2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
1
INFLUENCE LINES
PRELIMINARIES
Moving loads -- Loads applied to a structure with points of application (including their
magnitude) can vary as a function of positions on the structure. Examples of moving loads
include live load on buildings, traffic or vehicle loads on bridges, loads induced by wind and
earthquake, etc. In the analysis, the moving loads can be modeled as varying distributed
loads, a series of concentrated loads, or the combination of distributed loads and
concentrated loads.
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
2
Responses due to a moving unit load -- Quantities of interest at a particular point
within a given structure, e.g. internal forces, support reactions, deformations, displacements
and rotations, due to an applied moving unit load. The quantities are given in terms of
functions of a position of a moving unit load on the structure; these response functions are
termed as the influence functions and their graphical representations are known as the
influence lines.
Application of the influence functions (lines)
Let fA be a quantity of interest at a point A within a given structure due to applied distributed
load q and a series of concentrated loads {P1, P2, …, PN} and fAI denote the influence
function of the corresponding quantity at point A. By a method of superposition, we obtain the
relation of fA and fAI as
N
³ fAI q dx ¦ fAI (x i )Pi
fA
(1)
i 1
Figure 1
where the integral is to be taken over the region on which load q is applied and xi indicates
the location on which the load Pi is applied. For instance, assume that the influence line of
the support reaction at point A (RAI) of the beam is given as shown in the Figure 3a. The
support reaction at point A (RA) due to applied loads as shown in Figure 3b can then be
obtained using Eqn. (1) as follow:
L/2
A moving unit load -- a concentrated load of unit magnitude with its point of application
RA
varies as a function of position on the structure.
³R
AI
q dx R AI (L/4 )P R AI (3L/4 )2P
0
§ L/2
·
q ¨¨ ³1 - x/L dx ¸¸ 3/4 P 1/4 2P 3qL/8 5P/4
©0
¹
1
1
q
x
A
2P
B
RA
L/4
RAI
L
L/4
3/4
1-x/L
Figure 2
P
Area = 3L/8
Responses due to moving loads -- Quantities of interest that indicate the effect of the
moving loads on a structure, e.g. internal forces, support reactions, displacements and
rotations, deformations, etc.
RAI
0
L
Figure 3a
x
L/4
L/4
1/2
1/4
RAI
L
Figure 3b
x
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
3
In addition, the influence lines can also be used to predict the load pattern that maximizes
responses at a particular point of the structure. For instance, let consider a two-span
continuous beam subjected to both dead load (fixed load) and live load (varying load) as
shown in the figure below.
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
4
INFLUENCE LINES FOR DETERMINATE BEAMS
1
x
Live load
Dead load
A
B
GBI
RAI
B
To determine the maximum positive bending moment at points A, the maximum negative
moment B, and the maximum positive shear at point A due to these applied dead and live
loads, we construct first the influence lines MAI, MBI, and VAI as shown below.
MCI
C
TBI
D
Deformed state
Undeformed state
RDI
¾ Support reactions (e.g. RAI, RDI)
¾ Bending moment at a particular section (e.g. MCI)
¾ Shear force at a particular section (e.g. VCI)
1
x
A
VCI
¾ Deflection at a particular point (e.g. GBI)
A
B
¾ Rotation at a particular point (e.g. TBI)
Direct Methods for Constructing Influence Lines
¾ Treat a structure subjected to a moving unit load (as function of positions)
MAI
x
¾ Influence functions are obtained by considering all possible load locations
¾ Support reactions
MBI
x
-- Equilibrium equations of the entire structure
¾ Internal forces
-- Method of sections
VAI
x
-- Equilibrium equations of parts of the structure
¾ Displacement and rotations
-- Determining support reactions and internal forces from equilibrium
It is evident from the influence lines that the maximum positive bending moment at point A
occurs when the live load is placed only on the first span; the maximum negative moment at
point B occurs when the live load is placed on both spans; and the maximum positive shear
occurs when the live load is placed on the first half of the first span and on the second span.
The maximum value of the responses can then be obtained using Eqn.(1) for each
corresponding loading pattern. It is noted that the dead load is fixed and therefore it is
applied to both spans of the beam for all cases.
