Finite Element (Direct Stiffness) Kekakuan Langsung

8/18/2019 Finite Element (Direct Stiffness) Kekakuan Langsung

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Data beban

P1 = 50 kNP2 = 100 kN

q1 = 0 kN/m

q2 = 0 kN/m

Properties penampang

Bahan / material :

No. Element E G b (m) h (m)

1 25000000 0.5 0.5

2 25000000 0.4 0.4

3 25000000 0.5 0.5

4 25000000 0.4 0.4

5 25000000 0.5 0.5

25000000 0.4 0.4

! 25000000 0.5 0.5

" 25000000 0.4 0.4

# 25000000 0.4 0.4

10 25000000 0.4 0.4

PENYELESAIAN :

STEP 1 : Discretize and Global Degrees of reedom !D"#


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D" ses$ai s$mb$ global str$%t$r dan tent$%an &eban ata$ 'omen p

No.titik ($oint) %&'

1

1 1 0 ran. *

2 2 0 ran. +

3 ,1 0 -ot.

24 ,2 50 ran. *5 ,3 0 ran. +

,4 0 -ot.

3

! ,5 0 ran. *

" , 0 ran. +

# ,! 0 -ot.

4

10 10 0 ran. *

11 11 0 ran. +

12 ," 0 -ot.

5

13 ,# 0 ran. *

14 ,10 0 ran. +15 ,11 0 -ot.

1 ,12 0 ran. *

1! ,13 0 ran. +

1" ,14 0 -ot.

!

1# 1# 0 ran. *

20 20 0 ran. +

21 ,15 0 -ot.

"

22 22 0 ran. *

23 23 0 ran. +

24 ,1 0 -ot.

#

25 ,1! 50 ran. *

2 ,1" 100 ran. +

2! ,1# 0 -ot.

%i.Global i

Beban /omen

rahBeban/omen


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10

2" ,20 0 ran. *

2# ,21 100 ran. +

30 ,22 0 -ot.

Nomor +oint $nt$% setiap element

Element !i#(oint

Sudut (deg)n6 n6 k

1 1 2 #0

2 2 3 0

3 3 4 2!0

4 3 5 0

5 5 ! 2!0

5 0

! " 2!0

" 3 # #0

# # 10 0

10 10 5 2!0

Displacement titi% s$mb$ global str$%t$r pada tiap element

Element +

1 1 2 3 4

2 4 5 !

3 ! " # 10

4 ! " # 13

5 13 14 15 1#

13 14 15 1

! 1 1! 1" 22

" ! " # 25

# 25 2 2! 2"

10 2" 2# 30 13

,e%ap Elemen dalam 'atri%s -e%a%$an &atang

Batan6 EA / L 4.EI / L 2.EI / L 6.EI / L^2

1 152500 13020".333 5104.1! 4""2".125

2 .! 35555.555 1!!!!.!!!!!!" """"."""""""#

3 152500 13020".333 5104.1! 4""2".125

4 .! 35555.555 1!!!!.!!!!!!" """"."""""""#

5 152500 13020".333 5104.1! 4""2".125 .! 35555.555 1!!!!.!!!!!!" """"."""""""#

! 152500 13020".333 5104.1! 4""2".125

" 1000000 53333.3333 2.! 20000

# .! 35555.555 1!!!!.!!!!!!" """"."""""""#

10 1000000 53333.3333 2.! 20000

'eng)it$ng Ga.a dan 'omen /+$ng !A%ibat beban l$ar di sepan+ang b

ELE'ENT 1

7arena ti,ak a,a 6a8a ,ie9anan6 bentan6 reebo,8 maka

Ga.a dan 'omen /+$ng pada Element 1 :


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/nit : kNm

;a


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ELE'ENT 7

7arena ti,ak a,a 6a8a ,ie9anan6 bentan6 reebo,8 maka


/nit : kNm;a


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10 0 0 0 0

o>al i 1 2 3

Element ke 1 1 2 ,1

Element ke 2 ,2 ,3 ,4Element ke 3 ,5 , ,!

Element ke 4 ,5 , ,!

Element ke 5 ,# ,10 ,11

Element ke ,# ,10 ,11

Element ke ! ,12 ,13 ,14

Element ke " ,5 , ,!

Element ke # ,1! ,1" ,1#

Element ke 10 ,20 ,21 ,22

Element ?1

o>al ai 1 2 3 4

Global ai 1 2 3 4

%&' 0 0 ,1 ,2

Element ?2

o>al ai 1 2 3 4

Global ai 4 5 !

%&' ,2 ,3 ,4 ,5

Element ?3

o>al ai 1 2 3 4Global ai ! " # 10

%&' ,5 , ,! 0

Element ?4

o>al ai 1 2 3 4

Global ai ! " # 13

%&' ,5 , ,! ,#

Element ?5

o>al ai 1 2 3 4Global ai 13 14 15 1#

%&' ,# ,10 ,11 0

Element ?

o>al ai 1 2 3 4

Global ai 13 14 15 1

%&' ,# ,10 ,11 ,12

Element ?!

o>al ai 1 2 3 4

Global ai ! " # 25%&' ,12 ,13 ,14 0

7eeaian S& L"-AL element ,en6an S& GL"&AL dan D" ara) glob


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Element ?"

o>al ai 1 2 3 4

Global ai ! " # 25

%&' ,5 , ,! ,1!

Element ?#

o>al ai 1 2 3 4Global ai 25 2 2! 2"

%&' ,1! ,1" ,1# ,20

Element ?10

o>al ai 1 2 3 4

Global ai 2" 2# 30 13

%&' ,20 ,21 ,22 ,#

Element 1 :

0 0

0 0

- = 0 0 1 0

0 0 0

0 0 0

0 0 0 0

.125!E01! 1 0 0

1 .12E01! 0 0

@-A = 0 0 1 0

0 0 0 .125!42E01!

0 0 0 1

0 0 0 0

.125!E01! 1 0 0

1 .12E01! 0 0

0 0 1 0

0 0 0 .125!42E01!

0 0 0 1

0 0 0 0

Element 2 :

0 0

0 0

- = 0 0 1 0

0 0 0

0 0 00 0 0 0

'atri%s Transformasi ;,< dan ;,

T

< tiap element

>o α1

in α1

in α1

>o α1

>o α1

in α1

@-A =

>o α1

in α1

in α1

>o α1

>o α1

in α1


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1 0 0 0

0 1 0 0

@-A = 0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

Element 3 :

0 0

0 0

- = 0 0 1 0

0 0 0

0 0 0

0 0 0 0

1."3"E01 1 0 0

1 1."3"E01 0 0

@-A = 0 0 1 0

0 0 0 1."3!!2E01

0 0 0 1

0 0 0 0

1."3"E01 1 0 0

1 1."3"E01 0 0

0 0 1 0

0 0 0 1."3!!2E01

0 0 0 1

0 0 0 0

Element 4 :

0 0

0 0

- = 0 0 1 0

0 0 0

0 0 0

0 0 0 0

1 0 0 0

0 1 0 0@-A = 0 0 1 0

0 0 0 1

@-A =

>o α1

in α1

in α1 >o α1

>o α1

in α1

@-A =

>o α1

in α1

in α1

>o α1

>o α1

in α1


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0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 10 0 0 0

0 0 0 0

Element 5 :

0 0

0 0

- = 0 0 1 0

0 0 0

0 0 00 0 0 0

1."3"E01 1 0 0

1 1."3"E01 0 0

@-A = 0 0 1 0

0 0 0 1."3!!2E01

0 0 0 1

0 0 0 0

1."3"E01 1 0 0

1 1."3"E01 0 0

0 0 1 0

0 0 0 1."3!!2E01

0 0 0 1

0 0 0 0

Element 6 :

0 0

0 0

- = 0 0 1 00 0 0

0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

@-A = 0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

1 0 0 0

@-A =

>o α1

in α1

in α1

>o α1

>o α1

in α1

@-A =

>o α1

in α1

in α1

>o α1

>o α1

in α1


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0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

Element 7 :

0 0

0 0

- = 0 0 1 0

0 0 0

0 0 0

0 0 0 0

1."3"E01 1 0 0

1 1."3"E01 0 0@-A = 0 0 1 0

0 0 0 1."3!!2E01

0 0 0 1

0 0 0 0

1."3"E01 1 0 0

1 1."3"E01 0 0

0 0 1 0

0 0 0 1."3!!2E01

0 0 0 10 0 0 0

Element 8 :

0 0

0 0

- = 0 0 1 0

0 0 0

0 0 0

0 0 0 0

.125!E01! 1 0 0

1 .12E01! 0 0

@-A = 0 0 1 0

0 0 0 .125!42E01!

