Transcript Document

Stress Analysis -MDP N161
Deflection
of Beams
Chapter 12
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Deflection of beam is a design limit.
It is affected by the material
properties, cross-section properties
and the position of the supports.
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Elastic Curve
The elastic curve
is the
deflection
diagram of
the
longitudinal
axis
that
passes
through
the
centroid of each
cross-sectional
area
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Internal Moment and Deflection
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Elastic
Elastic
curve
curve
cancan
be be
expressed
expressed
as as
V =V f(x)
= f(x)
v v
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x' x
s 0
x
(   y )  
 lim
s 0

y

 x  lim

Since x = E x
and x = -Mz y/ Iz
Then
1
M
=
ρ
EI
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From
Calculus
and
after
approximation for sake of application
to engineering structures which
specify
small
deformation,
the
relation between the deflection and
the curvature is:
1
ρ
Since
M =
EI
≈
d2v
dx2
d2v
dx2
1
ρ
MDP: N161- Stress Analysis Fall 2009
=
M
EI
9
Slope and Deflection by Integration
d 2v
dx2
=
M
EI
To get the elastic curve v (x) for a beam of
constant cross-sectional area and made of
the same material, you have to know the
variation of the moment M with x and then
double integrate the above equation
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The
integrations
produces
two
constants, which could be evaluated
from the known values of deflection v
or the slope =dv/dx at certain
sections
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Boundary Conditions at supports
At roller or
supports;
deflection v =0
pin
the
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For fixed support v =0
and the slope  = dv/dx =0
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Example 8-1
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Solution
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