平板結構在工程上的應用很廣，舉凡機械、航太、船舶以及土木等工程領域隨處可見，而平板應用中往往會承受力的效應而影響其應力分佈、自然頻率及穩定性，因此本研究目的在探討平板承受隨機邊緣、非均佈平面剪力的隨機穩定性。本文將假設平板的材料為一等向、均質、彈性的矩形板，平板四邊為簡支撐(simply-supported)，且受到兩對平面剪力施加於平板的四邊。文中利用虎克定律(Hooke's law)獲得應力與應變之間的關係式，並利用牛頓第二運動定律(Newton's second law)及古典板理論推導平板的運動方程式，再分別用彈性力學的愛麗絲應力函數(Airy stress function)來求得平板的平面應力分佈，以及利用漢米爾頓原理(Hamilton’s principle)及延伸的里茲法(extended Ritz method) 計算平板平面振動的響應，利用位移求得平板的平面應力分佈，並比較兩種方法所得之平面應力分佈。接著利用葛樂金法(Galerkin’s method)將側向運動方程式離散化求出自然頻率，接著利用模態分析法將耦合的離散化運動方程式進行部份解耦合，最後再利用統計平均法(stochastic average method)得到響應振幅之伊東微分方程式(Ito differential equation)，再使用伊東微分法則求得響應振幅之均方穩定準則，從而得出平板的隨機穩定邊界。
數值結果顯示不同長寬比平板之應力合力分佈，只是把長寬比為1之應力合力分佈拉長或縮短。平板承受所有受分佈力形式中，均佈邊緣剪力的挫曲負荷值會最大，而集中力形式中，以施加在平板邊緣中間的挫曲負荷值會最大。在各種分佈剪力形式中，平板受到邊緣均佈隨機剪力的穩定區域會最大；在各種集中力形式中，受到施加於平板邊緣中間的集中力，其穩定區域為最大。

The plate structure in engineering is very broad, example: the engineering machinery, aerospace, shipbuilding and civil engineering, etc. Random stability analysis of rectangular plates subjected to random non-uniform in-plane edge shear excitations in this thesis. Assume that the material property of the rectangular plate is isotropic, the plate subjected to random non-uniform in-plane shear and simply-supported at the edge. First, this investigates the in-plane stresses of the plate for two methods in this thesis, one method use Airy stress function the other use Hamilton’s principle and extened Ritz method and then investigates the out-of-plane vibration and random stability of the plate. For the in-plane stresses of the plate use two methods, the Airy stress function is used to derived the in-plane stress distributions of the plane, the other use Hamilton’s principle derived in-plane equation of the motion, next it is used to obtain discretized equations, then the stress distributions of the plane subjected to non-uniform in-plane edge shear excitations are obtained.For the out-of-plane equation of the motion of the plate by Newton’s second law, next it is discretized by generalized Galerkin’s method, then the natural frequency and the mode shapes of the out-of –plane vibration are calculated, next the out-of-plane discretized equations of the motion of the plate is used to partially uncouple by modal analysis method.Finally it is used to derive Ito differential equation by stochastic average method, then use Ito differential rule to derive second order Ito differential equation and obtain stability criteria.
Form the numerical results show that the stress distributions of the plane in the different aspect ratios, the resultant stress distribution of aspect ratio of the plate is 1 lengthened or shortened, the critical buckling load of the plates subjected to uniform in-plane edge shear excitations is the largest in the in-plane non-uniform shear force form, the critical buckling load of the plates subjected to in-plane concentration shear force in the middle of the plate edge is the largest in-plane concentration shear force form, the stability in the region of the plates also.

參考文獻

[1] C. W. Lee, “A three-dimensional solution for simply supported thick rectangular plates,” Nuclear Engineering and Design, vol. 6, pp. 155-162, 1967.
[2] N. Tahan, M. N. Pavlović and M. D. Kotsovos, “Single fourier series solutions for rectangular plates under in-plane forces, with particular reference to the basic problem of collinear compression. Part 1: Closed-form solution and convergence study,” Thin-Walled Structures, vol. 53, pp. 291-303, 1993.
