Static and dynamic analysis of beam assemblies using a differential system on an oriented graph

Many systems in engineering, theoretical physics and other domains of natural sciences can be investigated using a linear mathematical model having the character of a differential system defined within a given network. This network may consist of one-dimensional elements characterised by local coordinate systems. These elements (recti- or curvilinear) are interconnected at nodes, through which energy, mass and stiffness properties of the elements are transmitted as a function of time. The system as a whole is generally determined by some boundary conditions or assumed to be interconnected with other subsystems. Elements of the system are considered to have continuously distributed parameters (mass, stiffness, conductivity, etc.). External energy may be supplied through boundary conditions or by excitation of elements at nodes. The problem of the system’s response, or a relevant eigenvalue problem, can be understood as a problem of a differential system on an oriented graph. This graph is a corresponding geometrical representation of the system investigated, where elements of the graph represent individual beams of the system. Therefore the physical part of the problem is fully included in the original differential system, but without any indication of its domain shape. As illustrations of this theoretical study, the conventional Slope Deflection Method (SDM), developed in the past for statics and later for dynamics of continuous frames are outlined in this paper, along with some illustrations from other branches. It should be noted that the character of the resulting algorithm is similar to the FEM if special macro-elements provided by direct solution of the relevant differential system are used. High numerical stability of the approach used here is a significant strength in comparison with other procedures. This follows from the principal attributes of the proposed method. Easy implementation of the theory into existing software packages is possible.