THE NEW HYPOTHESIS ANGULAR DEFORMATION AND FILLING OF DIAGRAMS IN BENDING WITH TORSION IN REINFORCED CONCRETE STRUCTURES

The authors considered a simple method for constructing bend-torsion functionals by grid methods. Analysis of the diagrams of angular deformations and shear stresses made it possible to develop a new hypothesis of angular deformations. The consequences of the hypothesis were in the form of expressions from the analysis of diagrams. The authors also obtained functionals for determining angular deformations, bending and torque moments from the compressed area of concrete and reinforcement. The projection ratios helped to determine the shear and normal stresses through deformations using diagrams. The filling of the diagrams was in the form of expressions using functionals. The authors recorded expressions for determining the filling of the diagrams, as well as the total bending and torque moments.


INTRODUCTION
Experimental and theoretical studies in the field of torsion with bending are associated with the need to develop a design scheme and take into account a number of new effects of deformation of reinforced concrete with spatial cracks. Some of the earliest research in the field of torsion with bending were presented in [1][2]. A large theoretical and experimental basis for the development of the theory of bending with torsion is described in -publications [3][4][5][6], [7][8][9]. The main task is to develop models that allow describing the behavior of structures at all stages of loading. Rectangular cross-sections are one of the most applicable, therefore the research of this type of cross-section is most important [10][11][12][13]. However, the complex stress state has not yet been sufficiently considered in scientific publications [14][15][16][17][18][19][20][21][22][23]. There is a problem of searching for a new hypothesis of linear and angular deformations for rectangular sections and determining the filling of curvilinear diagrams in bending with torsion. It is necessary to use the analytical functional and special functions for deplanation of the cross section to obtain practical models. Engineering proposals have not been found for projecting the coefficients of the stress-strain and elastoplastic state from the stress and strain diagram.

Research methods
Determination of deformations and stresses in a complex stress-strain state in a rectangular section can be obtained from the Timoshenko-Goodyer theory of elasticity [24] using a membrane analogy. The function of Timoshenko and Goodyear [24] can be represented as: φ A -torsion angle for cross-section in edge fibers of compressed concrete or tensile reinforcement; f * -complex series of Timoshenko and Goodyer in the theory of elasticity. This function is complex and time consuming to calculate. Its calculation is rather difficult for plastic regions and regions with cracks. Therefore, a simple new method from the families of the mesh method was found to develop deformation functionals for approximating any rectangular mean sections in compressed and stretched zones using special squares ( Fig. 1, a). A more frequent splitting of the cross-section was used with the use of other points to correct the values of the obtained function ( Fig.1, b). We obtained the analytical first functional f 5,* (y,z) after several adjustments: through function f 1,* (y) (horizontal parabola about the y-axis) and function f 2,* (z) (vertical parabola about the z-axis). · -transition between functions; signs "+" and "-" are adopted respectively for quadrants I, III and II, IV. We had received an error of up to 2% at the considered points and to 7% at any points of the cross section when applying our functional to find the values of the functions ( Fig. 1, b). The analytic undefined second functional is a function of three functions: The error is 15% for the first iteration and 2% for the second iteration. We have developed a new hypothesis and formulated the definition. The proposed new hypothesis of angular deformations -the kinematics between fibers for the relative transverse fiber upper and lower total shear strains of concrete and reinforcement (γ sum,b and γ sum,s ) to determine their ratios in distances from the neutral axis, which has a special geometric figure for the function f sum,y (signs "+", "-" taken for different quadrants), as well as the parameter between concrete in plastic and elastic areas to obtain an equation with deformation f b,el . Note: there is a special section 3-3, where the local corner regions do not have a kinematic connection between the outermost fibers through the neutral axis of the section. We have determined the corollaries of the hypothesis.
Corollary 2. Reduction of the zone of compressed concrete from the load (Fig. 2) has the form: Corollary 3. The coefficients γ 3 and γ 4 were found from two pairs of triangles: The distances z k and z k* were found: The coefficient γ 3 was expressed from the equations (10) and (11): We got after algebraic transformations in equation (12): The equation (18) can be solved by iterating. Thus, the corollary of the hypothesis has the form: Corollary 1. The proportion for a trapezoid of angular deformations is the ratio of a vector r 1 to any point A and its horizontal projection r 2 . The proportion is used in a trapezoid (section 2-2, y=b/8, b/4, 3b/8), in a triangle (section 1-1, y=0, r 2 =0), but is not used in a special section 3-3 (y=0,5b). The coefficient φ los was obtained in the form of a parabola (9). Corollary 2. The decrease in the distance from the neutral axis of the compressed concrete under load is the proportion (10). was obtained from the undefined functional f 5,* (y,z) obtained above using differentiation: 186 228 1200 25 (19) Where f 5,* -functional in (2); φ A -torsion angle for cross-section in edge fibers of compressed concrete or tensile reinforcement. The component of the relative angular deformations γ t,yx was obtained in a similar way.
The total shear deformations have the form: When passing from plastic to elastoplastic deformation at point 2, we get: Shear strains from torsion have jumps in the diagram (Fig. 2, b). The deformation function during crack formation f sum,γ,Δ1crc (jump 1 on the deformation diagram) is similar to the function from formula (22), only less by a coefficient k γ,sum : The function of deformations at the appearance of the second crack-trace has the form: When analyzing with the approximation of the branches of the graph of the Timoshenko-Goodyer function [24] and the first functional, we obtain an error of up to 7% (Fig. 3, a) and less than 1% (for Fig. 3, b). We also get the coefficients for projecting normal and shear stresses using deformations from the diagrams "σ x -ε x ", "τ-γ". When analyzing with the approximation of the branches of the graph of the Timoshenko-Goodyer function [24] and the first functional, we obtain an error of up to 7% (Fig. 3, a) and less than 1% (for Fig. 3, b). The authors obtained the coefficients and from points C, B, A, D using the projection of deformations and stresses for the Prandtl diagram with constraints: j=С, В, A, D; The coefficients φ ij for parameters with limited angular deformations at point C have the form:  The third definite functional before the formation of cracks has the form:

