Calculation of three-layer bent reinforced concrete elements considering fully transformed concrete deformation diagrams

The article deals with the method for calculating the three-layer bent reinforced concrete elements, taking into account the total deformation diagrams of different concrete layers. There are formulas and calculations in cases of presence and absence of cracks in the tension zone.


Introduction
Design of bent reinforced-concrete elements according to the procedures of norms providing for rectangular tension diagram in the compressed and stretched zones of concrete, respectively, for determining the strength and fracture toughness in some cases leads to substantial deviations of experimental and theoretical data [1][2][3][4][5]. Different ways to improve these techniques allow to bring together experimental and theoretical results, but they do not eliminate the main drawback of ignoring full of concrete deformation diagram. Noted above in some cases can lead to substantial deviations of the theoretical and experimental values [6][7][8][9]. As mentioned in a number of works of scientists, the most perspective direction for further improvement of calculation methods of reinforced concrete elements is the inclusion of formulas with complete descending branches concrete deformation diagrams in the compressed and stretched zones of the element. Using this approach, you can obtain analytical dependence describing the heavy-deformed state at all stages of the construction load. Also it provides a unified approach to the determination of the strength, toughness and crack resistance of concrete elements [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. It should be noted that the vast majority of works was devoted to the development of new and innovative calculation methods of reinforced concrete bent elements of solid section. There is not enough similar work relating to the elements of the laminate section.

