บทที่ 6 moment distribution methodcivil.in.th/cen321/cen321-06t.pdfmoment distribution...

Moment Distribution Method 1

FABBAAB ML

32LEI2

M

+= )(

FBABABA ML

32LEI2

M

+= )(

6 Moment Distribution Method Indeterminate Beam Rigid Frame Slope-Deflection

4 1) Fixed End Moments2) Rotation A 3) Rotation B 4) Relative Translation A B


Moment DistributionLocking Joint

Fixed End Moments (MFAE ,MFEA ,MFED ,.... )

Unlocking Unbalanced Moment E (A ,B ,C D ) Carry-Over Moment

2

B

A C

D

EMFAE

MFDE

MFCEMFEA

MFED

MFEA+MFEB+MFBC+MFED

B

A C

D

E


Moment Distribution

Moment Slope-Deflection Moment

Rotation

Fixed End Moments ( Slope-Deflection )

+ -

+-


Fixed-End MomentsP

L

L/28

PL8

PL WL12

WL2

12WL2

W

L

a

)( 2222

a3aL8L6L12

Wa+ )( a3L4

L12Wa

2

3

P

L

a2

2

LPab b

2

2

LbPa

ML

a)( a2b

LMb

2 b

)( ab2LMa

2

WL20

WL2

30WL2

W

L96WL5 2

96WL5 2

L/2

1

3

5

2

4

6

7


Distribution Factor Distribution Joint

MFEDM

B

A C

D

E MFEC

MFEB

MFEA

Frame

Joint E

()

E

M

B

A C

D

E

B

A C

D

E


Slope-Deflection

EEAEEAEA 4K4LEI

M == )(

EECEECEC 4K4LEI

M == )(

EEBEEBEB 4K4LEI

M == )(

EEDEEDED 4K4LEI

M == )(


Joint E

K4KKKK4MMMMM EEDECEBEAEEDECEBEA =+++=+++= )(

MK

KM

KK

MK

KM

KK

K4M EDECEBEA

E )()()()( +

+

+

=

=

EDECEBEA MMMMM +++=

Distributed Moments

Distribution Factor

DEA DEB DBC DED Joint E Distribution Factor


Carry-Over Factor Carry-Over Moment (A) (B)

Slope-Deflection

MBAMAB A BA

AAB LEI

4M = ABA LEI

2M =

ABAB M21

M =

Carry-Over Moment

Carry-Over Factor (CBA) (B) Fixed End


EX.1 ( 1Joint) Moment Distribution

B

A C

D

E3I

2I

4II

1.2 k/ft.

10 ft10 ft

10 ft

10 ft


1) Fixed End Moments (FEM)

2) Distributed Moments (DM)

Fixed End Moment

Distributed Moments

D

B

A CE10 k-ft. .

.ftk10

121021

12WL

FEMFEM22

AEEA =

===

100.1100.210

0.310

0.410

)( 0FEM =

EI2010EI2

LEI

KEI1010EI

LEI

K EBEA .;. ======

EI4010EI4

LEI

KEI3010EI3

LEI

K EDEC .;. ======

EIEI40EI30EI20EI10K =+++= ....

10EIEI10KKDF EAEA ././ ===

40DF30DF20DF EDECEB .;.;. ===


2) Distributed Moments (DM) - Unbalance Moment

E = (0.110) + (0.210) + (0.310) + (0.410) = -10 ()

0010DFUMDM AEAE === 11010DFUMDM EAEA === .

0010DFUMDM BEBE === 22010DFUMDM EBEB === .

0010DFUMDM CECE === 33010DFUMDM ECEC === .

0010DFUMDM DEDE === 44010DFUMDM EDED === .

DFMomentUnbalanceDM =


3) Carry-Over Moments (COM)COM = DM () / 2

4) ()

Carry-Over Moments

DF 0 0.1 0 0.2 0 0.3 0 0.4 AE EA BE EB CE EC DE ED1) FEM -10 10 0 0 0 0 0 02) DM 0 -1 0 -2 0 -3 0 -43) COM -0.5 0 -1 0 -1.5 0 -2 04) -10.5 9 -1 -2 -1.5 -3 -2 -4

B

A C

D

1 2

342

0.5

1.51

DF= 0.1+0.2+0.3+0.4 = 1


EX.2 Moment Distribution( EI )

B C

60 ft. 20 ft. 20 ft.

A1 k/ft. 20 k



Fixed End Moments2) Distributed Moments (DM)

- Distribution Factor (DF)

Distributed Moments

B CA300 k-ft. 300 k-ft. 100 k-ft. 100 k-ft.

