R-function Theory for Bending Problem of Shallow Spherical Shells with Polygonal Boundary
Vol. 6 No. 3 (2020): March
Research Articles
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The governing differential equations of the bending problem of simply supported shallow spherical shells on Winkler foundation are simplified to an independent equation of radial deflection. The independent equation of radial deflection is decomposed to two Laplace operators by intermediate variable. The R-function theory is applied to describe a shallow spherical shell on Winkler foundation with concave boundary, and then a quasi-Green's function is established by using the fundamental solution and the normalized boundary equation. The quasi-Green's function satisfies the homogeneous boundary condition of the problem. The Laplace operators of the problem are reduced to two simultaneous Fredholm integral equations of the second kind by the Green's formula. The singularity of the kernel of the integral equation is eliminated by choosing a suitable form of the normalized boundary equation. The integral equations are discretized into the homogeneous linear algebraic equations to proceed numerical computing. The singular term in the discrete equation is eliminated by the integral method. Some numerical examples are given to verify the validity of the proposed method in calculating simple boundary conditions and polygonal boundary conditions. A comparison with the ANSYS finite element (FEM) solution shows a good agreement, and it demonstrates the feasibility and efficiency of the present method.
[1] Kong, Fan-zhong, Xiao-ping Zheng, and Zhen-han Yao. "Numerical Simulation of 2D Fiber-Reinforced Composites Using Boundary Element Method.” Applied Mathematics and Mechanics 26, no. 11 (November 2005): 1515–1522. doi:10.1007/bf03246259..
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[3] Zhao, Wei-jia, Li-qun Chen, and W. Zu Jean. "A Finite Difference Method for Simulating Transverse Vibrations of an Axially Moving Viscoelastic String.” Applied Mathematics and Mechanics 27, no. 1 (January 2006): 23–28. doi:10.1007/s10483-006-0104-1.
[4] Rvachev, V.L. "Theory of R-function and Some of its Application.” Kiev: Nauk Dumka (1982):415-421. (in Russian)
[5] Li, Shan-qing, and Hong Yuan. "Quasi-Green's Function Method for Free Vibration of Clamped Thin Plates on Winkler Foundation.” Applied Mathematics and Mechanics 32, no. 3 (March 2011): 265–276. doi:10.1007/s10483-011-1412-x.
[6] Yuan, Hong, Shan-qing Li, and Ren-huai Liu. "Green Quasifunction Method for Vibration of Simply-Supported Thin Polygonic Plates on Pasternak Foundation.” Applied Mathematics and Mechanics 28, no. 7 (July 2007): 847–853. doi:10.1007/s10483-007-0701-y..
[7] Li, Shanqing, and Hong Yuan. "Green Quasifunction Method for Free Vibration of Clamped Thin Plates.” Acta Mechanica Solida Sinica 25, no. 1 (February 2012): 37–45. doi:10.1016/s0894-9166(12)60004-4.
[8] Li, Shan-qing, and Hong Yuan. "Quasi-Green's Function Method for Free Vibration of Simply-Supported Trapezoidal Shallow Spherical Shell.” Applied Mathematics and Mechanics 31, no. 5 (May 2010): 635–642. doi:10.1007/s10483-010-0511-7.
[9] Li, Shanqing, and Hong Yuan. "Green Quasifunction Method for Free Vibration of Simply-Supported Trapezoidal Shallow Spherical Shell on Winkler Foundation.” Acta Mechanica Solida Sinica 23, no. 4 (August 2010): 370–376. doi:10.1016/s0894-9166(10)60038-9.
[10] Paliwal, D.N., and S.N. Sinha. "Static and Dynamic Behaviour of Shallow Spherical Shells on Winkler Foundation.” Thin-Walled Structures 4, no. 6 (January 1986): 411–422. doi:10.1016/0263-8231(86)90038-8.
[11] Majeed, Aaqib, A. Zeeshan, and Shahid Mubbashir. "Vibration Analysis of Carbon Nanotubes Based on Cylindrical Shell by Inducting Winkler and Pasternak Foundations.” Mechanics of Advanced Materials and Structures 26, no. 13 (February 5, 2018): 1140–1145. doi:10.1080/15376494.2018.1430282.