0 0 0 1

0 0 0 0

.125!E01! 1 0 0

1 .12E01! 0 0

0 0 1 00 0 0 .125!42E01!

0 0 0 1

@-A =

>o α1

in α1

in α1

>o α1

>o α1

in α1

@-A =

>o α1

in α1

in α1

>o α1

>o α1

in α1

@-A =


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0 0 0 0

Element 9 :

0 0

0 0

- = 0 0 1 0

0 0 0

0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

@-A = 0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

Element 1 :

0 00 0

- = 0 0 1 0

0 0 0

0 0 0

0 0 0 0

1."3"E01 1 0 0

1 1."3"E01 0 0@-A = 0 0 1 0

0 0 0 1."3!!2E01

0 0 0 1

0 0 0 0

1."3"E01 1 0 0

1 1."3"E01 0 0

0 0 1 0

0 0 0 1."3!!2E010 0 0 1

0 0 0 0

>o α1

in α1

in α1

>o α1

>o α1

in α1

@-A =

>o α1 in α1 in α

1>o α

1

>o α1

in α1

@-A =


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===> DALA' S

Batan6 = 1

Elemen ,alam matrik kekakan batan6 :

E / 4.E / 2.E / .E / C2

152500 13020".333 5104.1! 4""2".125

D = >o theta = .125!42E01!

D8 = in theta = 1

o>al ai 1 2 3

1 152500 0 0

2 0 24414.025 4""2".125

3 0 4""2".125 13020".333333

4 152500 0 0

5 0 24414.025 4""2".125 0 4""2".125 5104.1!

1 2 ,1

Global ai 1 2 3

1 24414.025 #.421#1"E011 4""2".125

2 #.422E011 152500 2.##10"5E012

3 4""2".125 2.##10"5E012 13020".333333

4 24414.025 #.421#2E011 4""2".125

5 #.422E011 152500 2.##10#E012

4""2".125 2.##10"5E012 5104.1!

===> DALA' S

Batan6 = 2


E / 4.E / 2.E / .E / C2

.! 35555.555 1!!!!.!!!!!!" """"."""""""#

D = >o theta = 1

D8 = in theta = 0

o>al ai 1 2 3

1 .! 0 0

2 0 2#2.#2#2# """"."""""""#

3 0 """"."""""""# 35555.555555

4 .! 0 0

5 0 2#2.#2#3 """".""""""#

0 """"."""""""# 1!!!!.!!!!!!"

,2 ,3 ,4Global ai 4 5

4 .! 0 0

0it$ng matri%s %e%a%$an batang 1 ;S'1<

@1A =

@1

A =


@2A =


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0 """"."""""""# 1!!!!.!!!!!!"

,5 , ,!

Global ai ! " #

! .! 0 0

" 0 2#2.#2#2# """"."""""""#

# 0 """"."""""""# 35555.55555513 .! 0 0

14 0 2#2.#2#3 """".""""""#

15 0 """"."""""""# 1!!!!.!!!!!!"

===> DALA' S

Batan6 = 5


E / 4.E / 2.E / .E / C2

152500 13020".333 5104.1! 4""2".125

D = >o theta = 1."3!!2E01

D8 = in theta = 1

o>al ai 1 2 3

1 152500 0 0

2 0 24414.025 4""2".125

3 0 4""2".125 13020".333333

4 152500 0 0

5 0 24414.025 4""2".125 0 4""2".125 5104.1!

,# ,10 ,11

Global ai 13 14 15

13 24414.025 2."25!5E010 4""2".125

14 2."2!E010 152500 ".#!32E012

15 4""2".125 ".#!32E012 13020".333333

1# 24414.025 2."25"E010 4""2".125

20 2."2!E010 152500 ".#!3255E012

21 4""2".125 ".#!32E012 5104.1!

===> DALA' S

Batan6 =


E / 4.E / 2.E / .E / C2

.! 35555.555 1!!!!.!!!!!!" """"."""""""#

D = >o theta = 1

D8 = in theta = 0

o>al ai 1 2 3

1 .! 0 0

@4A =


@5A =

@5

A =

0it$ng matri%s %e%a%$an batang 6 ;S'6


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2 0 2#2.#2#2# """"."""""""#

3 0 """"."""""""# 35555.555555

4 .! 0 0

5 0 2#2.#2#3 """".""""""#

0 """"."""""""# 1!!!!.!!!!!!"

,# ,10 ,11Global ai 13 14 15

13 .! 0 0

14 0 2#2.#2#2# """"."""""""#

15 0 """"."""""""# 35555.555555

1 .! 0 0

1! 0 2#2.#2#3 """".""""""#

1" 0 """"."""""""# 1!!!!.!!!!!!"

===> DALA' S

Batan6 = !


E / 4.E / 2.E / .E / C2

152500 13020".333 5104.1! 4""2".125

D = >o theta = 1."3!!2E01

D8 = in theta = 1

o>al ai 1 2 3

1 152500 0 02 0 24414.025 4""2".125

3 0 4""2".125 13020".333333

4 152500 0 0

5 0 24414.025 4""2".125

0 4""2".125 5104.1!

,12 ,13 ,14

Global ai 1 1! 1"

1 24414.025 2."25!5E010 4""2".125

1! 2."2!E010 152500 ".#!32E0121" 4""2".125 ".#!32E012 13020".333333

22 24414.025 2."25"E010 4""2".125

23 2."2!E010 152500 ".#!3255E012

24 4""2".125 ".#!32E012 5104.1!

===> DALA' S

Batan6 = "


E / 4.E / 2.E / .E / C21000000 53333.3333 2.! 20000

@A =

@

A =


@!A =

@!

A =



16/332


17/332

Batan6 = 10


E / 4.E / 2.E / .E / C2

1000000 53333.3333 2.! 20000

D = >o theta = 1."3!!2E01

D8 = in theta = 1

o>al ai 1 2 3

1 1000000 0 0

2 0 10000 20000

3 0 20000 53333.3333333

4 1000000 0 0

5 0 10000 20000

0 20000 2.!

,20 ,21 ,22Global ai 2" 2# 30

2" 10000 1."1#345E010 20000

2# 1."1#E010 1000000 3.!545E012

30 20000 3.!545E012 53333.3333333

13 10000 1."1#35E010 20000

14 1."1#E010 1000000 3.!5445E012

15 20000 3.!545E012 2.!

IE,,",!?L"rtkan eai mb 6lobal/trktr ,en6an %&' 8an6 ,i,ahlkan

%&' : 1 2 ,1

%i.Global ai : 1 2 3

%itata lan6 manal ==F ,1 ,2 ,3

3 4 5

3 13020".333 4""2".125 2.##10#E012

4 4""2".125 #10"0.!2#1! #.421#1"E011

5 2.##1E012 #.421#1"E011 15542.#2# 5104.1! 4""2".125 """"."""""""#

! 0 .! 0

" 0 0 2#2.#2#3

# 0 0 """"."""""""#

12 0 0 0

13 0 0 0

14 0 0 0

15 0 0 0

1 0 0 0

1! 0 0 0

1" 0 0 0

21 0 0 0

24 0 0 0

@10

A =

@10

A =

'AT,I-S -E-A-/AN ST,/-T/, ;S'S<

@A =


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25 0 0 0

2 0 0 0

2! 0 0 0

2" 0 0 0

2# 0 0 0

30 0 0 0

1 4""2".125 24414.025 #.421#2E0112 2.##1E012 #.421#2E011 152500

10 0 0 0

11 0 0 0

1# 0 0 0

20 0 0 0

22 0 0 0

23 0 0 0

0it$ng 'atri%s &eban (oint* Nodal

Nodal forces:

a A%ibat beban di sepan+ang bentang !%ed$a $+$ng di%e

Element: 1

0

0

= 0

00

0

ranormai b lokal ke 6lobal ==F

.125!42E01!

1

= 0

0

00

Global ai

1

2

= 3

4

5

Element: 2

@A1

@A1

@A1


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0

0

= 0

0

0

0


1

0

= 0

0

0

0

Global ai

4

5

=

!

"

#

Element: 3

0

0

= 0

0

0

0


1."3!!2E01

1

= 0

0

0

0

Global ai

!

"

= #10

11

@A2

@A2

@A1

@A1

@A1

@

A1


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12

Element: 4

0

0

= 0

0

0

0


1

0

= 0

0

0

0

Global ai

!

"

= #

13

14

15

Element: 5

0

0

= 0

0

0

0


1."3!!2E01

1

= 0

0

0

0

Global ai

13

14

@A1

@A1

@A1

@A1

@A1


21/332

= 15

1#

20

21

Element: 6

0

0

= 0

0

0

0


1

0

= 0

0

0

0

Global ai

13

14

= 15

1

1!

