Tegangan Tarik, Tekan Dan Geser

BAHAN AJAR

MEKANIKA KEKUATAN MATERIAL

Dipersiapkan oleh Aswata, Ir, Drs, SE, MM, IPM

AKULT AKULTAS TEKNIK TEKN IK ! JURUSAN TEKNIK T EKNIK MESIN UNI"ERSITAS SULTAN A#EN# TIRTA$ASA %&'%

SATUAN A(ARA PERKULIAHAN )SAP*

Min++ ' % 0 2 3 5 6i7 ' 8 9 UTS : '& '' '% '0 '2 6i7 % '3 '5 UAS

& ' % 0 2

Na-a Ba. Pen/ahlan Te+an+an Tarik, Tekan, #eser Tin1aan Ele-en aki.at Te+an+an Aksial )Nor-al* Torsi )Pntiran* #a4a #eser /an Mo-en Lentr

3 Te+an+an /i Balok

5 8 9 :

Te+an+an /i Balok )Lan1tan* Analisa Te+an+an /an Re+an+an Penerapan Te+an+an Bi/an+ De;leksi Balok

'& Balok Statis Tak Tent '' Kolo-

Mata Kliah Mekanika Kekatan Material .er.o.ot 0 SKS )Satan Kre/it Se-ester* artin4a wakt kliah )tatap -ka* a/alah 0 1a- se-in++, /ita-.ah 0 1a- .ela1ar -an/iri< Jika -ahasiswa ti/ak terti. /ala- -en+atr wakt .ela1arn4a, -aka hasiln4a ti/ak akan -e-as kan< Ha-pir se-a 1ian /ilakkan /en+an =ara >pen Book, tetapi 4an+ .ar -e-.ka .kn4a saat 1ian, ha-pir pasti skar ntk lls< Perl .an4ak latihan /i lar wakt kliah< Mahasiswa .e.as -e-ilih kelas A ata B, ti/ak ter+antn+ /en+an NIM, .ahkan .oleh ikt /ala % kelas, A /an B< Be+it pla /ala- 6i7, ti/ak .isa /ilaran+ ntk ikt % kali< $an+ $an+ /inilai a/ala 4an+ ter.esar /en+an =atatan ti/ak a/a ke=ran+an< Ke11ran /ala- -en+er1akan soal san+at /i1n1n+ tin++i, sehin++a tin/akan .eker1a sa-a /ala -en+er1akan soal /apat -en4e.a.kan -ahasiswa -en/apat nilai E )Ti/ak Lls*, .aik .a+i 4an+ -en4ontek ata 4an+ /i=ontek, -eskipn 4an+ /i=ontek ti/ak -en4a/arin4a< Latihan soal /i r-ah )PR, Ho-e ?ork*, ?ork*, 1+a hars /iker1akan sen/iri, .ersi;at se.a+ai .kti keha/iran< Ha/ir tetapi ti/ak -en+er1akan soal /ian++ap a.sen, se/an+an a.sen tetapi -en4erah! kan hasil peker1aan r-ah tepat wakt /an /iker1akan sen/iri /ian++ap ha/ir< Da;tar Kepstakaan ' Mekanika Mekanika Bahan, Bahan, #ere #ere @ Ti-oshen Ti-oshenko, ko, Pener.it Pener.it Erlan++a, Erlan++a, Jili/ Jili/ ' @ % )Bk )Bk wa1i.* wa1i.*

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Min++ ' % 0 2 3 5 6i7 ' 8 9 UTS : '& '' '% '0 '2 6i7 % '3 '5 UAS

& ' % 0 2

Na-a Ba. Pen/ahlan Te+an+an Tarik, Tekan, #eser Tin1aan Ele-en aki.at Te+an+an Aksial )Nor-al* Torsi )Pntiran* #a4a #eser /an Mo-en Lentr

3 Te+an+an /i Balok

5 8 9 :

Te+an+an /i Balok )Lan1tan* Analisa Te+an+an /an Re+an+an Penerapan Te+an+an Bi/an+ De;leksi Balok

'& Balok Statis Tak Tent '' Kolo-

Mata Kliah Mekanika Kekatan Material .er.o.ot 0 SKS )Satan Kre/it Se-ester* artin4a wakt kliah )tatap -ka* a/alah 0 1a- se-in++, /ita-.ah 0 1a- .ela1ar -an/iri< Jika -ahasiswa ti/ak terti. /ala- -en+atr wakt .ela1arn4a, -aka hasiln4a ti/ak akan -e-as kan< Ha-pir se-a 1ian /ilakkan /en+an =ara >pen Book, tetapi 4an+ .ar -e-.ka .kn4a saat 1ian, ha-pir pasti skar ntk lls< Perl .an4ak latihan /i lar wakt kliah< Mahasiswa .e.as -e-ilih kelas A ata B, ti/ak ter+antn+ /en+an NIM, .ahkan .oleh ikt /ala % kelas, A /an B< Be+it pla /ala- 6i7, ti/ak .isa /ilaran+ ntk ikt % kali< $an+ $an+ /inilai a/ala 4an+ ter.esar /en+an =atatan ti/ak a/a ke=ran+an< Ke11ran /ala- -en+er1akan soal san+at /i1n1n+ tin++i, sehin++a tin/akan .eker1a sa-a /ala -en+er1akan soal /apat -en4e.a.kan -ahasiswa -en/apat nilai E )Ti/ak Lls*, .aik .a+i 4an+ -en4ontek ata 4an+ /i=ontek, -eskipn 4an+ /i=ontek ti/ak -en4a/arin4a< Latihan soal /i r-ah )PR, Ho-e ?ork*, ?ork*, 1+a hars /iker1akan sen/iri, .ersi;at se.a+ai .kti keha/iran< Ha/ir tetapi ti/ak -en+er1akan soal /ian++ap a.sen, se/an+an a.sen tetapi -en4erah! kan hasil peker1aan r-ah tepat wakt /an /iker1akan sen/iri /ian++ap ha/ir< Da;tar Kepstakaan ' Mekanika Mekanika Bahan, Bahan, #ere #ere @ Ti-oshen Ti-oshenko, ko, Pener.it Pener.it Erlan++a, Erlan++a, Jili/ Jili/ ' @ % )Bk )Bk wa1i.* wa1i.*

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% Stren+t Stren+thh o; Materi Material, al, LS Ne+i Ne+i,, Si+-a Si+-a Series, Series, Tata Tata M=!#ra M=!#raw w Hill, Hill, %&&9 %&&9

