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METODE MATRIK APLIKASI METODE MATRIK UNTUK ANALISA STRUKTUR BALOK

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Analisa Struktur Metode Matriks

Contoh Soal Analisa Matriks

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DIDAPAT DARI DOSEN UNIVERSITAS GUNADARMA SULARDI., ST., MT.

METODE MATRIK APLIKASI METODE MATRIK MATRIK UNTUK ANALISA STRUKTUR BALOK

1. PENGERTIAN UMUM

Metode matrik adalah suatu pemikiran baru pada analisa struktur, yang berkemban

bersamaan dengan populernya penggunaan pengguna an computer otomatis untuk operasi o perasi perhitunga aritmatika.

Hal utama dalam analisa untuk menenentukan baik itu deformasi ataupun stress pad

struktur, ialah sampai jauh mana sudah diketahui sifat karakteristik hubungan gaya da deformasi deformasi dari elemen-elem elemen-elemen en struktur struktur,, dan memaksakan memaksakan terpenuhiny terpenuhinyaa syarat-sy syarat-sy kompatibiliti dan kesetimbangan, ada tiga hal yang mendasari analisis ini, yaitu : 1. kes kesetim etimba bang ngan an Master your semester with Scribd Read For 30this Days Sign to vote on title 2. hubungan stress stress dan strain, strain, atau gaya dalam dalam dan daup nFree deformasi defor masi & The New York Times ti,atau  Useful  Not useful 3. kompatibili kompatibiliti,at au kontinuitas kontinuitas dari deformasi deformasi Special offer for students: Only $4.99/month.

Cancel anytime.

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Master your semester with Scribd & The New York Times Special offer for students: Only $4.99/month.

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dimana :

{ Q} # gaya yang timbul pada titik-titik diskrit akibat adanya lendutan. { D} # lendutan pada titik-titik diskrit

[ K ] # menyatakan kekakuan dari struktur

metode kekakuan ini juga disebut metode lendutan (displacement method), method), karena analis

dimulai dengan $ lendutan% sehingga dengan demikian urutan kerjanya secara garis bes adalah sebagai berikut :

1. kompabiliti kompabiliti&& yaitu yaitu mencari hubungan hubungan antara antara deformasi deformasi dengan dengan lendutan, lendutan, ata

secara secara tegasn tegasnya ya mencar mencarii deform deformasi asi apa yang yang terjad terjadii pada pada elemen elemen-el -eleme eme dititik-ti dititik-titik tik diskrit diskrit akibat diberikanny diberikannyaa lendutan lendutan pada struktur struktur dititik-ti dititik-ti tersebut. 2. persa persama maan an hubun hubunga gan n stre stress ss dan stra strain in,, yait yaitu u menca mencari ri hubung hubungan an menge menge

gaya-gaya gaya-gaya dalam yang timbul timbul sebagai sebagai akibat adanya deformasi deformasi pada eleme elemen pada struktur tersebut. 3. kesetimbang kesetimbangan, an, langkah langkah terakhir terakhir yang menyatakan menyatakan hubungan gaya luar luar diti

diskrit dengan gaya-gaya dalam atau mencari berapa besar gaya luar di ujun

elemen-elemen yang tepat diimbangi oleh gaya-gaya dalam elemen titik-titi diskrit.

Master your iala ialah h suat suScribd atu u cara cara untuk untuk anal analis isaa stru strukt ktur ur dima dimana na dala dalam m pr Metode Metosemester de kekakuan kekakuan with  Read Free Foron 30this Days Sign up to vote title  perumusan dari analisanya diambil lendutan di titik-titik diskrit besaraan $anu & The New York Times Not useful  Useful sebagai Cancel anytime.

