Contoh Soal Analisa Matriks (Portal Hal.186)

7/25/2019 Contoh Soal Analisa Matriks (Portal Hal.186)

1/101

CONTOH 4.3

Properties penampang

Batang 1 :

b = 0.30 m

h = 0.50 m

A = 0.15 m2

I = 0.003125 m4

L1 = 2.5 m

Batang 2 :

b = 0.30 m

h = 0.50 m

A = 0.15 m2I = 0.003125 m4

L2 = 5 m

Batang 3 :

b = 0.30 m

h = 0.50 m

A = 0.15 m2

I = 0.003125 m4

L3 = 2.5 m

Bahan / material :E = 23500000 kN/m2

Hitung matriks kekakuan batang 1 (SM1)

Akibat disla!ement arah 1 ""# $1 %translasi arah sb l&kal 1'

() 1.1 = E.A / L = 1410000

() 2.1 = 0 = 0

() 3.1 = 0 = 0


2/101

() 1.2 = 0 = 0

() 2.2 = 12.EI/L*3 = 5+400

() 3.2 = +.EI / L*2 = ,0500

Akibat disla!ement arah 3 ""# $3 %r&tasi arah sb l&kal 3'

() 1.3 = 0 = 0

() 2.3 = +.EI / L*2 = ,0500

() 3.3 = 4 EI / L = 11,500


() 1.4 = " E.A / L = "1410000

() 2.4 = 0 = 0

() 3.4 = 0 = 0


() 1.5 = 0 = 0

() 2.5 = "12.EI/L*3 = "5+400

() 3.5 = "+.EI / L*2 = ",0500

Akibat disla!ement arah + ""# $+ %r&tasi arah sb l&kal +'

() 1.+ = 0 = 0

() 2.+ = +.EI / L*2 = ,0500

() 3.+ = 2 EI / L = 5-,50

(snan matriks kekakan batang 1 %()1' :

()11 ()12 ()13

()21 ()22 ()23

()1 = ()31 ()32 ()33

()41 ()42 ()43

()51 ()52 ()53

()+1 ()+2 ()+3

(dt batang 1 : =1


3/101

0

= 0

0 0 1

0 0 0

0 0 0

0 0 0

rans&se matrik &tasi

1.00 0 0

= 0 1.00 0

0 0 1

0 0 0

0 0 0

0 0 0

)atriks kekakan strktr menadi :

1.00 0.00 0.00

0.00 1.00 0.00

0.00 0.00 1.00

()( 1 = 0.00 0.00 0.00

0.00 0.00 0.000.00 0.00 0.00

1410000 0 0

0 5+400 ,0500

0 ,0500 11,500

"1410000 0 0

0 "5+400 ",0500

0 ,0500 5-,50



() 1.1 = E.A / L = ,05000

() 2.1 = 0 = 0

() 3.1 = 0 = 0


!&s 1

sin 1

" sin 1

!&s 1

()( = . () .


4/101

() 1.2 = 0 = 0

() 2.2 = 12.EI/L*3 = ,050

() 3.2 = +.EI / L*2 = 1,+25


() 1.3 = 0 = 0

() 2.3 = +.EI / L*2 = 1,+25

() 3.3 = 4 EI / L = 5-,50


() 1.4 = " E.A / L = ",05000

() 2.4 = 0 = 0

() 3.4 = 0 = 0


() 1.5 = 0 = 0

() 2.5 = "12.EI/L*3 = ",050

() 3.5 = "+.EI / L*2 = "1,+25


() 1.+ = 0 = 0

() 2.+ = +.EI / L*2 = 1,+25

() 3.+ = 2 EI / L = 23,5


()11 ()12 ()13

()21 ()22 ()23

()2 = ()31 ()32 ()33

()41 ()42 ()43

()51 ()52 ()53

()+1 ()+2 ()+3

(dt batang 2 : =

ata =

)atriks trans&rmasi r&tasi :

2

2


5/101

0

= 0

0 0 1

0 0 0

0 0 0

0 0 0


0.80 -0.6 0

= 0.6 0.80 0

0 0 1

0 0 0

0 0 0

0 0 0


0.-0 "0.+0 0.00

0.+0 0.-0 0.00

0.00 0.00 1.00

()( 2 = 0.00 0.00 0.00

0.00 0.00 0.000.00 0.00 0.00

5+4000 "4230 "105,5

423000 5+40 14100

0 1,+25 5-,50

"5+4000 4230 105,5

"423000 "5+40 "14100

0 1,+25 23,5



() 1.1 = E.A / L = 1410000

() 2.1 = 0 = 0

() 3.1 = 0 = 0


() 1 2 = 0 = 0

!&s 1

sin 1

" sin 1

!&s 1

()( = . () .


6/101

() 2.2 = 12.EI/L*3 = 5+400

() 3.2 = +.EI / L*2 = ,0500


() 1.3 = 0 = 0

() 2.3 = +.EI / L*2 = ,0500

() 3.3 = 4 EI / L = 11,500


() 1.4 = " E.A / L = "1410000

() 2.4 = 0 = 0

() 3.4 = 0 = 0


() 1.5 = 0 = 0

() 2.5 = "12.EI/L*3 = "5+400

() 3.5 = "+.EI / L*2 = ",0500


() 1.+ = 0 = 0

() 2.+ = +.EI / L*2 = ,0500

() 3.+ = 2 EI / L = 5-,50


()11 ()12 ()13

()21 ()22 ()23

()3 = ()31 ()32 ()33

()41 ()42 ()43

()51 ()52 ()53

()+1 ()+2 ()+3

(dt batang 3 : =

ata =)atriks trans&rmasi r&tasi :

3

3


7/101

0

= 0

0 0 1

0 0 0

0 0 0

0 0 0


1.00 0 0

= 0 1.00 0

0 0 1

0 0 0

0 0 0

0 0 0


1.00 0.00 0.00

0.00 1.00 0.00

0.00 0.00 1.00

()( 3 = 0.00 0.00 0.00

0.00 0.00 0.00

0.00 0.00 0.00

1410000 0 0

0 5+400 ,0500

0 ,0500 11,500

"1410000 0 0

0 "5+400 ",0500

0 ,0500 5-,50

Susun MAT!"S "#"A"$AN ST$"T$

0 0 0

1 2 3

164106000 0 0

0 5+6400 ,06500

()( 1 = 0 ,06500 11,6500

"164106000 0 0

0 "5+6400 ",06500

0 ,06500 5-6,50

$1 0 $2

4 5 +

!&s 1

sin 1

" sin 1

!&s 1

()( = . () .


