The continuation method for calculation of continuous beams of arbitrary bending rigidity

Abstarct. This article presents an effective mathematical continuation method for the numerical implementation of the multipoint boundary value problem, to which the calculation of a beam of arbitrary rigidity at any of its supports is reduced. The problem can be treated as a direct one in the matter of constructing an optimal design based on beam systems. A test example of the calculation is given.


Introduction
In computational practice, it is often necessary to carry out a strength analysis of structural elements whose computational scheme can be reduced to a multispan beam of variable rigidity under a variety of conditions of its support and loading [1,2]. Obtaining an analytical solution of such problems is associated with certain, sometimes unresolved difficulties due to the fact that the differential equation of beam bending in this case has variable coefficients and, in addition, its solutions are not continuous. This paper presents the calculation of irregular beam systems based on the continuation method [3]. Matrix algebra allows various transformations with systems of linear algebraic and differential equations in a compact form and contributes to reducing the volume of calculations by putting them into a clear and simple form. The main advantages of the continuation method are as follows: wide variation of the type and number of supports, span lengths, and other parameters without having to repeat the calculation; calculation of complex redundant frameworks by a standard algorithm, without solving systems of symmetric equations, without reducing the calculation accuracy.
The calculation is based on the following assumptions: -the system has any number of spans of different lengths; -supports of any type -rigid or resilient ones; -the end supports can be restrained, hinged immovable, hinged movable, or simple; -the intermediate supports of the beam have horizontal mobility; -the system is subjected to load distributed along the length, concentrated forces, moments; -in general, the beam is of a stepped and variable cross-section; -the system is linear and deformable, i.e. the generalized Hooke's law is valid. Based on the above method, a program for the calculation of continuous beams of arbitrary bending rigidity is made, and the automated software package "ASSISTANT" [4,5], which includes the calculation of beams, shafts, stepped beams, and other elastic systems, is created.

Setting up a problem. Continuation method. Continuation formula
The continuation method is based on the differential equation of the problem and its solution with subsequent derivatives.
The differential equation for the beam bending is written as [1]: E -modulus of elasticity, where 0 Y -initial stress-strain state; K Y -stress-strain state in the n-th section of the system; P -impact (bending) matrix, which is the product of the corresponding span and transition matrices obtained by the continuation formula.
Let us consider the range of existence of the variable x ( (2) at specified boundary conditions -arbitrary vector function of a valid argument x ; i  -numbers of constant type. In general, the solution of equation (2) may have finite discontinuities at a number of points, but within intervals is continuous, bounded, and can be conceived of as [3]: where Y has discontinuity of the first kind, then it can be written as follows: The solution 1 l i  Y included in this formula can also be conceived of as: (7) Based on formulas (6) The relation (8) expresses the mathematical meaning of the continuation method and is called the continuation formula. The

 
ii Al section matrices included here are determined from differential equation (2)

Matrices of sections, jumps, and transition through intermediate supports
The   ii Al section matrices included in formula (8), as mentioned above, are determined from differential equation (2) The i F jump matrices are determined by considering the matching conditions of the limiting points of adjacent intervals. Since the partition of the beam into sections is carried out in such a way that all the external concentrated forces and moments fall into the j -th node points (limiting points of the sections), the static components of the i Y vectors undergo discontinuities when passing through these points, whereas: where j M , j P -external concentrated moments and forces. The stress-strain state vector at an arbitrary point j of the beam is written as: where j W -deflection at the point j , In short for any case: (15) Applying (14) to (15), we obtain: It follows therefrom that: When passing through the first rigid support, there will be a jump in the magnitude of the shear force, numerically equal to the reaction 1 S arising in this support. Let us also represent the reaction 1 S as a linear dependence on 2 X : of direct and alternating current circuits, three-phase circuit, etc. The package is a multiuser system. The use of such a system of automated teaching allows you to improve the quality of students' study of disciplines and improve student performance, as well as significantly facilitate the work of the professor and reduce the time spent on control.

Conclusion
The suggested continuation method is high accurately. Due to its versatility, the developed algorithm of the continuation method greatly simplifies and standardizes the process of analyzing complex systems with a high degree of static indeterminacy, and also allows a wide range of initial parameters of the basic system of the supporting structure in the process of calculations. The continuation method can be widely used for other elastic systems: shafts, space systems, frames, plates, shells, etc. for the main types of force action: bending, stability, vibrations, transverse-longitudinal bending.