Keywords: membrane structures, cable structures, flexible structures, shape finding, cutting pattern, dynamic relaxation method, optimization.

The membrane and cable structures, a kind of flexible structural system, are unstable initially because of the material property. These structures can be stable by introducing the initial stress, and thus the nonlinear analysis procedure considering geometric nonlinearity is needed. The first step is shape finding analysis to determine the initial equilibrium shape and the second step is stress-deformation analysis to investigate the behavior of structures under various service loads. The final step is cutting pattern of membrane. This paper is dealing with three categories.

The first one is shape finding method by modified dynamic relaxation method. There have been many researches for the shape finding such as force density method, finite element method and dynamic relaxation method. Dynamic relaxation method, proposed by Day, is similar with Finite Element Method, but the equilibrium equation is solved by introducing the kinetic damping and viscous damping. Dynamic relaxation method developed by Barnes is a method which removes the viscous damping, that is, introduces only the kinetic damping.

In this paper, a dynamic relaxation method, which the lumped stiffness, the unit lumped viscous, the unit lumped mass have been considered, is introduced.

The Newmark's assumption have been considered.

The second one is the cutting pattern method with optimization technique. The mathematical concept for optimum method can be expressed as follows:

• Control variable : nodal coordinates of 2-D cutting pattern
• Object function
  where, : Design stress, : Actual stress
• Restrictive condition
  : Equilibrium equation, : 3-D coordinates in actual equilibrium shape

The objective function and restrictive condition have to be expressed by control variable in order to get solutions of optimum problem. However, it is difficult to express everything only with control variable because the restrictive condition is obtained by actual equilibrium analysis and they are consisted by complicate simultaneous equations for variable , also, the value of in object function is determined by variable . To avoid these problems, other additional assumptions should be considered.

In this study, a new assumption 'membrane elements on modified cutting pattern have stresses same as design stress' is employed. This assumption is different from assumption(3), that is, assumption(3) is used in actual equilibrium analysis. It is also assumed that nodal displacements of z-direction for modified cutting pattern procedure are not occurred. Employing these 2 additional assumptions, initial displacement of modified cutting pattern is obtained. This procedure is newly defined as 'modified cutting pattern analysis'. It can be mathematically expressed as followings.

• Initial stress of cutting pattern = design stress
• Restrictive condition
• : 2-D equilibrium equation determining the shape of modified cutting pattern

Because the modified cutting pattern is subjected to initial stresses equivalent to given design stress, the modified cutting pattern analysis can be performed with only restrictive variable . On this state, stresses should be to zero if one force to free all nodes on boundaries except for a bench node and the initial strains are occurred. Next, an actual equilibrium analysis is performed with these changed 2-D coordinates. However, some errors should be occurred in actual membrane stress when the actual equilibrium analysis is performed with the data from modified cutting pattern. Here, the error is occurred. To reduce this error, the procedure is repeated until it converges. This is called as 'optimum cutting pattern analysis'.

The third one is some practical design examples focused on 2002 world cup stadiums covered with membrane roof in Korea are presented.

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