222 Mechanics of Materials 2 $8.3 B asn关 Bany y=nz Fig.8.2. A Z (normol to ABC) an B Body-force stresses Surface area ABC.△s introduced at centroid (shown removed from elemert for C simplification】 Fig.8.3.The state of stress on an inclined plane through any given point in a three-dimensional cartesian stress system. ABC and YZ are given by the angle between the normals to both planes n and x,etc.) For convenience,let the plane ABC initially be some perpendicular distance h from so that the cartesian stress components actually acting at can be shown on the sides of the tetrahdedron element ABCO so formed(Fig.8.3).In the derivation below the value of h will be reduced to zero so that the equations obtained will relate to the condition when ABC passes through O. In addition to the cartesian components,the unknown components of the stress on the plane ABC,i.e.Pn,Py and pin,are also indicated,as are the body-force field stress components which act at the centre of gravity of the tetrahedron.(To improve clarity of the diagram they are shown displaced from the element.) Since body-force stresses are defined as forces/unit volume,the components in the X,Y and Z directions are of the form F×△S年 where ASh/3 is the volume of the tetrahedron.If the area of the surface ABC,i.e.AS,is assumed small then all stresses can be taken to be uniform and the component of force in222 Mechanics of Materials 2 $8.3 I f 7.1-12 Fig. 8.2 A 2 t ;2 I FX Body- farce stresses introduced at centroid (shown rernced frwn element for simpificatm) L AK2 AS Fig. 8.3. The state of stress bn an inclined plane through any given point in a three-dimensional Cartesian stress system. ABC and YZ are given by the angle between the normals to both planes n and x, etc.) For convenience, let the plane ABC initially be some perpendicular distance h from Q so that the Cartesian stress components actually acting at Q can be shown on the sides of the tetrahdedron element ABCQ so formed (Fig. 8.3). In the derivation below the value of h will be reduced to zero so that the equations obtained will relate to the condition when ABC passes through Q. In addition to the Cartesian components, the unknown components of the stress on the plane ABC, i.e. pxn, pvn and pzn, are also indicated, as are the body-force field stress components which act at the centre of gravity of the tetrahedron. (To improve clarity of the diagram they are shown displaced from the element.) Since body-force stresses are defined as forces/unit volume, the components in the X, Y and Z directions are of the form F x AS; where Ash13 is the volume of the tetrahedron. If the area of the surface ABC, i.e. AS, is assumed small then all stresses can be taken to be uniform and the component of force in