7.2 Maximum Stresses
71
Fig. 7.1 To analysis of
stresses
F 1 = S cos α, F 2 = S cos β, F 3 = S cos γ,
(7.8)
where α, β ? γ mean angles between the normal line n (Fig. 7.1) to the considered
area and respective principal directions.
Let us calculate the square of resultants of all forces distributed over the pyramid
faces. It can be represented by the formula
(σ 1 F 1 )
2
+ (σ 2 F 2 )
2
+ (σ 3 F 3 )
2 ,
(7.9)
or as ((S) 2 , where is full stress on the area S. We will obtain
=
σ 2
1 cos 2 α + σ 2
2 cos 2 β + σ 2
3 cos 2 γ .
(7.10)
In what follows, for certainty reasons, we will assume that
σ 1 > σ 2 > σ 3 .
(7.11)
In these conditions, from Eq. (7.10) we have
2 σ 2
1 (cos 2 α + cos 2 β + cos 2 γ ) = σ 2
1 ,
2 σ 2
3 (cos 2 α + cos 2 β + cos 2 γ ) = σ 2
3 .
Hence it follows that principal stresses determine the maximum and minimal
values of full stress.
Normal stress σ in the section abc can be obtained if making up the sum of
projections of all forces applied to that pyramid to the normal line to its basis. We
obtain
σ 1 F 1 cos α + σ 2 F 2 cos β + σ 3 F 3 cos γ = σ S;
hence we have
σ = σ 1 cos
2 α + σ 2 cos
2 β + σ 3 cos
2 γ.
(7.12)