.ftk30012601

12WL

FEMFEM22

ABBA =

===

.ftk1008

40208

PLFEMFEM CBBC =

===

EI025040EI

KEI01667060EI

K BCBA .;. ====

EI041670EI0250EI016670K ... =+=

)(Fixed0DFAB =

200

BCA

80 12060 ft. 20 ft. 20 ft.

40EI041670EI016670KKDF BABA .././ ===

)(Fixed0DFCB =

60EI041670EI02570KKDF BCBc .././ ===



A ; (-300) = 300 ()

B ; (300-100) = -200 ()

C ; 100 = -100 ()

00300DFUMDM ABAAB ===

8040200DFUMDM BABBA === .

12060200DFUMDM BCBBC === .

00100DFUMDM CBCCB ===



4) ()

DF 0 0.4 0.6 0 AB BA BC CB1)FEM -300 300 -100 1002)DM 0 -80 -120 03)COM -40 0 0 -604) -340 220 -220 40

B CA340 220 220 40

B CA40 80 120 60

Carry-Over Moments


EX.3 A 0.0016 Rad. Moment Distribution ( EI = 10,000 k-ft2.)

B C

A

10 ft.

20 ft.

2II =0.0016 Rad.


1) = 0.0016 Rad. FEM

2) Distributed Moment (DM)


..., ftk4610

00160000104LEI4

FEMAB =

=

=

..., ftk2310

00160000102LEI2

FEMBA =

=

=

B

A

10 ft.=0.0016 Rad.

3.2

6.4

EI1020EI2

KEI1010EI

K BCBA .;. ==== EI20EI10EI10K ... =+=

)(Fixed0DFAB = 50EI20EI10KKDF BABA .././ ===

)(Fixed0DFCB = 50EI20EI10KKDF BCBC .././ ===



A ; 6.4 = -6.4 ()

B ; (3.2 - 0) = -3.2 ()

C ; 0

0046DFUMDM ABAAB === .

615023DFUMDM BABBA ... ===

615023DFUMDM BCBBC ... ===

0DFUMDM CBCCB ==



4) ()

DF 0 0.5 0.5 0 AB BA BC CB1)FEM 6.4 3.2 0 02)DM 0 -1.6 -1.6 03)COM -0.8 0 0 -0.84) 5.6 1.6 -1.6 -0.8

BC

A5.6

1.6 0.8


Joint Joint Moment Distribution

FEM (Locking) DM (Unlocking) COM ()

Joint 1 Joint Moment Distribution Carry-Over Moment

FEM DM COM COM ()


EX.4 Moment Distribution ( E I )

BC

4 m. 2 m. 2 m.

A 300 kg/m.800 kg

EI EI




BC

4 m. 2 m. 2 m.

A300 kg/m.

800 kg

EI EI

-400 400 -800 800 .mkg40012

430012WL

FEM22

AB =

==

.mkg8008

480012

43008PL

12WL

FEM22

BC =

==

.mkg40012

430012WL

FEM22

BA =

==

.mkg8008

480012

43008

PL12WL

FEM22

CB =

+

=+=

EI2504EI

KEI2504EI

K BCBA .;. ==== EI50EI250EI250K ... =+=


)(Hinge1DFCB = 50EI500EI250KKDF BCBc .././ ===



A ; 400 = -400 ()

B ; (400-800) = -400 ()

C ; 800 = -800 ()

00400DFUMDM ABAAB ===

20050400DFUMDM BABBA === .

20050400DFUMDM BCBBC === .

8001800DFUMDM CBCCB ===


DF 0 0.5 0.5 1FEM -400 400 -800 800DM 0 200 200 -800COM 100 0 -400 100DM 0 200 200 -100COM 100 0 -50 100DM 0 25 25 -100COM 12.5 0 -50 12.5DM 0 25 25 -12.5COM 12.5 0 -6.25 12.5DM 0 3.125 3.125 -12.5COM 1.563 0 -6.25 1.563DM 0 3.125 3.125 -1.563COM 1.563 0 -0.781 1.563DM 0 0.391 0.391 -1.563 -172 857 -857 0


4) ()

BA

300 kg/m.172 857

B C

300 kg/m.800 kg

857


EX.5 Moment Distribution (SFD) (BMD)( E I )

B

C

A3 m.

6 m.

2EIEI3 T/m.

2.5 T

3EI

6 m.

DE

6 T3 m.

3 m.

