Moving Load

8/8/2019 Moving Load

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2101-301 Structural Analysis I Influence LineDr. Jaroon Rungamornrat 5

Example1: Construct influence lines R AI , R BI , V CI , M CI , CI , CI of a simply supported beam

Solution Consider the beam subjected to a moving unit load as shown below.

Influence lines for reactions R AI , R BI

0 M B + 0 x)-(1)(L )(L)(R AI

L

x -LR AI

0 M A

+ 0 (1)(x) )(L)(R BI

L

x R BI

A C B

L/3 2L/3

AC

B

L/3 2L/3

1 x

A B

1 x

R AI R BI

R AI

x L0

R BI

x

1

1

L0

EI


Influence lines for shear and bending moment V CI , M CI

0 F

Y + 0 V 1R

CI AI

L

x 1R V AI CI

0 M C + 0 M x)-(1)(L/3 )(L/3)(R CI AI

3

2x x

3

1)L(R M AI

CI

0 F Y + 0 V R CI AI

L

x 1R V AI CI

0 M C + 0 M )(L/3)(R CI AI

L

x 1

3

L

3

LR M AI

CI

1

A B

x L/3

R AI R BI

C

A B

1

R AI R BI

C

x L/3

A

1 x L/3

R AI

C

V CI

M CI

A

R AI

C

V CI

M CI




Influence lines for deflection and rotation CI , CI

V CI

x L0

M CI

x

2/3

L0

L/3-1/3

2L/9

L/3

A B

x L/3

R AI R BI

C

x

1

L/3-x/3

BMDM

2x/3

x-L/3

A B

x L/3

R AI R BI

C

x

1

L/3-x/3

2x/3

L/3-x

A B

2/3 1/3C

x

1

2L/9

A B

1/L 1/LC

x

1/3

1

-2/3

Actual System I Actual S stem II

Virtual S stem I Virtual System II

BMD

M

BMDM

BMD

M

R AI

x L0

12/3


The deflection CI for x L/3 can be obtained using the unit load method along with the actualsystem I and the virtual system I; i.e.

dx EI

M M L

0

CI

9

2L

L

x

3

2 x

3

L

3

L x

2EI

1

9

2L

3

2

3

L

3

x L

2EI

1

9

2L

3

2

3

2L

3

2x

2EI

1

2 2 9x 5L81EI

x

The deflection CI for x L/3 can be obtained using the unit load method along with the actualsystem II and the virtual system I; i.e.

dx EI

M M L

0

CI

9

2L

2L

x

6

7

3

L x x

3

L

2EI

1

9

2L

3

2

3

L

3

x L

2EI

1

9

2L

3

2

3

2L

3

2x

2EI

1

2 2 9x 18Lx L162EI

L- x

CI

x L0

4L3 /243EI

L/3




The rotation CI for x L/3 can be obtained using the unit load method along with the actualsystem I and the virtual system II; i.e.

dx EI

M M L

0

CI

3

1

L

x

3

2 x

3

L

3

L x

2EI

1

3

1

3

2

3

L

3

x L

2EI

1

3

2

3

2

3

2L

3

2x

2EI

1

2 2 3x L18EIL

x

The rotation CI for x L/3 can be obtained using the unit load method along with the actualsystem II and the virtual system I; i.e.

dx EI

M M L

0

CI

3

2

2L

x

6

7

3

L x x

3

L

2EI

1

3

1

3

2

3

L

3

x L

2EI

1

3

2

3

2

3

2L

3

2x

2EI

1

2 2 3x 6Lx L18EIL

x -L

CI

x L0

-4L2 /162EI

L/3


Example2 : Construct influence lines R AI , M AI , V BI , M BI , BI , BI of a cantilever beam


