Müller-Breslau’s Principle
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Developed by Heinrich Müller-Breslau in 1886.
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Used to construct influence lines for forces
and moments quickly and graphically.
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Can be stated as follows:
The influence line for a force (or moment) response
function is given by the deflected shape of the released structure obtained by
removing the restraint corresponding to the response function from the original
structure and by giving the released structure a unit displacement (or
rotation) at the location and in the direction of the response function, so
that only the response function and the unit load perform external work.
Müller-Breslau’s Principle
- The influence
line for a force or moment response function is
determined by:
- The
deflected shape of the released structure obtained through
the following steps:
1.
Remove the restraint corresponding to the
response function from the original structure.
2.
Apply a unit displacement or unit rotation
at the same location,
§ In
the direction of the desired response.
3.
Ensure that only the response function
and the unit load perform external work.
- The
resulting deflected shape represents the influence line for
that particular force or moment.
Applicability:
- Applicable
to:
- Reactions
- Shear
forces
- Bending
moments
- Truss
member forces
- Not
applicable to:
- Deflections
(displacement-type responses)
Validity of Müller-Breslau’s Principle
1. Basic Concept
- Consider
a simply supported beam with a moving unit load.
- We
aim to derive the influence lines for:
- Vertical
reactions
and 
- Shear
- Bending
moment
- These
will be constructed using Müller-Breslau’s principle, and results
will be compared with equilibrium-based influence lines.
2. Influence Line for 
- Step
1 – Remove restraint:
Replace the hinge at A with
a roller → only horizontal reaction acts.
→ Point A is free to move vertically.
- Step
2 – Virtual displacement:
Give at point A, a unit virtual
displacement 
in the direction of .
→ Beam remains straight (no bending), as it becomes statically unstable.
- Step
3 – Virtual work principle:
According to virtual work for rigid
bodies:
hence,
where 
= displacement of the point
of application of the unit load.
- Interpretation:
The displacementat any position 
equals the ordinate of
the influence line for .
⇒ The deflected shape represents
the influence line for .
- Geometric
relation (similar triangles):
hence,
which is the same as Ay, which was derived by
equilibrium consideration.
3. Influence Line for 
- Similar
process:
- Remove
vertical restraint at C.
- Give
unit displacement upward at C.
- Resulting
deflected shape gives:
- Same
as derived by equilibrium.
4. Influence Line for Shear 
- Step
1 – Remove restraint:
Cut the beam at point B → separate
segments AB and BC.
Apply shear forces and moments at the cut section.
- Step
2 – Virtual displacement:
Apply a relative unit
displacement at B:
- Segment
AB moves downward by
- Segment
BC moves upward by
- So,
- Step
3 – Compatibility:
The values of and depend on the requirement that the rotations, θ, of the two portions AB
and BC be the same (i.e., the segments AB′ and B′′C in
the displaced position must be parallel to each other), so that the net work
done by the two moments MB is zero, and only the shear forces
SB and the unit load perform work.
- Step
4 – Virtual work:
⇒
→ Deflected shape represents influence line for shear.
- Geometry
(from similar triangles):
and
Solving gives:
(Same as found by equilibrium.)
5. Influence Line for Bending Moment 
- Step
1 – Remove restraint:
Insert a hinge at B →
portions AB and BC can rotate relative to each other.
- Step
2 – Virtual rotation:
Apply unit rotation 
at B:
(AB rotates
counter-clockwise)
(BC rotates clockwise)
with
- Step
3 – Virtual work:
Wve
= 
Wve
= 
Wve
= 
Wve
= 
→ Deflected shape gives influence line for .
- From
geometry:
∆ = aθ1
= (L-a) θ2
Or,
θ1
= 
θ2
Also,
Or,
Solving, 
Now, the expression,
Which is same as the equilibrium method result.
Conclusion
- In statically
determinate structures:
- Removing
a restraint → structure becomes a mechanism (unstable).
- When
displaced, members move as rigid bodies, not bending.
- Hence,
influence lines are straight.
- In statically
indeterminate structures:
- Removing
a restraint does not make it unstable.
- Members
deform (bend), producing curved influence lines.
Qualitative Influence Lines
Definition
- A qualitative
influence line shows the general shape or pattern of how a
response (reaction, shear, or moment) changes as a load moves across a
structure.
- It does
not include numerical values of the ordinates.
