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**Chapter 7**

**SHEAR STRESSES IN BEAMS**

(ii) Solid Circular Section

â€¢ The shear stress at a fibre on the plane of cross-section 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

- Shear stress in terms of y form N.A. is given by

- Average shear stress is given by

- Maximum shear stress occurs at the N.A. & is given by

Hence

- The distance from N.A. at which the average shear stress is equal to the local shear stress

(ii) Solid Circular Section

- The shear stress at a fibre on the plane of cross-section located at a distance y from neutral axis is given by

- Maximum shear stress occurs at the N.A. & is given by

- Average shear stress is given by

Hence

- The distance from N.A. at which the local shear stress is equal to average shear stress is given by

(iii) Triangular Section

- Shear stress at a distance y form vortex is given by

- Maximum shear stress exists at (at the middle of triangle) and is given by

- Average shear stress is given by

- Shear stress at N.A. form top is given by

(iv) Diamond Section

- Shear stress at level PQ is given by

- Shear stress at N.A.

- Average shear stress =

Hence t_{n.a}. = t_{avg}

- Maximum shear stress occurs at form top and bottom or form neutral axis

Hence

(V) I Section

- Shear Stress distribution in flange:

- Shear stress at the junction of flange & web, but within the flange.

- Shear stress distribution within the web

- Maximum shear stress exists at N.A. and is given by

- Shear stress at the junction of web and flange but within the web

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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