[3] N. Tahan, M. N. Pavlović and M. D. Kotsovos, “Single fourier series solutions for rectangular plates under in-plane forces, with particular reference to the basic problem of collinear compression. Part 2: Stress distribution,” Thin-Walled Structures, vol. 53, pp. 1-26, 1993.
[4] A. K. L. Srivastava, P. K. Datta and A. H. Sheikh, “Buckling and vibration of stiffened plates subjected to partial edge loading,” International Journal of Mechanical Sciences, vol. 45, pp. 73-93, 2003.
[5] X. Wang, L. Gan and Y. Wang, “A differential quadrature analysis of vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses,” Journal of Sound and Vibration, vol. 298, pp. 420-431, 2006.
[6] S. K. Panda and L. S. Ramachandra, “Buckling of rectangular plates with various boundary conditions loaded by non-uniform in-plane loads,” International Journal of Mechanical Sciences, vol. 52, pp. 819-828, 2010.
[7] S. A. Eftekhari and A. A. Jafari, “A mixed method for free and forced vibration of rectangular plates,” Applied Mathematical Modeling, vol. 36, pp. 2814-2831, 2012.
[8] D. A. Simons and A. W. Leissa, “Vibrations of rectangular cantilever plates subjected to in-plane acceleration loads,” Journal of Sound and Vibration, vol. 17, pp. 407-422, 1971.
[9] S. Srinivas and A. K. Rao, “An exact analysis of the free vibrations of simply-supported viscoelastic plates,” Journal of Sound and Vibration, vol. 19, pp. 251–259, 1971.
[10] A. W. Leissa, “The free vibration of rectangular plates,” Journal of Sound and Vibration, vol. 31, pp. 257-293, 1973.
[11] C. E. Gianetti, L. Diez and P. A. A. Laura, “Transverse vibrations of rectangular plates with elastically restrained edges and subject to in-plane shear forces,” Journal of Sound and Vibration, vol. 54, pp. 409-417, 1977.
[12] L. Diez, C. E. Giánetti and P. A. A. Laura, “A note on transverse vibrations of rectangular plates subject to in-plane normal and shear forces,” Journal of Sound and Vibration, vol. 59, pp. 503-509, 1978.
[13] H. C. Chan and O. Foo, “Vibration of rectangular plates subjected to in-plane forces by the finite strip method,” Journal of Sound and Vibration, vol. 64, pp. 583-588, 1979.
[14] R. H. Gutiérrez and P. A. A. Laura, “Fundamental frequency of orthotropic plates subject to in-plane shear forces,” Fibre Science and Technology, vol. 16, pp. 149-156, 1982.
[15] S. M. Dickinson and A. Di Blasio, “On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of the orthotropic rectangular plates,” Journal of Sound and Vibration, vol. 108, pp. 51-62, 1986.
[16] C. S. Kim, P. G. Young and S. M. Dickinson, “On the flexural vibration of rectangular plates approached by using simple polynomials in the Rayleigh-Ritz method,” Journal of Sound and Vibration, vol. 143, pp. 379-394, 1990.
[17] J. P. Singh and S. S. Dey, “Transverse vibration of rectangular plates subjected to inplane forces by a difference based variational approach,” International Journal of Mechanical Sciences, vol. 32, pp. 591-599, 1990.
[18] H. P. Lee and S. P. Lim, “Free vibration of Isotropic and Orthotropic Square Plates with square Cutouts Subjected to In-plane Forces,” Composite and Structures, vol. 43, pp. 431-437, 1992.
[19] K. M. Liew, Y. Xiang and S. Kitipornchi, “Transverse Vibration of Thick Rectangular Plates-1. Comprehensive Set of Boundary Conditions,” Computers and Structures, vol. 49, pp. 1-29, 1993.
[20] N. S. Bardell, R. S. Langley and J. M. Dunsdon, “On the free in-plane vibration of isotropic rectangular plates,” Journal of Sound and Vibration, vol. 191, pp. 459-467, 1996.
[21] K. Hyde, J. Y. Chang, C. Bacca and J. A. Wickert, “Parameter studies for plane stress in-plane vibration of rectangular plates, ” Journal of Sound and Vibration, vol. 247, pp. 471–487, 2001.
