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Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

Document Description: Shear Stresses in Beams for Mechanical Engineering 2022 is part of Strength of Materials (SOM) for Mechanical Engineering SSC JE (Technical) preparation. The notes and questions for Shear Stresses in Beams have been prepared according to the Mechanical Engineering exam syllabus. Information about Shear Stresses in Beams covers topics like and Shear Stresses in Beams Example, for Mechanical Engineering 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Shear Stresses in Beams.

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SHEAR STRESSES IN BEAMS
(ii) Solid Circular Section
Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
• The shear stress at a fibre on the plane of cross-section located at a distance y from
neutral axis is given by
Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering 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

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  • Average shear stress is given by

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Hence Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

(ii) Solid Circular Section

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  • Average shear stress is given by

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Hence Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
(iii) Triangular Section
Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  • Maximum shear stress exists at Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering (at the middle of triangle) and is given by 

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  • Average shear stress is given by

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  •  Shear stress at N.A. Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering form top  is given by

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical EngineeringShear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
(iv) Diamond Section
Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical EngineeringShear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  • Shear stress at level PQ is given by

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical EngineeringShear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  • Shear stress at N.A.

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  • Average shear stress = Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

Hence tn.a. = tavg

  • Maximum shear stress occurs at Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering form top and bottom or Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering form neutral axis

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
HenceShear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
(V) I Section
Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  • Shear Stress distribution in flange:

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

  • Shear stress distribution within the web

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Shear stress distribution in some other section: 

Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering
Shear Stresses in Beams Notes | Study Mechanical Engineering SSC JE (Technical) - Mechanical Engineering

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

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