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Reciprocal Method of Work for Solving the Bending of Rectangular Thin Plate on Elastic Foundation under the Combined Action of Lateral Loads and Forces in Middle Plane of the Plate

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Abstract

The reciprocal method of work is used to solve the bending problems of thin plate on elastic foundation under the combined action of lateral loads and forces in middle plane of the plate. The basic system is a simply supported thin plate on elastic foundation under combined action of lateral loads and forces in the middle plane of the plate. Two expressions of the deflection surface equation of the basic system are derived, the first is the double trigonometric function, and the second is the mixture of hyperbolic function and trigonometric function. The practical systems consist of thin plates on elastic foundation constrained by different boundary conditions. The deflection surface equation of the practical system is solved by using the reciprocal method of work between the practical system and the basic system. Compared with the finite element method with ANSYS software, the solution error is less than 5%, which satisfies the requirement of maximum error. This indicates that application of the reciprocal method of work is correct for solving the problem.

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REFERENCES

  1. R. Li, Y. Zhong, and M. Li, “Analytic bending solutions of free rectangular thin plates resting on elastic foundations by a new symplectic superposition method,” Proc. Roy. Soc. A: Math. Phys. Eng. Sci. 469 (2153), 20120681 (2013). https://doi.org/10.1098/rspa.2012.0681

  2. O. Fergani, I. Lazoglu, A. Mkaddem, et al., “Analytical modeling of residual stress and the induced deflection of a milled thin plate,” Int. J. Adv. Manuf. Technol. 75 (1), 455–463 (2014). https://doi.org/10.1007/s00170-014-6146-3

    Article  Google Scholar 

  3. B. Wang, P. Li, and R. Li, “Symplectic superposition method for new analytic buckling solutions of rectangular thin plates,” Int. J. Mech. Sci. 119, 432–441 (2016). https://doi.org/10.1016/j.ijmecsci.2016.11.006

    Article  ADS  Google Scholar 

  4. S. Yang, X. Hua, X. Liu, and C. Li, “Analysis of stability factors of roadway roof and determination of unsupported roof distance,” Shock Vibr. 2021, 2271257 (2021). https://doi.org/10.1155/2021/2271257

  5. O. C. Zienkiewicz and Y. K. Cheung, “The finite element method for analysis of elastic isotropic and orthotropic slabs,” Proc. Inst. Civil Eng. 28 (4), 471–488 (1964). https://doi.org/10.1680/iicep.1964.10014

    Article  Google Scholar 

  6. Y. K. Cheung and O. C. Zinkiewicz, “Plates and tanks on elastic foundations—an application of finite element method,” Int. J. Solids Struct. 1 (4), 451–461 (1965). https://doi.org/10.1016/0020-7683(65)90008-9

    Article  Google Scholar 

  7. A. A. Komlev and S. A. Makeev, “The calculation of rectangular plates on elastic foundation the finite difference method,” Int. J. Phys.: Conf. Ser. 944 (1), 012056 (2018). https://doi.org/10.1088/1742-6596/944/1/012056

  8. G. Bezine, “A new boundary element method for bending of plates on elastic foundations,” Int. J. Solids Struct. 24 (6), 557–565 (1988). https://doi.org/10.1016/0020-7683(88)90057-1

    Article  MATH  Google Scholar 

  9. A. El-Zafrany, S. Fadhil, and K. Al-Hosani, “A new fundamental solution for boundary element analysis of thin plates on Winkler foundation,” Int. J. Num. Meth. Eng. 38 (6), 887–903 (1995). https://doi.org/10.1002/nme.1620380602

    Article  MATH  Google Scholar 

  10. A. Jahanpour and F. Roozbahani, “An applicable formula for elastic buckling of rectangular plates under biaxial and shear loads,” Aerosp. Sci. Technol. 56, 100–111 (2016). https://doi.org/10.1016/j.ast.2016.07.005

    Article  Google Scholar 

  11. E. Betti, “Teoria della elasticita,” Il Nuovo Cimento (1869–1876), 7 (1), 69–97 (1872).

  12. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959)

    MATH  Google Scholar 

  13. M. K. Huang and H. D. Conway, “Bending of a uniformly loaded rectangular plate with two adjacent edges clamped and the others either simply supported or free,” J. Appl. Mech. 19 (4), 451–460 (1952). https://doi.org/10.1115/1.4010542

