Calculation methodology of reinforced concrete elements based on calculated resistance of reinforced concrete

Calculation methodology of reinforced concrete elements based on the calculated resistance of reinforced concrete is presented. The basic dependence which allows setting the strength of bending sections and noncentral compressed elements is obtained. The proposed method for calculating reinforced concrete elements is based on the use of nonlinear diagrams of material deformation, the hypothesis of flat sections and deformation criteria for the destruction of materials. The basic equations of strength are reduced to dimensionless quantities and are tabulated. When compiling the tables, the formula proposed in Euroсode 2 was adopted as the diagram of concrete deformation, and for the reinforcement two linear Prandtl diagram was used. The calculated formulas of the proposed method fully correspond to the formulas of the classical resistance of materials, and make it possible to solve the most frequently encountered problems in the practice of modern construction. The reliability of the dependencies is experimentally confirmed. There are calculation examples of bending and non-central compressed elements by the developed methodology. 1 Review of recent studies and publications Design of reinforced concrete elements based on nonlinear dependencies are listed in the standards of many countries [1, 2, 3], has set a question about their practical calculation for most of the engineers. This is because the usage of the formulas of these rules without a computer [1, 2, 3] is practically impossible. It is generally accepted that any calculation by using computers should be checked or evaluated through classical and practical techniques. Creating such a technique would allow making an assessment, verification and analysis conducted computer calculations. Simplified and approximate methods [4, 5, 6, 7] not correspond to exact solution that is obtained by nonlinear dependencies in most of cases. Consider the calculation of resistibility of sections of flexural reinforced concrete elements with a single reinforcement. Carrying force of these elements must be determined by following conditions: * Corresponding author: dim7@ukr.net DOI: 10.1051/ , 02020 (2017) 71160202 116 MATEC Web of Conferences matecconf/201 Transbud-2017 0 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). 1) achievement of extremum of function of carrying force 0 / = c d dM ε in limits cu c ε ε − 1 and occurrence of fluidity in the stretched reinforcement; 2) achievement of boundary deformation of concrete compressed brink cu ε and limits of fluidity in the stretched reinforcement in the absence of extremum of function of carrying force; 3) occurrence of extremum of function of carrying force 0 / = c d dM ε in limits cu 1 c ε ε − without achievement fluidity of reinforcement; 4) achievement of boundary deformation of concrete cu ε without achievement fluidity of reinforcement and in the absence of extremum of function of carrying force in limits cu 1 c ε ε − ; 5) achievement deformations in the stretched reinforcement with value of cu ε . 2 Basic material and results Calculation of resistibility performed by using the equation of equilibrium of external forces and internal forces, deformation diagrams of concrete and reinforcement and the function of changes of deformation by height section. As a function of diagrams of concretes deformation should be taken like function of stresses in the concrete which would correspond to the conditions of nonlinear deformation of concrete. The following dependences are proposed for calculations of nonlinear structures in the standards [1]: formula of rules Euroсode 2 (3.4), fifth degree polynomial (3.5), twoand three a linear dependence between the stresses and deformations. The dependence of stress-deformation for reinforcement is taken as two linear Prandtl diagrams. The distribution of deformation by height section at the moment of the destruction taking in a linear dependence like: , 1 , 1 1 x r or r x or x r ε ε ε = = = (1) where ε – relative deformation of the material at a distance х from neutral line, r / 1 – curvature of element in section. In most cases, diagrams of concrete deformation are function of related strength and deformation characteristics: the calculated resistance of concrete to axial junction c f , concrete deformation module c E and limits of concrete deformation 1 c ε , cu ε . This can be expressed in the following functional dependence ). , , , ( 1 cu c c c c E f f ε ε σ = (2) There are many formulas that associate deformation and strength characteristics of concrete. The following expressions can be written by its summarizing ), ( c c f f E = ), ( 1 c c f f = ε ). ( c cu f f = ε (3) With taking into account (3) the final dependence (2) will take the following form ). ( c c f f = σ (4) The function laid down in the current norms [1, 2] will be examined for further research. It is also called Eurocode function DOI: 10.1051/ , 02020 (2017) 71160202 116 MATEC Web of Conferences matecconf/201 Transbud-2017 0


