文档介绍：LOGO STRESS ON OBLIQUE PLANE THROUGH A POINT Now it is possible to determine the stresses on any other oblique plane passing through the point using the direction cosines. Now it is possible to determine the stresses on any other oblique plane passing through the point using the direction cosines. LOGO ? In practice, the state of stress at a point with reference to the ? general Cartesian co-ordinate system is not very significant. ?? Because of that: ?? The failure of a structure or a body is due to fracture or plastic yielding. ? The failure may occur on a different plane. ? The failure plane is always inclined to the three co-ordinate axes. ?? Therefore, in this section the method of determining the ? stresses on an oblique plane will be described. O ? Considering the equilibrium of the tetrahedron OABC in ? x , ? xy , ? xz be stresses on plane OBC Let ? y , ? yx , ? yz be stresses on plane OAC Let ? z , ? zx , ? zy be stresses on plane OAB and ? nx , ? ny , ? nz be stresses on plane ABC 321,0 Al Al Al AF zx yx x nx x?????????????() where ?A is the area of ABC Dividing Eq.() by ?A, 321lll zx yx x nx???????() LOGO Similar expressions for ? ny and ? nz can be obtained. Expressing these relations in matrix form: ??????????????????????????????? 3 2 1l l l z yz xz zyy xy zx yx x nz ny nx????????????} ]{[}{l n???() () Eq.() are gives the Cartesian components of stress on an inclined plane. LOGO obtain normal and shear stresses on this inclined plane , consider a set of axes x', y', z' . Let x' axis coincide with the normal n to the plane ABC in Fig. . The direction cosines can be written as follows: LOGO ng these direction cosines, 321'lll nz ny nx x???????() Substituting for ? nx, ? ny and ? nz from Eq.() 133221 23 22 21'222lllllllll zx yz xy zyxx?????????????() Similarly, 321''mmm nz ny nx yx???????() or )(.)(. )(. 311