Abstract: The present investigation extends the application of the newlydeveloped finite element method of lines (FEMOL) to nonlinear problems bysolving a series of representative nonlinear model problems in a fashion ofunified formulation and solution. This is the first paper in this series,and isconcerned with the FEMOL solution of a geometrically nonlinear problem--large deflection of membranes.FEMOL formulation for membranes on arbitrary domains is derived. Numerical results computed from a number of membranes of different shapes show an important property in this class of problems, i.e. the existence of a limit deformation state.The solution accuracy, reliability and efficiency exhibit that the method is highly effective and powerful for this kind of geometrically nonlinear problem.