Müller Breslau Principle

Module 6 : Influence Lines
Lecture 5 : Mller-Breslau Principle

Objectives
In this course you will learn the following
The Mller-Breslau principle for influence lines.
Derivation of the principle for different types of internal forces.
Example of application of this principle.
6.5 Mller-Breslau Principle

The Mller-Breslau principle uses Betti's law of virtual work to construct influence lines. To illustrate the
method let us consider a structure AB (Figure 6.7a). Let us apply a unit downward force at a distance x
and
at supports A and B ,
from A , at point C . Let us assume that it creates the vertical reactions
respectively (Figure 6.7b). Let us call this condition System 1. In System 2 (figure 6.7c), we have the
. Here
is the deflection at point C .
same structure with a unit deflection applied in the direction of
Figure 6.7 (a) Given system AB , (b) System 1, structure under a unit load, (c) System 2, structure with a
unit deflection corresponding to
According to Betti's law, the virtual work done by the forces in System 1 going through the corresponding
displacements in System 2 should be equal to the virtual work done by the forces in System 2 going
through the corresponding displacements in System 1. For these two systems, we can write:
The right side of this equation is zero, because in System 2 forces can exist only at the supports,
corresponding to which the displacements in System 1 (at supports A and B ) are zero. The negative sign
accounts for the fact that it acts against the unit load in System 1. Solving this equation we get:
before
In other words, the reaction at support A due to a unit load at point C is equal to the displacement at point
C when the structure is subjected to a unit displacement corresponding to the positive direction of support
reaction at A . Similarly, we can place the unit load at any other point and obtain the support reaction due
.
to that from System 2. Thus the deflection pattern in System 2 represents the influence line for
Following the same general procedure, we can obtain the influence line for any other response parameter
as well. Let us consider the shear force at point C of a simply-supported beam AB (Figure 6.8a). We apply
a unit downward force at some point D as shown in System 1 (Figure 6.8b). In system 2 (Figure 6.8c), we
. Note that the displacement at point C is
apply a unit deflection corresponding to the shear force,
applied in a way such that there is no relative rotation between AC and CB . This will avoid any virtual work
) going through the rotation in System 2. Now, according to
done by the bending moment at C (
Betti's law:
Figure 6.8 (a) Given system AB , (b) System 1, structure under a unit load, (c) System 2, structure with a
unit deflection corresponding to
, (d) System 2, structure with a unit deflection corresponding to
Thus, the deflected shape in System 2 represents the influence line for shear force
want to find the influence line for bending moment
. Similarly, if we
, we obtain System 2 (Figure 6.8d) by applying a
unit rotation at point C (that is, a unit relative rotation between AC and CB ). However, we do not want any
going
relative displacement (between AC and CB ) at point C in order to avoid any virtual work done by
through the displacements in System 2. Betti's law provides the virtual work equation:
So, as we have seen earlier, the displaced shape in System 2 represents the influence line for the response
parameter
.
Construction of System 2 for a given response function is the most important part in applying the MllerBreslau principle. One must take care that other than the concerned response function no other force (or
moment) in System 1 should do any virtual work going through the corresponding displacements in System
2. So we make all displacements in System 2 corresponding to other response functions equal to zero. For
,
and
are equal to zero. Example 6.4
example, in Figure 6.8c, displacements corresponding to
illustrates the construction of influence lines using Mller-Breslau principle.
Example 6.4 Construct influence lines for
and
for the beam AB in Fig. E6.4.
Solution:
System 2 for
: (Note that there is no bending moment at D , i.e.
System 2 for
System 2 for
at point D
: (Note that only

,
contributes to virtual work because even though there is rotation

)
The deflected shape in each system 2 provides the influence line for the corresponding response function.
Recap
In this course you have learnt the following
The Mller-Breslau principle for influence lines.
Derivation of the principle for different types of internal forces.
Example of application of this principle.

Müller Breslau Principle

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What is the Müller-Breslau principle?

How is System 2 constructed for different response functions?

Module 6 : Influence Lines

Lecture 5 : Mller-Breslau Principle

Derivation of the principle for different types of internal forces.

Example of application of this principle.

6.5 Mller-Breslau Principle

unit deflection corresponding to

, (d) System 2, structure with a unit deflection corresponding to

, we obtain System 2 (Figure 6.8d) by applying a

Example 6.4 Construct influence lines for

for the beam AB in Fig. E6.4.

: (Note that there is no bending moment at D , i.e.

: (Note that only

contributes to virtual work because even though there is rotation

Derivation of the principle for different types of internal forces.

Example of application of this principle.

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