1"

Element: 7

0

0

= 0

0

0

0


1."3!!2E01

1

= 0

00

0

@A1

@A1

@A1

@A1

@A1

@A1


22/332

Global ai

1

1!

= 1"

22

2324

Element: 8

0

0

= 0

0

0

0


.125!42E01!

1

= 0

0

0

0

Global ai

!

"

= #

25

2

2!

Element: 9

0

0

= 0

0

0

0


1

0

@A1

@A1

@A1

@A1

@A1


23/332

= 0

0

0

0

Global ai

252

= 2!

2"

2#

30

Element: 1

0

0

= 0

0

0

0


1."3!!2E01

1

= 0

0

0

0

Global ai

2"

2#

= 30

13

14

15

S$s$n matri%s beban e%iFalen

$oint b Global Element 1

1 1 0

1 2 0

1 3 0

2 4 0

e = 2 5 0 2 0

3 ! 0

@A1

@A1

@A1

@A1

@A1


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3 " 0

3 # 0

4 10 0

4 11 0

4 12 0

5 13 0

5 14 05 15 0

1 0

1! 0

1" 0

! 1# 0

! 20 0

! 21 0

" 22 0

" 23 0

" 24 0

# 25 0

# 2 0

# 2! 0

10 2" 0

10 2# 0

10 30 0

e = 0 1 1

0 2 2

0 3 ,1

0 4 ,20 5 ,3

0 ,4

0 ! ,5

0 " ,

0 # ,!

0 10 10

0 11 11

0 12 ,"

0 13 ,#

0 14 ,10

0 15 ,11

0 1 ,12

0 1! ,13

0 1" ,14

0 1# 1#

0 20 20

0 21 ,15

0 22 22

0 23 23

0 24 ,1

0 25 ,1!0 2 ,1"

0 2! ,1#


25/332

0 2" ,20

0 2# ,21

0 30 ,22

b A%ibat &eban di +oint !%ed$a $+$ng di%e%ang#

$oint b Global1 1 0

1 2 0

1 3 0

2 4 50

= 2 5 0

2 0

3 ! 0

3 " 0

3 # 0

4 10 0

4 11 0

4 12 0

5 13 0

5 14 0

5 15 0

1 0

1! 0

1" 0

! 1# 0

! 20 0

! 21 0" 22 0

" 23 0

" 24 0

# 25 50

# 2 100

# 2! 0

10 2" 0

10 2# 100

10 30 0

c &eban %ombinasi ata$ gab$ngan !Ae C A+#

$oint b Global %&'

1 1 1

1 2 2

1 3 ,1

2 4 ,2

> = 2 5 ,3

2 ,4

3 ! ,5

3 " ,3 # ,!

4 10 10


26/332

4 11 11

4 12 ,"

5 13 ,#

5 14 ,10

5 15 ,11

1 ,12

1! ,13 1" ,14

! 1# 1#

! 20 20

! 21 ,15

" 22 22

" 23 23

" 24 ,1

# 25 ,1!

# 2 ,1"

# 2! ,1#

10 2" ,20

10 2# ,21

10 30 ,22

ata lan6 > : %i9la>ement /

,1 0

,2 50

,3 0

,4 0 > = ,5 0

, 0

,! 0

," 0

,# 0

,10 0

,11 0

,12 0

,13 0

,14 0

,15 0

,1 0

,1! 50

,1" 100

,1# 0

,20 0

,21 100

,22 0

1 0

2 0

10 011 0

1# 0


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20 0

22 0

23 0

0.00001!4 2.!15E005 2.2#41#E00" !."#443#E00!

2.!15E005 #.34"E005 1.4415"E00! 1.5"105E005

2.2#4E00" 1.44E00! .3#3"!!E00! .3#2"0E00"

!."#44E00! 1.5"1E005 .3#2"0E00" 0.0000102!#

2."1E005 #.225E005 1.454151E00! 1.55!"#E005

1."#3E00" 1.021E00" 0.000000001 4.551323E00"

3.244E00 #.#!#E00 4.12432E00" #.#4"111E00!

".42##E00 2.#1E005 3.3#0#1E00" 5.344#E00

2.5#E005 #.151E005 1.444!"2E00! 1.544!4E005

1.1001E011 !."4#E00# 1.5"345E010 5.#0#00E00#

2."#5#E00 #.#!0E00 1.55#4!E00" 1."02E00

2.51E005 #.124E005 1.440501E00! 1.5401E005

4.1"5!E00" 1.441E00! 2.2#2E010 2.4323"1E00"

4.4!5!E00 1.540E005 2.4323"E00" 2.5##"5E00

".524E00 2.#33E005 4.3"20E00" 4.#4#!2!E00

!.!051E00 2.51E005 4.1"5#E00" 4.4!5!40E00

3.40"E005 0.000115"4 2.13353E00! 0.00001"!1

2.#"2E00" 2.522E00" 1.0#!"1"E00# 4.0!2244E00"

5.24!"E00! 1."#E00 0.000000001 3.!2!!!E00!

3.40"E005 0.000115"4 2.133#25E00! 0.00001"!22

1.0#03E00" 2.!5"E00" 2.5"5"!E010 1.11"21E00#!.141"E00! 2.10!E00 1.0!!14E00" 1.522405E00!

-ea>tion:

GAYAGAYA DALA' /(/NG ELE'EN

&atang: 1

%eormai trktr ,alam arah B G&B ,i batan6

entkan no. b 6lobal ,ahl

1 1 0.0000002 2 0.000000

3 ,1 0.00300

4 ,2 0.0104

5 ,3 0.00001"

,4 0.001!30

%eormai trktr ,alam arah B &7 ,i batan6 1H

.12E01! 1 01 .125!42E01! 0

0 0 1

@%A1 =

M 1 = R . DMS 1

∆M 1


28/332

0 0 0

0 0 0

0 0 0

b lokal %&'

0.000000 1 1

0.000000 2 20.00300 3 ,1

0.00001" 4 ,2

0.0104 5 ,3

0.001!30 ,4

Ga.a $+$ng batang

0 1525000 0

0 0

0 152500

0 0

0 0

0 2!.##!"522#

0 21.35#!25521

0 4.232E0140 2!.##!"522"#

0 21.35#!2552

0 ".543"#020"5

&atang: 2



4 ,2 0.0104

5 ,3 0.00001"

,4 0.001!30! ,5 0.010423

" , 0.000054

# ,! 0.001443


1 0 0

0 1 0

0 0 10 0 0

0 0 0

∆M 1

AM 1 = AML 1 + SM 1 . M 1

@1A =

@1A =

@%A2 =

M 2 = R . DMS 2

∆M 2


29/332

0 0 0

b lokal %&'

0.0104 1 ,2

0.00001" 2 ,3

0.001!30 3 ,4

0.010423 4 ,50.000054 5 ,

0.001443 ,!

Ga.a $+$ng batang

0 .!

0 0

0 00 .!

0 0

0 0

0 2".3402!44!#

0 2!.##!"522#

0 ".543"#020"

0 2".3402!44"

0 2!.##!"522"#0 "1.44324!15

&atang: 3



! ,5 0.010423

" , 0.000054

# ,! 0.001443

10 10 0.000000

11 11 0.00000012 ," 0.0031"!


1."3"E01 1 0

1 1."3!!2E01 0

0 0 1

0 0 0

0 0 00 0 0

∆M 2

AM 2 = AML 2 + SM 2 . M 2

@2A =

@2A =

@%A3 =

M 3 = R . DMS 3

∆M !


30/332

b lokal %&'

0.000054 1 ,5

0.010423 2 ,

0.001443 3 ,!

0.000000 4 10

0.000000 5 11

0.0031"! ,"

Ga.a $+$ng batang

0 152500

0 0

0 0

0 152500

0 00 0

0 "4.05204"20#3

0 2".3"0#5#3#11

0 113.523"3!54

0 "4.05204"20#

0 2".3"0#5#3#1

0 1.!05303E013

&atang: 4



! ,5 0.010423

" , 0.000054

# ,! 0.001443

13 ,# 0.0103"

14 ,10 0.0000!4

15 ,11 0.001435


1 0 0

0 1 0

0 0 1

0 0 0

0 0 0

0 0 0

b lokal %&'

0.010423 1 ,5

∆M !

AM 3 = AML 3 + SM 3 . M 3

@3A =

@3A =

@%A4 =

M 4 = R . DMS 4

∆M 4


31/332

0.000054 2 ,

0.001443 3 ,!

