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We note that, if an area under the (MyEI) diagram is located
above the x axis, its first moment with respect to a vertical axis will
be positive; if it is located below the x axis, its first moment will be
negative. We check from Fig. 9.41, that a point with a positive tan-
gential deviation is located above the corresponding tangent, while
a point with a negative tangential deviation would be located below
that tangent.
*9.10 APPLICATION TO CANTILEVER BEAMS AND
BEAMS WITH SYMMETRIC LOADINGS
We recall that the first moment-area theorem derived in the preceding
section defines the angle u
DyC
between the tangents at two points C
and D of the elastic curve. Thus, the angle u
D
that the tangent at D
forms with the horizontal, i.e., the slope at D, can be obtained only
if the slope at C is known. Similarly, the second moment-area theo-
rem defines the vertical distance of one point of the elastic curve from
the tangent at another point. The tangential deviation t
DyC
, therefore,
will help us locate point D only if the tangent at C is known. We
conclude that the two moment-area theorems can be applied effec-
tively to the determination of slopes and deflections only if a certain
reference tangent to the elastic curve has first been determined.
In the case of a cantilever beam (Fig. 9.42), the tangent to the
elastic curve at the fixed end A is known and can be used as the ref-
erence tangent. Since u
A
5 0, the slope of the beam at any point D
is u
D
5 u
DyA
and can be obtained by the first moment-area theorem.
On the other hand, the deflection y
D
of point D is equal to the tan-
gential deviation t
DyA
measured from the horizontal reference tangent
at A and can be obtained by the second moment-area theorem.
D
=
D/A
y
D
= t
D/A
Reference tangent
Tangent at D
D
P
Fig. 9.42 Application of moment-area
method to cantilever beams.
9.10 Application to Cantilever Beams and
Beams with Symmetric Loadings
C
C
B
y
max
t
B/C
A
B
A
P
Horizontal
Reference tangent
(a)
(b)
B/CB
C
B
D
t
D/C
t
B/C
y
D
A
Reference tangent
(c)
D/CD
P
Fig. 9.43 Application of moment-area
method to simply supported beams with
symmetric loadings.
In the case of a simply supported beam AB with a symmetric
loading (Fig. 9.43a) or in the case of an overhanging symmetric beam
with a symmetric loading (see Sample Prob. 9.11), the tangent at the
center C of the beam must be horizontal by reason of symmetry and
can be used as the reference tangent (Fig. 9.43b). Since u
C
5 0, the
slope at the support B is u
B
5 u
ByC
and can be obtained by the first
moment-area theorem. We also note that |y|
max
is equal to the tan-
gential deviation t
ByC
and can, therefore, be obtained by the second
moment-area theorem. The slope at any other point D of the beam
(Fig. 9.43c) is found in a similar fashion, and the deflection at D can
be expressed as y
D
5 t
DyC
2 t
ByC
.
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