Review of recent studies and publications
Design of reinforced concrete elements based on nonlinear dependencies are listed in the standards of many countries [1,2,3], has set a question about their practical calculation for most of the engineers. This is because the usage of the formulas of these rules without a computer [1,2,3] is practically impossible. It is generally accepted that any calculation by using computers should be checked or evaluated through classical and practical techniques. Creating such a technique would allow making an assessment, verification and analysis conducted computer calculations. Simplified and approximate methods [4,5,6,7] not correspond to exact solution that is obtained by nonlinear dependencies in most of cases.
Consider the calculation of resistibility of sections of flexural reinforced concrete elements with a single reinforcement. Carrying force of these elements must be determined by following conditions:

Basic material and results
Calculation of resistibility performed by using the equation of equilibrium of external forces and internal forces, deformation diagrams of concrete and reinforcement and the function of changes of deformation by height section. As a function of diagrams of concretes deformation should be taken like function of stresses in the concrete which would correspond to the conditions of nonlinear deformation of concrete. The following dependences are proposed for calculations of nonlinear structures in the standards [1]: formula of rules Euroсode 2 (3.4), fifth degree polynomial (3.5), two-and three a linear dependence between the stresses and deformations. The dependence of stress-deformation for reinforcement is taken as two linear Prandtl diagrams. The distribution of deformation by height section at the moment of the destruction taking in a linear dependence like: where ε -relative deformation of the material at a distance х from neutral line, r / 1curvature of element in section.
In most cases, diagrams of concrete deformation are function of related strength and deformation characteristics: the calculated resistance of concrete to axial junction c f , concrete deformation module c E and limits of concrete deformation 1 c ε , cu ε . This can be expressed in the following functional dependence The bending reinforced concrete elements at a single reinforcement are proposed to consider. After putting equilibrium equation and conducting simple transformation with considering the hypothesis of flat sections we will get: -for non-overreinforced Both sides of equation (7) і (8) are divided into c f and taking into account (6), we will get: -for non-overreinforced The name parameter ω is mechanical reinforcement coefficient [4,5]. The formulas (9) and (10) with introduced notations (11) will take the following form: -for non-overreinforced The notation is introduced to both equations ( ) Thereby -for non-overreinforced Introduced parameter z k generally depends on the mechanical reinforcement coefficient ϖ and deformation characteristics of concrete. There are more details about the impact of deformation characteristics of concrete on bearing capacity of bending elements. Here is a submission of parameter dependence schedule z k depending on the mechanical reinforcement coefficient ϖ for different kinds of concrete (Fig.1). The impact of deformation characteristics of concrete (parameters on bearing capacity of bending reinforced concrete elements for considered kinds of concrete varies with differences within 10% (Fig. 2). This error is completely allowed by normative coefficients of variation of strength for such elements. This makes possible to offer slightly simplified force model of reinforced concrete elements calculation. The impact of deformation characteristics of strength sections is ignored and common functional dependence is taken ) (ϖ f k z = for all kinds of concrete. This dependence is true not only for different kinds of concrete and reinforcement and even for different duration of the load. Formulas for non-central compression can be derived by similar arguments. Dependence of parameter z k from the mechanical reinforcement coefficient ϖ is presented in tables. where zM f − estimated resistance of reinforced concrete on bend, which depends on kinds of concrete and reinforcement, section shapes and percent of reinforcement section that defined by the expression (17). Parameter z k in expression (7) is calculated by

Conclusions
The methodology of practical calculation of strength of reinforced concrete elements based on deformation model is proposed. The tables are compiled based on this methodology allowing to quickly and easily perform calculations of strength of overreinforced and nonoverreinforced beams by deformation methodology without using computer programs. To simplify the work of designers, engineers-practitioners and students of construction specialties it is appropriate to include the table into design standards of concrete elements.