slope deflection bergoyang

44
BAN NO.3 CONTOH SOAL  Kontr uksi Bergoya ng I I 1). Hitung Nilai K EI 15 2 EI 12  EI 10 2). Menentukan Momen Frimer (MF) 12 X 9 X 6  2 15  2 -12  X 9  2 X 6 15  2 24 X 12 8 = -  = 0 3). Tentukan nilai R ( ) 15 10 30 30 30 = 17.28 = -25.92 MF DC = = 3 R R BC R CD = MF CD X = 0 = MF AB = 2 R 3 K BC = x 30 R AB  = x 30 = K CB  = SOLUSI 5 K BD =  K DB  = X = 36 Kip ft Kip ft K AB =  K BA  = X = 2 Kip ft MF BC = - MF CB  = = MF BA SLOFE DEFLECTION 2I  10 M 24 k  A B C D 6 M 6 M 12 k 6 M 9 M

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Page 1: Slope Deflection Bergoyang

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BAN NO.3

CONTOH SOAL   Kontr uksi Bergoyang 

I

I

1). Hitung Nilai K 

EI

15

2 EI

12

 EI

10

2). Menentukan Momen Frimer (MF)

12 X 9 X 6   2

15   2

-12   X 9   2 X 6

15   2

24 X 12

8

= -   = 0

3). Tentukan nilai R ( ∆ )

15

10

30

30

30

= 17.28

= -25.92

MFDC

=

=

3 R 

R BC

R CD

=

MFCD

X =

0

=

MFAB =

2 R 

3

K BC =

x 30

R AB   = x 30

=

K CB   =

SOLUSI

5

K BD =   K DB   = X =

36 Kip ft

Kip ft

K AB =   K BA   = X = 2

Kip ft

MFBC = - MFCB   = =

MFBA

SLOFE DEFLECTION

2I

 

10 M

24 k

 A

B C

D

6 M 6 M

12 k

6 M

9 M

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Masukkan kembali nilai yang didapat kedalam rumus Slope Deflection

= + K ( -2 ΦA   - ΦB   + 2 R )

= + 2 { ( -2 x ) - ( + (   2 x   ) }

= + 2 ( + )

= + 2 ( )

= +

=

= + K ( -2 ΦB   - ΦA   + 2 R )

= + 2 { ( -2 x ) - ( ) +   2 x   ) }

= + 2 ( + )

= + 2 ( )

= +

=

= + K ( -2 ΦB   - ΦC   + 0 R )

= + 5 { ( -2 x ) - ( ) +   0 x   ) }

= + 5 ( + )

= + 5 ( )

= +

=

= + K ( -2 ΦC   - ΦB   + 0 R )

= + 5 { ( -2 x ) - ( ) +   0 x   ) }

= + 5 ( + )

= + 5 ( )

= +

=

= + K ( -2 ΦC   - ΦD   + 3 R )

= + 3 { ( -2 x ) - ( ) +   3 x   ) }

= + 3 ( + )= + 3 ( )

= +

=

= + K ( -2 ΦD   - ΦC   + 3 R )

= + 3 { ( -2 x ) - ( ) +   3 x   ) }

= + 3 ( + )

= + 3 ( )

= +

=

-MFDC

57.311

27.28

19.104

0.0 0

0.00

0.0 -54.55

0.0 0.00

57.311

27.275 2.76

-2.76

-1.99

19.104

38.207

19.104

38.207

1.99

73.545

-36 -5.521

2.7604

19.104

0

19.104

0

19.104

1.99

-27.28

-25.92 -3.976

-25.92

-7.51

-36 2.7604

-36

MDC

17.28 -8.64

17.28 -17.28

0.00

-25.92

0.0 -5.521

0.0

-25.92 23.603

-2.32

36

17.28 22.43

1.9881

17.28 -44.86 -1.99

2.76

36 -3.976 -2.76

MCD   MFCD

MBC   MFBC

1.9881 22.43

-22.43

MCB   -MFCB

36 -6.74

36

24.51

73.54

0.0

MAB   MFAB

2.32

MBA   -MFBA

-33.68

11.80

-36 -37.54

-73.54

0.0

0.0

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8). Lakukaan pemotongan pada masing-masing titik ( Free body )

B 2I C

I

I

a). potongan A & B

Kf 

HB = Kip

HA = Kip

1. Reaksi perletakan

= 0

-   x 15 + P x   9 + = 0

- 15 HB   + 12 x 9 +   = 0- 15 HB   + +   = 0

- 15 HB   + = 0

=   /   -15

=   Kip ( )

