Lithospheric plane strain deformation caused by a vertical dip-slip fault

Abstract

A detailed solution for the lithospheric plane strain deformation of a two-layer composition consisting of a homogeneous, elastically isotropic (HEI) stratum of finite thickness lying on a homogeneous, elastically orthotropic substratum (OSS) produced by two-dimensional faulting placed in the isotropic stratum has been obtained. We find the deformation field in the integral form employing Airy’s stress function method, assuming the upper surface to be traction free. The denominator is approximated as a finite sum of exponential terms (FSET) using the least square method to calculate the integrals analytically. The deformation field for a vertical dip-slip (VDS) line dislocation is elaborated. The displacement field in the stratum assuming the OSS as Olivine or Barytes has been compared with the isotropic case numerically for the continental crust model (CCM).

Research highlights

We obtain the analytical solution for an HEI stratum of finite thickness lying on a homogeneous elastically OSS by a VDS line dislocation placed in the stratum. We choose the two-dimensional approximation, and the stresses and displacement have been obtained in the integral form by applying Airy’s stress function approach. To evaluate the integrals analytically, we approximate the denominator as FSET using least square approximation and then the integrals are evaluated analytically using the standard integral tables. The expressions for displacement field has been obtained and plotted numerically for the CCM.

Appendices

Appendix A

$$ \begin{aligned} C_{1} & = - \frac{1}{k\Delta }\left[ {S_{1}^{ - } e^{ - kh} \left\{ {\left( {a - b} \right) - N_{3} - N_{4} - N_{5} - 2M_{3} kH + M_{1} e^{2kH} } \right\}} \right. \\ & \quad + S_{2}^{ - } e^{ - kh} \left\{ { - 2M_{4} + M_{3} kh + \left( {N_{1} - N_{2} } \right)kH + \left( {N_{3} + N_{4} } \right)k\left( {H - h} \right)} \right. + 2M_{3} k^{2} H\left( {H - h} \right)\left. { + M_{1} khe^{2kH} } \right\} \\ & \quad + S_{1}^{ + } e^{kh} \left\{ {\left( {a - b} \right) + N_{1} - N_{2} - N_{5} + 2M_{3} kH + M_{2} e^{ - 2kH} } \right\} \\ & \quad - S_{2}^{ + } e^{kh} \left\{ {2M_{4} + M_{3} kh - \left( {N_{1} - N_{2} } \right)k\left( {H - h} \right) - \left( {N_{3} + N_{4} } \right)kH} \right. + 2M_{3} k^{2} H\left( {h - H} \right)\left. {\left. { + M_{2} khe^{ - 2kH} } \right\}} \right] \\ \end{aligned} $$

$$ \begin{aligned} C_{2} & = - \frac{1}{k\Delta }\left[ {S_{1}^{ - } e^{ - kh} \left\{ {2M_{3} + 2M_{1} e^{2kH} } \right\} - S_{2}^{ - } e^{ - kh} \left\{ {M_{3} + N_{1} - N_{2} + 2kM_{3} \left( {H - h} \right)\left. { + M_{1} \left( {1 - 2kh} \right)e^{2kH} } \right\}} \right.} \right. \nonumber \\ & \quad + S_{1}^{ + } e^{kh} \left\{ {2\left( {N_{1} - N_{2} } \right) + 4M_{3} kH} \right\} + S_{2}^{ + } e^{kh} \left\{ {\delta^{2} \left( {a - b} \right) + 2N_{2} - N_{3} + N_{4} - N_{5} } \right. \nonumber \\ & \left. {\left. {\quad + 2k\left( {N_{1} - N_{2} } \right)\left( {H - h} \right) - 2\left( {M_{3} - N_{3} - N_{4} } \right)kH + 4M_{3} k^{2} H(H - h) + M_{2} e^{ - 2kH} } \right\}} \right] \end{aligned} $$

$$ \begin{aligned} C_{3} & = - \frac{1}{k\Delta }\left[ {S_{1}^{ - } e^{ - kh} \left\{ {\delta^{2} \left( {a - b} \right) + N_{1} + N_{2} + 2N_{4} - N_{5} } \right.} \right. + 2M_{3} kH + 2\left( {N_{1} - N_{2} + N_{3} + N_{4} } \right)kH \\ & \quad + 4M_{3} k^{2} H^{2} + \left. {M_{2} e^{ - 2kH} } \right\} + S_{2}^{ - } e^{ - kh} \left\{ {2M_{4} + M_{3} kh - (N_{1} - N_{2} )k(H - h) - (N_{3} + N_{4} )kH} \right. \\ & \quad - 4M_{4} kh + 2M_{3} k^{2} H\left( {h - H} \right) + 2k^{2} \left( {N_{1} - N_{2} + N_{3} + N_{4} } \right)Hh + 4M_{3} k^{3} H^{2} h \\ & \quad + \left. {M_{2} khe^{ - 2kH} } \right\} - S_{1}^{ + } e^{kh} \left\{ {M_{3} + N_{1} - N_{2} } \right. + 2M_{3} kH + \left. {M_{2} e^{ - 2kH} } \right\} + S_{2}^{ + } ke^{kh} \left\{ {2M_{4} } \right. \\ & \quad + M_{3} kh - \left( {N_{1} - N_{2} } \right)k\left( {H - h} \right) - \left( {N_{3} + N_{4} } \right)kH + 2M_{3} k^{2} H\left( {h - H} \right) + \left. {M_{2} \left. {khe^{ - 2kH} } \right\}} \right] \\ \end{aligned} $$

$$ \begin{aligned} C_{4} & = \frac{1}{k\Delta }\left[ {S_{1}^{ - } e^{ - kh} \left\{ {2\left( {N_{3} + N_{4} } \right) + 4M_{3} kH} \right\}} \right. + S_{2}^{ - } e^{ - kh} \left\{ {\left( {a - b} \right) - N_{3} - N_{4} - N_{5} } \right. \\ & \left. {\quad + 2\left( {N_{3} + N_{4} } \right)kh - 2M_{3} kH + 4M_{3} k^{2} Hh + M_{2} e^{ - 2kH} } \right\} \\ & \quad - 2S_{1}^{ + } e^{kh} \left\{ {M_{3} + M_{2} e^{ - 2kH} } \right\} + S_{2}^{ + } e^{kh} \left\{ {\left( {a - b} \right) - N_{3} - N_{4} - N_{5} + 2kM_{3} \left( {h - H} \right)} \right. \\ & \left. {\left. {\quad + M_{2} \left( {2kh + 1} \right)e^{ - 2kH} } \right\}} \right] \\ \end{aligned} $$

$$ \begin{aligned} A_{1} & = \frac{{ - 2e^{akH} }}{\alpha k\Delta }\left[ {S_{1}^{ - } e^{ - kh} \left\{ { - O_{1} e^{ - kH} + \left( {O_{1} - \frac{2}{\alpha }\left( {1 + b} \right) + 2kHO_{2} } \right)e^{kH} } \right\}} \right. \\ & \quad + S_{2}^{ - } e^{ - kh} \left\{ {\left( {\frac{{O_{1} - O_{2} }}{2} + \frac{1}{\alpha }\left( {1 - b} \right) + k\left( {H - h} \right)O_{1} } \right)e^{ - kH} } \right. \\ & \quad \left. { + \left( {\frac{1}{\alpha }\left( {1 + b} \right) - \frac{{O_{1} - O_{2} }}{2} + k(O_{1} h - O_{2} H) - \frac{2}{\alpha }\left( {1 + b} \right)hk + 2k^{2} HhO_{2} } \right)e^{kH} } \right\} \\ & \quad + S_{1}^{ + } e^{kh} \left\{ {\left( {O_{2} - \frac{2}{\alpha }\left( {1 - b} \right) - 2kHO_{1} } \right)e^{ - kH} - O_{2} e^{kH} } \right\} \\ & \quad + S_{2}^{ + } e^{kh} \left\{ {\left( {\frac{1}{\alpha }\left( {1 - b} \right) + \frac{{O_{1} - O_{2} }}{2} + (O_{1} H - O_{2} h)k + \frac{2}{\alpha }\left( {1 - b} \right)hk + 2k^{2} HhO_{1} } \right)e^{ - kH} } \right. \\ & \quad \left. {\left. { + \left( {\frac{{O_{2} - O_{1} }}{2} + \frac{1}{\alpha }\left( {1 + b} \right) - k\left( {H - h} \right)O_{2} } \right)e^{kH} } \right\}} \right] \\ \end{aligned} $$

$$ \begin{aligned} A_{2} & = \frac{{2e^{bkH} }}{\alpha k\Delta }\left[ {S_{1}^{ - } e^{ - kh} \left\{ { - O_{3} e^{ - kH} + \left( {O_{3} - \frac{2}{\alpha }\left( {1 + a} \right) + 2kHO_{4} } \right)e^{kH} } \right\}} \right. \nonumber \\ & \quad + S_{2}^{ - } e^{ - kh} \left\{ {\left( {\frac{{O_{3} - O_{4} }}{2} + \frac{1}{\alpha }\left( {1 - a} \right) + k\left( {H - h} \right)O_{3} } \right)e^{ - kH} } \right. \nonumber \\ & \quad \left. { + \left( {\frac{1}{\alpha }\left( {1 + a} \right) - \frac{{O_{3} - O_{4} }}{2} + (O_{3} h - O_{4} H)k - \frac{2}{\alpha }\left( {1 + a} \right)hk + 2k^{2} HhO_{4} } \right)e^{kH} } \right\} \nonumber \\ & \quad + S_{1}^{ + } e^{kh} \left\{ {\left( {O_{4} - \frac{2}{\alpha }\left( {1 - a} \right) - 2kHO_{3} } \right)e^{ - kH} - O_{4} e^{kH} } \right\} \nonumber \\ & \quad + S_{2}^{ + } e^{kh} \left\{ {\left( {\frac{1}{\alpha }\left( {1 - a} \right) + \frac{{O_{3} - O_{4} }}{2} + (O_{3} H - O_{4} h)k + \frac{2}{\alpha }\left( {1 - a} \right)hk + 2k^{2} HhO_{3} } \right)e^{ - kH} } \right. \nonumber \\ & \left. {\left. {\quad + \left( {\frac{{O_{4} - O_{3} }}{2} + \frac{1}{\alpha }\left( {1 + a} \right) - k\left( {H - h} \right)O_{4} } \right)e^{kH} } \right\}} \right] \end{aligned} $$

where

$$ N_{1} = \frac{2\mu }{\alpha }\left( {r_{1} - r_{2} + ar_{2} - br_{1} } \right),\quad \quad N_{2} = \frac{2\mu }{\alpha }\left( {s_{1} - s_{2} + as_{2} - bs_{1} } \right), $$

$$ N_{3} = \frac{2\mu }{\alpha }\left( {r_{1} - r_{2} - ar_{2} + br_{1} } \right),\quad \quad N_{4} = \frac{2\mu }{\alpha }\left( {s_{1} - s_{2} - as_{2} + bs_{1} } \right), $$

$$ N_{5} = 2\mu \left\{ {s_{1} - s_{2} + ar_{2} - br_{1} + 2\mu \left( {r_{1} s_{2} - r_{2} s_{1} } \right)} \right\},\quad \quad \delta = \frac{2}{\alpha } - 1, $$

$$ M_{1} = \delta \left( {a - b} \right) + N_{3} - N_{4} + N_{5} ,\quad M_{2} = \delta \left( {a - b} \right) - N_{1} - N_{2} + N_{5} , $$

$$ M_{3} = a - b - N_{5} ,\quad M_{4} = \frac{1}{2}\left\{ {\frac{1}{2}\left( {1 - \delta^{2} } \right)\left( {a - b} \right) - \left( {N_{2} + N_{4} } \right)} \right\}, $$

$$ O_{1} = 1 + b - 2\mu r_{2} - 2\mu s_{2} ,\quad \quad O_{2} = 1 - b - 2\mu r_{2} + 2\mu s_{2} , $$

$$ O_{3} = 1 + a - 2\mu r_{1} - 2\mu s_{1} ,\quad \quad O_{4} = 1 - a - 2\mu r_{1} + 2\mu s_{1} , $$

$$ \begin{aligned} \Delta & = M_{1} e^{2kH} + M_{2} e^{ - 2kH} + \left\{ {\left( {1 + \delta^{2} } \right)\left( {a - b} \right) + N_{1} + N_{2} - N_{3} + N_{4} - 2N_{5} } \right. \\ & \quad + 2\left( {N_{1} - N_{2} + N_{3} + N_{4} } \right)kH + 4\left. {M_{3} k^{2} H^{2} } \right\}. \\ \end{aligned} $$

