Tugas Analisis Struktur V MEKTEK Sloop D

(1)

SOAL: SISTEM PORTAL 2 DIMENSI QUIZ

Penyelesaian :

A. ELEMEN AB(1) Diketahui :

A : 0.1 cm²

E : 200000 kg/cm² = 200 t/cm²

I : 500 cm α : 45 ° L

1 : (x+y) = 10 m L

2 : (z) = 8 m L

3 : (x) = 1 m q : (y+z) = 17 t/m P : (x+z) = 9 ton

(2)

(3)

Cos27

0 270Sin 0 0 0 0

-Sin27

0 Cos270 0 0 0 0

0 0 1 0 0 0

[ T ] = 0 0 0 Cos 270 Sin 270 0

0 0 0 - Sin270 Cos 270 0

0 0 0 0 0 1

0 1 0 0 0 0

-1 0 0 0 0 0

0 0 1 0 0 0

[ T ]

T₌ ₀ ₀ ₀ ₀ ₁ ₀

0 0 0 -1 0 0

0 0 0 0 0 1

[

Kg1

]

= [ T ]

T

. [

KL1

] . [ T ]

[

Kg

1

]

₌

0 1- 0 0 0 0 20 0 0 20- 0 0

1 0 0 0 0 0 0 0.001 3 0 0.001- 3

0 0 1 0 0 0

x

0 3 400 0 -3 100

0 0 0 0 -1 0 -20 0 0 20 0 0

0 0 0 1 0 0 0 0.001- -3 0 0.001 -3

0 0 0 0 0 1 0 3 200 0 -3 400

0 1 0 0 0 0

-1 0 0 0 0 0

x

0 0 1 0 0 0

0 0 0 0 1 0

0 0 0 -1 0 0

(4)

[

Kg1

]

=

B. ELEMEN

(5)

[ KL 2 ] =

−0,1.200

800 0 0

0,1.200

800 0 0

0

−12.200.500 8003

−6.200.500

8002 0

12.200.500 8003

−6.200.500 8002

0

6.200.500 8002

2.200.500

800 0

−6.200.500 8002

4.200.500 800

Cos 0 Sin 0 0 0 0 0

-Sin 0 Cos 0 0 0 0 0

0 0 1 0 0 0

[ T ]

= 0 0 0 Cos 0 Sin 0 0

0 0 0 - Sin 0 Cos 0 0

0 0 0 0 0 1

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

[ T ]

T₌ ₀ ₀ ₀ ₁ ₀ ₀

0 0 0 0 1 0

0 0 0 0 0 1

[

Kg2

]

= [ T ]

T

. [

KL

2

_{] . [ T ]}

[

Kg 2

]

₌

1 0 0 0 0 0 25 0 0 -25 0 0

0 1 0 0 0 0 0 0.002 1 0 0.002- 1

0 0 1 0 0 0

x

0 1 500 0 -1 125

0 0 0 1 0 0 -25 0 0 25 0 0

0 0 0 0 1 0 0

-0.00

2 -1 0 0.002 -1

(6)

X

[

Kg

2

]

₌

C. ELEMEN CD (3)

sinα=CO

CD

sin 45= 9

CD

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

UB VB  B UC VC  C

25 0 0 -25 0 0 UB

0 0 1 0 0 1 VB

0 1 500 0 -1 125 B

-25 0 0 25 0 0 UC

0 0 -1 0 0 -1 VC

(7)

CD= 9 1272,83

6.200.500

1272,82 0

−12.200.500 1272,83

6.200.500 1272,82

0

6.2000.500 1272,82

4.200.500

1272,8 0

−6.200.500 1272,82

200.500 1272,83

−6.200.500

1272,82 ₀

12.200.500 1272,83

−6.200.500 1272,82

0

6.200.500 1272,82

2.200.500

1272,8 0

−6.2000.500 1272,82

(8)

(9)

