Civil Engineering Reference
In-Depth Information
concentrated loads is a comparatively simple practical case. An alternative approach
is to introduce so-called
singularity
or
half-range
functions. Such functions were first
applied to beam deflection problems by Macauley in 1919 and hence the method is
frequently known as
Macauley's method
.
We now introduce a quantity [
x
−
a
] and define it to be zero if (
x
−
a
)
<
0, i.e.
x
<
a
, and
to be simply (
x
a
] is known as a singularity or half-range
function and is defined to have a value only when the argument is positive in which
case the square brackets behave in an identical manner to ordinary parentheses. Thus
in Ex. 13.6 the bending moment at a section of the beam furthest from the origin for
x
may be written as
−
a
)if
x
>
a
. The quantity [
x
−
M
=
R
A
x
−
W
[
x
−
a
]
This expression applies to both the regions AC and CB since
W
[
x
a
] disappears for
x
<
a
. Equations (iii) and (iv) in Ex. 13.6 then become the single equation
−
EI
d
2
v
d
x
2
=
R
A
x
−
W
[
x
−
a
]
which on integration yields
x
2
2
EI
d
v
W
2
[
x
a
]
2
d
x
=
R
A
−
−
+
C
1
and
x
3
6
W
6
[
x
a
]
3
EI
v
=
R
A
−
−
+
C
1
x
+
C
2
Note that the square brackets
must be retained
during the integration. The arbitrary
constants
C
1
and
C
2
are found using the boundary conditions that
v
=
0 when
x
=
0
a
]
3
is zero for
x
<
a
,we
and
x
=
L
. From the first of these and remembering that [
x
−
have
C
2
=
0. From the second we have
R
A
L
3
W
6
[
L
a
]
3
0
=
6
−
−
+
C
1
L
in which
R
A
=
W
(
L
−
a
)
/
L
.
Substituting for
R
A
gives
Wa
(
L
−
a
)
C
1
=−
(2
L
−
a
)
6
L
Then
6
L
a
)
x
W
a
)
x
3
a
]
3
EI
v
=−
−
(
L
−
+
L
[
x
−
+
a
(
L
−
a
)(2
L
−
The deflection of the beam under the load is then
Wa
2
(
L
a
)
2
−
v
C
=−
3
EIL
as before.