Influence lines for reactions R AI , M AI

0 M A + 0 (1)(x)M AI

x M AI

0 F Y

+ 0 1R AI

1R AI

1 x

A B

1 x

R AI

M AI

M AI

x L0

R AI

x

-L

1

L0

A B

L/2

EI

L/2

A B

L/2

EI

L/2

1




Influence lines for shear and bending moment V CI , M CI

0 F Y + 0 V BI

0 V BI

0 M B + 0 M BI

0 M BI

0 F Y + 0 V R BI AI

1R V AI BI

0 M B + 0 M M )(L/2)(R BI AI AI

2

L x M

2

LR M AI

AI BI

1

A

x L/2

R AI

M AI B

BV BI M BI

A

R AI

B

V BI M BI

1

A

x L/2

R AI

BM AI M AI


Influence lines for deflection and rotation BI , BI

V BI x L0

M BI x

1

L0

L/2

-L/2

L/2

A

x L/2

R AI

B

x

1

BMDM

-x

Actual S stem I Actual System II

Virtual System I Virtual S stem II

1

M AI A

x L/2

R AI

B

1

BMDM

M AI

A

1

B

x

1

BMD

M

-L/2

-L/2 A

0 B

x

1

BMD

M

1

1

1

-L/2+x

L/2

-L/2

x

-x L/2-x

L/2

M AI x L0

R AI x

11

L0

-L/2




The deflection BI for x L/2 can be obtained using the unit load method along with the actualsystem I and the virtual system I; i.e.

dx EI

M M L

0

BI

2

L

2

1

2

L x -

EI

1

2

L

3

1

2

L

2

L

2EI

1

2

L

3L

2x

3

1 x

2

L x

2

L

2EI

1

2x 3L12EI

x 2

The deflection BI for x L/2 can be obtained using the unit load method along with the actualsystem II and the virtual system I; i.e.

dx EI

M M L

0

CI

2

L

2

1

2

L x -

EI

1

2

L

3

1

2

L

2

L

2EI

1

L6x 48EI

L2

BI

x L0

L3 /24EI

L/2

5L3 /48EI


The rotation BI for x L/2 can be obtained using the unit load method along with the actualsystem I and the virtual system II; i.e.

dx EI

M M L

0

CI

12

L x -

EI

11

2

L

2

L

2EI

1

1 x

2

L x

2

L

2EI

1

2EI

x 2

The rotation BI for x L/2 can be obtained using the unit load method along with the actualsystem II and the virtual system I; i.e.

dx EI

M M L

0

CI

12

L x -

EI

11

2

L

2

L

2EI

1

4x L8EI

L

BI

x L0

-L2 /8EI

L/2

-3L2 /8EI




Example3: Construct influence lines R AI , M AI , R BI , V CI , V BLI , M BLI , V BRI , M BRI , V DI , and M DI of abeam shown below


