Elastoplastic Stress Analysis and Residual Stresses in Cylindrical Bar Under Combined Bending and Torsion

Abstract

The excessive stresses during the coiling of helical springs could lead to breakage of the rod. Moreover, the high level of residual stress in the formed helical spring reduces considerably its fatigue life. For the practical estimation of residual and coiling stresses in the helical springs the analytical formulas are necessary. In this paper the analytical solution of the problem of elastic–plastic deformation of cylindrical bar under combined bending and torsion moments is found for a special nonlinear stress–strain law. The obtained solution allows the analysis of the active stresses during the combined bending and twist. Moreover, the residual stresses in the bar after springback are also derived in closed analytical form. The results of this analysis are applied to the actual engineering problem of determination of stresses during the manufacturing of helical coiled springs. A practically important example, describing the manufacturing of helical coiled spring is worked out to illustrate the simplicity achieved in determining the plasticization process and residual stresses. The obtained results match the reported measured values. The developed method does not require numerical simulation and is perfectly suited for programming of coiling machines, estimation of loads during manufacturing of cold-wounded helical springs and for dimensioning and wear calculation of coiling tools.