Contoh Soal Analisa Matriks (Portal Hal.172)

7/25/2019 Contoh Soal Analisa Matriks (Portal Hal.172)

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CONTOH 4.2

Properties penampang

Batang 1 (KOLOM) :

b1 = 0.35 m

h1 = 0.35 m

A1 = 0.1225 m2

I1 = 0.0012505208 m4

L1 = 3.5 m

Batang 2 (BALOK) :

b2 = 0.20 m

h2 = 0.45 m

A2 = 0.09 m2

I2 = 0.00151875 m4

L2 = 5 mBahan / mate!a" :

# = 21000000 $%/m2

&!t'ng mat!$ $e$a$'an batang 1 (M1)

A$!bat *!+"a,ement aah 1 -- 1 (tan"a! aah b "$a" 1)

M 1.1 = #.A / L = 735000

M 2.1 = 0 = 0

M 3.1 = 0 = 0


M 1.2 = 0 = 0

M 2.2 = 12.#I/L3 = 7350

M 3.2 = .#I / L2 = 1282.5

A$!bat *!+"a,ement aah 3 -- 3 (ta! aah b "$a" 3)


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M 1.3 = 0 = 0

M 2.3 = .#I / L2 = 1282.5

M 3.3 = 4 #I / L = 30012.5


M 1.4 = - #.A / L = -735000

M 2.4 = 0 = 0

M 3.4 = 0 = 0


M 1.5 = 0 = 0

M 2.5 = -12.#I/L3 = -7350

M 3.5 = -.#I / L2 = -1282.5

A$!bat *!+"a,ement aah -- (ta! aah b "$a" )

M 1. = 0 = 0

M 2. = .#I / L2 = 1282.5

M 3. = 2 #I / L = 1500.25

''nan mat!$ $e$a$'an batang 1 (M1) :

M11 M12 M13 M14

M21 M22 M23 M24

M1 = M31 M32 M33 M34

M41 M42 M43 M44

M51 M52 M53 M54

M1 M2 M3 M4

'*'t batang 1 : = 90

Mat!$ tanma! ta! :

0 0

= 0 0

0 0 1 0

0 0 0

0 0 0

0 0 0 0

1

, 1

!n 1

- !n 1

, 1

, 1

- !n 1


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an+e mat!$ ta! 6

0.00 -1 0 0

= 1 0.00 0 0

0 0 1 0

0 0 0 0.000 0 0 1

0 0 0 0

Mat!$ $e$a$'an t'$t' mena*! :

0.00 -1.00 0.00 0.00

1.00 0.00 0.00 0.00

0.00 0.00 1.00 0.00M 1 = 0.00 0.00 0.00 0.00

0.00 0.00 0.00 1.00

0.00 0.00 0.00 0.00

4.50242#-011 -7350 -1282.5 -4.502#-011

735000 4.502#-013 7.8792#-013 -735000

0 1282.5 30012.5 0

-4.5024#-011 7350 1282.5 4.502#-011

-735000 -4.502#-013 -7.879#-013 735000

0 1282.5 1500.25 0

&!t'ng mat!$ $e$a$'an batang 2 (M2)


M 1.1 = #.A / L = 378000

M 2.1 = 0 = 0

M 3.1 = 0 = 0


M 1.2 = 0 = 0

M 2.2 = 12.#I/L3 = 301.8

M 3.2 = .#I / L2 = 754.5

A$!bat *!+"a,ement aah 3 -- 3 (ta! aah b "$a" 3)

M 1.3 = 0 = 0

M = . M .


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M 2.3 = .#I / L2 = 754.5

M 3.3 = 4 #I / L = 25515


M 1.4 = - #.A / L = -378000

M 2.4 = 0 = 0

M 3.4 = 0 = 0


M 1.5 = 0 = 0

M 2.5 = -12.#I/L3 = -301.8

M 3.5 = -.#I / L2 = -754.5

A$!bat *!+"a,ement aah -- (ta! aah b "$a" )

M 1. = 0 = 0

M 2. = .#I / L2 = 754.5

M 3. = 2 #I / L = 12757.5

''nan mat!$ $e$a$'an batang 2 (M2) :

M11 M12 M13 M14

M21 M22 M23 M24

M2 = M31 M32 M33 M34

M41 M42 M43 M44

M51 M52 M53 M54

M1 M2 M3 M4

'*'t batang 2 : = 0

Mat!$ tanma! ta! :

