# Influence Lines - ust.hk

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25 Νοε 2013 (πριν από 4 χρόνια και 7 μήνες)

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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

Concentrated

Concentrated

Concentrated

w

P

V

M

Analysis

w, P

V, M

Design

Note:

-

V
max

and
M
max

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

V

M

x

V

and
M

at different locations of the beam

x

V

and
M

at a fixed location

V

M

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

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

A

B

C

D

Concentrated

Concentrated

Concentrated

Application of Influence Lines

A

B

C

D

S
B

M
B

1

Find maximum forces

w
d

w
l

Concentrated

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?