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TARIK, TEKAN 1. TE#AN#AN TARIK,

/an #ESER

'<' Pen+antar Mekanika Bahan< Dala- -ata kliah Mekanika Strktr telah /i.ahas +a4a!+a4a 4an+ .eker1a pa/a .en/a< Dari +a4a!+a4a terse.t ke-/ian /iperhitn+kan reaksi t-pann4a, sehin++a ke-/ian .i /iperiksa kon/isi kesei-.an+ann4a< Pa/a -ata kliah Mekanika Kekatan Material, reaksi reak si t-pan lar ini ti/ak perl /i=ari, karena 4an+ akan le.ih /iperhatikan a/alah kea/aan /i /ala- strktr .en/a it sen/iri< Kea/aan /i /ala- strktr .en/a ini 1ika ke-/ian /ih.n+kan /en+an kekatan -aterialn sen/iri )Il- Lo+a-*, -aka akan /iperoleh kon/isi a-an ata ti/ak a-an< Untk .isa -elakkan analisa /i /ala- sat .en/a, .iasan4a /ian++ap .ahwa .en/an4a ho-o+en, artin4a kepa/atann4a -erata, ko-posisin4a 1+a sera+a-< '<% Te+an+an Te+an+an /an Re+an+an Nor-al< $an+ /i-aks/ /en+an istilah Nor-al a/alah arah te+an+an /an re+an+an ts. a/alah searah s earah /en+an +aris s-. pan1an+ .en/a< Maka 4an+ ter1a/i a/alah te+an+an tarik ata tekan< A P P Pan/an+ ton+kat /i sa-pin+ /en+an las pena-pan+ A< L Ton+kat /itarik /en+an #a4a Nor-al P Maka /ir-skan, P  C )tarik* )'!'* A P P te+an+an nor-al A Lδ Satan te+an+an a/alah +a4a per satan las Serin+ ntk pen4e/erhanaan, tan/a  ntk te+an+an tarik, tan/a ! ntk te+an+an tekan< P Be/akan antara te+an+an /en+an tekanan Tekanan a/alah +a4a /ari sat .en/a 4an+ .era/ .er a/ /en+an .en/a lain Jika ton+kat /itekan ke lantai /en+an +a4a P, P p C )tekan* A Lantai -aka tekanan 4an+ .eker1a pa/a lantai A Re+an+an Nor-al< Pa/a +a-.ar /i atas terlihat .ahwa karena +a4a P -aka pan1an+ ton+kat -en1a/i L  δ δ Maka /i/e;inisikan .esarn4a re+an+an nor-al  ε C )tarik* )'!%* L Karena satan F /an satan L a/alah sa-a, -aka G ti/ak -e-iliki satan< Sa-a seperti haln4a /en+an te+an+an, G .isa /itlis  )-e-an1an+* ata ! )-e-en/ek*< Jika +a4a aksial )nor-al* ts. -en+hasilkan te+an+an nor-al tarik ata tekan 4an+ ter.a+i rat -aka +a4a nor-al ts. akan .eker1a -elali titik psat .erat, ata 1ika -aterialn4a ho-o+en -aka akan .eker1a pa/a titik .erat area )las* pena-pan+<



(ontoh, se.atan+ ton+kat .a1a pan1an+ L 2& - /an /ia-eter / 9 -- /i.e.ani ? ',3 L (ari r-s te+an+an te+an+an 4an+ 4an+ ter1a/i ter1a/i sepan1an+ sepan1an+ L, /an =ari =ari te+an+an te+an+an -aksi--n4a< -aksi--n4a< Misa Mi sallkan kan te+ te+an an+a +ann 4an 4an++ ter ter11a/i a/i .er .era/ a/aa /i /i pos posiisi  /ar /arii pan pan+k +kal al ton+k on+kat at<< Maka pena-pan+ A akan -en+ala-i +a4a tarik se.esar se.es ar ?  )L!*2 /% 

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TARIK, TEKAN 1. TE#AN#AN TARIK,

/an #ESER

?

/i -ana  a/alah -asa 1enis .a1a 89&& k+-0< Maka te+an+an -aksi-- ter1a/i pa/a  C & se.esar )?  L2 / %*  2 / %< '<0 Besaran Mekanis Bahan< Dia+ra- tarik .a1a lnak< Titikk li-it proporsional propors ional  p , >A +aris lrs Hk- Hooke  A Titi D $iel/ stren+th, te+an+an llh, 4 B $iel/ E B ( Bata Batass plas plasti tisi sita tass se-p se-prn rnaa ( A D Ulti-ate stren+th,  E raktr, p pts G' /aerah linier, hk- Hooke  C G E & G G% /aerah plastisitas se-prna G' G% G0 G2 G0 /aerah strain har/enin+ )perkerasan karena per.ahan .ent G2 /aerah Ne=kin+ )pe-.entkan leher* '<2 Daerah ker1a peren=anaan ele-en!ele-en -esin< Daerah ker1a peren=anaan -esin nor-aln4a ter1a/i /i /aerah >A, karena .ersi;at elastis se-prna< Jika .e.ann4a /ihilan+kan, -aka pan1an+ .en/a akan ke-.ali ke pan1an+ se-la< Jika /aerah ker1a lewat /ari A, -aka 1ika .e.ann4a /ihilan+kan, pe-lran plastis 4an+ s/a ter1a/i ti/ak akan hilan+< Jika ter1a/i .erlan+!lan+, -aka kran .en/a akan .er.ah< Setiap .ahan )-aterial* -e-iliki karakteristik /ia+ra- tarik 4an+ .er.e/a!.e/a )khas*< '<3 Hk- Hooke, Poisson Ratio< Hk- Hooke  C G E , /i -ana  a/alah te+an+an )N-%* )'!0* G a/alah re+an+an P P E a/alah -o/ls Elastisitas , Mo/ls $on+ $on+ )N-%* P

