Shear Stresses in Beams Notes | EduRev

Strength of Materials (SOM)

Mechanical Engineering : Shear Stresses in Beams Notes | EduRev

The document Shear Stresses in Beams Notes | EduRev is a part of the Mechanical Engineering Course Strength of Materials (SOM).
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Shearing Stress

In the theory of bending, presence of shear and the distortion of plane sections was neglected because the effect of it is not on bending stress is not of practical importance.

  • But, it is important to consider the shear stresses for their own importance.
  • Here, only the shearing stresses in the transverse palnes parallel to the shearing force and the complimentary shear stresses in the longitudinal planes parallel to the axis of the beam.
Shear stress distribution over rectangular section
F​ig. shows the rectangular cross-section of the beam, over which we have to determine the distribution of shear stress. Consider a layer ab at any above the N.A.

Shear Stresses in Beams Notes | EduRev

Shearing stress on a layer JK of beam at distance y from neutral axis.
Shear Stresses in Beams Notes | EduRev

Shear Stresses in Beams Notes | EduRev

Where,

  • V = Shearing force
  • Shear Stresses in Beams Notes | EduRevFirst moment of area
    Shear Stresses in Beams Notes | EduRev
Shear stress in Rectangular Beam
  • Suppose, we have to determine the shear stress at the longitudinal layer having y distance from neutral axis.
    Shear Stresses in Beams Notes | EduRev
    Shear Stresses in Beams Notes | EduRev
    Shear Stresses in Beams Notes | EduRev
    Shear Stresses in Beams Notes | EduRevShear Stresses in Beams Notes | EduRev
Circular Beam
  • Centre of gravity of semi-circle lies at distance from centre or base line. As it is symmetrical above neutral axis, hence at neutral axis shear stress will be maximum.
    Shear Stresses in Beams Notes | EduRev
  • For τmax substituting y = 0
    Shear Stresses in Beams Notes | EduRev
    Shear Stresses in Beams Notes | EduRevShear Stresses in Beams Notes | EduRev

Shears Stress in Hollow Circular Cross-Section

  • In hollow circular cross-section, if we have to calculate τ at neutral axis by the formula
    Shear Stresses in Beams Notes | EduRev

Shear Stress in Triangular Section

  • In a triangular cross-section, if we have to calculate τ at neutral axis, then in formula
    Shear Stresses in Beams Notes | EduRevShear Stresses in Beams Notes | EduRev

Shear Stress in I-section

Shear Stresses in Beams Notes | EduRev
Shear Stresses in Beams Notes | EduRev

Combined Stresses

Under combination of  

  • Direct Stress
    σd = P/A

where P = axial thrust, A = area of cross-section

  • Bending Stress
    σb = My/l

where M = bending moment, y- distance of fibre from neutral axis, I = moment of inertia.

  • Shear stresses
    Τ = Tr/J

where T = torque, r = radius of shaft, J = polar moment of inertia.

Combined Stress is: 

  • Equivalent Torsional Moment:-
    The equivalent torsional moment is defined as the torsional moment, which when acting alone, will produce the same torsional shear stress in the shaft as under the combined action of bending moment (Mb) and torsional moment (Mt)
    Shear Stresses in Beams Notes | EduRev
  • Equivalent Bending Moment:-
    The equivalent Bending moment is defined as the bending moment, which when acting alone, will produce the same bending stresses (tensile and compressive) in the shaft as under the combined action of bending moment (Mb) and torsional moment (Mt)
    Shear Stresses in Beams Notes | EduRev
    Example:- A shaft of diameter 8 cm is subjected to a bending moment of 3000 Nm and a twisting moment of 4000 Nm. The maximum normal stress induced in the shaft is equal to
    Solution:-
    Shear Stresses in Beams Notes | EduRev
  • Stress due to combined direct and bending load:
    Suppose a beam under direct compressive and bending load as shown in the diagram. where A = Area,
    y = distance of extreme fiber from Neutral axis and I = moment of Inertia
    Shear Stresses in Beams Notes | EduRevDirect compressive stress σ = F/A
    Bending stress will vary linearly from center to extreme fiber
    σb = My/I
    so, total stress at upper fibre = Direct compressive stress+ tensile stress due to a bending load
    = (-σ)+σb
    Shear Stresses in Beams Notes | EduRev
    Same for lower extreme fiber = Direct compressive stress+ Compressive stress due to bending load
    =(-σ)+(-σb)
    Shear Stresses in Beams Notes | EduRev
    Example-
    For the component loaded with a force F as shown in the figure, the axial stress at the corner point P is
    Shear Stresses in Beams Notes | EduRevSolution:-
    At point P two types of stress are acting, bending & axial tensile load

    So, bending stress =
    Shear Stresses in Beams Notes | EduRev
    Axial tensile stress =
    Shear Stresses in Beams Notes | EduRev
    Total axial stress at P = σb + σa
    Shear Stresses in Beams Notes | EduRev

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