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thus satisfying the identity l2p + m2p + n2p = 1.

Substitution of any principal stress value, again say 01, into the above equations together with the given cartesian stress components allows solution of the determinants and yields values for a 1, b\ and c\, hence k\ and hence l\, m\ and n\, the desired eigen vectors. The process can then be repeated for the other principal stress values o2 + 03.

### 8.19. Octahedral planes and stresses

Any complex three-dimensional stress system produces three mutually perpendicular principal stresses o\,o2, and 03. Associated with this stress state are so-called octahedral planes each of which cuts across the corners of a principal element such as that shown in Fig. 8.24 to produce the octahedron (8-sided figure) shown in Fig. 8.25. The stresses acting on the octahedral planes have particular significance.

The normal stresses acting on each of the octahedral planes are equal in value and tend to compress or enlarge the octahedron without distorting its shape. They are thus said to be hydrostatic stresses and have values given by

Similarly, the shear stresses acting on each of the octahedral planes are also identical and tend to distort the octahedron without changing its volume.

Fig. 8.25. Principal stress system showing the eight octahedral planes forming an octahedron.

The value of the octahedral shear stressesis given by

*23 and ri3 being the maximum shear stresses in the 1-2, 2-3 and 1-3 planes respectively.

Thus the general state of stress may be represented on octahedral planes as shown in Fig. 8.26, the direction cosines of the octahedral planes being given by l = m = n = ±1/^12 + 12 + 12 = ±1/V5 (8.58)

The values of the octahedral shear and direct stresses may also be obtained by the graphical construction of §8.9 since they are represented by a point in the shaded area of the three-dimensional Mohr's circle construction of Figs. 8.8 and 8.9.

* AJ. Durelli, E.A. Phillips and C.H. Tsao. Anahsis of Stress and Strain, chap. 3. p. 26, McGraw-Hill, New York. 1958.

Octahedral planes

Octahedral planes

Fig. 8.26. Representation of a general state of stress on the octahedral planes.

Principal stresses Octahedral normal Octahedral sheor stresses stresses

Fig. 8.26. Representation of a general state of stress on the octahedral planes.

The octahedral shear stress has a particular significance in relation to the elastic failure of materials. Whilst its value is always smaller than the greatest numerical (principal) shear stress, it nevertheless has a value which is influenced by all three principal stress values and has been shown to be a reliable criterion for predicting yielding under complex loading conditions.

The maximum octahedral shear stress theory of elastic failure thus assumes that yield or failure under complex stress conditions will occur when the octahedral shear stress has a value equal to that obtained in the simple tensile test at yield. Now for uniaxial tension, a2 = a3 = 0 and 0\ — oy and from eqn. (8.56)

0 0