Shear Stresses in Beams | Mechanical Engineering SSC JE (Technical) PDF Download

SHEAR STRESSES IN BEAMS
(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

 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.