= 0

HA   x 15   - P x 6 + = 0

15   HA   - 12 x 6 + = 0

15   HA   - +

15   HA   = 0

= / 15

=   Kip ( )

2). Momen dan Lintang

= x X = x X   - P ( X - 9 )

= X = X -   X +

= X +

= =

73.542.32

4.65

2.32

7.35

69.7

4.65

2.32

0

ΣHA

HB

0

12

2.32

73.54

-7.35   108

-7.35

108 2.32

110.3

2.32

2.32

72 2.32

-110.3

4.65

-69.7

ΣHB

Mx   4.65

Interval 0 ≤ x ≤ 9

Mx

108

Interval 9 ≤ x ≤ 15

DxDx   4.65

2.32

0

4.65

7.35

4.65

RHA

RHA

RHB

RHB

D

 A

24 k 

6 6

12 k 

6 ft

9 ft

 

B

 

C

10 ft

12 k

6 ft

9 ft

 A

B

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Tabel Momen dan Lintang pada potongan A & B

b. Potongan B & C

24 K 

B C

= K = K 

1. Reaksi perletakan

=   0-   x 12 + P x   6 + - = 0

- 12 RC   + 24 x 6 +   - = 0

- 12 RC   + +   - = 0

- 12 RC   +   = 0

=   /   -12

= K  ( )

=   0

R B   x 12 - P x   6 - + =

12   R B   - 24 x 6 -   + = 0

12   R B   - -   + = 0

12   R B   = 0

=   /

= K  ( )

2. Momen dan Lintang

= x X - =

= X -

73.54

73.54

6

2.32

2.32

2.32

 Nilai Momen

Kip ft

41.81

Kip ft

23.23

6.06

6

 Nilai LintangX Rumus Momen Rumus Lintang

0   0.00

9

Kip

Kip ft

Kip

2   9.29 Kip ft 4.65   Kip

1

Kip ft 4.65   Kip

4.65

13.94

Kip ft

Kip ft 4.65

37.16 Kip ft 4.65

7   32.52

4.65

4.65

4   18.58

Kip

Kip

Kip

4.65

4.65

4.65

Kip

6   27.87 Kip ft

13   12.39 Kip ft

9   41.81 Kip ft

8

10

12

Kip ft

Kip ft

27.10

4.65

Kip

Kip ft

Kip

Kip ft   Kip

Kip ft

Kip

-7.35

Kip

-7.35

-2.32

Kip

Kip ft   Kip

-7.35

11

14

3

5

-7.35   Kip

-7.35

-7.35

Kip

-7.35

-72.8

Interval 0 ≤ x ≤ 6

RC 73.54

2.32

15

6.06

Dx2.32

17.94

2.32

72.77 12

ΣMC

6.06 2.32

Mx   RBV

2.32   73.54

17.94R BV   6.06

R CV

R BV

73.54

2.32 73.54

144

R CV

73.54

2I

34.46

19.75

5.04

ΣMB

R CV   -215.23

215.23

144

R BV

7,35 X + 108 7,35

 

4,65 X 4,65

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= x X   - P ( X - 6 )   -

= X -   X +   -

= X +

=

Tabel Momen dan Lintang pada potongan A & B

C). Potongan C & D

= K 

= K 

1. Reaksi perletakan

= 0

x 10 -   = 0

=   /   10

=   kip ( )

= 0

-   HD   x 10   - = 0

HD   = / -10

HD =   kip ( )

2). Momen dan Lintang

= x X

= X

=

-7.35

Dx   7.35

7.35

Interval 0 ≤ x ≤ 10

Mx   7.35

73.5

Hc

Hc 7.35

73.5

ΣHD

ΣHC

Hc

73.5

73.5

0

73.5

Interval 6 ≤ x ≤ 12

Mx   RBV

6.06 24 144

0

1

Dx   -17.9

X Rumus Momen

Kip ft 6.06

3.75 Kip ft 6.06

-17.9   141.68

2   9.81 Kip ft 6.06

Kip

 Nilai Momen Nilai Lintang

Kip-2.32

4   21.94 Kip ft 6.06

Kip

3   15.88 Kip ft 6.06   Kip

6   34.07 Kip ft 6.06

Kip

5   28.00 Kip ft 6.06   Kip

7   16.13

8   -1.80

-17.94   Kip

Kip

Kip

Kip ft -17.94

11   -55.61 Kip ft

Kip

12   -73.54

10

Kip ft -17.94

6

Kip ft -17.94

9   -19.74

Kip ft -17.94

-37.67

Kip ft

Kip

Kip ft -17.94   Kip

Kip-17.94

34.07

Kip

2.32

2.32

Rumus Lintang

7.35

Hc 7.35

Hc

6,06 X - 2,32 6,06

-17,9X + 141,68 -17,9

 