P Jika se.ah .alok /itarik /en+an +a4a P, -aka akan ter1a/i pe-lran pan1an+, /ala- saat 4an+ sa-a ter1a/i pen+e=ilan pena-pan+< Hal ini ter1a/i karena hk- kekekalan ol-e pa/a as-si ti/ak a/a per.aha kepa/atan .ahan< Jika -o/ls elastisitas pa/a se-a arah sa-a, -aka /ise.t isotropik< Pen+e=ilan pena-pan+ karena pe-lran aial strain G akan -en+hasilkan lateral strain G < Maka /i/e;inisikan lateral strain G Poisson Ratio  C C )'!2* aial strain G Karena 1ika aial strain -e-an1an+ )* akan -en+hasilkan lateral strain -en+kert )!* , -ak .iasa 1+a /itlis  G  C !  G Jika se.ah k.s /en+an sisi a, ., = /itarik sisi a!n4a, -aka sisi a akan -lr /en+an Ga< Sisi . /an = akan -en+ala-i re+an+an ! G sehin++a sisi . akan -e-en/ek /en+an ! .G< I/e-, sisi = -e-en/ek /en+an ! =G<

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1. TE#AN#AN TARIK, TEKAN

/an #ESER

Maka ol-e .ar 4an+ ter1a/i, "' C a.= )'  G*)'!G*)'!G* C "o )'  G*)'!G*)'!G* Den+an -en+a.aikan hasil perkalian G%, -aka /iperoleh  "' C "o )'  G ! % G* sehin++a per.ahan ol-en4a, O" C "' ! "o C "o G )' ! %* Per.ahan "ol-e Satan e 

eC

O" C "o

G )' ! %*

C

σ

E

)' ! %*

535

)'!3* )'!5*

Besaran e ini 1+a /ikenal /en+an istilah /elatasi (ontoh  Se.ah pipa .a1a /en+an pan1an+ L 2 ;t, /ia-eter lar 5 in, /ia-eter /alaP 2,3 in, /i.eri .e.an tekan P '2& kp< E C 0&<&&& ksi /an Poisson ratio  C &,0 /' Hitn+  a* Perpen/ekan F L .* Re+an+an lateral G  /% =* Perta-.ahan /ia-ater lar O/% /an /ia-eter /ala- O/' /* Per.ahan )perta-.ahan* te.al /in/in+ Ot e* Per.ahan )perta-.ahan* ol-e ;* Delatasi Pen4elesaian  Las pena-pan+ pipa , A C 2 )/%% ! /'% * C '%,052 in% Te+an+an nor-al, σ C PA C ! '',0%0 ksi )tekan* Te+an+an llh .ahan )lihat ta.el* , .a1a strktral, σ4 C 05 ksi Karena te+an+an 4an+ ter1a/i -asih /i .atas proporsional, -aka .erlak hk- Hooke<  C G E , ata G C E C ! 088,2  '&!5 )ne+atip karena -e-en/ek* a* Perpen/ekan F C G L C ! &,&'9 in< .* Re+an+an lateral, G C ! G  C ''0,%  '& !5 )positip, artin4a -e-.esar* /% C 5<&&'E&& in =* Perta-.ahan /ia-eter lar O/% C G /% C &,&&&58: in Perta-.ahan /ia-eter /ala- O/' C G /' C &,&&&3&:5 in /' C 2<3&&E&& in /* Perta-.ahan te.al /in/in+ Ot C G )/% ! /'*% C &,&&&&92:0 in C %<:8%E!&2 in e* Per.ahan ol-e .isa /ihitn+ /en+an =ara -en+hitn+ ol-e .ar ! ol-e la-a< "ol-e la-a C 2 )/%% ! /'%* L C 3:0,25& in0 "ol-e .ar C 3:0<3'2 in0 Maka, per.ahan )perta-.ahan* ol-e O" C &<&32 in0 C &<&&: Jika -en++nakan persa-aan )'!3* -aka /iperoleh har+a pen/ekatan O" Per.ahan ol-en4a, O" C "' ! "o C "o G )' ! %* C &<&:& in0 )(atatan  A/an4a per.e/aan hasil perhitn+an /an 4an+ tertlis /i .k -e-.erikan pe-aha-an, .eker1alah /en+an r-s 4an+ /i4akini .enar, /en+an =ara 4an+ .enar< Lakkan lan+kah pe-eriksaan, se.a. .k 1+a .isa -en4a1ikan kesalahan =etak<* ;* Delatasi e C G )' ! %* C '<3'&E!&2 ata C &<&'3  )har+a pen/ekatan* '<5 Te+an+an /an Re+an+an #eser< Jika arah +a4a, te+an+an /an re+an+an nor-al a/alah se1a1ar ata searah /en+an +aris s-.


/an #ESER

-aka +a4a, te+an+an /an re+an+an +eser arahn4a te+ak lrs terha/ap arah +aris s-.< Jika kersakan aki.at re+an+an nor-al a/alah .en/a 4an+ -e-an1an+ ata -e-en/ek, -ak kersakan aki.at re+an+an +eser a/alah .en/an4a pts, patah per+eseran< Kesa-aan 4an+ ter1a/i pa/a .i/an+ +eser< Pan/an+ se.ah pen 4an+ -en+ala-i pe-.e.anan +a4a lintan+ )+eser* " " Maka /i/e;inisikan, " A te+an+an +eser Q C A A C las pena-pan+ pe Maka satan te+an+an +eser C satan +a4a  satan las a Pan/an+ .i/an+!.i/an+ +eser pa/a ske-a kotak a.= /i sa-pin+ = /en+an arah s-.!s-. ,4,7 !n4a< Q p . Karena a/a kesa-aan kesei-.an+an,  +a4a /an -o-en C &  Q -aka arah Q pa/a tiap sisi 4an+ .erha/apan akan salin+ .erlawa an arah, /an pa/a tiap sisi 4an+ .erse.elahan -aka arah Q akan Q 4 salin+ .erte- ata salin+ -en1ahi sisi perte-an .i/an+< s  r Aki.at /ari arah!arah te+an+an ts. -aka kotak a.= akan .er.ah 7 Se+i!2 prs akan .er.ah -en1a/i 1a1aran +en1an+, sisi sp /an sr Q Q akan -irin+ sehin++a -e-.entk s/t se.esar % /en+an +aris p  s-. te+ak ata -en/atar, ata s/t psr akan -en1a/i % !  % /an s/t sr akan .er.ah -en1a/i %   Re+an+an +eser< Q s r S/t  4an+ ter1a/i /i/e;inisikan se.a+ai re+an+an +eser< % Karena .erpa s/t, -aka satann4a a/alah satan ra/ian< Konensi )per1an1ian* tan/a ntk te+an+an /an re+an+an +eser< Te+an+an +eser 4an+ .eker1a pa/a sisi positip sat ele-en )kotak* akan .ertan/a positip 1ika arahn4a searah /en+an arah positip s-. koor/inat< Sisi positip a/alah 1ika sisi ts. -en arah sesai /en+an arah positip sat +aris s-.< Maka akan a/a 0 sisi positip /an 0 sisi ne tip /ari se.ah kotak< Jika sisi it -en+ha/ap ke arah s-.  positip, -aka sisi ts. /ise.t sisi positip< Te+an+an +eser 4an+ ter1a/i pa/a .i/an+ ts. akan -en+ikti arah s-. 4 /an 7< Tan/a te+an+an +eser akan ter+antn+ pa/a arah positip ata ne+atip s-. koor/inatn4a< Maka /ala- ilstrasi +a-.ar /i atas, te+an+an +eser 4an+ ter1a/i se-an4a .ertan/a positip< Te+an+an +eser 4an+ .eker1a pa/a sisi ne+atip akan .ertan/a positip 1ika .erlawanan arah /en+an arah positip s-. koor/inat< A/apn tan/a ntk re+an+an +eser a/alah positip 1ika s/t antara /a -ka positip )ata % -ka ne+atip* .erkran+< Se.alikn4a akan .ertan/a ne+atip 1ika .erlak ke.alikann4a< Maka pa/a +a-.ar ilstrasi /i atas, se-a tan/a re+an+an a/alah .ertan/a positip< Hk- Hooke ntk kea/aan +eser< Seperti pa/a kea/aan tarik, hk- Hooke +eser a/alah  QC# )'!8* /i -ana # a/alah -o/ls elastisitas +eser, satann4a sa-a /en+an satan E

