Nonlinear deformations of clamped annular membranes subjected to uniformly distributed axisymmetric loading

Nonlinear axisymmetric deformations of clamped annular membranes have been investigated. A single nonlinear governing differential equation in terms of radial stress was used. Literature review indicates that no attempt has been made to obtain the numerical solution of the annular membrane problem using a single nonlinear governing differential equation. The solution of the single equation which governs the response of the annular membrane was solved using the finite difference scheme in conjunction with the Newton-Raphson technique. The equations representing the boundary conditions were incorporated into the equations representing the stresses at the interior regions of the membrane. This resulted in a condensed tridiagonal matrix for the Jacobian of the system, which was solved using a tridiagonal reduction scheme. The structure and sparsity of the Jacobian was exploited resulting in economy of computer storage requirements and permitted the partitioning of the membrane into a larger number of regions which yielded an improved solution. The tridiagonal reduction scheme also resulted in an increase in computational speed. To verify the validity of the present method, a classical problem which involves a clamped circular membrane subjected to uniformly distributed loading was analyzed. The results obtained by the present method agreed extremely well with the classical solution. However, in the case of an annular membrane, the literature reveals a conflict between the numerical results of two groups of researchers. One group asserts that - as the hole radius decreases - the stress concentration factor approaches an upper limit of two, while results from another group of investigators show this value to be much higher. The results obtained by using the present results indicate that the stress concentration factor is indeed bounded by a value of two. The present method exhibited excellent convergence characteristics. In conclusion, the use of a single nonlinear differential equation in terms of radial stress which governs the response of an annular membrane, in conjunction with the Newton-Raphson method, appears to be an ideal choice for the solution of the nonlinear annular membrane problem.