70
7 Stressed State in a Body Point
where
I 1 = σ x + σ y + σ z ,
I 2 = σ x σ y + σ y σ z + σ z σ x − τ
2
xy − τ
2
yz − τ
2
zx ,
I 3 = σ x σ y σ z + 2τ xy τ yz τ zx − σ x τ
2
yz − σ y τ
2
zx − σ z τ
2
xy .
(7.4)
By connecting this body at the defined load with any other system of coordinates,
we will change the stress coordinates σ x , . . . , τ zx . However, principal stresses will
not depend on the selection of the coordinate system. Consequently, the coefficients
of Eq. (7.3) that define principal stresses keep constant values not depending on the
selection of coordinate axes. Such values are called invariants.
Equation (7.3) has at least one real root. If we know it, we can determine the
principal direction from Eqs. (2.2) and (7.1). Consequently, the direction of the area
where only normal stress acts can be found. Assume that we selected the axis Oz
to be coinciding with this direction. According to the Bezout theorem, Eq. (7.3) can
be represented as
(σ − σ z )[σ
2
− σ (σ x − σ y ) − σ x σ y + τ xy ] = 0.
(7.5)
By opening this equation and grouping the members at σ and σ 2 ,, we will obtain a
certain view of Eq. (7.3) where τ zx = τ zy = 0, since the axis Oz is principal. In this
case, two other principal stresses are defined by the formulas
σ 1,2 =
σ x + σ y
2
±
σ x − σ y
2
2
+ τ 2
xy , (σ 3 = σ z ).
(7.6)
The angle between principal stresses and the axis Ox is determined from the first
Eq. (2.2) when substituting σ under formula (7.6), e. g.
tg nx =
1
2τ xy
σ y − σ x ±
(σ y − σ x ) 2 + 4τ 2
xy
.
(7.7)
The roots of Eq. (7.7) define two mutually orthogonal directions
n 1 x and
n 2 x.
7.2 Maximum Stresses
Let us consider some body subject to a homogeneous stressed state. Let us mentally
separate a pyramid from the body whose faces coincide with principal areas and the
basis is arbitrarily inclined (Fig. 7.1).
By writing the areas of faces F i , (i = 1, 2, 3), as respective projections of the
basis S, we obtain
7 Stressed State in a Body Point
where
I 1 = σ x + σ y + σ z ,
I 2 = σ x σ y + σ y σ z + σ z σ x − τ
2
xy − τ
2
yz − τ
2
zx ,
I 3 = σ x σ y σ z + 2τ xy τ yz τ zx − σ x τ
2
yz − σ y τ
2
zx − σ z τ
2
xy .
(7.4)
By connecting this body at the defined load with any other system of coordinates,
we will change the stress coordinates σ x , . . . , τ zx . However, principal stresses will
not depend on the selection of the coordinate system. Consequently, the coefficients
of Eq. (7.3) that define principal stresses keep constant values not depending on the
selection of coordinate axes. Such values are called invariants.
Equation (7.3) has at least one real root. If we know it, we can determine the
principal direction from Eqs. (2.2) and (7.1). Consequently, the direction of the area
where only normal stress acts can be found. Assume that we selected the axis Oz
to be coinciding with this direction. According to the Bezout theorem, Eq. (7.3) can
be represented as
(σ − σ z )[σ
2
− σ (σ x − σ y ) − σ x σ y + τ xy ] = 0.
(7.5)
By opening this equation and grouping the members at σ and σ 2 ,, we will obtain a
certain view of Eq. (7.3) where τ zx = τ zy = 0, since the axis Oz is principal. In this
case, two other principal stresses are defined by the formulas
σ 1,2 =
σ x + σ y
2
±
σ x − σ y
2
2
+ τ 2
xy , (σ 3 = σ z ).
(7.6)
The angle between principal stresses and the axis Ox is determined from the first
Eq. (2.2) when substituting σ under formula (7.6), e. g.
tg nx =
1
2τ xy
σ y − σ x ±
(σ y − σ x ) 2 + 4τ 2
xy
.
(7.7)
The roots of Eq. (7.7) define two mutually orthogonal directions
n 1 x and
n 2 x.
7.2 Maximum Stresses
Let us consider some body subject to a homogeneous stressed state. Let us mentally
separate a pyramid from the body whose faces coincide with principal areas and the
basis is arbitrarily inclined (Fig. 7.1).
By writing the areas of faces F i , (i = 1, 2, 3), as respective projections of the
basis S, we obtain