D. GLOBAL

UA VA ѲA UB VB ѲB UC VC ѲC UD VD ѲD

0 0 1 0 0 1 0 0 0 0 0 0 UA

0 20 0 0 -20 0 0 0 0 0 0 0 VA

1 0 400 -1 0 200 0 0 0 0 0 0 ѲA

0 0 -1 25 0 -1 -25 0 0 0 0 0 UB

0 -20 0 0 20 1 0 0 1 0 0 0 VB

1 0 100 -1 1 900 0 -1 125 0 0 0 ѲB

0 0 0 -25 0 0 33 8 0 -8 -8 0 UC

0 0 0 0 0 -1 8 8 -1 -8 -8 0 VC

0 0 0 0 1 250 0 -1 814 0 0 79 ѲC

0 0 0 0 0 0 -8 -8 0 8 8 0 UD

0 0 0 0 0 0 -8 -8 0 8 8 0 VD

0 0 0 0 0 0 0 0 157 0 0 314 ѲD

(10)

[

F

]

= [ K ]. [

U

]

{

F_A

GA M_A F_B G_B M_B FC GC MC F_D G_D

MD

} {

U_A VA θ_A U_B V_B θ_B UC VC θC U_D V_D θD

}

(11)

UB VB ѲB UC VC ѲC UA VA ѲA UD VD ѲD

0.9 25 0 -1 -25 0 0 0 0 -1 0 0 0 UB

-68 0 20 1 0 0 1 0 -20 0 0 0 0 VB

-89.856

7 -1 1 900 0 -1 125 1 0 100 0 0 0 ѲB

0 -25 0 0 33 8 0 0 0 0 -8 -8 0 UC

-68 0 0 -1 8 8 -1 0 0 0 -8 -8 0 VC

90.666

7 0 1 250 0 -1 814 0 0 0 0 0 79 ѲC

RHA+8

.1 0 0 1 0 0 0 0 0 1 0 0 0 0

RVA 0 -20 0 0 0 0 0 20 0 0 0 0 0

MA-7.29 -1 0 200 0 0 0 1 0 400 0 0 0 0

RHD 0 0 0 -8 -8 0 0 0 0 8 8 0 0

RVD 0 0 0 -8 -8 0 0 0 0 8 8 0 0

MD 0 0 0 0 0 157 0 0 0 0 0 314 0

REDUKSI

0.9 25 0 -1 -25 0 0 UB

-68 0 20 1 0 0 1 VB

-89.8567 -1 1 900 0 -1 125 ѲB

0 -25 0 0 33 8 0 UC

-68 0 0 -1 8 8 -1 VC

90.6667 0 1 250 0 -1 814 ѲC

INVERS

UB 226 0 0 226 -226 0 0.9

VB 0 0 0 0 0 0 -68

ѲB 0 0 0 0 0 0 -89.857

UC 226 0 0 226 -226 0 0

VC -226 0 0 -226 226 0 -68

(12)

RHA+8.

1 0 0 1 0 0 0

RVA 0 -20 0 0 0 0

MA-7.29 -1 0 200 0 0 0

RHD 0 0 0 -8 -8 0

RVD 0 0 0 -8 -8 0

MD 0 0 0 0 0 157

=

RAH = -21,69 + 8,1 = -13,59

RAV = 93,85

MA = -10340,12- 7,29 = -10347,41

RHD = 20,79

RVD = 42.15

MD = -9122,90

CROSS CHECK

ƩV = RAV + RDV – (q.l) = 93,85+ 42.15– (17 . 8) = 0

ƩH = RAH +RDH + P -21.69

(13)

= -13,59+20,79+9 = 0

SOAL: SISTEM RANGKA BATANG QUIZ Nama : Viorenza Everlyn

Npm : 123 110 198

P1 = 10 TON P2 = 5 TON P3 = 2 TON L = 10 m α = 60

Tentukan Kekakuan elemen Metode Matrix !

