74
7 Stressed State in a Body Point
The intensity of (tangential) stresses with the accuracy up to the constant
multiplier equals a mean square of tangential stresses defined on the sphere
(infinitely small radius in the case of a non-homogeneous stressed state). 1 This
means that The intensity of (tangential) stresses is proportional to the mean square
of tangential stresses for all possible areas going through this point of the body.
7.4 Some Properties of Tangential Stresses
By using formula (7.17), we can obtain
τ 12 + τ 23 = τ 31 .
(7.21)
Similar to the numbering of principal stresses (σ 1 > σ 2 > σ 3 ), (p. 71), let us
designate the smallest of principal tangential stresses as τ 1 , the middle one as τ 2 and
the maximum one as τ 3 . Then formula (7.21) will look like [1]
τ 3 = τ 1 + τ 2 .
(7.22)
Since we assumed that τ 2 > τ 1 , it follows from formula (7.22) that
τ 2 >
1
2
τ 3 .
(7.23)
Let us introduce designations
y =
τ o
τ 3
, x =
τ 2
τ 3
.
Formula (7.22) shows that τ 1 = τ 3 − τ 2 . By substituting this result into formula
(7.19), we obtain
y(x) =
2
3
(1 − x) 2 + x 2 + 1.
(7.24)
We use formula (7.24 to analyze the change in the intensity of tangential stresses
τ o as the tangential stress τ 2 changes, which we will conditionally call the middle
stress [2]. Taking into account the condition (7.23), we note that the values of the
function y(x) are interesting only at 0, 5 x 1. Atthe ends of this segment, we
will have
1 This property was proved by V.V. Novozhilov.