Thermal Bending of Circular Plates for Non-Axisymmetrical Problems

ABSTRACT

Using the Green function, the boundary integral formula and natural boundary integral equation for thermal elastic problems are obtained. Then based on bending solutions to circular plates subjected to the non-axi- symmetrical load, by utilizing the Fourier series and convolution formulae, the bending solutions under non-axisymmetrical thermal conditions have been obtained. The calculating process is simple. Examples show the discussed methods are effective.

Using the Green function, the boundary integral formula and natural boundary integral equation for thermal elastic problems are obtained. Then based on bending solutions to circular plates subjected to the non-axi- symmetrical load, by utilizing the Fourier series and convolution formulae, the bending solutions under non-axisymmetrical thermal conditions have been obtained. The calculating process is simple. Examples show the discussed methods are effective.

KEYWORDS

Thermal Bending Problems, Circular Plate, Boundary Integral Formula, Natural Boundary Integral Equation

Thermal Bending Problems, Circular Plate, Boundary Integral Formula, Natural Boundary Integral Equation

Cite this paper

nullZ. Dong, W. Peng, J. Li and F. Li, "Thermal Bending of Circular Plates for Non-Axisymmetrical Problems,"*World Journal of Mechanics*, Vol. 1 No. 2, 2011, pp. 44-49. doi: 10.4236/wjm.2011.12006.

nullZ. Dong, W. Peng, J. Li and F. Li, "Thermal Bending of Circular Plates for Non-Axisymmetrical Problems,"

References

[1] S. P. Timoshengko and S. Woinowsky-Krieger, “Theory of Plates and Shells,” 2nd Edition, McGraw-Hill, New York, 1959.

[2] R. Szilard, “Theory and Analysis of Plates-Classical and Numerical Methods,” Prentice Hall, New Jersey, 1974.

[3] J. L. Zhu, “The Boundary Element Method for Elliptic Boundary Value Problems,” Science Press, Beijing, 1988.

[4] J. L. Zhu, “The Boundary Integral Equation Method for Solving Dirichelt Problem of Plane Biharmonic Equation,” Journal of Computational Mathematics, Vol. 6, No. 3, 1984, pp. 278-288.

[5] K. Chandrashekhara, “Theory of Plates,” University Press, Hyderabad, 2001, pp. 147-182.

[6] P. F. Hou, L. J. Guo and W. Lu, “Simply Supported Circular Plate under Uniform Thermo-Mechanical Coupling Loading,” Journal of Zhejiang University (Engineering Science), Vol. 41, No. 1, January 2007, pp. 104-108.

[7] Z. G. Zhao and P. R. Wang, “The Laplace Transform Finite Element Method for Dynamic Coupled Thermoelastic Bending Problems of Thin Plates,” Acta Mechanica Solida Sinica, Vol. 18, No. 2, 1997, pp. 183-187.

[8] Y. Sun and W. X. Zhong, “Finite Element Surface Stress Calculation,” Chinese Journal of Computational Mechanics, Vol. 27, No. 2, April 2010, pp. 177-181.

[9] I. Babuska and T. Stroubolis, “The Finite Element and Its Reliability,” Oxford University Press, London, 2001.

[10] B. L. Fu, “The New Bending Theorem of the Thin Plates on Reciprocal Method,” Science Press, Beijing, 2003.

[11] A. W. Wang, “Solution to Asymmetric Bending of Circular Plates under Single Load by Using Point-Source Function,” Acta Mechanica Sinica, Vol. 24, No. 3, 1992, pp. 381-387.

[12] X. S. Wang, “δ Function and Its Application in Mechanics,” Science Press, Beijing, 1993.

[13] D. H. Yu, “Mathematics Theory of National Boundary Element Method,” Science Press, Beijing, 1993.

[14] D. H. Yu, “Natural Boundary Integral Equations and Related Computational Methods ,” Journal of Yanshan University, Vol. 28, No. 2, March 2004, pp. 111-113.

[15] S. C. Li and Z. Z. Dong, “Natural Boundary Element Method to the Bending Problem of the Circular Plate under the Non-Continuous Loads,” Journal of Guangdong Industrial University, Vol. 16, No. 2, 2004, pp.83-88.