C

D

 I 10 ft

 

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Tabel Momen dan Lintang pada potongan A & B

GAMBAR MOMEN, LINTANG DAN NORMAL

Kip ft

Kip ft

Kip ft

Kip ft

0

0

Gambar Momen

-2.32

10

9

1

X

Kip ft 7.35

 Nilai Momen Nilai Lintang

Kip

Kip73.54

Kip

66.19 Kip ft 7.35   Kip

8   58.84 Kip ft 7.35

Kip

Kip7   51.48 Kip ft 7.35

6   44.13 Kip ft 7.35

5   36.77 Kip ft

3   22.06 Kip ft

14.71 Kip ft 7.35

7.35

2

Rumus Momen Rumus Lintang

0

7.35   Kip

4

Kip ft 7.35   Kip

Kip ft 7.35

Kip

Kip

Kip

0.00

73.54

34.07

73.54

7.35

29.42 Kip ft 7.35

41.81

 

-7,35 X -7,35

 

 A

B

C

D

- 2,32 Kip ft

(+

(+

(-)

(-)

(+

(-)

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kip

7.35   kip

4.65 kip   -7.35 Kip

0   7.35   kip

kip

0

Gambar Lintang

Kip

kip

Gambar Normal

6.06

17.94

6.06

-17.94

 A

B

C

D

(+

(+

(-)(-) (-)

 A

B C

D

(-)

-

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Detail Asumsi Pembebanan

 ASUMSI PEMBEBANAN

q1 -   Beban Mati

Balok Ukuran 30 x 30 cm = x x =

q1   = x = 0.353   T/1.2 0.294

SLOPE DEFLECTION

SO US

0.35 0.35 2.4

4 M

q1

 A B

C

D

E

4.5 M 4.5 M

EI  EI

0.01 M Keramik

0.02 M Spesi K

0.12 M Plat Lan

0.35 x 0.35 M B

2.25 M

2.25 M

EI EI

 

q2

2.25 M

4.5 M

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q2 -   Beban Hidup + Beban Mati

-   Beban Hidup

Untuk Ruang Pertemuan =

Wu1   = x =

-   Beban Mati

~ Plat Lantai = x = T/m2

~ Spesi Keramik = x = T/m2

~ Keramik = x = /m +T/m

Wu2   = x =

Wu =   Wu1 +   Wu2   =

PENYELESAIAN 

Dari penyelesaian diatas dapat saya gambarkan sebagai berikut :

q2   = T/m' q2   = Wu x

=

=

q1   = T/m'

1). Menentukan Momen Frimer (MF) 

0.353

2.756

1.225

1.2 0.354 0.425

2.756

2 0.021 0.042

1 0.024 0.024

0.5 0.8   T/

T/

1.225   T/m

0.354

500 Kg/m

0.12 2.4 0.288

1.6

4 M

 A B

C

D

E

4.5 M 4.5 M

EI  EI

EI EI

2.25 M

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1 5

12 96

x   2 x

= +

= TM

1 5

12 96

x   2 x

= +

= TM

2). Rumus Slope Deflection 

M AB   = 1 EI x θB   -   MBD   = x (

= EI θB   - = EI θB

MBA   = 1 EI x 2 θB   +   MDB   = x (

4.5

= EI θB   + = EI θD

MBC   = 1 EI x 2 θB   MDE   = 1 EI x 2 θD

4 4

= EI θB   = EI θD

MCB   = 1 EI x θB   MED   = 1 EI x θD

= EI θB   = EI θD

3). Joint Condition 

MBA   +   MBC   +   MBD =   0

0.5 0.5

4 4

0.25 0.25

3.502 1 EI

4.5

0.44 3.502 0.44

3.502 1 EI

0.22 3.502 0.44

2.

9

0.5954 2.91

4.5 4.5

+

  5

3.502

+   qL2

=

  0.353 4.5

12

3.502

-   MFDB   =   MFBD   =   qL2

qL2

+

  5 2.

9=

  0.353 4.5

12

0.5954 2.91

=   qL2-   MF AB   =   MFBA    +

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= TM

MBD   = x + x -

= - -

= TM

MDB   = x + x +

= + +

= TM

MDE   = x

= TM

MED   = x

= TM

6). Free Body 

= T/m1

TM TM TM

= /m1

M

TM

4.5 4.

