Geometrical Nonlinear Analysis Of Framed Structures Based On The Rigid Body Rule

With the development of structural analysis,design,construction method and the improvement of material performance,the structures becoming more and more slender,which exhibit significant geometrical nonlinear characteristics,and it is important to perform accurate buckling and post-buckling analysis for structure design.The general geometrical nonlinear analysis method are based on finite element method.The geometrical nonlinear analysis method are based on the large deformation theory of continuum mechanics,with the use of virtual displacement principle or energy method,the linear and nonlinear component of strain are considered and then the displacement interpolation function is introduced to express element displacement,thus the elastic stiffness and geometric stiffness are obtained.Combined with incremental iteration algorithm,the geometrical nonlinear behaviors of structures can be analysed.In the traditional geometrical nonlinear analysis,the derivation of element stiffness matrix is complicated,and the physical meaning of derivation is not clear.The iteration direction and step size can not be effectively identified by the traditional incremental iteration method,thus may result in numerical divergence.Based on the above limitations of displacement and deformation description in traditional geometrical nonlinear analysis,as well as the limitation of numerical calculation methods,in the present research,the theory of rigid body rule and the generalized displacement control method(GDC)are presented first,In geometrical nonlinear analysis,the displacements of each element of a structure in each incremental step can be decomposed into two parts as the natural deformations and rigid displacements.In general,the rigid displacements constitute a great portion of the incremental displacements of each element of the structure.When a finite element is subjected to a rigid body motion,the initial forces acting on the element in equilibrium at 1C,i.e.,before the rigid body rotation,must rotate or translate with the rigid body motion,while their magnitudes remain unchanged,so as to preserve the equilibrium of the element at 2C,i.e.,after the rigid body rotation.This is the rigid body rule.The rigid body rule can also be used in the force recovery procedure to deal with the initial forces acting on the element in equilibrium,thus the force recovery procedure is accurate and simplified.Besides,the generalized displacement method is based on an reasonable constraint equation,the generalized stiffness parameter is used to detect the change of structure stiffness,the method has been demonstrated to be numerically stable at the critical points,effective in adjusting the step sizes,and self-adaptive in changing the loading directions,it is suitable for solving nonlinear problems with multiple limit points and snap-back points.The rigid-body qualified geometric stiffness matrix is derived from the virtual work equation for a rigid displacement field.Based on the rigid body rule and GDC method,benchmarks such as truss structures,frame structures and a practical circular arch are analyzed,the results shows that the geometric stiffness matrix is rigid body qualified,it has a simple form and can be used in the predictor stage which can change the loading directions effectively,reduce iteration numbers,thus improve the computation efficiency.The GDC method is suitable for solving nonlinear problems with multiple limit points and snap-back points.Based on the present geometrical nonlinear method,combined with the refined plastic hinge model,a simplified method for second-order elasto-plastic analysis of plane frames is presented.With the present method,the material nonlinear problem being coulped with the geometric nonlinear problem in the forming of the plastic hinge of the frame structures could be solved efficiently.