Bending Stress & Shear Stress in beams
Bending Stress & Shear Stress in beams
Bending (looking at radial sections)
Photographs
Normal Strain
Y measured from the neutral axis
Compress /Extend from Neutral Axis
Compress
Extend
Compress
Extend
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
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.
Neutral axis is at centroid of cross sectional area
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
Shear Stress in Bending
Shear Load, V, is distributed on cross sectional area
Visualizing Shear Stress
Splitting due to Shear Stress
Shear Stress Distribution
Shear Stress is largest at
neutral axis
Shear Stress is 0 at the
extremes from neutral axis
Shear Stress definitions
yAdytyQ
y
y
max_
1
)'()(
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
Shear stress is maximum at Neutral Axis (NA)
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
End