Penyelesain:

(14)

[KL1_{] =} AE 10

[

1 0 −1 0

0 0 0 0

−1 0 1 0

0 0 0 0

]

[T1_{] =}

[

cos 60 sin 60 0 0

−sin 60 cos 60 0 0

0 0 cos 60 sin 60

0 0 −sin 60 cos 60

]

ELEMEN AB (1)

[KG

1

_]

_{= [T}

1

_]

T

_{. [KL}

1

_{] . [T}

1

_]

0.5 0.866- 0 0 1 0 -1 0 0.5 0.866 0 0

0.8

66 0.5 0 0 AE 0 0 0 0 X

-0.86

6 0.5 0 0

0 0 0.5

-0.86

6 10 -1 0 1 0 0 0 0.5 0.866

0 0 0.866 0.5 0 0 0 0 0 0 0.866- 0.5

UA VA UB VB

0.25 0.433 -0.25 0.433- UA

0.433 0.75 0.433- -0.75 VA

-0.25 0.433- 0.25 0.433 UB -0.433 -0.75 0.433 0.75 VB

ELEMEN CD (5)

[KG

5

_]

_{= [T}

5

_]

T

_{. [KL}

5

_{] . [T}

5

_]

0.5 0.866025 0 0

-0.8660

25 0.5 0 0

0 0 0.5 0.866025

0 0

-0.866

(15)

0.5 0.866- 0 0 1 0 -1 0 0.5 0.866 0 0

0.8

66 0.5 0 0 AE 0 0 0 0 X

-0.86

6 0.5 0 0

0 0 0.5

-0.86

6 10 -1 0 1 0 0 0 0.5 0.866

0 0 0.866 0.5 0 0 0 0 0 0 0.866- 0.5

UC VC UD VD

0.25 0.433 -0.25 0.433- UC

0.433 0.75 0.433- -0.75 VC

-0.25 0.433- 0.25 0.433 UD -0.433 -0.75 0.433 0.75 VD

ELEMEN AC(2) = ELEMEN BD(4) = ELEMEN CE (6)

[KL2_{] =} AE 10

[

1 0 −1 0

0 0 0 0

−1 0 1 0

(16)

[T2_{] =}

[

cos 0 sin 0 0 0

−sin 0 cos 0 0 0

0 0 cos 0 sin 0

0 0 −sin 0 cos 0

]

=¿

Jadi [KL2_] ₌

Jadi [KL4_]=

Jadi [KL6_] ₌

ELEMEN BC (3) = ELEMEN DE (7)

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

UA VA UC VC

1 0 -1 0 UA

0 0 0 0 VA

-1 0 1 0 UC

0 0 0 0 VC

UB VB UD VD

1 0 -1 0 UB

0 0 0 0 VB

-1 0 1 0 UD

0 0 0 0 VD

UC VC UE VE

1 0 -1 0 UC

0 0 0 0 VC

-1 0 1 0 UE

(17)

[KL2_{] =} AE 10

[

1 0 −1 0

0 0 0 0

−1 0 1 0

0 0 0 0

]

[T2_{] =}

ELEMEN BC (3)

[KG

3

_]

_{= [T}

3

_]

T

_{. [KL}

3

_{] . [T}

3

_]

0.5 0.866 0 0 1 0 -1 0 0.5 -0.86

6 0 0

-0.86

6 0.5 0 0 AE 0 0 0 0 X 0.866 0.5 0 0

0 0 0.5 0.866 10 -1 0 1 0 0 0 0.5 -0.86

6

0 0

-0.86

6 0.5 0 0 0 0 0 0 0.866 0.5

UC VC UB VB

0.25 0.433- -0.25 0.433 UC -0.433 0.75 0.433 -0.75 VC

-0.25 0.433 0.25 0.433- UB

0.433 -0.75 0.433- 0.75 VB

ELEMEN DE (7)

[KG

7

_]

_{= [T}

7

_]

T

_{. [KL}

7

_{] . [T}

7

_]

0.5 0.866 0 0 1 0 -1 0 0.5 -0.86

6 0 0

- 0.5 0 0 A 0 0 0 0 X 0.8 0.5 0 0

0.5 0.866- 0 0 0.866 0.5 0 0

(18)

