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. A related notion, plane
strain, is often applicable to very thick members.
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 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 , , and , that is being stretched in the direction and
compressed in the direction, by pairs of opposite forces with magnitudes 10 N
and 20 N, respectively, uniformly distributed over the corresponding
faces. The stress tensor inside the block will be
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_stress Plane 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
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. 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 term arises from the Poisson's effect. This strain term
can be temporarily removed from the stress analysis to leave only the
in-plane terms, effectively reducing the analysis to two dimensions.
Stress transformation in plane stress and plane strain
Consider a point in a continuum under a state of plane stress, or plane strain,
with stress components and all other stress components equal to zero. From
static equilibrium of an infinitesimal material element at , the normal stress
and the shear stress on any plane perpendicular to the - plane passing
through with a unit vector making an angle of with the horizontal, i.e. is
the direction cosine in the direction, is given by:
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 , if one knows the stress components on any two perpendicular
directions at that point. It is important to remember that we are
considering a unit area of the infinitesimal element in the direction
parallel to the - plane. The principal directions, 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 equal to zero.
Thus we have: and we obtain
This equation defines two values which are apart. The same result can be
obtained by finding the angle which makes the normal stress a maximum, i.e.
The principal stresses and , or minimum and maximum normal stresses and ,
respectively, can then be obtained by replacing both values of into the
previous equation for . This can be achieved by rearranging the equations
for and , first transposing the first term in the first equation and squaring
both sides of each of the equations then adding them. Thus we have
where which is the equation of a circle of
radius centered at a point with coordinates , called Mohr's circle. But
knowing that for the principal stresses the shear stress , then we obtain from
this equation: When the infinitesimal element is
oriented in the direction of the principal planes, thus the stresses
acting on the rectangular element are principal stresses: and . Then the
normal stress and shear stress as a function of the principal stresses can
be determined by making . Thus we have Then the maximum shear stress occurs
when , i.e. : Then the minimum shear stress occurs
when , i.e. : See also
Plane strain References
^ a b c Meyers and Chawla: "Mechanical Behavior of Materials," 66-75.