Strain-based formulas for stresses in profiled center-wound rolls

A new solution to the problem of modeling elastic stresses in center-would rolls is proposed, Its premise is that residual strains form the web relax to create stresses within the roll. Stress terms - obtained with linear and non-linear elastic solutions - are weighted averages of web stresses in ce nter-would rolls. However, there are major differences between these calcul ated stresses and previous solutions. The solution here rigorously satisfie s two boundary conditions and is in closed form. The proposed approach cont ains four stress terms: Two are general stresses from the homogenous part o f the solution, while two new solutions come from the residual strains in t he roll generated by web tension stresses. The radial dependence of the res idual strains (i.e., the roll profile) is ver important, since the residual roll stress depend on the winding conditions. All winding profiles lead to complete, although algebraically complicated, elastic solutions in analyti c form. Nonlinear elastic solutions are also investigated using the effecti ve residual web stress and an stress-dependent radial elastic modulus. Nume rical nonlinear equations developed form expressions derived here have stra ightforward solutions. A selection of material constitutive laws and roll p rofiles are evaluated for comparison. Finally, the modeled roll stresses ar e compared with pull-tab-measured stresses from instrumented would rolls. T he radial plateau stresses in the midroll are further approximated with sev eral simple closed-form algebraic expressions.