Moment at elastic-plastic bending of steel sheet. Part 1. Parabolic approximation of steel’s hardening zone

Abstract

To calculate the power parameters of metallurgical machines in the production of pipes from steel sheet, it is extremely necessary to know the analytical dependence of the elastic-plastic hardening curve of steel and the numerical value of bending moment of steel sheet depending on the curvature’s radius of its neutral plane during bending. Otherwise, the power characteristics of links of metallurgical machines (driving mechanisms of punches, torsion shafts, power frames and so on) may exceed the permissible values, which will lead to their breakage. To obtain the mechanical characteristics of steel, a round or flat steel sample of standard dimensions is usually stretched and the stretching diagram is obtained in the coordinates “normal stress σ – relative elongation ε”, which is given as a graph. In the experimental stretching diagram, the yield strength σy (or σ0.5), the ultimate strength σu, the relative elongation at break δ and the relative narrowing ψ (not always determined experimentally) of the neck of the round sample at break are usually found (determined). The stretching curve has the maximum at the moment of the beginning of the neck formation in the round steel sample (at the relative deformation εu). For a number of steels (for example, for the low-carbon steels), the deformation curve may have the yield area, where the normal stress σ is practically unchanged when the relative longitudinal deformations ε change. At this moment, the internal restructuring of the steel microstructure occurs at the level of the steel grains — sliding at the grains’ boundary and changing the orientation of the grains relative to each other. The size of the yield area is significantly smaller than the steel hardening zone. Therefore, the yield area is usually neglected in the analytical and numerical calculations.