2EI




B

C

A3 m.

6 m.

2EIEI

3 T/m. 2.5 T

3EI

6 m.

DE

6 T3 m.

3 m.

2EI.// mT9126312WLFEM 22BD ===

..// mT548668PLFEMCB ===.// mT9126312WLFEM 22DB ===

..// mT548668PLFEMBC ===... mT57352PLFEMDE ===

0FEMFEM BAAB ==

EI50006EI3

KEI333306EI2

KEI166706EI

K BCBDBA .;.;. ======

EIEI5000EI3330EI1670K =++= ...

)(Fixed0DFAB =

1670KKDF BABA ./ == 5000KKDF BCBC ./ ==3330KKDF BDBD ./ ==

)(Fixed0DFCB = )(Cantiliver0DFED = )(Hinge1DFDB =


A B C D AB BA BD BC CB DB DEDF 0 0.167 0.333 0.5 0 1 0FEM 0 0 -9 4.5 -4.5 9 -7.5DM 0 0.75 1.5 2.25 0 -1.5 0

COM 0.35 0 -0.75 0 1.125 0.75 0DM 0 0.125 0.25 0.375 0 -0.75 0COM 0.063 0 -0.35 0 0.188 0.125 0DM 0 0.058 0.117 0.275 0 -0.125 0COM 0.029 0 -0.063 0 0.088 0.059 0DM 0 0.011 0.021 0.032 0 -0.059 0COM 0.006 0 -0.03 0 0.016 0.011 0DM 0 0.005 0.01 0.015 0 -0.011 0 0.45 0.95 -8.30 7.35 -3.08 7.5 -7.5


(BMD) (SFD)

B

C

AD E

0.45

-0.95-8.30

-3.08

-7.35

-7.50

(T.-m.)

B

C

A D E-0.23

-0.23

-3.71

9.13

-8.87

2.5

(T.)


Modified Stiffness Factor (K) (Far End) 4 1) Stiffness Factor Fixed ()

KAB = EI/L ( Stiffness Factor )

)( AAB 2LEI2

M = ALEI

4 = )(BA


2) Stiffness Factor Hinge (Modified Stiffness Factor K)

KAB = 3EI/4L (Modified Stiffness Factor Hinge)

3) Stiffness Factor (B = - A)

KAB = EI/2L (Modified Stiffness Factor B = - A)

)( BAAB 2LEI2

M +=BA

2 AB /=

)( BAAB 2LEI2

M = AL2EI1

4 = )(

AL4EI3

4 = )(

BABA


4) Stiffness Factor (B = A)

KAB = 3EI/2L (Modified Stiffness Factor B = A)

)( BAAB 2LEI2

M += AL2EI3

4 = )(

BA BA=


EX.6 4 Moment Distribution( E I )

BC

4 m. 2 m. 2 m.

A 300 kg/m.800 kg

EI EI


2) BC ( Hinge)

DF 0 0.571 0.429 1 AB BA BC CBFEM -400 400 -800 800DM 0 228.4 171.6 -800COM 114.2 0 -400 0DM 0 228.2 171.6 0COM 114.2 0 0 0DM 0 0 0 0 -171.6 856.8 -856.8 0

4EI

LEI

KBA == 16EI3

L4EI3

KBC ==

BC

4 m. 2 m. 2 m.A

300 kg/m.800 kg

EI EI-400 400 -800 800

3

16EI7

16EI3

4EI

K =+=


EX.7 Moment Distribution ( EI )

B C

A1.2 k/ft.

12 k

D

12 k

10 ft.10 ft. 10 ft. 10 ft.20 ft.


- Fixed End Moments (FEM)

DF 0 0.667 0.333 AB BA BCFEM -30 30 -40DM 0 6.67 3.33COM 3.33 0 0DM 0 0 0 -26.27 36.67 -36.67 36.67 -36.67 26.67

.// ftk30820128PLFEMAB ===

././ ftk4012202112WLFEM 22BC ===

./ ftk3082012FEMBA ==

./. ftk40122021FEM 2CB ==

.// ftk30820128PLFEMCD === ./ ftk3082012FEMDC ==

40EI

340EI

20EI

K20EI

KBA =+== ,



DC

1.2 k/ft. 1.2 k/ft. 1.2 k/ft.

A B E F

24 ft. 24 ft. 24 ft. 24 ft. 24 ft.



