Third Order Generalised Beam Theory and Its Complete Solution

摘要

The basic theoretical structure of the third-order Generalised Beam Theory (GBT) was firstly explained using the large strain theory of nonlinear continuum mechanics (Chiu, 2012). By introduced the two virtual works of longitudinal membrane bending moment and membrane shear strain into the 2nd order GBT system, two 3rd order terms ~（ijrk)υ_σand ~（ijrk)υ_τ were obtained. The complete expansions of the GBT 2nd and 3rd order terms, in the form of a series of large discretized iterated functions, were obtained by the author and used in the Newton step method as a set of tangent stiffness matrices. Three analytical Jacobian matrixes including the linear components of the 2nd order terms, the nonlinear components of the 2nd order terms and the complete components of the 3rd order terms were written. Finally, by leading out the membrane stresses as the 3rd order terms using advanced numerical techniques to find the complete solution of the highly nonlinear equation system, the GBT provides a rigorous and efficient numerical tool for geometric nonlinear analysis.