Civil Engineering Reference
In-Depth Information
In establishing the shear force and bending moment influence lines for the section K
of the beam in Fig. 20.1(a) we have made use of the previously derived relationships
for the support reactions,
R
A
and
R
B
. If only the influence lines for
S
K
and
M
K
had
been required, the procedure would have been as follows.
With the unit load between A and K
S
K
=
R
B
Now, taking moments about A
R
B
L
−
1
x
=
0
so that
x
L
R
B
=
Therefore
x
L
S
K
=
This, of course, amounts to the same procedure as before except that the calculation
of
R
B
follows the writing down of the expression for
S
K
. The remaining equations for
the influence lines for
S
K
and
M
K
are derived in a similar manner.
We note from Fig. 20.1 that all the influence lines are composed of straight-line seg-
ments. This is always the case for statically determinate structures. We shall therefore
make use of this property when considering other beam arrangements.
E
XAMPLE
20.1
Draw influence lines for the shear force and bending moment at
the section C of the beam shown in Fig. 20.2(a).
In this example we are not required to obtain the influence lines for the support reac-
tions. However, the influence line for the reaction
R
A
has been included to illustrate
the difference between this influence line and the influence line for
R
A
in Fig. 20.1(b);
the reader should verify the
R
A
influence line in Fig. 20.2(b).
Since we have established that influence lines for statically determinate structures
consist of linear segments they may be constructed by placing the unit load at different
positions, which will enable us to calculate the principal values.
S
C
influence line
With the unit load at A
S
C
=−
R
B
=
0
(by inspection)
With the unit load immediately to the left of C
S
C
=
R
B
(i)
