Beam Pure Bending
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Transcript of Beam Pure Bending
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Beams: Pure Bending 1
Beams: Pure Bending(4.1-4.5)MAE 314 Solid Mechanics
Xiaoning Jiang
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Beams: Pure Bending 2
Beams in Pure Bending Prismatic beams subject to equal and opposite couples acting in the
same plane are in pure bending.
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Beams: Pure Bending 3
Pure vs. Non-Uniform Bending Pure bending: Shear force (V) = 0 over the section
Non-uniform bending: V 0 over the section
Pure bending
Moment normal stresses
Non-uniform bending
Moment normal stresses Shear force shear stresses (Ch. 6)
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Beams: Pure Bending 4
Pure Bending: Assumptions Beam is symmetric about the x y plane
All loads act in the x y plane
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Beams: Pure Bending 5
Pure Bending:
Curvature Sections originally perpendicular to longitudinal
(y-z) axis remain plane and perpendicular:Plane sections remain plane.
Sign convention
Positive bending moment:beam bends towards +y direction
Negative bending moment:beam bends towards -y direction
Right angle
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Beams: Pure Bending 6
Pure Bending: Deformation Since angles do not change
(remain plane), there is no
shear stress.
The top part of the beam contracts
in the axial direction. The bottom part of the beam expands
in the axial direction.
There exists a line in the beam that
remains the same length called the
neutral line. Set y = 0 at the neutral line.
= radius of curvature
x< 0 for y > 0 and x> 0 for y < 0
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Beams: Pure Bending 7
Pure Bending: Axial Strain The length of DE is LDE=
The length of JK is LJK= (-y)
Axial strain at a distance y from
the neutral axis (x):
Maximum compressive strainoccurs on the upper surface.
Maximum tensile strainoccurs on the lower surface.
)( y
L
LL
LDE
DEJK
x
y
x
LDE
LJK
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Beams: Pure Bending 8
Pure Bending: Axial Strain
c = maximum distance between the neutral axis and the upper or
lower surface
When c is the distance to the surface in compression
When c is the distance to the surface in tension
cmax
cmax
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Beams: Pure Bending 9
Pure Bending: Transverse Strain Recall there are no transverse stresses since
the beam is free to move in the y and z directions.
However, transverse strains (in the y and z
directions) exist due to the Poissons ratioof the material.
' = radius of anticlastic curvature = /
xy
yy
xz
yz
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Beams: Pure Bending 10
Pure Bending: Normal Stress Let us now assume that the beam is made of a linear-elastic material.
The normal stress varies linearly with the distance from the neutralsurface.
EyE
xx
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Beams: Pure Bending 11
Pure Bending: Normal Stress Recollecting that the applied loading is a pure moment, we calculate
resultant loads on the cross-section.
The resultant axial force must be equal to zero.
0 AAx
A
x ydA
E
dAEdA
0 A
ydA
which is the definition of the centroid, so theneutral axis is just the centroid of the section.
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Beams: Pure Bending 12
Pure Bending: Normal Stress The resultant moment about the z-axis must be equal to the applied
moment M.
MdAyE
dAyEdAy
AA
x
A
x
2
Definition of the second moment of inertia, I
EIM
I
Myx
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13
Pure bending
Neutral line (surface)
Normal strain and stress (bending)
Moment of inertia
Reading for the next lecture: A.1-A.5, 4.6
Key Concepts