CIVL3310 STRUCTURAL ANALYSIS

Professor CC Chang

Chapter 6: Influence Lines for Statically Determinate

Structures

Why Influence Lines?

AB C D

Dead Load

Co

nce

ntr

ated

Liv

e L

oad

Co

nce

ntr

ated

Liv

e L

oad

Co

nce

ntr

ated

Liv

e L

oad

Distributed Live LoadDistributed Live LoadDistributed Live Load

w

P

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 (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 diagramsFixed loads

V 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

xA

M

y

y

B

10

11

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)11AW

0δW

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)11BW

0δW

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

M

V

V

M

D 1

D 2q 1

q 2

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

Dq 1 q 2

0f(x)1MW 21 qq

Include ONLY the reaction/force in the virtual work

= 1f(x)M

M

VM

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

AB C D

SB

MB

1 1P

P

P

SB MB

wl

dxwl

dxwyM lB a b

balconstw

ba lB ydxwdxwyM

l

y

Distributed Loads

AB C D

Dead Load

Co

nce

ntr

ated

Liv

e L

oad

Co

nce

ntr

ated

Liv

e L

oad

Co

nce

ntr

ated

Liv

e L

oad

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

Co

nce

ntr

ated

Liv

e L

oad

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?