FE Analysis of Membrane Systems Including Wrinkling and Coupling 99 tend to become quite little in comparison with the total size of the structure. It has been proved to be feasible [2]and [1]to describe correctly the forma- tion of the wrinkles using extensively mesh refinement procedures together with low order thin-shell elements.An analogous approach using higher order shells and a fixed reference mesh,joint with some comparison with experi- mental data can be found in [11].A key point to be taken in account is that this procedures need a mesh density inversely proportional to the expected size of the wrinkles.In other words the smaller are the wrinkles,the more el- ements are needed to correctly describe the phenomena.As in our structures, the thickness is very low compared to the other dimensions,the referenced approaches would become soon too expensive. An alternative procedure is based on the "enrichment"of the elements involved in the simulation.The idea is to renounce to a description of the single wrinkle and to focus the analysis of the average stress and displacement field.This allows to consider in the analysis elements of size bigger then the expected wrinkle size introducing the effect of the local instability in the calculation of the stress or strain field at integration point level.We would like to stress that this approach is not necessarily less "precise"than the former.Indeed no information on the wrinkling size is provided,however the global stress field is correctly described.It is as well important to highlight how the position of the wrinkles is never known given its strong dependence on the initial imperfections,therefore the only reliable result is the individuation of the "wrinkled zone"that can be correctly described by both methods. 3.1 Enriched Material Model Over the years many different proposals to perform the element enrichment were presented.Mainly two different approaches survived,one based on ma- nipulations of the gradient of deformations,the second connected with a re- definition of the constitutive model. The former,proposed by Roddeman in [6]and [7],is based on the def- inition of an effective deformation gradient obtained superimposing to the normal displacement field a term connected with the formation of wrinkles. This modification allows to describe correctly the shortening of the average plane of the membrane in presence of compressive stresses. The latter is based on a modification of the stress-strain relationship, meaning that the constitutive law is modified not to allow compressive stresses.The main advantage of this second techniques,is to make the imple- mentation completely independent from the element used,characteristic that makes them very attractive for the practical implementation. A "new"material model,based on the modification of a standard linear material is introduced in current section.This formulation is based on the penalization of the elastic characteristics of the material in the system of the principal stresses.In simple words,the material is softened in the direction of the compressive stresses and keeps its characteristics in the other direction.FE Analysis of Membrane Systems Including Wrinkling and Coupling 99 tend to become quite little in comparison with the total size of the structure. It has been proved to be feasible [2] and [1] to describe correctly the forma￾tion of the wrinkles using extensively mesh refinement procedures together with low order thin-shell elements. An analogous approach using higher order shells and a fixed reference mesh, joint with some comparison with experi￾mental data can be found in [11]. A key point to be taken in account is that this procedures need a mesh density inversely proportional to the expected size of the wrinkles. In other words the smaller are the wrinkles, the more el￾ements are needed to correctly describe the phenomena. As in our structures, the thickness is very low compared to the other dimensions, the referenced approaches would become soon too expensive. An alternative procedure is based on the ”enrichment” of the elements involved in the simulation. The idea is to renounce to a description of the single wrinkle and to focus the analysis of the average stress and displacement field. This allows to consider in the analysis elements of size bigger then the expected wrinkle size introducing the effect of the local instability in the calculation of the stress or strain field at integration point level. We would like to stress that this approach is not necessarily less ”precise” than the former. Indeed no information on the wrinkling size is provided, however the global stress field is correctly described. It is as well important to highlight how the position of the wrinkles is never known given its strong dependence on the initial imperfections, therefore the only reliable result is the individuation of the ”wrinkled zone” that can be correctly described by both methods. 3.1 Enriched Material Model Over the years many different proposals to perform the element enrichment were presented. Mainly two different approaches survived, one based on ma￾nipulations of the gradient of deformations, the second connected with a re￾definition of the constitutive model. The former, proposed by Roddeman in [6] and [7], is based on the def￾inition of an effective deformation gradient obtained superimposing to the normal displacement field a term connected with the formation of wrinkles. This modification allows to describe correctly the shortening of the average plane of the membrane in presence of compressive stresses. The latter is based on a modification of the stress-strain relationship, meaning that the constitutive law is modified not to allow compressive stresses. The main advantage of this second techniques, is to make the imple￾mentation completely independent from the element used, characteristic that makes them very attractive for the practical implementation. A ”new” material model, based on the modification of a standard linear material is introduced in current section. This formulation is based on the penalization of the elastic characteristics of the material in the system of the principal stresses. In simple words, the material is softened in the direction of the compressive stresses and keeps its characteristics in the other direction