Postbuckling Analysis of End Supported Rods under Self-Weight Using Intrinsic Coordinate Finite Elements
A formulation of postbuckling analysis of end supported rods under self-weight has been presented by the variational method. The variational formulation involving the strain energy due to bending and the potential energy of the self-weight, are expressed in terms of the intrinsic coordinates. The variational formulation is accomplished by introducing the Lagrange multiplier technique to impose the boundary conditions. The finite element method is used to derive a system of nonlinear equations resulting from the stationary of the total potential energy and then Newton-Raphson iterative procedure is applied to solve this system of equations. The numerical results demonstrate the postbluckled configurations of end supported rods under self-weight. This finite element method based on variational formulation expressed in term of intrinsic coordinate is highly recommended for postbuckling analysis of end-supported rods under self-weight.
[1] A. E. H. Love, Treatise on the mathematical theory of elasticity. Cambridge University Press: London, 1944.
[2] S. P. Timoshenko, J. N. Goodier, Theory of elasticity. McGraw-Hill: Singapore, 1982.
[3] S. S. Antman, Nonlinear problems of elasticity. Applied Mathematical Sciences 107. Springer Verlag. New York. USA, 1991.
[4] C. M. Wang, C. Y. Wang, J.N. Reddy, Exact solutions for buckling of structural members. CRC Press: boca Raton. Florida, 2005.
[5] D. Bigoni, Extremely deformable structures. Ed. Springer: Wien-New York, 2015.
[6] R. Haftka, W. Nachbar, “Post-buckling analysis of an elastically-restrained column,” Int. J. Solids Stuct., vol. 6, no. 11, pp. 1433-1449, Nov. 1970.
[7] C. M. Wang, K. K. Ang “Buckling capacities of braced heavy columns under an axial load,” Comput. Struc., vol. 28, no. 5, pp. 563-571, 1988.
[8] Y. H. Chai, C. M. Wang, “An application of differential transformation to stability analysis of heavy columns,” Int. J. Struc. Stab. Dyn., vol. 6, no. 3, pp. 317-332, 2006.
[9] S. B. Coskun, M. T. Atay, “Determination of critical buckling load for elastic columns of constant and variable cross-sections using variational iteration method,” Comput. Math. Appl., vol. 58, no. 11-12, pp. 2260-2266, Dec. 2009.
[10] D. J. Wei, S. X. Yan, Z. P. Zhang, X. F. Li, “Critical load for buckling of non-prismatic columns under self-weight and tip force,” Mech. Res. Commun., vol. 37, no. 6, pp. 554-558, Sep. 2010.
[11] Y. Huang, Q. Z. Luo, “A simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint,” Comput. Math. Appl., vol. 61, no. 9, pp. 2510-2517, May 2011.
[12] W. H. Duan, C. M. Wang, “Exact solution for buckling of columns including self-weight,” J. Eng. Mech., vol. 134, no. 1, pp. 116-119, Jan. 2008.
[13] B. Han, F. Li, C. Ni, Q. Zhang, C. Chen, T. Lu, “Stability and initial post-buckling of a standing sandwich beam under terminal force and self-weight,” Arch. Appl. Mech., vol. 86, no. 6, pp. 1063-1082, June 2016.
[14] C. Y. Wang, “Post-buckling of a clamped-simply supported elastic,” Int. J. Non-Linear Mech., vol. 32, no. 6, pp. 1115-1122, Nov. 1997.
[15] M. A. Vaz, D. F. C. Silva, “Post-buckling analysis of slender elastic rods subjected to terminal forces,” Int. J. Non-Linear Mech., vol. 38, no. 4, pp. 483-492, June 2003.
[16] L. N. Virgin, R. H. Plaut, “Postbuckling and vibration of linearly elastic and softening columns under self-weight,” Int. J. Solids Struct., vol. 41, no. 18-19, pp. 4989-5001, Sep. 2004.
[17] M. A. Vaz, G. H. W. Mascaro, “Post-buckling analysis of slender elastic vertical rods subjected to terminal forces and self-weight,” Int. J. Non-Linear Mech., vol. 40, no. 7, pp. 1049-1056, Sep. 2005.
[18] S. Li, Y. Zhou, “Post-buckling of a hinged-fixed beam under uniformly distributed follower forces,” Mech. Res. Commun., vol. 32, no. 4, pp. 359-367, July-Aug. 2005.
[19] J. Liu, Y. Mei, X. Dong “Post-buckling behavior of a double-hinged rod under self-weight,” Acta Mech. Solida Sinica., vol. 26, no. 2, pp. 197-204, Apr. 2013.
[20] B. K. Lee, S. J. Oh, “Elastica and buckling of simple tapered columns with constant volume,” Int. J. Non-Linear Mech., vol. 37, no. 18, pp. 2507-2518, May. 2000.
[21] K. Lee, “Post-buckling of uniform cantilever column under a combined load,” Int. J. Non-Linear Mech., vol. 36, no. 5, pp. 813-816, July 2001.
