Deflections of continuous reinforced concrete elements

The results of experimental investigations of deflections of continuous reinforced concrete elements are obtained, empirical dependences are deduced. Comparison of the deflection values with the results of calculations by the author’s method of NIISK (Bambura A.M.), adapted deformation model of OSACEA, what was realized with the help of the mathematical program MATLAB and also the algorithm of theoretical calculations of the finite element method was developed using the mathematical-graphical environment of the LIRA-SOFT software complex. The results of experimental investigations of deflections of continuous beams were compared with theoretical data (graphical representations of displacements in the form of isofields were obtained).The finite element calculation allowed to monitor the stress-strain state of the test beams at all stages of work, which made it possible to compare the obtained experimental values of the deflections with the designed ones. The performed experiments confirmed the feasibility of taking into account the shear deformations on the supporting sections in determining the deflections of reinforced concrete beams.


Introduction
Deformation calculation is of great importance when designing reinforced concrete elements and structures. It is known that the occurrence of one of the boundary states of structures, is characterized precisely by the development of excessive deformations (displacements) from static and dynamic loads.
The estimation of the curvature and displacement of the cross-section is also necessary in determining internal forces in statically indeterminate systems, both in the exploitation stage and before failure, and not only from force effects, but also from temperature fluctuations, shrinkage of concrete and displacement of supports.

Analysis of publications
The strength characteristics of reinforced concrete elements depend significantly on the method of their reinforcement and the composition of the concrete, which is confirmed by many studies of various building structures [1][2][3][4][5][6][7][8][9][10][11]. Investigation of continuous reinforced concrete beams was done by V.Ye. Babich [1][2][3][4][5], A.M. Bambura [6] and many others scientists. It is confirmed that the redistribution of forces in the first stages of loading is affected by cracks, and even before the destruction is significantly affected by plastic deformation in concrete and reinforcement.

The purpose of research
The purpose of the study is to improve the method of calculation of deflections of continuous reinforced concrete beams, to compare the experimental values of deflections of two-span reinforced concrete beams with its calculated values, calculated by the most common methods and the proposed advanced method of calculation.

Research results
Below are the results of experimental and theoretical studies.

Analysis of the results of the experiment
The deflections of the prototypes [7] were measured at six points: in the middle of spans, under concentrated forces and at the free ends of the beams ( Figure 1). The emergence of the first normal cracks and subsequent emergence inclined ones in the shear-span was accompanied by a sharp increase in deflections. The increase in the magnitude of the deflection during the occurrence of these cracks was not proportional to the increase in the external load. As the bearing capacity was exhausted, the deflection beams increased significantly even with a slight increase in the external load. This is due, on the one hand, to non-linear deformations of compressed concrete with a large percentage of longitudinal reinforcement or with a small amount of stretched reinforcement, and, on the other hand, to shear deformations in the shear span caused by the combined action of the bending moment M and V.
The research methodology, the main characteristics of the test specimens-beams and the design schemes of power units are given in [8,9].
According to the results of the experiment, new data were obtained, namely adequate mathematical models characterizing deflections of continuous beams in the middle of spans and under concentrated forces in the exploitation stage (1), (2) and before destruction (3) The magnitude of the deflections under concentrated forces (2), (4) is most affected by the relative shear-span X 1 . The deflections in the span and under the concentrated forces in the "exploitation stage" depend on the same factors (1), (2).Thus, they increase relative to their average of 1.53 and 0.274 with an increase in the relative shear-span from 1 to 3, respectively, by 15.7 and 314 %, with an increase in the amount of lower longitudinal reinforcement from 0.0101 to 0.0199 by 34 and 51 %, withan increase in the amount of upper longitudinal reinforcement from 0.0101 to 0.0199 by 24 and 29 %.
Bends in the middle of spans (3) before destruction increase relative to the average value of 3.02: with the increase of the shear-span from 1 to 3 -by 27 %, with the increase in the amount of lower longitudinal reinforcement from 0.0101 to 0.0199 -by 5.6 %.

An advanced method of calculation
The methods for determining the deflections of the investigated elements are based on the incomplete Mohr integral. It is recommended to define deflections by the formula: where, x M and x V -is the bending moment and the transverse force in the i-th section what occurred from the action of a single force applied in the direction of the desired displacement in the crosssection "i" for which the deflection is determined; where, x V -transverse force in section X from the action of external force; 2 b  -correction factor that takes into account the impact of long-term creep; b G -concrete shear module; crc  -coefficient that takes into account the effect of cracks on shear deformation. In the absence of cracks equal to 1, and in the presence of inclined 4.8.
In the presence of normal and inclined cracks this factor is determined by the formulas:   sр uf uf uf The indicated curvatures are determined by the average values of the relative deformations of reinforcement and concrete in the areas between the cracks. Before the appearance of the plastic hinge over the middle support (Figure 2a), the deflection caused by the moment can be determined by the simplified formula: where, after the appearance of the plastic hinge (Figure 2b): Deflections are then determined by formulas (9) and (10).
To determine the maximum deflection, a single force is applied in the middle of the spans. To determine deflections in the short-term action of loading in relatively short beams (L<10h) use formula (5), at L≥10h exposure to the second addition of formula (5), can be neglected. The results of the calculations of displacements in the LIRA-SOFT arepresented in the form of isofield in Figures 3-5. a) b)

Conclusions
The performed experiments confirmed the feasibility of taking into account the shear deformations on the supporting sections in determining the deflections of reinforced concrete beams, and their contribution is 25-30 % of the total deflection of the beam. Therefore, it is recommended to define the deflections as the sum of the deflections caused by the moment and shear deformations. In this case, before and after the appearance of the plastic hinge, it is necessary to take into account the indicator of the proportionality of the moments change and the curvature of the beam with constant cross-section along the all length without pre-stressing the reinforcement. Adequate mathematical models obtained from experimental researches characterize the deflections of the studied beams in the mid-spans and under concentrated forces in the maintenance stage and before failure.
The simulated deformation technique with the finite element method allows to predict sufficiently accurately the deformability of continuous beams in different span sections (υ=10…22 %), at the same time, the normative results of deflections show their poor convergence (υ=25…89 %).