-- Displacement and rotations are obtained from
9 Direct integration method
9 Moment area and conjugate structure methods
9 Energy methods, etc.
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
5
Example1: Construct influence lines RAI, RBI, VCI, MCI, GCI, TCI of a simply supported beam
C
A
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
6
Influence lines for shear and bending moment VCI, MCI
B
EI
L/3
2L/3
x ” L/3
1
A
Solution Consider the beam subjected to a moving unit load as shown below.
B
C
A
> 6FY
B
C
L/3
0
@
+
> 6M C
1
x
0
@
+
B
A
@
+
MCI
L- x
L
+
R AI 1 (R AI 1)L
x
3
2x
3
A
VCI
B
C
C
A
RAI
RBI
> 6FY
x
L
0
@
+
R AI VCI
VCI
0
1
R AI
1
RAI
L
x
> 6M C
0
@
+
(R AI )(L/3) MCI
1
MCI
RBI
0
0
(RBI )(L) (1)(x) 0
RBI
0
x
L
1
x • L/3
RAI
@
0
(R AI )(L) (1)(L - x) 0
R AI
0
MCI
RAI
(R AI )(L/3) (1)(L/3 - x) MCI
RBI
RAI
> 6M A
R AI 1 VCI
VCI
2L/3
Influence lines for reactions RAI, RBI
0
C
A
RBI
RAI
VCI
1
x
> 6M B
1
x ” L/3
L
x
x
L
0
R AI L
3
L§ x·
¨1 ¸
3© L¹
MCI
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
7
1
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
The deflection GCI for x ” L/3 can be obtained using the unit load method along with the actual
system I and the virtual system I; i.e.
2/3
RAI
0
x
L
L
2/3
³
GCI
VCI
0
Influence Line
8
L/3
-1/3
0
x
L
1 § L x ·§ L · ª 2 2L º 1 §
L ·§ L
· ª§ 2 x · 2L º
¨
¸¨ ¸ «
¨ x ¸¨ x ¸ «¨ ¸ »
»
2EI © 3 ¹© 3 ¹ ¬ 3 9 ¼ 2EI ©
3 ¹© 3
¹ ¬© 3 L ¹ 9 ¼
2L/9
MCI
0
L/3
x
L
Influence lines for deflection and rotation GCI, TCI
x ” L/3
A
RAI
x • L/3
C
A
B
RBI
L
BMD
M
³
GCI
x
0
L/3-x
Actual System I
1
B
1/3
A
1/L
C
MGM
dx
EI
1 § L x ·§ L · ª 2 2L º 1 § L
L · ª§ 7 x · 2L º
·§
¨
¸¨ ¸ «
¨ x ¸¨ x ¸ «¨ ¸
»
2EI © 3 ¹© 3 ¹ ¬ 3 9 ¼ 2EI © 3
3 ¹ ¬© 6 2L ¹ 9 »¼
¹©
Actual System II
1
C
The deflection GCI for x • L/3 can be obtained using the unit load method along with the actual
system II and the virtual system I; i.e.
L/3-x/3
x-L/3
2/3
B
RBI
2x/3
x
A
1
RAI
L/3-x/3
2x/3
BMD
M
1 § 2x ·§ 2L · ª 2 2L º
¨ ¸¨ ¸
2EI © 3 ¹© 3 ¹ «¬ 3 9 »¼
x
5L2 9x 2
81EI
1
C
MGM
dx
EI
B
1/L
1 § 2x ·§ 2L · ª 2 2L º
¨ ¸¨ ¸
2EI © 3 ¹© 3 ¹ «¬ 3 9 »¼
x-L 2
L 18Lx 9x 2
162EI
2L/9
1/3
BMD
GM
x
BMD
GM
x
-2/3
Virtual System I
Virtual System II
4L3/243EI
GCI
0
L/3
L
x
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
9
The rotation TCI for x ” L/3 can be obtained using the unit load method along with the actual
system I and the virtual system II; i.e.
L
³
TCI
0
Example2: Construct influence lines RAI, MAI, VBI, MBI, GBI, TBI of a cantilever beam
B
EI
A
L/2
x
L2 3x 2
18EIL
L/2
Solution Consider the beam subjected to a moving unit load as shown below.
