Stress (physics)

In physics, stress is the internal distribution of forces within a body that balance and react to the loads applied to it. Stress is a tensor quantity with nine terms, but can be described fully by six terms due to symmetry. Simplifying assumptions are often used to represent stress as a vector for engineering calculations.

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).

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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 the use of Mohr's circle.

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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.

Related Topics:
Christian Otto Mohr - 1882

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1. Draw two perpendicular axes with the horizontal axis representing normal stress, while the vertical axis the shear stress.

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2. Plot the state of stress on the x-plane as the point A, whose abscissa is the magnitude of the normal stress (tension is positive), and whose ordinate is the shear stress (counterclockwise shear is negative).

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3. Mark the magnitude of the normal stress σy on the horizontal axis (tension being positive).

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4. Mark the midpoint of the two normal stresses, O (Figure 3).

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5. Draw the circle with radius OA, centered at O (Figure 4).

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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

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the points where the circle crosses the horizontal axis, for they represent the magnitudes of the principal stresses (Figure 5).

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Mohr's circle may also be applied to three-dimensional stress. In this case, the diagram has three circles, two within a third.

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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.

Related Topics:
Brittle - Ductile

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