Appendix B

$$ \begin{aligned} \phi_{1} & = \left| { - \left\{ {2\delta M_{4} + \frac{{N_{3} + N_{4} - M_{3} }}{\alpha }} \right\}\frac{{q_{n} }}{{Q_{n}^{2} }}} \right. \\ {\kern 1pt} & \quad + \frac{1}{{Q_{n}^{2} }}\left\{ {M_{3} \left( {z - h + \frac{2H}{\alpha }} \right) - \left( {N_{1} - N_{2} } \right)\delta \left( {H - h} \right) - \left( {N_{3} + N_{4} } \right)\left( {z + \delta H} \right) - 4zM_{4} } \right\}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ {\kern 1pt} & \quad + \frac{{4q_{n} }}{{Q_{n}^{4} }}\left\{ { - M_{3} H\delta \left( {H - h} \right) + \left( {N_{1} - N_{2} } \right)(H - h)z + \left( {N_{3} + N_{4} - M_{3} } \right)Hz} \right\}\left( {3 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ {\kern 1pt} & \quad - \frac{{24M_{3} zH\left( {H - h} \right)}}{{Q_{n}^{4} }}\left( {1 - \frac{{8q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{8q_{n}^{4} }}{{Q_{n}^{4} }}} \right) - \left( {2M_{4} - \frac{{M_{3} + N_{1} - N_{2} }}{\alpha }} \right)\frac{{p_{n} }}{{P_{n}^{2} }} \\ {\kern 1pt} & \quad + \frac{1}{{P_{n}^{2} }}\left\{ {M_{3} \left( {z + \delta h - \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {z - H} \right) - \left( {N_{3} + N_{4} } \right)\left( {H - h} \right)} \right\}\left( {1 - \frac{{2p_{n}^{2} }}{{P_{n}^{2} }}} \right)\quad \quad \\ & \quad + \frac{{4M_{3} p_{n} \left( {H - h} \right)\left( {z - H} \right)}}{{P_{n}^{4} }}\left( {3 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) + \left( {2M_{4} - \frac{{M_{3} - N_{3} - N_{4} }}{\alpha }} \right)\frac{{t_{n} }}{{T_{n}^{2} }} \\ {\kern 1pt} & \quad - \frac{1}{{T_{n}^{2} }}\left\{ {M_{3} \left( { - z + h + \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {h - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {z - \frac{2h}{\alpha } - H} \right) - 4hM_{4} } \right\}\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & {\kern 1pt} \quad - \frac{{4t_{n} }}{{T_{n}^{4} }}\left\{ {M_{3} H\left( {z - H - \delta h} \right) + \left( {N_{1} - N_{2} } \right)Hh + \left( {N_{3} + N_{4} } \right)h\left( {H - z} \right)} \right\}\left( {3 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad + \frac{{24M_{3} hH\left( {H - z} \right)}}{{T_{n}^{4} }}\left( {1 - \frac{{8t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{8t_{n}^{4} }}{{T_{n}^{4} }}} \right) + \left( { - 2M_{4} + \frac{{M_{3} - N_{3} - N_{4} }}{\alpha }} \right)\frac{{u_{n} }}{{U_{n}^{2} }} \\ {\kern 1pt} & \quad - \frac{1}{{U_{n}^{2} }}\left\{ {M_{3} \left( {z + \delta h - \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - z} \right)} \right\}\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad - \left\{ {\frac{{4M_{3} u_{n} \left( {h - H} \right)\left( {z - H} \right)}}{{U_{n}^{4} }}} \right\}\left. {\left( {3 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right)} \right|_{n = 2} + \left| {\frac{{M_{2} q_{n} }}{{\alpha Q_{n}^{2} }}} \right. - \frac{{M_{2} (h - z)}}{{Q_{n}^{2} }}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad - \frac{{M_{2} t_{n} }}{{\alpha T_{n}^{2} }} - \frac{{M_{2} (h - z)}}{{T_{n}^{2} }}\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right)\left. { + \frac{{M_{2} u_{n} }}{{\alpha U_{n}^{2} }} - \frac{{M_{2} \left( {z + \delta h} \right)}}{{U_{n}^{2} }}\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right) - \frac{{4M_{2} hzu_{n} }}{{U_{n}^{4} }}\left( {3 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right)} \right|_{n = 4} \\ \end{aligned} $$

$$ \begin{aligned} \phi_{2} & = M_{1} \left| {\frac{{p_{n} }}{{\alpha P_{n}^{2} }}} \right. + \frac{{\left( {z + \delta h} \right)}}{{P_{n}^{2} }}\left( {1 - \frac{{2p_{n}^{2} }}{{P_{n}^{2} }}} \right) - \frac{{4hzp_{n} }}{{P_{n}^{4} }}\left. {\left( {3 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right)} \right|_{n = 2} + \left| { - \left( {2\delta M_{4} + \frac{{N_{3} + N_{4} - M_{3} }}{\alpha }} \right)} \right.\frac{{q_{n} }}{{Q_{n}^{2} }} \\ & \quad + \frac{1}{{Q_{n}^{2} }}\left\{ {M_{3} \left( {z - h + \frac{2H}{\alpha }} \right) - \left( {N_{3} + N_{4} } \right)\left( {z + \delta H} \right) - \left( {N_{1} - N_{2} } \right)\delta \left( {H - h} \right) - 4zM_{4} } \right\}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ {\kern 1pt} & \quad - \frac{{4q_{n} }}{{Q_{n}^{4} }}\left\{ {M_{3} H\delta \left( {H - h} \right) - \left( {N_{1} - N_{2} } \right)(H - h)z - \left( {N_{3} + N_{4} - M_{3} } \right)Hz} \right\}\left( {3 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad - \frac{{24M_{3} zH\left( {H - h} \right)}}{{Q_{n}^{4} }}\left( {1 - \frac{{8q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{8q_{n}^{4} }}{{Q_{n}^{4} }}} \right) - \left( {2M_{4} - \frac{{M_{3} + N_{1} - N_{2} }}{\alpha }} \right)\frac{{p_{n} }}{{P_{n}^{2} }} \\ & \quad + \frac{1}{{P_{n}^{2} }}\left\{ {M_{3} \left( {z + \delta h - \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {z - H} \right) - \left( {N_{3} + N_{4} } \right)\left( {H - h} \right)} \right\}\left( {1 - \frac{{2p_{n}^{2} }}{{P_{n}^{2} }}} \right) \\ & \quad + \frac{{4M_{3} p_{n} \left( {H - h} \right)\left( {z - H} \right)}}{{P_{n}^{4} }}\left( {3 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) + \left( {2M_{4} - \frac{{M_{3} - N_{3} - N_{4} }}{\alpha }} \right)\frac{{t_{n} }}{{T_{n}^{2} }} \\ & \quad - \frac{1}{{T_{n}^{2} }}\left\{ {M_{3} \left( {h - z + \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {h - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {z - \frac{2h}{\alpha } - H} \right) - 4hM_{4} } \right\}\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad - \frac{{4t_{n} }}{{T_{n}^{4} }}\left\{ {M_{3} H\left( {z - H - \delta h} \right) + \left( {N_{1} - N_{2} } \right)Hh + \left( {N_{3} + N_{4} } \right)h\left( {H - z} \right)} \right\}\left( {3 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ {\kern 1pt} & \quad + \frac{{24M_{3} hH\left( {H - z} \right)}}{{T_{n}^{4} }}\left( {1 - \frac{{8t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{8t_{n}^{4} }}{{T_{n}^{4} }}} \right) + \left( { - 2M_{4} + \frac{{M_{3} - N_{3} - N_{4} }}{\alpha }} \right)\frac{{u_{n} }}{{U_{n}^{2} }} \\ & \quad - \frac{1}{{U_{n}^{2} }}\left\{ {M_{3} \left( {z + \delta h - \frac{2H}{\alpha }} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - z} \right) + \left( {N_{1} - N_{2} } \right)\left( {H - h} \right)} \right\}\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad + \frac{{4M_{3} u_{n} \left( {H - h} \right)\left( {z - H} \right)}}{{U_{n}^{4} }}\left. {\left( {3 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right)} \right|_{n = 4} + \left| {\frac{{M_{2} q_{n} }}{{\alpha Q_{n}^{2} }}} \right. + \frac{{M_{2} (z - h)}}{{Q_{n}^{2} }}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) + \frac{{M_{2} u_{n} }}{{\alpha U_{n}^{2} }} \\ & \quad - \frac{{M_{2} \left( {z + \delta h} \right)}}{{U_{n}^{2} }}\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }} } \right) - \frac{{4M_{2} hzu_{n} }}{{U_{n}^{4} }}\left( {3 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) + \frac{{M_{2} (z - h)}}{{T_{n}^{2} }}\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) - \left. {\frac{{M_{2} }}{\alpha }\frac{{t_{n} }}{{T_{n}^{2} }}} \right|_{n = 6} \\ \end{aligned} $$

