Solution Of Rectangular Plates With Four Free Edges On Elastic Foundation By Variational Principle

In many fields of engineering,the analysis of many structures or elements of structures,such as raft foundation of buildings,flood gates, airport runways,rigid highway surfaces,and boatyard soleplates,can always be reduced to analyses and computations of the plates on elastic foundation and their components.The analysis of the plates on elastic foundation is about the coaction of the two media soil-structure, especially about the bending of rectangular plates with four free edges on elastic foundation,which has long been regarded as a difficult problem in the classical plate theory.The mechanics of elasticity generally resort to two descriptive methods in mathematics,namely the boundary value of differential equation,and the extrema of coefficients functional.These two methods are used in this thesis on the study of the rectangular plates with four free edges on elastic foundation.Pertaining to these two solutions,it is usually rather difficult to solve differential equations,while it is easier if getting an approximate solution from the variation of functional.According to the soil-structure coaction theory,Kirchhoff thin plate theory,Rayleigh-Ritz theory based on the least potential principle,by analyzing the constituting characteristics of exact analytical solutions, this thesis changes the bending of rectangular plates with four free edges on elastic foundation to energy function in deflection function,and then solves the extremal problem of coefficients functional.It also chooses the deflection function that satisfies all the boundary conditions and free corner conditions as the solution of variation principle,calculates the deformation and internal force of the bending of rectangular plates with four free edges on elastic foundation,judges the convergence property and the superposition property of the trial function using PKPM finite element software,and does the precision analysis of the results of the trial function.Through examples of numerical value,this deflection trial functions has the superposition property,and a highly precise result of deflectional deformation can be achieved.This variational solution principle is clear,and the method is simple(only with two unknown variables),so it has an extensive prospect of being applied in engineering projects.And thus the expression of the deflection trial functional is the explicit function with the value of the coordinate points x,y on the plate as the variables.The expression of the deflection function can not only work out the static force on the plate,but also work as the mathematical model of further studies on the free vibration and stability of the plate.At the same time,through the examples of numerical value,we find that Rayleigh-Ritz theory is proximate calculation,and the trial function is only part of admissible functions,so the extremum of functional is only partial instead of the whole.Therefore,the trial function is just an approximate solution,with the calculation of the internal force being the second differential of the deflection trial function,which causes the calculation of the internal force is not so perfect within the small areas around the action spot of the concentrated force.While because the deflectional trial function satisfies the conditions of the boundary force,the closer to the plate boundary,the more approximate is the internal force calculations to the finite element method.To summarize,taking advantage of the proximate elastic plate principle recommended by American Concrete Institute(ACI),the author transforms the bending of rectangular plates with four free edges on elastic foundation to axial symmetry of the polar coordinate in elastic mechanics,and figures out directly the boundary value of the control equation of the bending differential coefficient.Therefore,the internal force calculations within the small areas around the action spot of the concentrated force is approximate to the finite element method.