0.308

-3.85

-0.963 

3.365 3.776 4.084

0.3528

2.756

-1.712 0.14 3.50

5.351 

0.5 -3.85

-1.926 

0.25

0.62 3.502

0.22   -3.85 3.502

-4.084 

0.44 -3.85   0.22

0.274 0.86 3.50

0.154 

0.44 0.62

q1

q2

 A B q1

 

B

B

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R  AV   = - = T

-

ΣV   = 0R  AV   + R BV   = R 1   + R 2

+ = +=

b). Momen dan Lintang 

2. Titik potongan A dan B

= T/m

TM TM

= T/m1

T T

M M

M

~ Menghitung momen dari titik A ke titik BR 1   =

TM = T/m =

=

= T/m

R  AV   = T R 2   =

M

= X =

==

~ Bidang MomenMx = x X - - R  1   x - R 2   (

Interval 0 ≤ X ≥ 2,25

R  AV   3.365 X

2

0.3528

X

2.25q2'   2.76

2.25

3.365 2.756

x3.803   q2'

=2.25   2.76

3.803   R BV=   3.985

4.5

3.365 3.776

0.3528

17.112

2.25 2.25

7.788 7.788

2.756

R  AV=

3.803

4.5

3.803 3.985 1.59 6.201

q1

q2

 A

q1

q2

 A B

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= X - - X2

- X2

 (

= X - X2

- X3

-

~ Bidang Lintang

Dx = - X - X2

~ Menghitung momen dari titik B ke titik A 

TM

R BV   = T

M

~ Bidang Momen

Mx = - x -X - + R  1   x + R 2

= X - - X + X (

= X - X - X -

~ Bidang Lintang Dx = - f' (Mx) / Dx

Dx = + X + X2

3. Titik potongan B dan C

2.25

X Rumus Momen Rumus Lintang

0

-X

3.776

-3.985 0.3528 0.612

2

3.985 0.1764 0.204

x 3.985

2

3.985 3.776 0.353 0.612

Interval 0 ≤ X ≥ 2,25

R BV   3.776

2.25

2.25

3.776

0

3.803 0.3528 0.612

X Rumus Momen Rumus Lintang

0.1764 0.204

3.803 3.365 0.353 0.612

3.365

2

3.803

3.803 X - 0.1764 X2 - 0.204 X3- 3.365 3.803 - 0.3528 X - 0.612 X2

q1

q2

 A

3.985 X - 0.1764 X2 - 0.204 X3- 3.776 - 3.985 + 0.3528 X + 0.612 X2

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T

TM

4 M

TM

T

a). Reaksi Perletakan 

= 0- x 4 + + = 0

- 4 + = 0

- =

-

= 0

- x 4 + + = 0

- 4 + = 0

= T

b). Momen dan Lintang 

T

TM

T

4 M

T

TM

T

7.598

0.308

0.116

0.116

0.154

7.598

0.116

-4

0.462 0.116

4

0.462

=  -0.462

ΣMC

0.308 0.154

ΣMB0.154 0.308

0.462

=

7.598

0.308

0.154

7.598

B

C

RCH

RCH

RCH

B

C

RCH

RBH

RBH

RBH

RCH

RBH

RBH

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- R DV   = -

R DV   = - = T

-

ΣMD   = 0

R BV   x - R 1   x - R 2   x -R BV   - x - x -

R BV   - - - + = 0

R BV   - = 0

- R BV   = -

R BV   = - = T

-

ΣV   = 0R 

BV

  + R DV

  = R 1

  + R 2

+ = +

=

b). Momen dan Lintang 

= T/m

TM TM

= T/m

T T

M M

M

R 1TM = T/m

1

= T/m1

R BV   = T R 23.612   q2'

4.084 2.756

X

0.3528

x

3.612   R DV=   4.176

4.5

16.256 3.612

4.5

2.25 2.25

7.788 7.788

2.756

4.084

R BV =

3.612 4.176 1.59 6.201

5.351

0.3528

5.351

4.5 16.256

4.5 16.256

4.5 3.572 13.95 4.084

4.0844.5 1.59 2.25 6.201 2.25 4.