Special offer for students: $4.99/month. yang yang Only hendak hend ak dicari dicari.da .dalam lam proses proses mengana menganalis lisaa akan mengena mengenall bebera beberapa pa matri' matri' yan

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dimana :

{ Q} # gaya yang timbul pada titik-titik diskrit akibat adanya lendutan. { D} # lendutan pada titik-titik diskrit

[ K ] # menyatakan kekakuan dari struktur

metode kekakuan ini juga disebut metode lendutan (displacement method), method), karena analis

dimulai dengan $ lendutan% sehingga dengan demikian urutan kerjanya secara garis bes adalah sebagai berikut :

1. kompabiliti kompabiliti&& yaitu yaitu mencari hubungan hubungan antara antara deformasi deformasi dengan dengan lendutan, lendutan, ata

secara secara tegasn tegasnya ya mencar mencarii deform deformasi asi apa yang yang terjad terjadii pada pada elemen elemen-el -eleme eme dititik-ti dititik-titik tik diskrit diskrit akibat diberikanny diberikannyaa lendutan lendutan pada struktur struktur dititik-ti dititik-ti tersebut. 2. persa persama maan an hubun hubunga gan n stre stress ss dan stra strain in,, yait yaitu u menca mencari ri hubung hubungan an menge menge

gaya-gaya gaya-gaya dalam yang timbul timbul sebagai sebagai akibat adanya deformasi deformasi pada eleme elemen pada struktur tersebut. 3. kesetimbang kesetimbangan, an, langkah langkah terakhir terakhir yang menyatakan menyatakan hubungan gaya luar luar diti

diskrit dengan gaya-gaya dalam atau mencari berapa besar gaya luar di ujun

elemen-elemen yang tepat diimbangi oleh gaya-gaya dalam elemen titik-titi diskrit.

Master your iala ialah h suat suScribd atu u cara cara untuk untuk anal analis isaa stru strukt ktur ur dima dimana na dala dalam m pr Metode Metosemester de kekakuan kekakuan with  Read Free Foron 30this Days Sign up to vote title  perumusan dari analisanya diambil lendutan di titik-titik diskrit besaraan $anu & The New York Times Not useful  Useful sebagai Cancel anytime.

Special offer for students: $4.99/month. yang yang Only hendak hend ak dicari dicari.da .dalam lam proses proses mengana menganalis lisaa akan mengena mengenall bebera beberapa pa matri' matri' yan

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2. matr matrik ik kekok kekokoha ohan n int inter erne nen n [ S ] , suatu matri' yang memenuhi hokum hook dalam mana dinyatakan hubungan antara gaya dan deformasi :

{  } = [ S ]{ d }

!1.3"

dimana :

{  } # menyatakan gaya dalam elemen [ S ] # adalah matri' kekokohan intern elemen { d } # menyatan deformasi elemen

3. matr atri' sa satis [ ! ] , suatu matri' yang menyatakan kesetimbangan antara gay luar dan gaya dalam :

{ Q} # [ ! ]{  }

!1.("

dimana :

{ Q} # menytakan gaya luar yang bekerja dititik diskrit [ ! ] # matri' statis {  } # gaya dalam elemen Maka ketiga matri' di atas digabungkan, maka akan didapatkan hubungan :

{Q} = [ !]{  } {Q} = [ !][] [ S ]{ d } {Q} = [ !][ S ]( [ A]{ d } ) {Q} = [ !][ S ][][ A]{ D} {Q} = [ K ]{ D}


!1

-"

Read Free Foron 30this Days Sign up to vote title

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[ K ] = [ !][][ S ][][ A]

!1.1/"

0adi 0adi salah salah satu satu tujuan tujuan termi terminal nal yang yang pentin penting g adalah adalah proses proses analisa analisa ini ialah dap

menurunkan matrik kekakuan struktur [ K ] menurut persamaan !1.1/". elanjutnya aka mudah dicapai tujuan akhir, yaitu analisa lendutan dan gaya dalam elemen.