[22] H. Akhavan, Sh. H. Hashemi, H. R. D. Taher, A. Alibeigloo and Sh. Vahabi, “Exact solutions for rectangular Mindlin plates under in-plane loads resting on Pasternak elastic foundation. Part II: Frequency analysis,” Computational Materials Science, vol. 44, pp. 951-961, 2009.
[23] Y. F. Xing and B. Liu, “Exact solutions for the free in-plane vibrations of rectangular plates,” International Journal of Mechanical Sciences, vol. 51, pp. 246–255, 2009.
[24] B. Liu and Y. F. Xing, “Comprehensive exact solutions for free in-plane vibrations of orthotropic rectangular plates,” European Journal of Mechanics A/Solids, vol. 30, pp. 383–395, 2011.
[25] L. Dozio, “On the use of the Trigonometric Ritz method for general vibration analysis of rectangular Kirchhoff plates,” Thin-Walled Structures, vol. 49, pp. 129-144, 2010.
[26] S. M. Dickinson, “Lateral vibration of rectangular plates subject to in-plane forces,” Journal of Sound and Vibration, vol. 16, pp. 465-472, 1971.
[27] C. Mei and T. Y. Yang, “Free vibrations of finite element plates subjected to complex middle-plane force systems,” Journal of Sound and Vibration, vol. 23, pp. 145-156, 1972.
[28] A. W. Leissa and E. F. Ayoub, “Vibration and buckling of a simply supported rectangular plate subjected to a pair of in-plane concentrated forces,” Journal of Sound and Vibration, vol. 127, pp. 155-171, 1988.
[29] S. H. shen, “Postbuckling behavior of rectangular plates under combined loading,” Thin-Walled Structures, vol. 8, pp. 203-216, 1989.
[30] X. Wang, C. W. Bert and A. G. Striz, “Differential Quadrature Analysis of Deflection, Buckling, and Free Vibration of Beams and Rectangular Plates,” Computers and Structures, vol. 48, pp. 473-479, 1993.
[31] X. Wang, X. Wang and X. Shi, “Differential quadrature buckling analyses of rectangular plates subjected to non-uniform distributed in-plane loadings,” Thin-Walled Structures, vol. 44, pp. 837-843, 2006.
[32] X. Wang, L. Gan and Y. Zhang, “Differential quadrature analysis of the buckling of thin rectangular plates with cosine-distributed compressive loads on two opposite sides,” Advances in Engineering Software, vol. 39, pp. 497-504, 2008.
[33] X. Wang and J. Huang, “Elastoplastic buckling analyses of rectangular plates under biaxial loadings by the differential quadrature method,” Thin-Walled Structures, vol. 47, pp. 14-20, 2009.
[34] O. Civalek, A. Korkmaz and C. Demir, “Discrete singular convolution approach for buckling analysis of rectangular Kirchhoff plates subjected to compressive loads on two-opposite edges,” Advances in Engineering Software, vol. 41, pp. 557-560, 2010.
[35] Y. Tang and X. Wang, “Buckling of symmetrically laminated rectangular plates under parabolic edge compressions,” International Journal of Mechanical Sciences, vol. 53, pp. 91-97, 2011.
[36] A. V. Lopatin and E. V. Morozov, “Buckling of the SSCF rectangular orthotropic plate subjected to linearly varying in-plane loading,” Composite Structures, vol. 93, pp. 1900-1909, 2011.
[37] T. Q. Bui, M. N. Nguyen and Ch. Zhang, “Buckling analysis of Reissner–Mindlin plates subjected to in-plane edge loads using a shear-locking-free and meshfree method,” Engineering Analysis with Boundary Elements, vol. 35, pp. 1038-1053, 2011.
[38] A. J. Wilson and S. Rajasekaran, “Elastic stability of all edges simply supported, stepped and stiffened rectangular plate under Biaxial loading,” Applied Mathematical Modeling, (In Press, Corrected Proof), Available online 5 July 2013.
[39] L. Y. Chung and H. Reisman, “Dynamics of rectangular plates,” International Journal of Engineering Science, vol. 7, pp. 93-113, 1969.
