Application of one-dimensional mechanical formulations to model the sagging behavior of a polymer sheet

The deformation of a heated plastic sheet clamped on two opposite sides subject to sagging under its own weight was examined experimentally and then modeled using two separate one-dimensional approaches based upon cable (membrane) and beam formulations. The cable formulation neglects both bending and shear deformation, but includes a generalized Maxwell viscoelastic constitutive model to capture the time-dependent nature of sheet sag. The resulting equations are integrated using a Runge-Kutta technique and solved via a shooting method. The beam formulation is based upon the Timoshenko theory and thus includes shear deformation along with the flexural contributions. A finite element method is developed from application of the principle of virtual work for the beam written in curvilinear coordinates in order to include the effects of finite deformation. A generalized Maxwell model is again employed to account for the time-dependent material response. In both formulations, the method of reduced variables is used to describe the variation of material response with temperature. The effect of temperature and thermal relaxation is included. The particular case of a styrenic material is discussed in detail.