The Solution of Torsion Problem for the Bars with Irregular Cross Sections by Boundary Element Method

摘要

The objective of the present study is to determine the stress distribution in bars with irregular cross sections under torsion by boundary element method. For this purpose an integral equation is built via reciprocal theorem. This integral equation involves two elastostatic states for the same body. First one represents the problem to be solved while the second does a singular elastostatic state which arises due to a singular, line body force in an infinite medium. It is accepted that Saint Venant’s principle is valid. The unknown of the integral equation mentioned above is the boundary value of the torsion function. In boundary element method, boundary is divided to linear elements whose end points are named as nodal points and this equation is reduced to a system of linear algebraic equations. The unknowns of this system are nodal values of torsion function. All singularities are eliminated. After evaluation torsion function, stresses can be determined inside the region. But a different formulation is necessary for determination of the unknown stress component on the boundary. By the way the torsional rigidity of the cross-section is also determined. Three sample problems are solved. In the first case cross-section is selected to be a rectangular to check the formulation. In the second sample problem a rectangular cross-section involving a notch is considered. The third sample problem is a rectangular reinforced concrete column with four rebars. For the first problem results are coincided with analytical solution. Interesting point is that the relative error has millionth order for both displacements and stresses while for torsional rigidity has .09 percent. The last problem is a mix-boundary value problem in a multiple connected region with two different materials.