SHEAR STRESSES-BEAM CLASS
NOTES IN MECHANICAL
Shear stress in a beam vary parabolically.
These shear stresses are maximum at
the centroidal axis and zero at the
extreme fibers. Further bending stresses
vary linearly. Bending stresses are
maximum at the extreme fibers and
zero at the centroidal axis.
Shear Stress Equation
Shear stress in any fiber is given by the equation
τy = V A’ Y’ /I B
τy is the shear stress in a fiber at distance y from the centroid axis
Normally V will be the absolute maximum shear force found from the shear force diagram ( Or V is the shear force at section being considered along the length of the beam and is found from the shear force diagram).
A’ is the area of cross section above the fiber being considered at distance y from the centroid axis
Y’ is the distance of centroid of area A’ from the centroid axis of the entire cross section
I is the moment of inertia of the entire cross section about the centroid axis
B is the width of the fiber at distance y from the centroid axis
The above equation is for horizontal shear stress. A shear stress is always accompanied by a complimentary shear stress. Thus the same equation applies to vertical shear stress too. A loaded beam is subjected variable bending moment ‘M’ and a shear force ‘V’. The relation between bending moment and shear force is V=d M /dx. It establishes that the shear stress is variable.
Fig. Shear stress distribution in typical cross sections
Fig. Variation Shear Stress I Section
Special Features of shear stresses in beams
1. There are longitudinal as well as vertical shear stresses. These are variable stresses. These are maximum at the centroid axis and zero at the outermost fiber.
2. These are complementary of each other. One of these is applied while the other is induced.
3. Both are equal in magnitude but opposite in nature i.e. one has a clockwise and other has an anti-clockwise rotational tendency.
4. Both vary in a parabolic manner.
5. Shear stress is maximum at the centroid axis and zero at the extreme fibers. Bending stresses are maximum at the extreme fibers and zero at the centroid axis. It is true in case of symmetrical sections like square, rectangular and circular.
6. Therefore, a beam can be designed independently on bending and shear stress basis.
7. Shear stress is maximum at half the height in case of a triangular section and not at the centroid axis. It is an exception.
8. For rectangular, circular and square sections, maximum bending stress becomes the maximum principal stress.
9. However in case of an I section, maximum principal stress occurs at the junction of web and flange.
10. The ratio of τmax/τav is as follows:
Shape of cross section
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