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Parametric dependencies of the post-ultimate strength behavior of a ship’s hull girder in waves

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Abstract

To rationally assess the consequence of a ship’s hull girder collapse, it is necessary to know the post-ultimate strength behavior of the hull girder including the global deformation and motions under extreme wave-induced loads. In the foregoing research, the authors proposed a numerical analysis system to predict the collapse behavior in waves including the post-ultimate strength behavior. The primary objective of the present paper is to clarify the parametric dependencies of the severity of the collapse in a rational manner. The parameters may include those related to load-carrying capacity and the extreme loads. To this end, an analytical solution to describe the post-ultimate strength behavior is derived. Assuming that a plastic hinge is formed at the midship during the collapse procedure, the whole ship is modeled as a two-rigid-bodies system connected to each other amidship via a nonlinear rotational spring, which represents the nonlinear relationship between the bending moment and the rotational angle. The relationship may be modeled as piece-wise linear curves. It is further assumed that large motions and elastic/plastic deformations of the hull girder may not affect the load evaluations, and that the hull girder is subjected to a large single wave. Some important parameters to predict the severity of the collapse are specified based on the analytical solution.

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Acknowledgments

This research was partly supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (A), (20246126), 2010.

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Authors and Affiliations

  1. Department of Naval Architecture and Ocean Engineering, Osaka University, Suita, Japan

    Weijun Xu, Kazuhiro Iijima & Masahiko Fujikubo

  2. College of Shipbuilding Engineering, Harbin Engineering University, Harbin, China

    Weijun Xu

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  1. Weijun Xu
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  2. Kazuhiro Iijima
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  3. Masahiko Fujikubo
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Correspondence to Weijun Xu.

Appendix

Appendix

The parameters in the solution of each path

  1. 1.

    Path OA

$$ \theta_{\text{OA}} = e^{ - nt} (C_{1} \cos \omega_{{{\text{d}}1}} t + C_{2} \sin \omega_{{{\text{d}}1}} t) + R_{1} \sin \omega_{0} t + T_{1} \cos \omega_{0} t + \frac{{M_{\text{s}} }}{{\omega_{1}^{2} I}} $$
$$ R_{1} = \frac{{M_{0} }}{I} \cdot \frac{{\omega_{1}^{2} - \omega_{0}^{2} }}{{(\omega_{1}^{2} - \omega_{0}^{2} )^{2} + 4n^{2} \omega_{0}^{2} }} \, $$
$$ T_{1} = - \frac{{M_{0} }}{I} \cdot \frac{{2n\omega_{0} }}{{(\omega_{1}^{2} - \omega_{0}^{2} )^{2} + 4n^{2} \omega_{0}^{2} }} $$
$$ C_{1} = \frac{{M_{0} }}{I} \cdot \frac{{2n\omega_{0} }}{{(\omega_{1}^{2} - \omega_{0}^{2} )^{2} + 4n^{2} \omega_{0}^{2} }} - \frac{{M_{\text{s}} }}{{\omega_{1}^{2} I}} $$
$$ C_{2} = \frac{{M_{0} \omega_{0} }}{I} \cdot \frac{{2n^{2} - \omega_{1}^{2} + \omega_{0}^{2} }}{{\sqrt {\omega_{1}^{2} - n^{2} } \left[ {(\omega_{1}^{2} - \omega_{0}^{2} )^{2} + 4n^{2} \omega_{0}^{2} } \right]}} $$
$$ \omega_{1}^{2} = \frac{{(2k_{{{\text{s}}1}} + \rho gV \cdot GM)}}{I} $$
$$ \omega_{{{\text{d}}1}}^{2} = \omega_{1}^{2} - n^{2} ,\quad n = \frac{C}{2I},\quad k_{{{\text{s}}1}} = k_{1} $$
  1. 2.