Influence lines for reactions R AI , M AI , R BI and shear force V CI

From FBD II, we obtain

0 M C + 0 )(L)(R BI

0 R BI

0 F Y + 0 R V BI CI

0 R V BI CI

A B

L/2 L L

1 x

A B

1 x L

R AI

M AI

A B

L L

C

C

R BI

C

R BI

B

C

V CI

FBD I FBD II

D

L/2

L/2 L/2

D


From FBD I, we obtain

0 F Y + 0 1R R BI AI

1R 1R BI AI

0 M A + 0 (1)(x) )(2L)(R M - BI AI

-x x L2R M BI AI

From FBD IV, we obtain

0 M C + 0 L)(1)(x )(L)(R BI

1L

x R BI

0 F Y + 0 1R V BI CI

L

x 2 R 1V BI CI

From FBD III, we obtain

0 F Y + 0 1R R BI AI

L

x 2 R 1R BI AI

A B

1 x L

R AI

M AI

R BI

C

R BI

B

C

V CI

FBD III FBD IV

1




0 M A + 0 (1)(x) )(2L)(R M - BI AI

2L x x -L2R M BI AI

M AI x 0

R AI x

-L

-1

1

L 2L

3L

L

0 L

2L 3L

1

R BI x

2

0 L 2L 3L

1

V CI x

-1

0 L

2L 3L

1


Influence lines for shear and bending moment V BLI , M BLI

0 F Y + 0 R V BI BLI

BI BLI -R V

0 M B + 0 M BLI

0 M BLI

0 F Y + 0 1R V BI BLI

BI BLI R 1V

0 M B + 0 2L)(1)(x M BLI

x 2LM BLI

A B

1 x L

R AI

M AI

R BI

C

R BI

BV BLI M BLI

A B

1 x 2L

R AI

M AI

R BI

C

R BI

BV BLI M BLI

1




Influence lines for shear and bending moment V BRI , M BRI

0 F Y + 0 V BRI

0 V BRI

0 M B + 0 M BRI

0 M BRI

R BI x

2

0 L 2L 3L

1

V BLI x

-1

0 L 2L 3L

-1

M BLI x

-L

0 L 2L 3L

A B

1 x L

R AI

M AI

R BI

C V BRI M BRI


0 F Y + 0 1V BRI

1V BRI

0 M B + 0 2L)(1)(x M BRI

x 2LM BRI

A B

1 x 2L

R AI

M AI

R BI

C V BRI M BRI

1

V BRI x

1

0 L 2L 3L

M BRI x

-L

0 L 2L 3L

1




Influence lines for shear and bending moment V DI , M DI

0 F Y + 0 R V BI DI

BI DI -R V

0 M B + 0 )(3L/2)(R M BI DI

/2 3LR M BI DI

0 F Y + 0 1R V BI DI

BI BLI R 1V

A B

1 x L/2

R AI

M AI

R BI

C

V DI M DI

D

B

R BI

C

D

A B

1 x L/2

R AI

M AI

R BI

C

V DI M DI

D

B

R BI

C

D

1


0 M B + 0 )(3L/2)(R L/2)(1)(x M BI DI

x L/2 /2 3LR M BI DI

Remarks

1. The influences lines of support reactions and internal forces (shear force and bending moment) for statically determinate beams are piecewise linear; i.e.they consists of only straight line segments.

2. The influence functions of the internal forces can be obtained in terms of theinfluence functions of the support reactions; therefore, the influence lines of internal forces can be readily obtained from those for support reactions.

3. The influence lines of the deflection and rotation at any points of the statically determinate beam generally consist of curve segments.

R BI x

2 1

V DI x

-1

0 L 2L

3L

1

M DI x

L/2

1

L/2

-L/2

0 L 2L3L

L/2

0 L 2L 3L








Influence Line for Bending Moment . Let assume that the influence line of the bending at point C, M CI , is to be determined. By applying the principle of virtual work to the actual system with a special choice of the virtual displacement as indicated in the virtual system

3a (the virtual displacement associated with the rigid body motion of the beam resulting from the release of the bending moment constraint at C) , we obtain

)()( x v x v 1 W E ; CI CI I M 1M W

I E W W

)( x v M CI (5)

Muller-Breslau Principle: “The influence line of the shear force at a particular point has anidentical shape to the virtual displacement obtained from releasing the shear constraint at that point and introducing a rigid body motion with unit relative virtual displacement between the two ends of the shear release with their slope remaining the same.”

1

Virtual System 3a

Actual system

x

v ( x )

R AI

R BI

M AI A B

C

V CI M CI

RELEASE moment constraint

1

M CI x 1

1


Example4: Use Muller-Breslau principle to construct influence lines R AI , R DI , R FI , V BI , V CLI ,V CRI , V DLI , V DRI ,V EI , M BI , M DI , and M EI of a statically determinate beam shown below

Solution The influence line of the support reaction R DI is obtained as follow: 1) release thedisplacement constraint at point D, 2) introduce a rigid body motion, 3) impose unitdisplacement at point D, and 4) the resulting virtual displacement is the influence line of R DI .

The value of the influence line at other points can be readily determined from the geometry,for instance,

3/2 L3L/2 1h2 )/())((

1/2 LL/2 1h3 )/())((

3/4L/2 L/43/2 h1 )/())((

A D

L/4 L/2 L/2

C B

L/4

E F

L/2

A

D

L/4 L/2 L/2

C B

L/4

E F

L/2

RELEASE displacement constraint

1

R DI x

1h1=3/4

h2 =3/2

h3=1/2




The influence line of the shear force V EI is obtained as follow: 1) release the shear constraintat point E , 2) introduce a rigid body motion, 3) impose unit relative displacement at point E and 4) the resulting virtual displacement is the influence line of V EI .