Literature review
Investigation of reinforced concrete columns with recessed longitudinal rods without transverse reinforcement is a very actually problem of modern building and constructions.  [2];Effective concrete for agricultural construction [3];Guidance on the computer calculation of concrete columns with mixed reinforcement [4]; Recommendations for the design of eccentrically compressed reinforced concrete three-layer structures with high reinforcement [5]; Threelayer load-bearing panel [6]; Double pre-tense reinforced concrete columns [7]; Flexible double layer evenly crimped concrete panels [8]; Impact of pre-tense uneven sections of reinforced concrete elements [9]; Calculation of two-layer pre-tense concrete panels [10]; Two-layer reinforced concrete panels with uneven compacted sections [11]; Strength concrete columns with high pre-compression fittings [12]; Features of the calculation of the girder with pre-compressed reinforcement of the upper zone s [13]; Automated calculation of bent elements combined pretension reinforcement [14].
-the diagram of concrete strain is taken as the initial, recommended by SEB-FIP [1], which in this case takes the form: (1) This estimated diagram (1) is considered to be right for compressed and stretched fibers of different concrete layers: -before cracking, sections remain plane during the deformation, so the hypothesis of plane sections is considered to be right; after cracking warping of sections is taken into account by the developed technique [1]; -description of the analytical chart of deformation of high strength steel and its changes caused by pretension are taken as described in [2][3][4][5][6]; -neutral axis diagrams of deformation and tension are the same, which is justified for a short time uploading (no time to manifest nonequilibrium deformation), and prolonged uploading is valid only for the elements in which the stiffness of the compressed and stretched zones are changed simultaneously [10].
The adoption of the prerequisites mentioned settlement allows with one voice to determine strength, toughness and crack resistance of concrete elements for any external operational impact and pretension. The calculation starts with the selection of the initial value of an external force, certainly less destructive. Each successive value of the efforts of the new stage of the calculation is determined from the expressionܰ ݈ = ܰ ‫1−ܭ‬ + ∆ܰ ‫ܭ‬ where ‫ܭ‬ = 1,2,3 …the forces number N.
In monotonic loading member there are two stages of the work. The first stage -is the work item without cracks in the tension zone. Due to the fact that the section is solid, extreme concrete fiber deformations are interconnected with the expressions: Moment of fracture is the limit state in the first stage of the work, in which the deformation of the stretched fiber and the tension At the moment of cracking, function‫ܯ‬ = ‫ܯ‬൫Ԑ ‫ݐܾ‬ ൯ reaches a maximum at a value of The second stage of work is characterized by the presence of cracks. The deformation of the stretched fibers of the crack is assumed to be‫ܭ‬ ߝ • ߛ ‫ݐܾߝ‬ • ߝ ‫ݐܾ‬ തതതത , thus, the descending branch transformed diagram is realized "ߪ ‫ݐܾ‬ − ߝ ‫ݐܾ‬ ". The ultimate state of the second stage of the work is the beginning of the destruction of the state in which the extreme deformations of compressed fibers reach the value ߝ ܾ = ߝ ‫ݑܾ‬ (ߝ ‫ݑܾ‬ > ‫ܭ‬ ߝܴ ߛ ߝܾ ߝ ܾܴ ), and the tension respectively, ߪ ܾ തതത = ߪ ‫ݑܾ‬ (ߪ ‫ݑܾ‬ < ‫ܭ‬ ܴ ߛ ܴܾ ߝ ܾ ). With the destruction, the function ‫ܯ‬ = ‫ߝ(ܯ‬ ܾ ഥ )reaches a maximum at the appropriate value of the deformation ߝ ܾ ഥ = ߝ ‫ݑܾ‬ , т.е. ‫ߝ݀/ܯ݀‬ ܾ ഥ = 0. After the beginning of the destruction, the element continues to be deformed by declining external force. The descending branch of the diagram appears "ܰ − ߝ ܾ ഥ " или " ܰ − ӕ".
In general, the static system of equations is written as follows: In the first stage we accept К ܴ = 1; К ߝܾ = 1; ߛ ܴܾ = 1; ߛ ܴܾ = 1 , that is, transformation diagram is not performed.
Due to the fact that the number of unknowns exceeds the number of equations, it was necessary to describe the relationship of unknowns using the following equations: Based on the hypothesis of plane sections we can write the equation of deformation relationship of steel with extreme deformations of concrete fibers: Solving the system of equations (5) ... (9) assuming an elastic reinforcement work, wherein: If the condition σ ‫ݏ‬ ≪ σ`݁ ݈ (where σ ‫ݏ‬ ≪ σ`݁ ݈ -the new value of the elastic limit) is not satisfied, the calculation is repeated. In this case, the equations (5) ... (9) are added equations relating tension and deformation of high-strength steel [8][9][10][11][12]. In the presence of pretension, the deformations are determined first ߝ sp , due to the efforts of pretension, and then, using the same method of stress σ ‫ݏ‬ . As a result, we obtain the solution of systems of equations ‫,ݔ‬ ‫,ݕ‬ ߝ ܾ , ߝ ‫ݐܾ‬ ,σ ‫ݏ‬ ,σ`‫.ݏ‬ At each stage of loading, the force is determined, perceived stretched concrete area concerning very short fibers: When the function ‫ܯ«‬ ‫ݐܾ‬ -ߝ ‫ݐܾ‬ » peaks The next step is performed the transformation of deformation diagrams of concrete layers, depending on the strain gradient (tension) and the effect of pretension. For this purpose, the tension diagram in concrete is used because of the pretension force, determined previously. Depending on the level and mark of the initial (pretension) and resign (from the external load) tension in the concrete, each concrete fiber ratios ߛ ܴܾ and ߛ Ԑܾ are determined in accordance with the established methodology [1-3].
Upon receipt of the new tension diagram in concrete, diagrams re-transformation "ߪ ܾ − ߝ ܾ " of all concrete layers is made and the original system of equations is solved again, the moments ‫ܯ‬ ‫ݐܾ‬ are defined and cracks formation is checked.
The calculation is repeated to achieve the desired convergence and then a new value is given by adding the increment efforts ∆ܰ ‫ܭ‬ .
Equations of equilibrium of internal and external forces in determining the effort of cracking and breaking force have an identical view, which provides a common approach to the assessment of the strength and fracture structures. This approach also makes it possible to determine the arching columns and deflections at all stages of the work.