Abstract
This paper is devoted to the analytical modelling of a sandwich beam. Three models of this beam are elaborated. Two nonlinear individual shear theories of deformation of a plane cross-sections are proposed. Based on Hamilton’s principle, two differential equations of motion for each model are obtained. The bending, buckling and free flexural vibration problems of the simply-supported sandwich beam considering these three models are studied. The results of these analytical investigations are presented in tables.Keywords:
shear deformation theory, deflection, critical load, fundamental natural frequency.References
1. Plantema F.J., Sandwich Construction: The Bending and Buckling of Sandwich Beams, Plates, and Shells, John Wiley & Sons, Inc., New York, London, Sydney, 1966.
2. Allen H.G., Analysis and Design of Structural Sandwich Panels, Pergamon Press, Oxford, London, Edinburgh, New York, Toronto, Sydney, Paris, Braunschweig, 1969.
3. Noor A.K, Burton W.S., Bert C.W., Computational models for sandwich panels and shells, Applied Mechanics Reviews, 49(3): 155–199, 1996, https://doi.org/10.1115/1.3101923
4. Frostig Y., Buckling of sandwich panels with a flexible cores – high-order theory, International Journal of Solids and Structures, 35(3–4): 183–204, 1998, https://doi.org/10.1016/S0020-7683%2897%2900078-4
5. Vinson J.R., Sandwich structures, Applied Mechanics Reviews, 54(3): 201–214, 2001, https://doi.org/10.1115/1.3097295
6. Icardi U., Applications of Zig-Zag theories to sandwich beams, Mechanics of Advanced Materials and Structures, 10(1): 77–97, 2003, https://doi.org/10.1080/15376490306737
7. Steeves C.A., Fleck N.A., Collapse mechanisms of sandwich beams with composite faces and a foam core, loaded in three-point bending. Part 1: analytical models and minimum weight design, International Journal of Mechanical Sciences, 46(4): 561–583, 2004, https://doi.org/10.1016/j.ijmecsci.2004.04.003
8. Yang M., Qiao P., Higher-order impact modeling of sandwich structures with flexible core, International Journal of Solids and Structures, 42(20): 5460–5490, 2005, https://doi.org/10.1016/j.ijsolstr.2005.02.037
9. Magnucka-Blandzi E., Magnucki K., Effective design of a sandwich beam with a metal foam core, Thin-Walled Structures, 45(4): 432–438, 2007, https://doi.org/10.1016/j.tws.2007.03.005
10. Carrera E., Brischetto S., A survey with numerical assessment of classical and refined theories for the analysis of sandwich plates, Applied Mechanics Reviews, 62(1): 010803, 2009, https://doi.org/10.1115/1.3013824
11. Kreja I., A literature review on computational models for laminated composite and sandwich panels, Central European Journal of Engineering, 1(1): 59–80, 2011, https://doi.org/10.2478/s13531-011-0005-x
12. Magnucka-Blandzi E., Dynamic stability and static stress state of a sandwich beam with a metal foam core using three modified Timoshenko hypothesis, Mechanics of Advanced Materials and Structures, 18(2): 147–158, 2011, https://doi.org/10.1080/15376494.2010.496065
13. Magnucka-Blandzi E., Mathematical modelling of a rectangular sandwich plate with a metal foam cores, Journal of Theoretical and Applied Mechanics, 49(2): 439–455, 2011.
