Weak Form Quadrature Element Analysis Of Reticulated Shell Stability Based On Geometrically Exact Beam Theory

The stability analysis of reticulated shell structures is of major research concern,which includes the global buckling of the whole structure and local buckling of members.It is affected by factors such as: initial imperfection of members,height-to-span ratio,loading forms,properties of material and so on.During the stability analysis,reticulated shells usually undergo obvious nonlinearity,and most of traditional linear methods are not valid anymore.Thus,tracing the nonlinear equilibrium path and acquiring the loaddisplacement curve during the whole process becomes essential in stability analysis of reticulated shells.The weak form quadrature element method(abbreviated as quadrature element method or QEM)is an efficient numerical tool which is based on variational principles,numerical integrations and the differential quadrature analogue.Displacement shape functions are not necessarily designated compared with the conventional finite element method.It has been shown that the method enjoys great advantages when dealing with problems with complex shapes(such as variable cross-section),loading conditions,nonhomogeneous materials and so forth.Geometrically exact beam theory is a rigorous description of beams subject to geometrically nonlinear effect.In the theory,section rotation is introduced into the configuration space of the beam and the deformation of the beam is measured in the section frame.Rotational quaternions can be used to describe 3D rotations.The theory has been applied to analysis of beams with large displacements and rotations without accumulated error due to iterations.A weak form quadrature beam element model is formulated based on geometrically exact beam theory.Incorporating quaternions for 3D finite rotations into the beam model,the whole-process load-displacement analysis of spatial beam structures considering both geometrical nonlinearity and material nonlinearity is conducted.The main factors for the stability of reticulated shells are discussed.The following efforts are made in the present work:(1)Elasto-plastic analysis of spatial beam structures is presented based on a spatial geometrically exact beam model and its corresponding quadrature element formulation.The fiber model is incorporated into the formulation to take into account material nonlinearity and the algorithm of preventing spurious unloading is also introduced to optimize the convergence.(2)Elasto-plastic analysis of spatial shear-rigid beam structures is presented based on a spatial geometrically exact shear-rigid beam model and its corresponding quadrature element formulation which is simplified through the incorporation of auxiliary Lagrange multipliers.The assumption of yield surface has also been introduced into the model.(3)With elasto-plastic quadrature element model for geometrically exact beams,influential factors on reticulated shell stability including yield strength of material,hardening modulus,height-to-span ratio,loading forms and so on are discussed for some typical spherical reticulated shells.Results are compared with those of finite element software and the effectiveness of the model is verified.(4)With elasto-plastic quadrature element model for geometrically exact curved beams,the influence of initial curvature on the load-carrying capacity of some typical spherical reticulated shells is investigated through parameter analysis.Results are compared with those from the consistent imperfection mode assumption and the technical specification for space frame structures.One of the deficiencies in current computational practice,the lack of sole influence of initial member curvature,is made up for.It is suggested that the yield strength of material and the specific form of a reticulated shell should be considered in the formula of the technical specification.