0 0

= 0 0

0 0 1 0

0 0 0

0 0 00 0 0 0

2

, 1

!n 1

- !n 1

, 1

, 1

- !n

1


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an+e mat!$ ta! 6

1.00 0 0 0

= 0 1.00 0 0

0 0 1 0

0 0 0 1.00

0 0 0 0

0 0 0 0

Mat!$ $e$a$'an t'$t' mena*! :

1.00 0.00 0.00 0.00

0.00 1.00 0.00 0.00

0.00 0.00 1.00 0.00

M 2 = 0.00 0.00 0.00 1.000.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00

378000 0 0 -378000

0 301.8 754.5 0

0 754.5 25515 0

-378000 0 0 378000

0 -301.8 -754.5 0

0 754.5 12757.5 0

Susun MATRIKS KEKAKUAN STRUKTUR

1

1 2 3 7

7350 0 -1283 -7350

0 735000 0 0

M 1 = -1283 0 30013 1283

-7350 0 1283 7350

0 -735000 0 0

-1283 0 1500 1283

1 2 3

7 8 9 4

378000 0 0 -378000

0 302 755 0

M 2 = 0 755 25515 0

-378000 0 0 378000

0 -302 -755 0

0 755 12758 0

MATRIKS KEKAKUAN STRUKTUR

M = . M .


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1 2 3

7 8 9

385350 0 1283 7

M = 0 73802 755 8

1283 755 55528 9

Batang 1

Gbr AM 1 =

Batang 2

Gbr AM 2 =

Transformasi dari sumbu lokal ke sumbu global struktur

Batang 1

0 1

0 2

AM 1 = 0 3

0 4

0 5

0 6

SB LOKAL

0.00 -1.00 0.00 0.00

1.00 0.00 0.00 0.00

= 0.00 0.00 1.00 0.00

0.00 0.00 0.00 0.00

Hitung rekasi di ujung batang / freebody (AML) --- Beban dimasukkan ke!uali beban di joint

AML1

AML2

AML3

AML4

AML5

AML6

AML1

AML2

AML3

AML4

AML5AM

L6

RT


7/70

0.00 0.00 0.00 1.00

0.00 0.00 0.00 0.00

0 2 1

0 1 2

= 0 3 30 5 7 D1

0 4 8 D2

0 9 D3

SB GLOBAL

Batang 2

0 1 7

150 2 8AM 2 = 125 3 9

0 4 4

150 5 5

-125 6

SB LOKAL

1.00 0.00 0.00 0.00

0.00 1.00 0.00 0.00= 0.00 0.00 1.00 0.00

0.00 0.00 0.00 1.00

0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00

0 1 7 D1

150 2 8 D2

= 125 3 9 D3

0 4 4150 5 5

-125 6

SB GLOBAL

BEBAN EKIVALEN GABNGAN !AE"

B 1 B 2

1 0 0

2 0 0

3 0 0

4 0 0

5 0 150

AMS1

RT

AMS2


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-A# = 0 ; -125

1 7 0 0

2 8 0 150

3 9 0 125

SB GLOBAL

B#b$% &$'$ ()*%+ ,$%/% !A("

1 0

2 0

3 0

4 0

A = 5 0

0

1 7 40

2 8 03 9 0

SB GLOBAL

BEBAN OINT GABNGAN !A"

A = A( AE

1 0 0

2 0 0

3 0 04 0 0

A, = 5 0 ; -150 =

0 125

7 40 0

8 0 -150

9 0 -125

1 402 -150

3 -125

"#$%&MA' 'T&*T&

&!t'ng +e+!n*ahan

2.1529#-00 .292#-009 -.07#-007

= .29192#-009 1.357#-00 -1.885#-007 'at!n:

ea,t!n A++"!e*

40 ; 0

0 ; -1500 ; -125

&1 ; 0 =

E1 ; 0

M1 ; 0

&3 ; 0

E3 ; -150

M3 ; 125

an*a (;) (-) *!tent'$an "eh aa

THEOR10

1

2

1=

2=

eF!te 1: $G*1=1-$G*2

*1=!nee($)G(1-$G*2)

