Equilibrium Displacement and Stress Distribution in a Two-Dimensional, Axially Moving Web Under Transverse Loading

Von Karman nonlinear plate equations are modified to describe the motion of a wide, axially moving web with small flexural stiffness under transverse loading. The model can represent a paper web or plastic sheet under some conditions. Closed-form solutions to two nonlinear, coupled equations governing the transverse displacement and stress function probably do not exist. The transverse forces arising from the bending stiffness are much smaller than those arising from the applied axial tension except near the edges of the web. This opens the possibility that boundary layer and singular perturbation theories can be used to model the bending forces near the edges of the web when determining the equilibrium solution and stress distribution. The present analysis is applied to two examples: (I) a web deflecting under its own uniformly distributed weight; (II) a web deflecting under a transverse load whose distribution is described by the product of sine functions in the axial and width directions. Membrane theory and linear plate theory solutions are used to characterize the importance of the web deformation solutions.