Research And Finite Element Verification On Torsion And Vibration Theory Of Channel Cross-Section Beam Based On Plate-Beam Theory

At present,there are few researches on the bending and torsional theory of slot section with single axisymmetric opening.In order to be used as a theoretical reference in practical engineering,based on Professor Zhang Wenfu’s "plate-beam theory",this paper carries out a theoretical study on the combined torsion and vibration of a single material and functionally graded composite beam with a trough section.At the same time,ANSYS finite element software is used to establish the corresponding model for static and dynamic analysis and verification.Functionally gradient material is a new type of material composed of two or more kinds of materials,whose composition,structure and properties are distributed according to a certain gradient and change continuously.Because of the particularity of this material,it is mostly used in extreme conditions to ensure normal and stable work.For functionally gradient beam members,the material composition,density,elastic modulus and other characteristics can be gradient along the length,thickness and width of the member.Currently,there are more researches on the static and dynamic characteristics of FGM beam whose material changes along the axial direction(beam length direction),but few researches on the mechanical characteristics of the material changes along the gradient direction of the thickness and width of each plate.Therefore,this study focuses on the static and dynamic characteristics of a trough section beam with a continuous distribution of material along the thickness and width.The main research contents of this paper are as follows:(1)Based on the "plate-beam theory",the equations of total strain energy and total potential energy of a cantilever beam with a single steel and functionally graded material are derived respectively.Based on energy variational model and differential equation model,the twist Angle formula of cantilever beam end is obtained.The functionally graded material varies along the thickness and along the width.It is worth noting that three unknowns need to be introduced into the longitudinal displacement of the centroid of each plate in the derivation of the centroid of a beam with a groove section,and the centroid coordinates need to be solved by means of the displacement-deformation coordination condition and the force balance condition.At the same time,the warpage stiffness and free torsional stiffness of the beam with the groove section are obtained,and the theoretical value of torsional Angle is compared with the calculation result of the model established by the finite element software ANSYS,so as to verify the correctness of the "plate-beam theory".(2)Based on the "plate-beam theory",the pseudo-static analysis and research of simply supported beams with trough section under torsional vibration also start from a single material and FGM composite material distributed along the thickness and width direction.The equations of total torsional kinetic energy and total potential energy of a simply supported beam with a trough section are derived.Based on the two models and simply supported boundary conditions,the analytical solutions of each order vibration frequency are obtained.At the same time,the corresponding finite element model is established to solve the problem,and the natural vibration frequency solutions of each order obtained from the theoretical derivation are compared with the finite element simulation solutions,and the corresponding mode diagram is obtained,so as to verify the accuracy of the "plate-beam theory".(3)Based on the "plate-beam theory",the mechanical properties of the simply-supported beam with a trough section under flexural and torsional vibration are analyzed,also starting from a single material and two distribution forms of FGM composite materials.The equations of total bending and twisting energy and total potential energy of simply supported beam with groove section are derived.Based on the two models,boundary conditions,Euler equation and differential operator method,the higher-order differential equation is solved,and the analytical solutions of each order frequency of flexural and torsional vibration are obtained.The numerical frequency solution obtained from the corresponding finite element model is compared with the theoretical frequency solution of each order of natural vibration,so as to verify the applicability of the "plate-beam theory".