tailieunhanh - Elasticity Part 6

Tham khảo tài liệu 'elasticity part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Sadd Elasticity Final Proof 6 13pm page 139 8 Two-Dimensional Problem Solution The previous chapter developed the general formulation for the plane problem in elasticity. This formulation results in two types of in-plane problems plane strain and plane stress. It was further shown that solution to each of these problem types could be conveniently handled using the Airy stress function approach. This scheme reduces the field equations to a single partial differential equation and for the case of zero body forces this result was the biharmonic equation. Thus the plane elasticity problem was reduced to finding the solution to the biharmonic equation in a particular domain of interest. Such a solution must also satisfy the given boundary conditions associated with the particular problem under study. Several general solution techniques were briefly discussed in Section . These include the use of power series or polynomials and Fourier methods. We now pursue the solution to several two-dimensional problems using these methods. Our formulation and solution is conducted using both Cartesian and polar coordinate systems. In many cases we use MATLAB software to plot the stress and displacement field distributions in order to better understand the nature of the solution. Plane problems can also be solved using complex variable theory and this powerful method is discussed in Chapter 10. Cartesian Coordinate Solutions Using Polynomials We begin the solution to plane elasticity problems with no body forces by considering problems formulated in Cartesian coordinates. When taking boundary conditions into account this formulation is most useful for problems with rectangular domains. The method is based on the inverse solution concept where we assume a form of the solution to the biharmonic equation @f 2 @f @f and then try to determine which problem may be solved by this solution. The assumed solution form for the Airy stress function is taken to be a general .