Patching Asymptotics Solution of a Cable with a Small Bending Stiffness
Publication: Journal of Structural Engineering
Volume 139, Issue 2
Abstract
The analysis of a cable with a small bending stiffness is a problem encountered in many engineering applications such as the fatigue assessment of stay cables, the modeling of pipeline laying operation, or the determination of bending stresses in drillpipe assemblies. Because this phenomenon is modeled by a singularly perturbed equation, standard numerical techniques fail to solve these problems efficiently. As an alternative, provided the complexity of the analytical developments does not preclude their application, these problems may be tackled with appealing analytical procedures such as matching asymptotics or multiple scales. Otherwise, advanced numerical simulations combining patching asymptotics within a numerical framework are the only possible approach for problems where the governing equations are too complex. Patching asymptotics also feature a number of merits such as the possibility of using a boundary layer with a finite extent. Aiming at a better understanding of this latter technique, the purpose of this paper was to determine the solution of a cable with a small bending stiffness. Interesting details about patchability conditions and about how to restore higher derivative continuity are included. The accuracy of the patching asymptotics approach is also compared with that of matched asymptotics.
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Acknowledgments
The idea of considering a patching asymptotics approach for the solution of that problem emerged during a stay of V. Denoël at the Commonwealth Scientific and Industrial Research Organization (Perth, Australia). The authors thank Professor E. Detournay for interesting discussions and interactions.
References
Antman, S. S. (1995). Nonlinear problems in elasticity. Applied mathematical sciences, Springer, Berlin.
Bieniasz, L. K. (2008). “Adaptive solution of bvps in singularly perturbed second-order odes, by the extended numerov method combined with an iterative local grid h-refinement.” Appl. Math. Comput., 198(2), 665–682.
Burgess, J. J. (1993). “Bending stiffness in a simulation of undersea cable deployment.” Int. J. Offshore Polar Eng., 3(3), 197–204.
Cluni, F., Gusella, V., and Ubertini, F. (2007). “A parametric investigation of wind-induced cable fatigue.” Eng. Struct., 29(11), 3094–3105.
Denoël, V., and Detournay, E. (2010). “Multiple scales solution for a beam with a small bending stiffness.” J. Eng. Mech., 136(1), 69–77.
Denoël, V., and Detournay, E. (2011). “Eulerian formulation of constrained elastica.” Int. J. Solids Struct., 48(3–4), 625–636.
Esipova, V. (1975). “The asymptotic behavior of solutions of the general boundary value problem for singularly perturbed systems of ordinary differential equations of conditionally stable type.” Differ. Equations, 11(11), 1457–1465.
Euler, L. (1744). Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti. Additamentum 1. 〈http://eulerarchive.org/ E065〉.
Hinch, E. J. (1991). Perturbation methods, Vol. 1, Cambridge University Press, Cambridge, U.K.
Hong, K. J., Kiureghian, A. D., and Sackman, J. L. (2005). “Bending behavior of helically wrapped cables.” J. Eng. Mech., 131(5), 500–511.
Irvine, H. M. (1975). “Statics of suspended cables.” J. Eng. Mech., 101(3), 187–205.
Jain, M. K., Iyengar, S. R. K., and Subramanyam, G. S. (1984). “Variable mesh methods for the numerical-solution of 2-point singular perturbation problems.” Comput. Methods Appl. Mech. Eng., 42(3), 273–286.
Kevorkian, J., and Cole, J. D. (1996). Multiple scale and singular perturbation methods. Applied mathematical sciences, Springer, Berlin.
Kumar, M., Kumar Mishra, H., and Singh, P. (2009). “A boundary value approach for a class of linear singularly perturbed boundary value problems.” Adv. Eng. Softw., 40(4), 298–304.
Levien, R. (2008). “From spiral to spline: Optimal techniques in interactive curve design,” Ph.D. thesis, University of California–Berkeley, Berkeley, CA.
Nayfeh, A. H. (1973). Perturbation methods, Wiley, New York.
Papailiou, K. O. (1997). “On the bending stiffness of transmission line conductors.” IEEE Trans. Power Deliv., 12(4), 1576–1583.
Quarteroni, A., and Valli, A. (1994). Numerical approximation of partial differential equations, Springer, Berlin.
Rao, S. C. S., and Kumar, M. (2007). “B-spline collocation method for nonlinear singularly-perturbed two-point boundary-value problems.” J. Optim. Theory Appl., 134(1), 91–105.
Reddy, Y. N., and Chakravarthy, P. P. (2004). “Numerical patching method for singularly perturbed two-point boundary value problems using cubic splines.” Appl. Math. Comput., 149(2), 441–468.
Rienstra, S. W. (1987). “Analytical approximations for offshore pipelaying problems.” Proc., Int. Conf. on Industrial and Applied Mathematics, Centrum voor Wiskunde en Informatica, Amsterdam, Netherlands.
Schmeiser, C., and Weiss, R. (1986). “Asymptotic analysis of singular singularly perturbed boundary-value-problems.” SIAM J. Math. Anal., 17(3), 560–579.
Stynes, M., and Oriordan, E. (1986). “A finite-element method for a singulary perturbed boundary-value problem.” Numerische Mathematik, 50(1), 1–15.
Timoshenko, S. P., and Goodier, J. N. (1987). Theory of elasticity, 3rd Ed., McGraw Hill, New York.
Traeger, W., and Kollegger, J. (2010). “Fatigue testing of stay cables at resonant frequency.” IASMAS 2010, Bridge Maintenance, Safety, Management and Life Cycle Optimization, CRC Press, Boca Raton, FL, 3589–3592.
Wang, C. Y. (1986). “A critical review of the heavy elastica.” Int. J. Mech. Sci., 28(8), 549–559.
Wang, W., Kosor, R., and Bai, Y. (2011). “Fatigue analysis on multi-spanning pipelines.” J. Harbin Eng. Univ., 32(5), 560–564.
Wei, X., and Qiang, S. (2011). “Fatigue performance of anchorage zone for long-span single pylon cable-stayed bridge.” J. Southwest Jiaotong Univ., 46(6), 940–945.
Wolfe, P. (1991). “Asymptotic analysis of a rod with small bending stiffness.” Q. Appl. Math., 49(1), 53–65.
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Published In
Journal of Structural Engineering
Volume 139 • Issue 2 • February 2013
Pages: 180 - 187
Copyright
© 2013 American Society of Civil Engineers.
History
Received: Jan 6, 2012
Accepted: Apr 24, 2012
Published online: Apr 26, 2012
Published in print: Feb 1, 2013
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