 | Stress physics: Encyclopedia II - Stress physics - Plane stress
Stress physics - Plane stress
Plane stress is a two-dimensional state of stress (Figure 2). This 2-D state models well the state of stresses in a flat, thin plate loaded in the plane of the plate. Figure 2 shows the stresses on the x- and y-faces of a differential element. Not shown in the figure are the stresses in the opposite faces and the external forces acting on the material. Since moment equilibrium of the differential element shows that the shear stresses on the perpendicular faces are equal, the 2-D state of stresses is characterized by three independent stress components (σx, σy, τxy).
Stress physics - Principal stresses
Cauchy was the first to demonstrate that at a given point, it is always possible to locate two orthogonal planes in which the shear stress vanishes. These planes are called the principal planes, while the normal stresses on these planes are the principal stresses. The common technique for doing this is by use of Mohr's circle.
Principal stresses are the maximum and minimum values of the normal stresses. Eigenvalues of a stress tensor show the principal stresses, and the eigenvectors show the direction of the principal stresses.
Stress physics - Mohr's circle
A graphical representation of any 2-D stress state was proposed by Christian Otto Mohr in 1882. Consider the state of stress at a point P in a body (Figure 2). The Mohr's circle may be constructed as follows.
1. Draw two perpendicular axes with the horizontal axis representing normal stress, while the vertical axis the shear stress.
2. Plot the state of stress on the x-plane as the point A, whose abscissa (x value) is the magnitude of the normal stress, σx (tension is positive), and whose ordinate (y value) is the shear stress (clockwise shear is positive).
3. Mark the magnitude of the normal stress σy on the horizontal axis (tension being positive).
4. Mark the midpoint of the two normal stresses, O (Figure 3).
5. Draw the circle with radius OA, centered at O (Figure 4).
6. A point on the Mohr's circle represents the state of stresses on a particular plane at the point P. Of special interest are the points where the circle crosses the horizontal axis, for they represent the magnitudes of the principal stresses (Figure 5).
Mohr's circle may also be applied to three-dimensional stress. In this case, the diagram has three circles, two within a third.
Engineers use Mohr's circle to find the planes of maximum normal and shear stresses, as well as the stresses on known weak planes. For example, if the material is brittle, the engineer might use Mohr's circle to find the maximum component of normal stress (tension or compression); and for ductile materials, the engineer might look for the maximum shear stress.
Other related archives1882, 2-D, 3-D, Augustin Louis Cauchy, Christian Otto Mohr, Eigenvalues, Hooke's law, MPa, N, Nails, Pascal's law, Poisson contraction, Poisson's ratio, Residual stresses, SI, Strain tensor, Stress concentration, Stress-energy tensor, Stress-strain diagrams, abscissa, area, behavior, bolt, brittle, compressive strength, coordinates, cross section, diameter, ductile, eigenvectors, engineering, fluid, force, fracture, hard, katana, magnitude, martensite, measurement, mm, nitrogen, normal, normal stress, ordinate, pascal, physics, plastic, plastic deformation, plasticity, pressure, prestressed concrete, proportional, psi, rifling, rubber, scalar, shear stress, steel, strain, tensile strength, tensile test, tensor, tensors, torque, toughened glass, trace, vector, viscosity, von Mises stress, wire, yield
 Adapted from the Wikipedia article "Plane stress", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
