Abstract
This study investigates the effects of fluid–structure and soil–structure interaction on the free vibration response of functionally graded sandwich plates. To this aim, an exemplary problem is analyzed, whereas a metal/ceramic sandwich plate is placed at the bottom of a tank filled in with fluid. Two cases are considered: (i) soft core, i.e., a sandwich plate with metal core and ceramic skins, and (ii) hard core, i.e., a sandwich plate with ceramic core and metal skins. In both cases, the skins are modelled as suitable functionally graded materials (FGMs). The soil is modelled as a Pasternak foundation. The free vibration analysis is carried out according to the extended higher order sandwich plate theory (EHSAPT). The fluid is assumed to be inviscid, incompressible, and irrotational. Hamilton’s principle is exploited to deduce the governing equations and the corresponding boundary conditions. The Rayleigh–Ritz method with two-variable orthogonal polynomials is used to compute the natural frequencies of the sandwich plate. The adopted approach is first validated through comparison with results published in the literature. Then, the effects are studied of several parameters on the dynamic response of the system.
Similar content being viewed by others
References
Sharma N, Mahapatra TR, Panda SK, Katariya P (2020) Thermo-acoustic analysis of higher-order shear deformable laminated composite sandwich flat panel. J Sandw Struct Mater 22(5):1357–1385
Katariya PV, Panda SK, Mahapatra TR (2019) Prediction of nonlinear eigenfrequency of laminated curved sandwich structure using higher-order equivalent single-layer theory. J Sandw Struct Mater 21(8):2846–2869
Katariya PV, Panda SK (2019) Numerical frequency analysis of skew sandwich layered composite shell structures under thermal environment including shear deformation effects. Struct Eng Mech 71(6):657–668
Katariya PV, Panda SK, Mahapatra TR (2018) Bending and vibration analysis of skew sandwich plate. Aircr Eng Aerosp Technol 90:885–895
Katariya PV, Panda SK (2020) Numerical analysis of thermal post-buckling strength of laminated skew sandwich composite shell panel structure including stretching effect. Steel Compos Struct 34(2):279–288
Katariya PV, Panda SK (2019) Numerical evaluation of transient deflection and frequency responses of sandwich shell structure using higher order theory and different mechanical loadings. Eng Comput 35(3):1009–1026
Katariya PV, Panda SK (2019) Frequency and deflection responses of shear deformable skew sandwich curved shell panel: a finite element approach. Arab J Sci Eng 44(2):1631–1648
Singha TD, Rout M, Bandyopadhyay T, Karmakar A (2020) Free vibration analysis of rotating pretwisted composite sandwich conical shells with multiple debonding in hygrothermal environment. Eng Struct 204:110058
Singh S, Harsha S (2020) Thermo-mechanical analysis of porous sandwich S-FGM plate for different boundary conditions using Galerkin Vlasov’s method: a semi-analytical approach. Thin-Walled Struct 150:106668
Miyamoto Y, Kaysser W, Rabin B, Kawasaki A, Ford RG (2013) Functionally graded materials: design, processing and applications. Springer, Berlin
Udupa G, Rao SS, Gangadharan K (2014) Functionally graded composite materials: an overview. Proced Mater Sci 5:1291–1299
Eltaher MA, Alshorbagy AE, Mahmoud F (2013) Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams. Compos Struct 99:193–201
Ramteke PM, Mehar K, Sharma N, Panda S (2020) Numerical prediction of deflection and stress responses of functionally graded structure for grading patterns (power-law, sigmoid and exponential) and variable porosity (even/uneven). Sci Iran. https://doi.org/10.24200/sci.2020.55581.4290
Ramteke PM, Patel B, Panda SK (2020) Time-dependent deflection responses of porous FGM structure including pattern and porosity. Int J Appl Mech 12:2050102
Akbaş Ş, Fageehi Y, Assie A, Eltaher M (2020) Dynamic analysis of viscoelastic functionally graded porous thick beams under pulse load. Eng Comput. https://doi.org/10.1007/s00366-020-01070-3
Swaminathan K, Sangeetha D (2017) Thermal analysis of FGM plates: a critical review of various modeling techniques and solution methods. Compos Struct 160:43–60
Kanu NJ, Vates UK, Singh GK, Chavan S (2019) Fracture problems, vibration, buckling, and bending analyses of functionally graded materials: a state-of-the-art review including smart FGMS. Part Sci Technol 37(5):583–608
Abo-bakr H, Abo-bakr R, Mohamed S, Eltaher M (2021) Multi-objective shape optimization for axially functionally graded microbeams. Compos Struct 258:113370
