Applications of the Integral Operators Method in Beam-Column Element

An integral operators method has been advocated for computing stiffness matrix and load matrix of special nonlinear or non-uniform beam-column element. In the method, the complicated integrating course over the entire element are divided into some regularized integral operators, and the internal forces of the element excited by nodal displacements or distributing element loads is explicitly expressed with these integral operators. And then, the stiffness matrix and load matrix can be directly obtained. The method is derived from basic structural mechanics without any hypotheses. Its result is an analytical solution whose accuracy is just determined by the accuracy of the computational methods of the integral operators. The method provides a possible way to accurately compute material nonlinear or non-uniform beam-column element, especially for load matrix. The method can overcome the deficiencies of the common methods in solving practical engineering problems. The common methods, which are generally based on interpolating shape functions and numerical integration, have some intrinsic shortages, i.e. 1) The shape functions are artificially assumed. The selected functions that lead to exact solutions for uniform member can not be simply extended to the non-uniform member. 2) The precondition of obtaining exact solutions by numerical integration is that the integrated expression is the continuous polynomial. However, this precondition is not satisfied in most engineering problems, which eventually leads to systemic errors. 3) The points of numerical integration are finite, and their sites are relatively stationary. We can not freely densify or move them to adapt mutating stiffness. 4) The load matrix is difficult to be exactly computed. With the displacement shape function, the mutating stiffness is primarily neglected in the course of integrating load matrix. And, with the force shape function, only very simple element load can be treated. In this paper, some applications of the integral operator method are to be introduced. The computation of prestressed member is a typical problem about load matrix. The proposed method is successfully used in this field, especially for varied beam or polygonal cable style which can not be solved by the common methods. The fiber element is widely used to analyses nonlinear capabilities of reinforced concrete member. The traditional methods in this element encounter all the previously described problems. In order to fully embody the influence of the non-uniform stiffness, the flexibility method with force shape function has been employed. However, this treatment was not sufficient to compute common distributing load, especially the important unbalanced internal forces. The proposed integral operator method can solve these problems.