Volume 13, Issue 4
Analytical Solution for Vibration of Continuously Varying-Thickness Beams Resting on Pasternak Elastic Foundations

Zhiyuan Li, Yepeng Xu & Dan Huang

Adv. Appl. Math. Mech., 13 (2021), pp. 850-866.

Published online: 2021-04

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  • Abstract

In this paper, the vibration characteristics of beams with arbitrarily and continuously varying thickness and resting on Pasternak elastic foundations were analytically studied based on the elasticity theory directly. The general expression of stress function, which exactly satisfies the governing differential equations and the boundary conditions, was derived. The frequency equation governing the free vibration of varying-thickness beams resting on a Pasternak elastic foundation can be obtained by using the Fourier series expansion of the boundary conditions on the upper and lower surfaces of the beam. Convergence and comparison studies were conducted to demonstrate the high accuracy and efficiency of the present method. Application of the proposed analytical method to some typical beams with different geometry, Poisson's ratio, elastic coefficients of foundation were conducted further, and some new results are reported which may be used as an alternative of benchmark or standard solutions for numerical or other approximate results.

  • Keywords

Varying-thickness beam, free vibration, Pasternak elastic foundation, analytical solution, Fourier series expansion.

  • AMS Subject Headings

70J30, 74B05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-13-850, author = {Zhiyuan Li , and Yepeng Xu , and Dan Huang , }, title = {Analytical Solution for Vibration of Continuously Varying-Thickness Beams Resting on Pasternak Elastic Foundations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2021}, volume = {13}, number = {4}, pages = {850--866}, abstract = {

In this paper, the vibration characteristics of beams with arbitrarily and continuously varying thickness and resting on Pasternak elastic foundations were analytically studied based on the elasticity theory directly. The general expression of stress function, which exactly satisfies the governing differential equations and the boundary conditions, was derived. The frequency equation governing the free vibration of varying-thickness beams resting on a Pasternak elastic foundation can be obtained by using the Fourier series expansion of the boundary conditions on the upper and lower surfaces of the beam. Convergence and comparison studies were conducted to demonstrate the high accuracy and efficiency of the present method. Application of the proposed analytical method to some typical beams with different geometry, Poisson's ratio, elastic coefficients of foundation were conducted further, and some new results are reported which may be used as an alternative of benchmark or standard solutions for numerical or other approximate results.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0284}, url = {http://global-sci.org/intro/article_detail/aamm/18754.html} }
TY - JOUR T1 - Analytical Solution for Vibration of Continuously Varying-Thickness Beams Resting on Pasternak Elastic Foundations AU - Zhiyuan Li , AU - Yepeng Xu , AU - Dan Huang , JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 850 EP - 866 PY - 2021 DA - 2021/04 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2019-0284 UR - https://global-sci.org/intro/article_detail/aamm/18754.html KW - Varying-thickness beam, free vibration, Pasternak elastic foundation, analytical solution, Fourier series expansion. AB -

In this paper, the vibration characteristics of beams with arbitrarily and continuously varying thickness and resting on Pasternak elastic foundations were analytically studied based on the elasticity theory directly. The general expression of stress function, which exactly satisfies the governing differential equations and the boundary conditions, was derived. The frequency equation governing the free vibration of varying-thickness beams resting on a Pasternak elastic foundation can be obtained by using the Fourier series expansion of the boundary conditions on the upper and lower surfaces of the beam. Convergence and comparison studies were conducted to demonstrate the high accuracy and efficiency of the present method. Application of the proposed analytical method to some typical beams with different geometry, Poisson's ratio, elastic coefficients of foundation were conducted further, and some new results are reported which may be used as an alternative of benchmark or standard solutions for numerical or other approximate results.

Zhiyuan Li, Yepeng Xu & Dan Huang. (1970). Analytical Solution for Vibration of Continuously Varying-Thickness Beams Resting on Pasternak Elastic Foundations. Advances in Applied Mathematics and Mechanics. 13 (4). 850-866. doi:10.4208/aamm.OA-2019-0284
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