[22] V. V. Kuznetsov, S. V. Levyakov “Complete solution of the stability problem for elastica of Euler’s column.” Int. J. Non-Linear Mech., vol. 37, no.6, pp. 1003-1009, Sep. 2002.
[23] O. Sepahi, M. R. Forouzan, P. Malekzadeh “Differential quadrature application in post-buckling analysis of a hinged-fixed elastica under terminal forces and self-weight,” J. Mech. Sci. Technol., vol. 24, no. 1, pp. 331-336, Jan. 2010.
[24] C. Y. Wang, “Large post-buckling of heavy tapered elastica cantilevers and its asymptotic analysis,” Arch. Mech., vol. 64, no. 2, pp. 207-220, Jan. 2012.
[25] K. Cai, D. Y. Gao, Q. H. Qin, “Postbuckling analysis of a nonlinear beam with axial functionally graded material,” J. Eng. Math., vol. 88, no. 1, pp. 121-136, Oct. 2014.
[26] T. Huang, D. W. Dareing, “Buckling and lateral vibration of drill pipe,” J. Eng. Ind., vol. 90, no. 4, pp. 613-619, Nov. 1968.
[27] T. Huang, D. W. Dareing, “Buckling and frequencies of long vertical pipes,” J. Eng. Mech. Div., vol. 95, no. 1, pp. 167-182, 1969.
[28] A. Lubinski, “A study of the buckling of rotary drilling stings,” API Drilling and Production Practice., pp. 178-214, 1950.
[29] T. Kokkinis, M. M. Bernitsas, “Post-buckling analysis of heavy columns with application to marine risers,” SNAME J. Ship Res., vol. 29, no. 3, pp. 162-169, Sep. 1985.
[30] M. A. Vaz, M. H. Patel, “Analysis of drill strings in vertical and deviated holes using the Galerkin technique,” Eng. Struct. vol. 17, no. 6, pp. 437-442, July 1995.
[31] M. A. Vaz, M. H. Patel, “Initial post-buckling of submerged slender vertical structures subjected to distributed axial tension,” Appl. Ocean Res. vol. 20, no. 6, pp. 325-335, Dec. 1998.
[32] H. L. Langhaar, Energy methods in applied mechanics. New York: John Wiley & Sons, 1962.
[1] A. E. H. Love, Treatise on the mathematical theory of elasticity. Cambridge University Press: London, 1944.
[2] S. P. Timoshenko, J. N. Goodier, Theory of elasticity. McGraw-Hill: Singapore, 1982.
[3] S. S. Antman, Nonlinear problems of elasticity. Applied Mathematical Sciences 107. Springer Verlag. New York. USA, 1991.
[4] C. M. Wang, C. Y. Wang, J.N. Reddy, Exact solutions for buckling of structural members. CRC Press: boca Raton. Florida, 2005.
[5] D. Bigoni, Extremely deformable structures. Ed. Springer: Wien-New York, 2015.
[6] R. Haftka, W. Nachbar, “Post-buckling analysis of an elastically-restrained column,” Int. J. Solids Stuct., vol. 6, no. 11, pp. 1433-1449, Nov. 1970.
[7] C. M. Wang, K. K. Ang “Buckling capacities of braced heavy columns under an axial load,” Comput. Struc., vol. 28, no. 5, pp. 563-571, 1988.
[8] Y. H. Chai, C. M. Wang, “An application of differential transformation to stability analysis of heavy columns,” Int. J. Struc. Stab. Dyn., vol. 6, no. 3, pp. 317-332, 2006.
[9] S. B. Coskun, M. T. Atay, “Determination of critical buckling load for elastic columns of constant and variable cross-sections using variational iteration method,” Comput. Math. Appl., vol. 58, no. 11-12, pp. 2260-2266, Dec. 2009.
[10] D. J. Wei, S. X. Yan, Z. P. Zhang, X. F. Li, “Critical load for buckling of non-prismatic columns under self-weight and tip force,” Mech. Res. Commun., vol. 37, no. 6, pp. 554-558, Sep. 2010.
[11] Y. Huang, Q. Z. Luo, “A simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint,” Comput. Math. Appl., vol. 61, no. 9, pp. 2510-2517, May 2011.
[12] W. H. Duan, C. M. Wang, “Exact solution for buckling of columns including self-weight,” J. Eng. Mech., vol. 134, no. 1, pp. 116-119, Jan. 2008.
[13] B. Han, F. Li, C. Ni, Q. Zhang, C. Chen, T. Lu, “Stability and initial post-buckling of a standing sandwich beam under terminal force and self-weight,” Arch. Appl. Mech., vol. 86, no. 6, pp. 1063-1082, June 2016.
[14] C. Y. Wang, “Post-buckling of a clamped-simply supported elastic,” Int. J. Non-Linear Mech., vol. 32, no. 6, pp. 1115-1122, Nov. 1997.