- Stiffness Factor (K) Distribution Factor (DF) B

C

.././ ftk65712242112WLFEMFEM 22BAAB ====

96EI7

24EI

32EI

K24EI

LEI

K32EI

L4EI3

K BCBA =+===== ,,

.././ ftk65712242112WLFEMFEM 22DCCD ====

.././ ftk65712242112WLFEMFEM 22FEEF ====

16EI

48EI

24EI

K48EI

242EI1

K24EI

LEI

K CDCB =+===== ,,

5710KKDF4290KKDF BCBCBABA ./;./ ====

3330KKDF6670KKDF CDCDCBCB ./;./ ====


DF 1 0.429 0.571 0.667 0.333FEM -57.6 57.6 0 0 -57.6DM 57.6 -24.7 -32.9 38.4 19.2COM 0 28.8 19.2 -16.5 0DM 0 -20.6 -27.4 -11 5.5COM 0 0 5.5 -13.7 0DM 0 -2.36 -3.14 9.14 4.56COM 0 0 4.57 -1.57 0DM 0 -1.96 -2.61 1.05 0.52COM 0 0 0.53 -1.31 0DM 0 -0.23 -0.30 0.87 0.44COM 0 0 0.44 -0.15 0DM 0 -0.19 -0.25 0.10 0.05COM 0 0 0.05 -0.13 0DM 0 -0.02 -0.03 0.09 0.04 0 36.3 -36.3 27.3 -27.3


Moment Distribution Joint

Joint(=0)

Frame Joint

Frame (Sway)


Superposition ( Joint Translation) 1) Joint Translation R Joint- Moment Distribution R 2) R R- Joint

R

R=

+


R Joint Joint Translation

R Joint Translation

1) Joint Translation Joint- F.E.M. Lock Joint

2) Joint Rotation 1)- Unlock Joint Moment Distribution

Note - Moment Distribution F.E.M. Joint Translation

=

+

R1

R2


EX.9 Moment Distribution ( E I )

3EIa

Wb

L

EIc

L


1) Joint Translation

ac ca cb bcDF 0 0.75 0.25 0FEM -0.0833 0.0833 -0.0833 0.0833 WL2

DM 0 0 0 0 -0.0833 0.0833 -0.0833 0.0833 WL2

R

wL/2wL/2

w

wL/2

-0.0833wL2 wwL/2 wL/2

0.0833wL20.0833wL2

bccbcaac FEMFEMFEMFEM ===

222 WL121

12242112WL === /./

LEI4

KLEI

KLEI3

K cbca === ;;

R3EIa

Wb

L

EIc

L

)(Fixed0DFac =

750LEI4LEI3DFca .)//()/( ==

)(Fixed0DFbc =

250LEI4LEIDFcb .)//()/( ==

- Fixed End Moment (FEM)

- Stiffness Factor (K)



WL3EIa W b

L

EIc

L

22ac LEI18

LEI36

FEM

=

=)(

2) R Joint-Translation Fixed-End Moment Moment Distribution

- Fixed End Moment (FEM) ac ca cb bcDF 0 0.75 0.25 0FEM -18 -18 6 6 EI/L2

COM 4.5 0 0 1.5DM 0 0 0 0 -13.5 -9 9 7.5 EI/L2

22ca LEI18

LEI36

FEM

=

=)(

2cb LEI6

FEM

= 2bc LEI6

FEM

=

3EI

a b

L

EIcL


Fy = 0

ac ca cb bc -13.5 -9 9 7.5 EI/L2 = 1 / 39(1) -0.0833 0.0833 -0.0833 0.0833(2) -0.3462 -0.2308 0.2308 0.1923

M -0.4295 -0.1475 0.1475 0.2756 WL2

WL3EI

a22.5EI/L2

b

L

EIc

L

16.5EI/L22LEI516522WL /)..( +=

39WL

LEI 2

2 =


EX.10 Joint Moment Distribution( E I )

9 kN.

EI

8/3EI

6 m.

6 m.

A

B C


- 9 kN. B

- C BC Roller

- Fixed End Moment (FEM)

=1.

6 m.

6 m.

RC

A

B

6 m.

6 m.

