Buckling Analysis Of Single-layer Reticulated Shells Based On Discrete Element Method

In recent decades,large-span spatial reticulated structures have been widely used in the field of engineering structures because of their reasonable force,low cost,beautiful shape and large span.The research on the stability of reticulated shells has captured the interest of many researchers.The finite element method is the most commonly used method for buckling analysis,but it is necessary to integrate and correct the tangent stiffness matrix when considering geometric nonlinearity.When it is difficult to converge non-linear equation iterations or integrate stiffness matrixes,it is necessary to modify the method itself.However,it is not necessary for the discrete element method to deliberately distinguish small deformation or large deformation.Without assembling stiffness matrixes and iterations during structural geometric nonlinear analysis,it has advantages in analyzing complex problems such as large deformation,nonlinearity and discontinuity.Therefore,based on member discrete element method(DEM),the elastic and elastic-plastic buckling analysis of single-layer reticulated shells is realized by theoretical derivation and numerical simulation in this paper,providing a new approach for tracking post-buckling behaviors of structures.In this paper,the basic idea and theory of the member discrete element method are introduced,which lays a theoretical foundation for the study of structural geometric nonlinearity and material nonlinearity.However,there are few literatures about analyzing the whole process of structural buckling by the discrete element method.Therefore,the discrete element force control method and the discrete element displacement control method are proposed to track the whole process of structural instability.The applicability and calculation characteristics of the two methods are expounded.In addition,the application of the discrete element displacement control method for different load conditions is discussed.By analyzing the elastic buckling of several classical examples and large single-layer reticulated shells,it is proved that the discrete element method can effectively track the member instability,local instability and overall instability of the structures.It can capture the multiple instability of the structures and the complex post-buckling behaviors,which reveals the macroscopic mechanism of structural elastic buckling.How to consider the plastic development of members on the basis of structural geometric nonlinear analysis is the key to the elastic-plastic buckling analysis of the single-layer reticulated shell structure.The basic idea of the discrete element method is to divide the structure into a set of discrete elements,which satisfy the relationship between force and displacement.The motion of each unit follows Newton’s second law,and uses the dynamic relaxation method to calculate the movement state of each unit at each time step,and then the movement of the whole structure is obtained.When solving the structural elastic-plastic problems,except the yield equation used to determine the unit force state and elastic-plastic constitutive to calculate internal force increment,the other solution process are unchanged.Therefore,in the discrete element calculation program,the elastic-plastic analysis of the element can be developed into a separate module,which can be called when needed.The discrete element method is simple,clear and has a good accuracy.In this paper,two kinds of discrete element elastic-plastic analysis method,discrete element plastic hinge method and discrete element plastic zone method are proposed.Discrete element plastic hinge method does not consider the cross-section plastic development,directly consider the full cross-yield,and the computation amount is small.The internal force yield function,the elastic-plastic constitutive model and the loading and unloading criterion are established.The calculation and analysis flow of the discrete element plastic hinge method is also given.Discrete element plastic zone method divides the cross-section into several small areas.The stress state of small areas is analyzed to form the stress state of the whole section.This method is more accurate than the discrete element plastic hinge method.In this paper,the calculation formula of the cross-section strain increment,and the elasto-plastic constitutive equation under the three-dimensional stress-strain state are deduced.The loading and unloading criterion and the integral formula of the cross-section internal force are also established.Numerical examples show that the discrete element plastic zone method can reasonably consider the plastic deformation during the large displacement analysis of the structure,but it will not bring a significant increase in the computation amount.Based on above theory,a geometrical and material nonlinear analysis program for spatial structures is coded by using Fortran95.By analyzing several truss structures,planar bar structures,single-layer reticulated shell structures and the K6 large-scale test model,it is proved that the discrete element method is suitable for the whole process analysis of elastic and elastic-plastic buckling of large-scale single-layer reticulated shell structures,expanding the application of discrete element method in structural engineering and provides a new approach for the structural stability analysis.