Abo-Bakr RM, Eltaher MA, Attia MA (2020) Pull-in and freestanding instability of actuated functionally graded nanobeams including surface and stiffening effects. Eng Comput. https://doi.org/10.1007/s00366-020-01146-0
Akbaş ŞD, Bashiri AH, Assie AE, Eltaher MA (2020) Dynamic analysis of thick beams with functionally graded porous layers and viscoelastic support. J Vib Control. https://doi.org/10.1177/1077546320947302
Ramteke PM, Panda SK, Sharma N (2019) Effect of grading pattern and porosity on the eigen characteristics of porous functionally graded structure. Steel Compos Struct 33(6):865–875
Burlayenko VN, Sadowski T (2020) Free vibrations and static analysis of functionally graded sandwich plates with three-dimensional finite elements. Meccanica 55(4):815–832
Liu J, Hao C, Ye W, Yang F, Lin G (2021) Free vibration and transient dynamic response of functionally graded sandwich plates with power-law nonhomogeneity by the scaled boundary finite element method. Comput Methods Appl Mech Eng 376:113665
Eltaher M, Abdelrahman A, Al-Nabawy A, Khater M, Mansour A (2014) Vibration of nonlinear graduation of nano-Timoshenko beam considering the neutral axis position. Appl Math Comput 235:512–529
Katariya PV, Mehar K, Panda SK (2020) Nonlinear dynamic responses of layered skew sandwich composite structure and experimental validation. Int J Non-Linear Mech. https://doi.org/10.1016/j.ijnonlinmec.2020.103527
Katariya PV, Panda SK, Mehar K (2020) Theoretical modelling and experimental verification of modal responses of skewed laminated sandwich structure with epoxy-filled softcore. Eng Struct 228:111509
El Meiche N, Tounsi A, Ziane N, Mechab I (2011) A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate. Int J Mech Sci 53(4):237–247
Frostig Y, Baruch M, Vilnay O, Sheinman I (1992) High-order theory for sandwich-beam behavior with transversely flexible core. J Eng Mech 118(5):1026–1043
Frostig Y, Baruch M (1996) Localized load effects in high-order bending of sandwich panels with flexible core. J Eng Mech 122(11):1069–1076
Frostig Y, Thomsen OT (2004) High-order free vibration of sandwich panels with a flexible core. Int J Solids Struct 41(5–6):1697–1724
Frostig Y, Thomsen OT (2008) Non-linear thermal response of sandwich panels with a flexible core and temperature dependent mechanical properties. Compos B Eng 39(1):165–184
Khalili S, Mohammadi Y (2012) Free vibration analysis of sandwich plates with functionally graded face sheets and temperature-dependent material properties: a new approach. Eur J Mech -A/Solids 35:61–74
Karimi M, Khorshidi K, Dimitri R, Tornabene F (2020) Size-dependent hydroelastic vibration of FG microplates partially in contact with a fluid. Compos Struct 244:112320
Han D, Liu G, Abdallah S (2020) An Eulerian-Lagrangian-Lagrangian method for 2D fluid-structure interaction problem with a thin flexible structure immersed in fluids. Comput Struct 228:106179
Khorshidi K, Karimi M (2019) Analytical modeling for vibrating piezoelectric nanoplates in interaction with inviscid fluid using various modified plate theories. Ocean Eng 181:267–280
Omiddezyani S, Jafari-Talookolaei R-A, Abedi M, Afrasiab H (2018) The size-dependent free vibration analysis of a rectangular Mindlin microplate coupled with fluid. Ocean Eng 163:617–629
Ramian A, Jafari-Talookolaei R-A, Valvo PS, Abedi M (2020) Free vibration analysis of sandwich plates with compressible core in contact with fluid. Thin-Walled Struct 157:107088
Watts G, Pradyumna S, Singha M (2018) Free vibration analysis of non-rectangular plates in contact with bounded fluid using element free Galerkin method. Ocean Eng 160:438–448
Canales F, Mantari J (2017) Laminated composite plates in contact with a bounded fluid: free vibration analysis via unified formulation. Compos Struct 162:374–387
Canales F, Mantari J (2018) Discrepancy on the free vibration of laminated composite plates coupled to a compressible and incompressible fluid domain. Ocean Eng 167:267–281
Cheung Y, Zhou D (2000) Coupled vibratory characteristics of a rectangular container bottom plate. J Fluids Struct 14(3):339–357
Cheung Y, Zhou D (2002) Hydroelastic vibration of a circular container bottom plate using the Galerkin method. J Fluids Struct 16(4):561–580
Pecker A (2007) Soil Structure Interaction. In: Pecker A (ed) Advanced earthquake engineering analysis. Springer, Vienna, pp 33–42. https://doi.org/10.1007/978-3-211-74214-3_3
Malekzadeh P (2009) Three-dimensional free vibration analysis of thick functionally graded plates on elastic foundations. Compos Struct 89(3):367–373
Shahsavari D, Shahsavari M, Li L, Karami B (2018) A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation. Aerosp Sci Technol 72:134–149