    Path AB

$$ \theta_{\text{AB}} = D_{1} e^{{\omega_{21} t}} + D_{2} e^{{\omega_{22} t}} + R_{2} \sin \omega_{0} t + T_{2} \cos \omega_{0} t + \frac{{2k_{2} \theta_{\text{OA}} (t_{1} ) - 2M_{\text{u}} + M_{\text{s}} }}{{\omega_{2}^{2} I}} $$
$$ R_{2} = \frac{{M_{0} }}{I} \cdot \frac{{\omega_{2}^{2} - \omega_{0}^{2} }}{{(\omega_{2}^{2} - \omega_{0}^{2} )^{2} + 4n^{2} \omega_{0}^{2} }},\quad T_{2} = - \frac{{M_{0} }}{I} \cdot \frac{{2n\omega_{0} }}{{(\omega_{2}^{2} - \omega_{0}^{2} )^{2} + 4n^{2} \omega_{0}^{2} }} \, $$
$$ D_{1} = \frac{{\omega_{22} R^{\prime }_{2} - T^{\prime }_{2} }}{{(\omega_{22} - \omega_{21} )e^{{\omega_{21} t_{1} }} }} $$
$$ D_{2} = \frac{{\omega_{21} R^{\prime }_{2} - T^{\prime }_{2} \, }}{{(\omega_{21} - \omega_{22} )e^{{\omega_{22} t_{1} }} }} $$
$$ \omega_{2}^{2} = \frac{{2k_{2} + \rho gV \cdot GM}}{I} $$
$$ \omega_{21} = - n + \sqrt {n^{2} - \omega_{2}^{2} } ,\quad \omega_{22} = - n - \sqrt {n^{2} - \omega_{2}^{2} } $$
$$ \begin{aligned} R^{\prime }_{2} & = e^{{ - nt_{1} }} (C_{1} \cos \omega_{d1} t_{1} + C_{2} \sin \omega_{{{\text{d}}1}} t_{1} ) + (R_{1} - R_{2} )\sin \omega_{0} t_{1} \\ & \quad + (T_{1} - T_{2} )\cos \omega_{0} t_{1} - \frac{{2k_{2} \theta_{\text{OA}} (t_{1} ) - 2M_{\text{u}} + M_{\text{s}} }}{{\omega_{2}^{2} I}} + \frac{{M_{\text{s}} }}{{\omega_{1}^{2} I}} \\ \end{aligned} $$
$$ \begin{aligned} T_{2}^{\prime } & = - ne^{{ - nt_{1} }} (C_{1} \cos \omega_{{{\text{d}}1}} t_{1} + C_{2} \sin \omega_{{{\text{d}}1}} t_{1} ) \\ & \quad + e^{{ - nt_{1} }} ( - C_{1} \omega_{{{\text{d}}1}} \sin \omega_{{{\text{d}}1}} t_{1} + C_{2} \omega_{{{\text{d}}1}} \cos \omega_{{{\text{d}}1}} t_{1} ) \\ & \quad + (R_{1} - R_{2} )\omega_{0} \cos \omega_{0} t_{1} - (T_{1} - T_{2} )\omega_{0} \sin \omega_{0} t_{1} \\ \end{aligned} $$
  1. 3.

    Path BC

$$ \theta_{\text{BC}} = e^{ - nt} (E_{1} \cos \omega_{{{\text{d}}3}} t + E_{2} \sin \omega_{{{\text{d}}3}} t) + R_{3} \sin \omega_{0} t + T_{3} \cos \omega_{0} t + \frac{{2k_{3} \theta_{\text{AB}} (t_{2} ) - 2M_{\text{BC}} + M_{\text{s}} }}{{\omega_{3}^{2} I}} $$
$$ R_{3} = \frac{{M_{0} }}{I} \cdot \frac{{\omega_{3}^{2} - \omega_{0}^{2} }}{{(\omega_{3}^{2} - \omega_{0}^{2} )^{2} + 4n^{2} \omega_{0}^{2} }},\quad T_{3} = - \frac{{M_{0} }}{I} \cdot \frac{{2n\omega_{0} }}{{(\omega_{3}^{2} - \omega_{0}^{2} )^{2} + 4n^{2} \omega_{0}^{2} }} \, $$
$$ E_{1} = R^{\prime }_{3} \cos \omega_{{{\text{d}}3}} t_{2} - T^{\prime }_{3} \sin \omega_{{{\text{d}}3}} t_{2} $$
$$ E_{2} = R^{\prime }_{3} \sin \omega_{{{\text{d}}3}} t_{2} + T^{\prime }_{3} \cos \omega_{{{\text{d}}3}} t_{2} $$
$$ \omega_{{{\text{d}}3}}^{2} = \omega_{3}^{2} - n^{2} ,\quad \omega_{3}^{2} = \frac{\rho gV \cdot GM}{I} $$
$$ \begin{aligned} R^{\prime }_{3} & = D_{1} e^{{(\omega_{21} + n)t_{2} }} + D_{2} e^{{(\omega_{22} + n)t_{2} }} + e^{{nt_{2} }} (R_{2} - R_{3} )\sin \omega_{0} t_{2} \\ & \quad + e^{{nt_{2} }} (T_{2} - T_{3} )\cos \omega_{0} t_{2} + e^{{nt_{2} }} \left[ {\frac{{2k_{2} \theta_{\text{OA}} (t_{1} ) - 2M_{\text{u}} + M_{\text{s}} }}{{\omega_{2}^{2} I}} + \frac{{2M_{\text{BC}} - M_{\text{s}} }}{{\omega_{3}^{2} I}}} \right] \\ \end{aligned} $$
$$ \begin{aligned} T^{\prime }_{3} & = \frac{{\omega_{21} + n}}{{\omega_{{{\text{d}}3}} }}D_{1} e^{{(\omega_{21} + n)t_{2} }} + \frac{{\omega_{22} + n}}{{\omega_{{{\text{d}}3}} }}D_{2} e^{{(\omega_{22} + n)t_{2} }} \\ & \quad + e^{{nt_{2} }} (R_{2} - R_{3} )\left( {\frac{{\omega_{0} }}{{\omega_{{{\text{d}}3}} }}\cos \omega_{0} t_{2} + \frac{n}{{\omega_{{{\text{d}}3}} }}\sin \omega_{0} t_{2} } \right) \\ & \quad - e^{{nt_{2} }} (T_{2} - T_{3} )\left( {\frac{{\omega_{0} }}{{\omega_{{{\text{d}}3}} }}\sin \omega_{0} t_{2} - \frac{n}{{\omega_{{{\text{d}}3}} }}\cos \omega_{0} t_{2} } \right) \\ & \quad + \frac{{ne^{{nt_{2} }} }}{{\omega_{{{\text{d}}3}} }}\left( {\frac{{2k_{2} \theta_{\text{OA}} (t_{1} ) - 2M_{\text{u}} + M_{\text{s}} }}{{\omega_{2}^{2} I}} + \frac{{2M_{\text{BC}} - M_{\text{s}} }}{{\omega_{3}^{2} I}}} \right) \\ \end{aligned} $$
  1. 4.

