Tegangan Tarik dan Tegangan Tekan
1
BAB I TEGANGAN TARIK DAN TEGANGAN TEKAN
1.1. Pendahuluan
Mekanika bahan (mechanics (mechanics of material ) adalah cabang mekanika terapan yang mengkaji tentang kelakuan benda-benda pejal yang dikenakan berbagai jenis pembebanan. Bidang studi ini dikenal pula dengan beberapa nama, seperti “kekuatan bahan” ( strength of material ) dan “mekanika benda terdeformasi” (mechanis of deformable bodie). bodie). Benda-benda pejal yang ditinjau mencakup batang yang dibebani secara aksial, aksia l, poros yang mengalami puntiran berbagai jen is balok, tiang atau kolom. Biasanya tujuan tu juan analisisnya adalah menentukan tegangan ( stress), stress), regangan ( strain), strain), dan lendutan (deflection (deflection)) yang dihasilkan oleh berbagai jenis beban. Tegangan merupakan perluasan dari pelajaran ilmu gaya (statika), tetapi terdapat perbedaan yang nyata antara keduanya. Pada dasarnya, statika membahas hubungan antara gaya yang bekerja pada benda kaku dan benda dalam keadaan seimbang, sedangkan kekuatan bahan meliputi hubungan antara gaya luar yang bekerja dan pengaruhnya terhadap gaya dalam benda tersebut. Di samping itu, pada analisis analis is kekuatan, benda tidak t idak lagi dianggap kaku, sehingga deformasi kecil keci l tetap diperhitungkan. Jadi, kekuatan bahan adalah kemampuan suatu benda untuk menahan gaya yang terjadi padanya sampai pada batas yang ditentukan, sedangkan tegangan adalah reaksi dalam (gaya dalam) persatuan luas penampang. Gaya dalam adalah reaksi bagian dalam benda terhadap gaya luar. Tegangan dilambangkan dengan huruf Yunani :
(sigma), misalnya untuk kasus benda yang
ditarik aksial (seperti pada uji tarik). F
A F
F A
, dimana :
=
tegangan tarik (N/m2 = Pa)
F
=
gaya normal yang besarnya sama dengan gaya luar (N)
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Tegangan Tarik dan Tegangan Tekan
A =
2
luas penampang yang menahan gaya 2
luar (m ) Jenis tegangan pada dasarnya hanya terdiri dari tegangan normal dan tegangan geser. Tegangan normal adalah tegangan yang diakibatkan oleh gaya luar yang arahnya tegak t egak lurus penampang. F
F F
Penampang batang
A
Gambar 1.1. Gaya normal normal pada batang
Tegangan geser tegangan yang diakibatkan oleh gaya luar yang arahnya sejajar penampang yang menahan F
Penampang batang
Gambar 1.2. Gaya geser pada pelat Adapun jenis tegangan yang lain adalah merupakan pengembangan dari kedua jenis tegangan tersebut, yaitu tegangan tarik, tegangan tekan, tegangan lentur, tegangan puntir, yang kesemuanya itu ditinjau berdasarkan pembebanannya. Tegangan tarik adalah tegangan yang diakibatkan oleh beban tarik, sedangkan tegangan tekan diakibatkan oleh beban tekan. Bila benda ditarik akan terjadi pemanjangan dan bila ditekan ditekan akan terjadi pemendekan.
