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SHEAR STRESSES IN BEAMS
(ii) Solid Circular Section
• The shear stress at a fibre on the plane of crosssection located at a distance y from
neutral axis is given by
moment of the area above
EF y = distance form neutral axis.
I = moment of inertia about
N.A. b = width of the beam at the level EF
Hence
(ii) Solid Circular Section
Hence
(iii) Triangular Section
(iv) Diamond Section
Hence t_{n.a}. = t_{avg}
Hence
(V) I Section
Shear stress distribution in some other section:
CORE OF SECTIONS OF DIFFERENT SHAPES
1. Rectangular Section
In order that tension may not develop, we have the condition
wherek = radius of gyration of the section with respect to the NA
d = depth of the section
Thus, for not tension in the section, the eccentricity must not exceed
For a rectangular section of width b and depth d.
and A = b.d.
Hence
Substituting this value of k, we get
or
Thus the stress will be wholly compressive throughout the section, if the line of action of P falls within the rhombus (as shaded portion of figure), the diagonals of which are of length d/3 and b/3 respectively. This rhombus is called the core or kern of the rectangular section.
2. Solid Circular Section
The core of a solid circular section is a circle, with the same centre, and diameter d/4.
3. Hollow Circular Section
For a hollow circular section,
Hence the core for a hollow circular section is a concentric circle of diameter
where d = inner diameter,
D = outer diameter.
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