Study On The Torsion Constant And Shear Stress Of Arbitrary Cross-section

Torsion is usually appeared in our life and production, torsion deformation、stressdistribution and torsional rigidity is important in cross-section design and globalstructure analysis. For simple shape cross-section, Poisson function can get theanalytical solution, and then according to the torsion equation of elasticity, we can solvethe torsional rigidity、 shear stress and rate of twist and so on. But for the torsion ofcomplex or multi-connected cross-section, it’s difficult to achieve analytical solution.The nature of Torsion is to solve two-order partial differential equation, i.e. Poissonequation. The analysis methods of Torsion includes Boundary Element Method andFinite Element Method based on wrapping function of torsion theory, Finite ElementMethod based on Prandtl function and the numerical method based on thin-walled rodtheory. Because the Finite Element Method is popular in solving partial differentialequation in engineer, so we adopt the FEM Method to analyze torsion ofmulti-connected cross-section structure.This paper focusing on Finite Difference Method and Finite Element Methodprecedes the following analysis and study:Summarize two main methods of torsion: the Analysis solution method and thenumerical solution method. the Analysis solution method involves wrapping functionmethod、 conjugate function method and so on; The numerical solution methodconsists of Finite Difference Method、Finite Element Method、Temperature fieldanalogy for torsion stress solution function and Boundary Element Method etc.Discuss essential equation of torsion, which lays the foundation for the application ofFinite Element Theory.This paper utilizes triangle element to mesh gridding, first, programs the FiniteDifference Method code for torsion, and then, tests and verifies the simply connectedcross-section, thus proving the validity and accuracy of TFDM program.Based on Torsion stress function, This paper discusses and improves the equivalentfunctional of the difference function and definite conditions of solution according tothe stress theory of multi-connected cross-section rod.On the foundation, it deduces theFinite Element matrix that can be used to solve the multi-connected section, convertsthe definite conditions to amass load added to the inner-boundary, then apply master node and slave node method to achieve the condition that nodes of the sameinner-boundary have the same value, thus directly analyzing multi-connected region,realizing the immediate solution of torsion stress function in multi-connectedcross-section.This method simplifies numbers of virtual elements and nodes, reducesthe compute cost, improves the efficiency. These numerical project cases demonstratethat this method has supernal precision in solving multi-connected and simplyconnected cross-section, comparing with the former Finite Element Methods; this onehas much more superiority.