Mathematical modeling for dynamic stability of sandwich beam with variable mechanical properties of core. (English) Zbl 1367.74021
Summary: The paper is devoted to mathematical modelling of static and dynamic stability of a simply supported three-layered beam with a metal foam core. Mechanical properties of the core vary along the vertical direction. The field of displacements is formulated using the classical broken line hypothesis and the proposed nonlinear hypothesis that generalizes the classical one. Using both hypotheses, the strains are determined as well as the stresses of each layer. The kinetic energy, the elastic strain energy, and the work of load are also determined. The system of equations of motion is derived using Hamilton’s principle. Finally, the system of three equations is reduced to one equation of motion, in particular, the Mathieu equation. The Bubnov-Galerkin method is used to solve the system of equations of motion, and the Runge-Kutta method is used to solve the second-order differential equation. Numerical calculations are done for the chosen family of beams. The critical loads, unstable regions, angular
frequencies of the beam, and the static and dynamic equilibrium paths are calculated analytically and verified numerically. The results of this study are presented in the forms of figures and tables.
MSC:
| 74H55 | Stability of dynamical problems in solid mechanics |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 70K20 | Stability for nonlinear problems in mechanics |
Keywords:
mathematical modelling; dynamic stability; metal foam core with variable mechanical property; static and dynamic equilibrium path; angular frequencyReferences:
| [1] | Libove, C. and Butdorf, S. B. A General Small-Deflection Theory for Flat Sandwich Plates, NASA TN 1526, NASA, Washington, D. C. (1948) |
| [2] | Plantema, F. J. Sandwich Construction: the Bending and Buckling of Sandwich Beams, Plates and Shells, John Wiley and Sons, New York (1966) |
| [3] | Reissner, E. Finite deflections of sandwich plates. Journal of the Aeronautical Science,15(7), 435-440 (1948) · doi:10.2514/8.11610 |
| [4] | Banhart, J. Manufacture, characterization and application of cellular metals and metal foams. Progress in Material Science,46, 559-632 (2001) · doi:10.1016/S0079-6425(00)00002-5 |
| [5] | Magnucki, K. and Szyc, W. Strength and Stability of Sandwich Beams and Plates with Aluminum Foam Core (in Polish), Poznan University of Technology Publishing House, Poznaó (2012) |
| [6] | Magnucki, K., Smyczyóski, M., and Jasion, P. Deflection and strength of a sandwich beam with thin binding layers between faces and a core. Archives of Mechanics,65(4), 301-311 (2013) · Zbl 1291.74124 |
| [7] | Dębowski, D., Magnucki, K., and Malinowski, M. Dynamic stability of a metal foam rectangular plate. Steel Composite Structures,10(2), 151-168 (2010) · doi:10.12989/scs.2010.10.2.151 |
| [8] | Cao, C. Y. and Zhong, Y. Dynamic response of a beam on a Pasternak foundation and under a moving load. Journal of Chongqing University (English Edition),7(4), 311-316 (2008) |
| [9] | Pradhan, M., Dash, P. R., and Pradhan, P. K. Static and dynamic stability analysis of an asymmetric sandwich beam resting on a variable Pasternak foundation subjected to thermal gradient. Meccanica,51, 725-739 (2015) · Zbl 1339.74014 · doi:10.1007/s11012-015-0229-6 |
| [10] | Douville, M. A. and Le Grognec, P. Exact analysis solutions for the local and global buckling of sandwich beam-columns under various loadings. International Journal of Solids and Structures,50, 2597-2609 (2013) · Zbl 1299.68079 · doi:10.1016/j.ijsolstr.2013.04.013 |
| [11] | Huang, H. and Kardomateas, G. A. Buckling and initial post buckling behavior of sandwich beams including transverse shear. AIAA Journal,40, 2331-2335 (2002) · doi:10.2514/2.1571 |
| [12] | Jasion, P., Magnucka-Blandzi, E., Szyc, W., and Magnucki, K. Global and local buckling of sandwich circular and beam-rectangular plates with metal foam core. Thin-Walled Structures,61, 154-161 (2012) · doi:10.1016/j.tws.2012.04.013 |
| [13] | Jasion, P., Magnucki, K., and Wasilewicz, P. Global Buckling of Sandwich Beam-Column with Physically Nonlinear Core, Stability of structures XIIIth symposium, Department of Strength of Materials and Structures, Technical University of Lodz, Lodź, 301-308 (2012) |
