Space diagonal and octahedral plane
As will be seen in Chapter 13, there are occasions when we must think in terms of threedimensional stress systems. In order to do this, it is general practice to use the stress space formed between the three principal stress axes, Octi,0cti,0cr3. For example in Fig. 3.35a, point P represents the threedimensional stress state (<7i, 02,03).
If we consider all the points where o\= a2 = (73, it is found that they lie on a straight line which passes through the origin and makes the same angle (cos1 1/^3) with each of the three axes. This line is known as the hydrostatic stress axis or the space diagonal. A plane that is normal to the space diagonal is known as an octahedral plane. Obviously there are an infinite number of octahedral planes on a space diagonal but we are usually only interested in the one corresponding to the stress system being considered.
The normal stress acting at point P on the octahedral plane is given the symbol t70Ct and is called the octahedral normal stress. The shear stress acting on the octahedral plane at point P is given the symbol r^t and is called the octahedral shear stress. The expressions for <7<>ct and roct are derived in most stress analysis text books and are:
Toct = 3 \/[(<71 ~ 02 f ~ (<72  <73 f ~ (<73  <7] )2]
By inserting a\ = <7i — u, etc., it can easily be shown that <7oCt = <70Ct and
To express triaxial test results the formulae must be changed to: _ (<7a + 2<7r)
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