!+"a,ement: '2 = 0.0001795029

2 = -0.000179714

+h!2 = -0.002279435

ea,t!n: &1 = 0

E1 = 0

M1 = 0

&3 = 0

E3 = 150M3 = -125

ata' 27.852077784

132.089777923

-31.724472274

-7.852077785

17.910222077

-155.3089103

ata' 27.852077784

132.089777923

-31.724472274


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-7.852077785

17.910222077

-155.3089103

ata' 27.852077784

132.089777923

-31.724472274-7.852077785

17.910222077

-155.3089103

Ata'

Bent'$ mat!$ A, *ah'"' : 1

2

A/ A3 Ae A = 3

4

5

7

8

9

b g"ba"

0.0001795029m

*! = -0.000179714 m

-0.002279435 a*

0 27.852077784

0 132.089777923

0 ; -31.724472274 =

ea,t = 0 -7.852077785

150 17.910222077

-125 -30.308910357


45/70

@atatan a = %/m2

Batang 2 :

0.20

0.45

9.00#-02

# = 2.10#;07

1.52#-03

5

0

E!emen "a!am matri#s #e#a#uan atang

12.EI L>3 EA L 4.EI L

7.35#;03 3.78#;05 2.55#;04

.#I / L2 12.#I / L3 theta

1.29#;04 7.35#;03 90 *eg7.5#;03 3.0#;03 0 *eg

AM SUM$U STRUKTUR)*(O$A(

12.#I / L3

7.35#;03

b2=

h2=

A2=

I2=

L2=

theta2=


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'2 2 +h!2 ,e$ :

-7350 -4.45739#-011 -1282.5 7350

-4.45739#-011 -735000 7.88#-13 0

1.29#;04 -7.87923#-013 1.50#;04 -1283

7.35#;03 4.45739#-011 1.29#;04 -7350

4.45739#-011 7.35#;05 -7.87924#-013 01.29#;04 -7.87923#-013 3.00#;04 -1283

AM SUM$U STRUKTUR)*(O$A(

12.#I / L3

3.0#;03

'3 3 +h!3 ,e$ :

-378000 0 0 378000

0 -301.8 7.5#;03 0

0.00#;00 -754.5 1.28#;04 0

3.78#;05 0 0.00#;00 -378000

0 3.0#;03 -754.5 0

0.00#;00 -754.5 2.55#;04 0

0 0 0 0 0 0

'1 1 +h!1 '3 3 +h!3

-7350 -4.45739#-011 1282.5 -378000 0 0

-4.45739#-011 -735000 -7.87924#-013 0 -301.8 754.5

-1282.5 7.87923#-013 1500.25 0 -754.5 12757.5

7350 4.45739#-011 -1282.5 0 0 0

4.45739#-011 735000 7.87923#-013 0 0 0

-1282.5 7.87923#-013 30012.5 0 0 00 0 0 378000 0 0

0 0 0 0 301.8 -754.5

0 0 0 0 -754.5 25515

6K < C*D


47/70

H,e ! a,t!ng *FnFa*

'2 2 +h!2 '1 1

'2 385350 4.45739#-011 1282.5 -7350 -4.45739#-011

2 4.45739#-011 73801.8 754.5 -4.45739#-011 -735000+h!2 1282.5 754.5 55527.5 -1282.5 7.87923#-013

'1 -7350 -4.45740#-011 -1282.5 7350 4.45739#-011

1 -4.45739#-011 -735000 7.879#-013 4.45739#-011 735000

+h!1 1282.5 -7.87924#-013 1500.25 -1282.5 7.87923#-013

'3 -378000 0 0 0 0

3 0 -301.8 -754.5 0 0

+h!3 0 754.5 12757.5 0 0

b g"ba"

= $ $ G *1

$ $ *2

$G*1;$G*2

$G*1;$G*2

m

m

a*

; 27.852077784 = 27.8520778 $%

; 132.089777923 = 132.089778 $%

; -31.724472274 = -31.7244723 $%.m

; -7.852077785 = -7.8520778 $%

; 17.910222077 = 17.910222 $%; -30.308910357 = -155.30891 $%.m

-

-

-

-

-

-

-

-

-


48/70

-

-

-

-

-

--

-

-

0 1 0

0 2 0

0 -Ae = 3 0

40 4 0

0 5 150

0 125

0 7 0

0 8 150

0 9 -125

b g"ba"

