THE PROBLEM OF SLOPE OPTIMIZATION OF THE REINFORCED CONCRETE PRESTRESSED BEAM

Abstract

The solution of the theoretical problem of slope optimization of the reinforced concrete beam and selection of the optimal height on the support is given in the article. A single-span hinged prestressed double-sloped beam of rectangular cross-section, which is loaded with a uniformly distributed linear load, is considered. The position of the cross section of the beam in length is characterized by a variable abscissa. The height of the beam at a certain distance from the left support is calculated in accordance with a certain slope. The working height of the beam in this place is determined at a given distance from the lower stretched face of the beam to the centre of gravity of the stretched pre-stressed reinforcement. The condition of the strength of the beam in the normal cross section with a single reinforcement and the equilibrium equation are obtained based on static dependences. The bearing capacity of the section at a known distance from the left support is expressed in terms of the ultimate bending moment, which is a linear function of the coordinate along the span. At the first stage of solving the problem, the required height of the beam on the support is determined from the condition of strength at a given slope of the upper shelf and the cross-sectional area of the pre-stressed reinforcement. It is proved that when touching the graphs of bearing capacity and bending moment from the load on the beam, the strength of the normal cross-sections is provided along the entire span. On this basis, it is concluded that the working height of the beam on the support can be determined from the equations of equality of moments and their first derivatives. In the second stage of solving the problem, the slope of the upper belt is determined, at which the volume of concrete for the beam will be the smallest. Since the cross-sectional area of the longitudinal reinforcement is assumed to be constant, it is sufficient to minimize only this volume. To investigate the function of volume at the extremum, there was found its first derivative of the slope and equated to zero. The optimization problem was solved with certain control source data. A graph of the volume function was obtained, which has a minimum at a certain value of the slope. The technique of performance of check on restriction of height of a compressed zone of concrete at an optimum slope is developed and resulted. If the restriction condition is not met, it is recommended to take a smaller cross-sectional area of the pre-stressed reinforcement. It is proved that in this case the height of the beam increases and the relative height of the compressed zone and deflection decrease.