    Path CD′

$$ \theta_{{{\text{CD}}^{\prime } }} = e^{ - nt} (F_{1} \cos \omega_{{{\text{d}}4}} t + F_{2} \sin \omega_{{{\text{d}}4}} t) + R_{4} \sin \omega_{0} t + T_{4} \cos \omega_{0} t + \frac{{2k_{4} \theta_{\text{BC}} (t_{3} ) - 2M_{\text{BC}} + M_{\text{s}} }}{{\omega_{4}^{2} I}} $$
$$ R_{4} = \frac{{M_{0} }}{I} \cdot \frac{{\omega_{4}^{2} - \omega_{0}^{2} }}{{(\omega_{4}^{2} - \omega_{0}^{2} )^{2} + 4n^{2} \omega_{0}^{2} }} $$
$$ T_{4} = - \frac{{M_{0} }}{I} \cdot \frac{{2n\omega_{0} }}{{(\omega_{4}^{2} - \omega_{0}^{2} )^{2} + 4n^{2} \omega_{0}^{2} }} \, $$
$$ F_{1} = R^{\prime }_{4} \cos \omega_{{{\text{d}}4}} t_{3} - T^{\prime }_{4} \sin \omega_{{{\text{d}}4}} t_{3} $$
$$ F_{2} = R^{\prime }_{4} \sin \omega_{{{\text{d}}4}} t_{3} + T^{\prime }_{4} \cos \omega_{{{\text{d}}4}} t_{3} $$
$$ \omega_{{{\text{d}}4}}^{2} = \omega_{4}^{2} - n^{2} ,\quad \omega_{4}^{2} = \frac{{2k_{4} + \rho gV \cdot GM}}{I},\quad k_{4} = k_{1} $$
$$ \begin{aligned} R^{\prime }_{4} & = (E_{1} \cos \omega_{{{\text{d}}3}} t_{3} + E_{2} \sin \omega_{{{\text{d}}3}} t_{3} ) + e^{{nt_{3} }} (R_{3} - R_{4} )\sin \omega_{0} t_{3} \\ & \quad + e^{{nt_{3} }} (T_{3} - T_{4} )\cos \omega_{0} t_{3} - e^{{nt_{3} }} \left[ {\frac{{2M_{\text{BC}} - M_{\text{s}} }}{{\omega_{3}^{2} I}} + \frac{{2k_{4} \theta_{\text{BC}} (t_{3} ) - 2M_{\text{BC}} + M_{\text{s}} }}{{\omega_{4}^{2} I}}} \right] \\ \end{aligned} $$
$$ \begin{aligned} T^{\prime }_{4} & = \left( { - E_{1} \frac{{\omega_{{{\text{d}}3}} }}{{\omega_{{{\text{d}}4}} }}\sin \omega_{{{\text{d}}3}} t_{3} + E_{2} \frac{{\omega_{{{\text{d}}3}} }}{{\omega_{{{\text{d}}4}} }}\cos \omega_{{{\text{d}}3}} t_{3} } \right) \\ & \quad + e^{{nt_{3} }} (R_{3} - R_{4} )\left( {\frac{{\omega_{0} }}{{\omega_{{{\text{d}}4}} }}\cos \omega_{0} t_{3} + \frac{n}{{\omega_{{{\text{d}}4}} }}\sin \omega_{0} t_{3} } \right) \\ & \quad - e^{{nt_{3} }} (T_{3} - T_{4} )\left( {\frac{{\omega_{0} }}{{\omega_{{{\text{d}}4}} }}\sin \omega_{0} t_{3} - \frac{n}{{\omega_{{{\text{d}}4}} }}\cos \omega_{0} t_{3} } \right) \\ & \quad - \frac{{ne^{{nt_{3} }} }}{{\omega_{{{\text{d}}4}} }}\left( {\frac{{2M_{\text{BC}} - M_{\text{s}} }}{{\omega_{3}^{2} I}} + \frac{{2k_{4} \theta_{\text{BC}} (t_{3} ) - 2M_{\text{BC}} + M_{\text{s}} }}{{\omega_{4}^{2} I}}} \right) \\ \end{aligned} $$
  1. 5.

