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Finite Element Method - Shells a special case of three - dimensional analysis - reissner - mindlin assumptions _08 This monograph presents in detail the novel "wave" approach to finite element modeling of transient processes in solids. Strong discontinuities of stress, deformation, and velocity wave fronts as well as a finite magnitude of wave propagation speed over elements are considered. These phenomena, such as explosions, shocks, and seismic waves, involve problems with a time scale near the wave propagation time. Software packages for 1D and 2D problems yield significantly better results than classical FEA, so some FORTRAN programs with the necessary comments are given in the appendix. The book is written for researchers, lecturers, and advanced students interested in problems of numerical modeling of non-stationary dynamic processes in deformable bodies and continua, and also for engineers and researchers involved designing machines and structures, in which shock, vibro-impact, and other unsteady dynamics and waves processes play a significant role.
Shells as a special case of three-dimensional analysis Reissner-Mindlin assumptions 8.1 Introduction In the analysis of solids the use of isoparametric, curved two- and three-dimensional elements is particularly effective, as illustrated in Chapters and and presented in Chapters and 10 of Volume It seems obvious that use of such elements in the analysis of curved shells could be made directly simply by reducing their dimension in the thickness direction as shown in Fig 8.1 Indeed, in an axisymmetric situation such an application is illustrated in the example of Fig 9.25 of Volume With a straightforward use of the three-dimensional concept, however, certain difficulties will be encountered In the first place the retention of displacement degrees of freedom at each node leads to large stiffness coefficients from strains in the shell thickness direction This presents numerical problems and may lead to ill-conditioned equations when the shell thickness becomes small compared with other dimensions of the element The second factor is that of economy The use of several nodes across the shell thickness ignores the well-known fact that even for thick shells the ‘normals’ to the mid-surface remain practically straight after deformation Thus an unnecessarily high number of degrees of freedom has to be carried, involving penalties of computer time In this chapter we present specialized formulations which overcome both of these difficulties The constraint of straight ‘normals’ is introduced to improve economy and the strain energy corresponding to the stress perpendicular to the mid-surface is ignored to improve numerical ~onditioning.’-~ With these modifications an efficient tool for analysing curved thick shells becomes available The accuracy and wide range of applicability of the approach is demonstrated in several examples 8.2 Shell element with displacement and rotation parameters The reader will note that the two constraints introduced correspond precisely to the socalled Reissner-Mindlin assumptions already discussed in Chapter to describe the Shell element with displacement and rotation parameters 267 Fig 8.1 Curved, isoparametric hexahedra in a direct approximation to a curved shell behaviour of thick plates The omission of the third constraint associated with the thin plate theory (normals remaining normal to the mid-surface after deformation) permits the shell to experience transverse shear deformations - an important feature of thick shell situations The formulation presented here leads to additional complications compared with the straightforward use of a three-dimensional element The elements developed here are in essence an alternative to the processes discussed in Chapter 5, for which an independent interpolation of slopes and displacement are used with a penalty function imposition of the continuity requirements The use of reduced integration is useful if thin shells are to be dealt with - and, indeed, it was in this context that this procedure was first discovered.4-’ Again the same restrictions for robust behaviour as those discussed in Chapter become applicable and generally elements that perform well in plate situations will well in shells 268 Shells as a special case 8.2.1 Geometric definition of an element Consider a typical shell element illustrated in Fig 8.2 The external faces of the element are curved, while the sections across the thickness are generated by straight lines Pairs of points, itopand ibottom,each with given Cartesian coordinates, prescribe the shape of the element Let