835


H.n+an anatar # /en+an E a/alah 

# C

/an #ESER

E % )'  *

Te+an+an i7in /an .e.an i7in< Telah /isin++n+ /i -ka .ahwa --n4a, te+an+an ker1a terletak /i +aris te+an+an propor onal a+ar ti/ak ter1a/i pe-lran 4+ per-anen< Tetapi, .esaran σ p ini ti/ak .iasa /isa-paika $an+ .iasa /isa-paikan a/alah te+an+an 4iel/ ata σ4< Maka, a+ar ter=ipta sat kon/isi 4an a-an, perl a/a ;aktor kea-anan 4an+ .esarn4a le.ih /ari '< Besarn4a an+ka ;aktor kea-an! an ini ter+antn+ .an4ak hal, -isaln4a 1enis, -a=a- pe-.e.anan, 1+a peratran sete-pat< σ4 Di/e;inisikan, ;aktor kea-anan )Sa;et4 a=tor* S C σ4 C σ ata  i7in S σi7in Mar+in kea-anan )Sa;et4 -ar+in* /ir-skan se.a+ai S ! ' Se/an+kan .e.an i7in a/alah hasil kali antara te+an+an i7in /en+an las pena-pan+< Tent sa1a, .e.an i7in ts. a/alah 4an+ le.ih a-an terha/ap te+an+an nor-al ata +eser, sehin++a hars a-an ntk ke/an4a< (ontoh '!8< Diketahi +antn+an .a1a seperti ilstrasi /i sa-pin+< Le.ar .' C ',3 in /an .% C 0 in, te.al t C &,3 in< / Hitn+ +a4a .e.an P 1ika  a* Te+an+an i7in pen++antn+ /i .' C '5<&&& psi t .' .%

P

.* Te+an+an i7in pen++antn+ /i .i/an+ irisan pen ''<&&& =* Te+an+an i7in pen++antn+ /i .% C %5<&&& psi /* Te+an+an +eser i7in /i pen C 53&& psi

'<9 Peren=anaan ntk .e.an nor-al /an +eser< A (ontoh soal '!9< Diketahi papan penn1k arah ? 3,2 kN /i+antn+ pa/a .atan+ AB(< Jarak AB % -, (D &,9 -, DE ',9 -, EB &,2 -< ? /i ten+a DE< Ren=anakan /ia-eter pen ( /an las pena-pan+ .atan+ AB 1ika te+an+an i7in tarik '%3 Mpa /an te+an+an i7in +eser 23 Mpa (4 B Batan+ B( /iapit oleh kpin+an (< A.aikan .erat .atan+< ( Mla!-la hars /i+a-.ar DBB!n4a /l< MB C & ( D E  B (B (4 )B(* C ?)BE  DE%* (4 C ?)BE  DE%*B( C %<02& kN ? 3,2 kN M( C & B )B( sin * C ?)(D  DE%* tan  C A(B(  C ATAN A(B( C &<399 ra/ B C ?)(D  DE%*B( sin  C 3<3'5 kN Diperoleh B4 C B sin  C 0<&5& kN 4 C (4  B4 ! ? C & kN ).enar* ( C B C B =os  C 2<3:& kN ( C (  (4 C 3<'3% kN

935


/an #ESER

/ % C %(Q a* Di-eter pen (  Q C  (  %)2 / % * / C 9<32& -A C B   C 22<'0% --% .* Las pena-pan+ .atan+ AB  C B  A (atatan  Karena .e.an ? .eker1a /i ten+ah DE, -aka lan+sn+ sa1a .e.an ts. /ipin/ahkan ke ten+ah DE< Ja/i ti/ak la+i /iperlkan -e-.a+i ? -en1a/i D /an E< Dala- perhitn+an /i atas, satan hars /iperhatikan, /isetin+katkan /l< Latihan soal!soal  '<9!3 2 ;t 2 ;t A ( 0 ;t

B

Batan+ AB /i1epit oleh % .atan+ D( /en+an .at (< Jika te+an+an i7in +eser .at a/alah '0<3&& psi, .erapa kran /ia-eter .at / -ini-- V P C %5&& l.<

P

D '<9!'2 A

Batan+ A( /an B( /ih.n+kan en+sel /i (, /i.eri .e.an P< Bahan .atan+ A( /anB( a/alah sa-a, pan1an+ B( a/alah L )tetap*, se/an+kan pan1an+ A( .er.ah!.ah /en+an =ara -e! naik!trnkan A sehin++a s/t  1+a .er.ah<  ( Jika s/t  .erkran+, -aka pan1an+ A( .erkran+, tetapi las B L P pena-pan+ .atan+ .erta-.ah a+ar .atan+ tetap kat )a-an* karena +a4a aksialn4a .erta-.ah< Se.alikn4a 1ika  .erta-.ah, -aka pan1an+ A( 1+a .er! ta-.ah, tetapi las pena-pan+n4a .isa .erkran+< (ari r-s h.n+an antara s/t  /en+an .erat .atan+ , /an =ari  a+ar .erat .atan+ -ini--<

'<8!''