1.2 DERAJAT KETIDAKTENTUAN KINEMATIS KINEMATIS

ntuk analisa ini akan dimulai dengan mengambil lendutan di titik-titik diskrit sebaga sasaran yanmg harus dihitung.

ntuk mengetahui dimana harus $dipasang% besaran lendutan yang akan dicari tersebu maka maka harus harus diketa diketahui hui dahulu dahulu bebera beberapa pa deraja derajatt ketida ketidak k tentua tentuan n kinema kinemati tiss atau atau ist lainnya derajat kebebasan (de"ree of freedom) dari freedom) dari struktur.

erajat ketidak-tentuan kinematis ialah suatu besaran yang menytakan jumlah kompone

bebas dari lendutan dititik diskrit yang mungkin terjadiyang berhubungan denga

diberikannya diberikannya suatu pembebanan pembebanan pada struktur struktur.. i baah ini diberikan diberikan beberapa beberapa maca struktur bidang yang akan ditujukkan berapa derajat ketidak-tentuan kinematisnya.



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You're Reading a Preview Unlock full access with a free trial.

Download With Free Trial



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1.! DASAR PER"ITUNGAN

alam pasal ini, akan dijelaskan secara mendetail urut-urutan analisa dari

sua

konstruksi bidang !dua dimensi" dengan berdasarkan pada metode kekakuan.

ekarang terlihat satu konstruksi seperti seperti ditunjukkan pada gambar 2.! selanjutnya akan diikuti urutan dari proses analisa.

!a" gambar konstruksi statis tak tentu

You're Reading a Preview !b" derajat ketidak-tentuan kinematis : 3

Unlock full access with a free trial.



!c" diagr


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a luar eki5alen Q yang koresponding dengan lenduta

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!e" diberikan D1 # 1 satuan

!f" diberikan D2 #1 satuan

You're Reading a Preview full access with a free trial. !g" diberikan D3Unlock #1 satuan




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7onstruksi ini ialah balok menerus di atas empat perletakan, satu jepit dan tiga send

merupakan suatu konstruksi dengan derajat ketidak-tentuan kinematis sebesar 3 !gamba 2.b" 8angkah pertama ialah menyelidiki kompatibilitas dari struktur, dengan ja

memberikan berturut-turut lendutan D1 = 1, D2 = 1 dan D3 = 1 !gambar 2.e, 2.f, dan 2.g Mudah dapat kita lihat, baha : d 2 = d 3 = D1 d ( = d - = D2 d , = D3 d 1 = /

atau disusun secara sistematis : d 1 d 2 d 3 d ( d d 

= / = D1 = D1 a Preview = You're DReading 2 = Unlock D full2 access with a free trial. = D3

Download With Free Trial bila dinyatakan dalam hubungan matri' :

 d 1  / 1 / d  1 / / Master your semester with  2  Scribd   D  d 3  1 / /  1  & The New York Times d  = / 1 /  D2    D  Special offer for students: Only $4.99/month.  (   d -  / 1 /  3     


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!1.11"

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/ 1  1  { A} =  / /  / ↑ 



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/  ← d 1

/

  /  ← d 3  /  ← d 3 /  ← d (  1  ← d ↑ / ← d 2

/ / 1 1 /

↑

!1.13"

D1 = 1 D2 = 1 D3 = 1

8angkah keduaialah menyelidiki hubungan gaya dalam dan deformasi dengan melih tiap-tiap elemen sebagai bagian yang diskrit, seperti pada gambar 2.h. ari sifat elastis elemen, didapatkan hubungan : d 1 =

1  1 %1 3 #$ 1

d 2 = −

−

1  1 %1 , #$ 1

1  2 %1 , #$ 1

+

!1.1("

1  2 %1 3 #$ 1

dimana :

You're Reading a Preview d 1 # menyatakan deformasi yang terjadi di ujung elemen Unlock full access with a free trial.

 # menyatakan gaya dalam yang ada di ujung elemen, dalam hal ini momen lentur Download With Free Trial

sebenarnya pers.! 1.1( " ini sudah bukan hal yang asing lagi karena sudah sering dijump dalam analisa struktur dengan metode perputaran sudut !sloop deflection method".

