,\ldots,N - 1 ),\\ &f_{3sc} = g_{iN}^{ ( 1 )}g_{Nj}^{ ( 1 )} \ ( i,j = 2,\ldots,N - 1 ),\\ &I_{sc} = \bigl[ [ 0 ]_{N - 2 \times 1}I_{N - 2} \bigr] \end{aligned}$$
(s–s):
$$M_{b} = \left [ \begin{array}{c@{ \ }c@{ \ }c@{ \ }c@{ \ }c@{ \ }c@{ \ }c@{ \ }c} - \lambda I_{N - 2} & 0 & 0 & 0 & - \frac{h}{a} ( g_{4ss}^{ ( 1 )} + \frac{1}{\overline{r}_{i}}I_{ss} ) & - \frac{mh}{a}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 & 0 \\ 0 & 0 & 0 & - \frac{h}{a}g_{3ss}^{ ( 1 )} & \frac{1}{\overline{C}_{55}}I_{N} & 0 & 0 & - \frac{h\overline{e}_{5}}{a\overline{C}_{55}}g_{3ss}^{ ( 1 )} \\ 0 & 0 & 0 & \frac{mh}{a}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 & \frac{1}{\overline{C}_{44}}I_{N - 2} & 0 & \frac{mh\overline{e}_{4}}{a\overline{C}_{44}}\frac{1}{\overline{r}_{i}}I_{N - 2} \\ s_{1}I_{N - 2} & s_{2}g_{4ss}^{ ( 1 )} + s_{3}\frac{1}{\overline{r}_{i}}I_{ss} & s_{4}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 & 0 & 0 & s_{5}I_{N - 2} & 0 \\ s_{6}g_{3ss}^{ ( 1 )} + s_{7}\frac{1}{\overline{r}_{i}}I_{ss}^{T} & M_{52} & M_{53} & 0 & - \lambda I_{N} & 0 & s_{19}g_{3ss}^{ ( 1 )} + s_{20}\frac{1}{\overline{r}_{i}}I_{ss}^{T} & 0 \\ s_{21}\frac{1}{\overline{r}_{i}}I_{N - 2} & M_{62} & M_{63} & 0 & 0 & - \lambda I_{N - 2} & s_{30}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 \\ 0 & 0 & 0 & 0 & s_{31} ( g_{4ss}^{ ( 1 )} + \frac{1}{\overline{r}_{i}}I_{ss} ) & s_{32}\frac{1}{\overline{r}_{i}}I_{N - 2} & - \lambda I_{N - 2} & M_{78} \\ s_{35}I_{N - 2} & s_{36}g_{4ss}^{ ( 1 )} + s_{37}\frac{1}{\overline{r}_{i}}I_{ss} & s_{38}\frac{1}{\overline{r}_{i}}I_{N - 2} & 0 & 0 & 0 & s_{39}I_{N - 2} & 0 \end{array} \right ] $$
where
$$\begin{aligned} &\begin{aligned}M_{52} &= s_{8} \bigl( g_{1ss}^{ ( 2 )} - f_{ss} \bigr)\\ &\quad + s_{11} \biggl[ [ 0 ]_{N \times 1} \biggl[ \frac{1}{\overline{r}_{j}}g_{3ss}^{ ( 1 )} \biggr] [ 0 ]_{N \times 1} \biggr]\\ &\quad + \left [ \begin{array}{l} {[ s_{13}\frac{1}{\overline{r}_{1}}g_{1j}^{ ( 1 )} ]} \\ {[ s_{12}\frac{1}{\overline{r}_{i}}g_{4ss}^{ ( 1 )} ]} \\ {[ s_{13}\frac{1}{\overline{r}_{N}}g_{Nj}^{ ( 1 )} ]} \end{array} \right ] + \left [ \begin{array}{l} {[ s_{15}\frac{1}{\overline{r}_{1}^{2}} ]} [ 0 ]_{1 \times N - 1} \\ {[ s_{14}\frac{1}{\overline{r}_{i}^{2}}I_{ss} ]} \\ {[ 0 ]_{1 \times N - 1}} {[ s_{15}\frac{1}{\overline{r}_{N}^{2}} ]} \end{array} \right ]\end{aligned}\\ &M_{53} = s_{16}\frac{1}{\overline{r}_{j}}g_{3ss}^{ ( 1 )} + s_{17}\frac{1}{\overline{r}_{i}}g_{3ss}^{ ( 1 )} + s_{18}\frac{1}{\overline{r}_{i}^{2}}I_{ss}^{T},\\ &M_{62} = s_{22}\frac{1}{\overline{r}_{j}}g_{4ss}^{ ( 1 )} + s_{23}\frac{1}{\overline{r}_{i}}g_{4ss}^{ ( 1 )} + s_{24}\frac{1}{\overline{r}_{i}^{2}}I_{ss} \\ &M_{63} = s_{25}g_{2ss}^{ ( 2 )} + s_{26}\frac{1}{\overline{r}_{j}}g_{2ss}^{ ( 1 )} + s_{27}\frac{1}{\overline{r}_{i}}g_{2ss}^{ ( 1 )} + s_{28}\frac{1}{\overline{r}_{i}^{2}}I_{N - 2},\\ & M_{78} = s_{32} \biggl( g_{2ss}^{ ( 2 )} + \frac{1}{\overline{r}_{i}}g_{2ss}^{ ( 1 )} \biggr) + s_{34} \frac{1}{\overline{r}_{i}^{2}}I_{N - 2} \\ &g_{2ss}^{ ( 1 )} = g_{ij}^{ ( 1 )} \ ( i,j = 2, \ldots,N - 1 ),\\ & g_{3ss}^{ ( 1 )} = g_{ij}^{ ( 1 )} \ ( i = 1,\ldots,N,j = 2,\ldots,N - 1 ) \\ &g_{4ss}^{ ( 1 )} = g_{ij}^{ ( 1 )} \ ( i = 2, \ldots,N - 1,j = 1,\ldots,N ),\\ &g_{1ss}^{ ( 2 )} = g_{ij}^{ ( 2 )} \ ( i,j = 1,\ldots,N ) \\ &g_{2ss}^{ ( 2 )} = g_{ij}^{ ( 2 )} \ ( i,j = 2, \ldots,N - 1 ),\\ &f_{ss} = g_{i1}^{ ( 1 )}g_{1j}^{ ( 1 )} + g_{iN}^{ ( 1 )}g_{Nj}^{ ( 1 )} \ ( i,j = 1, \ldots,N ),\\ &I_{ss} = \bigl[ [ 0 ]_{N - 2 \times 1}I_{N - 2} [ 0 ]_{N - 2 \times 1} \bigr] \end{aligned}$$
where
$$\begin{aligned} &M_{52} = s_{8}g_{ij}^{ ( 2 )} + s_{11}\frac{1}{\overline{r}_{j}}g_{ij}^{ ( 1 )} + s_{12}\frac{1}{\overline{r}_{i}}g_{ij}^{ ( 1 )} + s_{14}\frac{1}{\overline{r}_{i}^{2}}I_{N},\\ & M_{53} = s_{16}\frac{1}{\overline{r}_{j}}g_{ij}^{ ( 1 )} + s_{17}\frac{1}{\overline{r}_{i}}g_{ij}^{ ( 1 )} + s_{18}\frac{1}{\overline{r}_{i}^{2}}I_{N} \\ &M_{62} = s_{22}\frac{1}{\overline{r}_{j}}g_{ij}^{ ( 1 )} + s_{23}\frac{1}{\overline{r}_{i}}g_{ij}^{ ( 1 )} + s_{24}\frac{1}{\overline{r}_{i}^{2}}I_{N},\\ &M_{63} = s_{25}g_{ij}^{ ( 2 )} + s_{26} \frac{1}{\overline{r}_{j}}g_{ij}^{ ( 1 )} + s_{27} \frac{1}{\overline{r}_{i}}g_{ij}^{ ( 1 )} + s_{28} \frac{1}{\overline{r}_{i}^{2}}I_{N} \\ &M_{78} = s_{32} \biggl( g_{ij}^{ ( 2 )} + \frac{1}{\overline{r}_{i}}g_{ij}^{ ( 1 )} \biggr) + s_{34} \frac{1}{\overline{r}_{i}^{2}}I_{N}, \\ &s_{1} = \frac{\overline{k}_{3}}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}},\quad s_{2} = - \frac{h}{a} \biggl( \frac{\overline{k}_{3}\overline{C}_{13} + \overline{e}_{1}\overline{e}_{3}}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr),\\ & s_{3} = - \frac{h}{a} \biggl( \frac{\overline{k}_{3}\overline{C}_{23} + \overline{e}_{2}\overline{e}_{3}}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr),\\ & s_{4} = - \frac{mh}{a} \biggl( \frac{\overline{k}_{3}\overline{C}_{23} + \overline{e}_{2}\overline{e}_{3}}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr)\\ &s_{5} = \frac{\overline{e}_{3}}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}},\quad s_{6} = - \frac{h}{a} \biggl( \frac{\overline{k}_{3}\overline{C}_{13} + \overline{e}_{1}\overline{e}_{3}}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr),\\ & s_{7} = - \frac{h}{a} \biggl( \frac{\overline{k}_{3} ( \overline{C}_{13} - \overline{C}_{23} ) + \overline{e}_{3} ( \overline{e}_{1} - \overline{e}_{2} )}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr) \\ &\begin{aligned} s_{8} &= - \frac{h^{2}}{a^{2}} \biggl( \overline{C}_{11} \\ &\quad+ \frac{\overline{e}_{1} ( \overline{e}_{1}\overline{C}_{33} - \overline{e}_{3}\overline{C}_{13} ) - \overline{C}_{13} ( \overline{k}_{3}\overline{C}_{13} + \overline{e}_{1}\overline{e}_{3} )}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr)\end{aligned}\\ &s_{9} = \frac{h^{2}}{a^{2}} \biggl( \frac{\overline{e}_{1} ( \overline{e}_{1}\overline{C}_{33} - \overline{e}_{3}\overline{C}_{13} ) - \overline{C}_{13} ( \overline{k}_{3}\overline{C}_{13} + \overline{e}_{1}\overline{e}_{3} )}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr),\\ &s_{10} = \frac{h^{2}\overline{C}_{11}}{a^{2}} \\ &\begin{aligned}s_{11} &= - \frac{h^{2}}{a^{2}} \biggl( \overline{C}_{12} \\ &\quad + \frac{\overline{e}_{1} ( \overline{e}_{2}\overline{C}_{33} - \overline{e}_{3}\overline{C}_{23} ) - \overline{C}_{13} ( \overline{k}_{3}\overline{C}_{23} + \overline{e}_{2}\overline{e}_{3} )}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr)\end{aligned} \end{aligned}$$
$$\begin{aligned} &s_{12} = \frac{h^{2}}{a^{2}} \biggl( \overline{C}_{12} - \overline{C}_{11} + \frac{ ( \overline{C}_{13} - \overline{C}_{23} ) ( \overline{k}_{3}\overline{C}_{13} + \overline{e}_{1}\overline{e}_{3} ) - ( \overline{e}_{1} - \overline{e}_{2} ) ( \overline{e}_{1}\overline{C}_{33} - \overline{e}_{3}\overline{C}_{13} )}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr),\quad s_{13} = \frac{h^{2}\overline{C}_{12}}{a^{2}} \\ &s_{14} = \frac{h^{2}}{a^{2}} \biggl( m^{2} \overline{C}_{66} + \overline{C}_{22} - \overline{C}_{12} + \frac{ ( \overline{C}_{13} - \overline{C}_{23} ) ( \overline{k}_{3}\overline{C}_{23} + \overline{e}_{2}\overline{e}_{3} ) - ( \overline{e}_{1} - \overline{e}_{2} ) ( \overline{e}_{2}\overline{C}_{33} - \overline{e}_{3}\overline{C}_{23} )}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr) \\ &s_{15} = \frac{h^{2}}{a^{2}} \bigl( m^{2} \overline{C}_{66} + \overline{C}_{22} \bigr),\quad s_{16} = - \frac{mh^{2}}{a^{2}} \biggl( \overline{C}_{12} + \frac{\overline{e}_{1} ( \overline{e}_{2}\overline{C}_{33} - \overline{e}_{3}\overline{C}_{23} ) - \overline{C}_{13} ( \overline{k}_{3}\overline{C}_{23} + \overline{e}_{2}\overline{e}_{3} )}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr),\quad s_{17} = - \frac{mh^{2}\overline{C}_{66}}{a^{2}} \\ &s_{18} = \frac{mh^{2}}{a^{2}} \biggl( \overline{C}_{66} + \overline{C}_{22} - \overline{C}_{12} + \frac{ ( \overline{C}_{13} - \overline{C}_{23} ) ( \overline{k}_{3}\overline{C}_{23} + \overline{e}_{2}\overline{e}_{3} ) - ( \overline{e}_{1} - \overline{e}_{2} ) ( \overline{e}_{2}\overline{C}_{33} - \overline{e}_{3}\overline{C}_{23} )}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr) \end{aligned}$$
$$\begin{aligned} &s_{19} = - \frac{h}{a} \biggl( \frac{\overline{e}_{3}\overline{C}_{13} - \overline{e}_{1}\overline{C}_{33}}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr),\\ & s_{20} = - \frac{h}{a} \biggl( \frac{\overline{e}_{3} ( \overline{C}_{13} - \overline{C}_{23} ) - \overline{C}_{33} ( \overline{e}_{1} - \overline{e}_{2} )}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr),\\ & s_{21} = \frac{mh}{a} \biggl( \frac{\overline{k}_{3}\overline{C}_{23} + \overline{e}_{2}\overline{e}_{3}}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr)\\ &s_{22} = \frac{mh^{2}\overline{C}_{66}}{a^{2}},\\ &\begin{aligned} s_{23} &= \frac{mh^{2}}{a^{2}} \biggl( \overline{C}_{12} \\ &\quad + \frac{\overline{e}_{2} ( \overline{e}_{1}\overline{C}_{33} - \overline{e}_{3}\overline{C}_{13} ) - \overline{C}_{23} ( \overline{k}_{3}\overline{C}_{13} + \overline{e}_{1}\overline{e}_{3} )}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr)\end{aligned}\\ &\begin{aligned} s_{24} &= \frac{mh^{2}}{a^{2}} \biggl( \overline{C}_{22} + 2 \overline{C}_{66}\\ &\quad + \frac{\overline{e}_{2} ( \overline{e}_{2}\overline{C}_{33} - \overline{e}_{3}\overline{C}_{23} ) - \overline{C}_{23} ( \overline{k}_{3}\overline{C}_{23} + \overline{e}_{2}\overline{e}_{3} )}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr),\end{aligned}\\ & s_{25} = - \frac{h^{2}\overline{C}_{66}}{a^{2}},\quad s_{26} = \frac{h^{2}\overline{C}_{66}}{a^{2}}\\ &s_{27} = - \frac{2h^{2}\overline{C}_{66}}{a^{2}},\\ &\begin{aligned} s_{28} &= \frac{h^{2}}{a^{2}} \biggl( m^{2}\overline{C}_{22} + 2 \overline{C}_{66} \\ &\quad + \frac{m^{2}\overline{e}_{2} ( \overline{e}_{2}\overline{C}_{33} - \overline{e}_{3}\overline{C}_{23} ) - m^{2}\overline{C}_{23} ( \overline{k}_{3}\overline{C}_{23} + \overline{e}_{2}\overline{e}_{3} )}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr)\end{aligned}\\ &s_{29} = \frac{mh}{a} \biggl( \frac{\overline{e}_{3}\overline{C}_{23} - \overline{e}_{2}\overline{C}_{33}}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr),\quad s_{30} = - \frac{h\overline{e}_{5}}{a\overline{C}_{55}},\\ & s_{31} = - \frac{mh\overline{e}_{4}}{a\overline{C}_{44}},\quad s_{32} = \frac{h^{2}}{a^{2}} \biggl( \frac{\overline{k}_{1}\overline{C}_{55} + \overline{e}_{5}^{2}}{\overline{C}_{55}} \biggr),\\ & s_{33} = - \frac{h^{2}\overline{e}_{5}^{2}}{a^{2}\overline{C}_{55}}\\ &s_{34} = - \frac{m^{2}h^{2}}{a^{2}} \biggl( \frac{\overline{k}_{2}\overline{C}_{44} + \overline{e}_{4}^{2}}{\overline{C}_{44}} \biggr),\\ & s_{35} = \frac{\overline{e}_{3}}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}},\\ & s_{36} = \frac{h}{a} \biggl( \frac{\overline{e}_{1}\overline{C}_{33} - \overline{e}_{3}\overline{C}_{13}}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr),\\ & s_{37} = \frac{h}{a} \biggl( \frac{\overline{e}_{2}\overline{C}_{33} - \overline{e}_{3}\overline{C}_{23}}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr) \\ &s_{38} = \frac{mh}{a} \biggl( \frac{\overline{e}_{2}\overline{C}_{33} - \overline{e}_{3}\overline{C}_{23}}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \biggr),\quad s_{39} = - \frac{\overline{C}_{33}}{\overline{k}_{3}\overline{C}_{33} + \overline{e}_{3}^{2}} \end{aligned}$$
Appendix B
$$\begin{aligned} K &= \biggl( \frac{k_{w}h}{C_{11}} + \frac{m^{2}k_{g}h}{a^{2}C_{11}}\frac{1}{\overline{r}_{i}^{2}} \biggr)I_{N - 2} - \frac{k_{g}h}{a^{2}C_{11}}\frac{1}{\overline{r}_{i}}g_{ij}^{ ( 1 )} \\ &\quad - \frac{k_{g}h}{a^{2}C_{11}}g_{ij}^{ ( 2 )} \quad ( i,j = 2,\ldots,N - 1 ) \end{aligned} $$