4343 hh2 Lh2 Lh )//()//(

1/2 h12hhh1hh 444434 )(

1/2 hh 43

1/2 L/2 L/2 hh 32 )/())((

1/4L/2 L/4hh 2 1 )/())((

A D

L/4 L/2 L/2

C B

L/4

E F

L/2

RELEASE shear

constraint

V EI x

1h1=1/4

h2 =1/2 h4=1/2

1

h3=-1/2


The influence line of the bending moment M EI is obtained as follow: 1) release the bendingmoment constraint at point E , 2) introduce a rigid body motion, 3) impose unit relative rotationat point E without separation and 4) the resulting virtual displacement is the influence line of

M EI .


L/4h12 Lh2 Lh 333 )//()//(

L/4L/2 L/2 hh 32 )/())((

L/8 L/2 L/4hh 2 1 )/())((

The rest of the influence lines can be determined in the same manner and results are givenbelow.

A D

L/4 L/2 L/2

C B

L/4

E F

L/2

RELEASE moment constraint

M EI x

h1=-L/8

h2 =-L/4

h3=L/4

1

1




A D

L/4 L/2 L/2

C

B

L/4

E F

L/2

R AI x

1

R FI x

1/2

1

1/2

1/2 1/4

V BI x

1/2

-1/2

V CLI x

-1-1/2

V CRI x

-1-1/2

V DLI x

-1

-1/2

-1

V DRI x

1/2 1/4

1

M BI x

L/8

M DI x

L/4

L/8

1 x


Example5 : Use Muller-Breslau principle to construct influence lines R AI , M AI , R DI , V BI , V CI ,V DI ,V ELI ,V ERI , M BI , M DI , and M EI of a statically determinate beam shown below.

Solution By Muller-Breslau principle, we obtain the influence lines as follow: 1) release theconstraint associated with the quantity of interest, 2) introduce a rigid body motion, 3) imposeunit virtual displacement/rotation in the direction of released constraint, and 4) the resulting

virtual displacement is the influence line to be determined. It is noted that values at points onthe influence line can be readily determined from the geometry.

A D

L/4 L/2 L/2

C B

L/4

E F

L/2

A D

L/4 L/2 L/2

C B

L/4

E F

L/2

1 x

R AI x

1 1

1/2

-1/2

M AI x

L/2

L/4

-L/4

L/4

1

x R EI

1

1/2

3/2

1

1/2

-1/2

1

V BI x

V CI x

1

1/2

-1/2




A D

L/4 L/2 L/2

C B

L/4

E F

L/2

1 x

-1/2

V DI x

1/2

-1/2

x

-1/2 -1

-1/2

V ELI

x

1

V ERI

1

x

-L/4

-L/8

L/8

M BI

x M DI

L/4

-L/4

x

-L/2

M EI


Example6 : Use Muller-Breslau principle to construct influence lines R AI , M AI , R DI , R FI , V BI ,V CI , V DLI , V DRI ,V EI , M BI , and M DI of a statically determinate beam shown below.

Solution By Muller-Breslau principle, we obtain the influence lines as follow: 1) release theconstraint associated with the quantity of interest, 2) introduce a rigid body motion, 3) imposeunit virtual displacement/rotation in the direction of released constraint, and 4) the resulting

virtual displacement is the influence line to be determined. It is noted that values at points onthe influence line can be readily determined from the geometry.

A D

L/4 L/2 L/2

C B

L/4

E F

L/2

A D

L/4 L/2 L/2

C B

L/4

E F

L/2

1 x

R AI x

1 1

-1

M AI x

L/2

-L/2

L/4

1

x R EI

1

2

x

R EI

1




A D

L/4 L/2 L/2

C B

L/4

E F

L/2

1 x

V BI x

1

-1

V CI x

1

-1

1

V DLI x

-1 -1

V DRI x

1 1

V EI x

1

M BI x

-L/4

L/4

M BI

-L/2

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