Chapter 6:
Influence Lines for Statically Determinate Structures
CIVL3310 STRUCTURAL ANALYSIS
Professor CC Chang
Distributed
LiveLive
Load
Distributed
Load Distributed Live Load
w
Concentrated
Live Load
Concentrated
Live Load
Concentrated
Live Load
Why Influence Lines?
P
Dead Load
A
B
M
V
w, P
Analysis
D
C
V, M
Design
Note: loads can vary- LIVE LOADS
Vmax and Mmax under dead & live loads?
Can my bridge survive?
What is Influence Line ?
1 2 6 (1 6 8 )
8 5 (1 2 7 )
Force
1 (4 3 )
4 2 (8 4 )
Influence Lines Measurement
KSM Influence Lines Measurement
KSM Influence Lines Measurement
Influence Lines
Shear force and moment diagrams
Fixed loads
V M
x
V and M at different locations of the beam
x
Influence Lines
Load moves along the beam
V M
x
x
V and M at a fixed location
Influence Lines
• Influence line:
A graph of a response function
(such as reactions or internal
forces) of a structure as a function
of the position of a downward unit
load moving across the structure
Location of downward unit load
Constructing Influence Lines
• Point-by-point calculation
• Influence-line equation
• Graphical approach: Müller Breslau
Principle
Point-by-point calculation
• Construct the influence line for Ay
Influence-line equation
M
B
0
 A y (10 )  (10  x )( 1)  0
Ay  1 
1
x
10
Linear function of x
Influence Lines
• All statically determinate structures have
influence lines that consist of straight line
segments
• More examples!
Müller Breslau Principle
• Influence line for any action (reaction,
internal shear/moment) in a structure
is equal to the deflection curve when
we remove the action and replace it
with a corresponding unit
displacement or rotation
Influence line = properly disturbed shape
Müller Breslau Principle
Virtual work principle for a rigid-body system
F1
 r1
F
i
0
 r3
F3
i
 r2
F2
r
Virtual deformation
A structure in equilibrium:  Fi
i
Virtual work:
F4
0
 r4
imaginary deformation
Resultant force=0


 W    Fi    r  0
 i

 W    Fi   ri   0
i
Virtual work principle for a rigid-body system
Müller Breslau Principle
Influence line = properly disturbed shape
x
1
By
Ay
1
1
1
Ay
f(x)
Ay
By
Virtual work
δW  0
Influence line of Ay
 W  A y  1  1  f(x)  0
A y  f(x)
Müller Breslau Principle
Influence line = properly disturbed shape
1
x
Ay
By
1
1
1
f(x)
Ay
Virtual work
δW  0
By
By
Influence line of By
 W  B y 1  1  f(x)  0
B y  f(x)
Müller Breslau Principle
Influence line = properly disturbed shape
Include ONLY the reaction/force in the virtual work
1
x
V
Ay
V
By
1
M
1
D2
q1
q2
D1
Ay
f(x)
By
M
V
 W  V   D 1  D 2   M   q 2  q1   1  f(x)  0
V  f(x)
1
0
Müller Breslau Principle
Influence line = properly disturbed shape
Include ONLY the reaction/force in the virtual work
1
x
M
Ay
q1
M V
By
1
1
V D
q2
f(x)
By
Ay
 W  M   q1  q 2   1  f(x)  0
M  f(x)
1
Müller Breslau Principle
Influence line = properly disturbed shape
Deflect the structure such that only the force
which influence line that you are looking for
and the downward unit force contribute to the
virtual work due to the imaginary deflection.
All other forces that act on the virtually
deflected structure should not contribute to the
virtual work.
Influence Lines for Floor Girders
• Draw the influence line for the shear in
panel CD of the floor girder
Influence Lines for Trusses
Example 6.15
• Draw the influence line for member force GB
Example 6.15
• Draw the influence line for the member force GB
Application of Influence Lines
P
1
1
SB
A
w l  dx
wl
MB
B
D
C
P
SB
y
P
MB
a
b
M B  y  w l  dx
b
b
M B  a y  w l  dx      w l  a ydx
w  const
l
Distributed
LiveLive
Load
Distributed
Load Distributed Live Load
Concentrated
Live Load
Concentrated
Live Load
Concentrated
Live Load
Distributed Loads
Dead Load
A
B
C
D
1
A
SB
MB
Concentrated
Live Load
Application of Influence Lines
B
P
Distributed Live Load
wl
Given dead load
and live loads
Find maximum forces
wd
C
D
Application of Influence Lines
• Max shear force at C?
Application of Influence Lines
• Max shear force at C?
Case 1 :
(V C ) 1  4 . 5 ( 0 . 75 )  18 ( 0 . 625 )  18 ( 0 . 5 )  23 . 63 kN
Case 2 :
(V C ) 2  4 . 5 (  0 . 125 )  18 ( 0 . 75 )  18 ( 0 . 625 )  24 . 19 kN
Case 3 :
(V C ) 3  4 . 5 ( 0 )  18 (  0 . 125 )  18 ( 0 . 75 )  11 . 25 kN
6. Influence Lines
• What is influence line?
• Müller Breslau Principle
• What is the use of influence line?