Deflections
The
conjugate
conjugate
–
beam method
beam method
was developed by
Otto Mohr in 1860.
The method is based on the similarity between
the relationships for loading and shear, and
shear and moment.
( )
dV
w x
dx
= −
dM
V
dx
=
( )
V
w x dx
= −
∫
( )
M
w x dx dx
= −
∫∫
2
2
d M
w
dx
⇒
= −
Deflections
The previous expressions relate the internal
shear and moment to the applied load.
The slope and deflection of the elastic curve
are related to the internal moment by the
following expressions
d
M
dx
EI
θ
=
2
2
d y
M
dx
EI
=
M
dx
EI
θ
=
∫
M
y
dx dx
EI
=
∫∫
Deflections
Let’s compare expressions for shear,
V
,
and the slope,
θ
( )
dV
w x
dx
= −
d
M
dx
EI
θ
=
What do you see?
If you replace
w
with the term –
M/EI
the
expressions for shear force and slope are
identical
Deflections
Let’s compare expressions for bending
moment,
M
, and the displacement,
y
What do you see?
Just as before, if you replace
w
with the
term –
M/EI
the expressions for bending
moment and displacement are identical
2
2
d M
w
dx
= −
2
2
d y
M
dx
EI
=
Deflections
We will use this relationship to our advantage
by constructing a beam with the same length as
the real beam referred to as the
conjugate
beam
.
The conjugate beam is loaded with the
M/EI
diagram, simulating the external load
w
.
Deflections
w = w(
x
)
x
w
x
M
EI
Real beam
with applied loading.
Determine the bending moment
(draw the bending moment
diagram)
Conjugate beam
Conjugate beam
where the
applied loading is bending
moment from the real beam
Note the sign of loading
w
and
the
M/EI
on the conjugate
beam.
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 Fall '10
 GURLEY
 Second moment of area, conjugate beam, Otto Mohr

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