[12] Gao, Kang, Wei Gao, Di Wu, and Chongmin Song. "Nonlinear Dynamic Stability of the Orthotropic Functionally Graded Cylindrical Shell Surrounded by Winkler-Pasternak Elastic Foundation Subjected to a Linearly Increasing Load.” Journal of Sound and Vibration 415 (February 2018): 147–168. doi:10.1016/j.jsv.2017.11.038.
[13] Banić, Damjan, Michele Bacciocchi, Francesco Tornabene, and Antonio Ferreira. "Influence of Winkler-Pasternak Foundation on the Vibrational Behavior of Plates and Shells Reinforced by Agglomerated Carbon Nanotubes.” Applied Sciences 7, no. 12 (November 28, 2017): 1228. doi:10.3390/app7121228.
[14] Wang, Xiancheng, Wei Li, Jianjun Yao, Zhenshuai Wan, Yu Fu, and Tang Sheng. "Free Vibration of Functionally Graded Sandwich Shallow Shells on Winkler and Pasternak Foundations with General Boundary Restraints.” Mathematical Problems in Engineering 2019 (March 13, 2019): 1–19. doi:10.1155/2019/7527148.
[15] Shojaeefard, M.H., M. Mahinzare, H. Safarpour, H. Saeidi Googarchin, and M. Ghadiri. "Free Vibration of an Ultra-Fast-Rotating-Induced Cylindrical Nano-Shell Resting on a Winkler Foundation Under Thermo-Electro-Magneto-Elastic Condition.” Applied Mathematical Modelling 61 (September 2018): 255–279. doi:10.1016/j.apm.2018.04.015.
[16] Hussain, Muzamal, Muhammad Nawaz Naeem, and Mohammad Reza Isvandzibaei. "Effect of Winkler and Pasternak Elastic Foundation on the Vibration of Rotating Functionally Graded Material Cylindrical Shell.” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 232, no. 24 (January 22, 2018): 4564–4577. doi:10.1177/0954406217753459.
[17] Patra, Suchismita, Sudhakar Chaudhary, and V.V.K. Srinivas Kumar. "Approximation of the First Eigenpair of the -Laplace Operator Using WEB-Spline Based Finite Element Method.” Engineering Analysis with Boundary Elements 104 (July 2019): 277–287. doi:10.1016/j.enganabound.2019.03.035.
[18] Burman, Erik, Peter Hansbo, Mats G. Larson, Karl Larsson, and André Massing. "Finite Element Approximation of the Laplace–Beltrami Operator on a Surface with Boundary.” Numerische Mathematik 141, no. 1 (July 14, 2018): 141–172. doi:10.1007/s00211-018-0990-2.
[19] Prikazchikov, V. G. "Discrete Spectrum of the Laplace Operator for an Arbitrary Triangle with Different Boundary Conditions.” Cybernetics and Systems Analysis 55, no. 4 (July 2019): 570–580. doi:10.1007/s10559-019-00166-z.
[20] Mbehou, M. "The Euler-Galerkin Finite Element Method for Nonlocal Diffusion Problems with a p-Laplace-Type Operator.” Applicable Analysis 98, no. 11 (February 27, 2018): 2031–2047. doi:10.1080/00036811.2018.1445227.
[21] Ortner, Norbert. "Regularisierte Faltung von Distributionen. Teil 2: Eine Tabelle von Fundamentallösungen.” Zeitschrift Für Angewandte Mathematik Und Physik ZAMP 31, no. 1 (January 1980): 155–173. doi:10.1007/bf01601710.
[22] Kurpa, Lidiya, Tatiana Shmatko, and Galina Timchenko. "Free Vibration Analysis of Laminated Shallow Shells with Complex Shape Using the R-Functions Method.” Composite Structures 93, no. 1 (December 2010): 225–233. doi:10.1016/j.compstruct.2010.05.016.
