The Barycentric Rational Interpolation Galerkin Method For Beam Bending Problems

Beam is a kind of important engineering structural members, has been widely used in such fields as civil engineering, mechanical engineering, control engineering and aerospace structures and so on, the main deformation is bending, so the study of beam bending deformation under various loads is of great significance. Mathematical model of the beam bending problem can be boiled down to the solution of the differential equation under certain boundary conditions and initial conditions. Analytical solutions of beam bending problems can be obtained only when the load is relatively simple, such as uniformly distributed load, end concentrated load and so on, the deformation of the beam is analyzed by analytical method is quite complicated even is impossible when the beam with multiple support, variable cross-section and the load on the beam is more complex, it needs to write the governing equation of each beam segment, then piecewise integral, determine a series of integral constants. Therefore, this case needs the help of numerical method.The barycentric rational interpolation Galerkin method as a numerical calculation method for solving differential equations, has advantages of simple computational formulas, easy to programming, good node adaptability, convenient application of boundary conditions and connection conditions and high precision. It is well known that rational interpolation sometimes gives better approximations than polynomial interpolation,especially for large sequences of points. Using the rational function as the interpolation basis function, not only can improve the precision of interpolation, but also can effectively overcome the instability problem of interpolation. In classical rational function interpolation, there is no control over the occurrence of poles in the interval of interpolation. Berrut and Mittelmann suggested that it might be possible to avoid poles by using rational functions of higher degree. Floater and Hormann proposed a family of barycentric rational interpolants that have no real poles and arbitrarily high approximation orders on any real interval and the interpolation function with infinite time smoothness. Barycentric rational interpolation not only has higher accuracy in the special distribution interpolation node, but also has high interpolation accuracy for equidistant nodes.In this paper,we study two numerical methods for beam bending problems:one is based on the barycentric rational interpolation function as a trial function, using the generalized function to express the governing equation of beam bending deformation and the integrated property of Delta function, the barycentric interpolation Galerkin method is proposed to solve the problem of beam bending deformation; the second, differential matrices are obtained on the element in accordance with uncontinuous intervals to divide the computing elements and approximating unknown functions on the element in term of barycentric rational interpolation, the global matrix is obtained by assembled matrices on elements and by replacement method to apply the boundary conditions and connection conditions, the barycentric rational interpolation element Galerkin method is proposed to solve beam bending problem under complex loads. The beam angle, bending moment and shear force can be obtained directly by using the differential matrix after calculated the deflection of beam,.The barycentric rational interpolation Galerkin method is applied to solve the beam bending problems, such as the concentrated force, part of the uniform load, stiffness uncontinuous, and continuous beam and so on, high precision numerical solutions can be obtained, the numerical results are presented to demonstrate the effectiveness and accuracy of the method.The barycentric rational interpolation element Galerkin method is applied to solve the beam bending problems, such as the concentrated force couple, concentrated force and the beam under complex loads, continuous beam under uniformly distributed load, the middle hinged girder, varying stiffness beam, the middle sliding beam and so on, high precision numerical solutions can be obtained with small amount of nodes, the numerical results are presented to demonstrate the effectiveness and accuracy of the method.The numerical examples demonstrated that the barycentric rational interpolation Galerkin method has advantages of simple computational formulas, easy to programming, good node adaptability, convenient application of boundary conditions and connection conditions and high precision.