Chapter 7
Stressed State in a Body Point
7.1 Principal Stresses
Let us select an arbitrary system of Cartesian coordinates Oxyz. Let the stress
components relative to the selected coordinate systems be known in some point of
the body. Let us consider the problem of finding such areas that have no tangential
stress. As we know (for example, see [3, p. 67]), such areas are called principal,
and the values of normal stresses on these areas are called principal stresses.
The cosines of angles between the normal line to the principal area and
coordinate axes can be found from Eqs. (2.14), to which another condition must
be added
cos
2
nx + cos
2
ny + cos
2
nz = 1.
(7.1)
Equations (2.2) and (7.1) form a system of 4 equations relative to unknown
cosines and stresses σ . Since cosines in Eq. (7.1) cannot simultaneously turn to zero,
the system determinant (2.2) must turn to zero, e. g.
σ x − σ τ xy
τ xz
τ yx σ y − σ τ yz
τ zx
τ zy σ z − σ
= 0.
(7.2)
From Eq. (7.2), normal stresses (σ ) are determined in principal directions, e. g.
principal stresses. By opening the determinant, the last equation can be made to look
as follows:
σ
3
− I 1 σ
2
+ I 2 σ − I 3 = 0,
(7.3)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_7
69
Stressed State in a Body Point
7.1 Principal Stresses
Let us select an arbitrary system of Cartesian coordinates Oxyz. Let the stress
components relative to the selected coordinate systems be known in some point of
the body. Let us consider the problem of finding such areas that have no tangential
stress. As we know (for example, see [3, p. 67]), such areas are called principal,
and the values of normal stresses on these areas are called principal stresses.
The cosines of angles between the normal line to the principal area and
coordinate axes can be found from Eqs. (2.14), to which another condition must
be added
cos
2
nx + cos
2
ny + cos
2
nz = 1.
(7.1)
Equations (2.2) and (7.1) form a system of 4 equations relative to unknown
cosines and stresses σ . Since cosines in Eq. (7.1) cannot simultaneously turn to zero,
the system determinant (2.2) must turn to zero, e. g.
σ x − σ τ xy
τ xz
τ yx σ y − σ τ yz
τ zx
τ zy σ z − σ
= 0.
(7.2)
From Eq. (7.2), normal stresses (σ ) are determined in principal directions, e. g.
principal stresses. By opening the determinant, the last equation can be made to look
as follows:
σ
3
− I 1 σ
2
+ I 2 σ − I 3 = 0,
(7.3)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_7
69