$$\eqalign{ & {\phi _3} = {M_1}\left| {{1 \over {\alpha P_n^2}}} \right.\left( {{{2p_n^2} \over {P_n^2}} - 1} \right) - {{2\left( {z + \delta h} \right){p_n}} \over {P_n^4}}\left( {{{4p_n^2} \over {P_n^2}} - 3} \right) + {{12hz} \over {P_n^4}}{\left. {\left( {1 - {{8p_n^2} \over {P_n^2}} + {{8p_n^4} \over {P_n^4}}} \right)} \right|_{n = 2}} \cr & \quad \quad + \left| {{1 \over {Q_n^2}}\left( {2\delta {M_4} + {{{N_3} + {N_4} - {M_3}} \over \alpha }} \right)\left( {1 - {{2q_n^2} \over {Q_n^2}}} \right)} \right. \cr & \quad \quad + {{2{q_n}} \over {Q_n^4}}\left\{ {{M_3}\left( {z - h + {{2H} \over \alpha }} \right) - \left( {{N_1} - {N_2}} \right)\delta \left( {H - h} \right) - \left( {{N_3} + {N_4}} \right)\left( {z + \delta H} \right) - 4{M_4}z} \right\}\left( {3 - {{4q_n^2} \over {Q_n^2}}} \right) \cr & \quad \quad - {6 \over {Q_n^4}}\left\{ { - {M_3}H\delta \left( {H - h} \right) + \left( {{N_1} - {N_2}} \right)(H - h)z + \left( {{N_3} + {N_4} - {M_3}} \right)Hz} \right\}\left( {1 - {{8q_n^2} \over {Q_n^2}} + {{8q_n^4} \over {Q_n^4}}} \right) \cr & \quad \quad - {{96{M_3}zH\left( {H - h} \right){q_n}} \over {Q_n^6}}\left( {5 - {{20q_n^2} \over {Q_n^2}} + {{16q_n^4} \over {Q_n^4}}} \right) + {1 \over {P_n^2}}\left( {2{M_4} - {{{M_3} + {N_1} - {N_2}} \over \alpha }} \right)\left( {1 - {{2p_n^2} \over {P_n^2}}} \right) \cr & \quad \quad + {{2{p_n}} \over {p_n^4}}\left\{ {{M_3}\left( {z + \delta h - {{2H} \over \alpha }} \right) + \left( {{N_1} - {N_2}} \right)\left( {z - H} \right) - \left( {{N_3} + {N_4}} \right)\left( {H - h} \right)} \right\}\left( {3 - {{4p_n^2} \over {p_n^2}}} \right) \cr & \quad \quad - {{12{M_3}\left( {H - h} \right)\left( {z - H} \right)} \over {p_n^4}}\left( {1 - {{8p_n^2} \over {P_n^2}} + {{8p_n^4} \over {P_n^4}}} \right) - {1 \over {T_n^2}}\left( {2{M_4} - {{{M_3} - {N_3} - {N_4}} \over \alpha }} \right)\left( {1 - {{2t_n^2} \over {T_n^2}}} \right) \cr & \quad \quad - {{2{t_n}} \over {T_n^4}}\left\{ {{M_3}\left( { - z + h + {{2H} \over \alpha }} \right) + \left( {{N_1} - {N_2}} \right)\left( {h - H} \right) + \left( {{N_3} + {N_4}} \right)\left( {z - {{2h} \over \alpha } - H} \right) - 4h{M_4}} \right\}\left( {3 - {{4t_n^2} \over {T_n^2}}} \right) \cr & \quad \quad + {6 \over {T_n^4}}\left\{ {{M_3}H\left( {z - H - \delta h} \right) + \left( {{N_1} - {N_2}} \right)Hh + \left( {{N_3} + {N_4}} \right)h\left( {H - z} \right)} \right\}\left( {1 - {{8t_n^2} \over {T_n^2}} + {{8t_n^4} \over {T_n^4}}} \right) \cr & \quad \quad + {{96{M_3}hH\left( {H - z} \right){t_n}} \over {T_n^6}}\left( {5 - {{20t_n^2} \over {T_n^2}} + {{16t_n^4} \over {T_n^4}}} \right) + {1 \over {U_n^2}}\left( {2{M_4} - {{{M_3} - {N_3} - {N_4}} \over \alpha }} \right)\left( {1 - {{2u_n^2} \over {U_n^2}}} \right) \cr & \quad \quad - {{2{u_n}} \over {U_n^4}}\left\{ {{M_3}\left( {z + \delta h - {{2H} \over \alpha }} \right) + \left( {{N_1} - {N_2}} \right)\left( {H - h} \right) + \left( {{N_3} + {N_4}} \right)\left( {H - z} \right)} \right\}\left( {3 - {{4u_n^2} \over {U_n^2}}} \right) \cr & \quad \quad + {{12{M_3}\left( {h - H} \right)\left( {z - H} \right)} \over {U_n^4}}{\left. {\left( {1 - {{8u_n^2} \over {U_n^2}} + {{8u_n^4} \over {U_n^4}}} \right)} \right|_{n = 4}} + \left| { - {{{M_2}} \over {\alpha Q_n^2}}} \right.\left( {1 - {{2q_n^2} \over {Q_n^2}}} \right) - {{2{M_2}(h - z){q_n}} \over {Q_n^4}}\left( {3 - {{4q_n^2} \over {Q_n^2}}} \right) \cr & \quad \quad - {{2{M_2}(h - z){t_n}} \over {T_n^4}}\left( {3 - {{4t_n^2} \over {T_n^2}}} \right) + {{{M_2}} \over {\alpha T_n^2}}\left( {1 - {{2t_n^2} \over {T_n^2}}} \right) - {{{M_2}} \over {\alpha U_n^2}}\left( {1 - {{2u_n^2} \over {U_n^2}}} \right) \cr & \quad \quad - {{2{M_2}\left( {z + \delta h} \right){u_n}} \over {U_n^4}}\left( {3 - {{4u_n^2} \over {U_n^2}}} \right) + {{12{M_2}hz} \over {U_n^4}}{\left. {\left( {1 - {{8u_n^2} \over {U_n^2}} + {{8u_n^4} \over {U_n^4}}} \right)} \right|_{n = 6}} \cr} $$

$$ \begin{aligned} \phi_{4} & = M_{1} \left| { - \frac{{2p_{n} }}{{\alpha P_{n}^{4} }}} \right.\left( {3 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) - \frac{{6\left( {z + \delta h} \right)}}{{P_{n}^{4} }}\left( {1 - \frac{{8p_{n}^{2} }}{{P_{n}^{2} }} + \frac{{8p_{n}^{4} }}{{P_{n}^{4} }}} \right) + \frac{{48hzp_{n} }}{{P_{n}^{6} }}\left. {\left( {5 - \frac{{20p_{n}^{2} }}{{P_{n}^{2} }} + \frac{{16p_{n}^{4} }}{{P_{n}^{4} }}} \right)} \right|_{n = 2} \\ & \quad + \left| {\frac{{2q_{n} }}{{Q_{n}^{4} }}\left( {2\delta M_{4} + \frac{{N_{3} + N_{4} - M_{3} }}{\alpha }} \right)} \right.\left( {3 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad - \frac{6}{{Q_{n}^{4} }}\left\{ {M_{3} \left( {z - h + \frac{2H}{\alpha }} \right) - \left( {N_{1} - N_{2} } \right)\delta \left( {H - h} \right) - \left( {N_{3} + N_{4} } \right)\left( {z + \delta H} \right) - 4zM_{4} } \right\}\left( {1 - \frac{{8q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{8q_{n}^{4} }}{{Q_{n}^{4} }}} \right) \\ & \quad - \frac{{24q_{n} }}{{Q_{n}^{6} }}\left\{ { - M_{3} H\delta \left( {H - h} \right) + \left( {N_{1} - N_{2} } \right)(H - h)z + \left( {N_{3} + N_{4} - M_{3} } \right)Hz} \right\}\left( {5 - \frac{{20q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{16q_{n}^{4} }}{{Q_{n}^{4} }}} \right) \\ & \quad + \frac{{480M_{3} zH\left( {H - h} \right)q_{n} }}{{Q_{n}^{6} }}\left( {1 - \frac{{18q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{48q_{n}^{4} }}{{Q_{n}^{4} }} - \frac{{32q_{n}^{6} }}{{Q_{n}^{6} }}} \right) + \frac{{2p_{n} }}{{P_{n}^{4} }}\left( {2M_{4} - \frac{{M_{3} + N_{1} - N_{2} }}{\alpha }} \right)\left( {3 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) \\ & \quad - \frac{6}{{P_{n}^{4} }}\left\{ {M_{3} \left( {z + \delta h - \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {z - H} \right) - \left( {N_{3} + N_{4} } \right)\left( {H - h} \right)} \right\}\left( {1 - \frac{{8p_{n}^{2} }}{{P_{n}^{2} }} + \frac{{8p_{n}^{4} }}{{P_{n}^{4} }}} \right) \\ & \quad - \frac{{48M_{3} \left( {H - h} \right)\left( {z - H} \right)p_{n} }}{{P_{n}^{6} }}\left( {5 - \frac{{20p_{n}^{2} }}{{P_{n}^{2} }} + \frac{{16p_{n}^{4} }}{{P_{n}^{4} }}} \right) - \frac{{2t_{n} }}{{T_{n}^{4} }}\left( {2M_{4} - \frac{{M_{3} - N_{3} - N_{4} }}{\alpha }} \right)\left( {3 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad + \frac{6}{{T_{n}^{4} }}\left\{ {M_{3} \left( { - z + h + \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {h - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {z - \frac{2h}{\alpha } - H} \right) - 4hM_{4} } \right\}\left( {1 - \frac{{8t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{8t_{n}^{4} }}{{T_{n}^{4} }}} \right) \\ & \quad + \frac{{24t_{n} }}{{T_{n}^{6} }}\left\{ { - M_{3} H\left( {z - H + \delta h} \right) + \left( {N_{1} - N_{2} } \right)Hh + \left( {N_{3} + N_{4} } \right)h\left( {H - z} \right)} \right\}\left( {5 - \frac{{20t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{16t_{n}^{4} }}{{T_{n}^{4} }}} \right) \\ & \quad - \frac{{480M_{3} hH\left( {H - z} \right)t_{n} }}{{T_{n}^{6} }}\left( {1 - \frac{{18t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{48t_{n}^{4} }}{{T_{n}^{4} }} - \frac{{32t_{n}^{6} }}{{T_{n}^{6} }}} \right) + \frac{{2u_{n} }}{{U_{n}^{4} }}\left( {2M_{4} - \frac{{M_{3} - N_{3} - N_{4} }}{\alpha }} \right)\left( {3 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad + \frac{6}{{U_{n}^{4} }}\left\{ {M_{3} \left( {z + \delta h - \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - z} \right)} \right\}\left( {1 - \frac{{8u_{n}^{2} }}{{U_{n}^{2} }} + \frac{{8u_{n}^{4} }}{{U_{n}^{4} }}} \right) \\ & \quad + \frac{{48M_{3} \left( {h - H} \right)\left( {z - H} \right)u_{n} }}{{U_{n}^{6} }}\left. {\left( {5 - \frac{{20u_{n}^{2} }}{{U_{n}^{2} }} + \frac{{16u_{n}^{4} }}{{U_{n}^{4} }}} \right)} \right|_{n = 4} + \left| {\frac{{ - 2M_{2} q_{n} }}{{\alpha Q_{n}^{4} }}} \right.\left( {3 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad + \frac{{6M_{2} (h - z)}}{{Q_{n}^{4} }} + \frac{{6M_{2} (h - z)}}{{T_{n}^{4} }}\left( {1 - \frac{{8t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{8t_{n}^{4} }}{{T_{n}^{4} }}} \right) + \frac{{2M_{2} t_{n} }}{{\alpha T_{n}^{4} }}\left( {3 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad - \frac{{2M_{2} u_{n} }}{{\alpha U_{n}^{4} }}\left( {3 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) + \frac{{6M_{2} \left( {z + \delta h} \right)}}{{U_{n}^{4} }}\left( {1 - \frac{{8u_{n}^{2} }}{{U_{n}^{2} }} + \frac{{8u_{n}^{4} }}{{U_{n}^{4} }}} \right) + \frac{{48M_{2} hzu_{n} }}{{U_{n}^{6} }}\left. {\left( {5 - \frac{{20u_{n}^{2} }}{{U_{n}^{2} }} + \frac{{16u_{n}^{4} }}{{U_{n}^{4} }}} \right)} \right|_{n = 6} \\ \end{aligned} $$