@e$:

-

-

-

@e$:

27.852077784 $% &1 -

132.089777923 $% E1 -

-31.724472274 $%.m M1 -

-7.852077785 $% &3 -

17.910222077 $% E3 -

-155.3089103 $%.m M3 -


49/70

m

m

m2

$%/m2

m4

m

*egee

2.EI L 6.EI L>2 12.EI L>3

1.28#;04 7.5#;03 3.0#;03


50/70

0 -1283 -7350 0 -1283

735000 0 0 -735000 0

0 30013 1283 0 1500

0 1283 7350 0 1283

-735000 0 0 735000 00 1500 1283 0 30013

0 0 -378000 0 0

302 755 0 -302 755

755 25515 0 -755 12758

0 0 378000 0 0

-302 -755 0 302 -755

755 12758 0 -755 25515

,e$ :

*1 385350 4.45740#-011 1282.5

*2 4.45739#-011 73801.8 754.5

*3 1282.5 754.5 55527.5

0

0

00

0

0


51/70

+h!1 '3 3 +h!3

1282.5 -378000 0 0 '2

-7.87923#-013 0 -301.8 754.5 21500.25 0 -754.5 12757.5 +h!2

-1282.5 0 0 0 < 0

7.87923#-013 0 0 0 0

30012.5 0 0 0 0

0 378000 0 0 0

0 0 301.8 -754.5 0

0 0 -754.5 25515 0

,e$:

0.000179503

-0.000179714

-0.00227944

,e$:

132.09

-27.85

-31.72

-7.85

17.91-155.31

b "$a"


52/70

0 0 1

0 0 2

Ae = 0 A, = 0 3

0 40 4

-150 -150 5

-125 -125

0 0 7

-150 -150 8

125 125 9


53/70


54/70

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

385350 0 1283 - - -

0 73802 755 - - -

1283 755 55528 - - -


55/70

-

-

-

27.8520778 -

132.09 -

-31.72 -

-7.85 -

17.91 --155.31 -

b g"ba"


56/70

40 *1 4

-150 *2 5

A, = -125 *3

*1 0 1

*2 0 2

*3 0 3

0 7

-150 8

125 9


57/70

Properties penampang

Bahan / mate!a" :

# = 2.10#;07 $%/m2

Batang 1 :

0.35m

0.35m

1.23#-01 m2

# = 2.10#;07 $%/m2

1.25#-03 m4

3.5m

90 *egee

E!emen "a!am matri#s #e#a#uan atang

EA L 4.EI L 2.EI L 6.EI L>2

7.35#;05 3.00#;04 1.50#;04 1.29#;04

Re#ap E!emen "a!am Matri#s Ke#a#uan $atangBatang #A / L 4.#I / L 2.#I / L

1 7.35#;05 3.00#;04 1.50#;04

2 3.78#;05 2.55#;04 1.28#;04

%%%& su"a' DA(

u1 v1 phi1

u1 7350 0 -12862.5

v1 0 735000 0

phi1 -12862.5 0 30012.5

u2 -7350 0 12862.5

b1=

h1=

A1=

I1=

L1=

theta1=

Hitung matri#s #e#a#uan atang + ,SM+-

6M1 =


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v2 0 -735000 0

phi2 -12862.5 0 15006.25

%%%& su"a' DA(

u2 v2 phi2

u2 378000 0 0

v2 0 3061.8 7654.5

phi2 0 7654.5 25515

u3 -378000 0 0

v3 0 -3061.8 -7654.5

phi3 0 7654.5 12757.5

u2 v2 phi2u2 385350 0 12862.5

v2 0 738061.8 7654.5

phi2 12862.5 7654.5 55527.5

u1 -7350 0 -12862.5

v1 0 -735000 0

phi1 12862.5 0 15006.25

u3 -378000 0 0

v3 0 -3061.8 -7654.5

phi3 0 7654.5 12757.5

Hitung matri#s #e#a#uan atang 2 ,SM2-

6M2 =

MATRIKS KEKAKUAN STRUKTUR ,SMS-


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@atatan a = %/m2

Batang 2 :