1.2. Tegangan dan Regangan
Konsep
dasar
tegangan
dan
regangan
dapat
dijelaskan
dengan
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Tegangan Tarik dan Tegangan Tekan
3
Gambar 1.3. Batang prismatik yang mengalami tarik Akibat gaya aksial F akan timbul regangan (internal ( internal stress). stress). Untuk menghitung tegangan perlu dilakukan pemotongan khayal pada penampang m – n dengan arah tegak lurus sumbu batang. Potongan sebelah kanan dipisahkan sebagai benda bebas (lihat gambar). Pada ujung kanan bekerja gaya F (aksi), sedangkan pada ujung kiri timbul gaya reaksi yang besarnya sa ma dengan gaya aksinya. Tegangan normal yang terjadi dapat dihitung :
F A
dimana,
= tegangan normal normal tarik (N/m)
F = gaya tarik (N) A = luas penampang batang (m2) Akibat gaya tarik F, batang akan mengalami pemanjangan, sedangkan apabila beban F berupa gaya tekan batang akan mengalami pemendekan. Perubahan panjang batang baik yang mengalami pemanjangan maupun pemendekan dinyatakan dengan huruf Yunani
(delta) yang digambarkan dalam
gambar di atas untuk batang dengan beban tarik. Perbandingan antara perubahan panjang ( ) dengan panjang semula (L) disebut sebagai regangan (strain) yang dinyatakan dengan huruf Yunani
= (epsilon)
Dalam bentuk persamaan :
L
dimana,
= regangan
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Tegangan Tarik dan Tegangan Tekan
4
Regangan untuk kasus di atas dinamakan regangan normal (normal ( normal strain) strain) karena beban yang diberikan arahnya tegak lurus penampang. Regangan merupakan besaran tak berdimensi, karena regangan merupakan perbandingan antara dua ukuran panjang. Jika batang mengalami pemanjangan atau tarikan, maka regangannya disebut regangan tarik (tensile ( tensile strain) strain) dan regangannya dinyatakan berharga positif, sedangkan bila batang mengalami pemendekan atau tekanan, maka regangannya disebut regangan tekan (compressive ( compressive strain) strain) dan regangannya dinyatakan berharga negatif. negat if.
1.3. Diagram Tegangan – Regangan
Hubungan antara tegangan dan regangan dapat dinyatakan dengan diagram tegangan-regangan. Diagram ini sangat penting karena kita dapat mengetahui berbagai sifat bahan atau material dari diagram tersebut. Di bawah ini diberikan contoh diagram tegangan-regangan untuk baja karbon rendah ( low carbon steel ). ).
Gambar 1.4. Diagram tegangan-tegangan
O – A = daerah elastis A – C = daerah daerah plastis C – D = daerah perkuatan regangan regangan D – E = daerah daerah konstraksi luasan luasan
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Tegangan Tarik dan Tegangan Tekan
5
Keterangan :
Batas kesebandingan ( proportional limit ) merupakan batas atas daerah regangan yang tegangannya berbanding lurus dengan rega ngan.
Titik luluh ( yield point ) adalah suatu titik yang dicapai bila bahan dibebani, maka akan terjadi deformasi plastis/permanen, artinya benda tidak akan kembali ke bentuk semula bila beban dihilangkan.
Titik patah adalah titik tempat terjadinya patahan. Regangan patah dapat dipakai sebagai ukuran untuk sifat mampu bentuk suatu bahan. Suatu bahan dengan regangan patah yang besar akan lebih mudah dibentuk tanpa mengalami kerusakan.
Kekuatan tarik (ultimate (ultimate strength) strength) adalah tegangan maksimum yang dapat ditahan oleh material. mater ial. Bahan dengan kekuatan tarik yang lebih tinggi disebut lebih kuat dan
sebaliknya. Tegangan patah adalah tegangan sebenarnya (true ( true stress) stress) yang terjadi tepat pada aat benda akan patah. Bila kita perhatikan diagram tegangan regangan garis OA adalah garis lurus, ini menandakan bahwa pertambahan tegangan berbanding lurus dengan pertambahan regangan. Hal ini menggambarkan kelakuan elastis bahan yang diuji. Artinya apabila tegangan yang terjadi pada benda akan kembali ke bentuk semula pada saat beban dihilangkan. Setelah melewati titik A, tegangan dan regangan tidak lagi berbanding lurus. Oleh sebab itu tegangan dititik A disebut batas kesebandingan ( proportional limit ). ). Dengan benda diberi beban di atas batas kesebandingan, maka regangan r egangan akan bertambah lebih cepat daripada tegangan. Keadaan tersebut menandakan kalau bahan sudah mulai mengalami keluluhan/ kemuluran ( yielding ). ). Tegangan yang terjadi disebut tegangan luluh ( yielding stress) stress) atau titik luluh C, ditempat benda mengalami tegangan plastis, artinya benda akan mengalami perubahan bentuk permanen setelah batang melewati titik C. Pembebanan lebih lanjut akan menyebabkan tegangan naik hingga mencapai titik D yang merupakan batas maksimum kekuatan suatu bahan. Selanjutnya, akan disebut kekuatan tarik ( ultimate stress). stress).
