Text-book of mechanics . is referred. Conceive now a differential element, dx dy dz, in astressed body as represented in Fig. 67. Let the obliquestresses across its various faces be resolved into com-ponents parallel to the axes of reference, then the sixpossible oblique stresses must be represented by theeighteen normal and shearing stresses shown in Fig. 67. The notation used in designating these numerousstresses is as follows: Note first that a double subscript is employed. Thefirst of these subscripts always agrees with the name ofthe axis to which the plane across which the stressoccurs i
Text-book of mechanics . is referred. Conceive now a differential element, dx dy dz, in astressed body as represented in Fig. 67. Let the obliquestresses across its various faces be resolved into com-ponents parallel to the axes of reference, then the sixpossible oblique stresses must be represented by theeighteen normal and shearing stresses shown in Fig. 67. The notation used in designating these numerousstresses is as follows: Note first that a double subscript is employed. Thefirst of these subscripts always agrees with the name ofthe axis to which the plane across which the stressoccurs is perpendicular; the second subscript always STRESS, STRAIN, AND ELASTIC FAILURE 133 denotes the axis to which the stress is parallel. Thusthe stresses across face 3 have as first subscript x, fortheir plane dydz is perpendicular to the X-axis (thus,qx-, qx-, px~); the second subscript indicating the axesto which the stresses are parallel is now added (thus,Ixy, pxx). In the case of the normal stress pxx, as. / x U <lz. dxdydz P + Fig. 67 there is no need of the second subscript, there being noother px_, it is usually omitted, so that pxx is written px. The reader should now write the names of the stresseson faces 1 and 2 and compare his results with those inFig. 67. Consider the stresses on the face 6. The normalstress on this face is denoted by px. But as it acts ata differential distance dx from face 3, this stress willnot in general be equal to the stress px on face 3, for itmay have changed by the partial differential of px with 134 MECHANICS OF MATERIALS respect to x,(-^dx), and it should be denoted by px + -^ dx. In the figure this added differential is denoted dx & only by a plus sign. Similarly, qxz + denotes q„ + ^-zdx, pv + denotes pv + {^Ady, qzy + denotes qzy +dx \ayf %*»&, etc. dZ As the scope of this text precludes the discussion ofthe theory of stress in three dimensions, we shall use theabove notation and Fig. 67 simple to establish a funda-mental theorem
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