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. M. R. Khalili, K. Malekzadeh, and R. K. Mittal, “A new approach to static and dynamic analysis of composite plates with different boundary conditions,” Compos. Struct. 69 (2), 149–155 (2005). https://doi.org/10.1016/j.compstruct.2004.06.006

    Article  Google Scholar 

  15. C. W. Lim and X. S. Xu, “Symplectic elasticity: theory and applications,” Appl. Mech. Rev. 63 (5), 050802 (2010). https://doi.org/10.1115/1.4003700

  16. W. Z. Chien and K. Y. Ye, Elastic Mechanics (Science Press, 1980).

    Google Scholar 

  17. B. L. Fu, “Applications of the reciprocal theorem to solving the equations of the deflection surface of the rectangular plates with various edge conditions,” Appl. Math. Mech. (Eng. Ed.) 3 (3), 353–364 (1982).

  18. B. L. Fu, “Modified theorem of reciprocal works,” J. Yanshan Univ. 029 (003), 189–195 (2005).

    Google Scholar 

  19. B. L. Fu, “Corrected reciprocal theorem of works for bending thin plates and its application,” Appl. Math. Mech. 35 (11), 1197–1209 (2014).

    Google Scholar 

  20. B. L. Fu, “Corrected reciprocal theorem for 3D linear elasticity and its application,” Appl. Math. Mech. 36 (5), 523–538 (2015).

    Google Scholar 

  21. B. L. Fu, “Energy theorems for solving equations of deflections,” Appl. Math. Mech. 2 (6), 697–707 (1981).

    Google Scholar 

  22. Y. B. Zhu and B. L. Fu, “Further research on the bending of the cantilever rectangular plates under a concentrated load,” Appl. Math. Mech. 7 (10), 975–986 (1986).

    Article  Google Scholar 

  23. L. Z. Chen and B. L. Fu, “The bending of thin rectangular plates with mixed supported segments of straight edges,” Appl. Math. Mech. 16 (1), 47–58 (1995).

    Article  MATH  Google Scholar 

  24. B. L. Fu and W. F. Tan, “Reciprocal theorem method for solving the problems of bending of thick rectangular plates,” Appl. Math. Mech. 16 (4), 391–403 (1995).

    Article  MATH  Google Scholar 

  25. N. Li and S. Mirza, “Buckling analysis of clamped sandwich plates by the reciprocal theorem method,” Comput. Struct. 51 (2), 137–141 (1994). https://doi.org/10.1016/0045-7949(94)90044-2

    Article  ADS  MATH  Google Scholar 

  26. B. L. Fu and N. Li, “The method of the reciprocal theorem of forced vibration for the elastic thin rectangular plates (III) — Cantilever rectangular plates,” Appl. Math. Mech. 12 (7), 663–680 (1991).

    Article  MathSciNet  Google Scholar 

  27. F. F. Zhang, Elastic Plate, 2nd ed. (Science Press, 1984)

    Google Scholar 

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Funding

This research was supported by the National Natural Science Foundation of China (No. 51904173), the Shandong Province Higher School Science and Technology Plan Project (No. J18KA307), the Major Scientific and Technological Innovation Projects in Shandong Province (No. 2019SDZY01).

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Authors and Affiliations

  1. College of Computer Science and Engineering, Shandong University of Science and Technology, Qingdao, Shandong, China

    Dan Li, Xinming Lu & Xingli Zhang

  2. Shandong Lionking Software Co., Ltd., Taian, Shandong, China

    Xinming Lu & Longquan Zhou

  3. College of Intelligent Equipment, Shandong University of Science and Technology, Taian, Shandong, China

    Hongjuan Wang

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  1. Dan Li
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  2. Xinming Lu
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  3. Longquan Zhou
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  4. Hongjuan Wang
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  5. Xingli Zhang
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Correspondence to Dan Li.

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Li, D., Lu, X., Zhou, L. et al. Reciprocal Method of Work for Solving the Bending of Rectangular Thin Plate on Elastic Foundation under the Combined Action of Lateral Loads and Forces in Middle Plane of the Plate. Mech. Solids 58, 659–669 (2023). https://doi.org/10.3103/S0025654422601744

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