( )
The torsion angle φ A,i for each point A i of the cross-section has the form: The definite fourth functional for the torque is obtained after integrating function f 5,* (y,z)(2): . .

RESULTS
The bending and torque moments for deformation or stresses were determined, as well as the filling area of the deformation and stress diagrams. The indefinite bending moment and the definite bending moment for the small square are of the form: and The filling area of the diagram ω γ,i (y,z) have the form: The filling area ω γ,i (x,y,z) of the diagram and the distance z b,i for indefinite torque have a similar shape. We have elastic, plastic regions and cracks (lateral, normal, etc.) in compressed concrete and reinforcement, Figure 4. The total bending moments with cracks from small squares in compressed and stretched zones has the form: Where n -total number of small squares; m -the number of squares of the compressed area longitudinal reinforcement; k -transverse reinforcement with normal cracks and lateral cracks; j -cross-sections 1-6; ω σ -filling area for stress diagram. The total torque with cracks has the form:  inforcement; k -transverse reinforcement with normal cracks and lateral cracks; j -cross-sections 1-6; Y 2 -see (35); φ A,i (z,y) -see (36); ω γ,i -filling area for diagram of shear deformations; z η,b,i , b s,i , b s',i , z η,σ,i , z η,σ,i,* -distance from point O* to any point.

1.
A simple method from a family of mesh methods was found for developing linear and angular deformation functionals by approximating rectangular sections in compressed and stretched regions. 2. We analyzed diagrams of angular deformations and shear stresses, defined functionals, obtained a new hypothesis of angular deformations and corollaries from the hypothesis. 3. The bending and torque moments were presented using new functionals, the projection coefficients of normal and tangential stresses were determined using diagrams of compressed concrete. 4. The areas of filling of the diagram ω ε (x,y,z) (ω σ (x-,y,z)) and ω γ (x,y,z) (ω τ (x,y,z)) were obtained from the functionals of the bending and torque moments. 5. The analysis of the new functionality and functions of Timoshenko-Goodyer has been carried out. The error in finding the value of the functional considered is 2% at the points considered and 7% at any points of the cross section.