1
x
1 § 2x ·§ 2L · ª 2 2 º
¨ ¸¨ ¸ 2EI © 3 ¹© 3 ¹ «¬ 3 3 »¼
Influence Line
10
MGM
dx
EI
1 § L x ·§ L · ª 2 1 º 1 §
L ·§ L
· ª§ 2 x · 1 º
¨
¸¨ ¸ «
¨ x ¸¨ x ¸ «¨ ¸ »
»
2EI © 3 ¹© 3 ¹ ¬ 3 3 ¼ 2EI ©
3 ¹© 3
¹ ¬© 3 L ¹ 3 ¼
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
B
EI
A
L/2
L/2
Influence lines for reactions RAI, MAI
The rotation TCI for x • L/3 can be obtained using the unit load method along with the actual
system II and the virtual system I; i.e.
L
³
0
RAI
MGM
dx
EI
> 6M A
1 § L x ·§ L · ª 2 1 º 1 § L
L ·ª § 7 x · 2 º
·§
¨
¸¨ ¸ «
¨ x ¸¨ x ¸ « ¨ ¸
»
2EI © 3 ¹© 3 ¹ ¬ 3 3 ¼ 2EI © 3
3 ¹ ¬ © 6 2L ¹ 3 »¼
¹©
> 6FY
L/3
@
+
M AI (1)(x) 0
0
@
+
x
R AI 1 0
R AI 1
0
0
0
M AI
1 § 2x ·§ 2L · ª 2 2 º
¨ ¸¨ ¸ 2EI © 3 ¹© 3 ¹ «¬ 3 3 »¼
L- x 2
L 6Lx 3x 2
18EIL
B
A
MAI
TCI
1
x
L
x
L
-L
1
TCI
2
-4L /162EI
x
MAI
1
RAI
0
L
x
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
11
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence lines for shear and bending moment VCI, MCI
Influence Line
12
L
MAI 0
x
-L/2
1
x ” L/2 1
MAI
RAI
MBI VBI
B
A
1
0
x
L
1
B
1
RAI
VBI
> 6FY
0
> 6M B
0
@
@
+
+
VBI
0
VBI
0
M BI
MBI
0
L/2
L
0
L/2
L
x
-L/2
Influence lines for deflection and rotation GBI, TBI
0
MAI
x ” L/2 1
A
1
x ” L/2
B
A
MAI
B
RAI
RAI
M BI
x
L/2
0
BMD
M
-x
L/2
x
BMD
M
-x
x
L/2-x
-L/2
1
x • L/2
MAI
A
MAI
B
RAI
MBIVBI
A
Actual System I
Actual System II
B
RAI
1
> 6FY
0
@
+
R AI VBI
-L/2
0
B
A
A
1
0
1
1
VBI
> 6M B
0
@
+
R AI
1
(R AI )(L/2) M AI M BI
M BI
BMD
GM
-L/2
0
R AI L
M AI
2
L·
§
¨ x ¸
2
¹
©
x
1
B
1
BMD
GM
x
-L/2+x
Virtual System I
Virtual System II
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
13
The deflection GBI for x ” L/2 can be obtained using the unit load method along with the actual
system I and the virtual system I; i.e.
L
³
G BI
0
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
14
The rotation TBI for x ” L/2 can be obtained using the unit load method along with the actual
system I and the virtual system II; i.e.
L
MGM
dx
EI
³
TCI
0
1 § L ·§ L · ª 1 L º 1
§ L ·ª 1 L º
- x ¨ ¸ «
¨ ¸¨ ¸ »
2EI © 2 ¹© 2 ¹ «¬ 3 2 »¼ EI
© 2 ¹¬ 2 2 ¼
1 § L ·§ L ·
1
§L·
¨ ¸¨ ¸>1@ - x ¨ ¸>1@
2EI © 2 ¹© 2 ¹
EI
©2¹
1 § L
·§ L
· ª § 1 2x · L º
¨ x ¸¨ x ¸ « ¨ ¸ »
2EI © 2
¹© 2
¹ ¬ © 3 3L ¹ 2 ¼
x2
3L 2x 12EI
L
³
0
L
MGM
dx
EI
TCI
³
0
MGM
dx
EI
1 § L ·§ L · ª 1 L º 1
§ L ·ª 1 L º
- x ¨ ¸ «
¨ ¸¨ ¸ «
»
»
2EI © 2 ¹© 2 ¹ ¬ 3 2 ¼ EI
© 2 ¹¬ 2 2 ¼
1 § L ·§ L ·
1
§L·
¨ ¸¨ ¸>1@ - x ¨ ¸>1@
2EI © 2 ¹© 2 ¹
EI
©2¹
L2
6x L 48EI
L
L 4x 8EI
L /24EI
L/2
0
TBI
3
0
x2
2EI
The rotation TBI for x • L/2 can be obtained using the unit load method along with the actual
system II and the virtual system I; i.e.
