Acta Mechanica

, Volume 230, Issue 10, pp 3807–3821

# Accurate bending analysis of rectangular thin plates with corner supports by a unified finite integral transform method

• Jinghui Zhang
• Chao Zhou
• Salamat Ullah
• Yang Zhong
• Rui Li
Original Paper

## Abstract

In this paper, for the first time, the double finite integral transform method is introduced to explore the analytical bending solutions of rectangular thin plates with corner supports. Introducing the concept of generalized simply supported edge, the general form analytical solutions for bending of the plates under consideration are obtained by applying the finite integral transform to the governing equation and some of the boundary conditions. The analytical bending solutions of plates under specific boundary conditions are then obtained elegantly by imposing the remaining boundary conditions. Compared with the conventional inverse/semi-inverse methods, the present method is straightforward without any predetermination of solution forms, which makes it very attractive for exploring new analytical bending solutions of plates with complex boundary conditions. Another advantage of the method is that the analytical solutions obtained converge rapidly due to utilization of the sum functions. Comprehensive analytical results obtained in this paper illuminate the validity and accuracy of the present method by comparison with those from the literature and the finite element method.

## Notes

### Acknowledgements

The authors gratefully acknowledge the support from the LiaoNing Revitalization Talents Program (XLYC1807126, XLYC1802020), the National Natural Science Foundation of China (11825202), the Young Elite Scientists Sponsorship Program by CAST (No. 2015QNRC001) and the Fundamental Research Funds for the Central Universities of China (No. DUT18GF101).

