Shear Stresses in Beams | Civil Engineering SSC JE (Technical) - Civil Engineering (CE) PDF Download

Shear Stresses In Beams

SHEAR STRESS DISTRIBUTION

(i) Rectangular Beam

• The intensity of shear at a fiber on the plane of cross-section located.the level of EF at a distance y form the neutral axis is given by,

Where S = SF at the section
 = 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. and 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 t avg= t local

(iii) Triangular Section

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

• Maximum shear stress exists at y = h/2 (at the middle of triangle) and is given by

• Average shear stress is given by

• Shear stress at N.A.

is given by

(iv) Diamond Section

• Shear stress at level PQ is given by

• Shear stress at N.A.

• Average shear stress =

Hence ζ_n.a. = ζ_avg

• Maximum shear stress occurs at 3/8 d form top and bottom or d/8 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.