14. Baba B.O., Free vibration analysis of curved sandwich beams with face/core debond using theory and experiment, Mechanics of Advanced Materials and Structures, 19(5): 350–359, 2012, https://doi.org/10.1080/15376494.2010.528163
15. Phan C.N., Frostig Y., Kardomateas G.A., Analysis of sandwich beams with a compliant core and with in-plane rigidity–extended high-order sandwich panel theory versus elasticity, ASME: Journal of Applied Mechanics, 79(4): 041001–1-11, 2012, https://doi.org/10.1115/1.4005550
16. Magnucki K., Jasion P., Szyc W., Smyczynski M., Strength and buckling of a sandwich beam with thin binding layers between faces and a metal foam core, Steel and Composite Structures, 16(3): 325–337, 2014, https://doi.org/10.12989/scs.2014.16.3.325
17. Sayyad A.S., Ghugal Y.M., Bending, buckling and free vibration of laminated composite and sandwich beams: a critical review of literature, Composite Structures, 171: 486–504, 2017, https://doi.org/10.1016/j.compstruct.2017.03.053
18. Magnucka-Blandzi E., Bending and buckling of a metal seven-layer beam with crosswise corrugated main core – Comparative analysis with sandwich beam, Composite Structures, 183: 35–41, 2018, https://doi.org/10.1016/j.compstruct.2016.11.089
19. Czechowski L., Jankowski J., Kotełko M., Jankowski M., Experimental and numerical three-point bending test for sandwich beams, Journal of KONES Powertrain and Transport, 24(3): 53–62, 2017, https://doi.org/10.5604/01.3001.0010.3071
20. Birman V., Kardomateas G.A., Review of current trends in research and applications of sandwich structures, Composites Part B: Engineering, 142: 221–240, 2018, https://doi.org/10.1016/j.compositesb.2018.01.027
21. Sayyad A.S., Ghugal Y.M., Modeling and analysis of functionally graded sandwich beams: A review, Mechanics of Advanced Materials and Structures, 26(21): 1776–1795, 2019, https://doi.org/10.1080/15376494.2018.1447178
22. Zhen W., Yang C., Zhang H., Zheng X., Stability of laminated composite and sandwich beams by a Reddy-type higher-order zig-zag theory, Mechanics of Advanced Materials and Structures, 26(19): 1622–1635, 2019, https://doi.org/10.1080/15376494.2018.1444228
23. Magnucki K., Bending of symmetrically sandwich beams and I-beams – analytical study, International Journal of Mechanical Sciences, 150: 411–419, 2019, https://doi.org/10.1016/j.ijmecsci.2018.10.020
24. Magnucki K., Magnucka-Blandzi E., Generalization of a sandwich structure model: Analytical studies of bending and buckling problems of rectangular plates, Composite Structures, 255: 112944, 2021, 112944, https://doi.org/10.1016/j.compstruct.2020.112944
2. Allen H.G., Analysis and Design of Structural Sandwich Panels, Pergamon Press, Oxford, London, Edinburgh, New York, Toronto, Sydney, Paris, Braunschweig, 1969.
3. Noor A.K, Burton W.S., Bert C.W., Computational models for sandwich panels and shells, Applied Mechanics Reviews, 49(3): 155–199, 1996, https://doi.org/10.1115/1.3101923
4. Frostig Y., Buckling of sandwich panels with a flexible cores – high-order theory, International Journal of Solids and Structures, 35(3–4): 183–204, 1998, https://doi.org/10.1016/S0020-7683%2897%2900078-4
5. Vinson J.R., Sandwich structures, Applied Mechanics Reviews, 54(3): 201–214, 2001, https://doi.org/10.1115/1.3097295
6. Icardi U., Applications of Zig-Zag theories to sandwich beams, Mechanics of Advanced Materials and Structures, 10(1): 77–97, 2003, https://doi.org/10.1080/15376490306737
7. Steeves C.A., Fleck N.A., Collapse mechanisms of sandwich beams with composite faces and a foam core, loaded in three-point bending. Part 1: analytical models and minimum weight design, International Journal of Mechanical Sciences, 46(4): 561–583, 2004, https://doi.org/10.1016/j.ijmecsci.2004.04.003
8. Yang M., Qiao P., Higher-order impact modeling of sandwich structures with flexible core, International Journal of Solids and Structures, 42(20): 5460–5490, 2005, https://doi.org/10.1016/j.ijsolstr.2005.02.037
9. Magnucka-Blandzi E., Magnucki K., Effective design of a sandwich beam with a metal foam core, Thin-Walled Structures, 45(4): 432–438, 2007, https://doi.org/10.1016/j.tws.2007.03.005
10. Carrera E., Brischetto S., A survey with numerical assessment of classical and refined theories for the analysis of sandwich plates, Applied Mechanics Reviews, 62(1): 010803, 2009, https://doi.org/10.1115/1.3013824
11. Kreja I., A literature review on computational models for laminated composite and sandwich panels, Central European Journal of Engineering, 1(1): 59–80, 2011, https://doi.org/10.2478/s13531-011-0005-x
12. Magnucka-Blandzi E., Dynamic stability and static stress state of a sandwich beam with a metal foam core using three modified Timoshenko hypothesis, Mechanics of Advanced Materials and Structures, 18(2): 147–158, 2011, https://doi.org/10.1080/15376494.2010.496065
13. Magnucka-Blandzi E., Mathematical modelling of a rectangular sandwich plate with a metal foam cores, Journal of Theoretical and Applied Mechanics, 49(2): 439–455, 2011.