# 8 ;t

Ran+ka .atan+ /ih.n+kan sen/i< Batan+ AB -e-iliki las pena-pan+ 0,'0 in% te+an+an i7inn4a a/alah '%<&&& psi< Han4a /en+an -enin1a te+an+an /i .atan+ AB, .erapa .e.an i7in P V

A '2 ;t

D

B E 0K

5 ;t '<9 ! '0

 0K 5 ;t

L% A

P ( 0K 5 ;t L%

W

W (

Da .atan+ A( /an B( ter.at /ari .ahan 4an+ sa-a, hars -e-ikl .e.an P /i (< Jarak AB C L ti/ak /apat B /ir.ah, tetapi pan1an+ .atan+ /apat .er.ah /en+an =ar -er.ah s/t W< Jika W -en+e=il, -aka .atan+ akan -e-en/ek, tetapi

:35


/an #ESER

P las pena-pan+n4a akan -e-.esar karena hars -enah +a4a aksial 4an+ le.ih .esar, sehin++a .erat .atan+pn ikt terpen+arh< Tentkan h.n+an W /en+an .e.an P, /an tentkan W 4an+ /apat -e-.erikan .erat -ini- Be.an -ai-- .atan+ hars ti/ak .oleh -ele.ihi .e.an i7in< Berat .atan+ /ia.aikan<

'&35

ELEMEN STRUKTUR DIBEBANI AKSIAL

%<' Pen+antar Ele-en -esin 4an+ han4a -en+ala-i tarik ata tekan /ise.t se.a+ai ele-en 4an+ /i.e.ani se=ara aksial< (ontoh, silin/er hi/rolik, .atan+ trsses, pe+as spiral, /ll< Pe+as spiral se=ara th -e-an+ han4a -eneri-a .e.an aksial, tetapi .ahan pen4snn4a 4an+ .erpa kawat akan -eneri-a .e.an +eser 1+a< E;ek seperti inilah 4an+ akan /itin1a< %<% Per.ahan pan1an+< Misalkan pe+as /en+an pan1an+ ala-i L, pan1an+ tanpa a/a .e.an< L Karena a/a .e.an tarik P, -aka pan1an+ pe+as -en1a/i L  F< P Jika .e.an P ts. -asih /ala- area elastis linier, -aka .erlak hk δ L Hooke, .e.an /an perpan1an+an akan se.an/in+< P C k F , /i -ana k a/alah konstanta kekakan pe+as )%!'* Batan+ pris-atis< Pa/a /aerah elastis se-prna, -aka .erlak Hk- Hooke,  C G E δ C perta-.ahan pan1an+ Karena σ C PA /an ε C δ L , -aka PL FC )%!%* EA E C -o/ls elastisitas .ahan Hasil kali EA /ise.t ri+i/itas aksial )kekakan aksial*< aktor konstanta kekakan .atan+, P C k δ ata P EA kC C )%!0* δ L lei.ilitas ; a/alah ke.alikan /ari kekakan k, ' L ;C C )%!2* k A '' in B -aka EA (ontoh %<' ? :,3 in Batan+ kak AB( /ipasan+ /en+an en+sel /i B sehin++a titi -r ( A .isa naik trn, titik ( .era4n kiri kanan< pe+as Pe+as /ipasan+ /i ( -la!-la /ala- kea/aan .e.an ? C & Posisi penn1k ( /iatr /en+an =ara -en4etel -r< Kekakan pe+as, k C 2,% l.in< Mr -e-iliki pit=h p C ''5 in, artin4a ntk -a1-n/r se.esar ' in, hars /iptar '5 ptaran< Jika /i.eri .e.an ? se.esar % l., .erapa ptaran -r hars /i.erikan a+ar posisi panah ( tetap -enn1k ke posisi se-la V Jawa., MB C & -aka +a4a tarik pe+as  P )B(* C ? )AB* (ontoh %<% P 23& 6 %%3 R ata P C ? )ABB(* ? Jika ? C % l., -aka P C % )''  :,3* C %<0'39 l. 29& 5&& Karena a/a +a4a tarik P ts., -aka pe+as akan -lr A C '&%& S A C 3%& --% se.esar, δ C Pk C &<33'2 in --% T Jika ' ptaran -r akan .er+eser ''5 in, -aka -r perl /iptar 9<9%%' ptaran

11 / 56


12 / 56

Batan+ P6R kak< Batan+ 6S /an RT -e-iliki E C %&3 #pa< Hitn+ ? 1ika penrnan P /i.atasi han4a ' --< %<0 Per.ahan pan1an+ .atan+ 4an+ ti/ak sera+a-< Se.ah .atan+ pris-atik .isa sa1a /i.e.ani +a4a aksial ti/ak han4a pa/a 1n+!1n+n4a sa1 N' N' C ! PB  P(  PD N'L' A F'C

L'

PB

B

L%

(

B

PB

(

P(

N%

N% C P(  PD N0

( P(

P(

L0 D

D PD

PD

(ontoh %<0 A %&

D

%9

D PD

N0 C PD

F%C F0C

EA N%L% EA N0L0

EA F C F'  F%  F0 N L Jika E /an A .er.e/a, FC i i PD Ei Ai

Berapa penrnan titik ( 1ika las pena-pan+ AAB C &,%3 in%, AB( C &,'5 in%, -o/ls elastisitas .ahan E C %:,&  '& 5 psi %3 

Pen4elesaian B E Karena .atan+ BDE, .atan+ AB( -en+ala-i +a4a aial B ke atas D 0& 39&&X se.esar 3'89<5 X sehin++a +a4a tekan /i A %:%9<5 X A %:%9<5 X N'L' ( !3938' F'C C C !&<&&9&899  %& EA %%3& X 3'89<5 X 8E&5 ' N%L% B 583&& F%C C C &<&'23282  EA% 0& 3E&5 Sehin++a penrnan ( C &<&&52595  ( %%3& X %<2 Strktr Statis Tak Tent Reaksi t-pan pa/a kea/aan statis tak tent ti/ak /apat /i=ari han4a /en+an -en+an/alka r-s M C & /an  C &< Perl a/a ta-.ahan s4arat .atas, kon/isi lain 4an+ spesi;ik< R A A (ontoh, .atan+ AB /i sa-pin+ /i1epit 1n+!1n+n4a< #a4a nor-al P /i.erikan /i (, -aka R A /an R B ti/ak /apat se+era /i=ari< a Tetapi /iperoleh sat persa-aan  R A  R B C P (

