tailieunhanh - Elasticity Part 7

Tham khảo tài liệu 'elasticity part 7', 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 14pm page 169 Using the general stress solution these boundary conditions give the following four equations 2a2 2a21 0 - 6 - 2b21 0 2a2 2a21 a6 0 a6 2b21 s These are easily solved for the unknown constants giving s s s s a2 a6 -- a21 v b21 - 8 2 8 4 Back substituting these results determines the stress field solution sr ị 26 cos 26 sin 26 2 2 2 Sy 2 2 _ 26 _ 2 cos 26 sin 26 s Try 2 1 cos 26 sin 26 It has been pointed out that this problem has an apparent inconsistency in the shear stress component at the origin that is txy tyx at x y 0. To investigate this further let us reformulate the problem in Cartesian coordinates. The stress function can be expressed as . 21 y2 x2 y2 . -1 y 2 xy f Vs------------------------2 tal x - V - y 2 The shear stress is then given by ỡ2f Sy2 txy dxdy x2 y2 Excluding the origin this expression tends to zero for y 0 and to s for x 0 and thus has the proper limiting behavior for r 0. However it has been shown by Barber 1992 that the stress gradients in the tangential direction are of order O r 1 . Half-Space Examples Let us next consider the solution to several half-space examples with a domain specified by a 0 and b . We shall investigate examples with uniform loadings over portions of the boundary surface and also cases with concentrated forces. Half-Space Under Uniform Normal Stress Over x 0 The problem of a half space with uniform normal stress over the negative x-axis is shown in Figure 8-18. For the particular angles of a and b that create the half-space domain the general Airy stress function solution form can be reduced to Two-Dimensional Problem Solution 169 TLFeBOOK Sadd Elasticity Final Proof 6 14pm page 170 FIGURE 8-18 Half space under uniform loading over half of free surface. f a6r26 b21r2 sin 26 The hoop and shear stresses corresponding to this function are Ơ0 2a66 2b21 sin 26 Tr6 a6 2b21 cos 26 .