14. Baba B.O., Free vibration analysis of curved sandwich beams with face/core debond using theory and experiment, Mechanics of Advanced Materials and Structures, 19(5): 350–359, 2012, https://doi.org/10.1080/15376494.2010.528163
15. Phan C.N., Frostig Y., Kardomateas G.A., Analysis of sandwich beams with a compliant core and with in-plane rigidity–extended high-order sandwich panel theory versus elasticity, ASME: Journal of Applied Mechanics, 79(4): 041001–1-11, 2012, https://doi.org/10.1115/1.4005550
16. Magnucki K., Jasion P., Szyc W., Smyczynski M., Strength and buckling of a sandwich beam with thin binding layers between faces and a metal foam core, Steel and Composite Structures, 16(3): 325–337, 2014, https://doi.org/10.12989/scs.2014.16.3.325
17. Sayyad A.S., Ghugal Y.M., Bending, buckling and free vibration of laminated composite and sandwich beams: a critical review of literature, Composite Structures, 171: 486–504, 2017, https://doi.org/10.1016/j.compstruct.2017.03.053
18. Magnucka-Blandzi E., Bending and buckling of a metal seven-layer beam with crosswise corrugated main core – Comparative analysis with sandwich beam, Composite Structures, 183: 35–41, 2018, https://doi.org/10.1016/j.compstruct.2016.11.089
19. Czechowski L., Jankowski J., Kotełko M., Jankowski M., Experimental and numerical three-point bending test for sandwich beams, Journal of KONES Powertrain and Transport, 24(3): 53–62, 2017, https://doi.org/10.5604/01.3001.0010.3071
20. Birman V., Kardomateas G.A., Review of current trends in research and applications of sandwich structures, Composites Part B: Engineering, 142: 221–240, 2018, https://doi.org/10.1016/j.compositesb.2018.01.027
21. Sayyad A.S., Ghugal Y.M., Modeling and analysis of functionally graded sandwich beams: A review, Mechanics of Advanced Materials and Structures, 26(21): 1776–1795, 2019, https://doi.org/10.1080/15376494.2018.1447178
22. Zhen W., Yang C., Zhang H., Zheng X., Stability of laminated composite and sandwich beams by a Reddy-type higher-order zig-zag theory, Mechanics of Advanced Materials and Structures, 26(19): 1622–1635, 2019, https://doi.org/10.1080/15376494.2018.1444228
23. Magnucki K., Bending of symmetrically sandwich beams and I-beams – analytical study, International Journal of Mechanical Sciences, 150: 411–419, 2019, https://doi.org/10.1016/j.ijmecsci.2018.10.020
24. Magnucki K., Magnucka-Blandzi E., Generalization of a sandwich structure model: Analytical studies of bending and buckling problems of rectangular plates, Composite Structures, 255: 112944, 2021, 112944, https://doi.org/10.1016/j.compstruct.2020.112944