5L3/48EI
GBI
1 § L
·§ L
·
¨ x ¸¨ x ¸>1@
2EI © 2
2
¹©
¹
The deflection GBI for x • L/2 can be obtained using the unit load method along with the actual
system II and the virtual system I; i.e.
GCI
MGM
dx
EI
L
L/2
L
x
-L2/8EI
x
-3L2/8EI
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
15
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Example3: Construct influence lines RAI, MAI, RBI, VCI, VBLI, MBLI, VBRI, MBRI, VDI, and MDI of a
beam shown below
From FBD I, we obtain
> 6FY
C
D
A
Influence Line
16
0
@
+
R AI RBI 1 0
B
R AI
L/2
L/2
L
0
@
+
Solution Consider the beam subjected to a moving unit load as shown below.
- M AI (RBI )(2L) (1)(x) 0
M AI
2RBI L x
-x
1
x
C
D
L/2
1
L
> 6M A
A
1 RBI
B
L/2
L
x•L
L
MAI
Influence lines for reactions RAI, MAI, RBI and shear force VCI
A
1
VCI
B
C
RAI
1
B
C
RBI
RBI
FBD III
x”L
MAI
A
FBD IV
From FBD IV, we obtain
1
VCI
B
C
RAI
B
C
RBI
> 6MC
RBI
FBD I
0
@
+
RBI
FBD II
From FBD II, we obtain
> 6FY
> 6MC
0
@
+
0
@
+
(RBI )(L) 0
RBI
(RBI )(L) (1)(x L) 0
x
1
L
VCI RBI 1 0
VCI
1 RBI
0
2
x
L
From FBD III, we obtain
> 6FY
0
@
+
VCI RBI
VCI
> 6FY
0
RBI
0
0
@
+
R AI RBI 1 0
R AI
1 RBI
2
x
L
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
17
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
18
Influence lines for shear and bending moment VBLI, MBLI
> 6M A
0
@
+
- M AI (RBI )(2L) (1)(x) 0
M AI
2RBI L - x
x 2L
x”L
MAI
A
1
C
RAI
MAI 0
L
MBLI VBLI
B
B
RBI
RBI
L
2L
x
3L
> 6FY
-L
0
@
VBLI RBI
+
VBLI
> 6M B
RAI
1
1
0
L
2L
3L
0
L
@
M BLI
+
M BLI
x
-RBI
0
0
-1
2
1
RBI
0
0
2L
MAI
x
3L
A
1
x • 2L
C
MBLIVBLI
B
B
RAI
RBI
> 6FY
0
@
+
RBI
VBLI RBI 1 0
1
2L
VCI
0
3L
VBLI
x
L
-1
> 6M B
0
@
+
1 RBI
M BLI (1)(x 2L) 0
M BLI
2L x
1
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
19
0
2L
L
Influence Line
20
2
1
RBI
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
x
3L
MAI
A
1
x • 2L
C
B
RAI
VBLI
0
L
3L
2L
MBLI
0
L
RBI
x
> 6FY
-1
-1
3L
2L
MBRIVBRI
0
@
+
VBRI 1 0
VBRI
x
1
-L
> 6M B
0
@
+
M BRI (1)(x 2L) 0
M BRI
2L x
Influence lines for shear and bending moment VBRI, MBRI
x”L
MAI
A
1
MBRI VBRI
C
1
B
RAI
RBI
> 6FY
> 6M B
0
0
@
@
+
+
VBRI
0
VBRI
0
M BRI
0
M BRI
0
1
VBRI 0
L
2L
3L
0
L
2L
3L
MBRI
x
x
-L
1
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
21
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
22
> 6M B
Influence lines for shear and bending moment VDI, MDI
0
@
+
M DI (1)(x L/2) (RBI )(3L/2) 0
M DI
3LRBI /2 L/2 x
x ” L/21
MAI
D
A
C
B
RAI
RBI
RBI
MDI VDI
C
0
B
D
0
@
+
VDI RBI
VDI
> 6M B
0
@
+
MAI
A
L/2
L
3L
x
2L
MDI
0
L/2
L/2
L
2L
3L
x
-L/2
M DI (RBI )(3L/2) 0
3LRBI /2
Remarks
C
1. The influences lines of support reactions and internal forces (shear force and
bending moment) for statically determinate beams are piecewise linear; i.e.