Master your semester with Scribd 9ila pers. ! 1.1( " diin5erskan, akan didapat : & The New York Times ( #$ 2 #$  = d + d 1

1 Special offer for students: Only $4.99/month. %1

1

1

%1

2


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d ,

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9ila hubungan ini dinyatakan dalam bentuk matri', maka :

 ( #$ 1 2 #$ 1 /  % % 1  1 2 #$ 1 ( #$ 1  /   1   %1 %1     ( #$ 2  2  / /   3   %2  = 2 #$ 2   (   / / %2   -        ,  / / /   / /  / 

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0adi sebenarnya matri' [ S ] ialah suatu matri' yang menyatakan berapa besar gaya dalam

[  ]

yang timbul diujung elemen bila di titik-titik tersebut diberikan satu sat

deformasi { d } . 8angkah ketiga adalah menyelidiki tentang kesetimbangan gaya luar dan gaya dalam : Melihat gambar Q1 =  2 +  Q2 =  ( +  -

!1.2/"

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9ila dinyatakan secara matrik :

  1     2 Q1  / 1 1 / / /       3    Q2  = / / / 1 1 /  Q  / / / / / 1  (   3          ,  

!1.21"


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{ Q} # menyatakan gaya-gaya luar yang bekerja di titik-titik diskrit. { D} = menyatakan lendutan di titik bersangkutan yang berkoresponding denga gaya { Q} . ari persamaan ! 1.13" dan ! 1.23", ternyata didapatkan :

[ ! ] = [ A] &

!1.2"

persamaan ! 1.2"" ini dapat dibuktikan dengan prinsip kerja 5irtual.

a.

gaya luar 5irtual


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lendutan aktuil

Master your semester with 4amabar 1.3 konstruksi balok Scribd menerus pada mana dikerjakan gaya 5irtual. Read Free Foron 30this Days Sign up to vote title & The New York Times  Useful  Not useful Special offer for students: Only $4.99/month.

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{ d } = [ A]{ D} {Q ∗ } = [ ! ]{  ∗ }

{Q ∗ }

&

= {  } [ ! ] &

&

!1.2*" !1.2)" !1.3/"

maka persamaan ! " bisa ditulis &

{  ∗ } [ ! ] { D} = {  ∗ } [ A]{ D} &

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&

!1.31"

9ila disederhanakan, akan memberikan :

[ !] & = [ A] [ !] = [ A] & !1.32" !1.33"

engan demikian persamaan ! 1.33", bisa ditulis : You're Reading a Preview &

[ K ] = [ A] [ S ][ A]

!1.3("


engan demikian persamaan !Download 1.1/" telah yaitu untuk menurunka Withdipermudahkan, Free Trial

matri' kekakuan [ K ] , cukup hanya menurunkan dua matrik penbentuknya, yaitu matr deformasi [ A] dan matri' kekokohan intern elemen [ S ] .

Master your semester with Scribd Read Free Foron 30this Days Sign up to vote title & The Newntuk York Times menghitung gaya dalam digunakan hubungan  : Useful  Not useful Special offer for students: Only $4.99/month. [  ] = [ S ]{ d }

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!1.3"

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ontoh 3.1

ibaah ini akan dibahas secara singkat analisa dengan metode kekakuan dengan deraj ketidak-tentuan kinematik tingkat 1. 6

// kg;m

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!a" konstruksi yang akan dianalisa

!b" konstruksi dasar yang dikekang



!c" mopmen primer (fixed'end moment)

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2

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!e" gaya luar eki5alen dititik diskrit yang koresponding dengan lendutan D1 . Q1 = -/// − 32//! (" .m"

!f" diberikan D1 = 1 satuan




!g" diagram H - d


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Melihat gambar 1.( !f", dengan mudah akan didapatkan :

/ ← d 1 1 ← d   2 [ A] =   d ← 1 3    /  ← d ( ↑

D 1

=1

gari gambar 1.( !g" :


2 #$  ( #$ / Free Trial  1/ Download With 1/  2 #$ ( #$ /  with Scribd Master your semester 1/ Read Free Foron 30this Days Sign up to vote title [ S ] =  1/ #$ (Not & The New York Times useful  Useful   / / Special offer for students: Only $4.99/month.  *  Cancel anytime.