8/101

4536,3- 335601+ "1065,5

335601+ 25-6312 146100

()( 2 = "1065,5 146100 5-6,50

"4536,3- "335601+ 1065,5

"335601+ "25-6312 "146100

"1065,5 146100 263,5

$3 0 $4

, -

164106000 0 0

0 5+6400 ,06500

()( 3 = 0 ,06500 11,6500

"164106000 0 0

0 "5+6400 ",06500

0 ,06500 5-6,50

MAT!"S "#"A"$AN ST$"T$

$1 $2 $3

4 + ,

16-+36,3- "1065,5 "4536,3-

()( = "1065,5 1,+6250 1065,5

"4536,3- 1065,5 16-+36,3-

"1065,5 263,5 1065,5

Batang 1

Gbr AM 1 =

Batang 2

Gbr AM 2 =

Hitung rekasi di ujung batang / freebody (AML) ---> Beban dimasukkan kecuai beban di joi


9/101

Batang !

Gbr AM 3 =

"ransformasi dari sumbu oka ke sumbu goba struktur

Batang 1

0 1 1

30 2 2

AM 1 = 12.5 3 3

0 4 4

30 5 5

-12.5 6 6

SB LOKAL SB GLOBAL

1.00 0.00 0.00

0.00 1.00 0.00

= 0.00 0.00 1.00

0.00 0.00 0.00

0.00 0.00 0.00

0.00 0.00 0.00

0 1 1

30 2 2

= 12.53

3

0 4 4

30 5 5

-12.5 + 6

SB LOKAL SB GLOBAL

Batang 2

22.5 1

25.3125 2

AM 2 = 35.15625 3

37.5 4

RT

AMS

1


10/101

54.6875 5

-58.59375 6

SB LOKAL

0.80 -0.60 0.000.60 0.80 0.00

= 0.00 0.00 1.00

0.00 0.00 0.00

0.00 0.00 0.00

0.00 0.00 0.00

2.8125 4

33.75 5

= 35.15625 6

-2.8125 7

66.25 8

-58.59375 9

SB GLOBAL

Batang !

0 1

60 2AM 3 = 25 3

0 4

60 5

-25 6

SB LOKAL

1.00 0.00 0.00

0.00 1.00 0.00

= 0.00 0.00 1.00

0.00 0.00 0.00

0.00 0.00 0.00

0.00 0.00 0.00

0 7

60 8

= 25 9

0 10

60 11

RT

AMS2

RT

AMS3


11/101

SB GLOBAL

BEBAN EKIVALEN GABUNGAN AE!

B7 1

0 1 0

0 2 30

0 3 12.5

$1 4 0

0 5 30

"AE = $2 + "12.5 8

$3 , 0

0 - 0

$4 0

0 10 0

0 11 0

0 12 0

SB GLOBAL

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

0 1 0

0 2 0

0 3 0

$1 4 0

A = 0 5 0$2 + 0

$3 , 0

0 - 0

$4 0

0 10 0

0 11 0

0 12 0

SB GLOBAL

BEBAN /OINT GABUNGAN A!

A = A' AE

1 0 0

2 0 "30

3 0 "12.5

4 0 "2.-125

A! = 5 0 8 "+3.,5

+ 0 "22.+5+25

, 0 2.-125- 0 "12+.25

0 33.53,5


12/101

10 0 0

11 0 "+0

12 0 25

$1 "2.-125

$2 "22.+5+25

$3 2.-125

$4 33.53,5

#$%&'MA "'*+"*'

9itng erindahan

5.,05+5E"00, 2.22125E"00- 1.3-+5524+4-E"00,

= 2.22125E"00- 5.-3-15E"00+ "2.22124-1+3-E"00-

1.3-+55E"00, "2.22125E"00- 5.,05+4+11++,E"00,

2.22125E"00- ".,03+0E"00, "2.22124-1+3-E"00-

,AA-,AA #ALAM *.*, $L$M$

Batang 1

"(r#) *r*r +# #r# SUMBU GLOBAL &) b#*#$, 1 bb

0 1 0

0 2 0

0 0 3 0

-0.000001 4 1

0.000000 5 0

-0.000165 6 2

"(r#) *r*r +# #r# SUMBU LOKAL &) b#*#$, 1 bb

1 0 0

0 1 0

0 0 0 1

0 0 0

0 0 0

0 0 0

G## '$, b#*#$,

B"b#$ '()$* #$, "+#r# O: A:;

!

