Elastic Lateral-torsional Buckling Of Double Symmetric I-section Cantilevers Subjected To Combined Loads

For the sake of the requirement of the design of lateral-torsional buckling?LTB?in cantilevers with double-symmetric section,a novel practical formula of critical moment M_crr is proposed,which is applicable to the loading cases of a single load or combined loads.In current standards and literatures,for cantilevers,the methods of the calculation of M_cr were presented merely for some cases with a single load,including a moment at the free end?EM?,a concentrated load at the tip?CLT?and a uniformly distributed load?UDL?.Generally,those three loading cases are applied at the top flange,the shear center,or the bottom flange.In this paper,utilizing a method to calculate critical moment M_cr,named the combined moment factor C_b and the coefficient?_y method?C_b-?_y method?,this practical formula of M_cr is proposed,which is verified the accuracy and feasibility through theoretical analysis and finite element analysis?FEA?.Firstly,based on the governing equations of LTB,the 3-factor formula in combined moment factor C_b and the coefficient?_y are obtained by theoretical analysis.And a precondition of this process is given.There must be only one term in the approximate function of twist?,and can be certain terms for lateral displacement u.In terms of the derivation of the 3-factor formula,on the base of the total potential energy and Rayleigh-Ritz method,the theoretical formula of C_b,?_y and the 3 factors are presented by n-term-basis-function approximate function of u and 1-term basis function for?.Secondly,the reason is analyzed,that it is difficult to obtain the formula of the 3factors by theoretical derivation.According to the boundary condition?BC?at the tip of a cantilever,the relationship between this BC,including the first and third derivative of?,and torsion parameter K is presented.Therefore,if employing the Rayleigh-Ritz method to derivate the 3-factor formula theoretically,it is challenging to attain the result in high accuracy by the approximate function with fixed BC.Additionally,the effect of the second moment of area about the weak axis I_y on C_b is investigated by the value of C_b determined by the critical moment M_cr from FEA.It is indicated that the effect of I_y on C_b can be neglected compared with that of K.Thirdly,a novel FEA program of LTB is written which can analyze the critical moment M_cr in batch.Based on the element stiffness matrix involving the term of concentrated loads,the batch calculation of M_cr is implemented in MATLAB and its APIs for Excel.The precision of this program is verified by comparing its result with the counterpart from tests and ABAQUS in current literatures.Besides,with the application of the APIs for Python in ABAQUS,the Python scripts with fast modelling and analysis are written.Finally,the practical formulae of C_1,i and C_2,i in the 3 factors are proposed which is applicable to 4 kinds of single-loading cases in cantilevers.Also,those of the correlation factor C_1,ij and the combined moment factor C_b are presented which is applicable to the loading cases consist of two single loads.Moreover,for the single-loading cases,the accuracy and feasibility of current methods are verified by the comparison of the results from the practical formula in this paper,the current methods and FEA.It is indicated that there are significant errors or limited feasible ranges in the majority of current literatures.However,the practical formulae in this paper are quadratic polynomials in a concise form,as well as high accuracy and feasibility.Furthermore,the C_b-?_y method in this paper is suitable for the combined-loading cases in cantilevers.