ISSN: 2475-4811 (print)
ISSN: 2475-482X (online)
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland

Email: pawel.olejnik@p.lodz.pl

C. Steve Suh (editor)

Texas A&M University, USA

Email: ssuh@tamu.edu

Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China

Email: tuoxianguo@suse.edu.cn

The Non-Classical Problem of Thermoelastic Stability of an Elastically Restrained Orthotropic Plate of Variable Thickness

Journal of Vibration Testing and System Dynamics 6(2) (2022) 195--206 | DOI:10.5890/JVTSD.2022.06.002

Razmik M. Kirakosyan, Seyran P. Stepanyan

Institute of Mechanics NAS of RA, 24/2 Marshal Baghramyan ave, 0019, Yerevan, Armenia Yerevan State\addressNewline University, 1 Alex Manougian str, 0070, Yerevan, Armenia

Abstract

In this paper elastically clamping conditions are given for the bending problem of rectangular plate. It is assumed that these conditions and physical and mechanical properties of the plate material do not depend on temperature. Within the framework of the momentless theory, expressions are obtained for compressive forces, arising as a result of a uniform increase in temperature. A wide range of elastic pinching conditions is considered. In the general case, both a system of differential equations of partial derivatives with unknown variable coefficients and corresponding boundary conditions are obtained to solve the addressed problem. Unknown functions are represented by multiples. A homogeneous system of algebraic equations with respect to unknown coefficients of polynomials is obtained. For certain problems, introducing dimensionless quantities, a technique to calculate the critical temperature by the collocation method is described.