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Tegangan Tarik dan Tegangan Tekan
6
besarnya beban sampai akhirnya material yang diuji patah pat ah yaitu pada titik t itik E (titik patah). Untuk bahan yang rapuh (getas), pada saat patah regangan atau pengurangan luas penampang yang terjadi kecil. Di bawah ini dapat diberikan contoh diagram tegangan regangan dari berbagai macam bahan.
Gambar 1.5. Perbandingan diagram tegangan-regangan untuk berbagai bahan.
Tegangan dan regangan di atas menggambarkan kelakukan dari berbagai beban yang mengalami pembebanan. Bila pada saat beban dihilangkan sedikit demi sedikit, tegangan dan regangan bergerak dari titik A ke titik O secara linear (pada kurva tegangan-regangan), maka sifat bahan seperti ini disebut elastis. Apabila suatu bahan yang mengalami pembebanan, dan beban diambil bahan tidak bisa kembali seperti semula, maka sifat bahan tersebut disebut sifat plastisitas (plasticity) dan bahannya disebut sudah plastis (sudah mengalami perubahan permanen). Hal ini ditunjukkan pada diagram tegangan regangan pada daerah A hingga D
1.4. Hukum Hooke
Apabila suatu bahan yang mengalami pembebanan menunjukkan suatu hubungan linier antara tegangan dan regangan, maka bahan dikatakan elastis, sedangkan perbandingan antara tegangan dan regangan pada daerah elastis linier
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Tegangan Tarik dan Tegangan Tekan
E =
7
2
dimana, E = modulus elastisitas (N/m )
= tegangan normal (N/m )
= regangan
2
Modulus elastisitas seringkali disebut modulus young (young’s modulus) yang dinamakan untuk menghargai ilmuwan kebangsaan Inggris bernama Thomas Young yang hidup pada tahun (1773-1829). Modulus elastisitas adalah kemiringan dari diagram tegangan-regangan dalam daerah linier, yang harganya tergantung pada bahan yang digunakan. Persamaannya yaitu : tan
= E =
, yang dikenal sebagai Hukum
Hooke. Hukum ini diberikan untuk menghargai ilmuwan berkebangsaan Inggris Robert Hooke yang hidup pada tahun (1635-1703). Makin besar sudut kemiringan atau modulus elastisitas (E) berarti bahan akan sulit untuk dibentuk, sebaliknya kalau sudut kemiringan makin kecil, maka bahan akan mudah untuk dibentuk. F Hukum Hooke : E = Karena, Maka, E =
F A
L L
dan
L
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Tegangan Tarik dan Tegangan Tekan
8
Perpanjangan ( ) sebanding dengan panjang bahan (L) Perpanjangan ( ) berbanding terbalik dengan luas penampang (A) Perpanjangan ( ) berbanding terbalik dengan modulus elastisitas (E)
Contoh soal
Sebuah batang silinder baja yang modulus elastisitasnya (E) 2,1 . 10 6 kg/cm2, mempunyai panjang 400 cm, sedangkan diameternya adalah 25 mm, diberi beban 4500 kg pada ujung bawahnya. Tentukan besarnya : a. Tegangan normal b. Regangan c. Pertambahan panjang
Penyelesaian
Luas penampang batang (A) = a. Tegangan () =
F
b. Regangan ( ) =
A
E
=
=
4
. d 2 =
4500 4,909
4
2 . 2,5 2 = 4,909 cm
= 916,68 kg/cm2
916,68 2,1 . 10 6
= 0,0004365
c. Pertambahan panjang ( ) = . L = 0,0004365 . 4000 = 0,174 cm
1.5. Faktor Keamanan (n)
Faktor keamanan adalah angka yang menjamin agar benda yang dipakai atau
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Tegangan Tarik dan Tegangan Tekan
Ketidaktelitian dalam struktur
Variasi dalam sifat-sifat bahan
Keburukan yang disebabkan kondisi atau efek lingkungan yang lain.