$$ \begin{aligned} \phi_{5} & = M_{1} \left| {\frac{{p_{n} }}{{\alpha P_{n}^{2} }}} \right. + \frac{{\left( {z + \delta h} \right)}}{{P_{n}^{2} }}\left( {1 - \frac{{2p_{n}^{2} }}{{P_{n}^{2} }}} \right) - \frac{{4hzp_{n} }}{{P_{n}^{4} }}\left. {\left( {3 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right)} \right|_{n = \beta } + \left| { - \left( {2\delta M_{4} + \frac{{N_{3} + N_{4} - M_{3} }}{\alpha }} \right)} \right.\frac{{q_{n} }}{{Q_{n}^{2} }} \\ & \quad + \frac{1}{{Q_{n}^{2} }}\left\{ {M_{3} \left( {z - h + \frac{2H}{\alpha }} \right) - \left( {N_{3} + N_{4} } \right)\left( {z + \delta H} \right) - \left( {N_{1} - N_{2} } \right)\delta \left( {H - h} \right) - 4zM_{4} } \right\}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad + \frac{{4q_{n} }}{{Q_{n}^{4} }}\left\{ { - M_{3} H\delta \left( {H - h} \right) + \left( {N_{1} - N_{2} } \right)(H - h)z + \left( {N_{3} + N_{4} - M_{3} } \right)Hz} \right\}\left( {3 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad - \frac{{24M_{3} zH\left( {H - h} \right)}}{{Q_{n}^{4} }}\left( {1 - \frac{{8q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{8q_{n}^{4} }}{{Q_{n}^{4} }}} \right) - \left( {2M_{4} - \frac{{M_{3} + N_{1} - N_{2} }}{\alpha }} \right)\frac{{p_{n} }}{{P_{n}^{2} }} \\ & \quad + \frac{1}{{P_{n}^{2} }}\left\{ {M_{3} \left( {z + \delta h - \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {z - H} \right) - \left( {N_{3} + N_{4} } \right)\left( {H - h} \right)} \right\}\left( {1 - \frac{{2p_{n}^{2} }}{{P_{n}^{2} }}} \right) \\ & \quad + \frac{{4M_{3} p_{n} \left( {H - h} \right)\left( {z - H} \right)}}{{P_{n}^{4} }}\left( {3 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) + \left( {2M_{4} - \frac{{M_{3} - N_{3} - N_{4} }}{\alpha }} \right)\frac{{t_{n} }}{{T_{n}^{2} }} \\ {\kern 1pt} & \quad - \frac{1}{{T_{n}^{2} }}\left\{ {M_{3} \left( { - z + h + \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {h - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {z - \frac{2h}{\alpha } - H} \right) - 4hM_{4} } \right\}\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ {\kern 1pt} & \quad - \frac{{4t_{n} }}{{T_{n}^{4} }}\left\{ {M_{3} H\left( {z - H - \delta h} \right) + \left( {N_{1} - N_{2} } \right)Hh + \left( {N_{3} + N_{4} } \right)h\left( {H - z} \right)} \right\}\left( {3 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad + \frac{{24M_{3} hH\left( {H - z} \right)}}{{T_{n}^{4} }}\left( {1 - \frac{{8t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{8t_{n}^{4} }}{{T_{n}^{4} }}} \right) + \left( { - 2M_{4} + \frac{{M_{3} - N_{3} - N_{4} }}{\alpha }} \right)\frac{{u_{n} }}{{U_{n}^{2} }} \\ & \quad - \frac{1}{{U_{n}^{2} }}\left\{ {M_{3} \left( {z + \delta h - \frac{2H}{\alpha }} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - z} \right) + \left( {N_{1} - N_{2} } \right)\left( {H - h} \right)} \right\}\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad - \frac{{4M_{3} u_{n} \left( {h - H} \right)\left( {z - H} \right)}}{{U_{n}^{4} }}\left. {\left( {3 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right)} \right|_{n = \beta + 2} + \left| {\frac{{M_{2} q_{n} }}{{\alpha Q_{n}^{2} }}} \right. + \frac{{M_{2} (z - h)}}{{Q_{n}^{2} }}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad + \frac{{M_{2} u_{n} }}{{\alpha U_{n}^{2} }} - \frac{{M_{2} \left( {z + \delta h} \right)}}{{U_{n}^{2} }}\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right) + \frac{{4M_{2} hzu_{n} }}{{U_{n}^{4} }}\left( {3 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) + \frac{{M_{2} (z - h)}}{{T_{n}^{2} }}\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) - \left. {\frac{{M_{2} }}{\alpha }\frac{{t_{n} }}{{T_{n}^{2} }}} \right|_{n = \beta + 4} \\ \end{aligned} $$

$$ \begin{aligned} \phi_{6} & = M_{1} \left| {\frac{1}{{\alpha P_{n}^{2} }}} \right.\left( {\frac{{2p_{n}^{2} }}{{P_{n}^{2} }} - 1} \right) - \frac{{2\left( {z + \delta h} \right)p_{n} }}{{P_{n}^{4} }}\left( {\frac{{4p_{n}^{2} }}{{P_{n}^{2} }} - 3} \right) + \frac{12hz}{{P_{n}^{4} }}\left. {\left( {1 - \frac{{8p_{n}^{2} }}{{P_{n}^{2} }} + \frac{{8p_{n}^{4} }}{{P_{n}^{4} }}} \right)} \right|_{n = \beta } \\ & \quad + \left| {\frac{1}{{Q_{n}^{2} }}\left( {2\delta M_{4} + \frac{{N_{3} + N_{4} - M_{3} }}{\alpha }} \right)\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right)} \right. \\ & \quad + \frac{{2q_{n} }}{{Q_{n}^{4} }}\left\{ {M_{3} \left( {z - h + \frac{2H}{\alpha }} \right) - \left( {N_{1} - N_{2} } \right)\delta \left( {H - h} \right) - \left( {N_{3} + N_{4} } \right)\left( {z + \delta H} \right) - 4M_{4} z} \right\}\left( {3 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad - \frac{6}{{Q_{n}^{4} }}\left\{ { - M_{3} H\delta \left( {H - h} \right) + \left( {N_{1} - N_{2} } \right)(H - h)z + \left( {N_{3} + N_{4} - M_{3} } \right)Hz} \right\}\left( {1 - \frac{{8q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{8q_{n}^{4} }}{{Q_{n}^{4} }}} \right) \\ & \quad - \frac{{96M_{3} zH\left( {H - h} \right)q_{n} }}{{Q_{n}^{6} }}\left( {5 - \frac{{20q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{16q_{n}^{4} }}{{Q_{n}^{4} }}} \right) + \frac{1}{{P_{n}^{2} }}\left( {2M_{4} - \frac{{M_{3} + N_{1} - N_{2} }}{\alpha }} \right)\left( {1 - \frac{{2p_{n}^{2} }}{{P_{n}^{2} }}} \right) \\ & \quad + \frac{{2p_{n} }}{{p_{n}^{4} }}\left\{ {M_{3} \left( {z + \delta h - \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {z - H} \right) - \left( {N_{3} + N_{4} } \right)\left( {H - h} \right)} \right\}\left( {3 - \frac{{4p_{n}^{2} }}{{p_{n}^{2} }}} \right) \\ & \quad - \frac{{12M_{3} \left( {H - h} \right)\left( {z - H} \right)}}{{p_{n}^{4} }}\left( {1 - \frac{{8p_{n}^{2} }}{{P_{n}^{2} }} + \frac{{8p_{n}^{4} }}{{P_{n}^{4} }}} \right) - \frac{1}{{T_{n}^{2} }}\left( {2M_{4} - \frac{{M_{3} - N_{3} - N_{4} }}{\alpha }} \right)\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad - \frac{{2t_{n} }}{{T_{n}^{4} }}\left\{ {M_{3} \left( { - z + h + \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {h - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {z - \frac{2h}{\alpha } - H} \right) - 4hM_{4} } \right\}\left( {3 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad + \frac{6}{{T_{n}^{4} }}\left\{ {M_{3} H\left( {z - H - \delta h} \right) + \left( {N_{1} - N_{2} } \right)Hh + \left( {N_{3} + N_{4} } \right)h\left( {H - z} \right)} \right\}\left( {1 - \frac{{8t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{8t_{n}^{4} }}{{T_{n}^{4} }}} \right) \\ & \quad + \frac{{96M_{3} hH\left( {H - z} \right)t_{n} }}{{T_{n}^{6} }}\left( {5 - \frac{{20t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{16t_{n}^{4} }}{{T_{n}^{4} }}} \right) + \frac{1}{{U_{n}^{2} }}\left( {2M_{4} - \frac{{M_{3} - N_{3} - N_{4} }}{\alpha }} \right)\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad - \frac{{2u_{n} }}{{U_{n}^{4} }}\left\{ {M_{3} \left( {z + \delta h - \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - z} \right)} \right\}\left( {3 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad + \frac{{12M_{3} \left( {h - H} \right)\left( {z - H} \right)}}{{U_{n}^{4} }}\left. {\left( {1 - \frac{{8u_{n}^{2} }}{{U_{n}^{2} }} + \frac{{8u_{n}^{4} }}{{U_{n}^{4} }}} \right)} \right|_{n = \beta + 2} + \left| { - \frac{{M_{2} }}{{\alpha Q_{n}^{2} }}} \right.\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) - \frac{{2M_{2} (h - z)q_{n} }}{{Q_{n}^{4} }}\left( {3 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad - \frac{{2M_{2} (h - z)t_{n} }}{{T_{n}^{4} }}\left( {3 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) + \frac{{M_{2} }}{{\alpha T_{n}^{2} }}\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) - \frac{{M_{2} }}{{\alpha U_{n}^{2} }}\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad - \frac{{2M_{2} \left( {z + \delta h} \right)u_{n} }}{{U_{n}^{4} }}\left( {3 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) + \frac{{12M_{2} hz}}{{U_{n}^{4} }}\left. {\left( {1 - \frac{{8u_{n}^{2} }}{{U_{n}^{2} }} + \frac{{8u_{n}^{4} }}{{U_{n}^{4} }}} \right)} \right|_{n = \beta + 4} \\ \end{aligned} $$

