Weak Form Quadrature Element Analysis Of Geometrically Exact Shells With Drilling Rotations

Numerical integration is an important numerical computational tool.With reference to Chebyshev's equal weight quadrature,a modified Chebyshev quadrature is proposed after incorporating the end points of the interval.The practical application of Chebyshev's equal weight quadrature is restrictive since the order of quadrature with mere real abscissas cannot exceed nine.The modified Chebyshev quadrature alleviates the restriction considerably and renders it applicable in weak form quadrature element analysis.It is applied to the evaluation of integrals and linear and non-linear weak form quadrature element analysis of rods and beams.Results are compared with analytical solutions and those obtained using Lobatto quadrature,verifying the accuracy and the effectiveness of the proposed quadrature.Classical shell theories are established by disregarding the rotation around the normal to the shell surface,which makes it difficult t o tackle problems of non-smooth shell structures or provide a compatible connection with beams.Ibrahimbegovi? presented a geometrically nonlinear shell formulation including drilling rotation by extending the early work if Reissner that introduced non-symmetric strain measures conjugate to Biot stress.Fox and Simo introduced the polar decomposition of deformation gradient as constraint conditions to identify full rotations of shell model,which was further improved by Rebel with a normalization scheme as a remedy for the mesh dependence problem.The ways of adding the drilling rotations in Ibrahimbegovi? and Rebel's nonlinear shell theories are discussed and the resulting differences are addressed.The weak form Quadrature Element Method(QEM)is a high-order numerical method.It is based on variational principles and the utilization of numerical integrations and the differential quadrature analogue.The geometrically exact model is established based on nonlinear continuum mechanics and finite rotation theory,which provides an objective strain-configuration relationship under large deformation and fini te rotations.A quadrature element formulation for geometrically exact shells with drilling degrees of freedom based on Rebel's work is developed and used to analyze geometrically nonlinear problems of open channel-section beams,manifesting its merits in dealing with non-smooth shell problems.The kinematic constraints for adding drilling rotation in Rebel's theory bring out shear locking problems for low order finite elements.Owing to the high-order property of the QEM,this problem is overcome satisfactorily,and the advantages and wide applicability of the QEM in geometric nonlinear analysis are highlighted.