Nonlinear Dynamic Finite Element Analysis of Composite Cylindrical Shells Considering Large Rotations

Abstract

A new nonlinear finite element formulation is developed for the nonlinear dynamics of shell structures. Using a vectorial approach, the kinematics that can be used to describe very large displacements and large rotations are derived. Green strain measures and second Piola Kirchhoff stress measures are used in the determination of the total potential. A displacement-based eight-noded cylindrical shell element with a total of 36 degrees of freedom is developed. Several numerical examples are dynamically analyzed to observe the characteristics of large displacements and large rotations. A convergence study was carried out to establish the numerical accuracy of the model. The most efficient and accurate model was used. HE nonlinear vibration analysis of shells has been the focus of research for the past few years. Thin shells subjected to dynamic loads could encounter deflections of the order of the thick- ness of the shell. Dynamic response of thin shells also could lead to the phenomena of dynamic snapping or dynamic buckling. Be- cause these kinds of responses cannot be determined accurately using small displacement and small rotation theories, large defor- mation and large rotation theories are required. The complex nature of these theories requires solving numerous simultaneous, nonlin- ear, differential equations. Even with simplifying assumptions, these equations remain complex and extensive. As background, a few relevant past efforts are reviewed. Earlier work regarding the nonlinear vibrations for isotropic shell structures can be attributed to Carr, 1 Yeh,2 Clough and Wilson,3 Belytschko and Marchertas,4 and Belytschko and Tsay.5 These researchers used flat plate elements. Saigal and Yang6 developed a curved shell element for isotropic shells. For composite shell structures, Simitses7 provided an elegant analytical solution of thin laminated shells subjected to sudden loads. He ignored through-the-thickness shear effects and used Donnell-type kinematic relations. Raouf and Palazotto8'9 developed a perturbation procedure to derive a set of asymptotically consistent nonlinear equations of motion for an arbi- trary laminated composite cylindrical shell in cylindrical bending. They also neglected transverse shear effects. Simo and Tarnow10 de- veloped an energy and momentum conservating algorithm for shell analysis to analyze shells undergoing large rigid-body motions. Palazotto and Dennis11 presented the simplified large rotation (SLR) theory to analyze shells experiencing moderately large ro- tations. The SLR theory determines the equilibrium path of or- thotropic shells using a total Lagrangian approach and includes a parabolic transverse shear stress distribution. This approach cap- tures the appropriate kinematics through displacement polynomi- als. Green strain displacement relations are used, and all of the final displacement functions are carried into the expression for the total potential without making any attempt at separating the rigid- body movement. Using this approach, the features classically pre- sented for, including through-the-thickness shear (for example), can be exploited.12 Smith and Palazotto13 discussed eight higher-order shear theories to the nonlinear finite element analysis of compos- ite shells. They also used the Green-Lagrange strains and second Piola Kirchhoff stresses. These theories were successful for large displacements and moderately large rotations. A reason for the fail- ure of these theories in the modeling of large rotation can be at- tributed to the approximations used in the kinematics. Gummadi and Palazotto14 modified the kinematics incorporated in the SLR theory and successfully carried out the solutions to problems ex- hibiting the characteristics of very large displacements and rota- tions. In their work, a vectorial approach was followed to derive the appropriate kinematics that can capture the characteristics of large displacements and large rotations. Using appropriate assumptions, the kinematics for the SLR theory can be obtained as a special case. The large rotation kinematics was introduced through the use of sine and cosine functions. A 36-degree-of-freedom composite shell element was developed based on this theory. Based on the SLR theory, Tsai and Palazotto15 formulated a nonlinear dynamic finite element analysis procedure for composite cylindrical shells. In their work, a linear mass matrix was incorpo- rated along with the nonlinear stiffness matrix that was originally developed in SLR theory. This theory again has the same drawbacks as the SLR theory in terms of realizing the large displacements and large rotations. In the present paper, a theory that can capture the large displacement and large rotations is discussed. The present work is an extension of the authors' earlier work on large rotation theory for static analysis. A new mass matrix is generated using Hamilton equations. The mass matrix thus generated is a nonlin- ear mass matrix. A fi-M method15 is used to solve the nonlinear algebraic equations resulting from the finite element technique. Se- lected numerical examples are solved to determine the validity of the developed theory.