    Path D′D

$$ \theta_{{{\text{D}}^{\prime } {\text{D}}}} = e^{ - nt} (G_{1} \cos \omega_{{{\text{d}}5}} t + G_{2} \sin \omega_{{{\text{d}}5}} t) + \frac{{2k_{5} \theta_{\text{BC}} (t_{3} ) - 2M_{\text{BC}} + M_{\text{s}} }}{{\omega_{5}^{2} I}} $$
$$ G_{1} = R^{\prime }_{5} \cos \omega_{{{\text{d}}5}} t_{4} - T^{\prime }_{5} \sin \omega_{{{\text{d}}5}} t_{4} \, $$
$$ G_{2} = R^{\prime }_{5} \sin \omega_{{{\text{d}}5}} t_{4} + T^{\prime }_{5} \cos \omega_{{{\text{d}}5}} t_{4} $$
$$ \omega_{{{\text{d}}5}}^{2} = \omega_{5}^{2} - n^{2} ,\quad \omega_{5}^{2} = \frac{{2k_{5} + \rho gV \cdot GM}}{I},\quad k_{5} = k_{4} = k_{1} $$
$$ \begin{aligned} R^{\prime }_{5} & = (F_{1} \cos \omega_{{{\text{d}}4}} t_{4} + F_{2} \sin \omega_{{{\text{d}}4}} t_{4} ) + e^{{nt_{4} }} R_{4} \sin \omega_{0} t_{4} \\ & \quad + e^{{nt_{4} }} T_{4} \cos \omega_{0} t_{4} + e^{{nt_{4} }} \left[ {\frac{{2k_{4} \theta_{\text{BC}} (t_{3} ) - 2M_{\text{BC}} + M_{\text{s}} }}{{\omega_{4}^{2} I}} - \frac{{2k_{5} \theta_{\text{BC}} (t_{3} ) - 2M_{\text{BC}} + M_{\text{s}} }}{{\omega_{5}^{2} I}}} \right] \\ \end{aligned} $$
$$ \begin{aligned} T_{5}^{\prime } & = \left( { - F_{1} \frac{{\omega_{{{\text{d}}4}} }}{{\omega_{{{\text{d}}5}} }}\sin \omega_{{{\text{d}}4}} t_{4} + F_{2} \frac{{\omega_{{{\text{d}}4}} }}{{\omega_{{{\text{d}}5}} }}\cos \omega_{{{\text{d}}4}} t_{4} } \right) \\ & \quad + e^{{nt_{4} }} R_{4} \left( {\frac{{\omega_{0} }}{{\omega_{{{\text{d}}5}} }}\cos \omega_{0} t_{4} + \frac{n}{{\omega_{{{\text{d}}5}} }}\sin \omega_{0} t_{4} } \right) \\ & \quad - e^{{nt_{4} }} T_{4} \left( {\frac{{\omega_{0} }}{{\omega_{{{\text{d}}5}} }}\sin \omega_{0} t_{4} - \frac{n}{{\omega_{{{\text{d}}5}} }}\cos \omega_{0} t_{4} } \right) \\ & \quad + \frac{{ne^{{nt_{4} }} }}{{\omega_{{{\text{d}}5}} }}\left( {\frac{{2k_{4} \theta_{\text{BC}} (t_{3} ) - 2M_{\text{BC}} + M_{\text{s}} }}{{\omega_{4}^{2} I}} - \frac{{2k_{5} \theta_{\text{BC}} (t_{3} ) - 2M_{\text{BC}} + M_{\text{s}} }}{{\omega_{5}^{2} I}}} \right) \\ \end{aligned} $$

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Xu, W., Iijima, K. & Fujikubo, M. Parametric dependencies of the post-ultimate strength behavior of a ship’s hull girder in waves. J Mar Sci Technol 17, 203–215 (2012). https://doi.org/10.1007/s00773-012-0158-1

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