A%

0

:= S

::-1.A

:;S

::-1= SMS -1

:

#M1

M 1 = R . MS1

M 1


13/101

0 1410000

30 0

AM 1 0 12.5 0

0 -141000030 0

-12.5 0

Batang 2

"(r#) *r*r +# #r# SUMBU GLOBAL &) b#*#$, 2 bb

-9.7180E-07 4 1

0 5 0

0 -1.6499E-04 6 29.7180E-07 7 3

0 8 0

2.1799E-04 9 4


0.8 0.6 0-0.6 0.8 0

0 0 0 1

0 0 0

0 0 0

0 0 0

G## '$, b#*#$,

22.5 705000

25.3125 0

AM 2 0 35.15625 0

37.5 -705000

54.6875 0

-58.59375 0

Batang !"(r#) *r*r +# #r# SUMBU GLOBAL &) b#*#$, 3 bb

AM 1 = AML1 SM 1 . M 1

#M2

M 2 = R . MS2

M 2

AM 2 = AML2 SM 2 . M 2


14/101

9.7180E-07 7 3

0 8 0

0 2.1799E-04 9 4

0 10 0

0 11 0

0 12 0


1.0 0.0 0.0

0.0 1.0 0.0

0 0.0 0.0 1.0

0.0 0.0 0.00.0 0.0 0.0

0.0 0.0 0.0

G## '$, b#*#$,

0 1410000

60 0

AM 3 0 25 0

0 -1410000

60 0

-25 0

#M!

M 3 = R . MS3

M !

AM 3 = AML3 SM 3 . M 3


15/101

%ata beban

= 100 kN

;1 = 24 kN/m

;2 = 4- kN/m

#&emen 'a&am matriks kekakuan batang

Batang EA / L 4.EI / L 2.EI / L +.EI / L*2

1 1410000 11,500 5-,50 ,0500

2 ,05000 5-,50 23,5 1,+25

3 1410000 11,500 5-,50 ,0500

9bngan antar batang

N(. B#*#$, U'$, ) U'$, '

