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Research Papers: SPECIAL SECTION PAPERS

Analytical and Semi-Analytical Methods for the Evaluation of Dynamic Thermo-Elastic Behavior of Structures Resting on a Pasternak Foundation

[+] Author and Article Information
Xu Liang, Zeng Cao, Xing Zha, Jianxing Leng

Ocean College,
Zhejiang University,
Hangzhou 310058, Zhejiang, China

Hongyue Sun

Ocean College,
Zhejiang University,
Hangzhou 310058, Zhejiang, China
e-mail: shy@zju.edu.cn

1Corresponding author.

Contributed by the Pressure Vessel and Piping Division of ASME for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received August 9, 2017; final manuscript received December 9, 2017; published online December 14, 2018. Assoc. Editor: Fabrizio Paolacci.

J. Pressure Vessel Technol 141(1), 010908 (Dec 14, 2018) (10 pages) Paper No: PVT-17-1146; doi: 10.1115/1.4038724 History: Received August 09, 2017; Revised December 09, 2017

An analytical method and a semi-analytical method are proposed to analyze the dynamic thermo-elastic behavior of structures resting on a Pasternak foundation. The analytical method employs a finite Fourier integral transform and its inversion, as well as a Laplace transform and its numerical inversion. The semi-analytical method employs the state space method, the differential quadrature method (DQM), and the numerical inversion of the Laplace transform. To demonstrate the two methods, a simply supported Euler–Bernoulli beam of variable length is considered. The governing equations of the beam are derived using Hamilton's principle. A comparison between the results of analytical method and the results of semi-analytical method is carried out, and it is shown that the results of the two methods generally agree with each other, sometimes almost perfectly. A comparison of natural frequencies between the semi-analytical method and the experimental data from relevant literature shows good agreements between the two kinds of results, and the semi-analytical method is validated. Different numbers of sampling points along the axial direction are used to carry out convergence study. It is found that the semi-analytical method converges rapidly. The effects of different beam lengths and heights, thermal stress, and the spring and shear coefficients of the Pasternak medium are also investigated. The results obtained in this paper can serve as benchmark in further research.

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Figures

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Fig. 1

The geometry of a thermo-elastic beam on a Pasternak foundation

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Fig. 2

The time histories of peak beam deflection at the selected position, x = l/2, in the thermal environment by the analytical and the semi-analytical method: (a) case 1, (b) case 2, (c) case 3, and (d) case 4

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Fig. 3

The time histories of peak beam deflection at a selected point, x = l/2, of the beam, for four different numbers (N) of sampling points along the x-axis: (a) case 1, (b) case 2, (c) case 3, and (d) case 4

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Fig. 4

The time histories of peak beam deflection at two selected positions, x = l/2 and x = l/5, for different boundary conditions: (a) at the point x = l/2 and (b) at the point x = l/5

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Fig. 5

The time histories of peak beam deflection at two selected positions, x = l/2 and x = l/5, for different beam lengths and fixed height: (a) at the point x = l/2 and (b) at the point x = l/5

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Fig. 6

The time histories of peak beam deflection at two selected points, x = l/2 and x = l/5, for a given beam length and different heights: (a) at the point x = l/2 and (b) at the point x = l/5

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Fig. 7

The time histories of peak beam deflection at two different selected points, x = l/2 and x = l/5, for different surrounding temperatures: (a) at the point x = l/2 and (b) at the point x = l/5

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Fig. 8

The time histories of peak beam deflection at two different selected points, x = l/2 and x = l/5, with different spring coefficients: (a) at the point x = l/2 and (b) at the point x = l/5

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Fig. 9

The time histories of peak beam deflection at two different selected points x = l/2 and x = l/5 with different shear coefficients: (a) at the point x = l/2 and (b) at the point x = l/5

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