| Elastic rectangular thin plate is a common component of highway engineering structures,such as bridge deck,pavement slab and open culvert.In actual engineering,rectangular thin plate structures will be subjected to bending,buckling,vibration instability and other failures under static load,dynamic load or passing load.With the increasing geometric size of engineering structures,the boundary conditions and stress state of thin plate structures tend to be increasingly complex.According to the damage settings of rectangular thin plate structure in actual engineering,how to solve its static and dynamic problems has always been the focus of academic and engineering research.It is very hard to solve such kind of problems analytically with the mechanics of rectangular thin plates,due to the difficulty existing in solving methods and mathematical method.In order to solve the complex calculation and analysis of this kind of thin plate problem,our study takes the thin plate structure(Kirchhoff thin plate)as the research object,and adopted the improved Fourier series solution to solve the buckling,thermal buckling,bending and vibration of thin plate structures under complex boundary conditions,thus providing a reference for engineering design and theoretical calculation.The main research and contributions of this paper are as follows:(1)According to the current buckling instability of rectangular thin plate structure components in engineering,for the uniaxial buckling of anisotropic thin plates under the boundary conditions that one pair of edges were free and the other pair of edges were clampsupported/simply supported,two-dimensional sine-cosine series were adopted as the trial function of displacement function.Based on the improved Fourier series solution,the expressions of various high-order partial differential terms of displacement function were obtained by means of Stokes transformation.After bringing it into the governing equation,we got the expression of the forward transform of Fourier coefficient of displacement function,and then obtained the analytic solution expression of problems with unknown undetermined coefficient.A series of simultaneous linear algebraic equations can be obtained by satisfying corresponding boundary conditions.Corresponding values of undetermined coefficient through calculation were determined,and finally the analytic solution of this problem was derived.The uniaxial buckling under different boundary conditions,different transverse-longitudinal ratios and different thin plate properties were analyzed and studied.The critical load coefficient and corresponding buckling mode graph of isotropic/anisotropic thin plates were calculated.All analytical results were well fit with finite element results,which can be used as a benchmark solution to verify the numerical solution and experimental results.(2)According to the thermal buckling background of rectangular thin plate structures under the impact of temperature,for the thermal buckling problem of thin plates under the boundary conditions that combined clamp-supported with simply supported,we combined thermoelastic theory with Kirchhoff’s small deflection thin plate theory,deduced a governing equation for thermal buckling problems of anisotropic thin plates,and selected twodimensional sine series as the trial function of displacement function.Based on the improved Fourier series solution,we obtained the accurate results of the critical temperature and corresponding thermal buckling modes of isotropic/anisotropic thin plates under CCCC,CCSC,CSSC,CSCS and CSSS boundary conditions,by taking the same solving process as for the uniaxial buckling problem,analyzed and studied the thermal buckling results under different boundary conditions and different thin plate properties.The results were well fit with finite element results,and thus verified the effectiveness of the solving method.They can provide a theoretical support for optimizing the structural design of pavement and bridge deck.(3)The bending of cement concrete pavement slab with rectangular thin plate structure as the component was analyzed,and the supporting condition of dowel bar was simplified as the homogeneous spring.Based on the improved Fourier series solution,the analytic solutions of thin plate buckling under the conditions of free edges v.s.elastically rotationally restricted edges(RFRF)and four elastically rotationally restricted edges(RRRR)were solved analytically.In this derivation process,the two-dimensional sine-cosine series and twodimensional sine series were selected as trial functions,with the same solving process as for the uniaxial buckling and thermal buckling problems,the solution to problems was obtained.The deflection and internal force under different length-width ratios and different loads were analyzed and calculated.At the same time,by changing spring parameters,we also obtained the accurate results of buckling problems of isotropic/anisotropic thin plates under SFSF,CFSF,CFCF,CCCC,CCSC,CCSS,SCSC and CSSS.Through a comparison of calculation examples,it was found that the results obtained by this solving method fully satisfied boundary conditions,and were well fit with finite element results and literature results.The results obtained by the transformation of the problem to be solved can not only be used to verify other numerical solutions,but also serve as a theoretical reference to optimize the structural design of pavement.(4)Given the vibration effect of thin plates and the complexity of boundary conditions in engineering,for free vibration problems of isotropic/anisotropic thin plates under the constraint conditions that two adjacent edges were free and the other two edges rotated elastically,twodimensional half-sinusoidal series was selected as the trial function of displacement function,and the natural frequency and corresponding natural vibration mode graph were obtained using a two-dimensional improved Fourier series solution.In addition,by adjusting the rotational fixation coefficient that was led in,a variety of classical boundary conditions can be simulated,and the exact analytic solutions of free vibration problems of isotropic/anisotropic thin plates under CCFF,CSFF and SSFF boundary conditions can be obtained without the need to further deduce and program.At the same time,the vibration characteristics of isotropic/anisotropic thin plates under different width-length ratios were analyzed.The vibration law of RRFF square slabs with different rotational fixation coefficients was studied.At last,finite element numerical calculation was used to analyze and verify,more than 400 synthetic analytic solutions obtained by this method were compared with the results of other analytical and numerical methods.Good consistency was found,which proved the accuracy and effectiveness of the proposed method in this paper.Our results provide a reference for the analysis of buckling,bending and vibration behaviors of rectangular thin plate structures.They can be used as a benchmark solution to verify other numerical solutions and corresponding experimental results and lay a theoretical foundation for the optimization of concrete pavement slab,bridge deck and other structures. |
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