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 / .(  / .2 [ S ] =  /   /

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 /   /.2-   /. -  /

dari persamaan !1.3(" :

[ K ] = [ A] & [ S ][ A] # {/

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[ K ] = [ /.) #$ ] [ K ] − =  1 1

You're Reading a Preview  /.) #$ Unlock full access with a free trial.

engan mengubah gaya < menjadi gaya titik eki5alen di ujung elemen !gambar 1.(.c da Download With Free Trial e" dan dengan melihat persamaan !1.2" :

{ D} = [ K ] − { Q} 1

 1 {1*// Master your semester }  [ D ] = with  /.) #$ Scribd  2/// & The New York Times D = 1

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Special offer for students: Only $4.99/month. #$

dari persamaan !1.3" :


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  1   (//      *//   2    =    3  1///   (   -//   1  2  3  (

= (// = *// = 1/// = -//

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4ambar 1. istribusi gaya dalam You're Reading a Preview Unlock full access with a free trial.

hasil yang ditunjukkan oleh gambar 1. ialah menytakan besarnya momen lentur !dala

With Free Trial titik" yang didistribusikan hal ini sebagai momen batang,Download bukan sebagai momen

batang elemen 69 dan 9 sesuai dengan kekakuan masing-masing . jadi gaya dala

{  }

didapat dari hasil perhitungan ini bukan merupakan memen lentur yan Master your yang semester with Scribd  Read Free Foron 30this Days Sign up to vote title sebenarnya bekerja.  & The NewMomen Yorklentur Times Useful  Not useful  yang sebenarnya bekerja bisa diperoleh dengan mengurangi gaya dala Special offer for students: Only $4.99/month. {  } dengan momen primer elemen struktur.

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ebagai contoh kedua akan dibahas suatau konstruksi kinematis tertentu seperti pad gambar 1. !a".

!a" konstruksi yang akan dianalisa dengan beban Q

!b" struktur dasar yang dikekang


Download !c" derajat ketidak-tentuan kinematis : 2 With Free Trial


!d" diberikan D # 1 satuan


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!f" diagram H-d


!g" diagram kesetimbangan

Download With Free Trial 4ambar 1. balok di atas 2 tumpuan

Master your semester with Scribd  Read Free For 30this Days Sign up to vote title 8angkah pertama yang dilakukan ialah menganggap konstruksi ini on terdiri atas duaeleme & The Newdiskrit. York Times Not useful  Useful 6 dan 9 ! gambar 3. b". titik  segai titik diskrit mempunyai dua deraja Special offer for students: Only $4.99/month.

kebebasan, yaitu translasi dan rotasi.

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1  −  ,  1 − , =   1  ( You're Reading a Preview  1  Unlock full access with a free trial.  ( Download With Free Trial

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( , 2  [ S ] = #$  , /   / 

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

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!a" konstruksi yang akan dianalisa, dengan satu perletakan elastis dimana k # /. =>



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!c" deberikan D1 # 1 satuan


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ersoalan pada contoh ini sebenarnya sama dengan contoh 1.2, karena memunyai eleme batang yang sama dengan derajat kebebasan yang sama pula . maka proses analisa akan mendetail dibahas lagi disini, dan langsung akan matrik kekakuan :

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− 1  ,  1 − {  } #=>  ,  3  *  3   *

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!b" portal menerus menerus tanpa tanpa pergoyangan pergoyangan



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!a" portal bidang yang akan dianalisa, dengan bentuk konstruksi dan system pembebana yang simetris



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!e" gaya eki5alen dititik yang koresponding dengan lendutan 

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!g" diberikan D2 # 1 satuan



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engan memperhatikan gambar 1.3 akan didapatkan :

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METODE MATRIK APLIKASI METODE MATRIK UNTUK ANALISA STRUKTUR BALOK METODE MATRIK selvia

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ekarang akan dibahas analisa portal pada gambar !1.(" di baah ini :

!a" ortal yang dianalisa





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) +! = − ) !+ = − 1 .1///.( = −-//(" .m *

!c" Momen primer



Master your semester with Scribd Read Free Foron 30this Days Sign up to vote title & The New!d"York erajat Times ketidak-tentuan kinematsi : 2 !deformasi aksial diabaikan"  Useful  Not useful Special offer for students: Only $4.99/month.

