In-plane Nonlinear Elastic Stability Of Catenary Arches Incorporating Shear Deformation

The arch structure is widely used in bridges since it has beautiful shape and excellent mechanical properties.The stability problem of long-span arches is rather prominent,and the buckling is usually accompanied with obvious geometric nonlinearity.Because of the clear and practical Euler-Bernoulli beam theory,it is universally used in a study of the stability for arches.This theory has neglected an effect of shear deformation so that overestimates the critical load of in-plane nonlinear buckling of arches.This influence is particularly obvious for a superimposed arch or a small slenderness arch.The researches in the present that the influence of shear deformation on the in-plane nonlinear stability of non-circular arches only depend on numerical analysis.To explore the in-plane nonlinear buckling law of non-circular arches incorporating shear deformation,this paper took catenary fixed arches with arbitrary symmetrical sections for example.Based on the Timoshenko beam theory and the principle of virtual work,the approximate closed-form solutions of in-plane nonlinear stability of catenary fixed arches incorporating shear deformation are derived.The main work and results in this paper are:(1)The in-plane nonlinear strain-displacement expressions of non-circular arches incorporating shear deformation are derived.Based on the Timoshenko beam theory,the in-plane nonlinear strain-displacement expressions of non-circular arches in the Cartesian coordinate system are derived.According to a change of length,curvature and section rotation angle of the curve element before and after deformation,the nonlinear strain-displacement expressions of compression,bending and shear are derived.Since the deduction is based on an arbitrary curved element,the three nonlinear strains obtained in this paper are suitable for arbitrary non-circular arches in Cartesian rectangular coordinates.(2)A numerical method for determining nonlinear equilibrium paths of a non-circular arch based on the shooting method is proposed.Based on the in-plane nonlinear strain-displacement expression of non-circular arches with shear deformation,the virtual work principle is used to derive the in-plane nonlinear equilibrium differential equations of catenary.The numerical solution of the in-plane nonlinear equilibrium path of catenary arches incorporating shear deformation is obtained by using the shooting method to solve equilibrium differential equations and the bisection method to search for nonlinear equilibrium paths.Compared with the results of finite element,the numerical method is verified.The influence of shear deformation on the nonlinear equilibrium path of catenary arches is analyzed.The result demonstrates that neglecting the shear deformation yields an overestimation of critical load for snap-through buckling.(3)The approximate closed-form solution of the in-plane nonlinear stability of catenary arches incorporating shear deformation is derived.Based on the in-plane nonlinear approximate strain-displacement expression,the virtual work principle is used to derive the in-plane equilibrium differential equation of catenary arches incorporating shear deformation.The approximate curve integration method is used to deduce the in-plane nonlinear equilibrium path and buckling behavior path of arches based on the compressive strain and material formulas.The in-plane buckling equilibrium differential equations are deduced by using the virtual work principle.By solving the eigenvalues of the coefficient matrix of the general solution of equations,the two buckling internal force conditions and the approximate closed-form solution of critical load for bifurcation buckling are obtained.According to the geometric characteristics of the buckling behavior curve,the critical load equation of the snap-through buckling is deduced by employing extreme value theorem.Compared with the finite element results,all approximate closed-form solution in this paper is verified.(4)The approximate closed-form solution of the in-plane nonlinear buckling law of catenary arches considering shear deformation is derived.Based on expressions of vertical displacement and in-plane nonlinear equilibrium equation,the in-plane nonlinear buckling laws of catenary arches considering shear deformation are obtained.The requirements and the sequence of occurrence of the two in-plane nonlinear buckling are expounded in detail,and the critical slenderness ratios of different buckling modes of fixed arches are given.The load-displacement curve and buckling behavior of the arch with critical slenderness ratio are investigated,all approximate closed-form solution of buckling laws is verified.(5)The shear effect slenderness ratio parameter is proposed to demarcate the applied range of Euler-Bernoulli beam theory and Timoshenko beam theory in the analysis of nonlinear stability of non-circular arches.The critical loads of snap-through buckling and bifurcation buckling of arches incorporating and neglecting shear deformation are calculated.The results show that neglecting shear deformation overestimates the critical load of snap-through buckling and bifurcation buckling.Through the analysis of the relative difference between the two buckling critical loads of catenary fixed arches with different shear effect slenderness ratios and different rise-span ratios,it is found that the shear effect slenderness ratio has a significant effect on the buckling critical load of arches.Since this parameter is related to the slenderness ratio of arches,shear correction factor,and the Poisson’s ratio,it can be used to more reasonably define whether shear deformation needs to be considered.