| [14] | Jasion, P. and Magnucki, K. Global buckling of sandwich column with metal foam core. Journal of Sandwich Structures and Materials,15(6), 718-732 (2013) · doi:10.1177/1099636213499339 |
| [15] | Javidinejad, A. Buckling of beams and columns under combined axial and horizontal loading with various axial load in application locations. Journal of Theoretical and Applied Mechanics,42(4), 19-30 (2012) · Zbl 1330.74067 |
| [16] | Magnucki, K. and Stasiewicz, P. Elastic buckling of a porous beam. Journal of Theoretical and Applied Mechanics,42(4), 859-868 (2004) · Zbl 1195.74097 |
| [17] | Magnucki, K. Strength and buckling of sandwich beams-columns (in Polish). Modelowanie Inźynierskie,42, 249-258 (2011) |
| [18] | Grygorowicz, M., Magnucki, K., and Malinowski, M. Elastic buckling of a sandwich beam with variable mechanical properties of the core. Thin-Walled Structures,87, 127-132 (2015) · doi:10.1016/j.tws.2014.11.014 |
| [19] | Grygorowicz, M. Analytical and FEM studies on buckling of sandwich beams. Proceedings of 23rd Australasian Conference on the Mechanics of Structures and Materials (ACMSM23) (ed., Smith, S. T.), Byron Bay, Australia (2014) |
| [20] | Rivallant, S., Ferrero, J. F., and Barrau, J. J. Dynamic buckling of foam stabilized composite skin. Composite Structures,72, 486-493 (2006) · doi:10.1016/j.compstruct.2005.01.032 |
| [21] | Tagarielli, V. L., Deshpande, V. S., and Fleck, N. A. The dynamic response of composite sandwich beams to transverse impact. International Journal of Solid and Structures,44, 2442-2457 (2007) · doi:10.1016/j.ijsolstr.2006.07.015 |
| [22] | Wang, Z., Jing, L., Ning, J., and Zhao, L. The structural response of clamped sandwich beams subjected to impact loading. Composite Structures,93, 1300-1308 (2011) · doi:10.1016/j.compstruct.2010.05.011 |
| [23] | Tan, Z. H., Luo, H. H., Long, W. G., and Han, X. Dynamic response of clamped sandwich beam with aluminum alloy foam core subjected to impact loading. Composites: Part B,46, 39-45 (2013) · doi:10.1016/j.compositesb.2012.10.044 |
| [24] | Manalo, A. C. Behaviour of fibre composite sandwich structures under short and asymmetrical beam shear tests. Composite Structures,99, 339-349 (2013) · doi:10.1016/j.compstruct.2012.12.010 |
| [25] | Grygorowicz, M. and Magnucka-Blandzi, E. Stability of sandwich beams with variable properties of the core with dynamic loads. Proceedings of BIT’s 3rd Annual World Congress of Advanced Materials-2014, World Science Press, Chongqing (2014) · Zbl 1367.74021 |
| [26] | Aly, M. F., Hamza, K. T., and Farag, M. M. A materials selection procedure for sandwiched beams via parametric optimization with applications in automotive industry. Materials and Design, 56, 219-226 (2014) · doi:10.1016/j.matdes.2013.10.075 |
| [27] | Ashjari, M. and Khoshravan, M. R. Mass optimization of functionally graded plate for mechanical loading in the presence of deflection and stress constraints. Composite Structures,110, 118-132 (2014) · doi:10.1016/j.compstruct.2013.11.025 |
| [28] | Brighenti, R. Smart behaviour of layered plates through the use of auxetic materials. Thin-Walled Structures,84, 432-442 (2014) · doi:10.1016/j.tws.2014.07.017 |
| [29] | Zingoni, A. Group-theoretic insights on the vibration of symmetric structures in engineering. Philosophical Transaction of Royal Society A,372, 1-24 (2014) · Zbl 1353.74037 |
| [30] | Hadji, L., Atmane, H. A., Tounsi, A., Mechab, I., and Adda Bedia, E. A. Free vibration of functionally graded sandwich plates using four-variable refined plate theory. Applied Mathematics and Mechanics (English Edition),32(7), 925-942 (2011) DOI 10.1007/s10483-011-1470-9 · Zbl 1237.76061 · doi:10.1007/s10483-011-1470-9 |
| [31] | Magnucka-Blandzi, E. Dynamic stability and static stress state of a sandwich beam with a metal foam core using three modified timoshenko hypotheses. Mechanics of Advanced Materials and Structures,18(2), 147-158 (2011) · doi:10.1080/15376494.2010.496065 |
| [32] | Życzkowski, M. Technical Mechanics, Strength of Construction Elements (in Polish), Publishing PWN, Warszawa (1988) |
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