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1
Guiding Rules
in the
Conjugate Beam Method
(
Excerpt
from an
IJEE
Paper)
Courtesy:
Int. J. Engng. Ed.
, Vol. 26, No. 6, pp. 14221427, 2010
icjong@uark.edu
1.1 Pedagogy of the conjugate beam method.
The conjugate beam method is actually a natural
extension of the
momentarea theorems
. It is an elegant, efficient, and powerful method
published by Westergaard [1] some nine decades ago, although some considered Mohr (1868)
and Breslau (1865) to have prior influences. Elementary presentation of this method did appear
in early textbooks in mechanics of materials [2, 3]. For reasons unknown, this method is missing
in most such current textbooks. The pedagogy of the conjugate beam method lies in teaching and
applying the rules in this method [1, 11]. These rules are summarized as follows:
Rule 1:
The conjugate beam and the given beam are of the same
length
.
Rule 2
:
The load on the conjugate beam is the
elastic weight
, which is the bending
moment
M
in the given beam divided by the flexural rigidity
EI
of the given beam.
(This
elastic weight
is taken to act upward if the
bending moment
is positive
―
to cause top fiber in
compression
―
in beam convention.)
For each
existing
support condition
of the given beam, there is a
corresponding support
condition
for the conjugate beam. The correspondence is given by rules 3 through 7 as follows:
Existing support condition in the
given beam:
Corresponding support condition in the
conjugate beam:
Rule 3:
Fixed end
Free end
Rule 4:
Free end
Fixed end
Rule 5:
Simple support at the end
Simple support at the end
Rule 6:
Simple support
not
at the end
Unsupported hinge
Rule 7:
Unsupported hinge
Simple support
Rule 8
:
The conjugate beam is in static
equilibrium
.
Rule 9
:
The
slope
of the given beam at any cross section is given by the “
shear force
”
at that cross section of the conjugate beam.
(This
slope
is positive, or counterclockwise, if the “
shear force
” is positive
―
tending to rotate the beam
element clockwise
―
in beam convention.)
Rule 10
:
The
deflection
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