Complex bend of multilayered concrete rods

The problem of complex bend of multilayered rods based on concrete is considered. It is assumed that the rod of constant cross-section is of arbitrary shape and different brands of concrete are used in the cross-section of the rod layer by layer. The solution is sought by the small parameter method. The case of a complex bend of the rod pinched at both ends is considered as an example of this solution method. The distribution of bending moments and longitudinal forces in the zero and first approximations is determined.


Introduction
Rectilinear rods of various cross-section are widely used in different branches of mechanical engineering, aircraft and shipbuilding, in civil and industrial construction projects. During operation, they are affected by thermal, chemical and kinematic forces, and due to the production of the same type of serial elements, they are subject to increased requirements for reliable operation in short-term and long-term modes. In modern economic conditions, these requirements cannot be implemented with the use of traditional structural materials and therefore in recent decades, technologies have actively been developed to make heterogeneous and multilayered structures in which materials with different properties can work together to achieve a common goal: long-term and reliable operation at a reasonable cost of its maintenance. The paper deals with structures that belong to the category of layered rods with significantly different properties in the layers, which should be determined by special experiments and taken into account in the developed calculation method. Such method should be sufficiently reliable, relatively simple and flexible to take into account a wide range of possible variations of conditions of fastening, loading and structural forms of sections. The corresponding solutions for structures operating under flat bend conditions are obtained in the works [1], [2], [3], [4]. For complex bend conditions, the corresponding solutions are absent. We will consider concretes as the material of each layer. In calculations, we will take into account that concrete behaves almost as linearly elastic under tension, while under compression the diagrams show significant nonlinearity even at low loading levels.

Methods
As a solution method of the problem, we use the method of a small parameter which showed high efficiency in solving a wide class of problems [5], [6], [7], [8], [9]. We take the law of deformation for all materials as where A i , B i -are coefficients determined from experiments. The deformation ε for each layer should not exceed the limit values at tension ε + * i and compression ε − * i (figure 2), i.e. dependencies (1) should be determined on the segment The ratios [10] obtained from the analysis of the diagram (figure 2) can be used to determine the coefficients A i , B i .
We differentiate the ratios (1), then obtain where we get where E i is an elasticity modulus of concrete of i-th layer.
Considering that the point (-ε − * i ,-σ − * i ) in the diagram (figure 2) is the extremum point, we obtain from the ratios (1), (2), (4) Taking into account that the tensile strength σ + * i is much less than compression strength σ − * i , we consider the following ratios to be valid Thus, when using approximations (1) it is enough to have three traditional characteristics σ + * i , σ − * i , E. If there are diagrams of concrete deformation of each layer of the rod under tension and compression, the coefficients A i , B i in equations (1) can also be obtained by the method of least squares. Table 2 shows the values of the coefficients of equations (1), obtained by the formulas (4), (5), (6) and the method of least squares for concrete grades B10, B30, B50 on the basis of deformation diagrams obtained in experiments [11].   Comparisons of deformation diagrams constructed by approximating formulas (1), the coefficients of which are determined by the method of least squares and by the ratios (4), (5), (6), with experimental diagrams are shown in figures 3, 4. It is visible from the figures that experimental diagrams approach the solution well enough at deformations close to the limit.
We rewrite the deformation condition for i-th layer in the form where To solve this problem, we introduce a small dimensionless parameter δ, which is determined according to the ratio (7) as If we determine the parameter δ at i = 1 and assume that the first layer of the rod is made of concrete grade B10, then from 2 we obtain the value of the parameter δ ≈ 0, 0038.
We rewrite the ratios (7) in the form where Using the classical kinematic hypotheses of Kirchhoff-Lyav, for deformations we have expressions [12] ε(x, y, z) = ε 0 (x) − yκ z (x) + zκ y (x), where We consider that the values σ(x, y, z), u 0 (x), υ 0 (x), w 0 (x) depend on the specified parameter δ. We expand them into series of degrees of the specified parameter where the components at tension and movements get the index (n), corresponding to the degree of parameter δ. From ratios (10), (11), (12), we obtain the expression for deformations Substituting (12), (13) in the expression (9) and equating the ratios at identical degrees of δ, we obtain for the zero and first approximations Longitudinal forces and bending moments can be determined from the ratios Equilibrium equations have the form [12] where N is projections of the vector of distributed loading of the rod attached to the axis, M z , M y are projections of the vector of distributed moments on z and y axes. Substituting the expansion for stresses (16) in (12) we obtain where From equilibrium equations (17) and ratios (18), we obtain for the zero approximation From (13), (14), (19) we determine the longitudinal force and bending moments where From (17), (21) we obtain a system of linear differential equations to determine the movements in the zero approximation dx 2 and substitute in the first two. We obtain a system of two equations and two unknown functions where where Where we get from (24), (25) where the values of constants C 1 , C 2 , C 3 , C 4 , C 5 , C 6 , C 7 , C 8 can be determined from fastening conditions of the rod. From the last equation of the system (22), we obtain We substitute the found values υ (0) 0 , w (0) 0 into expression (26), and from the fastening conditions we get the expression for u (0) 0 . Then we can obtain an expression for deformations in the zero approximation We substitute the obtained value (27) in (15) and have We define the longitudinal force and bending moments in the first approximation where We substitute the obtained values (29), (30), (31) in equilibrium equations and receive a system of three differential equations to determine the movements dx 2 = 0. We rewrite the system (32) as where we get the expressions to determine the movements in the first approximation where , Having integrated the equations (34), (35), (36) and defined the integration constants from fastening conditions of the rod, we can determine the movements in the first approximation.
From equations (29), (30), (31), we can find the expressions for longitudinal force and bending moments in the first approximation.

Results
As an example, we consider the problem of complex bend of a rod pinched at both ends (figure 5) by distributed forces and moments where q 0x , q 0z , m 0 , α 1 , α 2 are constants. We assume that the cross-section of the rod has the form figure 6.  F(b 03 , h 3 ).