0.0103" 4 ,#

0.0000!4 5 ,10

0.001435 ,11

Ga.a $+$ng batang

0 .!

0 0

0 0

0 .!

0 0

0 0

0 24.#334"!!#4#

0 25.52"5"5421

0 !.5"121!#

0 24.#334"!!#5

0 25.52"5"54214

0 !.514!0034#

&atang: 5

%eormai trktr ,alam arah B G&B ,i batan6entkan no. b 6lobal ,ahl

13 ,# 0.0103"

14 ,10 0.0000!4

15 ,11 0.001435

1# 1# 0.000000

20 20 0.000000

21 ,15 0.0031!!


1."3"E01 1 0

1 1."3!!2E01 0

0 0 1

0 0 0

0 0 0

0 0 0

b lokal %&'

0.0000!4 1 ,#0.0103" 2 ,10

0.001435 3 ,11

∆M 4

AM 4 = AML 4 + SM 4 . M 4

@4A =

@4A =

@%A5 =

M 5 = R . DMS 5

∆M "

∆M "


32/332

0.000000 4 1#

0.000000 5 20

0.0031!! ,15

Ga.a $+$ng batang

0 152500

0 0

0 0

0 152500

0 0

0 0

0 115.""#4!22"0 2".34!"4!3#2

0 113.3#13"#5"

0 115.""#4!22!

0 2".34!"4!3#2

0 5."434E014

&atang: 6



13 ,# 0.0103"14 ,10 0.0000!4

15 ,11 0.001435

1 ,12 0.010354

1! ,13 0.00001"

1" ,14 0.001!02

%eormai trktr ,alam arah B &7 ,i batan6 H

1 0 00 1 0

0 0 1

0 0 0

0 0 0

0 0 0

b lokal %&'

0.0103" 1 ,#

0.0000!4 2 ,10

0.001435 3 ,110.010354 4 ,12

0.00001" 5 ,13

AM 5 = AML 5 + SM 5 . M 5

@5A =

@5A =

@%A =

M 6 = R . DMS 6

∆M 6

∆M 6


33/332

0.001!02 ,14

Ga.a $+$ng batang

0 .!0 0

0 0

0 .!

0 0

0 0

0 21.352204"

0 2".05335!52

0 "1.!#!131"5

0 21.352205

0 2".05335!51

0 ".540""25#

&atang: 7



1 ,12 0.010354

1! ,13 0.00001"

1" ,14 0.001!02

22 22 0.000000

23 23 0.000000

24 ,1 0.003031

%eormai trktr ,alam arah B &7 ,i batan6 !H

1."3"E01 1 0

1 1."3!!2E01 0

0 0 1

0 0 0

0 0 0

0 0 0

b lokal %&'

0.00001" 1 ,12

0.010354 2 ,13

0.001!02 3 ,14

0.000000 4 22

0.000000 5 230.003031 ,1

AM 6 = AML 6 + SM 6 . M 6

@A =

@A =

@%A! =

M 7 = R . DMS 7

∆M #

∆M #


34/332

Ga.a $+$ng batang

0 152500

0 0

0 0

0 152500

0 0

0 0

0 2".05335!51

0 21.352204"

0 ".540""25#1

0 2".05335!52

0 21.352205

0 1.!05303E013

&atang: 8



! ,5 0.010423

" , 0.000054

# ,! 0.001443

25 ,1! 0.01!"#3

2 ,1" 0.000135

2! ,1# 0.001044

%eormai trktr ,alam arah B &7 ,i batan6 "H

.12E01! 1 0

1 .125!42E01! 0

0 0 1

0 0 0

0 0 0

0 0 0

b lokal %&'

0.000054 1 ,5

0.010423 2 ,

0.001443 3 ,!

0.000135 4 ,1!

0.01!"#3 5 ,1"

0.001044 ,1#

Ga.a $+$ng batang

AM 7 = AML 7 + SM 7 . M 7

@!A =

@!A =

@%A" =

M 8 = R . DMS 8

∆M $

∆M $


35/332

0 1000000

0 0

0 0

0 1000000

0 0

0 0

0 "1.5"2!!!401"

0 24.#5041#!3"2

0 44.5!221!!#3

0 "1.5"2!!!402

0 24.#5041#!3"

0 55.22545!1!33

&atang: 9



25 ,1! 0.01!"#3

2 ,1" 0.000135

2! ,1# 0.001044

2" ,20 0.01!"5

2# ,21 0.0001#3

30 ,22 0.00104!

%eormai trktr ,alam arah B &7 ,i batan6 #H

1 0 0

0 1 0

0 0 1

0 0 0

0 0 0

0 0 0

b lokal %&'

0.01!"#3 1 ,1!

0.000135 2 ,1"

0.001044 3 ,1#

0.01!"5 4 ,20

0.0001#3 5 ,21

0.00104! ,22

Ga.a $+$ng batang

AM 8 = AML 8 + SM 8 . M 8

@"A =

@"A =

@%A# =

M 9 = R . DMS 9

∆M %

∆M %

AM 9 = AML 9 + SM 9 . M 9


36/332

0 .!

0 0

0 0

0 .!

0 0

0 0

0 25.04#5"021"

0 1".41!2225#"

0 55.22545!1!3

0 25.04#5"022

0 1".41!2225#"2

0 55.2!!"!"41

&atang: 1

%eormai trktr ,alam arah B G&B ,i batan6entkan no. b 6lobal ,ahl

2" ,20 0.01!"5

2# ,21 0.0001#3

30 ,22 0.00104!

13 ,# 0.0103"

14 ,10 0.0000!4

15 ,11 0.001435

%eormai trktr ,alam arah B &7 ,i batan6 10

1."3"E01 1 0

1 1."3!!2E01 0

0 0 1

0 0 0

0 0 0

0 0 0

b lokal %&'

0.0001#3 10.01!"5 2

0.00104! 3

0.0000!4 4

0.0103" 5

0.001435

Ga.a $+$ng batang

0 1000000

0 0

@#A =

@#A =

@%A10 =

M 10 = R . DMS 10

∆M 1&

∆M 1&

AM 10 = AML 10 + SM 10 . M 10


37/332

0 0

0 1000000

0 0

0 0

0 11".41!2225#"0 25.04#5"021"

0 55.2!!"!"411

0 11".41!222

0 25.04#5"022

0 44.#20442313

Element = 1 2

0asil 'an$al 2!.##" 2".34

21.3 2!.##"4.23E014 ".543#

2!.##" 2".34

21.3 2!.##"

".543# "1.4432

0asil SAP2 2!.##" 2".34

21.3 2!.##"

5."4E014 ".543#

2!.##" 2".34

21.3 2!.##"".543# "1.4432

TA&LE: Element orces rames

rame Station "$tp$tBase

et m et

1 0 %E%

1 4 %E%

2 0 %E%

2 %E%

3 0 %E%

3 4 %E%

4 0 %E%

4 %E%

5 0 %E%

5 4 %E%

0 %E%

%E%

! 0 %E%

! 4 %E%

" 0 %E%" 4 %E%

# 0 %E%

@10

A =

@10

A =


38/332


39/332

(m2) (m) (m4) 7eteran6an

2.500000E01 4 5.20"333E03 1 (5050)

1.00000E01 2.133333E03 2 (4040)

2.500000E01 4 5.20"333E03 1 (5050)

1.00000E01 2.133333E03 2 (4040)

2.500000E01 4 5.20"333E03 1 (5050)

1.00000E01 2.133333E03 2 (4040)

2.500000E01 4 5.20"333E03 1 (5050)

1.00000E01 4 2.133333E03 2 (4040)

1.00000E01 2.133333E03 2 (4040)

1.00000E01 4 2.133333E03 2 (4040)


40/332

Defined DOF

after boundary conditions

da Titi% (oint Str$%t$r dalam Ara) S$mb$ Global*S$mb$ Str$%t$r

TA&LE: (oint Boordinates

(oint BoordS.s BoordT.pe or,et et et m

1 G&B Darteian #

2 G&B Darteian #

3 G&B Darteian 3

4 G&B Darteian 3

5 G&B Darteian 3

G&B Darteian #

! G&B Darteian 3

" G&B Darteian #

# G&B Darteian 310 G&B Darteian 3

TA&LE: BonnectiFit. rame

rame (ointI (oint( IsB$rFed

et et et +e/No

1 1 2 No

2 2 3 No

3 4 3 No

4 3 5 No

5 ! 5 No

5 No

! " No


41/332

" 3 # No

# # 10 No

10 5 10 No

%

5 ==F Global ai

" # ==F Global ai

11 12 ==F Global ai

14 15 ==F Global ai

20 21 ==F Global ai

1! 1" ==F Global ai

23 24 ==F Global ai

2 2! ==F Global ai

2# 30 ==F Global ai

14 15 ==F Global ai

12.EI / L^! Sudut (deg) D D8

24414.025 #0 .125!4E01! 1

2#2.#2#2# 0 1 0

24414.025 2!0 1."3#!E01 1

2#2.#2#2# 0 1 0

24414.025 2!0 1."3#!E01 12#2.#2#2# 0 1 0

24414.025 2!0 1."3#!E01 1

10000 #0 .12323E01! 1

2#2.#2#2# 0 1 0

10000 2!0 1."3#!E01 1

entang dengan $+$ng di%e%ang*di+epit# !DALA' A,A0 S/'&/ L"-AL# :

rekai n6 terkekan6 eman8a = 0


42/332


43/332

rekai n6 terkekan6 eman8a = 0


44/332


45/332

5

2 2!

