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Bending Stress & Shear Stress in beams
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Bending (looking at radial sections)
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Photographs
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Normal Strain
Y measured from the neutral axis
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Compress /Extend from Neutral Axis
Compress
Extend
Compress
Extend
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Normal Bending Strain creates Normal Stress
E
y
y is measured from the neutral axis ρ is the radius of the curvature of the beam
The bending moment is all resisted by the sum of all normal bending stress. y
E
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Maximum Bending Stress, Maximum farthest from centroid (neutral axis)
x
MAX
xMAXMAXMAX
R
MAX
MAX
I
Mc
Ic
dAyc
ydAyc
ydAM
yc
cE
2)()(
The bending moment is all resisted by the normal bending stress over x-sectional area.
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Neutral axis is at centroid of cross sectional area
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To find the maximum bending stress • Draw shear & bending moment diagrams
• Find maximum moment, M, from bending moment diagram
• Calculate cross-section properties
– Centroid (neutral axis)
– Calculate Area Moment of Inertia about x-axis, Ix
– Find the farthest distance from neutral axis for cross section, c
• Max Bending Normal Stress = x
MAXI
Mc
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Shear Stress in Bending
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Shear Load, V, is distributed on cross sectional area
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Visualizing Shear Stress
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Splitting due to Shear Stress
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Shear Stress Distribution
Shear Stress is largest at
neutral axis
Shear Stress is 0 at the
extremes from neutral axis
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Shear Stress definitions
yAdytyQ
y
y
max_
1
)'()(
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centroiditsyaboveAreaQ
yAdytyQ
It
VQ
y
y
_*1__
)'()(
max_
1
V = shear force
I = area moment of inertia (2nd moment of area)
Q = first moment of area (above location y1)
t = thickness
y is measured from NA
Shear Stress at y1 above NA Shear Stress Max at y=0
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Shear stress is maximum at Neutral Axis (NA)
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What to remember about shear stress in bending?
• Shear stress is
– 0 at the points farthest from neutral axis
– maximum at the neutral axis
– It can be shown that :
A
V : sectioncross rrectangula a For
A
V : sectioncross circular a For
MAX
MAX
2
3
3
4
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End