1 1 2

2 2 3

3 3 4

(AL $AN (


16/101

() 4.2 = 0 =

() 5.2 = "12.EI/L*3 =

() +.2 = +.EI / L*2 =

() 4.3 = 0 =

() 5.3 = "+.EI / L*2 =

() +.3 = 2 EI / L =

() 4.4 = E.A / L =

() 5.4 = 0 =

() +.4 = 0 =

() 4.5 = 0 =

() 5.5 = 12.EI/L*3 =

() +.5 = "+.EI / L*2 =

() 4.+ = 0 =

() 5.+ = "+.EI / L*2 =

() +.+ = 4 EI / L =

""# !atatan matriks kekakan batang dalam arah smb l&kal %smb l&

1 2

()14 ()15 ()1+ 1410000 0

()24 ()25 ()2+ 0 5+400

()34 ()35 ()3+ = 0 ,0500

()44 ()45 ()4+ "1410000 0

()54 ()55 ()5+ 0 "5+400

()+4 ()+5 ()++ 0 ,0500

0deg


17/101

0 0 0

0 0 0 =

0 0 0

0

0

0 0 1

0 0 0

0 0 0

0 0 0

1.00 0 0

0 1.00 0

0 0 1

0.00 0.00 0.00 1410000 0

0.00 0.00 0.00 0 5+400

0.00 0.00 0.00 ? 0 ,0500

1.00 0.00 0.00 "1410000 0

0.00 1.00 0.00 0 "5+4000.00 0.00 1.00 0 ,0500

"1410000 0 0 1 0

0 "5+400 ,0500 0 1

0 ",0500 5-,50 ? 0 0

1410000 0 0 0 0

0 5+400 ",0500 0 0

0 ",0500 11,500 0 0

(AL $AN (


18/101

() 4.2 = 0 =

() 5.2 = "12.EI/L*3 =

() +.2 = +.EI / L*2 =

() 4.3 = 0 =

() 5.3 = "+.EI / L*2 =

() +.3 = 2 EI / L =

() 4.4 = E.A / L =

() 5.4 = 0 =

() +.4 = 0 =

() 4.5 = 0 =

() 5.5 = 12.EI/L*3 =

() +.5 = "+.EI / L*2 =

() 4.+ = 0 =

() 5.+ = "+.EI / L*2 =

() +.+ = 4 EI / L =


1 2

()14 ()15 ()1+ ,05000 0

()24 ()25 ()2+ 0 ,050

()34 ()35 ()3+ = 0 1,+25

()44 ()45 ()4+ ",05000 0

()54 ()55 ()5+ 0 ",050

()+4 ()+5 ()++ 0 1,+25

3+.-+-,+4+ deg 0.-

3+.-+-,+4+ deg 0.+


19/101

0 0 0

0 0 0 =

0 0 0

0

0

0 0 1

0 0 0

0 0 0

0 0 0

0.80 -0.6 0

0.6 0.80 0

0 0 1

0.00 0.00 0.00 ,05000 0

0.00 0.00 0.00 0 ,050

0.00 0.00 0.00 ? 0 1,+25

0.-0 "0.+0 0.00 ",05000 0

0.+0 0.-0 0.00 0 ",0500.00 0.00 1.00 0 1,+25

"5+4000 4230 "105,5 1 1

"423000 "5+40 14100 "1 1

0 "1,+25 23,5 ? 0 0

5+4000 "4230 105,5 0 0

423000 5+40 "14100 0 0

0 "1,+25 5-,50 0 0

(AL $AN (


20/101

() 5.2 = "12.EI/L*3 =

() +.2 = +.EI / L*2 =

() 4.3 = 0 =

() 5.3 = "+.EI / L*2 =

() +.3 = 2 EI / L =

() 4.4 = E.A / L =

() 5.4 = 0 =

() +.4 = 0 =

() 4.5 = 0 =

() 5.5 = 12.EI/L*3 =

() +.5 = "+.EI / L*2 =

() 4.+ = 0 =

() 5.+ = "+.EI / L*2 =

() +.+ = 4 EI / L =


1 2

()14 ()15 ()1+ 1410000 0

()24 ()25 ()2+ 0 5+400

()34 ()35 ()3+ = 0 ,0500

()44 ()45 ()4+ "1410000 0

()54 ()55 ()5+ 0 "5+400

()+4 ()+5 ()++ 0 ,0500

0 deg 1

0 deg 0


21/101

0 0 0

0 0 0 =

0 0 0

0

0

0 0 1

0 0 0

0 0 0

0 0 0

1.00 0 0

0 1.00 0

0 0 1

0.00 0.00 0.00 1410000 0

0.00 0.00 0.00 0 5+400

0.00 0.00 0.00 ? 0 ,0500

1.00 0.00 0.00 "1410000 0

0.00 1.00 0.00 0 "5+400

0.00 0.00 1.00 0 ,0500

"1410000 0 0 1 0

0 "5+400 ,0500 0 1

0 ",0500 5-,50 ? 0 0

1410000 0 0 0 0

0 5+400 ",0500 0 0

0 ",0500 11,500 0 0

$1 0 $2

4 5 + ==# EN)AN (B 7LBAL

"164106000 0 0 1 0

0 "5+6400 ,06500 2 0

0 ",06500 5-6,50 3 0

164106000 0 0 4 $1

0 5+6400 ",06500 5 0

0 ",06500 11,6500 + $2

$3 0 $4

, - ==# EN)AN (B 7LBAL

!&s 1

sin 1

" sin 1

!&s 1


22/101

"4536,3- "335601+ "1065,5 4 $1

"335601+ "25-6312 146100 5 0

1065,5 "146100 263,5 + $2

4536,3- 335601+ 1065,5 , $3

335601+ 25-6312 "146100 - 0

1065,5 "146100 5-6,50 $4

0 0 0

10 11 12 ==# EN)AN (B 7LBAL

"164106000 0 0 , $3

0 "5+6400 ,06500 - 0

0 ",06500 5-6,50 $4

164106000 0 0 10 0

0 5+6400 ",06500 11 0

0 ",06500 11,6500 12 0

$4

"1065,5 4 $1

263,5 + $2 =

1065,5 , $3

1,+6250 $4

0 1

30 2 kN

= 12.5 3 kN.m

0 4

30 5 kN

"12.5 + kN.m

SB LOKAL

22.5 1

25.3125 2

= 35.15+25 3

3,.5 4 a =

54.+-,5 5 b =

"5-.53,5 + L =

()("1

t

AML1

AML2

AML3

AML4

AML5

AML6

AML1 .!&s =

AML2 .sin =

AML3

AML4

AML5

AML6


23/101

SB LOKAL

0 1

+0 2 kN

= 25 3 kN.m

0 4

+0 5 kN

"25 + kN.m

SB LOKAL

0.00 0.00 0.00

0.00 0.00 0.00

0.00 0.00 0.00

1.00 0.00 0.00

0.00 1.00 0.00

0.00 0.00 1.00

0

0

0

1

0

2


24/101

0.00 0.00 0.000.00 0.00 0.00

0.00 0.00 0.00

0.80 -0.60 0.00

0.60 0.80 0.00

0.00 0.00 1.00

1

0

2

3

0

4

0.00 0.00 0.00

0.00 0.00 0.00

0.00 0.00 0.00

1.00 0.00 0.00

0.00 1.00 0.00

0.00 0.00 1.00

3

0

4

0

0

AMS2 = R

T

. AML2

AMS2

AMS3 = RT. AM

L3

AMS3


25/101

B7 2 B7 3

0 0 0

0 0 30

0 0 12.5

2.-125 0 2.-125

33.,5 0 = +3.,5

35.15+25 8 0 22.+5+25

"2.-125 0 "2.-125

++.25 +0 12+.25

"5-.53,5 25 "33.53,5

0 0 0

0 +0 +0

0 "25 "25

0 0

"30 0

"12.5 0

"2.-125 $1

= "+3.,5 0

"22.+5+25 $2

2.-125 $3"12+.25 0

33.53,5 $4


26/101

0 0

"+0 0

25 0

2.22125E"00- "2.-125 ".,1-0E"0, $1

".,03+E"00, ? "22.+5+25 = "1.+4E"04 $2

"2.2212E"00- 2.-125 .,1-0E"0, $3

5.-3-15E"00+ 33.53,5 2.1,E"04 $4

0 0 0 0

0 0 0 0

0 0 0 3 0 0

1 0 0 -9.7179607E-007

0 1 0 0

0 0 1 -0.0001649936


27/101

0 0 -1410000 0 0

56400 70500 0 -56400 70500

70500 117500 0 -70500 58750 3

0 0 1410000 0 0

-56400 -70500 0 56400 -70500

70500 58750 0 -70500 117500

0 0 0 -9.7179607E-0070 0 0 0

0 0 0 3 -0.0001649936 0

0.8 0.6 0 9.71796072E-007

-0.6 0.8 0 0

0 0 1 0.0002179852

0 0 -705000 0 0

7050 17625 0 -7050 17625

17625 58750 0 -17625 29375 3

0 0 705000 0 0

-7050 -17625 0 7050 -17625

17625 29375 0 -17625 58750


28/101

0.0 0.0 0.0 9.71796072E-007

0.0 0.0 0.0 0

0.0 0.0 0.0 3 0.0002179852 0

1.0 0.0 0.0 00.0 1.0 0.0 0

0.0 0.0 1.0 0

0 0 -1410000 0 0

56400 70500 0 -56400 70500

70500 117500 0 -70500 58750 3

0 0 1410000 0 0

-56400 -70500 0 56400 -70500

70500 58750 0 -70500 117500


29/101

12.EI / L*3

5+400

,050

5+400

)AN $AN AN$A %8' %"'

"1410000

0

0


30/101

0

"5+400

,0500

0

",0500

5-,50

1410000

0

0

0

5+400

",0500

0

",0500

11,500

al ata gl&bal menentkan en&m&ran dan tanda %8' %"'

