ct3109 : structural mechanics...

Ir J.W. Welleman page 1

CT3109 : Structural Mechanics 4

� 17-19 Influence Lines • Introduction

• Maxwell

• Static determinate structures

• Qualitative

• Quantitative

• Static indeterminate structures

• Qualitative

• Quantitative

• Special Applications

CIE3109 LECTURE 17


WHAT IS AN INFLUENCE LINE ?

To what extend does a quantity at a certain location change if a unit load moves along the structure.

What is the magnitude here of :

- Moment ? - Shear ?

1,0 kN

x

a


DEFINITION

An influence line gy is a graphical representation of the influence of a unit load at position x on the magnitude of a quantity at position a.

Relevant Quantities ???

• Moment

• Shear force

• Reactions

• displacement / rotation required tool : Maxwell’s Reciprocal Theorem

FORCE QUANTITIES

DISPLACEMENT QUANTITIES


WHY INFLUENCE LINES ?

Influence Lines are important to find the most unfavourable position of the loads for a certain quantity. Often used for moving loads on bridges and so on.

But also …..

Influence Lines can be used to find the most unfavourable position of the static distributed loads in structures like buildings.


SO …..

With an influence line we can directly read for a certain position inside the structure where we have to put the load in order to obtain the maximum value for the observed quantity.

FIXED POSITION

VARIABLE


EXAMPLE (STATIC DETERMINED)

1 Force Quantities - Reactions at A and B - Shear at C - Moment at C

2 Displacement Quantities - deflection in C - rotation in C required tool : Maxwell’s Reciprocal Theorem

1,0 kN

l = 10 m

B A C

2,5 m


Influence lines for the support reactions at A and B

1,0 kN

l =10 m

z-axis

x-axis

1,0

1,0

i-line

AV

i-line

BV

0,75

2,5 m

A B C

BV

AV


SIGN CONVENTION FOR SHEAR Positive shear force :

V

x

z

V

- Positive shear on a positieve sectional plane acts in the positive z-direction

- Positive shear on a negative sectional plane acts in the negative z-direction

positive sectional plane ( outer normal into the positive x-direction )

deformation symbol

for a positive shear

force


Influence line for the shear force at C

NEGATIVE

SHEAR FORCE

POSITIVE

SHEAR FORCE

1,0 kN

l =10 m

z-axis

x-axis

i-line

VC

0,75

2,5 m

-0,25

B A C

i-line

VC

0,75

0,25


WHAT ABOUT THE SIGNS ?? Note : Take care of these signs, see book page 83 !!!

1,0 kN

l =10 m

2,5(-) m

0,75 kN

0,25 kN

1,0 kN

0,75 kN 0,25 kN

B A

Just before the load passes C, the shear is –0,25, after

passage the shear jumps to + 0,75 !!


Influence line for the moment at C

- reproduce this yourself based on the reaction at A

1,0 kN

l=10 m

z-axis

x-axis

i-line

MC

2,5 m

1,875

B A C


Maxwell’s Reciprocal Theorem

ababbb21

aaa21 uFuFuFA ×+×+×=

babaaa21

bbb21 uFuFuFA ×+×+×=

Same amount of Work for both situations :

uaa uba

Fa Fb

ubb

uab

uaa

uba

Fa Fb

ubb uab

A B A B

bababa uFuF =


RESULT ?

Use:

bbbbbababa

bababaaaaa

FcuFcu

FcuFcu

==

==

Elaborate:

baab

ababbaba

cc

FcFFcF

=

=

Result:

bbbababbbab

babaaaabaaa

FcFcuuu

FcFcuuu

+=+=

+=+=

=

b

a

bbab

abaa

b

a

F

F

cc

cc

u

u

SYMMETRICAL FLEXIBILITY RELATION


Influence line for the deflection at C Check a number of points :

• Deflection at C t.g.v. 1,0 kN in C : wcc

• Deflection at C t.g.v. 1,0 kN in D : wcd = wdc

• Deflection at C t.g.v. 1,0 kN in E : wce = wec

CONCLUSION ???

1,0 kN

4×2,5 m

B A C D E

1,0 kN 1,0 kN

wcc wdc

wec

Apply

Maxwell !


Influence line for the rotation at C

How can we find the rotation in a simple manner ?

1,0 kN

4×2,5 m

B A C D E

deflection due to F at

C

1,0 kN 1,0 kN


D


E

ϕcc

ϕcd

ϕce


Apply Maxwell

=

d

d

c

c

44342414

34332313

24232212

14131211

d

d

c

c

T

F

T

F

cccc

cccc

cccc

cccc

w

w

ϕ

ϕ

THUS …….

• The displacement at C due to a unity couple (T=1.0) at C is equal to the

rotation at C due to a unity load (F=1.0) at C.


RESULT The deflection line of a unity couple at C is the influence line for

the rotation at C.

L = 10 m

B A C D E

i-line for the

rotation at C

TC = 1,0 kNm


SUMMARY : DISPLACEMENT QUANTITIES

• Influence line for the deflection at a certain point : o Put a unity load at the point of interest. o The deflection line due to the unity load is the required

influence line for the deflection at the point of observation.

• Influence line for the rotation at a certain point : o Put a unity couple at the point of interest. o The deflection line due to the couple is the required

influence line for the roation at the point of observation.

• Please find us a method as easy for the Influence Lines of Force Quantities ….


ELEGANT WAY TO FIND THE INFLUENCE LINES OF FORCE QUANTITIES

Müller-Breslau principle Release the degree of freedom which belongs to the observed force quantity (e.g. remove a suppport or add a hinge) and generate Negative Virtual Work with the observed force quantity by imposing a virtual unit displacement. The displaced structure (mechanism) is the required influence line.


EXAMPLE static determinate structure

Force Quantities

- reactions at A and B - shear force at C - moment at C

1,0 kN

l = 10 m

B A C 2,5 m


REACTIONS

+

+

1,0 kN

l = 10 m

B A C 2,5 m

B

AV

δ w =1,0 i-line for

AV

A

BV

δ w =1,0

i-line for

BV

x-axis

z-axis


MOMENT AT C

1,0 kN

l =10 m

z-axis

x-axis

i-line for

MC

2,5 m

B A C

S MC MC

δ w

δ θ =1,0

875,1

0,15,75,2

=

=+=

w

ww

δ

δδδθ


SHEAR FORCE AT C

1,0 kN

l = 10 m

z-as

x-as

i-line for

VC

2,5 m

B A C

“shear hinge”

VC

VC

δ w=1,0 +


SUMMARY

• Influence lines for the deflection at a certain point :

o Put a unit load at the point of interest. o The deflection line is the influence line for the deflection at the observed point.

• Influence line for the rotation at a certain point :

o Put a unit couple at the point of interest. o The deflection line is the inflence line for the rotation at the observed point.

• Influence line for force quantities, apply Müller-Breslau :

o Release the degree of freedom which belongs to the observed force quantity (e.g. remove a suppport or add a hinge) and generate Negative Virtual Work with the observed force quantity by imposing a virtual unit displacement or unit rotation.

o The position of the mechanism is the required influence line.

ct3109 : structural mechanics...

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