Formation of Moments Influence Matrix in Frame Based on Graph of its Design Scheme

Abstract. A procedure is proposed for the automated construction of the bending moments influence matrix in the frame based on the graph of its discrete model. Forming the incidence matrix of the graph characterizing the topological structure of the design scheme of the frame, through matrix transformations of the displacement vectors, at the transition from local coordinate systems to a global system, it is possible to establish the relationship between the nodal displacements and the displacement increments of individual sections in the direction of the axis of the segment and perpendicular to it. The composition of only three initial matrices of the frame structure, the incidence matrix of the graph, the node coordinate matrix and the matrix of the frame model internal rigidity, solves the problem of automatic formation of the bending moments influence matrix with the help of a PC. The procedure proposed for the construction of the influence matrix, makes it possible to find the forces in the frame structure caused by external load, in the matrix form.

During the intensive development of the finite element method in the second half of the 20 th century, a number of Russian scientists in the field of structural theory attempted to create alternative numerical methods to calculate frame structures with the help of computers [1] - [4]. In particular, A.R. Rzhanitsyn proposed to analyze the stress-strain state of frames in matrix form based on the duality principle of problems in structural theory. Representing any elastic frame structure in the form of a connected set of bar segments obtained by discretizing the system, a geometric [1] (kinematic [3]) matrix characterizing the deformed state of the structure, was taken as a key for solving the problem. In accordance with the duality principle, the stress state of the system in this case is determined by the equilibrium matrix, which is found by transposing the geometric matrix. As a rule, these authors limited themselves to the consideration of the bending deformations of frames, although the regard of the longitudinal deformations of the bars, as an example, presents no particular difficulties [4].
At that time, to create a fully automated method for the formation of a geometric matrix, it was sufficient to provide it with a description of the frame structure, its geometric parameters, as well as physical and mechanical properties, in matrix form. This description is completed in this work. As a result, the point at issue is the development of an automated method for frame structure calculation, which is an alternative to the finite element method.
Further on, according to the design scheme of the frame (Fig. 1, a), which was taken into consideration in article [1], along with the basic operations of the numerical method, including initial steps associated with the specification of the frame design scheme, structure and physical and mechanical properties in digital form, a calculation algorithm is presented, the ultimate goal of which is the formation of the frame bending moments  Referring to the discrete model of the frame (Fig. 1, b) and using the methods of graph theory, the topology of the frame structure can be described using the model graph ( Fig. 1, c) and the accompanying incidence matrix [5], [6]: Thus, to describe the geometric scheme of the frame structure, only two matrices are sufficient -the node coordinate matrix and the incidence matrix.
The physical and mechanical properties of frame elements can also be described via matrices. If bending deformations prevail in the system, then the stiffness cell (matrix) of any bar segment has the following form [ ] ( ).  The graph incidence matrix [S], along with the node coordinate matrix [K] and internal stiffness matrix [C], serves as the basis for the development of an algorithm for calculating frame structures, which is presented below. The ultimate goal of the algorithm is to form a bending moments influence matrix [ ] Λ in the frame. In contrast to [1], in this article the calculation process is proposed to be totally formalized so that all calculations can be performed automatically.
The first step of the algorithm for the numerical calculation of the frame is to form the quartic block matrices with the help of which the transition from global displacements of nodes Fig. 1, d).  Fig. 1, d). The numbers outside the bracket of the matrix [ ] Θ are noteworthy. The elements of the matrix located within the rows with Arabic numerals refer to the sections of the frame indicated in Figure 1 including the support ones (Fig. 2). When forming the geometric matrix of the frame under consideration, it is necessary to pay attention to the peculiarity that due to the presence of an independent parameter of the deformed state of the frame, namely, the angle of rotation of the middle node by the angle ϕ ζ ≡ 25 , which corresponds to the deformation pattern caused by the rotation of the node (Fig. 2, b). It should be remembered that the angular displacement index 25 ζ corresponds to the initial numbering of independent displacements of the problem under consideration in the global coordinate system. Due to the reduction in the number of components of the global displacement vector, at this stage, one more reindexing is performed, namely, now All operations concerning the matrix compilation can be fully automated and performed on a PC. The construction of the frame graph and the accompanying incidence matrix is also performed automatically [9]. The values of the bending moments in the design sections are found by the following formula

Conclusion
The application of geometric matrix [ ] Н , which is formed on the basis of the graph of the frame design scheme, makes it possible to form the bending moments influence matrix and, thereby, perform the frame structure calculations in the automatic mode. The proposed calculation method can be considered as an alternative to the finite element method.