0.86

6 E 66

0 0 0.5 0.866 10 -1 0 1 0 0 0 0.5 -0.86

6

0 0

-0.86

6 0.5 0 0 0 0 0 0 0.866 0.5

UE VE UD VD

0.25 0.433- -0.25 0.433 UE -0.433 0.75 0.433 -0.75 VE

-0.25 0.433 0.25 0.433- UD

0.433 -0.75 0.433- 0.75 VD

GLOBAL

UA VA UB VB UC VC UD VD UE VE

1.25 0.43 -0.25 -0.43 -1 0 0 0 0 0 UA

0.43 0.75 -0.43 -0.75 0 0 0 0 0 0 VA

-0.25 -0.43 1.50 0.00 -0.25 0.43 -1 0 0 0 UB

-0.43 -0.75 0.00 1.50 0.43 -0.75 0 0 0 0 VB

-1 0 -0.25 0.43 2.50 0.00 -0.25 -0.43 -1 0 UC

0 0 0.43 -0.75 0.00 1.50 -0.43 -0.75 0 0 VC

0 0 -1 0 -0.25 -0.43 1.50 0.00 -0.25 0.43 UD

0 0 0 0 -0.43 -0.75 0.00 1.50 0.43 -0.75 VD

0 0 0 0 -1 0 -0.25 0.43 0.25 -0.43 UE

(19)

KONDISI BATAS

[

F

]

= [ K ]. [

U

]

{

FA

GA F_B G_B F_C G_C FD GD F_E

G_E

}

{

RHA RVA P=10ton

0 0

P=−2ton

0

P=5ton

0

RVE

}

{

(20)

RE-ARRANGEMENT RHA -0.25

-0.4

3 -1 0 0 0 0 1.25 0.43 0 UA RVA -0.43

-0.7

5 0 0 0 0 0 0.43 0.75 0 VA RVE 0 0 0 0 0 -1 0 0 0 0.75 VE

REDUKSI MATRIX

(21)

0.43 0.75

0 0 0 -1 0 0.25 0.43 0.25- UE

INVERS

UB 1.875 -UD 1.375

(22)

-2.080 19 9.080

189

CROSS CHECK

ΣV= 0 ΣH= 0

RVA + RVE –P2 – P3 = 0 RHA + P1 = 0 -2.08019 + 9.080189 – 5 – 2 = 0 -10 + 10 = 0

0 = 0 OK! 0 = 0 OK!

GAYA BATANG

ELEMEN 1

i = A

j = B

{

FA

GA FB GB

}

= [T1_{] [KG}1_]

{

UA

VA UB VB

}

{

FA

GA FB GB

}

=

[

0,5 0,87 0 0

−0,87 0,5 0 0

0 0 0,5 0,87

0 0 −0,87 0,5

]

AE

5

[

0,25 0,435 −0,25 −0,435 0,435 0,757 −0,435 −0,757

−0,25 −0,435 0,25 0,435

−0,435 −0,757 0,435 0,757

]

5

AE

{

0 0 21,709

−9,760

}

¿

-2.40 0 2.40

(23)

ELEMEN 2

ELEMEN 3

(24)

ELEMEN 4

i = B

j = D

{

FB

GB FD GD

}

= [T4_{] [KG}4_]

{

UB VB UD VD

}

{

FB

GB FD GD

}

=

[

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

]

AE

5

[

1 0 −1 0

0 0 0 0

−1 0 1 0

0 0 0 0

]

5

AE

{

21.709

−9.760 14.111

−12.067

}

=

ELEMEN 5

i = C

j = D

{

FC

GC FD GD

}

(25)

{

FC GC FD GD

}

=

[

0,5 0,87 0 0

−0,87 0,5 0 0

0 0 0,5 0,87

0 0 −0,87 0,5

]

AE

5

[

0,25 0,435 −0,25 −0,435 0,435 0,757 −0,435 −0,757

−0,25 −0,435 0,25 0,435

−0,435 −0,757 0,435 0,757

]

5

AE

=

ELEMEN 6

i = C

j = E

{

FC GC FE

¿

}

= [T6_{] [KG}6_]

{

UC VC UE VE

}

-4.71

0 4.71

(26)

{

FC

ELEMEN 7