Research And Application Of A New Variable Section Lattice Beam Element

The lattice structure is a complex rod structure that is common in engineering practice.Among them,the variable cross-section lattice structure is widely used because it increases the utilization rate of materials,reduces the use of materials,and implements the principle of equal strength well.Nowadays,large-scale hoisting machinery is becoming taller,which increases the slenderness ratio of its structure.Under heavy load,the stability of its structure is particularly important.At present,there is no mature calculation formula like the constant section lattice structure,which brings great trouble to the engineering analysts,and the manual calculation is difficult to ensure the accuracy.Finite element analysis is composed of a large amount of heavy modeling and data,as well as modeling difficulties brought by actual working conditions.In order to analyze the axial pressure stability problem efficiently and accurately.This paper is based on the Euler-Bernoulli beam element,the variable beam lattice structure is equivalent to a beam element model with constant cross-sectional area and a second change of inertia moment in the axial direction.The main research contents and methods of this article are as follows:Firstly,based on the Euler-Bernoulli beam theory,a cubic spline interpolation function and a quadratic Lagrange interpolation function are used to construct the displacement field of a two-node Euler-Bernoulli beam element with a variable cross-section.The tangent stiffness matrix of the variable section lattice beam element,combined with the static condensation method,eliminates the node curvature freedom,and a new type of planar two-node variable section lattice beam element is obtained.When the taper coefficient ? of the variable section approaches 0,the stiffness matrix will degenerate into the corresponding uniform cross-section lattice beam element.Secondly,the torsional displacement field is introduced into the stiffness matrix of the space-variable beam element.The torsional displacement field and the axial displacement field of the element are constructed using the quadratic Lagrange interpolation function,and the lateral displacement field of the element is constructed using the cubic spline interpolation function.A Euler-Bernoulli beam element with three equal parts and 16 degrees of freedom,and its tangent stiffness matrix is derived.Then,the two degrees of freedom of the middle node are eliminated using the static agglomeration method,and the equivalent 12-degree-of-freedom space variable section of the two nodes Condensed beam element,the new variable section beam element has the same number of degrees of freedom and distribution as the traditional two-node beam element.Finally,the software MATHEMATICA is used to write a calculation program for the variable cross-section lattice beam structure.The polynomial integration,addition,and condensation are automatically generated and used to solve the stability and second-order effects of variable cross-section beam elements.Examples such as the variable-cut cotton cantilever beam,shuttle-shaped cross-section lattice structure,and portal frame structure were used to calculate the critical axial instability of the structure and the second-order effect deformation using the method of this paper and ANSYS software to verify the paper The correctness and effectiveness of the method.The analysis results show that the variable crosssection lattice plane and space stiffness matrix derived in this paper are correct and can be used for stability and second-order effect analysis.The method in this paper has extremely high accuracy and fast solution speed,and is an efficient analysis method.The new variable crosssection lattice beam element researched can not only be applied to the analysis of the axial compression stability of single members,but also can be used to analyze the axial compression of composite members with variable cross sections and equal sections.