Application Of The R-function Theory For The Problems Of Mechanics Of Plates And Shells, And Torsion

For the bending and vibration problems of thin plates and shallow spherical shells, and theelastic torsion problem, only few have analytical solution for the problems with a simpleboundary shape, such as the rectangular and circular shape. Using numerical methods, such asthe variational method, to solve the problems; only the trial function can be found for theproblems with a simple boundary shape. It is difficult to solve directly the problems with acomplicated boundary shape, and the R-function theory can solve the problems. This paperintroduces the R-function theory to the following problems with a complicated boundary shape:(1) The R-function theory and the variational method are used to study the elastic torsionproblem with a complicated cross section. When the variational method is used to solve theelastic torsion problem alone, the stress function can be set to meet the boundary condition, onlywith the simple cross section shape such as the rectangle and ellipse. For a complex cross sectionshape, it is hard to find a stress function to meet the boundary condition. The R-function theorycan solve the problem, the R-function theory can be used to describe a complex domain using aimplicit function. Introducing the R-function theory, the stress function can be easily constructedto meet the boundary condition with a complex cross section shape. The variational method isused to determine the stress function expression with a complex cross section shape, and then theper unit torsion angle and shear stress are obtained. Finally, numerical examples verify thefeasibility and validity of the present method.(2) The R-function theory and the quasi-Green's function method are utilized. The bendingand free vibration problems of simply supported polygonal shallow spherical shells (includingthat on elastic foundation) are studied. The governing differential equations of the problems aredecomposed into two simultaneous differential equations of lower order by utilizing anintermediate variable. A quasi-Green's function is established by using the fundamental solutionand the boundary equation of the problem. This function satisfies the homogeneous boundarycondition of the problem, but it does not satisfy the fundamental differential equation. The keypoint of establishing the quasi-Green's function consists in describing the boundary of theproblem by a normalized equation ω=0and the domain of the problem by an inequality ω>0.There are multiple choices for the normalized boundary equation. Based on a suitably chosen normalized boundary equation, a new normalized boundary equation can be established such thatthe singularity of the kernel of the integral equation is overcome. For any complicated area, anormalized boundary equation can always be found according to the R-function theory. Thus, theproblem can always be reduced to two simultaneous Fredholm integral equations of the secondkind without the singularity. Finally, the deflection is obtained by solving the discrete equations,but for the free vibration problems, the natural frequency is obtained by the condition that thereexists the nontrivial solution in the discrete equations. The numerical examples demonstrate thefeasibility and validity of the quasi-Green's function method.(3) The R-function theory and the quasi-Green's function method are applied. The freevibration of clamped thin plates with an arbitrary boundary shape and the bending problem ofclamped orthotropic thin plates (including that on Winkler foundation) are studied. For theclamped orthotropic thin plates, firstly the governing differential equation of the problem isreduced to the boundary value problem of the biharmonic operator by introducing the parameterstransformation. A quasi-Green's function is established by using the fundamental solution andthe boundary equation of the problem. This function satisfies the homogeneous boundarycondition of the problem, but it does not satisfy the fundamental differential equation. The keypoint of establishing the quasi-Green's function consists in describing the boundary of theproblem by a normalized equation ω=0and the domain of the problem by an inequality ω>0.There are multiple choices for the normalized boundary equation. Based on a suitably chosennormalized boundary equation, a new normalized boundary equation can be established such thatthe singularity of the kernel of the integral equation is overcome. For any complicated area, anormalized boundary equation can always be found according to the R-function theory. Thus, theproblem can always be reduced to a simultaneous Fredholm integral equation of the second kindwithout the singularity. Finally, the natural frequency is obtained by the condition that thereexists the nontrivial solution in the discrete equations, but for the bending problems, thedeflection is obtained by solving the discrete equations. Numerical examples are presented forrectangular, parallelogrammic and trapezoidal thin plates to demonstrate the feasibility andvalidity of the present method.