Dynamic response of buried fluid-conveying pipelines subjected to blast loading using shell theory

Abstract

In this study, the dynamic response of buried fluid-conveying pipelines subjected to blast loading using the Love shell theory has been investigated. The fluid is considered as ideal fluid, and the velocity potential is used to describe the fluid pressure acting on the pipeline. The governing equations of the buried fluid-conveying pipelines are derived through Hamilton’s principle. The modal superposition method and the Newmark integral method are used to analyze the dynamic response of the pipelines under blast loading. Results show that the displacement amplitudes of the pipelines are larger in the soil with a higher acoustic impedance. The Winkler foundation can enhance the stiffness of the pipelines. Moreover, the increase in the scaled distance leads to the decrease in the displacement amplitudes of the pipelines. The increase in the fluid velocity results in the rise of the displacement amplitudes of the pipelines. In addition, the maximum displacement increases first and then decreases with the increase in length-to-radius ratio of the pipelines. With the increase in thickness-to-radius ratio, the maximum displacement of the pipelines tends to decrease.

Appendix

The coefficients of Eqs. (36), (37) and (38) are given by the following:

$$ k_{11} = \frac{{A_{11} m^{2} \pi^{2} }}{{L^{2} }} + \frac{{A_{66} n^{2} }}{{R^{2} }} $$

$$ k_{12} = - \frac{{A_{12} mn\pi }}{LR} - \frac{{A_{66} mn\pi }}{LR} $$

$$ k_{13} = - \frac{{B_{11} m^{3} \pi^{3} }}{{L^{3} }} - \frac{{B_{12} mn^{2} \pi }}{{LR^{2} }} - \frac{{2B_{66} mn^{2} \pi }}{{LR^{2} }} - \frac{{A_{12} m\pi }}{LR} $$

$$ k_{21} = - \frac{{A_{12} mn\pi }}{LR} - \frac{{A_{66} mn\pi }}{LR} $$

$$ k_{22} = \frac{{A_{66} m^{2} \pi^{2} }}{{L^{2} }} + \frac{{A_{22} n^{2} }}{{R^{2} }} $$

$$ k_{23} = \frac{{B_{22} n^{3} }}{{R^{3} }} + \frac{{A_{22} n}}{{R^{2} }} + \frac{{B_{12} m^{2} n\pi^{2} }}{{L^{2} R}} + \frac{{2B_{66} m^{2} n\pi^{2} }}{{L^{2} R}} $$

$$ k_{31} = - \frac{{B_{11} m^{3} \pi^{3} }}{{L^{3} }} - \frac{{B_{12} mn^{2} \pi }}{{LR^{2} }} - \frac{{2B_{66} mn^{2} \pi }}{{LR^{2} }} - \frac{{A_{12} m\pi }}{LR} $$

$$ k_{32} = \frac{{B_{22} n^{3} }}{{R^{3} }} + \frac{{A_{22} n}}{{R^{2} }} + \frac{{B_{12} m^{2} n\pi^{2} }}{{L^{2} R}} + \frac{{2B_{66} m^{2} n\pi^{2} }}{{L^{2} R}} $$

$$ \begin{aligned} k_{33} &= \frac{{D_{11} m^{4} \pi^{4} }}{{L^{4} }} + \frac{{D_{22} n^{4} }}{{R^{4} }} + \frac{{2B_{22} n^{2} }}{{R^{3} }} + \frac{{A_{22} }}{{R^{2} }} \\ &\quad+ \frac{{2D_{12} m^{2} n^{2} \pi^{2} }}{{L^{2} R^{2} }} + \frac{{4D_{66} m^{2} n^{2} \pi^{2} }}{{L^{2} R^{2} }} + \frac{{2B_{12} m^{2} \pi^{2} }}{{L^{2} R}} \end{aligned} $$

where

$$ \left[ {\begin{array}{*{20}c} {A_{ij} } \\ {B_{ij} } \\ {D_{ij} } \\ \end{array} } \right] = \int_{{ - \frac{h}{2}}}^{\frac{h}{2}} {Q_{ij} \left[ {\begin{array}{*{20}c} {z^{0} } \\ {z^{1} } \\ {z^{2} } \\ \end{array} } \right]} \;dz\;\;\left( {i,j = 1,2,6} \right)\;\;\left\{ {\begin{array}{*{20}c} {Q_{11} = Q_{22} = {E \mathord{\left/ {\vphantom {E {\left( {1 - \nu^{2} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \nu^{2} } \right)}}} \\ {Q_{12} = Q_{21} = {{\nu E} \mathord{\left/ {\vphantom {{\nu E} {\left( {1 - \nu^{2} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \nu^{2} } \right)}}} \\ {Q_{66} = {E \mathord{\left/ {\vphantom {E {2(1 + \nu )}}} \right. \kern-\nulldelimiterspace} {2(1 + \nu )}}} \\ \end{array} } \right. $$