Civil Engineering Reference
In-Depth Information
seen in Section 20.1, the moment at a section K due to a unit load at a point C is
the influence line for the moment at K. Therefore, the
M
K
influence line may be
constructed by introducing a hinge at K and imposing a unit change in angle at K; the
displaced shape is then the influence line.
The argument may be extended to the construction of the influence line for the shear
force,
S
K
, at the section K. Suppose now that the virtual displacement,
v
C
, produces a
shear displacement,
v
S,K
, at K as shown in Fig. 20.3(c). Note that the direction of
v
C
is
now in agreement with the sign convention for shear force. Again, from the principle
of virtual work
1
v
C
=
S
K
v
S,K
If we choose
v
C
so that
v
S,K
=
1
S
K
=
v
C
(20.8)
Hence, since the shear force at the section K due to a unit load at any point C is the
influence line for the shear force at K, we see that the displaced shape in Fig. 20.3(c) is
the influence line for
S
K
when the displacement at K produced by the virtual displace-
ment at C is unity. A similar argument may be used to establish reaction influence
lines.
The Mueller-Breslau principle demonstrated above may be stated in general terms as
follows:
The shape of an influence line for a particular function (support reaction, shear force,
bending moment, etc.) can be obtained by removing the resistance of the structure to that
function at the section for which the influence line is required and applying an internal
force corresponding to that function so that a unit displacement is produced at the section.
The resulting displaced shape of the structure then represents the shape of the influence
line.
E
XAMPLE
20.2
Use the Mueller-Breslau principle to determine the shape of
the shear force and bending moment influence lines for the section C in the beam in
Ex. 20.1 (Fig. 20.2(a)) and calculate the values of the principal ordinates.
In Fig. 20.4(b) we impose a unit shear displacement at the section C. In effect we are
removing the resistance to shear of the beam at C by cutting the beam at C. We then
apply positive shear forces to the two faces of the cut section in accordance with the
sign convention of Section 3.2. Thus the beam to the right of C is displaced downwards
while the beam to the left of C is displaced upwards. Since the slope of the influence
line is the same on each side of C we can determine the ordinates of the influence line
by geometry. Hence, in Fig. 20.4(b)
c
1
e
c
1
a
1
=
c
1
f
c
1
b
1