[23] Awrejcewicz, J., L. Kurpa, and A. Osetrov. "Investigation of the Stress-Strain State of the Laminated Shallow Shells by R-Functions Method Combined with Spline-Approximation.” ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift Für Angewandte Mathematik Und Mechanik 91, no. 6 (February 1, 2011): 458–467. doi:10.1002/zamm.201000164.
[24] Kurpa, Lidiya, Tetyana Shmatko, and Jan Awrejcewicz. "Vibration Analysis of Laminated Functionally Graded Shallow Shells with Clamped Cutout of the Complex Form by the Ritz Method and the R-Functions Theory.” Latin American Journal of Solids and Structures 16, no. 1 (2019). doi:10.1590/1679-78254911.
[2] Š estak, Ivan, and BoŠ¡ko S. Jovanović. "Approximation of Thermoelasticity Contact Problem with Nonmonotone Friction.” Applied Mathematics and Mechanics 31, no. 1 (January 2010): 77–86. doi:10.1007/s10483-010-0108-6.
[3] Zhao, Wei-jia, Li-qun Chen, and W. Zu Jean. "A Finite Difference Method for Simulating Transverse Vibrations of an Axially Moving Viscoelastic String.” Applied Mathematics and Mechanics 27, no. 1 (January 2006): 23–28. doi:10.1007/s10483-006-0104-1.
[4] Rvachev, V.L. "Theory of R-function and Some of its Application.” Kiev: Nauk Dumka (1982):415-421. (in Russian)
[5] Li, Shan-qing, and Hong Yuan. "Quasi-Green's Function Method for Free Vibration of Clamped Thin Plates on Winkler Foundation.” Applied Mathematics and Mechanics 32, no. 3 (March 2011): 265–276. doi:10.1007/s10483-011-1412-x.
[6] Yuan, Hong, Shan-qing Li, and Ren-huai Liu. "Green Quasifunction Method for Vibration of Simply-Supported Thin Polygonic Plates on Pasternak Foundation.” Applied Mathematics and Mechanics 28, no. 7 (July 2007): 847–853. doi:10.1007/s10483-007-0701-y..
[7] Li, Shanqing, and Hong Yuan. "Green Quasifunction Method for Free Vibration of Clamped Thin Plates.” Acta Mechanica Solida Sinica 25, no. 1 (February 2012): 37–45. doi:10.1016/s0894-9166(12)60004-4.
[8] Li, Shan-qing, and Hong Yuan. "Quasi-Green's Function Method for Free Vibration of Simply-Supported Trapezoidal Shallow Spherical Shell.” Applied Mathematics and Mechanics 31, no. 5 (May 2010): 635–642. doi:10.1007/s10483-010-0511-7.
[9] Li, Shanqing, and Hong Yuan. "Green Quasifunction Method for Free Vibration of Simply-Supported Trapezoidal Shallow Spherical Shell on Winkler Foundation.” Acta Mechanica Solida Sinica 23, no. 4 (August 2010): 370–376. doi:10.1016/s0894-9166(10)60038-9.
[10] Paliwal, D.N., and S.N. Sinha. "Static and Dynamic Behaviour of Shallow Spherical Shells on Winkler Foundation.” Thin-Walled Structures 4, no. 6 (January 1986): 411–422. doi:10.1016/0263-8231(86)90038-8.
[11] Majeed, Aaqib, A. Zeeshan, and Shahid Mubbashir. "Vibration Analysis of Carbon Nanotubes Based on Cylindrical Shell by Inducting Winkler and Pasternak Foundations.” Mechanics of Advanced Materials and Structures 26, no. 13 (February 5, 2018): 1140–1145. doi:10.1080/15376494.2018.1430282.
[12] Gao, Kang, Wei Gao, Di Wu, and Chongmin Song. "Nonlinear Dynamic Stability of the Orthotropic Functionally Graded Cylindrical Shell Surrounded by Winkler-Pasternak Elastic Foundation Subjected to a Linearly Increasing Load.” Journal of Sound and Vibration 415 (February 2018): 147–168. doi:10.1016/j.jsv.2017.11.038.
