ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD
Abstract
In this investigation, we extend our search on the dynamic buckling loads of some elastic structures to that of a clamped column lying on a nonlinear (cubic) elastic foundation but impacted upon axially by a step load. In order to ensure a uniformly valid solution, we employ multi–scaling two–timing regular perturbation procedures in asymptotic expansions of the variables. It is shown that (a) clamped columns buckle at higher buckling loads than columns with simply–supported ends irrespective of whether the columns are loaded statically or dynamically and whether damped or undamped, (b) Specifically, the inequalities satisfied by the static buckling load and dynamic buckling load in the clamped case are respectively given as and as against and for simply–supported end conditions, (c) At low values of the static buckling load , there is no appreciable change in the values of the dynamic buckling load but at higher values of , increases sharply with increased static buckling load . However, the increase seems to decrease with increased damping. We are able to mathematically relate the dynamic buckling load to the static buckling load and thereby by–passing the labour of repeating the entire process for different imperfection parameters. Thus, given either or , we can predict either value without the actual knowledge of the size of the small imperfection parameter.
References
[2] Thompson, J. M. T. and Hunt, G. W.,A general theory of elastic stability, John Wiley and Sons Ltd., London. 1973.
[3] Hu, N. and Burguen ̃o, R.,Elastic postbuckling response of axially – loaded cylindrical shells with seeded geometric imperfection design; Thin – walled structures, 96, 256 – 268, 2015.
[4] Amazigo, J. C., Budiansky, B. and Carrier, G. F.,Asymptotic Analyses of the buckling of imperfect columns on nonlinear elastic foundations, Int. J. Solids Structures, 10, 1342 – 1356, 1970.
[5] Qiang, H., Zhang, S. and Yang, G.,The Asymptotic solution of a dynamic buckling problem in elastic column, Appl. Maths. and Mech. (English edition)20(8), 867 – 872, 1999.
[6] Amazigo, J. C. and Frank, D.,Dynamic buckling of an imperfect column on a nonlinear foundation, Quart. Appl. Math., 31(6), 1 – 9, 1973.
[7] Artem, H. S. and Aydin, L.,Exact solution and dynamic buckling analysis of a beam – column loading. Appl. Math. and Mech. 31(10), 1317 – 1324, 2010.
[8] Ette, A. M., Chukwuchekwa, J. U. and Udo – Akpan, I. U.,On the buckling of a clamped viscously damped column trapped by a step load. Int. J. of Appl. Sciences and Mathematics, 3 (2), 117 – 123, 2016.
[9] Udo-Akpan, I.U. and Ette, A.M.,On the dynamic buckling of a model structure with quadratic nonlinearity struck by a step load superposed on a quasi – static load, J. of the Nigerian Assoc. of Math. Physics, 35, 461-472, 2016.
[10] Ette, A.M. and Udo-Akpan, I.U.,On the buckling of a pre-statically loaded but viscously damped quadratic model structure trapped by a step load, Global J. of Mathematics, 7 (2), 761 – 770, 2016.
[11] Chukwuchekwa, J. U. and Ette, A. M.,Asymptotic analysis of an improved quadratic model structures subjected to static loading. J. of Nigerian Assoc. of Math. Physics, 32, 237-244, 2015.
[12] Chukwuchekwa, J. U. and Ette, A. M.,Perturbation technique in the buckling of a viscously damped elastic model structure pre – statically loaded but struck by a step load, J. of Nigerian Assoc. of Math. Physics, 37 – 70, 2016.
[13] Kolakowski, Z.,Static and dynamic interactive buckling regarding axial extension mode of thin – walled channels, J. of Theoritical and Appl. Mech. 48 (3), 703 – 714, 2010.
[14] Kowal – Michalska, K.,About some important parameters in dynamic buckling analysis of plate structures subjected to pulse loading, J. of Mech. And Mechanical Engng., 14 (2), 259 – 279, 2010.
[15] Mcshane, G.J., Pingle, S. M., Deshpande, V.S. and Flock, N.A.,Dynamic buckling of inclined structures, Int. J. of Solids and Struct., 49, 2830 – 2838, 2012.
[16] Belyaev, A. K. and gLim, D. N.,Stability of transverse vibration of rod under longitudinal step – wise loading, J. of Physics, Conference Series, 451, 1 – 6, 2012.
[17] Priyadarsini, R.S., Kalyanaraman, V. and Srinvasan, S.M.,Numerical and experimental study of buckling of advanced fibre composite cylinder under axial compression, Int. J. of Structural stability and Dynamics, 12 (4), 1 – 24, 2012.
[18] Budiansky, B. and Hutchinson, J.W.,Dynamic buckling of imperfection – sensitive structures,Proceedings of XI International Congr. Applied Mechanics. Springer – Verlag, Berlin. 1966.
[19] Ette, A. M. and Chukwuchekwa, J. U., On the static buckling of an externally pressurized finite circular cylindrical shell, J. of Nigerian Assoc. of Math. Physics, 11, 323 – 332, 2007.
[20] A. M. Ette, J. U. Chukwuchekwa, I. U. Udo-Akpan and Onuoha, N.O. (2020); A Multi-Timing Perturbation Analysis of the deformation and Dynamic Buckling of a Viscously Damped Toroidal Shell Segment Stressed by a Step Load, IOSR Journal of Mathematics (IOSR-JM), 16(4), 40-59.
[21] A. M. Ette, J. U. Chukwuchekwa, N.O. Onuoha, and I. U. Udo-Akpan (2020); On the Deformation and Static Buckling of a Toroidal Shell Segment using a two-term Fourier series imperfections, International Journal of Mathematics Trends and Technology (IJMTT), 66(8), 100-114
[22] G. Ozoigbo, A.M. Ette, J. Chukwuchekwa, W. Osuji, I.U. Udo-Akpan (2022); On the Analysis of a Pre-statically Loaded Nonlinear Cubic structure Pressurized by an Explicitly Time Dependent SlowlyVarying Load, American Journal of Mechanics and applications, 10(1), 1-15
[23] G.E. Ozoigbo, A.M. Ette, J.U. Chukwuchekwa, W.I. Osuji, I.U. Udo-Akpan (2023); Nonlinear Analytical Investigation of Dynamic Buckling of Spherical Shell Trapped Under a Periodic load, Mechanics of Solids, Springer Link, 58, 202-215.
[24] Obong, H.P. and UdoAkpan, I.U. (2023); A Review of Tensor Interaction in the Theory of Potential Models, Scientia Africana, 22(3), 11-22.
[25] O.G. Udoaka, U.J. Etim and I.U. UdoAkpan, (2024) Efficient Solution of Nonhomogeneous Linear Differential Equations with Constant coefficients: the method of undetermined coefficient approach, IEEE-SEM, 12(1), 21-36
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