9
Apabila pengambilan faktor keamanan sangat rendah maka kemungkinan kegagalan akan tinggi. Karena itu rancangan str ukturnya mungkin mungkin tidak diterima. diter ima. Sebaliknya, bila faktor keamanan sangat besar, maka pemakaian bahan akan boros dan struktur menjadi berat sehingga tidak cocok dari segi fungsi. Metode perencanaan yang lazim laz im adalah menggunakan faktor keamanan terhadap tegangan luluh maupun tegangan izin (allowable stress) atau tegangan kerja (working stress), yang tidak boleh dilampaui disetiap bagian dalam struktur. Tegangan yang diizinkan pada suatu benda. a. Tegangan izin oleh pembebanan tetap -
Pembebanan tetap dalam keadaan diam (stat is)
-
Pembebanan tetap dalam keadaan bergerak (dinamis), benda dinamis > beban statis, karena pembebanan dinamis selain menerima beban gaya luar, juga mengalami kelelahan akibat beban yang berubah-ubah. berubah-ubah.
b. Tegangan izin oleh pembebanan tidak tetap Pembebanan tidak tetap yang dimaksud adalah bebannya bergerak tetapi bendanya
sewaktu-waktu
mengalami
penambahan
beban
maupun
pengurangan beban. Oleh Oleh karena itu, n-nya kebih besar dari beban dinamis. c. Tegangan izin oleh pembebanan impak (kejut) Karena bebannya mengalami impak, maka n-nya lebih besar dari pada benda pembebanan tetap maupun tidak tetap. Hal ini disebabkan bendanya tidak
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Tegangan Tarik dan Tegangan Tekan
10
u n
Faktor keamanan terhadap tegangan batas (ultimate (ultimate stress) stress) harus lebih besar daripada tegangan luluh ( yield stress). stress). Hal ini disebabkan karena tegangan batas > tegangan luluh untuk semua bahan. Contoh soal
Sebuah kolom pendek berongga terbuat dari besi tulang, mempunyai garis tengah bagian luar 22 cm dan mendapat beban sebesar 45 ton. Kekuatan tekan dari dar i bahan adalah 7200 kgf/cm 2. Bila faktor keamanan adalah 4, hitunglah tegangan tekan yang diizinkan dan berapakah tebal dinding minimum yang diperlukan agar tegangan yang diizinkan jangan sa mpai dilampaui. p Penyelesaian
Kekuatan tekan :
u = 7200 kg/cm2
Tegangan tekan yang diizinkan ( ) t D
Kekuatan tekan faktor keamanan 7200 4
=
u n
= 1800 kg/cm2
Gambar 1.7. Contoh soal 1 Misalkan garis tengah lingkaran dalam adalah d cm, maka luas penampang kolom pendek adalah :
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Tegangan Tarik dan Tegangan Tekan
F A
≤ 1800
A≥
F 1800
≥
45000 1800
11
≥ 25 cm2
Dari persamaan 1 dan 2 didapat : 4
( 22 2
d 2 ) ≥ 25
2
d ≥ (222 – (222 – 1,27 . 25) d2 ≥ 452,17 d≥
452,17 ≥ 21,26 cm
Jadi tebal dinding minimum = t =
D d 2
=
22
21,26 2
= 0,368 cm
t = 3,68 mm Contoh soal
Suatu kabel baja menanggung beban 400 kg pada salah satu ujungnya. Jika 2
tegangan luluh kabel adalah 960 kg/cm dan faktor keamanannya 1,2. Berapakah diameter minimum yang diperlukan. Penyelesaian :
Tegangan izin kabel, d
=
y n 960 1,2