they consists of only straight line segments.
B
RAI
RBI
MDI VDI
1
C
B
D
> 6FY
1
x
3L
-1
1
D
1
2L
0
-RBI
M DI
x • L/2
L
RBI
VDI 0
> 6FY
2
1
0
@
+
RBI
VDI RBI 1 0
VBLI
1 RBI
2. The influence functions of the internal forces can be obtained in terms of the
influence functions of the support reactions; therefore, the influence lines of
internal forces can be readily obtained from those for support reactions.
3. The influence lines of the deflection and rotation at any points of the statically
determinate beam generally consist of curve segments.
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
23
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
24
¾ Virtual displacement due to release of shear constraint.
Muller-Breslau Principle
Actual Structure. Consider a statically determinate beam subjected to a moving unit load
as shown in the figure below.
x
1
1. Remove the shear constraint by introducing a shear release at point of
interest
2. The beam becomes statically unstable (partially or completely)
3. Introduce unit relative virtual displacement between the two ends of
the shear release with their slope remaining the same (provided that
the moment constraint exists at that point)
4. The virtual displacement at all other points results from the development
of the mechanism (or rigid body motion) of the entire beam.
1
Virtual System 2a
Virtual Displacement -- The fictitious and arbitrary displacement that is introduced to the
structure. For use further below, the following three types of virtual displacement for the
beam structure are considered:
RELEASE shear constraint
¾ Virtual displacement due to release of a support constraint.
1. Release a support constraint in the direction of interest
2. The beam becomes statically unstable (partially or completely)
3. Introduce unit virtual displacement (or unit virtual rotation if the
rotational constraint is released) in the direction that the support
constraint is released.
4. The virtual displacement at all other points results from the development
of the mechanism (or rigid body motion) of the entire beam
1
Virtual System 2b
RELEASE shear constraint
¾ Virtual displacement due to release of bending moment constraint.
1
Virtual System 1a
RELEASE displacement constraint
1. Remove the moment constraint by introducing a hinge at point of interest
2. The beam becomes statically unstable (partially or completely)
3. Introduce unit relative virtual rotation at the hinge without separation
(provided that the shear constraint exists at that point).
4. The virtual displacement at all other points results from the development
of the mechanism (or rigid body motion) of the entire beam.
1
1
Virtual System 1b
RELEASE rotational constraint
Virtual System 3a
RELEASE moment constraint
1
Virtual System 1c
1
RELEASE displacement constraint
RELEASE moment constraint
Virtual System 3b
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
25
Principle of Virtual Work: Consider a system or structure subjected to external applied
loads. The support reactions and internal forces at any locations within the structure are
in equilibrium with the applied loads if and only if the external virtual work (work done by
the external applied loads) is the same as the internal virtual work (work done by the
internal forces) for all admissible virtual displacements, i.e.
įWE
įWI
(2)
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Muller-Breslau Principle: “The influence line of a particular support reaction has an
identical shape to the virtual displacement obtained from releasing the support constraint
in the direction of the support reaction (under consideration) and introducing a rigid body
motion with unit displacement/unit rotation in the direction of the released constraint.”
