1.12 Homogeneous Stressed State
13
Fig. 1.6 To the law of
twoness of tangential stresses
between each other and directed oppositely. This trivial result can be considered as
a condition of homogeneity of a stressed state.
Now let us satisfy the third condition of equilibrium—zero equality of the sum of
moments. If we deem the size of the parallelepiped shown in Fig. 1.6 in the direction
perpendicular to the plane xy to be equal to zero, we find that tangential forces acting
on two horizontal areas are equal to τ x · b · 1. The moment of the pair of horizontal
forces will be M 1 = τ x · b · a. In the same manner, let us find the moment M 2 of
forces parallel to the axis y. For the directions of tangential forces given in Fig. 1.6,
the moment M 2 will have the sign opposite to the sign of M 1 , e. g. M 2 = −τ y · a · b.
By equating the sum of moments of all forces applied to the element to zero,
we get τ x = τ y . The acquired equality expresses the so-called law of twoness of
tangential stresses. For an arbitrary stressed state, it can be generalized as follows:
tangential stresses on perpendicular areas, perpendicular to the intersection line of
these areas, equal between each other.
These stresses are always directed as shown in Fig. 1.6 or opposite to the given
one, simultaneously for all four areas.
1.12 Homogeneous Stressed State
When studying the elongation of a material specimen previously, we already spoke
(p. 7) of the homogeneous stressed state. In a general case, let us mentally separate
from a body a rectangular parallelepiped whose faces are parallel to coordinate
planes (Fig. 1.7). The stress vector on any of its faces can be decomposed into
components in three directions coinciding with the direction of the coordinate axes
x, y, z.
One of the components will be directed along the normal line to the specified
face. This component of the stress vector is called normal, and its vector is called
normal stress. Let us designate normal stresses by the symbol σ k , (k ∼ x, y, z),
13
Fig. 1.6 To the law of
twoness of tangential stresses
between each other and directed oppositely. This trivial result can be considered as
a condition of homogeneity of a stressed state.
Now let us satisfy the third condition of equilibrium—zero equality of the sum of
moments. If we deem the size of the parallelepiped shown in Fig. 1.6 in the direction
perpendicular to the plane xy to be equal to zero, we find that tangential forces acting
on two horizontal areas are equal to τ x · b · 1. The moment of the pair of horizontal
forces will be M 1 = τ x · b · a. In the same manner, let us find the moment M 2 of
forces parallel to the axis y. For the directions of tangential forces given in Fig. 1.6,
the moment M 2 will have the sign opposite to the sign of M 1 , e. g. M 2 = −τ y · a · b.
By equating the sum of moments of all forces applied to the element to zero,
we get τ x = τ y . The acquired equality expresses the so-called law of twoness of
tangential stresses. For an arbitrary stressed state, it can be generalized as follows:
tangential stresses on perpendicular areas, perpendicular to the intersection line of
these areas, equal between each other.
These stresses are always directed as shown in Fig. 1.6 or opposite to the given
one, simultaneously for all four areas.
1.12 Homogeneous Stressed State
When studying the elongation of a material specimen previously, we already spoke
(p. 7) of the homogeneous stressed state. In a general case, let us mentally separate
from a body a rectangular parallelepiped whose faces are parallel to coordinate
planes (Fig. 1.7). The stress vector on any of its faces can be decomposed into
components in three directions coinciding with the direction of the coordinate axes
x, y, z.
One of the components will be directed along the normal line to the specified
face. This component of the stress vector is called normal, and its vector is called
normal stress. Let us designate normal stresses by the symbol σ k , (k ∼ x, y, z),