Influence Lines - ust.hk

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Nov 25, 2013 (3 years and 6 months ago)

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CIVL3310 STRUCTURAL ANALYSIS

Professor CC Chang

Chapter 6:

Influence Lines for Statically Determinate Structures

Why Influence Lines?

A

B

C

D

Dead Load

Concentrated


Live Load

Concentrated


Live Load

Concentrated


Live Load

Distributed Live Load

Distributed Live Load

Distributed Live Load

w

P

V

M

Analysis

w, P

V, M

Design

Note:

loads can vary
-

LIVE LOADS

V
max

and
M
max

under dead & live loads?

Can my bridge survive?

What is Influence Line ?

1 (43)
42 (84)
126 (168)
85 (127)
Force

Influence Lines Measurement

KSM Influence Lines Measurement

KSM Influence Lines Measurement

Influence Lines

Influence Lines

Shear force and moment diagrams

Fixed loads

V

M

x

V

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

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 A
y

Influence
-
line equation

x
A
x
A
M
y
y
B
10
1
1
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:

0


i
i
F
1
F
2
F
3
F
4
F
Resultant force=0

4
r

2
r

3
r

1
r

r

Virtual deformation

Virtual work:

0










r
F
W
i
i




0






i
i
i
r
F
W
Virtual work principle for a rigid
-
body system

imaginary deformation

0


i
i
F
Müller Breslau Principle

x

1

Influence line = properly disturbed shape

A
y

1

A
y

1

A
y

1

f(x)

f(x)
A
0
f(x)
1
1
A
W
0
δW
y
y








Influence line of A
y

Virtual work

B
y

B
y

Müller Breslau Principle

x

1

Influence line = properly disturbed shape

A
y

1

B
y

1

f(x)

Virtual work

f(x)
B
0
f(x)
1
1
B
W
0
δW
y
y








Influence line of B
y

B
y

A
y

1

B
y

Müller Breslau Principle

Influence line = properly disturbed shape

x

1

A
y

B
y

A
y

1

B
y

f(x)

1

V

M

V

V

M

D
1

D
2

q
1

q
2





0
f(x)
1
M
V
W
1
2
2
1



q

q


D

D



Include
ONLY

the reaction/force in the virtual work

1

0

f(x)
V

Müller Breslau Principle

Influence line = properly disturbed shape

x

1

A
y

B
y

A
y

1

B
y

f(x)

1

D

q
1

q
2



0
f(x)
1
M
W
2
1



q

q



Include
ONLY

the reaction/force in the virtual work

1

f(x)
M

M

V

M

V

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

A

B

C

D

S
B

M
B

1

1

P

P

P

S
B

M
B

w
l

dx
w
l

dx
w
y
M
l
B



a

b













b
a
l
const
w
b
a
l
B
ydx
w
dx
w
y
M
l
y

Distributed Loads

A

B

C

D

Dead Load

Concentrated


Live Load

Concentrated


Live Load

Concentrated


Live Load

Distributed Live Load

Distributed Live Load

Distributed Live Load

Application of Influence Lines

A

B

C

D

S
B

M
B

1

Given dead load

and live loads

Find maximum forces

w
d

Distributed Live Load

w
l

Concentrated


Live Load

P

Application of Influence Lines


Max shear force at C?

Application of Influence Lines


Max shear force at C?

kN
V
kN
V
kN
V
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?