Civil Engineering Reference
In-Depth Information
which gives
L
−
x
R
A
=
(20.1)
L
Hence
R
A
is a linear function of
x
and when
x
0;
both these results are obvious from inspection. The influence line (
IL
) for
R
A
(
R
A
IL
)
is then as shown in Fig. 20.1(b). Note that when the unit load is at C, the value of
R
A
is given by the ordinate cd in the
R
A
influence line.
=
0,
R
A
=
1 and when
x
=
L
,
R
A
=
R
B
influence line
The influence line for the reaction
R
B
is constructed in an identical manner. Thus,
taking moments about A
R
B
L
−
1
x
=
0
so that
x
L
R
B
=
(20.2)
Equation (20.2) shows that
R
B
is a linear function of
x
. Further, when
x
=
0,
R
B
=
0
and when
x
1, giving the influence line shown in Fig. 20.1(c). Again, with
the unit load at C the value of
R
B
is equal to the ordinate c
1
e in Fig. 20.1(c).
=
L
,
R
B
=
S
K
influence line
The value of the shear force at the section K depends upon the position of the unit
load, i.e. whether it is between A and K or between K and B. Suppose initially that the
unit load is at the point C between A and K. Then the shear force at K is given by
S
K
=
R
B
so that from Eq. (20.2)
x
L
S
K
=
(0
≤
x
≤
a
)
(20.3)
The sign convention for shear force is that adopted in Section 3.2. We could have
established Eq. (20.3) by expressing
S
K
in terms of
R
A
. Thus
S
K
=−
R
A
+
1
Substituting for
R
A
from Eq. (20.1) we obtain
L
−
x
x
L
S
K
=−
+
1
=
L
as before. Clearly, however, expressing
S
K
in the terms of
R
B
is the most direct
approach.
We see fromEq. (20.3) that
S
K
varies linearly with the position of the load. Therefore,
when
x
=
0,
S
K
=
0 and when
x
=
a
,
S
K
=
a
/
L
, the ordinate kg in Fig. 20.1(d), and is
