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After calculation of the three principal stress values, they can be placed in their normal conventional order of magnitude, viz. <T|, and 03.

The procedure is, in effect, the same as that of §8.13 but carried out in terms of the stress invariants.

8.18.1. Evaluation of direction cosines for principal stresses (eigen vectors)

Having determined the three principal stress values for a given three-dimensional complex stress state using the procedures of §8.13.1 or §8.18, above, a complete solution of the problem generally requires a determination of the directions in which these stresses act-as given by their respective direction cosines or eigen vector values.

The relationship between a particular principal stress op and the cartesian stress components is given by eqn (8.43)

If one of the known principal stress values, say o\, is substituted in the above equations together with the given cartesian stress components, three equations result in the three unknown direction cosines for that principal stress i.e. /], m\ and n\.

However, only two of these are independent equations and the additional identity l\ + m\ + n2 = 1 is required in order to evaluate l\m\ and n \.

The procedure can then be repeated substituting the other principal stress values 02 and <73, in turn, to produce eigen vectors for these stresses but it is tedious and an alternative matrix approach is recommended as follows:

Equation (8.43) above can be expressed in matrix form, thus:

with cofactors of the determinant on the elements of the first row of:

'■yz with the direction cosines or eigen vectors of the principal stresses given by:

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