,1" ,1#

5 2# 30

,21 ,22

5

14 15

,10 ,11

0 0

0 0

0 0

0

0

0 1

0 0

0 0

0 0

1 0

.125!42E01! 0

0 1

0 0

0 0

0 0

1 0

.125!42E01! 0

0 1

0 0

0 0

0 0

0

00 1

in α1

>o α1

in α1

>o α1


46/332

0 0

0 0

0 0

0 0

1 0

0 1

0 0

0 0

0 0

0 0

1 0

0 1

0 0

0 0

0 0

0

0

0 1

0 0

0 0

0 0

1 0

1."3!!2E01 0

0 1

0 0

0 0

0 0

1 0

1."3!!2E01 0

0 1

0 0

0 0

0 0

0

0

0 1

0 0

0 00 0

0 0

in α1

>o α1

in α1

>o α1


47/332

1 0

0 1

0 0

0 0

0 0

0 01 0

0 1

0 0

0 0

0 0

0

00 1

0 0

0 0

0 0

1 0

1."3!!2E01 0

0 1

0 0

0 0

0 0

1 0

1."3!!2E01 0

0 1

0 0

0 0

0 00

0

0 1

0 0

0 0

0 0

0 0

1 0

0 1

0 0

in α1

>o α1

in α1

>o α1


48/332

0 0

0 0

0 0

1 0

0 1

0 0

0 0

0 0

0

0

0 1

0 0

0 00 0

1 0

1."3!!2E01 0

0 1

0 0

0 0

0 0

1 0

1."3!!2E01 00 1

0 0

0 0

0 0

0

0

0 1

0 0

0 0

0 0

1 0

.125!42E01! 0

0 1

0 0

0 0

0 01 0

.125!42E01! 0

in α1

>o α1

in α1

>o α1


49/332

0 1

0 0

0 0

0 0

0

0

0 1

0 0

0 0

0 0

0 0

1 0

0 1

0 0

0 0

0 0

0 0

1 0

0 1

0 00 0

0 0

0 ,/'/S /'/'

0

0 1

0 0 0

0 0 = 00 0

1 0 0

1."3!!2E01 0 0

0 1

0 0

0 0

0 0

1 01."3!!2E01 0

0 1

in α1

>o α1

in α1

>o α1

u2

1

E/1

E/1

21

11

MS

MS

S

S


50/332

/'&/ L"-AL

12.E / C3

24414.025

4 5

152500 0 0

0 24414.025 4""2".125

0 4""2".125 5104.1!

152500 0 0

0 24414.025 4""2".1250 4""2".125 13020".33333

,2 ,3 ,4

4 5

24414.025 #.421#E011 4""2".125

#.421#2E011 152500 2.##10#E012

4""2".125 2.##11E012 5104.1!

24414.025 #.421#2E011 4""2".125

#.421#1"E011 152500 2.##10#E012

4""2".125 2.##11E012 13020".33333

/'&/ L"-AL

12.E / C3

2#2.#2#2#

4 5

.! 0 0

0 2#2.#2#3 """".""""""#

0 """"."""""# 1!!!!.!!!!!"

.! 0 0

0 2#2.#2#3 """"."""""#

0 """"."""""# 35555.55555

,5 , ,!! " #

.! 0 0

=

61

51

41

31

MS

MS

MS

MS

S

S

S


51/332

0 2#2.#2#3 """".""""""#

0 """"."""""# 1!!!!.!!!!!"

.! 0 0

0 2#2.#2#3 """"."""""#

0 """"."""""# 35555.55555

/'&/ L"-AL 3

3 13020".333333

4 4""2".125

5 2.##10#E012

12.E / C3 5104.1!

24414.025 ! 0

" 0

# 0

12 0

13 0

4 5 14 0152500 0 0 15 0

0 24414.025 4""2".125 1 0

0 4""2".125 5104.1! 1! 0

152500 0 0 1" 0

0 24414.025 4""2".125 21 0

0 4""2".125 13020".33333 24 0

25 0

10 11 ," 2 0

10 11 12 2! 0

24414.025 2."2E010 4""2".125 2" 02."25"E010 152500 ".#!32E012 2# 0

4""2".125 ".#!32E012 5104.1! 30 0

24414.025 2."25"E010 4""2".125

2."25!5E010 152500 ".#!32E012

4""2".125 ".#!32E012 13020".33333

/'&/ L"-AL

12.E / C3

2#2.#2#2#

4 5

.! 0 0

0 2#2.#2#3 """".""""""#

0 """"."""""# 1!!!!.!!!!!".! 0 0

0 2#2.#2#3 """"."""""#


52/332

0 """"."""""# 35555.55555

,# ,10 ,11

13 14 15

.! 0 0

0 2#2.#2#3 """".""""""#

0 """"."""""# 1!!!!.!!!!!".! 0 0

0 2#2.#2#3 """"."""""#

0 """"."""""# 35555.55555

/'&/ L"-AL

12.E / C3

24414.025

4 5

152500 0 0

0 24414.025 4""2".125

0 4""2".125 5104.1!

152500 0 0

0 24414.025 4""2".1250 4""2".125 13020".33333

1# 20 ,15

1# 20 21

24414.025 2."2E010 4""2".125

2."25"E010 152500 ".#!32E012

4""2".125 ".#!32E012 5104.1!

24414.025 2."25"E010 4""2".125

2."25!5E010 152500 ".#!32E012

4""2".125 ".#!32E012 13020".33333

/'&/ L"-AL

12.E / C3

2#2.#2#2#

4 5

.! 0 0


53/332

0 2#2.#2#3 """".""""""#

0 """"."""""# 1!!!!.!!!!!"

.! 0 0

0 2#2.#2#3 """"."""""#

0 """"."""""# 35555.55555

,12 ,13 ,141 1! 1"

.! 0 0

0 2#2.#2#3 """".""""""#

0 """"."""""# 1!!!!.!!!!!"

.! 0 0

0 2#2.#2#3 """"."""""#

0 """"."""""# 35555.55555

/'&/ L"-AL

12.E / C3

24414.025

4 5

152500 0 00 24414.025 4""2".125

0 4""2".125 5104.1!

152500 0 0

0 24414.025 4""2".125

0 4""2".125 13020".33333

22 23 ,1

22 23 24

24414.025 2."2E010 4""2".125

2."25"E010 152500 ".#!32E0124""2".125 ".#!32E012 5104.1!

24414.025 2."25"E010 4""2".125

2."25!5E010 152500 ".#!32E012

4""2".125 ".#!32E012 13020".33333

/'&/ L"-AL

12.E / C310000


54/332

4 5

1000000 0 0

0 10000 20000

0 20000 2.!1000000 0 0

0 10000 20000

0 20000 53333.333333

,1! ,1" ,1#

25 2 2!

10000 .045E011 20000

.044"E011 1000000 1.22515E012

20000 1.2251E012 2.!

10000 .044"E011 20000.044"5E011 1000000 1.22515E012

20000 1.2251E012 53333.333333

/'&/ L"-AL

12.E / C3

2#2.#2#2#

4 5

.! 0 0

0 2#2.#2#3 """".""""""#

0 """"."""""# 1!!!!.!!!!!"

.! 0 0

0 2#2.#2#3 """"."""""#

0 """"."""""# 35555.55555

,20 ,21 ,22

2" 2# 30

.! 0 0

0 2#2.#2#3 """".""""""#

0 """"."""""# 1!!!!.!!!!!"

.! 0 0

0 2#2.#2#3 """"."""""#

0 """"."""""# 35555.55555

/'&/ L"-AL


55/332

12.E / C3

10000

4 5

1000000 0 0

0 10000 20000

0 20000 2.!