$$ \begin{aligned} \phi_{7} & = M_{1} \left| { - \frac{{2p_{n} }}{{\alpha P_{n}^{4} }}} \right.\left( {3 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) - \frac{{6\left( {z + \delta h} \right)}}{{P_{n}^{4} }}\left( {1 - \frac{{8p_{n}^{2} }}{{P_{n}^{2} }} + \frac{{8p_{n}^{4} }}{{P_{n}^{4} }}} \right) + \frac{{48hzp_{n} }}{{P_{n}^{6} }}\left. {\left( {5 - \frac{{20p_{n}^{2} }}{{P_{n}^{2} }} + \frac{{16p_{n}^{4} }}{{P_{n}^{4} }}} \right)} \right|_{n = \beta } \\ & \quad + \left| {\frac{{2q_{n} }}{{Q_{n}^{4} }}\left( {2\delta M_{4} + \frac{{N_{3} + N_{4} - M_{3} }}{\alpha }} \right)} \right.\left( {3 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad - \frac{6}{{Q_{n}^{4} }}\left\{ {M_{3} \left( {z - h + \frac{2H}{\alpha }} \right) - \left( {N_{1} - N_{2} } \right)\delta \left( {H - h} \right) - \left( {N_{3} + N_{4} } \right)\left( {z + \delta H} \right) - 4zM_{4} } \right\}\left( {1 - \frac{{8q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{8q_{n}^{4} }}{{Q_{n}^{4} }}} \right) \\ & \quad - \frac{{24q_{n} }}{{Q_{n}^{6} }}\left\{ { - M_{3} H\delta \left( {H - h} \right) + \left( {N_{1} - N_{2} } \right)(H - h)z + \left( {N_{3} + N_{4} - M_{3} } \right)Hz} \right\}\left( {5 - \frac{{20q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{16q_{n}^{4} }}{{Q_{n}^{4} }}} \right) \\ & \quad + \frac{{480M_{3} zH\left( {H - h} \right)q_{n} }}{{Q_{n}^{6} }}\left( {1 - \frac{{18q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{48q_{n}^{4} }}{{Q_{n}^{4} }} - \frac{{32q_{n}^{6} }}{{Q_{n}^{6} }}} \right) + \frac{{2p_{n} }}{{P_{n}^{4} }}\left( {2M_{4} - \frac{{M_{3} + N_{1} - N_{2} }}{\alpha }} \right)\left( {3 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) \\ & \quad - \frac{6}{{P_{n}^{4} }}\left\{ {M_{3} \left( {z + \delta h - \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {z - H} \right) - \left( {N_{3} + N_{4} } \right)\left( {H - h} \right)} \right\}\left( {1 - \frac{{8p_{n}^{2} }}{{P_{n}^{2} }} + \frac{{8p_{n}^{4} }}{{P_{n}^{4} }}} \right) \\ & \quad - \frac{{48M_{3} \left( {H - h} \right)\left( {z - H} \right)p_{n} }}{{P_{n}^{6} }}\left( {5 - \frac{{20p_{n}^{2} }}{{P_{n}^{2} }} + \frac{{16p_{n}^{4} }}{{P_{n}^{4} }}} \right) - \frac{{2t_{n} }}{{T_{n}^{4} }}\left( {2M_{4} - \frac{{M_{3} - N_{3} - N_{4} }}{\alpha }} \right)\left( {3 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad + \frac{6}{{T_{n}^{4} }}\left\{ {M_{3} \left( { - z + h + \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {h - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {z - \frac{2h}{\alpha } - H} \right) - 4hM_{4} } \right\}\left( {1 - \frac{{8t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{8t_{n}^{4} }}{{T_{n}^{4} }}} \right) \\ & \quad + \frac{{24t_{n} }}{{T_{n}^{6} }}\left\{ {M_{3} H\left( {z - H - \delta h} \right) + \left( {N_{1} - N_{2} } \right)Hh + \left( {N_{3} + N_{4} } \right)h\left( {H - z} \right)} \right\}\left( {5 - \frac{{20t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{16t_{n}^{4} }}{{T_{n}^{4} }}} \right) \\ & \quad - \frac{{480M_{3} hH\left( {H - z} \right)t_{n} }}{{T_{n}^{6} }}\left( {1 - \frac{{18t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{48t_{n}^{4} }}{{T_{n}^{4} }} - \frac{{32t_{n}^{6} }}{{T_{n}^{6} }}} \right) + \frac{{2u_{n} }}{{U_{n}^{4} }}\left( {2M_{4} - \frac{{M_{3} - N_{3} - N_{4} }}{\alpha }} \right)\left( {3 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad + \frac{6}{{U_{n}^{4} }}\left\{ {M_{3} \left( {z + \delta h - \frac{2H}{\alpha }} \right) + \left( {N_{1} - N_{2} } \right)\left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - z} \right)} \right\}\left( {1 - \frac{{8u_{n}^{2} }}{{U_{n}^{2} }} + \frac{{8u_{n}^{4} }}{{U_{n}^{4} }}} \right) \\ & \quad + \frac{{48M_{3} \left( {h - H} \right)\left( {z - H} \right)u_{n} }}{{U_{n}^{6} }}\left. {\left( {5 - \frac{{20u_{n}^{2} }}{{U_{n}^{2} }} + \frac{{16u_{n}^{4} }}{{U_{n}^{4} }}} \right)} \right|_{n = \beta + 2} + \left| { - \frac{{2M_{2} q_{n} }}{{\alpha Q_{n}^{4} }}} \right.\left( {3 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad + \frac{{6M_{2} (h - z)}}{{Q_{n}^{4} }} + \frac{{6M_{2} (h - z)}}{{T_{n}^{4} }}\left( {1 - \frac{{8t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{8t_{n}^{4} }}{{T_{n}^{4} }}} \right) + \frac{{2M_{2} t_{n} }}{{\alpha T_{n}^{4} }}\left( {3 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad - \frac{{2M_{2} u_{n} }}{{\alpha U_{n}^{4} }}\left( {3 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) + \frac{{6M_{2} \left( {z + \delta h} \right)}}{{U_{n}^{4} }}\left( {1 - \frac{{8u_{n}^{2} }}{{U_{n}^{2} }} + \frac{{8u_{n}^{4} }}{{U_{n}^{4} }}} \right) + \left. {\frac{{48M_{2} hzu_{n} }}{{U_{n}^{6} }}\left. {\left( {5 - \frac{{20u_{n}^{2} }}{{U_{n}^{2} }} + \frac{{16u_{n}^{4} }}{{U_{n}^{4} }}} \right)} \right|_{n = \beta + 4} } \right] \\ \end{aligned} $$

$$ \begin{aligned} \psi_{1} & = \left| {\left\{ {2\delta M_{4} - \frac{{\left( {1 - \delta } \right)}}{2}\left( {N_{3} + N_{4} - M_{3} } \right)} \right\}} \right.\frac{y}{{Q_{n}^{2} }} \\ & \quad - 2\left\{ {M_{3} \left( {z - h} \right) - M_{3} H\left( {1 - \delta } \right) + \left( {N_{1} - N_{2} } \right)\delta \left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {\delta H - z} \right) - 4zM_{4} } \right\}\frac{{yq_{n} }}{{Q_{n}^{4} }} \\ {\kern 1pt} & \quad + \frac{4y}{{Q_{n}^{4} }}\left\{ {M_{3} H\delta \left( {H - h} \right) + \left( {N_{1} - N_{2} } \right)(H - h)z + \left( {N_{3} + N_{4} - M_{3} } \right)Hz} \right\}\left( {1 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ {\kern 1pt} & \quad + \frac{{96M_{3} zH\left( {H - h} \right)q_{n} y}}{{Q_{n}^{6} }}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) + \left\{ { - 2M_{4} + \frac{{\left( {1 - \delta } \right)}}{2}\left( {M_{3} + N_{1} - N_{2} } \right)} \right\}\frac{y}{{P_{n}^{2} }} \\ & \quad + 2\left\{ {M_{3} \left( {\left( {1 - \delta } \right)H + \frac{2h}{\alpha } - z} \right) - \left( {N_{1} - N_{2} } \right)\left( {z - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - h} \right)} \right\}\frac{{yp_{n} }}{{P_{n}^{4} }} \\ & \quad - \frac{{4M_{3} \left( {H - h} \right)\left( {z - H} \right)y}}{{P_{n}^{4} }}\left( {1 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) - \left\{ {2M_{4} - \frac{{\left( {1 - \delta } \right)}}{2}\left( {M_{3} - N_{3} - N_{4} } \right)} \right\}\frac{y}{{T_{n}^{2} }} \\ {\kern 1pt} & \quad - 2\left\{ {M_{3} \left( { - z + h + H\left( {1 - \delta } \right)} \right) + \left( {N_{1} - N_{2} } \right)\left( {h - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {z - h\left( {1 - \delta } \right) - H} \right) - 4hM_{4} } \right\}\frac{{t_{n} y}}{{T_{n}^{4} }} \\ {\kern 1pt} & \quad + \frac{4y}{{T_{n}^{4} }}\left\{ {M_{3} H\left( {z - H + \delta h} \right) + \left( {N_{1} - N_{2} } \right)Hh + \left( {N_{3} + N_{4} } \right)h\left( {H - z} \right)} \right\}\left( {1 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ {\kern 1pt} & \quad + \frac{{96M_{3} hH\left( {H - z} \right)t_{n} y}}{{T_{n}^{6} }}\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) + \left\{ {2M_{4} - \frac{{\left( {1 - \delta } \right)}}{2}\left( {M_{3} - N_{3} - N_{4} } \right)} \right\}\frac{y}{{U_{n}^{2} }} \\ {\kern 1pt} & \quad - 2\left\{ {M_{3} \left( {z - \delta h - H\left( {1 - \delta } \right)} \right) + \left( {N_{1} - N_{2} } \right)\left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - z} \right)} \right\}\frac{{u_{n} y}}{{U_{n}^{4} }} \\ & \quad + \frac{{4M_{3} \left( {h - H} \right)\left( {z - H} \right)y}}{{U_{n}^{4} }}\left. {\left( {1 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right)} \right|_{n = 2} + \left| {\frac{{M_{2} \left( {1 - \delta } \right)y}}{{2Q_{n}^{2} }}} \right. + \frac{{2M_{2} (h - z)q_{n} y}}{{Q_{n}^{4} }} \\ & \quad - \frac{{M_{2} \left( {1 - \delta } \right)y}}{{2U_{n}^{2} }} - \frac{{2M_{2} \left( {z - \delta h} \right)u_{n} y}}{{U_{n}^{4} }} + \frac{{4M_{2} hzy}}{{U_{n}^{4} }}\left( {1 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad - \frac{{2M_{2} (h - z)t_{n} y}}{{T_{n}^{4} }} + \left. {\frac{{M_{2} \left( {1 - \delta } \right)y}}{{2T_{n}^{2} }}} \right|_{n = 4} \\ \end{aligned} $$