[13] Banić, Damjan, Michele Bacciocchi, Francesco Tornabene, and Antonio Ferreira. "Influence of Winkler-Pasternak Foundation on the Vibrational Behavior of Plates and Shells Reinforced by Agglomerated Carbon Nanotubes.” Applied Sciences 7, no. 12 (November 28, 2017): 1228. doi:10.3390/app7121228.
[14] Wang, Xiancheng, Wei Li, Jianjun Yao, Zhenshuai Wan, Yu Fu, and Tang Sheng. "Free Vibration of Functionally Graded Sandwich Shallow Shells on Winkler and Pasternak Foundations with General Boundary Restraints.” Mathematical Problems in Engineering 2019 (March 13, 2019): 1–19. doi:10.1155/2019/7527148.
[15] Shojaeefard, M.H., M. Mahinzare, H. Safarpour, H. Saeidi Googarchin, and M. Ghadiri. "Free Vibration of an Ultra-Fast-Rotating-Induced Cylindrical Nano-Shell Resting on a Winkler Foundation Under Thermo-Electro-Magneto-Elastic Condition.” Applied Mathematical Modelling 61 (September 2018): 255–279. doi:10.1016/j.apm.2018.04.015.
[16] Hussain, Muzamal, Muhammad Nawaz Naeem, and Mohammad Reza Isvandzibaei. "Effect of Winkler and Pasternak Elastic Foundation on the Vibration of Rotating Functionally Graded Material Cylindrical Shell.” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 232, no. 24 (January 22, 2018): 4564–4577. doi:10.1177/0954406217753459.
[17] Patra, Suchismita, Sudhakar Chaudhary, and V.V.K. Srinivas Kumar. "Approximation of the First Eigenpair of the -Laplace Operator Using WEB-Spline Based Finite Element Method.” Engineering Analysis with Boundary Elements 104 (July 2019): 277–287. doi:10.1016/j.enganabound.2019.03.035.
[18] Burman, Erik, Peter Hansbo, Mats G. Larson, Karl Larsson, and André Massing. "Finite Element Approximation of the Laplace–Beltrami Operator on a Surface with Boundary.” Numerische Mathematik 141, no. 1 (July 14, 2018): 141–172. doi:10.1007/s00211-018-0990-2.
[19] Prikazchikov, V. G. "Discrete Spectrum of the Laplace Operator for an Arbitrary Triangle with Different Boundary Conditions.” Cybernetics and Systems Analysis 55, no. 4 (July 2019): 570–580. doi:10.1007/s10559-019-00166-z.
[20] Mbehou, M. "The Euler-Galerkin Finite Element Method for Nonlocal Diffusion Problems with a p-Laplace-Type Operator.” Applicable Analysis 98, no. 11 (February 27, 2018): 2031–2047. doi:10.1080/00036811.2018.1445227.
[21] Ortner, Norbert. "Regularisierte Faltung von Distributionen. Teil 2: Eine Tabelle von Fundamentallösungen.” Zeitschrift Für Angewandte Mathematik Und Physik ZAMP 31, no. 1 (January 1980): 155–173. doi:10.1007/bf01601710.
[22] Kurpa, Lidiya, Tatiana Shmatko, and Galina Timchenko. "Free Vibration Analysis of Laminated Shallow Shells with Complex Shape Using the R-Functions Method.” Composite Structures 93, no. 1 (December 2010): 225–233. doi:10.1016/j.compstruct.2010.05.016.
[23] Awrejcewicz, J., L. Kurpa, and A. Osetrov. "Investigation of the Stress-Strain State of the Laminated Shallow Shells by R-Functions Method Combined with Spline-Approximation.” ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift Für Angewandte Mathematik Und Mechanik 91, no. 6 (February 1, 2011): 458–467. doi:10.1002/zamm.201000164.
[24] Kurpa, Lidiya, Tetyana Shmatko, and Jan Awrejcewicz. "Vibration Analysis of Laminated Functionally Graded Shallow Shells with Clamped Cutout of the Complex Form by the Ritz Method and the R-Functions Theory.” Latin American Journal of Solids and Structures 16, no. 1 (2019). doi:10.1590/1679-78254911.
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