1000000 0 0

0 10000 20000

0 20000 53333.333333

,# ,10 ,1113 14 15

10000 1."1#3E010 20000

1."1#35E010 1000000 3.!545E012

20000 3.!545E012 2.!

10000 1."1#35E010 20000

1."1#345E010 1000000 3.!545E012

20000 3.!545E012 53333.333333

-/P!@D68@D@49:@(@54'ATB0!E@67@D@48:@(@48#ALSE##CIE,,",!?L""-/P!@

,2 ,3 ,4 ,5 , ,!

4 5 ! " #

,4 ,5 , ,! ," ,#

! " # 12 13

5104.1! 0 0 0 0 0

4""2".125 .! 0 0 0 0

""""."""""""# 0 2#2.#2#3 """".""""""# 0 015!3."""""# 0 """"."""""# 1!!!!.!!!!!" 0 0

0 13!!4!.3# 3.43302E010 2""2".125 4""2".125 .!

"""".""""""# 3.43302E010 25"425.#25# 1.22515E012 ".#!32E012 0

1!!!!.!!!!!!" 2""2".125 1.22515E012 25452.!!!!" 5104.1! 0

0 4""2".125 ".#!32E012 5104.1! 13020".333333 0

0 .! 0 0 0 13!!4!.3#5"3

0 0 2#2.#2#3 """"."""""# 0 4.45#21E010

0 0 """".""""""# 1!!!!.!!!!!" 0 2""2".125

0 0 0 0 0 .!

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 4""2".125

0 0 0 0 0 0


56/332

0 10000 .044"E011 20000 0 0

0 .045E011 1000000 1.2251E012 0 0

0 20000 1.22515E012 2.! 0 0

0 0 0 0 0 10000

0 0 0 0 0 1."1#35E010

0 0 0 0 0 20000

4""2".125 0 0 0 0 02.##10"5E012 0 0 0 0 0

0 24414.025 2."25"E010 4""2".125 4""2".125 0

0 2."2E010 152500 ".#!32E012 ".#!3255E012 0

0 0 0 0 0 24414.025

0 0 0 0 0 2."25"E010

0 0 0 0 0 0

0 0 0 0 0 0

%ang#

o>al ai

1

2

3

45

1 0 0 0 0

.125!42E01! 0 0 0 0

0 1 0 0 0

0 0 .125!4E01! 1 0

0 0 1 .125!4E01! 00 0 0 0 1

0

0

0

0

0

0

o>al ai

AMS 1 ' . AM

L 1


57/332

1

2

3

4

5

0 0 0 0 0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

0

0

0

0

0

0

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2

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0 0 0 0 1

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Element 2 Element 3 Element 4

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62/332

0 0 0

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63/332

%&'1 1

2 1

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0 0 0

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64/332

0 0 0

0 0 0

0 0 0

0 0 0

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0 0 0

0 0 00 0 0

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0 0 0

0 0 0

0 50 50

0 100 100

0 0 0

0 0 0

0 100 100

0 0 0

eormai trktr

,1 = 0.00300 1

,2 = 0.0104 2

,3 = 0.00001" 2

,4 = 0.001!30 2,5 = 0.010423 3

, = 0.000054 3

,! = 0.001443 3

," = 0.0031"! 4

,# = 0.0103" 5

,10 = 0.0000!4 5

,11 = 0.001435 5

,12 = 0.010354

,13 = 0.00001"

,14 = 0.001!02

,15 = 0.0031!! !

,1 = 0.003031 "

,1! = 0.01!"#3 #

,1" = 0.000135 #

,1# = 0.001044 #

,20 = 0.01!"5 10

,21 = 0.0001#3 10

,22 = 0.00104! 10

F S

FF*1.A

F+

%' =


65/332


66/332

.125!42E01! 1 0 0.0104

1 .125!4E01! 0 0.00001"

0 0 1 0.001!30

0 0 152500 0 024414.025 4""2".125 0 24414.025 4""2".125

4""2".125 13020".3333 0 4""2".125 5104.1!

0 0 152500 0 0

24414.025 4""2".125 0 24414.025 4""2".125

4""2".125 5104.1! 0 4""2".125 13020".333333

b lokal %&' / b 6lobal

2!.##!"5 1 1

21.35#!3 2 2

= 4.233E014 3 ,12!.##!"5 4 ,2

21.35#!3 5 ,3

".543"#0 ,4

H bb :

bb :

0 0 0 0.01045#"5!

0 0 0 1.!#1"3E005

0 0 0 0.001!301 0 0 0.010423

0 1 0 0.000054


67/332

0 0 1 0.001443

0 0 .! 0 0

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2#2.#2#3 """"."""""# 0 2#2.#2#3 """".""""""#

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2".3402! 1 ,2

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= ".543"#0 3 ,4

2".3402! 4 ,5

2!.##!"5 5 ,"1.44324! ,!

H bb :

bb :

0 0 0 0.01042343#!

0 0 0 5.3!#33E005

0 0 0 0.001443

1."3!!2E01 1 0 0.000000

1 1."3!!E01 0 0.0000000 0 1 0.0031"!


68/332

0 0 152500 0 0

24414.025 4""2".125 0 24414.025 4""2".125

4""2".125 13020".3333 0 4""2".125 5104.1!

0 0 152500 0 0

24414.025 4""2".125 0 24414.025 4""2".1254""2".125 5104.1! 0 4""2".125 13020".333333


"4.05204" 1 ,5

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= 113.523"3" 3 ,!

"4.05204" 4 10

2".3"0#5# 5 11

1.!0530E013 ,"

H bb :

bb :

0 0 0 0.01042343#!

0 0 0 5.3!#33E005

0 0 0 0.001443

1 0 0 0.0103"

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69/332

0 0 .! 0 0

2#2.#2#2# """"."""""# 0 2#2.#2#3 """"."""""""#

""""."""""""# 35555.5555 0 """"."""""# 1!!!!.!!!!!!"

0 0 .! 0 0

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""""."""""""# 1!!!!.!!!!" 0 """"."""""# 35555.555555


24.#334"" 1 ,5

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= !.5"12 3 ,!

24.#334"" 4 ,#

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!.514!0035 ,11

H bb :

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0 0 0 0.0103"03#4

0 0 0 !.41#3E005

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1."3!!2E01 1 0 0.000000

1 1."3!!E01 0 0.000000

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70/332

0 0 152500 0 0

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0 0 152500 0 0

24414.025 4""2".125 0 24414.025 4""2".125

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= 113.3#13#0 3 ,11

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5."43E014 ,15

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0 0 0 0.0103"03#40 0 0 !.41#3E005

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71/332


72/332

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24414.025 4""2".125 0 24414.025 4""2".125

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24414.025 4""2".125 0 24414.025 4""2".125

4""2".125 5104.1! 0 4""2".125 13020".333333


2".0533 1 ,12

21.35221 2 ,13

= ".540""3 3 ,14

2".0533 4 22

21.35221 5 23

1.!0530E013 ,1

H bb :

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0 0 0 0.01042343#!

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73/332

0 0 1000000 0 0

10000 20000 0 10000 20000

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0 0 1000000 0 0

10000 20000 0 10000 20000

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"1.5"2!!! 1 ,5

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= 44.5!222 3 ,!

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0 0 0 0.01!"#32#22

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74/332


75/332

20000 53333.33333 0 20000 2.!

0 0 1000000 0 0

10000 20000 0 10000 20000

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11".41!223 1 ,2025.04#5"0 2 ,21

= 55.2!!"!" 3 ,22

11".41!223 4 ,#

25.04#5"0 5 ,10

44.#204423 ,11

3 4 5

"4.052 24.#33 115.""#

2".3"1 25.52# 2".34"113.523" !.5" 113.3#14

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2".3"1 25.52# 2".34"

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"4.052 24.#33 115.""#

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2".3"1 25.52# 2".34"113.523" !.514! 113.3#14

BaseT.pe P ?2 ?3 T '2

et 7N 7N 7N 7Nm 7Nm

intati> 2!.##" 21.3 0 0 0

intati> 2!.##" 21.3 0 0 0

intati> 2".34 2!.##" 0 0 0

intati> 2".34 2!.##" 0 0 0

intati> "4.052 2".3"1 0 0 0

intati> "4.052 2".3"1 0 0 0

intati> 24.#33 25.52# 0 0 0

intati> 24.#33 25.52# 0 0 0

intati> 115.""# 2".34" 0 0 0

intati> 115.""# 2".34" 0 0 0

intati> 21.35 2".05 0 0 0

intati> 21.35 2".05 0 0 0

intati> 2".05 21.35 0 0 0

intati> 2".05 21.35 0 0 0

intati> "1.5"3 24.#5 0 0 0intati> "1.5"3 24.#5 0 0 0

intati> 25.05 1".41! 0 0 0


76/332

intati> 25.05 1".41! 0 0 0

intati> 11".41! 25.05 0 0 0

intati> 11".41! 25.05 0 0 0


77/332


78/332

Y H Special(t Global GlobalY GlobalHm m +e/No m m m

0 0 No # 0 0

0 4 No # 0 4

0 4 No 3 0 4

0 0 No 3 0 0

0 4 No 3 0 4

0 4 No # 0 4

0 0 No 3 0 0

0 0 No # 0 0

0 " No 3 0 "0 " No 3 0 "

Lengt) Bentroid BentroidY BentroidH

m m m m

4 # 0 2

0 4

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79/332


80/332


81/332


82/332


83/332


84/332


85/332


86/332


87/332


88/332


89/332

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90/332

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91/332


92/332


93/332

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10 11 ," ,# ,10 ,11

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0 0 0 0 0 0

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0 0 0 0 0 0

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3.!5445E012 25452.!!!!!" 0 """".""""""# 1!!!!.!!!!!!" 5104.1!