[40] H. Reisman and J. E. Greene, “Forced Motion of Circular Plates,” Div. of Interdisciplinary Studies and Research, School of Engineering, AFOSR Report no. 67- 0565, 1967.
[41] H. Reisman and Y. C. Lee, “Forced Motions of Rectangular Plates,”Theoretical and Applied Mechanics, vol. 3, pp. 1-69, 1968.
[42] K. Takahashi and Y. Konishi, “Dynamic stability of a rectangular plate subjected to distributed in-plane dynamic force,” Journal of Sound and Vibration, vol. 123, pp. 115-127, 1988.
[43] P. J. Deolasi and P. K. Datta, “Parametric instability characteristics of rectangular plates subjected to localized edge loading (compression or tension),” Computers & Structures, vol. 54, pp. 73-82, 1995.
[44] H. P. Lee and T. Y. Ng, “Dynamic stability of a moving rectangular plate subject to in-plane acceleration and force perturbations, ” Applied Acoustics, vol. 45, pp. 47–59, 1995.
[45] M. Ganapathi, B. P. Patel, P. Boisse and M. Touratier, “Non-linear dynamic stability characteristics of elastic plates subjected to periodic in-plane load, ” International Journal of Non-Linear Mechanics, vol. 35, pp. 467–480, 2000.
[46] A. K. L. Srivastava, P. K. Datta and A. H. Sheikh, “Buckling and vibration of stiffened plates subjected to partial edge loading,” International Journal of Mechanical Sciences, vol. 45, pp. 73-93, 2003.
[47] P. Jana and K. Bhaskar, “Stability analysis of simply-supported rectangular plates under non-uniform uniaxial compression using rigorous and approximate plane stress solutions,” Thin-Walled Structures, vol. 44, pp. 507-516, 2006.
[48] Y. G. Liu and M. N. Pavlović, “A generalized analytical approach to the buckling of simply-supported rectangular plates under arbitrary loads,” Engineering Structures, vol. 30, pp. 1346-1359, 2008.
[49] R. Daripa and M. K. Singha, “Stability analysis of composite plates under localized in-plane load,” Thin-Walled Structures, vol. 47, pp. 601–606, 2009.
[50] G. Ikhenazen, M. Saidani and A. Chelghoum, “Finite element analysis of linear plates buckling under in-plane patch loading,” Journal of Constructional Steel Research, vol. 66, pp. 1112-1117, 2010.
[51] R. L. Stratonovich, “Topics in the Theory of Random Noise,” vol. I, Gordon and Breach, New York, 1963.
[52] R. Z. Khas’minskii, “A limit theorem for the solutions of differential equations with random right-hand sides,”Theory of Probability and Its Applications, vol. 11, pp. 390-406, 1966.
[53] J. B. Roberts and P. D. Spanos, “Stochastic averaging:an approximate method of solving random vibration problems,” International Journal Nonlinear Mechanics, vol. 21, pp. 111-134, 1986.
[54] W. Q. Zhu, “Stochastic average method in random vibration,” Applied Mechanics Review, vol. 41, pp. 189-199, 1988.
[55] S. T. Ariaratnam and T. K. Srikantaiah, “Parametric instabilities in elastic structures under stochastic loading,” Journal of Structural Mechanics, vol. 6, pp. 349-365, 1978
[56] Y. Fujinrori, Y. K. Lin and S. T. Ariaratnam, “Rotor blade stability in turbulence flow. Part 2,”AIAA Journal, vol. 17, pp. 673-678, 1979
[57] S. T. Ariaratnam and W. C. Xie, “Lyapunov exponents and stochastic stability of coupled linear systems under real noise excitation,” Journal of Applied Mechanics, vol. 59, pp. 664-673, 1992.
[58] A. W. Leissa, Vibration of plates, NASA SP-160, 1969.
[59] X. D. Yang, L. Q. Chen, J. W. Zu, “Vibrations and Stability of an Axially Moving Rectangular Composite Plate, ” Journal of Applied Mechanics, vol. 78, pp. 011-018, 2011.