Influence Line for Shear Force. Let assume that the influence line of the shear force at
point C, VCI, is to be determined. By applying the principle of virtual work to the actual
system with a special choice of the virtual displacement as indicated in the virtual system
2a (the virtual displacement associated with the rigid body motion of the beam resulting
from the release of the shear constraint at C) , we obtain
It is important to note that the portion of the structure that undergoes virtual rigid body
motion (virtual displacement that produces no deformation) produces zero internal virtual
work.
Influence Line for Support Reactions. To clearly illustrate the strategy, let assume that
the influence line of the support reaction RAI is to be determined. By applying the principle
of virtual work to the actual system with a special choice of the virtual displacement as
indicated in the virtual system 1a (the virtual displacement associated with the rigid body
motion of the beam resulting from the release of the displacement constraint at A) , we
obtain
įWE
įWE
R AI ˜1 1 ˜ įv ( x ) R AI įv ( x )
;
įWI
įWE
1 ˜ įv ( x ) įv ( x )
įWE
įWI
Ÿ
0
MAI
R AI
įWI
;
(4)
1
MCI
A
VCI
B
Actual system
C
RAI
įv ( x )
VCI ˜1 VCI
įv ( x )
VCI
x
įWI
Ÿ
Influence Line
26
(3)
RBI
1
Gv(x)
x
1
A
MAI
Virtual System 2a
B
Actual system
RELEASE shear constraint
RAI
RBI
1
VCI
1
x
Gv(x)
Virtual System 1a
RELEASE displacement constraint
Muller-Breslau Principle: “The influence line of the shear force at a particular point has an
identical shape to the virtual displacement obtained from releasing the shear constraint at
that point and introducing a rigid body motion with unit relative virtual displacement
between the two ends of the shear release with their slope remaining the same.”
1
RAI
x
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
27
Influence Line for Bending Moment. Let assume that the influence line of the bending
at point C, MCI, is to be determined. By applying the principle of virtual work to the actual
system with a special choice of the virtual displacement as indicated in the virtual system
3a (the virtual displacement associated with the rigid body motion of the beam resulting
from the release of the bending moment constraint at C) , we obtain
įWE
1 ˜ įv ( x ) įv ( x )
įWE
įWI
įWI
;
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
28
Example4: Use Muller-Breslau principle to construct influence lines RAI, RDI, RFI, VBI, VCLI,
VCRI, VDLI, VDRI,VEI, MBI, MDI, and MEI of a statically determinate beam shown below
MCI
E
D
F
MCI ˜1 MCI
L/4
L/4
Ÿ
C
B
A
įv ( x )
(5)
L/2
L/2
L/2
Solution The influence line of the support reaction RDI is obtained as follow: 1) release the
displacement constraint at point D, 2) introduce a rigid body motion, 3) impose unit
displacement at point D, and 4) the resulting virtual displacement is the influence line of RDI.
1
x
MAI
A
1
MCI
E
F
D
B
Actual system
RELEASE displacement
constraint
C
RAI
C
B
A
VCI
RBI
1
L/4
L/2
L/4
L/2
L/2
Gv(x)
Virtual System 3a
h2=3/2
1
h1=3/4
RELEASE moment constraint
h3=1/2
RDI
x
1
MCI
1
x
Muller-Breslau Principle: “The influence line of the shear force at a particular point has an
identical shape to the virtual displacement obtained from releasing the shear constraint at
that point and introducing a rigid body motion with unit relative virtual displacement
between the two ends of the shear release with their slope remaining the same.”
The value of the influence line at other points can be readily determined from the geometry,
for instance,
h2
(1)(3L/2 ) /(L) 3/2
h3
(1)(L/2 ) /(L) 1/2
h1
(3/2 )(L/4 ) /(L/2 ) 3/4
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
29
The influence line of the shear force VEI is obtained as follow: 1) release the shear constraint
at point E, 2) introduce a rigid body motion, 3) impose unit relative displacement at point E
and 4) the resulting virtual displacement is the influence line of VEI.
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
30
The influence line of the bending moment MEI is obtained as follow: 1) release the bending
moment constraint at point E, 2) introduce a rigid body motion, 3) impose unit relative rotation
at point E without separation and 4) the resulting virtual displacement is the influence line of
MEI.