P . R B

B =

=

Karena .a+ian a -en+ala-i tarikan, -aka akan ter1a/i pe-lran, se.alikn4 .a+ian . -en+ala-i penekanan, -aka akan -en+ala-i pe-en/ekan< Tetapi .a+ai-anapn, pan1an+ totaln4a ti/ak akan .er.ah, sehin++a /iper! oleh ' persa-aan la+i  Fa  F . C FAB C & R A a ! R B . R A a R B . -aka =

=


a

AC

b

EA

Hasiln4a,

R A C P. L

CB

EA

R B C Pa L

13 / 56

! EA EA R A  R B C & Maka R A /an R B /apat /i=ari sehin++a F /apat /ihitn+<

(ontoh %!3 Ta.n+ ko-posit ter/iri /ari ta.n+ lar te-.a+a /an ta.n+ /ala- .a1a /itekan /en+an plat +a4a P< Las pena-pan+ ta.n+ .a1a )steel* As /an Mo/ls Elastisitas Es , se/an+kan te-.a+a )(opper* A= /an E=<

L

Tentkan .esaran!.esaran  a* #a4a tekan Ps /i ta.n+ .a1a /an P= /i ta.n+ te-.a+a .* Te+an+an tekan s /an = < =* Perpen/ekan ke/an4a< Pen4elesaian  a* #a4a 4an+ /iteri-a te-.a+a  .a1a C +a4a total, -aka P=  Ps C P Karena penekann4a /ari plat kak, -aka pe-en/ekann4a akan sa-a, F= C Fs P L P L P L ata P= P -aka P= L F= C = Fs C s C s C s A=E= AsEs A=E= AsEs A=E= AsE As Es As Es Ps C P ! P= Ps C P= P= C P ! P=  P= C P A= E= A= E= As Es A= E= P=  C P A= E= A= E= A= E= As Es P= C P Ps C P As Es  A= E= As Es  A= E= .* Te+an+an tekan P P E= P σ= C = C σs C s C A= As Es  A= E= As =* Pe-en/ekan P= L Ps L PL δ C C C A= E= As Es As Es  A= E=

P Es As Es



A= E=

%<3 Te+an+an pa/a poton+an -irin+< W

P P

A'

Jika a/a .i/an+ irisan -irin+ se.esar W /i ten+ah .atan+ P -aka te+an+an pa/a .i/an+ ts. /apat /iraikan -en1a/i +a4a nor-al N /an +a4a +eser ", /i -ana N N C P =os W /an " C P sin W " A/apn te+an+ann4a hars /ihitn+, karena las per-kaann4a /ari A s/ah .er.ah -en1a/i A' , /i -ana C A A


'

Per1an1ian tan/a< =os W Te+an+an nor-al σW .ertan/a positip 1ika .ersi;at tarik, ne+atip 1ika .ersi;at tekan< Te+an+an +eser QW .ertan/a positip 1ika arahn4a .erlawanan /en+an arah 1ar- 1a-, ne+atip 1ika searah /en+an 1ar- 1a-< Besarn4a te+an+an!te+an+an ts. a/alah  σW P =os% W P N A' QW C ! " C ! P sin W =os W σW C C QW A' A' A Karena PA C σ , =os% W C '% )'  =os %W* , /an sin W =os W C '% sin %W , -aka σW C σx cos2 W C '% σx ( 1 + cos 2W* QW C ! σx sin W =os W C ! '% σx sin 2W (ontoh %!'& P

P

A'

A

QW

σW

QW

A

A%

σW

QW

σY

σW

W σW

Se.ah .atan+ pris-atik -e-iliki las pena-pan+ A C '%&& --%, /itekan /en+an +a4a P C :& kN< Tentkan te+an+an 4an+ .eker1a pa/a W %3o /an Y ''3o Jawa.  %W C &<98%8 ra/ %Y C 2<&'20 ra/ Mla!-la tentkan arah!arah W, QW, Y /an QY  C !83 N--% σW C '% σx ( 1 + cos 2W* C !5'<5 N--% )MPa* QW C ! '% σx sin 2W C %9<8%8 N--% )MPa* QYC !%9<80 N--% )MPa* Besarn4a QYC QW han4a tan/an4a .erlawanan< σY C '% σx ( 1 + cos 2Y* C !'0<2 N--% )MPa*

σY

QW (ontoh %!'' P % .ah .atan+ .erpena-pan+ .1r san+kar /en+an sisi . hars /isa-.n+ /en+ le- pa/a posisi .i/an+ -irin+ φ C 2&o , /i.eri tekanan 9&&& l.< W Te+an+an i7in -aterial a/alah tarik C ''&& psi /an Q+eser C 5&& psi φ Se/an+kan te+an+an i7in ntk le- tarik C 83& psi /an Q+eser C 3&& psi Tentkan le.ar -ini-- . ntk .atan+ ts.< . Pen4elesaian  Jika s/t φ C 2&o, -aka s/t W C ! 3&o C !&<980 ra/ A/a % =ara pan/an+, se-a te+an+an /ilrskan, ata se-a /i-irin+kan< Dala- hal ini /ipilih, se-a te+an+an /ilrskan, ke-/ian /i=ari te+an+an terke=il a+ar /iperoleh las pena-pan+ ter.esar< Te+an+an nor-al perta-a a/alah σ C ! ''&& psi< Te+an+an +eser Q -ai-- ntk per-kaan rata a/alah '% σ , -aka σ C % Q C ! '%&& psi< Untk pelrsan te+an+an pa/a per-kaan -irin+ /i+nakan r-s 4an+ telah /iolah  Dari te+an+an nor-al -irin+ /iperoleh  C W  =os% W , ata σ C !'9'3 psi Dari te+an+an +eser -irin+ /iperolah  C QW  sin W =os W , ata σ C !'&'3 psi Karena /ari 2 .esaran te+an+an  4an+ terke=il a/alah '&'3 psi, -aka las pena-pan+

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4an+ tera-an a/alah , A C P , ata A C

15 / 56

8<9893 in% ata . ≥ %<9&5: in

Bahasan tentan+ E;ek ter-al, .e.an ke1t, .e.an ;atik, perilak non linier, ti/ak /i.erikan< Mahasiswa a+ar .ela1ar sen/iri /ari .k pe+an+an wa1i., Latihan soal  %<2!''