0 0 #10"0.!2#1! 2."25!5E010 4""2".125 0

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94/332

0 0 0 0 0 0

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0 0 0 0 0 00 0 0 0 0 0

0 0 0 0 0 0

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152500 ".#!3255E012 0 0 0 ".#!3255E012

0 0 24414.025 2."25"E010 4""2".125 0

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0

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95/332

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96/332


97/332

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98/332

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99/332


100/332

0 0 0

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101/332


102/332

0.00300 (oint "$tp$tBase BaseT.pe /1

0.0104 et et et m

0.00001" 1 %E% intati> 0

0.001!30 2 %E% intati> 0.01040.010423 3 %E% intati> 0.010423

0.000054 4 %E% intati> 0

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0.0031"! %E% intati> 0.010354

0.0103" ! %E% intati> 0

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103/332

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1."02E00 1.5401E005 2.4323"1E00" 2.5##"5E00 4.#4#!2!E00 4.4!5!40E00

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0.00000104# #.#!03#E00 1.55#4!2E00" 1."02E00 3.2254"0E00 2."#5"5E00

3.2254"0E00 0.00002#332 4.3"1#5E00" 4.#4#!2!E00 #.41#4#!E00 ".52445E00

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104/332


105/332


106/332

0.000054

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107/332

0.010423

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108/332

0.0000!4

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109/332


110/332

0.00001"

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111/332

0.000054

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112/332

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113/332


114/332

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115/332


116/332

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117/332


118/332


119/332


120/332


121/332


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123/332


124/332


125/332


126/332


127/332

12 13 14 15 1 1!

0 0 0 0 0 0


128/332


129/332


130/332


131/332

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1 1! 1" 1# 20 21

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0 0 0 0 0 0

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0 0 0 0 0 00 0 0 0 0 0

0 10000 .044"E011 20000 0 0

0 .044"E011 1000000 1.22515E012 0 0

0 20000 1.22515E012 2.! 0 0

0 0 0 0 0 0

0 0 0 0 10000 1."E010

0 0 0 0 1."1#35E010 1000000

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132/332

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133/332


134/332


135/332


136/332


137/332

Element " Element # Element 10

0 0 0

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138/332

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139/332


140/332

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m m -a,ian -a,ian -a,ian

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141/332

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0.00011212" 2.#411E00" 1.#001!E00 0.000112101 2.!"!"1E00" 2.1E00

2.12!03E00! .3!545"E00! 0.0000000#4 2.12#13E00! 1.14##1E00# ".4E00"

1.""3!4E005 2.154"#E00" 2.353#"E00! 1."!"#E005 4.!1#!43E00" 1.!2E00

3.4111E005 .!15335E010 ".3022!E00! 3.41"5E005 1.31444E00" .3E00"

0.000112101 2.!"!"12E00" 2.1211"E00 0.00011212" 2.#412E00" 1.#E00

2.12#13E00! 1.14##1E00# ".4100E00" 2.12!E00! .3!545"E00! #.40E00"

1."!""5E005 4.!1#!4E00" 1.!1##E00 1.""3!E005 2.154"#3E00" 2.4E00!

0.000115"352 2.!5"42E00" 2.10!4E00 0.000115"3!! 2.521#0E00" 1.#E00

2.133#3E00! 2.5"5"!E010 1.0!!13!E00" 2.134E00! 1.0#!"1"E00# #.5"E010

0.00001"!22 1.11"210E00# 1.522405E00! 0.00001"!1 4.0!224E00" 3.!3E00!

3.41"45E005 1.314442E00" .2#04E00" 3.411E005 .!1533E010 ".30E00!

3.40!!2E005 1.0#033E00" !.141!"2E00! 3.40"1E005 2.#"1"33E00" 5.25E00!

0.00024202!! 5.455"0"E00! 1.#003"E005 0.0002412" 5.45"24E00! 1.#E005

5.455"0"E00! 1.3!00E00 2.2#"0E00! 5.45"24E00! 2.41213E00# 2.2E00!

1.#003"E005 2.2#"0E00! 1.51332E005 1.""12E005 2.1#"4E00! 2.53E00!

0.0002412!# 5.45"235E00! 1.""125E005 0.00024202" 5.455"1E00! 1.#E005

5.45"24E00! 2.41213E00# 2.1#"4E00! 5.455"E00! 1.3!00E00 2.30E00!1.""125E005 2.1#"E00! 2.52"223E00! 1.#004E005 2.2#!#"E00! 1.52E005


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1" 21 24 25 2 2! 2" 2# 30

0 0 0 0 0 0 0 0 0


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22 23 ,1 ,1! ,1" ,1# ,20 ,21 ,22

22 23 24 25 2 2! 2" 2# 30

,22 1 2 10 11 1# 20 22 23

30 1 2 10 11 1# 20 22 23

0 4""2".13 2.##E012 0 0 0 0 0 0

0 24414.0 #.4E011 0 0 0 0 0 0

0 #.4E011 152500 0 0 0 0 0 00 4""2".13 2.##E012 0 0 0 0 0 0

0 0 0 24414.0 3E010 0 0 0 0

0 0 0 3E010 152500 0 0 0 0

0 0 0 4""2".13 ".#!E012 0 0 0 0

0 0 0 4""2".13 ".#!E012 0 0 0 0

20000 0 0 0 0 24414.0 2."E010 0 0

3."E012 0 0 0 0 3E010 152500 0 0

2.! 0 0 0 0 4""2".13 ".#!E012 0 0

0 0 0 0 0 0 0 24414.0 3E010

0 0 0 0 0 0 0 3E010 152500

0 0 0 0 0 0 0 4""2".13 ".#!E012

0 0 0 0 0 4""2".13 ".#!E012 0 0

0 0 0 0 0 0 0 4""2".13 ".#!E012


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0 0 0 0 0 0 0 0 0

""""."""# 0 0 0 0 0 0 0 0

1!!!!.!!" 0 0 0 0 0 0 0 0

20000 0 0 0 0 0 0 0 0

"""".""# 0 0 0 0 0 0 0 0

""""".""# 0 0 0 0 0 0 0 0

0 24414.03 #.42E011 0 0 0 0 0 00 #.42E011 152500 0 0 0 0 0 0

0 0 0 24414.0 2."3E010 0 0 0 0

0 0 0 2."3E010 152500 0 0 0 0

0 0 0 0 0 24414.0 2."3E010 0 0

0 0 0 0 0 2."3E010 152500 0 0

0 0 0 0 0 0 0 24414.0 2."3E010

0 0 0 0 0 0 0 2."3E010 152500


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0

0

0

0

00

0


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0

0

0

0

0

0

00

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0


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0 0.0030 &7

50 0.0104 &7

0 1.!#E005 &7

0 0.001!3 &7

0 0.010423 &7

0 5E005 &7

0 0.001443 &7

0 0.0031"! &7

0 0.0103" &7

0 !E005 &7

0 0.001435 &7

0 0.010354 &7

0 2E005 &7

0 0.001!02 &7

0 0.0031!! &7

0 0.003031 &7

50 0.01!"#3 &7

100 0.000135 &7

0 0.001044 &7

0 0.01!"5 &7

100 0.0001#3 &70 0.00104! &7


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3 ,1

4 ,2

5 ,3 ,4

! ,5

" ,

# ,!

12 ,"

13 ,#

14 ,10

15 ,11

1 ,12

1! ,13

1" ,14

21 ,15

24 ,1


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25 ,1!

