SHEAR STRESS
In additional to Axial (or normal) Stress and Strain (discussed in topic
2.1and 2.2), we may also have what is known as Shear Stress and Shear Strain.In Diagram 1
we have shown a metal rod which is solidly attached to the floor. We then exert a force,
F, acting at angle theta with respect to the horizontal, on the rod. The component of the
Force perpendicular to the surface area will produce an Axial Stress on the rod given by Force
perpendicular to an area divided by the area, or:
The component of the Force parallel to the area will also effect the
rod by producing a Shear Stress, defined as Force parallel to an area divided by the
area, or:
where the Greek
letter, Tau, is used to represent Shear Stress. The units of both Axial and Shear Stress
will normally be lb/in2 or N/m2.
Shear Strain:
Just as an axial stress results in an axial strain, which is the
change in the length divided by the original length of the member, so does shear stress
produce a shear strain. Both Axial Strain and Shear Strain are shown in
Diagram 2. The shear stress produces a displacement of the rod as indicated in the right
drawing in Diagram 2. The edge of the rod is displaced a horizontal distance,
from its initial position. This
displacement (or horizontal deformation) divided by the length of the rod L is equal to
the Shear Strain. Examining the small triangle made by
, L and the side
of the rod, we see that the Shear Strain,
/L , is also equal to the tangent of the
angle gamma, and since the amount of displacement is quite small the tangent of the
angle is approximately equal to the angle itself. Or we may write:
Shear Strain =
As with Axial Stress and Strain, Shear Stress and Strain are
proportional in the elastic region of the material. This relationship may be expressed as G = Shear Stress/Shear Strain, where
G is a property of the material and is called the Modulus of Rigidity (or at times, the
Shear Modulus) and has units of lb/in2. The Modulus of Rigidity for Steel is
approximately 12 x 106 lb/in2.
If a graph is made of Shear
Stress versus Shear Strain, it will normally exhibit the same
characteristics as the graph of Axial Stress
versus Axial Strain. There is an Elastic Region in which
the Stress is directly proportional to the Strain. The point at which the Elastic
Region ends is called the elastic limit, or the proportional limit. In
actuality, these two points are not quite the same. The Elastic Limit is the point
at which permanent deformation occurs, that is, after the elastic limit, if the force is
taken off the sample, it will not return to its original size and shape, permanent
deformation has occurred. The Proportional Limit is the point at which the
deformation is no longer directly proportional to the applied force (Hooke's Law no longer
holds). Although these two points are slightly different, we will treat them as the same
in this course. There is a Plastic Region, where a small increase in the Shear
Stress results in a larger increase in Shear Strain, and finally there is a Failure
Point where the sample fails in shear.
To summarize our shear stress/strain/Hooke's Law relationships up to
this point, we have:
While we will not go in any great depth, at this point, with respect to
Shear Stress and Strain, we will look at several relative easy examples. Please Select:
Example 1 - Shear Stress & Strain
Example 2 - Shear Stress & Strain.
or Select:
Topic 3: Stress, Strain
& Hooke's Law - Table of Contents
Strength of Materials Home Page