18.791 4.176

4.5

4.5 2.25 2.25

4.5 18.791

q1

q2

B

q1

q2

B D

q2'

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M= X

~ Bidang Momen

Mx = x X - - R  1   x - R 2

= X - - X2

- X2

= X - X2   - X3 -

~ Bidang Lintang

Dx = - X - X2

~ Menghitung momen dari titik B ke titik D

TM

R DV   = T

M

~ Bidang MomenMx = - x -X - + R  1   x + R 2

= X - - X2

+ X2

(

2

4.176 5.351 0.353 0.612

Interval 0 ≤ X ≥ 2,25

R DV   5.351 -X

2.25

2.25

5.351

x 4.176

0

X Rumus Momen Rumus Lintang

0.353 0.612

2

3.612 0.1764 0.204 4.084

Interval 0 ≤ X ≥ 2,25

3.612 0.3528 0.612

2

3.612 4.084

R BV   4.084 X

2.25q2'   2.8

2.25

2.8 2.25

3.612 X - 0.1764 X2 - 0.204 X3- 4.084 3.612 - 0.3528 X - 0.612 X2

q1

q2

D

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= X - X - X -

~ Bidang Lintang Dx = - f' (Mx) / Dx

Dx = + X + X2

5. Titik potongan B dan C

T

TM

4 M

TM

T

a). Reaksi Perletakan 

= 0

x 4 - - = 04 Rc - = 0

= T

= 0

x 4 - - = 0

4 - = 0

2.890 0.722

4

2.890

ΣME

1.926 0.963

ΣMD

0.963 1.9262.890

=

4.176

1.926

0.963

4.176

2.25

X Rumus Momen Rumus Lintang

0

5.351

-4.176 0.3528 0.612

2

4.176 0.1764 0.204

4.176 X - 0.3528 X2 - 0.612 X3- 5.351 - 4.176 - 0.3528 X - 0.612 X2

D

E

REH

RDH

REH

RDH

REH

RDH

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= T

b). Momen dan Lintang 

TM

TMT

4 M

T

TMTM

~ Bidang Momen

Mx = x X -

= X -

~ Bidang Lintang

Dx =

Rumus Lintang

TM

TM

TM

9) Gambar 

4   0.963 0.722

2.25   -0.301 0.722

1.926

0.722 1.926

Nilai Momen Nilai Lintan

0   -1.926 0.722

0.722

X Rumus Momen

4.176

1.9260.722

0.722

0.9634.176

Interval 0 ≤ X ≥ 4

0.722

4=

  2.890

0.722 X - 1.926

D

E

0.722

 RDH

RDH

REH

RDH

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TM TM

TM

TM

TM

TM

TM

Gambar Momen

T

T

T T

T

T

T

Gambar Lintang

-0.282

-0.116

-3.985

0.826

1.973

-0.154 0.

3.612

-0.091 -0.116

3.803

-3.365 -4.084

-3.776

0.308(+) (+)

(-) (-)(-)

(-)

(+)

(-)

(-)

(+)

A

B

C

A B

C

 

(+)

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T

Gambar Normal

7.598

(-)

A B

C

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 /m1

0.294

Garis leleh

rmik

tai

  alok

q1

q2

4.5 M

  4.5 M

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t

x

T/m'

2.25

0

2

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x   2

x   2

θD

2 θB   + θD   ) -

+ EI θD   -

2 θD   + θB   ) +

+ EI θB   +

3.502

3.502

0.22 3.502

0.22 3.502

56 4.5

6

56 4.5

6

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EI θD   - ) = 0

……….. Pers ( 1 )

……….. Pers ( 2 )

3.50 2

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4 M

TM

L

x = T

- = 0

- = 0

+ = 0

+ = 03.77665

3.776

76

4.5 6.201

3.365

3.365

5

0.963

C

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x

x

X

1 x al x t

20.5 x X x X

X x XX

X - 2 X )

3

2.25

0.6120.5 1.225

0.3528

2.76

0.3528 X

q1   X

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1 X )

3

 ( -X + 2 X )3

-1 X )

3

1.973 TM -0.091

-3.985   TTM

T

Nilai Momen Nilai Lintang

-3.776

1.973 TM -0.091 T

Nilai Momen Nilai Lintang

-3.365 TM 3.803   T

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x L

x = T

- = 0

- = 04.08451

4.5

4.084

4.5

6 6.201

L

g

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+ = 0+ = 0

= x

= x

= X

= 1 x al x t

0.3528

0.3528 X

q1   X

5.35184 5.351

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2

= 0.5 x X x X

= X x X

=   X

 ( X + 2 X )

3

  ( 1 X )

3

( -X + 2 X )

3

-1 X )

T

T0.826 TM -0.282

3.612

Nilai Lintang

-4.084 TM

Nilai Momen

0.5 1.225

0.612

2.8

2.25

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3

( )

0.826 TM -0.282

-4.176   TTM

T

Nilai Momen Nilai Lintang

-5.351

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( )

TM-5.351

g

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TM

TM

T

T

TM

0.722

0.722

-4.176

63

-1.926

(+)

(-)

(-)

(+)

(-)

D

E

D

E

 

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T

11.774

3.803

15.576

7.788

7.788

15.576

4.176

(-)

D

E