3 4 5 + sb l&kal

0 "1410000 0 0 1

,0500 0 "5+400 ,0500 2

11,500 0 ",0500 5-,50 3

0 1410000 0 0 4

",0500 0 5+400 ",0500 5

5-,50 0 ",0500 11,500 +

sb l&kal


31/101

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

0 "1410000 0 0 1

,0500 0 "5+400 ,0500 0

11,500 0 ",0500 5-,50 ? 0

0 1410000 0 0 0

",0500 0 5+400 ",0500 05-,50 0 ",0500 11,500 0

1

0 0 0 0 164106000

0 0 0 0 0

1 0 0 0 = 0

0 1 0 0 "164106000

0 0 1 0 0

0 0 0 1 0

)AN $AN AN$A %8' %"'

",05000

0

0


32/101

0

",050

1,+25

0

"1,+25

23,5

,05000

0

0

0

,050

"1,+25

0

"1,+25

5-,50


3 4 5 + sb l&kal

0 ",05000 0 0 1

1,+25 0 ",050 1,+25 2

5-,50 0 "1,+25 23,5 3

0 ,05000 0 0 4

"1,+25 0 ,050 "1,+25 5

23,5 0 "1,+25 5-,50 +

sb l&kal


33/101

0.8 0.6 0 0 0 0

-0.6 0.8 0 0 0 0

0 0 1 0 0 0

0 0 0 0.8 0.6 0

0 0 0 -0.6 0.8 0

0 0 0 0 0 1

0 ",05000 0 0 1

1,+25 0 ",050 1,+25 "1

5-,50 0 "1,+25 23,5 ? 0

0 ,05000 0 0 0

"1,+25 0 ,050 "1,+25 023,5 0 "1,+25 5-,50 0

4

0 0 0 0 4536,3-

0 0 0 0 335601+

1 0 0 0 = "1065,5

0 1 1 0 "4536,3-

0 "1 1 0 "335601+

0 0 0 1 "1065,5

)AN $AN AN$A %8' %"'

"1410000

0

0

0


34/101

"5+400

,0500

0

",0500

5-,50

1410000

0

0

0

5+400

",0500

0

",0500

11,500


3 4 5 + sb l&kal

0 "1410000 0 0 1

,0500 0 "5+400 ,0500 2

11,500 0 ",0500 5-,50 3

0 1410000 0 0 4

",0500 0 5+400 ",0500 5

5-,50 0 ",0500 11,500 +

sb l&kal


35/101

1.0 0.0 0 0 0 0

0.0 1.0 0 0 0 0

0 0 1 0 0 0

0 0 0 1.0 0.0 0

0 0 0 0.0 1.0 0

0 0 0 0 0 1

0 "1410000 0 0 1

,0500 0 "5+400 ,0500 0

11,500 0 ",0500 5-,50 ? 0

0 1410000 0 0 0

",0500 0 5+400 ",0500 0

5-,50 0 ",0500 11,500 0

,

0 0 0 0 164106000

0 0 0 0 0

1 0 0 0 = 0

0 1 0 0 "164106000

0 0 1 0 0

0 0 0 1 0


36/101

5.,05+E"00, 2.22124-E"00- 1.3-++E"00, 2.2212E"00-

2.2212E"00- 5.-3-151E"00+ "2.221E"00- ".,04E"00,

1.3-++E"00, "2.22125E"00- 5.,05+E"00, "2.221E"00-

2.2212E"00- ".,03+0E"00, "2.221E"00- 5.-3-2E"00+

-0 kN = b/L =

+0 kN =

=

3.125 m = a/L =

1.-,5 m =

5 m =

AML1

AML2 b2%3a8b'/L3=

AML3 ab2/L2=

AML4

AML5 a2%a83b'/L3=

AML6 "a2b/L2=


37/101

1.00 0.00 0.00 0.00 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 1.00 0.00

0.00 0.00 0.00 0.00 1.00

0.00 0.00 0.00 0.00 0.00


38/101

0.80 -0.60 0.00 0.00 0.000.60 0.80 0.00 0.00 0.00

0.00 0.00 1.00 0.00 0.00

= 0.00 0.00 0.00 0.80 -0.60

0.00 0.00 0.00 0.60 0.80

0.00 0.00 0.00 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 1.00 0.00 0.00