2 ,1"

2! ,1#

2" ,20

2# ,21

30 ,22

1 12 2

10 10

11 11

1# 1#

20 20

22 22

23 23


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,20

,21,22

,#

,10

,11


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Data beban

P1 = 20 kN

q1 = 10 kN/m

Properties penampang

Bahan / material :

No. Element E G b (m) h (m) (m2)

1 2!000 0.3 0. 1."00000E01

2 2!000 0.3 0. 1."00000E01

3 2!000 0.3 0. 1."00000E01

PENYELESAIAN :

STEP 1 : Discretize and Global Degrees of reedom !D"#

Defined DOF

after boundary


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D" ses$ai s$mb$ global str$%t$r dan tent$%an &eban ata$ 'omen pada Tit

%&'

11 1 0 ran. +

2 2 0 -ot.

23 3 0 ran. +

4 ,1 0 -ot.

35 5 0 ran. +

,2 0 -ot.

4! ,3 20 ran. +

" ,4 0 -ot.

Nomor +oint $nt$% setiap element Displacement titi% s$mb$ global str$%t$

Element !i#(oint

,t (,e6)(

n6 n6 k +

1 1 2 0 1 2

2 2 3 0 3 4

3 3 4 0 5

,e%ap Elemen dalam 'atri%s -e%a%$an &atang

Batan6 4.E / 2.E / .E / C2 12.E / C3 ,t (,e6)

1 !2.# 3.45 13."!5 3.41!1"!5 0

2 2#1. 145." 21".! 21".! 0

3 2#1. 145." 21".! 21".! 0

'eng)it$ng Ga.a dan 'omen /+$ng !A%ibat beban l$ar di sepan+ang bentan

ELE'ENT 1

7arena ti,ak a,a 6a8a ,ie9anan6 bentan6 reebo,8 maka rekai


/nit : kNm


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/nit : kNm

al ai 1 2 3 4

Global ai 1 2 3 4

%&' 0 0 ,3 ,4

===> DALA' S/'&/ L"

Batan6 = 1

Elemen ,alam matrik kekakan batan6 :4.E / 2.E / .E / C2 12.E / C3

!2.# 3.45 13."!5 3.41!1"!5

7eeaian S& L"-AL element ,en6an S& GL"&AL dan D" ara) global



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D = >o theta = 1

D8 = in theta = 0

o>al ai 1 2 3 4

1 3.41!1"!5 13."!5 3.41!1"!5 13."!5

2 13."!5 !2.# 13."!5 3.45

3 3.41!1"!5 13."!5 3.41!1"!5 13."!5

4 13."!5 3.45 13."!5 !2.#

1 2 3 ,1

Global ai 1 2 3 4

1 3.41!1"!5 13."!5 3.41!1"!5 13."!5

2 13."!5 !2.# 13."!5 3.45

3 3.41!1"!5 13."!5 3.41!1"!5 13."!5

4 13."!5 3.45 13."!5 !2.#

===> DALA' S/'&/ L"

Batan6 = 2


4.E / 2.E / .E / C2 12.E / C3

2#1. 145." 21".! 21".!

D = >o theta = 1

D8 = in theta = 0

o>al ai 1 2 3 4

1 21".! 21".! 21".! 21".!

2 21".! 2#1. 21".! 145."

3 21".! 21".! 21".! 21".!

4 21".! 145." 21".! 2#1.

3 ,1 5 ,2

Global ai 3 4 5

3 21".! 21".! 21".! 21".!

4 21".! 2#1. 21".! 145."

5 21".! 21".! 21".! 21".!

21".! 145." 21".! 2#1.

===> DALA' S/'&/ L"

Batan6 = 3


4.E / 2.E / .E / C2 12.E / C3

2#1. 145." 21".! 21".!

D = >o theta = 1

D8 = in theta = 0

@1A =

@1

A =


@2A =

@2

A =



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o>al ai 1 2 3 4

1 21".! 21".! 21".! 21".!

2 21".! 2#1. 21".! 145."

3 21".! 21".! 21".! 21".!

4 21".! 145." 21".! 2#1.

5 ,2 ,3 ,4

Global ai 5 ! "

5 21".! 21".! 21".! 21".!

21".! 2#1. 21".! 145."

! 21".! 21".! 21".! 21".!

" 21".! 145." 21".! 2#1.

IE,,",!?L""-/P!@D6

rtkan eai mb 6lobal/trktr ,en6an %&' 8an6 ,i,ahlkan

%&' : 1 2 3 ,1

%i.Global ai : 1 2 3 4

%itata lan6 manal ==F ,1 ,2 ,3 ,4

4 ! "

4 34.5 145." 0 0

145." 5"3.2 21".! 145."

! 0 21".! 21".! 21".!

" 0 145." 21".! 2#1.1 13."!5 0 0 0

2 3.45 0 0 0

3 205.03125 21".! 0 0

5 21".! 0 21".! 21".!

0it$ng 'atri%s &eban (oint* Nodal

Nodal forces:

a A%ibat beban di sepan+ang bentang !%ed$a $+$ng di%e%ang#

Element: 1

o>al ai

0 1

= 0 2

0 3

0 4

7arena 9a,a balok mener mb lokal batan6 =

Global ai

1 0

@3A =

@3

A =

'AT,I-S -E-A-/AN ST,/-T/, ;S'S<

@A =

@A1


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= 2 0

3 0

4 0

Element: 2

o>al ai

10 1

= 3.33333333 2

10 3

3.3333333 4


Global ai

3 10

= 4 3.3333333333

5 10

3.333333333

Element: 3

o>al ai

0 1

= 0 2

0 3

0 4


Global ai

5 0

= 0

! 0

" 0

S$s$n matri%s beban e%iFalen

$oint b Global Element 1 Element 2

1 1 0 0

1 2 0 0

2 3 0 10

2 4 0 3.3333333333

e = 3 5 0 10

3 0 3.333333333

4 ! 0 0

4 " 0 0

e = 0 1 10 2 2

10 3 3

@A1

@A2

@A2

@A3

@A3


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3.333333333 4 ,1

10 5 5

3.333333333 ,2

0 ! ,3

0 " ,4

b A%ibat &eban di +oint !%ed$a $+$ng di%e%ang#

$oint b Global %&'

1 1 0 1

1 2 0 2

2 3 0 3

2 4 0 ,1

= 3 5 0 5

3 0 ,2

4 ! 20 ,3

4 " 0 ,4

c &eban %ombinasi ata$ gab$ngan !Ae C A+#

$oint b Global %&'

1 1 1 0

1 2 2 0

2 3 3 10

2 4 ,1 3.333333333

> = 3 5 5 10

3 ,2 3.3333333333

4 ! ,3 04 " ,4 0

ata lan6 > : %i9la>ement / %eormai

,1 3.3333333

,2 3.33333333

,3 20

,4 0

> = 1 0

2 0

3 10

5 10

-ea>tion:

GAYAGAYA DALA' /(/NG ELE'EN

&atang: 1%eormai trktr ,alam arah B G&B ,i batan6 1H bb :



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1 1 0.000000

2 2 0.000000

3 3 0.000000

4 ,1 0.051440

%eormai trktr ,alam arah B &7 ,i batan6 1H bb :

7arena 9a,a balok mener mb lokal batan6 = mb 6lobal t

b lokal %&'

0.000000 1 1

0.000000 2 2

% 1 = 0.000000 3 3

0.051440 4 ,1

Ga.a $+$ng batang

0 3.41!1"!5 13."!5

0 13."!5 !2.#

0 3.41!1"!5 13."!5

0 13."!5 3.45

0 0.!03125

0 1."!5

0 0.!03125 =

0 3.!5

&atang: 2

%eormai trktr ,alam arah B G&B ,i batan6 1H bb :


3 3 0.000000

4 ,1 0.051440

5 5 0.000000

,2 0.15143



b lokal %&'

0.000000 1 3

0.051440 2 4

% 2 = 0.000000 3 5

0.15143 4 ,2

Ga.a $+$ng batang

@%A1 =

A' 1 = A'L 1 C S' 1 D' 1

@1A =

@1A =

@%A2 =


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10 21".! 21".!

3.33333333 21".! 2#1.

10 21".! 21".!

3.3333333 21".! 145."

10 21."!5

3.33333333 !.0"33333

10 21."!5 =

3.3333333 3.!

&atang: 3

%eormai trktr ,alam arah B G&B ,i batan6 1H bb :


5 5 0.000000

,2 0.15143

! ,3 0."!24

" ,4 0.425"12



b lokal %&'0.000000 1 5

0.15143 2

% 3 = 0."!24

Finite Element (Direct Stiffness) Kekakuan Langsung

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