# บทที่ 4 (Slope-Deflection) - Civil Method 3 4.1 สมการที่ใช ในวิธี Slope-Deflection Slope-Deflection Equations แสดงความสัมพันธ

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### Text of บทที่ 4 (Slope-Deflection) - Civil Method 3 4.1 สมการที่ใช...

• Slope-Deflection Method 1

4 Slope-Deflection Method

( ) Force Method

Slope-Deflection Displacement Method

Slope-Deflection Method 2

Slope-Deflection (Beam) (Frame) Prismatic

Prismatic

Non-Prismatic

• Slope-Deflection Method 3

4.1 Slope-DeflectionSlope-Deflection Equations

Mab = f (a , b , , Load)Mba = g (a , b , , Load)

(+)- (Mab , Mbc) - ( a , b ) - ()

Mab

Mba

ab

Slope-Deflection Method 4

Mab Mba 4 a

b

LEI2

ML

EI4M aba

aab

=

= .......

LEI4

ML

EI2M bba

bab

=

= .......

2ba2ab LEI6

MLEI6

M

=

= '''''' .......

Maba

a b Mba

b = 0 , = 0

Mab ba b Mbaa = 0 , = 0

Mab a b

Mbaa = 0 , b = 0

a = 0 , b = 0 = 0

• Slope-Deflection Method 5

Slope-Deflection

Fba2

baFbababababa

Fab2

baFababababab

MLEI6

LEI4

LEI2

MMMMM

MLEI6

LEI2

LEI4

MMMMM

+

+

=+++=

+

+

=+++=

'''

'''

Fbababa

Fabbaab

ML

32LEI2

M

ML

32LEI2

M

+

+=

+

+=

)(

)(

Slope-Deflection Method 6

Moment 1 ( a )

- a Hinge - b Fixed EndConjugate Beam Rotation a (a) Condition 1) Moment Conjugate Beam a = 0

a = 02) Moment Conjugate Beam b = 0

b = 0

baMab

a Mba

a

MabEI Mba

EI

• Slope-Deflection Method 7

(1) (2)

... ...

=

=

=

=

20L3L

LEI

M21

3L2

LEI

M21

0M

03L2

LEI

M21

3L

LEI

M21

0M

abaab

b

baab

a

..........)()(

;

)()(

;

= 1M2M baab .............

.......L

EI2M

LEI4

M abaa

ab

=

=

Slope-Deflection Method 8

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

• Slope-Deflection Method 9

EX. Slope-Deflection ABCD

EI EI/2A800 kg.

4 m.

B C D

300 kg./m.

2 m. 2 m. 1 m.

Slope-Deflection Method 10

1) Fixed End Moments

.. mkg40012

430012WL

M22

FAB =

=

=

.. mkg8008

480012

43008

PL12WL

M22

FBC =

=

=

EI EI/2A

800 kg.

4 m.

B C DW=300 kg./m.

2 m. 2 m. 1 m.

MABF MBAF MBCF MCBF

300 kg./m.A4 m.

BMABF MBAF

.. mkg40012WL

M2

FBA ==

300 kg./m.B CMBCF MCBF

800 kg.

2 m. 2 m... mkg800

8PL

12WL

M2

FCB =+=

• Slope-Deflection Method 11

2) Fixed End Moments Slope-Deflection () = 0

2 B C

4002EI

ML

32LEI2

M BF

ABBAAB =+

+= )(

400EIML

32LEI2

M BF

BABABA +=+

+= )(

80024EI

ML

32LEI2

M CBF

BCCBBC +=+

+= )()(

80024EI

ML

32LEI2

M CBF

CBCBCB ++=+

+= )()(

Slope-Deflection Method 12

3) (2 )

MBA+ MBC= 0 MCB= 150 kg.-m.

4) 2 3

BMBA MBC

FBD. B

A C CMCB 150 kg.-m.

FBD. C

B D

=+++=+ 1080024EI

400EI0MM CBBBCBA ..........])([][...

=++= 215080024EI

150M CBCB .................................])([...

• Slope-Deflection Method 13

(2)

(1)

(3) (4)

=+ 3EI6002

2 CB ...................................................,

=+ 4EI6001

6 CB .......................................................,

8005B = .

EI1120017

C

=

Slope-Deflection Method 14

5) 2

..,, mkg115001

400EI11

80052EI

4002EI

M BAB

===

..,, mkg1120010

400EI11

8005EI400EIM BBA =+=+=

800EI1120017

EI118005

24EI

80024EI

M CBBC =+= ),,()(

.., mkg11

20010

=

800EI1120017

2EI11

80054EI

80024EI

M CBCB +=++= ),,()(

.. mkg150 =

• Slope-Deflection Method 15

6) FBD.

MA = 1,500/11 kg.-m. RA = 4,425/11 kg.RB = 43,825/22 kg. RC = 24,325/22 kg.

