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In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular surface. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analysis is considerably simplified, as the stress state can be represented by a tensor of dimension 2 (representable as a 2 × 2 matrix rather than 3 × 3). A related notion, plane strain, is often applicable to very thick members.
Contents
- Mathematical definition
- Constitutive equations
- Plane stress in curved surfaces
- Plane strain strain matrix
- Stress transformation in plane stress and plane strain
- References
Plane stress typically occurs in thin flat plates that are acted upon only by load forces that are parallel to them. In certain situations, a gently curved thin plate may also be assumed to have plane stress for the purpose of stress analysis. This is the case, for example, of a thin-walled cylinder filled with a fluid under pressure. In such cases, stress components perpendicular to the plate are negligible compared to those parallel to it.
In other situations, however, the bending stress of a thin plate cannot be neglected. One can still simplify the analysis by using a two-dimensional domain, but the plane stress tensor at each point must be complemented with bending terms.
Mathematical definition
Mathematically, the stress at some point in the material is a plane stress if one of the three principal stresses (the eigenvalues of the Cauchy stress tensor) is zero. That is, there is Cartesian coordinate system in which the stress tensor has the form
For example, consider a rectangular block of material measuring 10, 40 and 5 cm along the
More generally, if one chooses the first two coordinate axes arbitrarily but perpendicular to the direction of zero stress, the stress tensor will have the form
and can therefore be represented by a 2 × 2 matrix,
Constitutive equations
See Hooke's law#Plane_stressPlane stress in curved surfaces
In certain cases, the plane stress model can be used in the analysis of gently curved surfaces. For example, consider a thin-walled cylinder subjected to an axial compressive load uniformly distributed along its rim, and filled with a pressurized fluid. The internal pressure will generate a reactive hoop stress on the wall, a normal tensile stress directed perpendicular to the cylinder axis and tangential to its surface. The cylinder can be conceptually unrolled and analyzed as a flat thin rectangular plate subjected to tensile load in one direction and compressive load in another other direction, both parallel to the plate.
Plane strain (strain matrix)
If one dimension is very large compared to the others, the principal strain in the direction of the longest dimension is constrained and can be assumed as zero, yielding a plane strain condition (Figure 7.2). In this case, though all principal stresses are non-zero, the principal stress in the direction of the longest dimension can be disregarded for calculations. Thus, allowing a two dimensional analysis of stresses, e.g. a dam analyzed at a cross section loaded by the reservoir.
The corresponding strain tensor is:
in which the non-zero
Stress transformation in plane stress and plane strain
Consider a point
These equations indicate that in a plane stress or plane strain condition, one can determine the stress components at a point on all directions, i.e. as a function of
The principal directions (Figure 8.3), i.e., orientation of the planes where the shear stress components are zero, can be obtained by making the previous equation for the shear stress
and we obtain
This equation defines two values
The principal stresses
where
which is the equation of a circle of radius
When
Then the maximum shear stress
Then the minimum shear stress