= 0.00 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


39/101

0

"30

"12.5

"2.-125

AE = "+3.,5

"22.+5+25

2.-125

"12+.25

33.53,5

0

"+0

25


40/101

-9.71796E-07

-1.64994E-04

9.71796E-07

2.17985E-04

0 1

0 2

0 3

-9.717961E-007 4

0 5

-0.0001649936 6


41/101

0 0 1.370

0 30 -11.632

0 0 12.5 -9.693 0

-9.717961E-007 0 -1.3700 30 11.632

-0.0001649936 -12.5 -19.387

-7.774369E-007 4 15.830776E-007 5 0

-0.0001649936 6 2

7.774369E-007 7 3

-5.830776E-007 8 0

0.0002179852 9 4

-7.774369E-007 22.5 -1.096

5.830776E-007 25.3125 0.942

0.000 0 35.15625 -3.270 0

0.000 37.5 1.096

0.000 54.6875 -0.942

0.000 -58.59375 7.980


42/101

9.717961E-007 7 3

0 8 0

0.0002179852 9 4

0 10 00 11 0

0 12 0

9.717961E-007 0 1.370

0 60 15.368

0.0002179852 0 25 25.613 0

0 0 -1.370

0 60 -15.368

0 -25 12.807


43/101


44/101


45/101

0 0 0 0 0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 00 0 0 0 1

2 3 4 5 + sb gl&bal

0 0 "164106000 0 0 1

5+6400 ,06500 0 "5+6400 ,06500 2

,06500 11,6500 0 ",06500 5-6,50 3

0 0 164106000 0 0 4

"5+6400 ",06500 0 5+6400 ",06500 5

,06500 5-6,50 0 ",06500 11,6500 +

sb gl&bal


46/101


47/101

1 0 0 0 0

1 0 0 0 0

0 1 0 0 0

0 0 1 1 0

0 0 "1 1 00 0 0 0 1

5 + , - sb gl&bal

335601+ "1065,5 "4536,3- "335601+ "1065,5 4

25-6312 146100 "335601+ "25-6312 146100 5

146100 5-6,50 1065,5 "146100 263,5 +

"335601+ 1065,5 4536,3- 335601+ 1065,5 ,

"25-6312 "146100 335601+ 25-6312 "146100 -

146100 263,5 1065,5 "146100 5-6,50

sb gl&bal


48/101


49/101

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

- 10 11 12 sb gl&bal

0 0 "164106000 0 0 ,

5+6400 ,06500 0 "5+6400 ,06500 -

,06500 11,6500 0 ",06500 5-6,50

0 0 164106000 0 0 10

"5+6400 ",06500 0 5+6400 ",06500 11

,06500 5-6,50 0 ",06500 11,6500 12

sb gl&bal


50/101

22.5 kN

25.3125kN

35.15+25kN.m

3,.5 kN

54.+-,5kN

"5-.53,5kN.m


51/101

0.00 0

0.00 30

0.00 > 12.5

0.00 0

0.00 30

1.00 -12.5


52/101

0.00 22.50.00 25.3125

0.00 > 35.15625

0.00 37.5

0.00 54.6875

1.00 -58.59375

0.00 0

0.00 60

0.00 > 25

0.00 0

0.00 60

1.00 -25


53/101


54/101


55/101

1.37 1 1.3,02 kN

18.37 2 1-.3+-0 kN

2.81 3 2.-0++ kN.m

-1.37 4 "1.3,02 kN41.63 5 41.+320 kN

-31.89 6 "31.--+,kN.m

21.40 1 1 21.403- kN

26.25 2 0 2+.254, kN

31.89 3 2 31.--+, kN.m

38.60 4 3 3-.5+2 kN

53.75 5 0 53.,453 kN

-50.61 6 4 "50.+133kN.m


56/101

1.37 1 3 1.3,02 kN

75.37 2 0 ,5.3+-0 kN

50.61 3 4 50.+133 kN.m

-1.37 4 0 "1.3,02 kN

44.63 5 0 44.+320 kN

-12.19 6 0 "12.134kN.m


57/101


58/101


59/101

0

0

0

$1

0

$2

@E>


60/101


61/101

$1

0

$2

$3

0

$4

@E>


62/101


63/101

$3

0

$4

0

0

0

@E>


64/101


65/101


66/101


67/101


68/101


69/101


70/101


71/101

Properties penampang

Bahan / material :

E = 2.35E80, kN/m2

Batang 1 :

0.30m

0.50m

1.50E"01m2

E = 2.35E80, kN/m2

3.13E"03m4

2.5m

0 degrees

#&emen 'a&am matriks kekakuan batang

EA / L 4.EI / L 2.EI / L +.EI / L*2

1.41E80+ 1.1-E805 5.--E804 ,.05E804

ekap #&emen 'a&am Matriks "ekakuan atangBatang EA / L 4.EI / L 2.EI / L

1 1.41E80+ 1.1-E805 5.--E804

2 ,.05E805 5.--E804 2.4E804

3 1.41E80+ 1.1-E805 5.--E804

1 2 3 4

* su'a+ %A,AM

Batang = 1

0 degrees

EA / L 4.EI / L 2.EI / L +.EI / L*21.41E80+ 1.1-E805 5.--E804 ,.05E804

b1=

h1=

A1=

I1=

L1=

theta1=

Hitung matriks kekakuan batang 1 /SM10

theta1=


72/101

? = !&s theta = 1

C = sin theta = 0

1 D1 hi1

1 1410000 0 0

D1 0 5+400 ,.05E804hi1 0 ,.05E804 1.1-E805

2 "1410000 0 0.00E800

D2 0 "5+400 ",0500

hi2 0 ,.05E804 5.--E804

* su'a+ %A,AM

Batang = 2

3+.-+-,+5 degrees

EA / L 4.EI / L 2.EI / L +.EI / L*2

,.05E805 5.--E804 2.4E804 1.,+E804

? = !&s theta = 0.-

C = sin theta = 0.+

2 D2 hi2

2 453,3- 33501+ "105,5

D2 33501+ 25-312 1.41E804

hi2 "105,5 1.41E804 5.--E804

3 "453,3- "33501+ 1.0+E804D3 "33501+ "25-312 "14100

hi3 "105,5 1.41E804 2.4E804

* su'a+ %A,AM

Batang = 3

0 degrees

EA / L 4.EI / L 2.EI / L +.EI / L*2

1.41E80+ 1.1-E805 5.--E804 ,.05E804

? = !&s theta = 1

C = sin theta = 0

3 D3 hi3

3 1410000 0 0

D3 0 5+400 ,.05E804

hi3 0 ,.05E804 1.1-E805

4 "1410000 0 0.00E800

D4 0 "5+400 ",0500

hi4 0 ,.05E804 5.--E804

()1 =


theta2=

()2 =


theta3=

()3 =

MAT!"S "#"A"$AN ST$"T$ /SM 0


73/101


74/101

BAAN7 3

&int 3 : , 0 reaksi ng titik 1 ""#

- +0 reaksi ng titik 1 ""#

25 reaksi ng titik 1 ""#

&int 4 : 10 0 reaksi ng titik 2 ""#

11 +0 reaksi ng titik 2 ""#12 "25 reaksi ng titik 2 ""#

Beban ekiDalen

&int (b 7l&bal Element 1

1 1 0

1 2 30

1 3 12.5

"Ae = 2 4 0

2 5 30

2 + "12.5

3 , 0

3 - 0

3 0

4 10 0

4 11 0

4 12 0

eban oint akibat beban &uar 'i oint

&int (b 7l&bal

1 1 01 2 0

A = 1 3 0

2 4 0

2 5 0

2 + 0

3 , 0

3 - 0

3 0

4 10 0

4 11 04 12 0

eban kombinasi atau gabungan (Ae 5 A)

