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In a Nut Shell: Bending stress, σ, directly depends
on the bending moment, M, at the
section at the beam
where the stress is to be calculated, on the distance, c, to the outer
fiber (might be the top
or bottom of the beam's section), and inversely on the moment
of inertia of area of
the entire cross section about its neutral (centroidal)
axis.
σ =
Mc / I
Common units for bending
stress are psi, ksi, MPa, N/mm2 (English/Metric)
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Strategy for analyzing bending 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 moment
distribution across the length of the beam.
This step involves plotting
the shear and bending moment distribution
along the length of
the beam. Pick the maximum bending
moment, M.
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Step 4 |
Determine the distance
from the neutral axis to the outer fibers
(top or bottom) of the
beam. Also determine which of the
outer
fibers are in tension
or compression based on the bending moment.
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Step 5
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Calculate the bending
stress for both the top and bottom fibers
using
σ = M ctop
/ I and σ = M cbottom
/ I
where M is the maximum bending moment at the
section
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Click here to return to
a discussion of bending members (beams).
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