## References

1. 1.
Timoshenko, S.P., Woinowsky-Krieger, S.: Theory of Plates and Shells. McGraw-Hill, Auckland (1959)
2. 2.
Bhaskar, K., Sivaram, A.: Untruncated infinite series superposition method for accurate flexural analysis of isotropic/orthotropic rectangular plates with arbitrary edge conditions. Compos. Struct. 83, 83–92 (2008)
3. 3.
Thai, H.-T., Kim, S.-E.: Analytical solution of a two variable refined plate theory for bending analysis of orthotropic Levy-type plates. Int. J. Mech. Sci. 54, 269–276 (2012)
4. 4.
Lee, S.L., Ballesteros, P.: Uniformly loaded rectangular plate supported at the corners. Int. J. Mech. Sci. 2, 206–211 (1960)
5. 5.
Wang, C.M., Wang, Y.C., Reddy, J.N.: Problems and remedy for the Ritz method in determining stress resultants of corner supported rectangular plates. Comput. Struct. 80, 145–154 (2002)
6. 6.
Lim, C.W., Yao, W.A., Cui, S.: Benchmark symplectic solutions for bending of corner-supported rectangular thin plates. IES J. Part A: Civ. Struct. Eng. 1, 106–115 (2008)Google Scholar
7. 7.
Lim, C.W., Xu, X.S.: Symplectic elasticity: theory and applications. Appl. Mech. Rev. 63, 050802 (2010)
8. 8.
Lim, C.W., Lü, C.F., Xiang, Y., Yao, W.: On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates. Int. J. Eng. Sci. 47, 131–140 (2009)
9. 9.
Li, R., Wang, B., Li, P.: Hamiltonian system-based benchmark bending solutions of rectangular thin plates with a corner point-supported. Int. J. Mech. Sci. 85, 212–218 (2014)
10. 10.
Li, R., Wang, B., Li, G.: Benchmark bending solutions of rectangular thin plates point-supported at two adjacent corners. Appl. Math. Lett. 40, 53–58 (2015)
11. 11.
Batista, M.: New analytical solution for bending problem of uniformly loaded rectangular plate supported on corner points. IES J. Part A: Civ. Struct. Eng. 3, 75–84 (2010)Google Scholar
12. 12.
Chowdhury, A.R., Wang, C.M.: Bending, buckling, and vibration of equilateral simply supported or clamped triangular plates with rounded corners. J. Eng. Mech. 142, 04016074 (2016)
13. 13.
Yshii, L.N., Lucena Neto, E., Monteiro, F.A.C., Santana, R.C.: Accuracy of the buckling predictions of anisotropic plates. J. Eng. Mech. 144, 04018061 (2018)
14. 14.
Thai, C.H., Ferreira, A.J.M., Wahab, M.A., Nguyen-Xuan, H.: A moving Kriging meshfree method with naturally stabilized nodal integration for analysis of functionally graded material sandwich plates. Acta Mech. 229, 2997–3023 (2018)
15. 15.
Thai, C.H., Nguyen, T.N., Rabczuk, T., Nguyen-Xuan, H.: An improved moving Kriging meshfree method for plate analysis using a refined plate theory. Comput. Struct. 176, 34–49 (2010)
16. 16.
Thai, C.H., Do, V.N.V., Nguyen-Xuan, H.: An improved moving Kriging-based meshfree method for static, dynamic and buckling analyses of functionally graded isotropic and sandwich plates. Eng. Anal. Bound. Elem. 64, 122–136 (2016)
17. 17.
Nguyen, T.N., Thai, C.H., Nguyen-Xuan, H.: A novel computational approach for functionally graded isotropic and sandwich plate structures based on a rotation-free meshfree method. Thin-Walled Struct. 107, 473–488 (2016)
18. 18.
Thai, C.H., Ferreira, A.J.M., Rabczuk, T., Nguyen-Xuan, H.: A naturally stabilized nodal integration meshfree formulation for carbon nanotube-reinforced composite plate analysis. Eng. Anal. Bound. Elem. 92, 136–155 (2018)
19. 19.
Thai, C.H., Ferreira, A.J.M., Carrera, E., Nguyen-Xuan, H.: Isogeometric analysis of laminated composite and sandwich plates using a layerwise deformation theory. Compos. Struct. 104, 196–214 (2013)
20. 20.
Thai, C.H., Ferreira, A.J.M., Bordas, S.P.A., Rabczuk, T., Nguyen-Xuan, H.: Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory. Eur. J. Mech. - A/Solids 43, 89–108 (2014)
21. 21.
Nguyen, T.N., Thai, C.H., Nguyen-Xuan, H.: On the general framework of high order shear deformation theories for laminated composite plate structures: A novel unified approach. Int. J. Mech. Sci. 110, 242–255 (2016)
22. 22.
Thai, C.H., Ferreira, A.J.M., Wahab, M.A., Nguyen-Xuan, H.: A generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates based on isogeometric analysis. Acta Mech. 227, 1225–1250 (2016)
23. 23.
Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method: Solid Mechanics. Butterworth-Heinemann, Oxford (2000)
24. 24.
Shi, G.: Flexural vibration and buckling analysis of orthotropic plates by the boundary element method. Int. J. Solids Struct. 26, 1351–1370 (1990)
25. 25.
Cheung, Y.K.: Finite Strip Method in Structural Analysis. Elsevier, Netherlands (2013)Google Scholar
26. 26.
Sneddon, I.N.: Application of Integral Transforms in the Theory of Elasticity. Springer, New York (1975)
27. 27.
Zhong, Y., Zhang, Y.S.: Free vibration of rectangular thin plate on elastic foundation with four edges free. J. Vib. Eng. 19, 566–570 (2006)Google Scholar
28. 28.
Li, R., Zhong, Y., Tian, B., Liu, Y.: On the finite integral transform method for exact bending solutions of fully clamped orthotropic rectangular thin plates. Appl. Math. Lett. 22, 1821–1827 (2009)
29. 29.
Tian, B., Zhong, Y., Li, R.: Analytic bending solutions of rectangular cantilever thin plates. Arch. Civ. Mech. Eng. 11, 1043–1052 (2011)
30. 30.
Tian, B., Li, R., Zhong, Y.: Integral transform solutions to the bending problems of moderately thick rectangular plates with all edges free resting on elastic foundations. Appl. Math. Model. 39, 128–136 (2015)
31. 31.
Zhang, S., Xu, L.: Bending of rectangular orthotropic thin plates with rotationally restrained edges: a finite integral transform solution. Appl. Math. Model. 46, 48–62 (2017)
32. 32.
Zhang, S., Xu, L.: Analytical solutions for flexure of rectangular orthotropic plates with opposite rotationally restrained and free edges. Arch. Civ. Mech. Eng. 18, 965–972 (2018)

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

## Authors and Affiliations

• Jinghui Zhang
• 1
• Chao Zhou
• 2
• Salamat Ullah
• 1
• Yang Zhong
• 2
• Rui Li
• 3
Email author
1. 1.Faculty of Infrastructure EngineeringDalian University of TechnologyDalianChina
2. 2.Department of Engineering MechanicsDalian University of TechnologyDalianChina
3. 3.State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, and International Research Center for Computational MechanicsDalian University of TechnologyDalianChina