300 kg./m.1,500/11 10,200/11

4,425/11 8,775/11

B8,775/11 26,275/22

300 kg./m.10,200/11 150

26,275/22 17,725/22

800 kg150

300A

1,500/11

RA RB

C17,725/22 300

RC

Slope-Deflection Method 16

EX. Slope-Deflection 1. 2. BMD. ABCD ( EI )

A

100 kg. B C D300 kg./m.

2 m. 2 m.

2 m.

50 kg.-m.

• Slope-Deflection Method 17

1) Fixed End Moments

,0MM FBAFAB == 0MM

FCB

FBC ==

.. mkg10012

230012WL

M22

FCD =

=

=

.. mkg10012

230012WL

M22

FDC =

==

300 kg./m.C2 m.

DMCDF MDCF

A

100 kg. B C DW

2 m. 2 m.

2 m.

50 kg.-m. MCDF MDCF

W = 300 kg./m.

Slope-Deflection Method 18

2) Fixed End Moments Slope-Deflection (= 0)

2 B C

BF

ABBAAB EIML32

LEI2

M =+

+= )(

BF

BABABA EI2ML32

LEI2

M =+

+= )(

)()( CBF

BCCBBC 2EIML32

LEI2

M +=+

+=

)()( CBF

BCCBCB 2EIML32

LEI2

M +=+

+=

100EI2ML

32LEI2

M CF

CDDCCD =+

+= )(

100EIML

32LEI2

M CF

CDDCDC +=+

+= )(

• Slope-Deflection Method 19

3) (2 ) B C

MBA+ MBC= 50 MCB + MCD = 0

CMCB

FBD. C

B DMCD

BMBA MBC

FBD. B

A

C100 kg.50 kg.-m.

Slope-Deflection Method 20

4) 2 3

(1) (2)

70C =

=++=+ 1502EIEI250MM CBBBCBA ....................)(...

=++=+ 20100EI22EI0MM CCBCDCB ..............)(...

• Slope-Deflection Method 21

5) B C 2

../)/( mkg320EI320EIEIM BAB ===

../)()( mkg3110EI3

70EI3

202EI2EIM CBBC =+=+=

../)/( mkg340EI320EI2EI2M BBA ===

../)()( mkg3160EI3

702

EI320

EI2EIM CBCB =+=+=

../ mkg3160100EI3

70EI2100EI2M CCD ===

../ mkg3370100EI3

70EI2100EIM CDC =+=+=

Slope-Deflection Method 22

6) FBD.

MA = 20/3 kg.-m. RA,x = 10 kg.RA,Y = 45 kg. RC = 310 kg.MD = 370/3 kg.-m. RA,x = 110 kg.RD,Y = 335 kg. 1

A

20/345

10

1045 40/3

B 10100 kg.

50 kg.-m.

40/3

110/3110

45

45

110/3110

45 160/3110

45C

RC=310

11011045 265 300 kg./m.

160/3 370/3265110

335110 D

• Slope-Deflection Method 23

7) (BMD.)

2

A

C DB

110/3

160/320/3

40/3

370/3

185/3

BMD(kg.-m.)

++

--

+-

Slope-Deflection Method 24

4.2 Slope-Deflection Joint Translation

Joint Translation

......

...Frame ...

=0

=0

• Slope-Deflection Method 25

Frame Symmetry () Load ( Sway )

=0

Centerline

Centerline

Centerline

Slope-Deflection Method 26

Joint Translation

...Frame ...

1

2

• Slope-Deflection Method 27

EX. Slope-Deflection ( EI )

A

P B C

D

10 ft.

20 ft.

Slope-Deflection Method 28

1) Side Sway - 1 B C (1 DOF Joint Translation)

2) Fixed End Moments

0MM0MM0MM FDCFCD

FCB

FBC

FBA

FAB ====== ,

A

P B C

D10 ft.

20 ft.

• Slope-Deflection Method 29

3) Fixed End Moments Slope-Deflection

3 B , C

)()( =++= 320200EI

ML

32LEI2

M BF

ABAB

BAAB

)()( =++= 3400200EI

ML

32LEI2

M BF

BAAB

BABA

)()( CBF

BCBC

CBBC 4080200EI

ML

32LEI2

M +=+

+=

)()( CBF

CBBC

CBCB 8040200EI

ML

32LEI2

M +=+

+=

)()( =++= 340200EI

ML

32LEI2

M CF

CDCD

DCCD

)()( =++= 320200EI

ML

32LEI2

M CF

DCCD

DCDC

Slope-Deflection Method 30

4) (3 ) B C

MBA+ MBC= 0 MCB + MCD = 0 1

FBD. B

BMBA

MBC

A

CP

0P2

MM2

MM DCCDBAAB =++

++ )()(20MM BAAB /)( + 20MM DCCD /)( +

A

P B C

DMAB MDC

FBD. C

BMCD

MCB

D

C

• Slope-Deflection Method 31

5) 3 4

=+

=+=+

6EI

P4001126060

5031204040340120

CB

CB

CB

.................................................,......................................................................................................................

=++=+ 104080200EI

340200EI

0MM CBBBCBA .......)()(...

=++=+ 20340200EI

8040200EI

0MM CCBCDCB .......)()(...

201

340200EI

320200EI

0P2

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