$$ \begin{aligned} \psi_{2} & = M_{1} \left| {\frac{{\left( {1 - \delta } \right)y}}{{2P_{n}^{2} }}} \right. - \frac{{2y\left( {\delta h + z} \right)p_{n} }}{{P_{n}^{4} }} - \frac{4hyz}{{P_{n}^{4} }}\left. {\left( {1 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right)} \right|_{n = 2} + \left| {\left\{ {2\delta M_{4} - \frac{{\left( {1 - \delta } \right)}}{2}\left( {N_{3} + N_{4} - M_{3} } \right)} \right\}} \right.\frac{y}{{Q_{n}^{2} }} \\ & \quad - 2\left\{ {M_{3} \left( {z - h} \right) - M_{3} H\left( {1 - \delta } \right) + \left( {N_{1} - N_{2} } \right)\delta \left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {\delta H - z} \right) - 4zM_{4} } \right\}\frac{{yq_{n} }}{{Q_{n}^{4} }} \\ & \quad + \frac{4y}{{Q_{n}^{4} }}\left\{ {M_{3} H\delta \left( {H - h} \right) + \left( {N_{1} - N_{2} } \right)(H - h)z + \left( {N_{3} + N_{4} - M_{3} } \right)Hz} \right\}\left( {1 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad + \frac{{96M_{3} zH\left( {H - h} \right)q_{n} y}}{{Q_{n}^{6} }}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) - \left\{ {2M_{4} - \frac{{\left( {1 - \delta } \right)}}{2}\left( {M_{3} + N_{1} - N_{2} } \right)} \right\}\frac{y}{{P_{n}^{2} }} \\ {\kern 1pt} & \quad + 2\left\{ {M_{3} \left( {\left( {1 - \delta } \right)H + \frac{2h}{\alpha } - z} \right) - \left( {N_{1} - N_{2} } \right)\left( {z - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - h} \right)} \right\}\frac{{yp_{n} }}{{P_{n}^{4} }} \\ {\kern 1pt} & \quad - \frac{{4M_{3} \left( {H - h} \right)\left( {z - H} \right)y}}{{P_{n}^{4} }}\left( {1 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) - \left\{ {2M_{4} - \frac{{\left( {1 - \delta } \right)}}{2}\left( {M_{3} - N_{3} - N_{4} } \right)} \right\}\frac{y}{{T_{n}^{2} }} \\ {\kern 1pt} & \quad - 2\left\{ {M_{3} \left( { - z + h + H\left( {1 - \delta } \right)} \right) + \left( {N_{1} - N_{2} } \right)\left( {h - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {z - h\left( {1 - \delta } \right) - H} \right) - 4hM_{4} } \right\}\frac{{t_{n} y}}{{T_{n}^{4} }} \\ & \quad + \frac{4y}{{T_{n}^{4} }}\left\{ {M_{3} H\left( {z - H + \delta h} \right) + \left( {N_{1} - N_{2} } \right)Hh + \left( {N_{3} + N_{4} } \right)h\left( {H - z} \right)} \right\}\left( {1 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ {\kern 1pt} & \quad + \frac{{96M_{3} hH\left( {H - z} \right)t_{n} y}}{{T_{n}^{6} }}\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) + \left\{ {2M_{4} - \frac{{\left( {1 - \delta } \right)}}{2}\left( {M_{3} - N_{3} - N_{4} } \right)} \right\}\frac{y}{{U_{n}^{2} }} \\ {\kern 1pt} & \quad - 2\left\{ {M_{3} \left( {z - \delta h - H\left( {1 - \delta } \right)} \right) + \left( {N_{1} - N_{2} } \right)\left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - z} \right)} \right\}\frac{{u_{n} y}}{{U_{n}^{4} }} \\ & \quad + \frac{{4M_{3} \left( {h - H} \right)\left( {z - H} \right)y}}{{U_{n}^{4} }}\left. {\left( {1 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right)} \right|_{n = 4} + \left| {\frac{{M_{2} \left( {1 - \delta } \right)y}}{{2Q_{n}^{2} }}} \right. + \frac{{2M_{2} (h - z)q_{n} y}}{{Q_{n}^{4} }} \\ & \quad - \frac{{M_{2} \left( {1 - \delta } \right)y}}{{2U_{n}^{2} }} - \frac{{2M_{2} \left( {z - \delta h} \right)u_{n} y}}{{U_{n}^{4} }} + \frac{{4M_{2} hzy}}{{U_{n}^{4} }}\left( {1 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) - \frac{{2M_{2} (h - z)t_{n} y}}{{T_{n}^{4} }} + \left. {\frac{{M_{2} \left( {1 - \delta } \right)y}}{{2T_{n}^{2} }}} \right|_{n = 6} \\ \end{aligned} $$

$$ \begin{aligned} \psi_{3} & = M_{1} \left| {\frac{{\left( {1 - \delta } \right)yp_{n} }}{{P_{n}^{4} }}} \right. + \frac{{2y\left( {\delta h + z} \right)}}{{P_{n}^{4} }}\left( {1 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) - \frac{{24yp_{n} hz}}{{P_{n}^{6} }}\left. {\left( {1 - \frac{{2p_{n}^{2} }}{{P_{n}^{2} }}} \right)} \right|_{n = 2} \\ & \quad + \left| {\left\{ {4\delta M_{4} - \left( {1 - \delta } \right)\left( {N_{3} + N_{4} - M_{3} } \right)} \right\}\frac{{yq_{n} }}{{Q_{n}^{4} }}} \right. \\ & \quad + \frac{2y}{{Q_{n}^{4} }}\left\{ {M_{3} \left( {z - h} \right) - M_{3} H\left( {1 - \delta } \right) + \left( {N_{1} - N_{2} } \right)\delta \left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {\delta H - z} \right)} \right.\left. { - 4zM_{4} } \right\}\left( {1 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad + \frac{{48q_{n} y}}{{Q_{n}^{6} }}\left\{ {M_{3} H\delta \left( {H - h} \right) + \left( {N_{1} - N_{2} } \right)(H - h)z + \left( {N_{3} + N_{4} - M_{3} } \right)Hz} \right\}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ {\kern 1pt} & \quad - \frac{96y}{{Q_{n}^{6} }}M_{3} zH\left( {H - h} \right)\left( {1 - \frac{{12q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{16q_{n}^{4} }}{{Q_{n}^{4} }}} \right) - \left\{ {4M_{4} - \left( {1 - \delta } \right)\left( {M_{3} + N_{1} - N_{2} } \right)} \right\}\frac{{p_{n} y}}{{P_{n}^{4} }} \\ & \quad - \frac{2y}{{P_{n}^{4} }}\left\{ {M_{3} \left( {\left( {1 - \delta } \right)H + \frac{2h}{\alpha } - z} \right) - \left( {N_{1} - N_{2} } \right)\left( {z - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - h} \right)} \right\}\left( {1 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) \\ & \quad - \frac{{48M_{3} \left( {H - h} \right)\left( {z - H} \right)p_{n} y}}{{P_{n}^{6} }}\left( {1 - \frac{{2p_{n}^{2} }}{{P_{n}^{2} }}} \right) - \left\{ {4M_{4} - \left( {1 - \delta } \right)\left( {M_{3} - N_{3} - N_{4} } \right)} \right\}\frac{{t_{n} y}}{{T_{n}^{4} }} \\ & \quad + \frac{2y}{{T_{n}^{4} }}\left\{ {M_{3} \left( { - z + h + H\left( {1 - \delta } \right)} \right) + \left( {N_{1} - N_{2} } \right)\left( {h - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {z - h\left( {1 - \delta } \right) - H} \right) - 4hM_{4} } \right\} \left( {1 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad + \frac{{48t_{n} y}}{{T_{n}^{6} }}\left\{ {M_{3} H\left( {z - H + \delta h} \right) + \left( {N_{1} - N_{2} } \right)Hh + \left( {N_{3} + N_{4} } \right)h\left( {H - z} \right)} \right\}\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad - \frac{{96M_{3} hH\left( {H - z} \right)y}}{{T_{n}^{6} }}\left( {1 - \frac{{12t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{16t_{n}^{4} }}{{T_{n}^{4} }}} \right) + \left\{ {4M_{4} - \left( {1 - \delta } \right)\left( {M_{3} - N_{3} - N_{4} } \right)} \right\}\frac{{u_{n} y}}{{U_{n}^{4} }} \\ {\kern 1pt} & \quad + \frac{2y}{{U_{n}^{4} }}\left\{ {M_{3} \left( {z - \delta h - H\left( {1 - \delta } \right)} \right) + \left( {N_{1} - N_{2} } \right)\left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - z} \right)} \right\}\left( {1 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad + \frac{{48M_{3} \left( {h - H} \right)\left( {z - H} \right)u_{n} y}}{{U_{n}^{6} }}\left. {\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right)} \right|_{n = 4} + \left| {\frac{{M_{2} \left( {1 - \delta } \right)q_{n} y}}{{Q_{n}^{4} }}} \right. - \frac{{2M_{2} (h - z)y}}{{Q_{n}^{4} }}\left( {1 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad - \frac{{M_{2} \left( {1 - \delta } \right)q_{n} y}}{{Q_{n}^{4} }} + \frac{{2M_{2} \left( {z - \delta h} \right)y}}{{U_{n}^{4} }}\left( {1 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) + \frac{{48M_{2} hzu_{n} y}}{{U_{n}^{6} }}\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad + \frac{{2M_{2} (h - z)y}}{{T_{n}^{4} }}\left( {1 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) + \left. {\frac{{M_{2} \left( {1 - \delta } \right)t_{n} y}}{{T_{n}^{4} }}} \right|_{n = 6} \\ \end{aligned} $$

