FE Analysis of Membrane Systems Including Wrinkling and Coupling 97 Any standard (non-linear)time integration technique can be theoretically used in conjunction with the proposed FE model for the study of dynamic response of the systems of interest.Some care should be however taken in the choice because the high geometric non-linearities tend to challenge the stability of the time-integration scheme chosen. Generally speaking,"statics"can be seen as the limit to which a dynamic process tends (under a given constant load).Dynamic systems show a "tran- sient"phase that vanishes in time to reach the so called "steady state";the presence of damping in the system reduces gradually the oscillations making the system tend to a constant configuration that is the "static"solution.The time needed for the system to reach this final configuration is controlled by the amount of damping.For values of system's damping exceeding a critical value,the transient phase disappears and the systems reaches directly the final solution without any oscillation. In many situations the main engineering interest is focused on "static" solutions rather than on the complete dynamic analysis of the system.The previews considerations suggests immediately that "statics"could be obtained efficiently by studying the dynamics of over damped systems.This could be obtained by simply adding a fictitious damping source to the "standard"dy- namic problem.The "only"problem is therefore the choice of an idoneous form for such damping.Unfortunately this choice is not trivial,however it possible to observe [10],[8 that the "steady state"solution of the system M+D+Kx=f (x) (57) is (statically)equivalent to that of the system D+Kx=f(x) (58) which can be seen as the previews for the case of zero density.The advantage of this equivalent system is that the inertia terms are always zero,consequently the system converges smoothly in time to its solution.This final solution is not affected by the particular choice of the damping,however in the author's experience,an effective choice is D=BM as proposed by [10]. Table (1)gives the details of the proposed solution procedure,making use of Newmark's integration scheme.The procedure described differs from a"real"dynamics simulation only on the choice of the damping and of the mass matrix.Any other choice is possible for the time integration scheme to be used.It is of interest to observe that the system described is highly dissipative,energy stability of the time integration scheme is therefore not crucial. 3 Wrinkling Simulation Given the lack of flexural stiffness,membrane systems are easily subjected to buckling in presence of any compressive load.The idea is that when aFE Analysis of Membrane Systems Including Wrinkling and Coupling 97 Any standard (non–linear) time integration technique can be theoretically used in conjunction with the proposed FE model for the study of dynamic response of the systems of interest. Some care should be however taken in the choice because the high geometric non–linearities tend to challenge the stability of the time–integration scheme chosen. Generally speaking, “statics” can be seen as the limit to which a dynamic process tends (under a given constant load). Dynamic systems show a “tran￾sient” phase that vanishes in time to reach the so called “steady state”; the presence of damping in the system reduces gradually the oscillations making the system tend to a constant configuration that is the “static” solution. The time needed for the system to reach this final configuration is controlled by the amount of damping. For values of system’s damping exceeding a critical value, the transient phase disappears and the systems reaches directly the final solution without any oscillation. In many situations the main engineering interest is focused on “static” solutions rather than on the complete dynamic analysis of the system. The previews considerations suggests immediately that “statics” could be obtained efficiently by studying the dynamics of over damped systems. This could be obtained by simply adding a fictitious damping source to the “standard” dy￾namic problem. The “only” problem is therefore the choice of an idoneous form for such damping. Unfortunately this choice is not trivial, however it possible to observe [10],[8] that the “steady state” solution of the system Mx¨ + Dx˙ + Kx = f (x) (57) is (statically) equivalent to that of the system Dx˙ + Kx = f (x) (58) which can be seen as the previews for the case of zero density. The advantage of this equivalent system is that the inertia terms are always zero, consequently the system converges smoothly in time to its solution. This final solution is not affected by the particular choice of the damping, however in the author’s experience, an effective choice is D = βM as proposed by [10]. Table (1) gives the details of the proposed solution procedure, making use of Newmark’s integration scheme. The procedure described differs from a “real” dynamics simulation only on the choice of the damping and of the mass matrix. Any other choice is possible for the time integration scheme to be used. It is of interest to observe that the system described is highly dissipative, energy stability of the time integration scheme is therefore not crucial. 3 Wrinkling Simulation Given the lack of flexural stiffness, membrane systems are easily subjected to buckling in presence of any compressive load. The idea is that when a