Research On Nonlinear Buckling And Free Vibration Of Thin-Walled Box Beam

Thin-walled box beam is widely used in crane construction machinery due to its high stiffness and wight ratio,which is suitable for large torsion and high bending stiffness.It is very meaningful to reduce the wight of crane box beam under given loads.As is known to all,the spatial buckling behavior of thin-walled box beams is very complex because of the coupling effect of deformation such as extension,bending,warping,torsion,distortion and shear lag.Therefore,nonlinear theory is needed to accurately predict the structural behavior of thin-walled box beams.In order to achieve the above purposes,this paper introduces in detail several energy methods which are most widely used,among which the Rayleigh-Ritz method and Galerkin method are analyzed and compared.On these basis of these theories,only the combination of bending and torsional deformation of thin-walled box beams is considered at first.Under such circumstances,the nonlinear post-buckling analysis is carried out.Then the influence of distortion deformation on the nonlinear behavior of thin-walled structure is considered.Third,the effect of shear lag on the nonlinear buckling of the structure is considered.Forth,the free vibration analysis of coupled bending and torsional deformation of thin-walled box beam is analyzed.Finally,the influence of distortion on free vibration is considered.The research contents of this paper mainly include:(1)The deformation characteristics of bending and torsion displacement fields of thin-walled box beams under the generalized coordinates are analyzed.And the general forms of axial extension,bending,warping and torsion displacement fields are expressed in the form of partial displacements.The equilibrium equations are derived on the premise of large displacement and small deformation,which makes it possible to study the nonlinear buckling and post-buckling of thin-walled box beams under concentrated or distributed loads.Based on Galerkin method,the governing differential equation is obtained,and three nonlinear equilibrium equations are solved by Newton-Rafson algorithm.For the bi-symmetric thin-walled box beam,the post-buckling path can be obtained only by considering the cubic term of the post-buckling equilibrium equation.Considering the pre-buckling,the vertical displacement of thin-walled box girder increases obviously,and the geometric nonlinear effect has a significant influence on the post-buckling.(2)The distortion displacement of thin-walled box beam is analyzed in detail by using the distortion generalized coordinate method in the beam-frame model.The characteristics of five kinds of deformation(extension,bending,warping,torsion and distortion)displacement fields under eccentric load are studied and analyzed.On the basis of Galerkin method,five strongly coupled equations of lateral buckling equilibrium for thin-walled box beams are established,and the quadric algebraic equation of critical bending moment is obtained.In the numerical example,the length-width ratio of the cross section and the change of the load height of the simply supported mono-symmetric thin-walled box beam are analyzed.By comparing and verifying the theoretical results with those produced by the shell finite element method(ANSYS)and F.Mohri theory,it can be seen that the nonlinear behavior of the section depends largely on the cross-section size and the length of the thin-walled box beam.As for the significant difference between the nonlinear model and linear model,the analysis shows that it is mainly caused by distortion deformation effect.The pre-buckling,buckling and post-buckling behaviors are closely related to load height parameters.(3)According to the generalized shear lag effect,six types of deformation(extension,bending,warping,torsion,distortion and shear lag)of thin-walled box beams are analyzed under the combined action of axial and eccentric loads.Six highly coupled nonlinear buckling equilibrium equations are established and solved by Newton-Rafson algorithm.In the examples,this theory is verified and compared with Meftah theory,Tan theory and finite element theory.In addition,the linear and nonlinear buckling of this theory and Tan theory are compared and analyzed with different load heights.It is concluded that unequal thickness of webs can effectively improve the bearing capacity of structures at the same condition.When thin-walled box beams are subjected to the combination of axial and eccentric vertical loads,these formulas are naturally suitable for capturing the responses of bending,torsion,distortion and shear lag.(4)On the basis of the second chapter,based on Benscoter theory and using the virtual work principle,the kinematics model of thin-walled box girder with bending and torsion is established,which includes the warping effect.Based on Hamiltonian principle,the coupling governing equations are derived to explain the motion of axial elongation,bending and torsion.And four coupling equations of free vibration are established.On the premise of simply supported boundary conditions,the natural frequency equation of bending and torsion free vibration of thin-walled box beam is derived.In numerical examples,the numerical calculation results of theoretical modal frequency are in good agreement with the calculation results of finite element method(ANSYS),and the bending-torsion coupling mode obtained is also relatively consistent,which verifies the effectiveness and accuracy of the theoretical method in this paper.Under the same conditions,the mechanical properties of thin-walled box beams with unequal webs are higher than those of box beams with equal webs.(5)On the basis of the third chapter,the influence of distortion on the free vibration of thin-walled box beam is considered.Under the condition of virtual displacement,five dynamic equilibrium equations are established.Under the simple supported boundary condition,a fourth order algebraic equation of natural frequency is obtained,from which the exact solution of the natural frequency of free vibration can be obtained.In the examples,the solutions are verified and compared with the finite element analysis results(FEA)and Proki(?)'s theoretical results.The calculated results of modal frequency are in good agreement with the predicted results of Proki(?) and Ansys.Considering the distortion effect,the kinematics model can perform the modal analysis of thin-walled box girder accurately and effectively.These modal frequency and mode results are of great significance to the structural design and performance analysis of thin-walled box girder.In this study,the deformation field of thin-walled box beam is analyzed according to the characteristics of the bearing deformation of thin-walled box girder.This paper aims at the nonlinear buckling and free vibration problems of thin-walled box beam.The corresponding kinematics model is established for the purpose of obtaining accurate solution,and the equilibrium equation is solved on the basis of variational method.This paper enriches the research methods and contents of nonlinear buckling and free vibration of thin-walled box beams,which is of positive significance to the nonlinear analysis of thin-walled structures and the development of vibration theory.