A! = Ae 8 A

1 0

A! = 2 "30

3 "12.5

4 "2.-125

5 "+3.,5+ "22.+5+25 8

, 2 -125


75/101

- "12+.25

33.53,5

10 0

11 "+0

12 25

)atri? e;ati&n:ea!ti&n Alied

0 "2.-125

0 "22.+5+25

0 2.-125

0 33.53,5

91 0

@1 "30 =

)1 "12.5

@2 "+3.,5

@3 "12+.25

94 0

@4 "+0

)4 25

$isla!ements:

2 = ".,1,+0,1+5E"00,

hi2 = "0.0001+43+

3 = .,1,+0,1+5E"00,

hi3 = 0.00021,-52

ea!ti&ns:91 = 0

@1 = "30

)1 = "12.5

@2 = "+3.,5

@3 = "12+.25

94 = 0

@4 = "+0

)4 = 25


76/101

atatan a = N/m2

Batang 2 :

0.30

0.50

1.50E"01

E = 2.35E80,

3.13E"03

5

3+.-+-,+4+

#&emen 'a&am matriks kekakuan batan

12.EI / L*3 EA / L 4.EI / L

5.+4E804 ,.05E805 5.--E804

+.EI / L*2 12.EI / L*3 theta

,.05E804 5.+4E804 0 deg

1.,+E804 ,.05E803 3+.-+-,+ deg

,.05E804 5.+4E804 0 deg

5 +

$M$ ST$"T$-,OA,

12.EI / L*35.+4E804

b2=

h2=

A2=

I2=

L2=

theta2=


77/101

2 D2 hi2 !ek :

"1410000 0 0 164106000

0 "5+400 ,.05E804 00.00E800 ",0500 5.--E804 0

1.41E80+ 0 0.00E800 "164106000

0 5.+4E804 ",0500 0

0.00E800 ",0500 1.1-E805 0

$M$ ST$"T$-,OA,

12.EI / L*3

,.05E803

3 D3 hi3 ek:

"453,3- "33501+ "105,5 4536,3-

"33501+ "25-312 1.41E804 335601+

1.0+E804 "14100 2.4E804 "1065,5

4.54E805 33501+ 1.0+E804 "4536,3-33501+ 2.5-E805 "14100 "335601+

1.0+E804 "14100 5.--E804 "1065,5

$M$ ST$"T$-,OA,

12.EI / L*3

5.+4E804

4 D4 hi4 ek:

"1410000 0 0 164106000

0 "5+400 ,.05E804 0

0.00E800 ",0500 5.--E804 0

1.41E80+ 0 0.00E800 "164106000

0 5.+4E804 ",0500 0

0.00E800 ",0500 1.1-E805 0


78/101

hi3 1 D1 hi1 D2 D3

"105,5 "1410000 0 0 33501+ "33501+

23,5 0 ,0500 5-,50 "5+400 "14100

105,5 0 0 0 "33501+ 33501+

1,+250 0 0 0 14100 5+4000 1410000 0 0 0 0

0 0 5+400 ,0500 "5+400 0

0 0 ,0500 11,500 ",0500 0

14100 0 "5+400 ",0500 314,12 "25-312

5+400 0 0 0 "25-312 314,12

0 0 0 0 0 0

",0500 0 0 0 0 "5+400

5-,50 0 0 0 0 ,0500

> ? FdG

kekang (keua&i beban &uar 'i oint)

gaCa h&riH&ntal / arah ? %sb strktr'

gaCa Dertikal /arah C %sb strktr'

m&men arah H %sb strktr'











79/101





gaCa Dertikal /arah C %sb strktr'm&men arah H %sb strktr'

Element 2 Element 3

0 0 0

0 0 30

0 0 12.5

2.-125 0 2.-125

33.,5 0 +3.,5

8 35.15+25 8 0 = 22.+5+25

"2.-125 0 "2.-125

++.25 +0 12+.25

"5-.53,5 25 "33.53,5

0 0 0

0 +0 +0

0 "25 "25

$

d1

d2

d3

d4

$

0 1 0

0 D1 "30

0 hi1 "12.5

0 2 "2.-125 d1

0 D2 "+3.,50 = hi2 "22.+5+25 d2

0 3 2 -125 d3


80/101

0 D3 "12+.25

0 hi3 33.53,5 d4

0 4 0

0 D4 "+0

0 hi4 25

2 hi2 3 hi3 1

2 1-+3,3- "105,5 "453,3- "105,5 "1410000

hi2 "105,5 1,+250 105,5 23,5 0

3 "453,3- 105,5 1-+3,3- 105,5 0

hi3 "105,5 23,5 105,5 1,+250 0

1 "1410000 0 0 0 1410000

D1 0 ,0500 0 0 0

hi1 0 5-,50 0 0 0

D2 33501+ "5+400 "33501+ 14100 0

D3 "33501+ "14100 33501+ 5+400 0

4 0 0 "1410000 0 0

D4 0 0 0 ",0500 0

hi4 0 0 0 5-,50 0

ek:

".,1,+0,1+5E"00, "

"0.0001+43+ "

.,1,+0,1+5E"00, "

