Stress distribution, wrinkling and parametric instability of thin rectangular webs

The first contribution of this study is the modification of von Karman's nonlinear plate equations to describe the motion of a wide, axially moving web with small flexural stiffness under transverse loading. The model can represent a web under some conditions. Closed form solutions to the 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 outer solution is developed at the middle of the web and the inner solutions are developed in the boundary layers. A uniform approximation of the deflection is obtained by summing the outer and inner solutions, and then subtracting the matching solutions. Membrane theory and linear plate theory solutions are used to characterize the importance of the web deformation solutions.; The second contribution is derivation of a criterion that predicts wrinkling of isotropic, compressible rectangular webs under uniform in-plane principal stresses. A web is termed wrinkled when one of the in-plane principal stresses is tensile and the other is sufficiently compressive. The compressive stress at impending wrinkling depends on the flexural stiffness, and it equals zero in the case of a membrane. A criterion predicting wrinkling is also derived using isotropic, incompressible membrane theory. This criterion predicts an infinite number of wrinkle waves in a wrinkled region. With small flexural stiffness, the number of wrinkle waves becomes finite at wrinkling and it is predictable along with the shape and the size of the wrinkled region. The number of the wrinkle waves increases as the aspect ratio of the rectangular web increases, as the in-plane principal tension increases, and as the flexural stiffness decreases.; The third contribution is the prediction of the onset of wrinkling in a web under nonlinearly distributed, in-plane, edge tension. Airy stress functions are determined in closed form for webs under in-plane edge loading whose distributions are described by sine functions. Superposition of those Airy functions permits the representation of the stress field in webs under arbitrarily prescribed edge loading. The loading at impending wrinkling and the corresponding wrinkling mode in the webs are predicted by the solution of an eigenvalue problem. When membrane theory solutions are contrasted with the web solutions, the importance of small flexural stiffness to the prediction of wrinkling is clearly demonstrated.; The fourth contribution is the prediction of the parametric stability of a rectangular web under constant, uniform, longitudinal tension plus periodic shear excitation. The magnitude of the shear stress is normally small compared to the longitudinal tension and is considered a perturbation to the stress field. Perturbation methods are used to determine the transition boundaries separating stable and unstable parametric regions. A combination resonance of difference-type cannot occur and a combination resonance of sum-type is the dominant instability mechanism. The excitation frequency of the smallest resonance decreases as the ratio of width to length increases. The lowest resonance zone enlarges as the flexural stiffness decreases. (Abstract shortened by UMI.)...