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(254)

The coefficients ${A}_{1},{B}_{2}$ are given by Eq. (241). Substitution of Eq. (241) into Eqs. (253), (254) yields

$\begin{array}{l}{u}_{r}^{A}=-\left[\xi \left(-2\xi {\zeta }_{S}\mathrm{sin}{\zeta }_{S}d\right)\mathrm{sin}{\zeta }_{P}z+\left(-\left({\xi }^{2}-{\zeta }_{S}^{2}\right)\mathrm{sin}{\zeta }_{P}d\right)\left(-{\zeta }_{S}\mathrm{sin}{\zeta }_{S}z\right)\right]{J}_{1}\hfill \\ {u}_{z}^{A}=\left[-2\xi {\zeta }_{S}\mathrm{sin}{\zeta }_{S}d{\zeta }_{P}\mathrm{cos}{\zeta }_{P}z+\xi \left(-\left({\xi }^{2}-{\zeta }_{S}^{2}\right)\mathrm{sin}{\zeta }_{P}d\right)\mathrm{cos}{\zeta }_{S}z\right]{J}_{0}\hfill \end{array}$ (255)

(255)

$\begin{array}{l}{\sigma }_{zz}^{A}=\mu \left\{\left({\xi }^{2}-{\zeta }_{S}^{2}\right)\left(-2\xi {\zeta }_{S}\mathrm{sin}{\zeta }_{S}d\right)\mathrm{sin}{\zeta }_{P}z+\left[2\xi \left({\xi }^{2}-{\zeta }_{S}^{2}\right)\mathrm{sin}{\zeta }_{P}d\right]{\zeta }_{S}\mathrm{sin}{\zeta }_{S}z\right\}{J}_{0}\hfill \\ {\sigma }_{rz}^{A}=-\mu \left\{2\xi \left(-2\xi {\zeta }_{S}\mathrm{sin}{\zeta }_{S}d\right){\zeta }_{P}\mathrm{cos}{\zeta }_{P}z+\left({\xi }^{2}-{\zeta }_{S}^{2}\right)\left[-\left({\xi }^{2}-{\zeta }_{S}^{2}\right)\mathrm{sin}{\zeta }_{P}d\right]\mathrm{cos}{\zeta }_{S}z\right\}{J}_{1}\hfill \end{array}$ (256)

(256)

Upon expansion and rearrangement, Eqs. (255), (256) become

$\begin{array}{l}{u}_{r}^{A}={\zeta }_{S}\hfill \end{array}$

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