# =

E!emen "a!am matri#s #e#a#ua

12.EI L>3 EA L

7.35#;03 3.78#;05

.#I / L2 12.#I / L3 theta

1.29#;04 7.35#;03 90 *eg

7.5#;03 3.0#;03 0 *eg

M SUM$U STRUKTUR)*(O$A(

u2 v2 phi2

-7350 0 -12862.5 7350

0 -735000 0 4.45740E-011

12862.5 0 15006.25 -12862.5

7350 0 12862.5 -7350

b2=

h2=

A2=

I2=

L2=

theta2=


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0 735000 0 -4.4574E-011

12862.5 0 30012.5 -12862.5

M SUM$U STRUKTUR)*(O$A(

u3 v3 phi3 u2

-3.78E+05 0 0 378000

0 -3061.8 7654.5 0

0 -7654.5 12757.5 0

3.78E+05 0 0 -378000

0 3061.8 -7654.5 0

0 -7654.5 25515 0

u1 v1 phi1 u3 v3-7350 0 12862.5 -378000 0

0 -735000 0 0 -3061.8

-12862.5 0 15006.25 0 -7654.5

7350 0 -12862.5 0 0

0 735000 0 0 0

-12862.5 0 30012.5 0 0

0 0 0 378000 0

0 0 0 0 3061.8

0 0 0 0 -7654.5


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0.20m

0.45m

9.00#-02m2

2.10#;07$%/m2

1.52#-03m4

5 m

0 *egee

atang

4.EI L 2.EI L 6.EI L>2 12.EI L>3

2.55#;04 1.28#;04 7.5#;03 3.0#;03

4.45740E-011 -12862.5 -7350 -4.4574E-011 -12862.5

735000 7.87924E-013 -4.4574E-011 -735000 7.88E-013

7.87924E-013 30012.5 12862.5 -7.8792E-013 15006.25

-4.4574E-011 12862.5 7350 4.45740E-011 12862.5


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-735000 -7.8792E-013 4.45740E-011 735000 -7.9E-013

7.87924E-013 15006.25 12862.5 -7.8792E-013 30012.5

v2 phi2 u3 v3 phi3

0 0 -378000 0 0

3061.8 7654.5 0 -3061.8 7654.5

7654.5 25515 0 -7654.5 12757.5

0 0 378000 0 0

-3061.8 -7654.5 0 3061.8 -7654.5

7654.5 12757.5 0 -7654.5 25515

phi30

7654.5

12757.5

0

0

0

0

-7654.5

25515


63/70

- -4.45740E-011 - - 4.46E-011 -

-4.45740E-011 - -7.8792360E-013 4.46E-011 - -7.9E-013

- -7.87924E-013 - - 7.88E-013 -

- 4.457396E-011 - - -4.5E-011 -


64/70

4.457396E-011 - 7.8792360E-013 -4.5E-011 - 7.88E-013

- -7.87924E-013 - - 7.88E-013 -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

385350 4.457396E-011 12862.5 -7350 -4.5E-011 12862.5

4.457396E-011 738061.8 7654.5 -4.5E-011 -735000 -7.9E-013

12862.5 7654.5 55527.5 -12862.5 7.88E-013 15006.25

-7350 -4.45740E-011 -12862.5 7350 4.46E-011 -12862.5

-4.45740E-011 -735000 7.8792360E-013 4.46E-011 735000 7.88E-013

12862.5 -7.87924E-013 15006.25 -12862.5 7.88E-013 30012.5

-378000 0 0 0 0 0

0 -3061.8 -7654.5 0 0 0

0 7654.5 12757.5 0 0 0


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-378000 0 0 - -4.457396E-011

0 -3061.8 7654.5 -4.45740E-011 -

0 -7654.5 12757.5 - -

0 0 0 - 4.4573964E-011

0 0 0 4.457396E-011 -

0 0 0 - 7.8792360E-013

378000 0 0 - -

0 3061.8 -7654.5 - -

0 -7654.5 25515 - -


67/70


68/70

- - 4.4573964E-011 - -

- 4.457396E-011 - 7.8792360E-013 -

- - -7.8792360E-013 - -

- - -4.4573964E-011 - -

-7.8792360E-013 -4.45740E-011 - -7.8792360E-013 -

- - -7.8792360E-013 - -

- - - - -

- - - - -

- - - - -


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- -

- -

- -

- -

- -

- -

- -

- -

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Documents

Contoh Soal Analisa Matriks (Portal Hal.172)