9 kNC

A

B

;)(6EI

LEI

K ABBA ==

6EI636EI2FEMFEM BAAB /)/(/ ===

9EI4

63EI8

LEI

K BCBC === )(

3EI

9EI4

43

KBC ==

0FEMFEM CBBC ==

2EI

3EI

6EI

K =+=



AB BA BC CBDF 0 0.333 0.667 1FEM -0.1667 -0.1667 0 0 EIDM 0 0.0556 0.1111 0COM 0.0278 0 0 0DM 0 0 0 0 -0.1389 -0.1111 0.1111 0 EI


)(Hinge1DFCB = 6670EI500EI3330KKDF BCBC .././ ===


- B (Fx=0)0.0417EI = 9 kN. EI = 216 ( EI )

AB BA BC CB

-0.1389 -0.1111 0.1111 0 EI = 216M -30 -24 24 0

9 kN.0.1111EI

0.1111EI

0.1389EIA

BC

0.0417EI

0.1417EI

0.1111EI

0.1389EIA

B0.1111EI

0.0185EI 0.0185EI

B C


EI = Moment-Distribution 9 kN. Factor

9 kN.24

24

30A

B C


EX.11 Moment Distribution( E I )

600 kg/m

EI

2EI

2.5 m.

1.5 m.300 kg

1.5 m.

1.0 m.A

B C

D

1.5EI



- Stiffness Factor (K) Distribution Factor (DF) B ; ;)(

3EI

LEI

K BABA ==

.. mkg51128

33008PL

FEMAB =

=

=

.. mkg51128

33008

PLFEMBA =

==

... mkg531212

5260012WL

FEM22

BC =

=

=

... mkg531212

5260012WL

FEM22

CB =

==

5EI4

52EI2

LEI

K BCBC === .)(

15EI17

5EI4

3EI

K =+=

600 kg/m

EI

2EI

2.5 m.

1.5 m.300 kg

1.5 m.

1.0 m.A

B C

D

1.5EI

)(Fixed0DFAB = 294015EI17

3EI

KK

DF BABA ./ === 7060

15EI17

5EI4

KK

DF BCBC ./ ===


C ;

1) Moment Distribution Joint Translation

600 kg/m

300 kg

A

B C

D

5EI4

52EI2

LEI

K CBCB === .)(

32EI9

44EI513

L4EI3

K CDCD =

==.)(

160EI173

32EI9

5EI4

K =+=

)(Hinge1DFDC =7400160EI173

5EI4

KK

DF CBCB ./ === 2600

160EI173

32EI9

KK

DF CDCD ./ ===


A B C D AB BA BC CB CD DCDF 0 0.294 0.706 0.740 0.260 1FEM -112.5 112.5 -312.5 312.5 0 0DM 0 58.8 141.2 -231.2 -81.3 0

COM 29.4 0 -115.6 70.6 0 0DM 0 34.0 81.6 -52.2 -18.4 0 COM 17 0 -26.1 40.8 0 0DM 0 7.67 18.43 -30.2 -10.6 0COM 3.84 0 -15.1 9.22 0 0DM 0 4.44 10.66 -6.82 -2.40 0COM 2.22 0 -3.41 5.33 0 0DM 0 1.00 2.41 -3.94 -1.39 0COM 0.50 0 -1.97 1.21 0 0DM 0 0.58 1.39 -0.80 -0.31 0COM 0.29 0 -0.45 0.70 0 0DM 0 0.13 0.32 -0.52 -0.18 0(1) -59.3 219.1 -219.1 114.6 -114.6 0


(H) = 300-96.7-28.7= 174.6 kg.

()

D 28.7

28.7600 kg/m

C701.8

B708.2

114.6

300

A

B

97.6

203.3

59.3

219.1


2) H Joint-Translation Fixed-End Moment

Moment Distribution


A

B C

D

3EI2

33

3EI2

FEMFEM BAAB

=

== )(

0FEMFEM CBBC ==

16EI9

43

4EI51

2FEMFEM DCCD

=

== ))(.(

174.6 kg

A

B C

D


AB BA BC CB CD DCDF 0 0.294 0.706 0.740 0.260 1FEM -0.6667 -0.6667 0 0 -0.5625 -0.5625 EIDM 0 0.196 0.471 0.4163 0.1463 0.5625

COM 0.098 0 0.2082 0.2355 0.2813 0DM 0 -0.0612 -0.1470 -0.3824 -0.1344 0COM -0.0306 0 -0.1912 -0.0735 0 0 DM 0 0.0562 0.1350 0.0544 0.0191 0COM 0.0281 0 0.0272 0.0675 0 0DM 0 -0.0080 -0.0192 -0.0500 -0.0175 0COM -0.0040 0 -0.0250 -0.0096 0 0DM 0 0.0074 0.0176 0.0071 0.0025 0COM 0.0037 0 0.0036 0.0088 0 0DM 0 -0.0011 -0.0025 -0.0065 -0.0023 0COM -0.0006 0 -0.0033 -0.0013 0 0DM 0 0.0010 0.0023 0.0010 0.0003 0(2) -0.5721 -0.4764 0.4764 0.2673 -0.2672 0 EI