- When
the numerical ordinates are also known, the diagram is called a quantitative
influence line.
Purpose
- Used
when only the trend or nature of variation of internal forces is
needed (not exact magnitudes).
- Helps
engineers quickly identify where maximum effects occur under moving
loads.
Use of Müller-Breslau’s Principle
- This
principle is most effective for drawing qualitative influence
lines.
Quantitative Determination
- If numerical
values of ordinates are needed, they are later found using the equilibrium
method.
In Short
Qualitative Influence Line: Only shows shape — for
understanding variation.
Quantitative Influence Line: Shows exact values — for numerical
analysis.
Müller-Breslau’s Principle: Ideal tool for constructing qualitative
influence lines.
Procedure for Constructing Influence Lines
(Using Müller-Breslau’s Principle and Equilibrium Method)
Step 1: Draw the General (Qualitative) Shape
Use Müller-Breslau’s Principle to determine the basic
shape of the influence line.
1.
Remove the corresponding restraint:
o Identify
the response function (e.g., reaction, shear, moment).
o Release
the restraint associated with that response (e.g., hinge → roller, cut → free
end, insert hinge → for moment).
2.
Apply a unit displacement or rotation:
o Apply
a small displacement (Δ = 1) or rotation (θ = 1) at the released
location in the positive direction of the response.
o The
resulting deflected shape represents the qualitative influence line.
3.
Sketch the deflected shape:
o The
shape must be consistent with the structure’s supports and continuity.
o For
statically determinate structures, the influence line will consist of straight-line
segments.
o If
only a qualitative influence line is required, the analysis stops here.
Step 2: Determine Numerical (Quantitative) Ordinates
If numerical values are needed, use the equilibrium
method.
1.
Apply a unit load on the original
(unreleased) structure at the position of the response function.
o Compute
the ordinate value (reaction, shear, or moment) using equilibrium
equations.
2.
Special case – Shear:
o Place
the unit load:
§ Just
to the left of the point of interest → gives 
§ Just
to the right → gives 
o The
difference gives the jump in the influence line at that point.
3.
If ordinate = 0 at the point of interest:
o Move
the unit load to the maximum or minimum point on the influence line.
o Determine
that value using equilibrium.
4.
Use geometry for remaining ordinates:
o Once
one ordinate is known, use linear geometry (straight-line proportion) to
find others at slope-change points.
Step 3: Advantages of This Combined Method
- Simplifies
construction of influence lines for any force or moment directly.
- No
need to first determine reaction influence lines.
- Especially
efficient for shear and bending moment influence lines.
In Short
Step 1: Use Müller-Breslau’s principle → Get the shape.
Step 2: Use equilibrium → Get the numerical values.
Result: Complete, accurate influence line (qualitative + quantitative).
Example
Given:
A simply supported beam 
Span lengths:
Required:
Draw the influence lines for
- Vertical
reactions
and 
- Shear
and bending moment 
at point C
(a) Influence Line for 
1.
Release and displacement:
o Remove
the roller support at B → released structure obtained.
o Give
point B a small upward displacement Δ in the direction of .
o The
deflected shape (dashed line) represents the qualitative influence line.
2.
Numerical ordinate at B:
o Place
a 1 kN unit load at point B on the original beam.
o Apply
moment equilibrium about D:
Hence, ordinate at B = 1.
3.
Ordinates at A and D (geometry):
o From
similar triangles:
Thus, ordinate at A = 
.
o Ordinate
at D = 0.
o Influence
line shown in Fig.(b).
Result:
(b) Influence Line for 
1.
Release and displacement:
o Remove
the hinge at D → released beam obtained.
o Give
at point D a small upward displacement Δ in the direction of .
o The
deflected shape gives the qualitative influence line.
2.
Numerical ordinate at D:
o Place
1 kN unit load at D:
o Ordinate
at D = 1.
o From
geometry, ordinate at C = 
, and A = 
,
Result:
(c) Influence Line for Shear 
1.
Release and displacement:
o Cut
the beam at C → released structure with two segments (AC and CD).
o Move
C of AC downward by Δ₁ and C of CD upward by
Δ₂ forming a unit relative displacement:
o The
deflected shape gives the qualitative influence line.
2.
Numerical ordinates (unit load positions):
o Calculate
Reactions:
kN
o Unit
load just right of C:
kN
3.