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!e" 4aya eki5alen @ dititik diskrit yang koresponding dengan lendutan 

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!h" iagram H-d

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) #+ = −323.+3 − ! −12-/" = )2,.2+.(" .m ) +# = −2,*.,( − ! +12-/" = −1-1*.,((" .m ) +! = −**.)* − !−-//" = (11./2 (" .m ) +*

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BIDANG DENGAN PERGO$ANGAN DIMANA Master your KONSTRUKSI semester PORTAL with Scribd  Read Free Foron 30this Days Sign up to vote title  & The New York Times DE%ORMASI AKSIAL DIABAIKAN  Useful  Not useful Special offer for students: Only $4.99/month.

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etelah pada pasal yang lalu dibahas analisa portal tanpa penggoyangan, sekarang aka

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Master your semester with Scribd  ersoalan kekakuan struktur pada contoh soal ini adalah sama dengan Read Free Foron 30this Days Sign up to vote titlecontoh 1, ja  & The New York Times Useful Not useful   proses menghitung kekakuan [ K ] adalah sama dengan contoh tersebut. Special offer for students: Only $4.99/month.

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 D1   -12   1   D2  = 1- #$  − (*  D   − (*  3

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 (.--   2-+./(    − +-+./( {  } =   − 2-+./(  +-+./(      +1.--   engan memperhatikan momen primer dari elemen-elemen struktur , maka ak didapatkan : ) A = (.--(" .m ) *A

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 1(/.)+3   121.-13    − 122.-1(   {  } =  1/1.-,/   3)*.(-1     2*(.+)/      Momen akhir :





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) A = 1(/.)+3

= 1(/.)+3(" .m ) *A = 121.-13 = 121.-13(" .m ) *# = − !−(//" = (//(" .m ) *D = − 221.-1( − !−-//" = 2+*.(*+(" .m ) D* = 1/1.-,/ − !+-//" = − 3*).((/(" .m ) D! = 3)*.(-1 = 3*).((/(" .m ) ! = 2*(.+)/ = 2*(.+)/(" .m



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dalam dan deformasi, baik gaya dalam maupun deformasi yang timbul hanyalah bersif aksial saja. ontoh terliha di baah ini.





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4amnbar 3.1 7onstruksi Bangka 9atang

Memperhatikan gambar 3.1, akan dengan mudah dapat ditentukan matrik [ A] , yait matrik yang menyatakan hubungan deformasi dan lendutan. ari gambar 3.1 e, untuk D1 = 1 d 1 d 2 d 3 d ( d -

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d 3

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/ / / ( ( -

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esuai dengan apa yang telah disinggung di bagian depan pada pasal ini, maka eleme

elemen pada konstruksi rangka batang ini hanya menderita deformasi aksial saja, yanm

dengan demikian hanya menimbulkan gaya dalam normal saja. 7arena disini membaha konstruksi yang elastis, maka hokum Hooke akan berlaku karenanya.

4ambar 3.1 9atang yang menderita gaya normal H dan mengalami deformasi aksial d Melihat 4ambar 3.1, d =

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engan melihat persamaan ! ", maka jelas dapat diketahui baha matrik [ S ] , aka Download With Free Trial terdirin dari elemen-elemen kekakuan aksial, yaitu :



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 A # / /  %  A #  / / You're Reading a Preview  %  A # [ S ] =  / Download With Free/Trial %   Master your semester / / Read Free Foron 30this Days  with/Scribd Sign up to vote title & The New York Times  Useful  Not useful  Special offer for students: Only $4.99/month.  / / / 1

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 D1  Q1   D     2 −1 Q2    = [ K ]    D3  Q3   D(  Q( 

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=/ 2 :  2 = / 3 :  3 = −1///(" ( :  ( = 2/*3.33(" ( :  - = −(1.+("

