Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

Mechanical Engineering SSC JE (Technical)

Mechanical Engineering : Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

The document Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev is a part of the Mechanical Engineering Course Mechanical Engineering SSC JE (Technical).
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SHEAR STRESSES IN BEAMS
(ii) Solid Circular Section
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
• The shear stress at a fibre on the plane of cross-section located at a distance y from
neutral axis is given by
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev 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

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  • Average shear stress is given by

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

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

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Hence Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

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

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

(ii) Solid Circular Section

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

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

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

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

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  • Average shear stress is given by

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Hence Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

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

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
(iii) Triangular Section
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

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

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  • Maximum shear stress exists at Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev (at the middle of triangle) and is given by 

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  • Average shear stress is given by

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Shear stress at N.A. Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev form top  is given by

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRevChapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
(iv) Diamond Section
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRevChapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  • Shear stress at level PQ is given by

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRevChapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  • Shear stress at N.A.

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  • Average shear stress = Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

Hence tn.a. = tavg

  • Maximum shear stress occurs at Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev form top and bottom or Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev form neutral axis

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
HenceChapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
(V) I Section
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  • Shear Stress distribution in flange:

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

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

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

  • Shear stress distribution within the web

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

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

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

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

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Shear stress distribution in some other section: 

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev

Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
CORE OF SECTIONS OF DIFFERENT SHAPES
1. Rectangular Section
In order that tension may not develop, we have the condition
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
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
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
For a rectangular section of width b and depth d.
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
and A = b.d.
Hence
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Substituting this value of k, we get
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
or Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
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
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
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, Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
Hence the core for a hollow circular section is a concentric circle of diameter
Chapter 7 Shear Stresses In Beams - Notes, Strength of Material, Mechanical Engineering Mechanical Engineering Notes | EduRev
where d = inner diameter,
D = outer diameter.

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