The Ritz Method with Discrete Approximation of Displacements for Analyzing Plates of Bridge Structures Supported by Stiffeners of Various Configurations

The article considers plates with stiffeners, the two opposite ends of which are fixed rigidly, and the other two are free. The stiffeners are located in the same direction, parallel to the free edges of the plate. The paper proposes a method for analyzing such structures by combining the reduction of the dimensionality of the problem using the L. V. Kantorovich method and the Ritz method with discrete approximation of displacements. The methods of defining stiffeners using unit column functions are considered. It is revealed that unit column functions are a convenient way to model stiffeners, even when it is necessary to consider stiffeners in boundary conditions. The stiffeners in the shapes of a box, a T-beam, an I-beam and a solid rectangular section are considered. Their geometric characteristics are given, which are used when constructing the functional of the total potential energy taking into account their discrete location. The calculation results obtained by the proposed method are compared with the solution obtained using the structural anisotropy method for the solution of the boundary value problem for ordinary differential equations. In a numerical experiment, the number of finite elements along the stiffener’s width is established, which is necessary and sufficient to solve the problem. Graphs of displacements of plate middle zones are printed, on the basis of which it is concluded that the proposed method has a fairly good convergence with the method of structural anisotropy. However, the plates calculated by these two methods deform differently and the Ritz method with discrete approximation gives a more accurate form of deformation: the plate deforms less in the area of the stiffeners and more between them. From the calculation results, it can be concluded that the presented method is the most optimal in terms of labor complexity and accuracy. However, if the ratio of the total width of the stiffeners to the width of the plate is 1:3 or more, the structural anisotropy method can be used as a simpler one.