$$ \begin{aligned} \psi_{4} & = M_{1} \left| {\frac{{\left( {1 - \delta } \right)y}}{{P_{n}^{4} }}\left( {\frac{{4p_{n}^{2} }}{{P_{n}^{2} }} - 1} \right)} \right. + \frac{{24yp_{n} \left( {\delta h + z} \right)}}{{P_{n}^{4} }}\left( {1 - \frac{{2p_{n}^{2} }}{{P_{n}^{2} }}} \right) + \left| {\frac{48hyz}{{P_{n}^{6} }}} \right.\left. {\left( {1 - \frac{{12p_{n}^{2} }}{{P_{n}^{2} }} + \frac{{16p_{n}^{4} }}{{P_{n}^{4} }}} \right)} \right|_{n = 2} \\ & \quad + \left| {\frac{y}{{Q_{n}^{4} }}\left\{ {4\delta M_{4} - \left( {1 - \delta } \right)\left( {N_{3} + N_{4} - M_{3} } \right)} \right\}} \right.\left( {\frac{{4q_{n}^{2} }}{{Q_{n}^{2} }} - 1} \right) \\ & \quad + \frac{{24q_{n} y}}{{Q_{n}^{6} }}\left\{ {M_{3} \left( {z - h} \right) - M_{3} H\left( {1 - \delta } \right) + \left( {N_{1} - N_{2} } \right)\delta \left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {\delta H - z} \right) - 4zM_{4} } \right\}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ {\kern 1pt} & \quad - \frac{48y}{{Q_{n}^{6} }}\left\{ {M_{3} H\delta \left( {H - h} \right) + \left( {N_{1} - N_{2} } \right)(H - h)z + \left( {N_{3} + N_{4} - M_{3} } \right)Hz} \right\}\left( {1 - \frac{{12q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{16q_{n}^{4} }}{{Q_{n}^{4} }}} \right) \\ {\kern 1pt} & \quad - \frac{{960M_{3} zH\left( {H - h} \right)q_{n} y}}{{Q_{n}^{8} }}\left( {3 - \frac{{16q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{16q_{n}^{4} }}{{Q_{n}^{4} }}} \right) + \frac{y}{{P_{n}^{4} }}\left\{ {4M_{4} - \left( {1 - \delta } \right)\left( {M_{3} + N_{1} - N_{2} } \right)} \right\}\left( {1 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) \\ {\kern 1pt} & \quad - \frac{{24p_{n} y}}{{P_{n}^{4} }}\left\{ {M_{3} \left( { - z + \frac{2h}{\alpha }} \right) + M_{3} H\left( {1 - \delta } \right) - \left( {N_{1} - N_{2} } \right)\left( {z - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - h} \right)} \right\}\left( {1 - \frac{{2p_{n}^{2} }}{{P_{n}^{2} }}} \right) \\ {\kern 1pt} & \quad + \frac{{48M_{3} \left( {H - h} \right)\left( {z - H} \right)y}}{{P_{n}^{6} }}\left( {1 - \frac{{12p_{n}^{2} }}{{P_{n}^{2} }} + \frac{{16p_{n}^{4} }}{{P_{n}^{4} }}} \right) + \frac{y}{{T_{n}^{4} }}\left\{ {4M_{4} - \left( {1 - \delta } \right)\left( {M_{3} - N_{3} - N_{4} } \right)} \right\}\left( {1 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ {\kern 1pt} & \quad + \frac{{24t_{n} y}}{{T_{n}^{6} }}\left\{ {M_{3} \left( { - z + h + H\left( {1 - \delta } \right)} \right) + \left( {N_{1} - N_{2} } \right)\left( {h - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {z - h\left( {1 - \delta } \right) - H} \right) - 4hM_{4} } \right\} \times \left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad - \frac{48y}{{T_{n}^{6} }}\left\{ {M_{3} H\left( {z - H + \delta h} \right) + \left( {N_{1} - N_{2} } \right)Hh + \left( {N_{3} + N_{4} } \right)h\left( {H - z} \right)} \right\}\left( {1 - \frac{{12t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{16t_{n}^{4} }}{{T_{n}^{4} }}} \right) \\ {\kern 1pt} & \quad - \frac{{960M_{3} hH\left( {H - z} \right)t_{n} y}}{{T_{n}^{8} }}\left( {3 - \frac{{16t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{16t_{n}^{4} }}{{T_{n}^{4} }}} \right) - \frac{y}{{U_{n}^{4} }}\left\{ {4M_{4} - \left( {1 - \delta } \right)\left( {M_{3} - N_{3} - N_{4} } \right)} \right\}\left( {1 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad + \frac{{24u_{n} y}}{{U_{n}^{6} }}\left\{ {M_{3} \left( {z - \delta h} \right) - M_{3} H\left( {1 - \delta } \right) + \left( {N_{1} - N_{2} } \right)\left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - z} \right)} \right\}\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad - \frac{{48M_{3} \left( {h - H} \right)\left( {z - H} \right)y}}{{U_{n}^{6} }}\left. {\left( {1 - \frac{{12u_{n}^{2} }}{{U_{n}^{2} }} + \frac{{16u_{n}^{4} }}{{U_{n}^{4} }}} \right)} \right|_{n = 4} + \left| {\frac{{M_{2} \left( {1 - \delta } \right)y}}{{Q_{n}^{4} }}} \right.\left( {\frac{{4q_{n}^{2} }}{{Q_{n}^{2} }} - 1} \right) \\ & \quad - \frac{{24M_{2} (h - z)q_{n} y}}{{Q_{n}^{6} }}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) + \frac{{M_{2} \left( {1 - \delta } \right)y}}{{U_{n}^{4} }}\left( {1 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) + \frac{{24M_{2} \left( {z - \delta h} \right)u_{n} y}}{{U_{n}^{6} }}\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad - \frac{{48M_{2} hzy}}{{U_{n}^{6} }}\left( {1 - \frac{{12u_{n}^{2} }}{{U_{n}^{2} }} + \frac{{16u_{n}^{4} }}{{U_{n}^{4} }}} \right) + \frac{{24M_{2} (h - z)t_{n} y}}{{T_{n}^{6} }}\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) - \left. {\frac{{M_{2} \left( {1 - \delta } \right)y}}{{T_{n}^{4} }}\left\{ {1 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right\}} \right|_{n = 6} \\ \end{aligned} $$

$$ \begin{aligned} \psi_{5} & = M_{1} \left| {\frac{{\left( {1 - \delta } \right)y}}{{2P_{n}^{2} }}} \right. - \frac{{2y\left( {\delta h + z} \right)p_{n} }}{{P_{n}^{4} }} - \frac{4hyz}{{P_{n}^{4} }}\left. {\left( {1 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right)} \right|_{n = \beta } \\ & \quad + \left| {\left\{ {2\delta M_{4} - \frac{{\left( {1 - \delta } \right)}}{2}\left( {N_{3} + N_{4} - M_{3} } \right)} \right\}} \right.\frac{y}{{Q_{n}^{2} }} \\ & \quad - 2\left\{ {M_{3} \left( {z - h} \right) - M_{3} H\left( {1 - \delta } \right) + \left( {N_{1} - N_{2} } \right)\delta \left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {\delta H - z} \right) - 4zM_{4} } \right\}\frac{{yq_{n} }}{{Q_{n}^{4} }} \\ {\kern 1pt} & \quad + \frac{4y}{{Q_{n}^{4} }}\left\{ {M_{3} H\delta \left( {H - h} \right) + \left( {N_{1} - N_{2} } \right)(H - h)z + \left( {N_{3} + N_{4} - M_{3} } \right)Hz} \right\}\left( {1 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad + \frac{{96M_{3} zH\left( {H - h} \right)q_{n} y}}{{Q_{n}^{6} }}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) - \left\{ {2M_{4} - \frac{{\left( {1 - \delta } \right)}}{2}\left( {M_{3} + N_{1} - N_{2} } \right)} \right\}\frac{y}{{P_{n}^{2} }} \\ & \quad + 2\left\{ {M_{3} \left( {\left( {1 - \delta } \right)H + \frac{2h}{\alpha } - z} \right) - \left( {N_{1} - N_{2} } \right)\left( {z - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - h} \right)} \right\}\frac{{yp_{n} }}{{P_{n}^{4} }} \\ & \quad - \frac{{4M_{3} \left( {H - h} \right)\left( {z - H} \right)y}}{{P_{n}^{4} }}\left( {1 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) - \left\{ {2M_{4} - \frac{{\left( {1 - \delta } \right)}}{2}\left( {M_{3} - N_{3} - N_{4} } \right)} \right\}\frac{y}{{T_{n}^{2} }} \\ {\kern 1pt} & \quad - 2\left\{ {M_{3} \left( { - z + h + H\left( {1 - \delta } \right)} \right) + \left( {N_{1} - N_{2} } \right)\left( {h - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {z - h\left( {1 - \delta } \right) - H} \right) - 4hM_{4} } \right\}\frac{{t_{n} y}}{{T_{n}^{4} }} \\ & \quad + \frac{4y}{{T_{n}^{4} }}\left\{ {M_{3} H\left( {z - H + \delta h} \right) + \left( {N_{1} - N_{2} } \right)Hh + \left( {N_{3} + N_{4} } \right)h\left( {H - z} \right)} \right\}\left( {1 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad + \frac{{96M_{3} hH\left( {H - z} \right)t_{n} y}}{{T_{n}^{6} }}\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) + \left\{ {2M_{4} - \frac{{\left( {1 - \delta } \right)}}{2}\left( {M_{3} - N_{3} - N_{4} } \right)} \right\}\frac{y}{{U_{n}^{2} }} \\ & \quad - 2\left\{ {M_{3} \left( {z - \delta h - H\left( {1 - \delta } \right)} \right) + \left( {N_{1} - N_{2} } \right)\left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - z} \right)} \right\}\frac{{u_{n} y}}{{U_{n}^{4} }} \\ & \quad + \frac{{4M_{3} \left( {h - H} \right)\left( {z - H} \right)y}}{{U_{n}^{4} }}\left. {\left( {1 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right)} \right|_{n = \beta + 2} + \left| {\frac{{M_{2} \left( {1 - \delta } \right)y}}{{2Q_{n}^{2} }}} \right. + \frac{{2M_{2} (h - z)q_{n} y}}{{Q_{n}^{4} }} \\ & \quad - \frac{{M_{2} \left( {1 - \delta } \right)y}}{{2U_{n}^{2} }} - \frac{{2M_{2} \left( {z - \delta h} \right)u_{n} y}}{{U_{n}^{4} }} + \frac{{4M_{2} hzy}}{{U_{n}^{4} }}\left( {1 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) - \frac{{2M_{2} (h - z)t_{n} y}}{{T_{n}^{4} }} + \left. {\frac{{M_{2} \left( {1 - \delta } \right)y}}{{2T_{n}^{2} }}} \right|_{n = \beta + 4} \\ \end{aligned} $$

