5. Stresses in Beams – Shear Formula
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Transcript of 5. Stresses in Beams – Shear Formula
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Mechanics of Solids (VDB1063)
Shear Formula
Lecturer: Dr. Montasir O. Ahmed
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Learning Outcomes
• To evaluate the shear stress by applying the shear formula
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LECTURE OUTLINES
Shear in Straight Members
The Shear Formula
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Copyright © 2011 Pearson Education South Asia Pte Ltd
• Transverse shear stress always has its associated longitudinal shear stress acting
along longitudinal planes of the beam.
Shear in Straight Members
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• Effects of Shear Stresses:
Shear in Straight Members
• As a result of shear stress, shear strain will be developed and these
will tend to distort the cross section in a complex manner (warping ).
• When beam is subjected to bending as well as shear, the cross
section will not remain plane as assumed in the application of the
flexure formula. However, for slender beams, this cross sectional
warping is small and can be neglected.
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The Shear Formula
Shear Formula for longitudinal and transverse shear stress:
Q =
𝐴′ = Is the area of the top/bottom portion of the member’s cross sectional area, above/below
the section plane where t is measured.
= Is the distance from the NA to the centroid of 𝐴′
𝜏 = 𝑉𝑄
𝐼𝑡
𝑦′
𝑦′ 𝐴′
𝑦′
where
𝜏 = shear stress in a member at point located a distance from
the neutral axis.
V = the internal resultant shear force, determined from the method of
sections and the equations of equilibrium.
I = the moment of inertia of the entire cross sectional area calculated about the NA.
t = width of the member’s cross sectional area, measured at the point where 𝜏 is to be determined.
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Flat Sections
7
• Limitation on the use of the shear formula:
The Shear Formula
• Flexure formula was used in the derivation of the shear formula. Therefore, it is
necessary that the material behave in a linear elastic manner.
Flange-web junction Cross section with an
irregular or
nonrectangular boundary
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The Shear Formula
Internal Shear• Section the member
perpendicular to its axis andobtain V
Section Properties
• Determine the location of NA
• Determine I for the entire cross section about the NA
• Pass an imaginary horizontal section through the point
where the 𝜏 is to be determined
• Measure t
• Determine A/ , Q = y-/ A/
Shear Stress• Apply the shear
stress formula
Procedure for application the shear formula
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EXAMPLE 1
A steel wide-flange beam has the dimensions shown in Fig. 7–11a.
If it is subjected to a shear of V = 80kN, plot the shear-stress
distribution acting over the beam’s cross-sectional area.
Copyright © 2011 Pearson Education South Asia Pte Ltd
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EXAMPLE 1 (cont)
• The moment of inertia of the cross-sectional area about the neutral axis is
• For point B, tB’ = 0.3m, and A’ is the dark
shaded area shown in Fig. 7–11c
Copyright © 2011 Pearson Education South Asia Pte Ltd
Solutions
4623
3
m 106.15511.002.03.002.03.012
12
2.0015.012
1
I
MPa 13.1
3.0106.155
1066.01080
m 1066.002.03.011.0''
6
33
'
''
33
'
B
BB
B
It
VQ
AyQ
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EXAMPLE 1 (cont)
• For point B, tB = 0.015m, and QB = QB’,
• For point C, tC = 0.015m, and A’ is
the dark shaded area in Fig. 7–11d.
• Considering this area to be composed of two rectangles,
• Thus,
Copyright © 2011 Pearson Education South Asia Pte Ltd
Solutions
MPa 6.22015.0106.155
1066.010806
33
B
BB
It
VQ
33 m 10735.01.0015.005.002.03.011.0'' AyQC
MPa 2.25015.0106.155
10735.010806
33
max
C
cC
It
VQ
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Important Points in this Lecture
• There is longitudinal shear stress associated with the transverse shear stresses.
• Shear stresses produce non uniform shear strain.
• The shear Formula for longitudinal and transverse shear stress is:
• The shear formula can’t be applied in predicting shear stress for:
1- Flat Sections
2- Flange-web junction
3- Cross section with an irregular or nonrectangular boundary
𝜏 = 𝑉𝑄
𝐼𝑡
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Next Class
Stress Caused by Combined Loadings
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Thank You