- EI (Fx =0)0.3495EI + 0.0668EI = 174.6EI = 174.6/0.4163 = 419.4 EI = 419.4 (2)

(1) -59.3 219.1 -219.1 114.6 -114.6 0(2) -0.5721 -0.4764 0.4764 0.2673 -0.2672 0 EI = 419.4(2) -239.9 -199.8 199.8 112.1 -112.1 0

AB BA BC CB CD DC(1)+(2) -299.2 19.3 -19.3 226.7 -226.7 0 kg.-m.

174.6 kg

A

BC

D 0.0668EI

0.3495EI



100 kg

A

B C

D

3 m. 5 m. 3 m.

4 m.

4 m.


1) Joint Translation- Fixed End Moments- H = 100 kg.

2) Joint Translation-

100 kg

A

B C

D

3 m. 5 m. 3 m.

4 m.

4 m.H=100 kg

100 kg

A

B C

D

Joint Translation

=

== EI256

53

5EI2

FEMFEM BAAB )(

0FEMFEM CBBC ==

=

== EI256

53

5EI2

FEMFEM DCCD

45

SIN

=

=



B ;

C ;


;)(5EI

LEI

K BABA == ;)( 5EI

LEI

K BCBC == 5EI2

5EI

5EI

K =+=

;)(5EI

LEI

K CBCB == ;)( 5EI

LEI

K CDCD == 5EI2

5EI

5EI

K =+=

)(Fixed0DFAB = 50EI400EI20DFDF BCBA ../. ===

)(Fixed0DFDC = 50EI400EI20DFDF CDCB ../. ===


AB BA BC CB CD DCDF 0 0.5 0.5 0.5 0.5 0FEM -0.24 -0.24 0 0 0.24 0.24DM 0 0.12 0.12 -0.12 -0.12 0COM 0.06 0 -0.06 0.06 0 -0.06DM 0 0.03 0.03 -0.03 -0.03 0COM 0.015 0 -0.015 0.015 0 -0.015DM 0 0.0075 0.0075 -0.0075 -0.0075 0 COM 0.0038 0 -0.0038 -0.0038 0 -0.0038DM 0 0.0019 0.0019 -0.0019 -0.0019 0 COM 0.0009 0 -0.0009 0.0009 0 -0.0009DM 0 0.0004 0.0005 -0.0005 -0.0004 0COM 0.0002 0 -0.0003 0.0003 0 -0.0002DM 0 0.00015 0.00015 -0.00015 -0.00015 0

-0.1601 -0.0801 0.0801 -0.0801 0.0801 0.1601 EI


- EI (Fx=0)100 = 0.0601EI 2 EI = 831.9 EI = 831.9

-0.1601 -0.0801 0.0801 -0.0801 0.0801 0.1601 EI = 831.9

-

AB BA BC CB CD DCM -133.2 -66.6 66.6 -66.6 66.6 133.2 kg.-m.

100 kg B CD

A0.0601EI

0.0601EI


Joint Translation 1 Degree of Freedom Moment Distribution

n DOF. Joint Translation n+1 case

Case 1 Joint Translation Rotation Case 2 (n+1) Joint Translation Case 1 DOF.

Lock Joint Translation Case


3 3 DOFS Joint Translation

F1

F2

F3

Case 1 Lock DOF


F1 = x1 + r12 x2 + r13 x3F2 = r21 x1 + x2 + r23 x3F3 = r31 x1 + r32 x2 + x3

r31 x1r21 x1

x1

Case 2 Lock DOF 1

r32 x2

r12 x2

x2

Case 3 Lock DOF 2

r23 x3r13 x3

x3

Case 4 Lock DOF 3

x1 , x2 x3


EX.13 Moment Distribution (EI )

50 k

100 k 20 ft.

20 ft.

10 ft.

6 ft.4 ft.

6 ft.4 ft.

A

B

C D

E

F


- 2 DOF Joint Translation 3 Case (n+1)

Case 1) DOF Joint Translation Lock - Fixed End Moments (FEM)

.ftk14410

64100L

PabFEM 2

2

2

2

BE =

==

0FEM =

.ftk9610

64100L

bPaFEM 2

2

2

2

EB =

==

.ftk7210

6450L

PabFEM 2

2

2

2

CD =

==

.ftk4810

6450L

bPaFEM 2

2

2

2

EB =

==

50 k

100 k 20 ft.