Geometry:
o From
similar triangles:
Ordinate at A = .
Result:
→ Shown in Fig. (d).
(d) Influence Line for Bending Moment 
1.
Release and rotation:
o Insert
a hinge at C to remove the moment restraint.
o Apply
a small unit rotation (θ = 1) at C:
§ Left
portion (AC) rotates counterclockwise
§ Right
portion (CD) rotates clockwise
2.
Numerical ordinate at C:
o Apply
1 kN unit load at C on the original beam.
o From
equilibrium:
Hence ordinate at C = 2 kN·m.
3.
Geometry:
o From
similar triangles:
Hence, the influence line is linear between A–C–D.
Result:
Example– Influence Lines Using Müller-Breslau’s Principle
Draw the influence lines for:
- Vertical
reactions at supports A and E,
- Reaction
moment at A,
- Shear
at point B, and
- Bending
moment at point D for the beam shown in Fig. (a).
1. Influence Line for
To determine the shape of the influence line for the
vertical reaction at A, remove the vertical restraint at A. This
converts the fixed support at A into a roller guide, which allows
vertical movement but restricts rotation and horizontal motion.
Now apply a small upward displacement 
at A. The beam
deflects as shown in Fig. (b):
- Portion
AC remains straight and horizontal (since rotation at A is
restrained).
- Portion
CF rotates about support E, which is on a roller and hence
fixed vertically.
- The
two rigid portions, AC and CF, of the beam remain straight
in the displaced configuration and rotate relative to each other at the
internal hinge at C, which permits such a rotation.
Hence, the deflected
shape represents the influence line for .
When a unit load (1 k) acts at A,
Other ordinates are
determined from geometry.
2. Influence Line for 
Next, remove the roller support at E and apply a
small vertical displacement at that point. The resulting deflected
shape is shown in Fig. (c).
Because A is fixed, the segment AC cannot translate or rotate,
and the rest of the beam bends accordingly.
When a unit load acts at E,
Other ordinates are
obtained from the geometry of the line.
3. Influence Line for 
To find the influence line for the moment reaction at A,
release the rotational restraint at A by replacing the fixed support
with a hinge. Then apply a small unit rotation (θ = 1 rad) in the
counterclockwise direction at A.
The deflected configuration shown in Fig.(d) represents the influence
line for 
.
Since the ordinate at A
is zero, the value at C can be determined using equilibrium:
Hence,
The other ordinates are
obtained from the geometry of the line.
4. Influence Line for
To construct the influence line for shear at B, cut
the beam at B and apply a unit upward displacement on the left
portion relative to the right, as in Fig. (e).
Support A remains fixed, so AB cannot rotate or translate.
Both portions AB and BC remain straight and parallel.
To find numerical
ordinates, place the unit load just to the left and right of B:
Intermediate ordinates
are determined by geometry.
5. Influence Line for 
For the bending moment at
D, insert a hinge at D and apply a unit rotation θ at that
location.
The deflected shape shown in Fig. (f) gives the qualitative influence
line for 
.
When a 1 k load is
applied at D,
Other ordinates are found
geometrically.
Example– Influence Lines for Reactions at Supports A and C
Given:
Draw the influence lines for the vertical
reactions at supports A and C for the beam shown in Fig. (a).
The beam is supported at A and C, with
internal hinges and roller supports at intermediate points (as shown).
1. Influence Line for
To obtain the influence
line for the vertical reaction at A,
- Remove the roller support at A so that
vertical movement is free but horizontal movement remains restricted.
- Apply a small upward displacement at A to the released beam.
The resulting deflected
shape, consistent with support conditions, is shown in Fig. (b).
This deflected shape directly represents the influence line for according to Müller-Breslau’s
Principle.
When a unit load (1
kN) is placed at A, the reaction is also 1 kN at A:
Hence, the ordinate at A
is 1.0.
Other ordinates (at B, C, D, etc.) can be determined geometrically.
2. Influence Line for
To obtain the influence
line for the vertical reaction at C,
- Remove the roller support at C so that it can
move vertically.
- Apply a small upward displacement at C.
The released beam
deflects as shown in Fig. (c).
This deflected shape represents the influence line for .
When a 1 kN load
is applied at C, the vertical reaction at C is also 1 kN:
The ordinates at B
and E are obtained from the geometry of the deflected shape.