[16] S. C. Li, Z. Z. Dong and W. H. Xie, “Application of Natural Boundary Element Method to the Bending Problem of the Elastic Thin Plate,” Journal of Xuzhou Normal University, Vol. 20, No. 4, 2002, pp. 12-15.

[17] S. C. Li, Z. Z. Dong and W. H. Xie, “The Analytical Formulas of Bending Deflection for Infinite Plates with a Unit Circle under the Boundary Loads,” Journal of Gansu Sciences, Vol. 16, No. 2, 2004, pp. 83-88.

[18] I. M. Gel’fand and G. E. Shilov, “Generalized Functions,” Academic Press, New York, 1964.

[19] D. H. Yu, “Natural Boundary Integral Method and Its Applications,” Science Press & Kluwer Academic Publishers, Beijing, 2002.

[1] S. P. Timoshengko and S. Woinowsky-Krieger, “Theory of Plates and Shells,” 2nd Edition, McGraw-Hill, New York, 1959.

[2] R. Szilard, “Theory and Analysis of Plates-Classical and Numerical Methods,” Prentice Hall, New Jersey, 1974.

[3] J. L. Zhu, “The Boundary Element Method for Elliptic Boundary Value Problems,” Science Press, Beijing, 1988.

[4] J. L. Zhu, “The Boundary Integral Equation Method for Solving Dirichelt Problem of Plane Biharmonic Equation,” Journal of Computational Mathematics, Vol. 6, No. 3, 1984, pp. 278-288.

[5] K. Chandrashekhara, “Theory of Plates,” University Press, Hyderabad, 2001, pp. 147-182.

[6] P. F. Hou, L. J. Guo and W. Lu, “Simply Supported Circular Plate under Uniform Thermo-Mechanical Coupling Loading,” Journal of Zhejiang University (Engineering Science), Vol. 41, No. 1, January 2007, pp. 104-108.

[7] Z. G. Zhao and P. R. Wang, “The Laplace Transform Finite Element Method for Dynamic Coupled Thermoelastic Bending Problems of Thin Plates,” Acta Mechanica Solida Sinica, Vol. 18, No. 2, 1997, pp. 183-187.

[8] Y. Sun and W. X. Zhong, “Finite Element Surface Stress Calculation,” Chinese Journal of Computational Mechanics, Vol. 27, No. 2, April 2010, pp. 177-181.

[9] I. Babuska and T. Stroubolis, “The Finite Element and Its Reliability,” Oxford University Press, London, 2001.

[10] B. L. Fu, “The New Bending Theorem of the Thin Plates on Reciprocal Method,” Science Press, Beijing, 2003.

[11] A. W. Wang, “Solution to Asymmetric Bending of Circular Plates under Single Load by Using Point-Source Function,” Acta Mechanica Sinica, Vol. 24, No. 3, 1992, pp. 381-387.

[12] X. S. Wang, “δ Function and Its Application in Mechanics,” Science Press, Beijing, 1993.

[13] D. H. Yu, “Mathematics Theory of National Boundary Element Method,” Science Press, Beijing, 1993.

[14] D. H. Yu, “Natural Boundary Integral Equations and Related Computational Methods ,” Journal of Yanshan University, Vol. 28, No. 2, March 2004, pp. 111-113.

[15] S. C. Li and Z. Z. Dong, “Natural Boundary Element Method to the Bending Problem of the Circular Plate under the Non-Continuous Loads,” Journal of Guangdong Industrial University, Vol. 16, No. 2, 2004, pp.83-88.

[16] S. C. Li, Z. Z. Dong and W. H. Xie, “Application of Natural Boundary Element Method to the Bending Problem of the Elastic Thin Plate,” Journal of Xuzhou Normal University, Vol. 20, No. 4, 2002, pp. 12-15.

[17] S. C. Li, Z. Z. Dong and W. H. Xie, “The Analytical Formulas of Bending Deflection for Infinite Plates with a Unit Circle under the Boundary Loads,” Journal of Gansu Sciences, Vol. 16, No. 2, 2004, pp. 83-88.

[18] I. M. Gel’fand and G. E. Shilov, “Generalized Functions,” Academic Press, New York, 1964.

[19] D. H. Yu, “Natural Boundary Integral Method and Its Applications,” Science Press & Kluwer Academic Publishers, Beijing, 2002.