References

1.  [1] Kirakosyan, R.M. (2000), Applied Theory of Orthotropic Plates of Variable Thickness, which Takes into Account the Effect of Transverse Shear Deformation, Yerevan. Gitutyun, 122p., (in Russian).
2.  [2] Kirakosyan, R.M. (2014), The Non-Classical Bending Problem of an Orthotropic Beam with an Elastically Clamped Support, DNAN RA, 114(2), 101-107, (in Russian).
3.  [3] Ambartsumyan, S.A. (1987), Theory of anisotropic plates, Moscow: Science, 360.
4.  [4] Kirakosyan, R.M. and Stepanyan, S.P. (2014), Stability of the Rod in the Presence of Elastic Clamped Support, NAS RA Reports, 114(4), 309-315.
5.  [5] Kirakosyan, R.M. and Stepanyan, S.P. (2014), Non-Classical Problem of Bending of an Orthotropic Beam of Variable Thickness with Elastically Clamped Support, NAS RA Reports, 114(3), 205-212, (in Russian).
6.  [6] Kirakosyan, R.M. (2015), On One Nonclasical Problem of a Bend of an Elastically Fastened Round Plate, NAS RA Reports, 115(4), 284-289, (in Russian).
7.  [7] Kirakosyan, R.M. and Stepanyan, S.P. (2016), The Non-Classical Boundary Value Problem of a Partially Loaded Round Orthotropic Plate, Elastically Pinch Along the Edge, Izv.NAS of Armenia. Mechanics, 69(3), 59-70, (in Russian). DOI:org/10.33018/69.3.12.
8.  [8] Kirakosyan, R.M. and Stepanyan, S.P. (2017), Non-Classical Problem of Bend of an Orthotropic Annular Plate of Variable Thickness with an Elastically Clamped Support, Proceedings of the YSU, Physical and Mathematical Sciences, 51(2), 168-176. https://doi.org/10. 46991/PYSU:A/2017.51.2.168.
9.  [9] Kirakosyan, R.M. and Stepanyan, S.P. (2017), Stability of the Rod with Allowance for the Reduction of the Compressive Force by an Elastically Clamped Support, Izv. NAS of Armenia. Mechanics, 70(3), 57-66, (in Russian). htt://doi.org/10.33018/70.3.5.
10.  [10] Kirakosyan, R.M. and Stepanyan, S.P. (2018), The Non-classical Problem of an Elastically Clamped Orthotropic Beam of Variable Thickness under the combined action of Compressive Forces and Transverse Load, Proceedings of the YSU, Physical and Mathematical Sciences, 52(2), p.101-108, https://doi.org/10.46991/ PYSU:A/2018.52.2.101.
11.  [11] Kirakosyan, R.M. and Stepanyan, S.P. (2019), The Non-classical Problem of an Orthotropic Beam of Variable Thickness with the simultaneous action of its own Weigt and Compressive Axial Forces, Proceedings of the YSU, Physical and Mathematical Sciences, 53(3), p.183-190, https://doi.org/10.46991/ PYSU:A/2019.53.3.183.
12.  [12] Stepanyan, S.P. (2021), Investigation of the Influence of an Intermediate Hinge Support in the Problem of Bending of an Elastically Restrained Orthotropic Beam, Proceedings of the YSU, Physical and Mathematical Sciences, 55(1), p.64-71. https://doi.org/10.46991/PYSU:A/2021.55.1.064.
13.  [13] Gevorgyan, G.Z. (2015), Vibrations of orthotropic strips of variable thickness taking into account the transverse shear under conditions of elastic joints, Proceedings of IV international conference Topical Problems of Continuum Mechanics'', Yerevan, pp.129-133, (in Russian).
14.  [14] Gomez, H. and de Lorenzis, L. (2016), The Variational Collocation Method, Computer Methods in Applied Mechanics and Engineering, 309, 152-181.
15.  [15] Alihemmati, J., Tadi Beni, Y., and Kiani, Y., (2021), Application of Chebyshev collocation method to unified generalized thermoelasticity of a finite domain, Journal of Thermal Stresses, 44(\ref{eq5}), 547-565. DOI.org/10.1080/01495739.2020.1867941.
16.  [16] Baghdasaryan, G.Y., Mikilyan, M.A., Vardanyan, I.A., Melikyan, K.V., and Marzocca, P. (2021), Thermoelastic non-linear flutter oscillations of rectangular plate, Journal of Thermal Stresses, 44(\ref{eq6}), 731-754, ttps://doi.org/10.1080/01495739.2020.1867941
17.  [17] Batista, M. (2019), Stability of clamped-elastically supported elastic beam subject to axial compression, International Journal of Mechanical Sciences, 155, 1-8, htt:// doi.org/10.1016/j.ijmecsci.2019.02.030.
18.  [18] Papargyri-Beskou, S., Tsepoura, K.G., Polyzos, D., and Beskos, D.E. (2003), Bending and stability analysis of gradient elastic beams, International Journal of Solids and Structures, (40), 385-400, http://doi.org/10.1016/ s0020-7683(02)0052-X.
19.  [19] Su, Z., Wang, L. Sun, K. et al, (2020), Transverse Shear and Normal Deformation Effects on Vibration Behaviors of Functionally Graded Micro-beams, Appl. Math. Mech.-Eng. Ed., 41(9), 1303-1320, http://doi.org/ 10.1007/s10483-020-2662-6.
20.  [20] Jin, Ya., Wang, W., and Djafari-Rouhani, B. (2020), Asymmetric Topological State in an Elastic Beam Based on Symmetry Principle, International Journal of Mechanical Sciences, 186, http://doi.org/10.1016/ j.ijmecsci.2020.105897.
21.  [21] Kirakosyan, R.M. and Stepanyan, S.P. (2015), Problem of Thermoelasticity for an orthotropic plate-strip of variable thickness with regard for transverse shear, Journal of Mathematical Sciences, 208(4), 417-424, Springer. https://doi.org/10.1007/s10958-015-2456-8.
22.  [22] Stepanyan, S.P. (2016), Thermo-Elasticity Problem of an Orthotropic Plate-Strip of Variable Thickness at the Presence of an Elastically Fastened Support, NAS RA Reports, 116(1), 26-33, (in Russian).
23.  [23] Xue, Z.N., Cao, G.Q., and Liu, J.L. (2021), Size-dependent thermoelasticity of a finite bi-layered nanoscale plate based on nonlocal dual-phase-lag conduction and Eringen's nonlocal elasticity, Applied Mathematics and Mechanics, 42, 1-16.
24.  [24] Nguyen, T.K., Thai, H.T., and Vo, T.P. (2020), A novel general higher-order shear deformation theory for static, vibration and thermal buckling analysis of the functionally graded plates, Journal of Thermal Stresses, 44(3), 377-394, https://doi.org/10.1080/01495739.2020.1869127.
25.  [25] Gaikwad, K.R. and Naner, Y.U. (2020), Naner Analysis of transient thermoelastic temperature distribution of a thin circular plate and its thermal deflection under uniform heat generation, Journal of Thermal Stresses, 44(1), 75-85, https://doi.org/10.1080/01495739.2020.1828009.
26.  [26] Birger, I.A. and Panovko, Y.G. (1964), Strength, stability, vibrations. Handbook, 3, Ed. Mechanical engineering, 564p., (in Russian).