Civil Engineering Reference
In-Depth Information
20.2 M
UELLER
-B
RESLAU
P
RINCIPLE
A simple and convenient method of constructing influence lines is to employ the
Mueller-Breslau principle which gives the shape of an influence line without the values
of its ordinates; these, however, are easily calculated for statically determinate systems
from geometry.
Consider the simply supported beam, AB, shown in Fig. 20.3(a) and suppose that a
unit load is crossing the beamand has reached the point C a distance
x
fromA. Suppose
also that we wish to determine the influence line for the moment at the section K, a
distance
a
from A. We now impose a virtual displacement,
v
C
, at C such that internal
work is done only by the moment at K, i.e. we allow a change in gradient,
θ
K
,atKso
that the lengths AK and KB rotate as rigid links as shown in Fig. 20.3(b). Therefore,
from the principle of virtual work (Chapter 15), the external virtual work done by the
unit load is equal to the internal virtual work done by the moment,
M
K
, at K. Thus
1
v
C
=
M
K
θ
K
If we choose
v
C
so that
θ
K
is equal to unity
M
K
=
v
C
(20.7)
i.e. the moment at the section K due to a unit load at the point C, an arbitrary distance
x
from A, is equal to the magnitude of the virtual displacement at C. But, as we have
l
A
K
B
C
x
a
L
(a)
y
c
u
k
(b)
y
S,K
y
c
F
IGURE
20.3
Verification of the
Mueller-Breslau
principle
(c)
