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In a Nut Shell: Shearing stress, τ , depends directly on
the shear force, V, at the section
of the beam where it is
to be calculated, on the first moment of area, Q, at the location of
the shearing stress, and
inversely on the moment of inertia of area, I, of the entire
cross section about its
neutral (centroidal) axis, and on the width, t,
of the cross-section.
τ =
VQ/It
Common units for shearing
stress are psi, ksi, MPa, N/mm2 (English/Metric)
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Strategy for analyzing shearing stress involves five key
steps as detailed in the
table below.
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Step 1 |
Determine the centroid of the beam's cross-section.
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Step 2 |
Determine the moment
of inertia of area of the beam's entire
cross-section about
its neutral (centroidal) axis. This step may
involve use of the
parallel axis theorem.
Recall for a
rectangular cross-section of width
b and height h
the moment of inertia
of area, I, about its centroidal axis is
(1/12) bh3
. Voids of area contribute
negative values of
moments of inertia.
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Step 3 |
Determine the
distribution of shear force across the length of the beam.
This step involves plotting
the shear force distribution along the length
of the beam. Pick the maximum shear force, V.
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Step 4 |
Calculate, Q, the
first moment of area at the location where the shearing
stress is to be
calculated. Q is the product of the
area, A, from where the
shearing stress is zero
to the location where the shearing stress is to be
calculated times the
distance, ybar, from the neutral
axis to the centroid of
of this area, A.
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Step 5
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Finally, calculate the
shearing stress using:
τ =
VQ/It
where t is the width of the beam's cross-section.
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Click here to return to
a discussion of shearing stress in beams.
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