Thai H-T, Park M, Choi D-H (2013) A simple refined theory for bending, buckling, and vibration of thick plates resting on elastic foundation. Int J Mech Sci 73:40–52
Hamed MA, Mohamed SA, Eltaher MA (2020) Buckling analysis of sandwich beam rested on elastic foundation and subjected to varying axial in-plane loads. Steel Compos Struct 34(1):75–89
Daikh AA, Houari MSA, Eltaher MA (2020) A novel nonlocal strain gradient Quasi-3D bending analysis of sigmoid functionally graded sandwich nanoplates. Compos Struct. https://doi.org/10.1016/j.compstruct.2020.113347
Mohamed N, Mohamed S, Eltaher M (2020) Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model. Eng Comput. https://doi.org/10.1007/s00366-020-00976-2
Chaduvula U, Patel D, Gopalakrishnan N (2013) Fluid-structure-soil interaction effects on seismic behaviour of elevated water tanks. Proced Eng 51:84–91
Kotrasová K, Harabinová S, Hegedüšová I, Kormaníková E, Panulinová E (2017) Numerical experiment of fluid-structure-soil interaction. Proced Eng 190:291–295
Hashemi SH, Karimi M, Taher HRD (2010) Vibration analysis of rectangular Mindlin plates on elastic foundations and vertically in contact with stationary fluid by the Ritz method. Ocean Eng 37(2–3):174–185
Sobhy M (2013) Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Compos Struct 99:76–87
Sobhy M (2016) An accurate shear deformation theory for vibration and buckling of FGM sandwich plates in hygrothermal environment. Int J Mech Sci 110:62–77
Ritz W (1909) Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. J für die reine und Angew Math (Crelles J) 1909(135):1–61
Aiello M, Ombres L (1999) Buckling and vibrations of unsymmetric laminates resting on elastic foundations under inplane and shear forces. Compos Struct 44(1):31–41
Abedi M, Jafari-Talookolaei R-A, Valvo PS (2016) A new solution method for free vibration analysis of rectangular laminated composite plates with general stacking sequences and edge restraints. Comput Struct 175:144–156
Bhat R (1987) Flexural vibration of polygonal plates using characteristic orthogonal polynomials in two variables. J Sound Vib 114(1):65–71
Liew K, Lam K, Chow S (1990) Free vibration analysis of rectangular plates using orthogonal plate function. Comput Struct 34(1):79–85
Liew K, Xiang Y, Kitipornchai S, Wang C (1993) Vibration of thick skew plates based on Mindlin shear deformation plate theory. J Sound Vib 168(1):39–69
Nallim LG, Martinez SO, Grossi RO (2005) Statical and dynamical behaviour of thin fibre reinforced composite laminates with different shapes. Comput Methods Appl Mech Eng 194(17):1797–1822
Kumar Y (2018) The Rayleigh-Ritz method for linear dynamic, static and buckling behavior of beams, shells and plates: a literature review. J Vib Control 24(7):1205–1227
Chakraverty S, Bhat R, Stiharu I (1999) Recent research on vibration of structures using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. Shock Vib Dig 31(3):187–194
Li Q, Iu V, Kou K (2008) Three-dimensional vibration analysis of functionally graded material sandwich plates. J Sound Vib 311(1–2):498–515
Frostig Y (2016) Shear buckling of sandwich plates–Incompressible and compressible cores. Compos B Eng 96:153–172
Xiang Y, Wang C, Kitipornchai S (1994) Exact vibration solution for initially stressed Mindlin plates on Pasternak foundations. Int J Mech Sci 36(4):311–316
Leissa AW (1973) The free vibration of rectangular plates. J Sound Vib 31(3):257–293
Acknowledgement
The first and second authors acknowledge the funding support of Babol Noshirvani University of Technology through Grant program No. BNUT/964113035/97.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A
The eleven governing equations of motion (three equations for the top face sheet, five equations for the core, and three equations for the bottom face sheet) for the FG sandwich plate resting on a Pasternak foundation in contact with fluid are as follows:
For the top face sheet:
For the core:
For the bottom face sheet:
Moreover, the corresponding boundary conditions are as follows:
At \(x = 0\) and \(x = a\):
At \(y = 0\) and \(y = a\):
Appendix B
The base functions \(\left( {\lambda_{i}^{k} \left( {k = u^{t} , v^{t} , w^{t} , \ldots , v_{1} } \right)} \right)\) can be considered as follows:
a)
where \(l=t,b,c\) and
b)
and
where \(l=t,b,c, k=t,b\) and
c)
and
where \(l=t,b,c, k=t,b\) and:
Rights and permissions
About this article
Cite this article
Ramian, A., Jafari-Talookolaei, RA., Valvo, P.S. et al. Fluid–structure–soil interaction effects on the free vibrations of functionally graded sandwich plates. Engineering with Computers 38 (Suppl 3), 1901–1921 (2022). https://doi.org/10.1007/s00366-021-01348-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00366-021-01348-0