20 ft.

10 ft.

6 ft.4 ft.

6 ft.4 ft.

A

B

CD

E

F


- Distribution Factor (DF) B ;

E ; C ;

D ;

Moment Distribution

;)(20EI

LEI

K BABA == ;20EI

KBC = 5EI1

10EI

20EI

20EI

K =++=;10EI

KBE =

500DF250DF250DF BEBCBA .;.;. ===

500DF250DF250DF EBEDEF .;.;. ===;)(

20EI

LEI

K BACB == ;10EI

KCD = 20EI3

10EI

20EI

K =+=

6670DF3330DF CDCB .;. ==

6670DF3330DF DCDE .;. ==

50 k

100 k

43.6

20.4

52.6

40.893.4

34.1

41.2

31.115.6

3.06 2.345

1.045

0.3372.5


Case 2) Joint B E Lock - Fixed End Moments (FEM)

- Distribution Factor (DF) Case 1 Moment Distribution

200EI3

203

20EI2

FEMFEMFEMFEM EFFEBAAB

=

==== )(

0FEM =

20 ft.

20 ft.

10 ft.

A

B

C D

E

F

x1200EI3

203

20EI2

FEMFEMFEMFEM DEEDCBBC

=

==== )(

2.07x10.455x1

2.07x1

0.272x1 0.273x1

0.273x10.273x1

2.70x1 2.70x1

0.21x12.49x12.07x1

2.49x1x1


Case 3) Joint C D Lock - Fixed End Moments (FEM)

- Distribution Factor (DF) Case 1 Moment Distribution

0FEM =

200EI3

203

20EI2

FEMFEMFEMFEM DEEDCBBC

=

==== )(20 ft.

20 ft.

10 ft.

A

B

CD

E

F

x2

4.90x2

1.19x2

4.90x2

0.095x2 0.095x2

0.635x20.635x2

1.27x2 1.27x2

3.82x25.08x2

2.07x25.08x2

x2


- 3

D 1.045 = -0.455X1 + X2 E 0.330 = -X1 + 1.19X2

X1 Case 2 X2 Case 3 Case 1 , Case 2 Case 3

2 1 3

21AB X6350X732420M ... +=

.. ftk2116M AB =

X1 = 1.99 kX2 = 1.95 k


EX.14 Moment Distribution1) B2) A , B C ( E I )

6 m. 4 m.

3EIA

20 kN

B EIC

3 m.


1) Joint Translation B- Fixed End Moments (FEM)


FEM


3EIA

20 kNB

6 m.EI C

4 m.

3 m.

0FEMFEM BAAB ==

.. mkN7534

3120L

bPaFEM 2

2

2

2

CB =

==

.. mkN25114

3120L

PabFEM 2

2

2

2

BC =

==

EI50KBA .= EI250KBC .=EI750K .=

6670DFBA .= 3330DFBC .=

PB C

Lba


AB BA BC CBDF 0 0.667 0.333 0FEM 0 0 -11.3 3.8DM 0 7.5 3.8 0

COM 3.8 0 0 1.9DM 0 0 0 0 3.8 7.5 -7.5 5.6

1 B

2 Joint Translation

BABCB VVR =

BCCBBCBC L320MMV /)( ++=

.).( mkN317RB =

ABBAABCB LMMV /)( +=

RB

VBA B VBC

A

RBB

C= EI50FEMAB .= EI50FEMBA .= EI3750FEMBC .= EI3750FEMCB .


AB BA BC CBDF 0 0.667 0.333 0

FEM -0.5 -0.5 0.375 0.375DM 0 0.083 0.042 0

COM 0.042 0 0 0.021DM 0 0 0 0 -0.458 -0.417 0.417 0.396 EI(2) -22.8 -20.7 20.7 19.7

(1)+(2) -19 -13.2 13.2 25.3

RB = 0.349EI EI = 17.3 / 0.349 = 49.7 / EI = 49.7 / EI

A = -19.0 kN-m. B = 13.2 kN-m. C = -25.3 kN-m.


EX.15 Moment Distribution( EI)

1.2 k/ft.

24 ft. 24 ft. 24 ft. 24 ft.

DCA B E


- C Joint C C= 0 Fixed Support


- Stiffness Factor (K) Distribution Factor (DF)

1.2 k/ft.

24 ft. 24 ft.