1
C
B
A
1
E
D
F
C
B
A
RELEASE shear
constraint
L/4
L/2
L/4
L/2
F
RELEASE moment
constraint
L/2
L/4
h2=1/2
E
D
L/2
L/4
L/2
L/2
h4=1/2
h1=1/4
h3=L/4
1
VEI
1
x
MEI
h3=-1/2
x
h1=-L/8
h2=-L/4
The value of the influence line at other points can be readily determined from the geometry,
for instance,
h3 /(L / 2 )
h4 /(L / 2 ) Ÿ h3
h4
h4 h3 1 Ÿ h4 (h4 ) 2h4 1 Ÿ h4 1/2
h3
h4
h2
(h3 )(L/2 ) /(L/2 ) 1/2
h1
(h2 )(L/4 ) /(L/2 ) 1/4
The value of the influence line at other points can be readily determined from the geometry,
for instance,
h3 /(L / 2 ) h3 /(L / 2 ) 1 Ÿ h3
h2
(h3 )(L/2 ) /(L/2 )
h1
(h2 )(L/4 ) /(L/2 )
1/2
L/4
L/4
L/8
The rest of the influence lines can be determined in the same manner and results are given
below.
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
31
x
E
D
Example5: Use Muller-Breslau principle to construct influence lines RAI, MAI, RDI, VBI, VCI,
VDI,VELI,VERI, MBI, MDI, and MEI of a statically determinate beam shown below.
F
C
L/4
Influence Line
32
1
B
A
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
L/2
L/4
L/2
L/2
C
B
A
E
D
F
1
1/2
L/4
L/4
RAI
L/2
L/2
L/2
x
1
1/2
RFI
x
1/4
1/2
Solution By Muller-Breslau principle, we obtain the influence lines as follow: 1) release the
constraint associated with the quantity of interest, 2) introduce a rigid body motion, 3) impose
unit virtual displacement/rotation in the direction of released constraint, and 4) the resulting
virtual displacement is the influence line to be determined. It is noted that values at points on
the influence line can be readily determined from the geometry.
1/2
1
x
VBI
x
C
B
A
E
D
F
-1/2
VCLI
x
L/4
1
1
-1/2
L/2
L/4
L/2
L/2
1
-1
1/2
VCRI
x
RAI
x
-1/2
-1/2
L/2
-1
VDLI
L/4
L/4
x
MAI
x
-1/2
-L/4
-1
-1
1
1
1/2
1/4
VDRI
1/2
x
REI
x
1
1
1/2
L/8
MBI
x
VBI
x
1
L/4
-1/2
1/2
L/8
MDI
3/2
VCI
x
x
-1/2
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
33
A
C
B
Influence Line
34
Example6: Use Muller-Breslau principle to construct influence lines RAI, MAI, RDI, RFI, VBI,
VCI, VDLI, VDRI,VEI, MBI, and MDI of a statically determinate beam shown below.
1
x
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
E
D
F
L/4
L/2
L/4
L/2
C
B
A
E
D
F
L/2
L/4
L/4
L/2
L/2
L/2
1/2
VDI
x
-1/2
-1/2
VELI
x
Solution By Muller-Breslau principle, we obtain the influence lines as follow: 1) release the
constraint associated with the quantity of interest, 2) introduce a rigid body motion, 3) impose
unit virtual displacement/rotation in the direction of released constraint, and 4) the resulting
virtual displacement is the influence line to be determined. It is noted that values at points on
the influence line can be readily determined from the geometry.
-1/2
-1/2
C
B
A
1
1
x
-1
E
D
F
1
VERI
x
L/4
1
1
L/2
L/4
L/2
L/2
1
L/8
MBI
x
RAI
x
-L/8
L/2
-L/4
L/4
MDI
-1
L/4
MAI
x
x
-L/2
-L/4
2
1
MEI
REI
x
x
-L/2
1
REI
x
2101-301 Structural Analysis I
Dr. Jaroon Rungamornrat
Influence Line
35
1
x
A
C
B
L/4
L/2
L/4
1
E
D
L/2
F
L/2
1
VBI
x
1
-1
VCI
x
-1
VDLI
x
-1
-1
1
1
VDRI
x
1
VEI
x
L/4
x
MBI
-L/4
MBI
-L/2