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Memperhatikan gambar di atas, akan didapat matrik matrik deformasi [ A]

Master your semester with Scribd & The New4ambar Yorkd,Times untuk D = 1 1

Special offer for students: Only $4.99/month. d 1 = /.*


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4ambar f, untuk

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D3

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=/ d 2 = /.3*d 3 = /.3*d ( = / d - = −1 d 1

4ambar g, untuk

D( = 1

=/ d 2 = /.)23 d 3 = −/.)23 d ( = / d - = / d 1

4ambar h, untuk

D-

=1

=/ d 2 = / d 3 = /./23 d ( = /. d - = / d 1

0adi atrik [ A] :  / .* / .  / /  [ A] =  / /  /.* − /.  1 / Special offer for students: Only $4.99/month.


Download With Free Trial /

/

/.3*-

/.)23

Master your semester with/.Scribd 3*- − /.)23 / / & The New York Times −1 / Matrik

 /   /.)23 Read Foron 30this Days Sign to vote title  upFree /.    Useful  Not useful Cancel anytime. /  /

[ S ] terdiri dari elemen-elemen kekakuan aksial, yaitu :

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 A1 # 1  / / / /  %  1   A # 2 2  / / / /    %2   A3 # 3 / / /  [ S ] =  / %3     A( # ( / / / /   % (   A #  / - - / / /  %-    2- #  2.1///   /  [ S ] =  /   /    /

   ,- # / / /  (.,-/  ,- # / / /   (.,-/ You're Reading a Preview  / / /  Unlock full access with a free trial.  -- #  / Download / With /Free Trial  ,.--/  /

/

/ / / / Master your semester2-with Scribd /  / -/ / / #$ & The New York Times  / / -/ / [ S ] = / 2///  Special offer for students: Only $4.99/month. /

  /

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     2/  / 33 33

/

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 / .*  /.   =  /.    /  /

 /.*  /   /   /.*  1

of 83

/

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/.*

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− / .

/.3*-

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− /.)23

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/  -.33  / 1* #   − 33.33 / [ K ] = 2///  /  /  12 −)

[ K ] −1

 

 1+2.  − 1/*.* 1  =  1)).* #   − 1/*.* − 21+.

 /   − 1  /  /  1


 Search document

/ 2- /  / -/ /  / / -/  / / /  / / /



 / /   #$ / /   2/// 2/  / 33.33 /

/

 /   /.)23   /.  /  /

− 33.33

 − )  / / (*.1/ ++.++   − (2./ / *-.1) − (2./ a Preview ++You're .++ Reading -2.  /

12


  2/-.* symetris  With Free Trial  − 1(-.2 Download 2*+.2 j  − 1(-.2 )(. 11*.2  − 2)/.( 1*).( 1*).( 3+*.2

Master your semester with Scribd & The New8endutan Yorkyang Times terjadi : Special offer for students: Only $4.99/month. { D} = [ K ] −1 { Q}



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elanjutnya :

{  } = [ S ][ A]{ D} 1/ / /  − 1+2.  2/  / / 1).2(.1/   1/*.*     #   1////  / = / 1).2− (.1- (.1- − 1)).* 2///  #  − 1/ / 1-   1/*.*   2/   33.33  / / /   21+ .  − 33.33  

 − 1)//  1 -*+(.* 2   {  } = -*+(.* 3 − )1// (     (-33  -

0adi dapat gaya-gaya &  1 = − )1/// (" !te(an"

You're Reading a Preview "  2 = -*+(.* (" !tari( " full access with a free trial.  3 = -*+(.* (" !tari( Unlock

 ( = − )1// (" !te(an"

Download With Free Trial "  - = (-33 (" !tari(



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Soa* +

iketahui portal bidang tanpa pergoyangan kesamping pada gambar di atas, dimana deformasi aksial diabaikan dari elemen-elemennya. Petan-aan +

Hitung momen akhir dari masing-masing elemen, dengan methode matrik You're Reading a Preview Unlock full access with a free trial.




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