CAB

../ ftk65712WLFEMFEM 2BAAB ===

96EI7

24EI

32EI

K24EI

LEI

K32EI

L4EI3

K BCBA =+===== ,,

../ ftk65712WLFEMFEM 2DCCD ===

5710KKDF4290KKDF BCBCBABA ./;./ ====

B ;


AB BA BC CB CD DC DE EDDF 1 0.429 0.571 0FEM -57.6 57.6 -57.6 57.6DM 57.6 0 0 0COM 0 28.8 0 1.9DM 0 -12.36 -16.44 0COM 0 28.8 0 -8.22DM 57.6 0 0 0 0 74.04 -74.04 49.38 -49.38 74.04 -74.04 0


-

1.2 k/ft.

24 ft. 24 ft. 24 ft. 24 ft.

DCA B E1.2 k/ft.

CB CD 0 Simply Support




1.2 k/ft.

DCA B E24 ft. 24 ft. 24 ft. 24 ft.

1.2 k/ft.

CB CD 0

../ ftk65712WLFEMFEM 2CBBC ===

../ ftk65712WLFEMFEM 2DCCD ===

1.2 k/ft.

24 ft. 24 ft.

CAB

16EI

32EI

32EI

K32EI

L4EI3

K32EI

L4EI3

K BCAB =+===== ;;

50161321DFDF CBBC .)//()/( ===

B ;

)(Hinge1DFAB =


AB BA BC CB CD DC DE EDDF 0.6 0.46 0.23 1FEM 0 0 -57.6 57.6DM 0 28.8 28.8 0COM 0 0 0 0DM 0 0 0 0 0 28.8 -28.8 57.6 -57.6 -28.8 28.8 0


EX.16 Moment Distribution( ACI)

2 k/ft. CA B4 k/ft. 2 k/ft.

4 k/ft.4 k/ft. 2 k/ft.

D

EF G

H

I J K L

EI

EI

EIEI

EI EI

EI

EI

2EI

2EI2EI

2EI2EI

2EI

16 ft. 16 ft. 16 ft.

12 ft.

12 ft.


-



2 k/ft. CA B4 k/ft. 2 k/ft.

D

EEI

2EI 2EI2EI

16 ft. 16 ft. 16 ft.

12 ft.FEI

GEI

HEI

../ ftk674212162FEMFEMFEMFEM 2DCCDBAAB =====

60DF24EI5

8EI

12EI

K8EI

16EI2

K12EI

K ABABAE .;; ==+====

460481381DFBA .)//()/( ==

A ;

../ ftk338512164FEMFEM 2CBBC ===

48EI13

K12EI

K16EI

32EI2

K8EI

K BFBCBA ===== ;;; B ;

2304813161DFBC .)//()/( ==


AB BA BCDF 0.6 0.46 0.23FEM -42.67 42.67 -85.33 DM 25.60 19.62 9.81COM 9.81 12.80 0DM -5.89 -5.89 -2.49COM -2.49 -2.49 0DM 1.76 1.35 0.68COM 0.68 0.88 0DM -0.41 -0.4 -0.40 -14.06 69.09 -77.98


14.06 k-ft.(MAF)

Joint [68.09+(-77.98)] = 9.89 k-ft.(MBF)

AB BA BCM -14.06 69.09 -77.98

7.03

14.06

14.06 68.09

9.89

77.98

4.95

77.98

- 1/2 Carry-Over Unlock




../ ftk674212162FEMFEMFEMFEM 2DCCDBAAB =====

StiffnessColumn12EI

KKKK FJBFEIAE ====

../ ftk338512164FEMFEM 2CBBC ===

4 k/ft. GE F 2 k/ft.4 k/ft.

H

IEI

2EI 2EI2EI

16 ft. 16 ft. 16 ft.

12 ft.JEI

KEI

LEI

A

EI

B

EI EI EI 12 ft.

C D

StiffnessBeam16EI

K8EI

K FGEF == ;;


EF FE FGDF 0.43 0.35 0.18FEM -85.33 85.33 -42.67 DM 36.69 -14.93 -7.68COM -7.47 18.35 0DM 3.21 -6.42 -3.30COM -3.21 1.16 0DM 1.38 -0.56 -0.29 -54.73 83.38 -53.94


- Joint E 2 Stiffness Factor (K)

..).( ftk3727735421

MM EIEA ===

..).( ftk713372721

MM IEAE ===

..).( ftk7214442921

MM FJFB ===

..).( ftk47721421

MM JFBF ===

บทที่ 6 moment distribution methodcivil.in.th/cen321/cen321-06t.pdfmoment distribution...

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