[15] M. A. Vaz, D. F. C. Silva, “Post-buckling analysis of slender elastic rods subjected to terminal forces,” Int. J. Non-Linear Mech., vol. 38, no. 4, pp. 483-492, June 2003.
[16] L. N. Virgin, R. H. Plaut, “Postbuckling and vibration of linearly elastic and softening columns under self-weight,” Int. J. Solids Struct., vol. 41, no. 18-19, pp. 4989-5001, Sep. 2004.
[17] M. A. Vaz, G. H. W. Mascaro, “Post-buckling analysis of slender elastic vertical rods subjected to terminal forces and self-weight,” Int. J. Non-Linear Mech., vol. 40, no. 7, pp. 1049-1056, Sep. 2005.
[18] S. Li, Y. Zhou, “Post-buckling of a hinged-fixed beam under uniformly distributed follower forces,” Mech. Res. Commun., vol. 32, no. 4, pp. 359-367, July-Aug. 2005.
[19] J. Liu, Y. Mei, X. Dong “Post-buckling behavior of a double-hinged rod under self-weight,” Acta Mech. Solida Sinica., vol. 26, no. 2, pp. 197-204, Apr. 2013.
[20] B. K. Lee, S. J. Oh, “Elastica and buckling of simple tapered columns with constant volume,” Int. J. Non-Linear Mech., vol. 37, no. 18, pp. 2507-2518, May. 2000.
[21] K. Lee, “Post-buckling of uniform cantilever column under a combined load,” Int. J. Non-Linear Mech., vol. 36, no. 5, pp. 813-816, July 2001.
[22] V. V. Kuznetsov, S. V. Levyakov “Complete solution of the stability problem for elastica of Euler’s column.” Int. J. Non-Linear Mech., vol. 37, no.6, pp. 1003-1009, Sep. 2002.
[23] O. Sepahi, M. R. Forouzan, P. Malekzadeh “Differential quadrature application in post-buckling analysis of a hinged-fixed elastica under terminal forces and self-weight,” J. Mech. Sci. Technol., vol. 24, no. 1, pp. 331-336, Jan. 2010.
[24] C. Y. Wang, “Large post-buckling of heavy tapered elastica cantilevers and its asymptotic analysis,” Arch. Mech., vol. 64, no. 2, pp. 207-220, Jan. 2012.
[25] K. Cai, D. Y. Gao, Q. H. Qin, “Postbuckling analysis of a nonlinear beam with axial functionally graded material,” J. Eng. Math., vol. 88, no. 1, pp. 121-136, Oct. 2014.
[26] T. Huang, D. W. Dareing, “Buckling and lateral vibration of drill pipe,” J. Eng. Ind., vol. 90, no. 4, pp. 613-619, Nov. 1968.
[27] T. Huang, D. W. Dareing, “Buckling and frequencies of long vertical pipes,” J. Eng. Mech. Div., vol. 95, no. 1, pp. 167-182, 1969.
[28] A. Lubinski, “A study of the buckling of rotary drilling stings,” API Drilling and Production Practice., pp. 178-214, 1950.
[29] T. Kokkinis, M. M. Bernitsas, “Post-buckling analysis of heavy columns with application to marine risers,” SNAME J. Ship Res., vol. 29, no. 3, pp. 162-169, Sep. 1985.
[30] M. A. Vaz, M. H. Patel, “Analysis of drill strings in vertical and deviated holes using the Galerkin technique,” Eng. Struct. vol. 17, no. 6, pp. 437-442, July 1995.
[31] M. A. Vaz, M. H. Patel, “Initial post-buckling of submerged slender vertical structures subjected to distributed axial tension,” Appl. Ocean Res. vol. 20, no. 6, pp. 325-335, Dec. 1998.
[32] H. L. Langhaar, Energy methods in applied mechanics. New York: John Wiley & Sons, 1962.
@article{"International Journal of Architectural, Civil and Construction Sciences:79290", author = "C. Juntarasaid and T. Pulngern and S. Chucheepsakul", title = "Postbuckling Analysis of End Supported Rods under Self-Weight Using Intrinsic Coordinate Finite Elements", abstract = "A formulation of postbuckling analysis of end supported rods under self-weight has been presented by the variational method. The variational formulation involving the strain energy due to bending and the potential energy of the self-weight, are expressed in terms of the intrinsic coordinates. The variational formulation is accomplished by introducing the Lagrange multiplier technique to impose the boundary conditions. The finite element method is used to derive a system of nonlinear equations resulting from the stationary of the total potential energy and then Newton-Raphson iterative procedure is applied to solve this system of equations. The numerical results demonstrate the postbluckled configurations of end supported rods under self-weight. This finite element method based on variational formulation expressed in term of intrinsic coordinate is highly recommended for postbuckling analysis of end-supported rods under self-weight.
", keywords = "Variational method, postbuckling, finite element method, intrinsic coordinate.", volume = "13", number = "11", pages = "693-5", }
