THREE-DIMENSIONAL STRESS Machinery's Handbook, 31st Edition
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Fig. 12. x, y, z- Coordinate System Showing Positive Directions of Stresses The von Mises effective tensile stress method can be used to evaluate three-dimensional stress failure of ductile materials. If stress analysis is based on the maximum-shear-stress theory of failure, the triaxial stress cubic equation is used first to calculate the three princi- pal stresses; true maximum shear stress can then be determined. The following procedure provides the principal maximum normal tensile and compressive stresses and the true maximum shear stress at any point on a body subjected to any combination of loads. The basis for the procedure is the stress cubic equation S 3 − AS 2 + BS − C = 0 in which: A = S x + S y + S z B = S x S y + S y S z + S z S x − S xy 2 − S yz 2 − S zx 2 C = S x S y S z + 2 S xy S yz S zx − S x S yz 2 − S y S zx 2 − S z S xy 2 and S x , S y , etc. are as shown in Fig. 12. The x , y , z coordinate system in Fig. 12 shows the positive directions of the normal and shear stress components on an elementary cube of material. Only six of the nine compo nents shown are needed for the calculations: the normal stresses S x , S y , and S z on three of the faces of the cube; and the three shear stresses S xy , S yz , and S zx . The remaining three shear stresses are known because S yx = S xy , S zy = S yz , and S xz = S zx . The normal stresses S x , S y , and S z are shown as positive (tensile) stresses; the opposite direction is negative (compressive). The first subscript of each shear stress identifies the coordinate axis per- pendicular to the plane of the shear stress; the second subscript identifies the axis to which the stress is parallel. Thus, S xy is the shear stress in the yz -plane to which the x -axis is perpendicular, and the stress is parallel to the y -axis. Step 1. Draw a diagram of the hardware to be analyzed, and show the applied loads P , T , and any others. Step 2. For any point at which the stresses are to be analyzed, draw a coordinate diagram similar to Fig. 12 and show the magnitudes of the stresses resulting from the applied loads (these stresses may be calculated by using standard basic equations from strength of mate rials, and should include any stress concentration factors). Step 3. Substitute the values of the six stresses S x , S y , S z , S xy , S yz , and S zx , including zero values, into the formulas for the quantities A through K . The quantities I , J , and K represent the principal normal stresses at the point analyzed. As a check, if the algebraic sum I + J + K equals A , within rounding errors, then the calculations up to this point should be correct.
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