Luas penampang penampang kabel : A =
2
= 800 kg/cm 4
d 2
F Gambar 1.8. Contoh soal 2 Agar kabel aman, artinya tidak mengalami perubahan penampang bila beban
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Tegangan Tarik dan Tegangan Tekan
12
Apabila sebuah batang ditarik (seperti pada gambar), maka pemanjangan aksial akan diikuti dengan kontraksi lateral dalam arah tegak lurus arah kerja gaya. Pada gambar,
garis putus-putus
menyatakan pembebanan F
bentuk dan
menyatakan
sebelum garis
bentuk
tebal setelah
pembebanan. Gambar 1.9. Batang ditarik Kontraksi
lateral
dapat
dilihat
dengan
mudah
pada
karet
yang
direnggangkan, tetapi pada logam perubahan dimensi lateral biasanya sangat kecil untuk dapat dilihat, walaupun dapat diamati dengan peukur. Perbandingan regangan lateral (melintang) dengan regangan aksial (memanjang) dikenal sebagai angka nisbah (perbandingan) Poisson’s (poisson’s ratio), dan dinyatakan dinyatakan dengan huruf Yunani ν (nu)
regangan lateral
=
=
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Tegangan Tarik dan Tegangan Tekan
13
Pada gambar diperlihatkan sebuah eleman
kecil
dari
bahan
yang
dipotong pada sebuah batang yang mengalami beban tarik. Bentuk awal dari
elemen
jajaran
diperlihatkan
genjang
siku-siku
oleh seperti
diperlihatkan garis putus-putus pada gambar, sedangkan bentuk akhir
Gambar 1.10. Perubahan bentuk dari sebuah elemen dalam keadaan tarik
elemen diperlihatkan dengan garis tebal.
Pemanjangan elemen dalam arah pembebanan adalah a, regangan aksial. Karena regangan lateral berkurang dengan b1 . ν .
; c1 . ν .
α
= ν .
dengan adalah
, dimensi-dimensi lateral
berturut-turut dalam arah y dan z. Jadi
dimensi terakhir dari elemen adalah a1 (1 + ) ; b1 (1 - ν . ) ; dan c1 (1 - ν - ν . ) dan volume akhirnya adalah : Vf = a1 b1 c1 (1 + ) (1 - ν . ) (1 - ν - ν . )
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Tegangan Tarik dan Tegangan Tekan
14
Contoh soal
Sebuah batang baja yang panjangnya 2,5 m berpenampang bujur sangkar yang panjang rusuknya = 100 mm dikenakan sebuah gaya aksial tarik F = 1300 kN. Dengan mengasumsikan bahwa modulus elastisitasnya = 200 G Pa dan angka poisson’snya adalah 0,3. Tentukan : a) Pemanjangan batang b) Pengurangan penampang c) Perubahan volume Gambar 1.11. Contoh soal 4 Penyelesaian 9
9
2
2
1 G Pa = 1 . 10 Pa = 1 . 10 N/m = 1 kN/mm a) Pemanjangan batang :
F . L A . E
=
1300 . 2,5 . 10 3 100 . 100 . 200
Tegangan tarik yang terjadi F
1300
= 1,625 mm
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Tegangan Tarik dan Tegangan Tekan
15
Vf = V + V = (2,5 . 10 7 + 6500) mm3 -6
3
= (0,025 + 6,5 . 10 ) m = 0,0250065 m
3
1.8. Tegangan Batang Akibat Beban Aksial
Pada umumnya, untuk menentukan besarnya tegangan yang diakibatkan beban aksial aks ial berat batang sering diabaikan, tetapi ada kalanya berat batang harus har us diperhitungkan seperti pada perencanaan alat.