Se.ah .atan+ kak AB( /i+antn+ /en+an 0 tali< Tali A pan1an+ %h, tali B /an ( pan1an+ h< %h  Bahan /an /ia-eter tali sa-a< h h A B ( Jarak!1arak seperti terlihat /i +a-.ar< L% L% Tentkan 1arak  a+ar posisi .alok tetap horisontal P 1ika +a4a P .eker1a /i ten+ah .atan+< Pen4elesaian  Ini a/alah kea/aan statis tak tent, karena a/a 0 t-pan< Misalkan reaksi /i AB( a/alah R A, R B /an R ( -aka persa-aann4a RA  R B  R ( C P Jika .alok tetap horisontal, -aka FA C FB C F( Ini 1+a 4an+ -en4e.a.kan kea/aan statis tak tent, karena a/a 0 .esaran 4an+ ti/ak /i! ketahi, 4ait R A, R B /an R (, se/an+kan .an4akn4a persa-aan han4a %< Untk pe-e=ahann4a perl a/a r-s 4an+ -en+h.n+kan antara +a4a R /en+an δ δ σ FCGL R-s -lran  C C Diperoleh h.n+an G C CGE Hk- Hooke ntk tarikan L E AE R L R L R L L C δB C B B C δC C ( ( δA C A A δ C AE AE AE AE R %h R h R h δA C A C δB C B C δC C ( % R A C R B C R ( AE AE AE R A  R B  R ( C P R A  % R A  % R A C P 3 RA C P R A C P3 RB C R ( C % R A C %P3 Men+hitn+ 1arak   MA C & R B )* ! P )L%*  R ( )L* C &

C

PL% ! R(L C R B

'2 L

%<2!'3 h

A a

(

Diketahi .atan+ kak AB /itahan % tali /en+an .ahan %h sa-a tapi .e/a pan1an+< #a4a P .eker1a /i 1n+ B< D B U1n+ A en+sel< a* Hitn+ +a4a tarik tali T( /an TD . L P .* Hitn+ len/tan FB

Jawa.  Batan+ AB kak, -aka F(  FD  FB C a  .  L ( C G( E C T(k  C G E C +a4alas pena-pan+

/i -ana k C 2 /%


F( C G( h C (E h C T( hkE % T a F(  FD C a  . T( h  TD %h C a  . T( C D FD C GD h C DE h C TD %hkE .  MA C & T( )a*  TD ).* C P )L* PL P.L TD C C .  % a%. .%  % a% % T a %PaL T( C D C . .%  % a% F(  FB C a  L Len/tan /i B  ata FB C F( La %PaLh % P L % h F( C T( hkE F C F La C C % B ( ).  % a% * KE ).%  % a% * KE %<2!'5 A

&<%3

Berapa .esar P a+ar si-pan+an .atan+ AD ti/ak B ( D -ele.ihi %o V P Jawa.  Si-pan+an δD -a C 3&& sin %o C '8<23 -&<% k C '3 kNk C 03 kN- Karena si-pan+an ts., +a4a D C 5'&<82 N Si-pan+an δA -a C 9<8%2: --, -aka +a4a A C '0&<98 N MB C & P )&,%* C A )&,%3*  D )&,3* P C '5:&<2 N ( D %<2!'9 Batan+ kak AB /itahan % tali (E /an DB< ',3 h h Berapa +a4a tarik pa/a tali V Jawa.  (A%  AE% C (E% θ A E  B ',%3 h% C &,%3 L% LChZ3 ( E P Len/tan .atan+ δB = 2 δ = 2 ! θ δ"B = δB L% L2 L2 Ke-lran tali #! = δC = δ sin θ = &<5558 δ E  )(E* L δC = E E C EA F(E C &<2222 EA δ δ= θ E EA EA (A h B )DB* EA δ δ = B C EA FDB C θ E % "B  E EA DB h B C 2<3 E MA C & E4 )L%*  B )L* C P )0L2* E4 C E sin W C &<5558 E '<0000 E  '9 E C 0P E C &<'33% P B C &<5:90 P 0.5

A %<2!': ',3 . θ

B

φ

(

Batan+ B(D kak /itahan tali A( /an AD< Hitn+ +a4a tali TA( /an TAD serta penrnan δD D Jawa. 

16 / 56


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1$ / 56

.

P W C atan ABB( C &<5203 ra/ A( C %<3 . φ C atan ABBD C &<2505 ra/ AD C 0<032' . δC C %0 δ" Penrnan titik  δC % δ" = 2 % 3 EA F( Perpan1an+an tali  FA( C δC sin θ = &<5 δC  C EA F C &<%2 A( LA( A( . L FC FAD C δ" sin φ = &<228% δ" EA FD EA AD C EA FAD C &<'000 LAD . δC C %0 δ"

AD4 C AD sin φ = &<228% AD A(4 C A( sin θ = &<5 A( C MB C &

δA" =

AD LAD

EA

.

&<8% AD

C 0<5'82

AD C A( C

'<&893 P '<%:2% P

P. EA δ" = δA" /sin φ =

%<2!%' A

EA FD

A( C '<% AD

A(4 )%.*  AD4 )0.* C P )0.* '<22 AD  '<02'5 AD C 0P

LAD C 0<032' .

A( C &<'5

9<&999

P. EA

Batan+ kak ED /itahan % tali, AB /an (D< ( Hitn+ +a4a tarik pa/a tali /an s/t rotasi .atan+< Jawa.  S/t W C &<5203 ra/ 5. E θ D θ B Pan1an+ AB C '3 . Pan1an+ (D C '& . P Persa-aan 4an+ a/a  ME C & A 9. 2. E D B AB4 )'%.*  (D4 )9.* C P )'%.* Ke-irin+an .atan+ EDB  D B DD  BB C δ" % δB = 9  '% C %  0 FB C BB C 0% DD C 0% FD 9. 2. θ B Maka % persa-aan 4an+ a/a '% AB4  9 (D4 C '% P AB FB C BB C 0% DD C 0% FD θ θ AB4 Perl /i=ari h.n+ann4a antara δ /en+an +a4a B B AB 0.

