INTERPRETING INTEGRATION CONSTANTS IN ELASTIC BEAM THEORY

Abstract

The linear elastic theory of bending is used to derive the differential equation of the elastic curve of a bent beam. Solving this second-order differential equation requires two integration constants for each beam section between supports. The direct integration method, together with the Clebsch procedure, is used to obtain the elastic curve equation. This paper explains the physical meaning of these integration constants. Various loading positions and scenarios are considered for several overhanging elastic beams, and the shape of the elastic curve is presented for each case. The beam bending stiffness is kept of a constant value for a given cross-section and material of the beam, and we show its correlation with the integration constants. We apply the criterion of the ultimate bending strength (flexural strength) of steel for dimensioning beams. Thus, the characteristic dimension of the profile is determined according to the maximum value of the bending moment for each type of beam, considering the values and distribution of the load. In this context, the selected examples have the same cross-section profiles but different stiffnesses, as each is dimensioned individually according to the changing loading values and distribution. We also consider overhanging beams, and how the deflection orientation correlates with the integration constants, which depend on the position and magnitude of the load. Our findings show that even slight changes in the magnitude of concentrated forces can significantly affect deflection behaviour. This effect, referred to as "dog's tail movement," was especially noticeable at the free ends of overhanging beams. This underscores the crucial need for precise determination of load factors in practical applications.