1.8.1. Batang penampang tetap
Batang penampang tetap adalah suatu batang yang berpenampang kontinyu untuk sepanjang batang. Contoh Soal
Sebuah
batang
panjangnya diameternya
baja
yang
m
dan
0,9 20
mm,
mengalami beban tarik 20 kN. Gambar 1.12 Contoh soal soal 4
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Tegangan Tarik dan Tegangan Tekan
16
Gambar 1.13. 1.13. Contoh soal 5 Penyelesaian Penyelesaian :
F diuraikan menjadi Fx sejajar sumbu dan Fy tegak lurus sumbu. Fx = F cos
= 6000 cos 30
F Fx A
A
=
6000 cos 30 = 16,6 N/mm2 . 20 2 4
1.8.2. Batang penampang berlubang
Batang berlubang yaitu batang pada sepanjang sumbu batang terdapat satu atau beberapa lubang. Analisa tegangannya dimaksudkan untuk mengetahui bagian yang kritis pada batang tersebut, semakin besar lubang maka luas
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Tegangan Tarik dan Tegangan Tekan
A1 = =
= luas penampang dalam
( D 2 t ) 2 =
4
( D 2 . D ) 2 8 4
2 (3 . D ) 2 = (9 . D ) 4 16 4 4
Gambar 1.14. 1.14. Contoh soal soal 6 Luas penampang batang = A – A 1 =
D 2 4
-
D 2 ) ( 9 . 4 16
17
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Tegangan Tarik dan Tegangan Tekan
18
Untuk menentukan diameter disembarang titik sepanjang batang (d x), yaitu dengan teori geometri. D d 2 d x
d
L x
2
d x d D d 2 L 2 x (dx – d) L = (D – d) x dx – d =
( D d ) x L
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Tegangan Tarik dan Tegangan Tekan
Gambar 1.16. 1.16. Contoh soal 7
19
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Tegangan Tarik dan Tegangan Tekan
20
Contoh soal
Sebuah batang kuningan ( = 540 kN/m3) dengan panjang 1,2 m berpenampang bujur sangkar yang sebagian sisinya berlubang (lihat gambar). Tentukan besarnya tegangan maksimum yang terjadi akibat berat benda itu sendiri.
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21
1.10. Tegangan akibat perubahan temperatur temperatur
Perubahan temperatur dari suatu benda akan menghasilkan perubahan dimensi. Misalnya sebuah balok seperti pada gambar dari suatu bahan yang homogen dapat memuai secara bebas ke semua arah. Jika bahan dipanaskan, maka rusuk-rusuk balok akan memanjang. Pada gambar garis tebal menunjukkan benda sebelum se belum mengalami pemanasan, pe manasan, sedangkan
garis
putus-putus
menunjukkan benda setelah mengalami pemanasan.
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Tegangan Tarik dan Tegangan Tekan
E L
L
. E t . E
= (t1 – t2) E Dengan, = tegangan normal yang terjadi (N/mm (N/ mm2) 2
E = modulus elastisitas (N/mm )
22
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Tegangan Tarik dan Tegangan Tekan
23
pengaruh temperatur dan pengaruh gaya. Karena kedua ujung dijepit, salah satu batang akan mengalami pemendekan yang sama besarnya dengan pemanjangan batang yang lainnya. a. Pemanjangan : Karena pengaruh temperatur
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Tegangan Tarik dan Tegangan Tekan
2.
24
Suatu batang tergantung dengan panjang L = 5 [m], diameter d = 5 [mm], berat jenis bahan = 10 . 9,81 [N/dm3], lihat gambar.
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Tegangan Tarik dan Tegangan Tekan
25
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Tegangan Tarik dan Tegangan Tekan
8.
26
Dua buah plat t = 20 [mm], disambung dengan las kampuh homogen dan tegangannya lebih kecil dari tegangan plat. Lebar dan tebal kampuh 80 [mm], 20 [mm], serta sudut kampu terhadap vertikal 15° ( Kampu V 30°). a). Hitung tegangan geser pada kampuh las bila F = 7500 [N]. Alas kampuh 5 [mm]
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Tegangan Tarik dan Tegangan Tekan
27