0.00021,-52 "

ek:8 1.3,02324+1 = 1.3,02324+ 1.3,02324+1

8 "11.+3204+0- = "41.+3204+ 1-.3+,530

8 ".+33,2424- = "22.133,24

8 11.,2-03+-04 = "52.0210+3

8 15.2,10+31+ = "110.,-04

8 "1.3,02324+1 = "1.3,02324+

8 "15.3+,53002 = ",5.3+,531

8 12.-0++2,5,52 = 3,.-0++2,+


81/101

%ata beban

= 100 kN

;1 = 24 kN/m;2 = 4- kN/m

Batang 3 :

m

m

m2

kN/m2 E =

m4

m tan theta 0.,5 3+.-+-,+

degrees 0.,5

#&emen 'a&am matriks kekak

2.EI / L +.EI / L*2 12.EI / L*3 EA / L

2.4E804 1.,+E804 ,.05E803 1.41E80+

b3=

h3=

A3=

I3=

L3=

theta3=


82/101

0 0 "164106000 0 0 "

5+6400 ,06500 0 "5+6400 ,06500 ",06500 11,6500 0 ",06500 5-6,50 "

0 0 164106000 0 0 "

"5+6400 ",06500 0 5+6400 ",06500 "

,06500 5-6,50 0 ",06500 11,6500 "

335601+ "1065,5 "4536,3- "335601+ "1065,5 "

25-6312 146100 "335601+ "25-6312 146100 "

146100 5-6,50 1065,5 "146100 263,5 "

"335601+ 1065,5 4536,3- 335601+ 1065,5 ""25-6312 "146100 335601+ 25-6312 "146100 "

146100 263,5 1065,5 "146100 5-6,50 "

0 0 "164106000 0 0 "

5+6400 ,06500 0 "5+6400 ,06500 "

,06500 11,6500 0 ",06500 5-6,50 "

0 0 164106000 0 0 "

"5+6400 ",06500 0 5+6400 ",06500 "

,06500 5-6,50 0 ",06500 11,6500 "


83/101

4 D4 hi4

0 0 0

0 0 0

"1410000 0 0

0 ",0500 5-,500 0 0

0 0 0

0 0 0

0 0 0

0 "5+400 ,0500

1410000 0 0

0 5+400 ",0500

0 ",0500 11,500

ek:

16-+36,3- "1065,5 "4536,3- "1065,5 " "

"1065,5 1,+6250 1065,5 263,5 " "

"4536,3- 1065,5 16-+36,3- 1065,5 " "

"1065,5 263,5 1065,5 1,+6250 " "

tanda%8'6%"' ditentkan arah gaCa th smb






tanda%8'6%"' ditentkan arah gaCa th smb .!&s a = -0

tanda%8'6%"' ditentkan arah gaCa th smb .sin a = +0


tanda%8'6%"' ditentkan arah gaCa th smb a = 3.125

tanda%8'6%"' ditentkan arah gaCa th smb b = 1.-,5

tanda%8'6%"' ditentkan arah gaCa th smb L = 5


84/101

0

"30

"12.5

"2.-125

"+3.,5

Ae = "22.+5+25

2.-125

"12+.25

33.53,5

0

"+0

25

ata lang A!

$

2 "2.-125 d1

A! = hi2 "22.+5+25 d2

3 2.-125 d3

hi3 33.53,5 d4

1 0D1 "30

hi1 12 5


85/101

D2 "+3.,5

D3 "12+.25

4 0

D4 "+0

hi4 25

D1 hi1 D2 D3 4 D4 hi4

0 0 33501+ "33501+ 0 0 0

,0500 5-,50 "5+400 "14100 0 0 0

0 0 "33501+ 33501+ "1410000 0 0

0 0 14100 5+400 0 ",0500 5-,50

0 0 0 0 0 0 0

5+400 ,0500 "5+400 0 0 0 0

,0500 11,500 ",0500 0 0 0 0

"5+400 ",0500 314,12 "25-312 0 0 0

0 0 "25-312 314,12 0 "5+400 ,0500

0 0 0 0 1410000 0 0

0 0 0 "5+400 0 5+400 ",0500

0 0 0 ,0500 0 ",0500 11,500

"


86/101

0.30m

0.50m

1.50E"01 m2

2.35E80,kN/m2

3.13E"03 m4

2.5m

0 degrees

uan batang

4.EI / L 2.EI / L +.EI / L*2 12.EI / L*3

1.1-E805 5.--E804 ,.05E804 5.+4E804


87/101

" " " " "

" " " " "" " " " "

" " " " "

" " " " "

" " " " "

" " " " "

" " " " "

" " " " "

" " " " "" " " " "

" " " " "

" " " " "

" " " " "

" " " " "

" " " " "

" " " " "

" " " " "


88/101

" "

" "

" "

" "

kN 7aCa lateral = b/L =

kN 7aCa a?ial =

=

m = a/L =

m =

m =

A)L1

A)L2 b2%3a8b'/L3=

A)L3 ab2/L2=A)

L4

A)L5 a2%a83b'/L3=

A)L+ "a2b/L2=


89/101


90/101

2

hi2

3

hi3

0

? 0

0

0

0

0

0

0


91/101


92/101


93/101

22.5 kN a?ial %arah sb l&kal' ""# dibah arah sb gl&bal

25.3125 kN lateral %arah sb l&kal' ""# dibah arah sb gl&bal

35.15+25kN.m m&men %arah sb l&kal'

3,.5 kN a?ial %arah sb l&kal' ""# dibah arah sb gl&bal

54.+-,5 kN lateral %arah sb l&kal' ""# dibah arah sb gl&bal

"5-.53,5kN.m m&men %arah sb l&kal'


94/101


95/101


96/101


97/101


98/101

? C

: 1- 13.5

: "15.1-,5 20.25

: 30 22.5

: "32.-125 43.,5


99/101


100/101


101/101

+0

sin 0.-++0254 +0

cos 0.5 +0

1.,32050-1 +0

tan theta 0.,5 3+.-+-,+

0.,5

Documents

Contoh Soal Analisa Matriks (Portal Hal.186)