$$ \begin{aligned} \psi_{6} & = M_{1} \left| {\frac{{\left( {1 - \delta } \right)yp_{n} }}{{P_{n}^{4} }}} \right. + \frac{{2y\left( {\delta h + z} \right)}}{{P_{n}^{4} }}\left( {1 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) - \frac{{24yp_{n} hz}}{{P_{n}^{6} }}\left. {\left( {1 - \frac{{2p_{n}^{2} }}{{P_{n}^{2} }}} \right)} \right|_{n = \beta } \\ & \quad + \left| {\left\{ {4\delta M_{4} - \left( {1 - \delta } \right)\left( {N_{3} + N_{4} - M_{3} } \right)} \right\}\frac{{yq_{n} }}{{Q_{n}^{4} }}} \right. \\ & \quad + \frac{2y}{{Q_{n}^{4} }}\left\{ {M_{3} \left( {z - h} \right) - M_{3} H\left( {1 - \delta } \right) + \left( {N_{1} - N_{2} } \right)\delta \left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {\delta H - z} \right) - 4zM_{4} } \right\}\left( {1 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad + \frac{{48q_{n} y}}{{Q_{n}^{6} }}\left\{ {M_{3} H\delta \left( {H - h} \right) + \left( {N_{1} - N_{2} } \right)(H - h)z + \left( {N_{3} + N_{4} - M_{3} } \right)Hz} \right\}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad - \frac{96y}{{Q_{n}^{6} }}M_{3} zH\left( {H - h} \right)\left( {1 - \frac{{12q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{16q_{n}^{4} }}{{Q_{n}^{4} }}} \right) - \left\{ {4M_{4} - \left( {1 - \delta } \right)\left( {M_{3} + N_{1} - N_{2} } \right)} \right\}\frac{{p_{n} y}}{{P_{n}^{4} }} \\ & \quad - \frac{2y}{{P_{n}^{4} }}\left\{ {M_{3} \left( {\left( {1 - \delta } \right)H + \frac{2h}{\alpha } - z} \right) - \left( {N_{1} - N_{2} } \right)\left( {z - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - h} \right)} \right\}\left( {1 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) \\ & \quad - \frac{{48M_{3} \left( {H - h} \right)\left( {z - H} \right)p_{n} y}}{{P_{n}^{6} }}\left( {1 - \frac{{2p_{n}^{2} }}{{P_{n}^{2} }}} \right) - \left\{ {4M_{4} - \left( {1 - \delta } \right)\left( {M_{3} - N_{3} - N_{4} } \right)} \right\}\frac{{t_{n} y}}{{T_{n}^{4} }} \\ {\kern 1pt} & \quad + \frac{2y}{{T_{n}^{4} }}\left\{ {M_{3} \left( { - z + h + H\left( {1 - \delta } \right)} \right) + \left( {N_{1} - N_{2} } \right)\left( {h - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {z - h\left( {1 - \delta } \right) - H} \right) - 4hM_{4} } \right\}\left( {1 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad + \frac{{48t_{n} y}}{{T_{n}^{6} }}\left\{ {M_{3} H\left( {z - H + \delta h} \right) + \left( {N_{1} - N_{2} } \right)Hh + \left( {N_{3} + N_{4} } \right)h\left( {H - z} \right)} \right\}\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad - \frac{{96M_{3} hH\left( {H - z} \right)y}}{{T_{n}^{6} }}\left( {1 - \frac{{12t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{16t_{n}^{4} }}{{T_{n}^{4} }}} \right) + \left\{ {4M_{4} - \left( {1 - \delta } \right)\left( {M_{3} - N_{3} - N_{4} } \right)} \right\}\frac{{u_{n} y}}{{U_{n}^{4} }} \\ & \quad + \frac{2y}{{U_{n}^{4} }}\left\{ {M_{3} \left( {z - \delta h - H\left( {1 - \delta } \right)} \right) + \left( {N_{1} - N_{2} } \right)\left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - z} \right)} \right\}\left( {1 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad + \frac{{48M_{3} \left( {h - H} \right)\left( {z - H} \right)u_{n} y}}{{U_{n}^{6} }}\left. {\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right)} \right|_{n = \beta + 2} + \left| {\frac{{M_{2} \left( {1 - \delta } \right)q_{n} y}}{{Q_{n}^{4} }}} \right. - \frac{{2M_{2} (h - z)y}}{{Q_{n}^{4} }}\left( {1 - \frac{{4q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad - \frac{{M_{2} \left( {1 - \delta } \right)q_{n} y}}{{Q_{n}^{4} }} + \frac{{2M_{2} \left( {z - \delta h} \right)y}}{{U_{n}^{4} }}\left( {1 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) + \frac{{48M_{2} hzu_{n} y}}{{U_{n}^{6} }}\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad + \frac{{2M_{2} (h - z)y}}{{T_{n}^{4} }}\left( {1 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) + \left. {\frac{{M_{2} \left( {1 - \delta } \right)t_{n} y}}{{T_{n}^{4} }}} \right|_{n = \beta + 4} \\ \end{aligned} $$

$$ \begin{aligned} \psi_{7} & = M_{1} \left| {\frac{{\left( {1 - \delta } \right)y}}{{P_{n}^{4} }}\left( {\frac{{4p_{n}^{2} }}{{P_{n}^{2} }} - 1} \right)} \right. + \frac{{24yp_{n} \left( {\delta h + z} \right)}}{{P_{n}^{4} }}\left( {1 - \frac{{2p_{n}^{2} }}{{P_{n}^{2} }}} \right) + \left| {\frac{48hyz}{{P_{n}^{6} }}} \right.\left. {\left( {1 - \frac{{12p_{n}^{2} }}{{P_{n}^{2} }} + \frac{{16p_{n}^{4} }}{{P_{n}^{4} }}} \right)} \right|_{n = \beta } \\ & \quad + \left| {\frac{y}{{Q_{n}^{4} }}\left\{ {4\delta M_{4} - \left( {1 - \delta } \right)\left( {N_{3} + N_{4} - M_{3} } \right)} \right\}} \right.\left( {\frac{{4q_{n}^{2} }}{{Q_{n}^{2} }} - 1} \right) \\ & \quad + \frac{{24q_{n} y}}{{Q_{n}^{6} }}\left\{ {M_{3} \left( {z - h} \right) - M_{3} H\left( {1 - \delta } \right) + \left( {N_{1} - N_{2} } \right)\delta \left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {\delta H - z} \right) - 4zM_{4} } \right\}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) \\ & \quad - \frac{48y}{{Q_{n}^{6} }}\left\{ {M_{3} H\delta \left( {H - h} \right) + \left( {N_{1} - N_{2} } \right)(H - h)z + \left( {N_{3} + N_{4} - M_{3} } \right)Hz} \right\}\left( {1 - \frac{{12q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{16q_{n}^{4} }}{{Q_{n}^{4} }}} \right) \\ & \quad - \frac{{960M_{3} zH\left( {H - h} \right)q_{n} y}}{{Q_{n}^{8} }}\left( {3 - \frac{{16q_{n}^{2} }}{{Q_{n}^{2} }} + \frac{{16q_{n}^{4} }}{{Q_{n}^{4} }}} \right) + \frac{y}{{P_{n}^{4} }}\left\{ {4M_{4} - \left( {1 - \delta } \right)\left( {M_{3} + N_{1} - N_{2} } \right)} \right\}\left( {1 - \frac{{4p_{n}^{2} }}{{P_{n}^{2} }}} \right) \\ & \quad - \frac{{24p_{n} y}}{{P_{n}^{4} }}\left\{ {M_{3} \left( { - z + \frac{2h}{\alpha }} \right) + M_{3} H\left( {1 - \delta } \right) - \left( {N_{1} - N_{2} } \right)\left( {z - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - h} \right)} \right\}\left( {1 - \frac{{2p_{n}^{2} }}{{P_{n}^{2} }}} \right) \\ & \quad + \frac{{48M_{3} \left( {H - h} \right)\left( {z - H} \right)y}}{{P_{n}^{6} }}\left( {1 - \frac{{12p_{n}^{2} }}{{P_{n}^{2} }} + \frac{{16p_{n}^{4} }}{{P_{n}^{4} }}} \right) + \frac{y}{{T_{n}^{4} }}\left\{ {4M_{4} - \left( {1 - \delta } \right)\left( {M_{3} - N_{3} - N_{4} } \right)} \right\} \left( {1 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ {\kern 1pt} & \quad + \frac{{24t_{n} y}}{{T_{n}^{6} }}\left\{ {M_{3} \left( { - z + h + H\left( {1 - \delta } \right)} \right) + \left( {N_{1} - N_{2} } \right)\left( {h - H} \right) + \left( {N_{3} + N_{4} } \right)\left( {z - h\left( {1 - \delta } \right) - H} \right) - 4hM_{4} } \right\} \left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) \\ & \quad - \frac{48y}{{T_{n}^{6} }}\left\{ {M_{3} H\left( {z - H + \delta h} \right) + \left( {N_{1} - N_{2} } \right)Hh + \left( {N_{3} + N_{4} } \right)h\left( {H - z} \right)} \right\}\left( {1 - \frac{{12t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{16t_{n}^{4} }}{{T_{n}^{4} }}} \right) \\ & \quad - \frac{{960M_{3} hH\left( {H - z} \right)t_{n} y}}{{T_{n}^{8} }}\left( {3 - \frac{{16t_{n}^{2} }}{{T_{n}^{2} }} + \frac{{16t_{n}^{4} }}{{T_{n}^{4} }}} \right) - \frac{y}{{U_{n}^{4} }}\left\{ {4M_{4} - \left( {1 - \delta } \right)\left( {M_{3} - N_{3} - N_{4} } \right)} \right\}\left( {1 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad + \frac{{24u_{n} y}}{{U_{n}^{6} }}\left\{ {M_{3} \left( {z - \delta h} \right) - M_{3} H\left( {1 - \delta } \right) + \left( {N_{1} - N_{2} } \right)\left( {H - h} \right) + \left( {N_{3} + N_{4} } \right)\left( {H - z} \right)} \right\}\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad - \frac{{48M_{3} \left( {h - H} \right)\left( {z - H} \right)y}}{{U_{n}^{6} }}\left. {\left( {1 - \frac{{12u_{n}^{2} }}{{U_{n}^{2} }} + \frac{{16u_{n}^{4} }}{{U_{n}^{4} }}} \right)} \right|_{n = \beta + 2} + \left| {\frac{{M_{2} \left( {1 - \delta } \right)y}}{{Q_{n}^{4} }}} \right.\left( {\frac{{4q_{n}^{2} }}{{Q_{n}^{2} }} - 1} \right) \\ & \quad - \frac{{24M_{2} (h - z)q_{n} y}}{{Q_{n}^{6} }}\left( {1 - \frac{{2q_{n}^{2} }}{{Q_{n}^{2} }}} \right) + \frac{{M_{2} \left( {1 - \delta } \right)y}}{{U_{n}^{4} }}\left( {1 - \frac{{4u_{n}^{2} }}{{U_{n}^{2} }}} \right) + \frac{{24M_{2} \left( {z - \delta h} \right)u_{n} y}}{{U_{n}^{6} }}\left( {1 - \frac{{2u_{n}^{2} }}{{U_{n}^{2} }}} \right) \\ & \quad - \frac{{48M_{2} hzy}}{{U_{n}^{6} }}\left( {1 - \frac{{12u_{n}^{2} }}{{U_{n}^{2} }} + \frac{{16u_{n}^{4} }}{{U_{n}^{4} }}} \right) + \frac{{24M_{2} (h - z)t_{n} y}}{{T_{n}^{6} }}\left( {1 - \frac{{2t_{n}^{2} }}{{T_{n}^{2} }}} \right) - \left. {\frac{{M_{2} \left( {1 - \delta } \right)y}}{{T_{n}^{4} }}\left\{ {1 - \frac{{4t_{n}^{2} }}{{T_{n}^{2} }}} \right\}} \right|_{n = \beta + 4} \\ \end{aligned} $$

where

$$ \begin{aligned} & P_{n}^{2} = y^{2} + p_{n}^{2} ,\quad p_{n} = nH + h + z, \\ & Q_{n}^{2} = y^{2} + q_{n}^{2} ,\quad q_{n} = nH - h + z, \\ & T_{n}^{2} = y^{2} + t_{n}^{2} ,\quad t_{n} = nH + h - z, \\ & U_{n}^{2} = y^{2} + u_{n}^{2} ,\quad u_{n} = nH - h - z, \\ & \left| {f(n)} \right|_{n = \beta }\,=\,f(\beta ). \\ \end{aligned} $$