CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 6: Influence Lines for Statically Determinate Structures
Feb 24, 2016
CIVL3310 STRUCTURAL ANALYSISProfessor CC Chang
Chapter 6: Influence Lines for Statically Determinate
Structures
Why Influence Lines?
AB C D
Dead Load
Con
cent
rate
d L
ive
Load
Con
cent
rate
d L
ive
Load
Con
cent
rate
d L
ive
Load
Distributed Live LoadDistributed Live LoadDistributed Live Load
wP
V M
Analysisw, P 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 (4 3) 42 (84)126 (168) 85 (127 )
Force
Influence Lines Measurement
KSM Influence Lines Measurement
KSM Influence Lines Measurement
Influence Lines
Influence LinesShear force and moment diagrams
Fixed loadsV M
xV and M at different locations of the beam
x
V and M at a fixed location
V M
Load moves along the beam
x
x
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
Res
pons
e
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
xA
xAM
y
y
B
1011
0)1)(10()10(0
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
A structure in equilibrium: 0i
iF
1F
2F
3F
4F
Resultant force=0
4r2r
3r1r
rVirtual deformation
Virtual work: 0
rFWi
i
0 i
ii rFW
Virtual work principle for a rigid-body system
imaginary deformation
0i
iF
Müller Breslau Principle
x 1
Influence line = properly disturbed shape
Ay
1
Ay
1
Ay
1
f(x)
f(x)A
0f(x)11AW0δW
y
y
Influence line of Ay
Virtual work
By
By
Müller Breslau Principle
x 1
Influence line = properly disturbed shape
Ay1
By
1
f(x)
Virtual work
f(x)B
0f(x)11BW0δW
y
y
Influence line of By
By
Ay
1
By
Müller Breslau PrincipleInfluence line = properly disturbed shape
x 1
Ay By
Ay
1
By
f(x)
1
V
MV
VM
D1
D2q1
q2
0f(x)1MVW 1221 qqDD
Include ONLY the reaction/force in the virtual work
= 1 = 0f(x)V
Müller Breslau PrincipleInfluence line = properly disturbed shape
x 1
Ay By
Ay
1
By
f(x)
1
Dq1 q2
0f(x)1MW 21 qq
Include ONLY the reaction/force in the virtual work
= 1f(x)M
M
VM
V
Müller Breslau PrincipleInfluence 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
AB C D
SB
MB
1 1P
P
P
SB MB
wldxwl
dxwyM lB a b
balconstw
ba lB ydxwdxwyM
l
y
Distributed Loads
AB C D
Dead Load
Con
cent
rate
d L
ive
Load
Con
cent
rate
d L
ive
Load
Con
cent
rate
d L
ive
Load
Distributed Live LoadDistributed Live LoadDistributed Live Load
Application of Influence Lines
AB C D
SB
MB
1
Given dead loadand live loads
Find maximum forces
wd
Distributed Live Loadwl
Con
cent
rate
d L
ive
Load
P
Application of Influence Lines• Max shear force at C?
Application of Influence Lines• Max shear force at C?
kNV
kNV
kNV
C
C
C
25.11)75.0(18)125.0(18)0(5.4)(:3 Case
19.24)625.0(18)75.0(18)125.0(5.4)(:2 Case
63.23)5.0(18)625.0(18)75.0(5.4)(:1 Case
3
2
1
6. Influence Lines• What is influence line?• Müller Breslau Principle• What is the use of influence line?