BB C FAB C δB sin θ =

AB LAB

EA

AB4 LAB

EA sin θ

δAB = δB sin θ BB C BB sin θ AB4 C AB sin θ AB C AB4  sin θ AB4 LAB '3 . FB C AB4 C EA sin % θ EA sin % θ (D4 L(D '& . C


D

(D4

EA sin % θ EA sin % θ FB C 0% FD '3 . 0& . AB4 C (D4 % % AB4 C (D4 EA sin θ %EA sin θ '% AB4  9 (D4 C '% P AB4 C (D4 C &<5 P AB C (D C AB4  sin θ = ' P FB S/t ke-irin+an .atan+  '3 . )&,5 P* P %3 P φ C C C % ''% C '% . EA sin % θ EB EA '% EA

1& / 56

P U NT I RA N ) T > R S I *

1' / 56

0<' Pen+antar Selain a/a +a4a tarik, tekan ata +eser, -aka -asih a/a la+i .entk +a4a lain 4an+ spesi;ik, 4ait +a4a pntir 4an+ -en+hasilkan -o-en pntir )torsi*< Bentk kersakan aki.at torsi ini san+at .er.e/a /en+an kersakan tarik ata tekan, se.a+ian -irip /en+an kersakan +eser< 0<% De;or-asi torsional .atan+ silin/ris< φ B A Karena terpntir, -aka +aris AB 4an+ se-la lrs ke-/ian .elok, T -e-ntir -en1a/i AB , sehin++a -e-.entk s/t pntir W< B!  Mo-en pntir .isa /i+a-.arkan /en+an anak panah 4an+ -en+ikti L hk- ptaran sekrp, -a1 1ika /iptar ke kanan, /an se.alikn4a< A

Pan/an+ sepoton+ ele-en /< T S/t +eser  /in4atakan se.a+ai  C BB  AB B!  Jika 1ari!1ari silin/er a/alah r, -aka BB C r /φ r /φ  C / sehin++a / /φ  / a/alah per.ahan s/t pntir φ terha/ap pan1an+ < /φ Maka s/t pntir persatan pan1an+ W C / r /φ Sehin++a r-s re+an+an +eser  C C rW / Pa/a kass pntiran -rni, -aka +aris BB a/alah lrs, ata  .kan ;n+si /ari < Tetapi 1ika silin/er .eron++a,  /i klit lar akan le.ih .esar /arpa/a  /i klit /ala- sesai /en+an per.an/in+an 1arak /ari titik psatn4a< -a -a R ata  C r  C -in -a -in r R )in R ata se=ara --, re+an+an +eser  r se.ah ele-en se1arak  /ari psat    C -a silin/er .er1ari!1ari R a/alah  R 

/φ B

0<0 Batan+ lin+karan elastis linier< Q-a a Q .  Q . Q  W Q r = / Q =

Karena .ersi;at linier, -aka

Jika a/a .i/an+ a.=/ /i klit lar, -aka karena ter! kena pntiran, ter1a/i per+eseran titik .= ke .=< Maka re+an+an pntir  akan -en+aki.atkan te+an+ +eser Q 4an+ .esarn4a a/alah Q C # /i -ana # a/alah -o/ls +eser karena -a C r W , -aka Q-a C r # W 


-a

r R-s Torsi< Q-a Q r



/A

20 / 56

Lihat ele-en /A se1arak  /ari psat, -aka ele-en Torsi /M C Q  /A ata % Q /A /M C Q  /A C -a r Q Q Jika /iinte+ralkan ntk T C -a % /A C -a IP selrh pena-pan+ r r /i -ana

IP C % /A

a/alah -o-en inersia polar Ban/in+kan /en+an M4 σ-a C I r-s te+an+an tarik

Sehin++a /iperoleh Q C T r -a IP ata se=ara -te+an+an +eser /i sat titik

QC

T IP

/i -ana  a/alah 1arak ra/is /ari titik psat

S/t Pntir W Dari r-s Q C #  W /iperoleh W C Q  #  Dari r-s T /iperoleh Q T QC C IP IP 

-aka

W C

T #IP

Karena W .erariasi sepan1an+ .atan+, -aka s/t pntir total /i 1n+ .atan+

φ=WL

(ontoh 0<' Diketahi se.ah .atan+ silin/ris pan1an+ 32 in, /ia-eter ',3 in /i.eri .e.an pntir %3& l.!;t Jika # C '',3  '& 5 psi .erapa te+an+an +eser -a /an s/t pntir -a V Jika te+an+an +eser i7in a/alah 5&&& psi /an s/t pntir i7in %,3o, .erapa Torsi i7in V Jawa.  Mo-en inersia polar I p C  /2  0% Q-a C '5 T C 23%:<2 psi sehin++a C &<2:59 in2 /0 S/t pntir -a TL φ=θL= C &<&%92 ra/ C '<5%29 o #IP Torsi i7in .er/asar te+an+an +eser i7in,

TC

/0Q

C 00'<'8 l.!;t '5 Torsi i7in .er/asar s/t pntir i7in, φ # Ip TC C 092<58 l.!;t L Dia-.il nilai 4an+ terke=il, T C 00'<'8 l.!;t 0<2 Torsi tak sera+aA/akalan4a se.ah poros -en+ala-i .e.erapa -a=a- torsi sepan1an+ poros< Maka =ara perhitn+ann4a akan se-a=a- .a. se.el- ini ntk .e.an nor-al tak sera+a-<


3

A T'

T2

B T% (

T'

Untk -en+etahi torsi pa/a setiap .a+iann4a -aka /ilakkan pe-oton+an ntk setiap se+-ent<

D Poton+an (D  Torsi 4an+ .eker1a a/alah T(D C T0 ! T% ! T'

3

A

21 / 56

B T% ( Poton+an B(  Torsi 4an+ .eker1a a/alah TB( C ! T% ! T'

A T'

B T% ( Poton+an AB  Torsi 4an+ .eker1a a/alah TAB C ! T'

A T'

φ = φ1 + φ2 + φ3 + . . . + φn S/t pntir 4an+ ter1a/i a/alah Jika .entk pena-pan+n4a ti/ak silin/ris, tetapi .erariasi -isaln4a ker=t -aka hars /en+an =ara inte+rasi<

(ontoh 0<2 T' T%

T0

Se.ah .atan+ .a1a soli/ AB(DE .erptar .e.as /i 1n+ A /an E Ro/a +i+i pen++erak ( -en++erakkan ro/a B /an D /en+an torsi T% C 23& N-< Torsi penahan T' C %83 N- /an T 0 C '83 N-

D L%

Dia-eter / C 0& --, B( C 3&& --, (D C 2&& --< # C 9& #p Hitn+ te+an+an +eser -ai-- /i setiap se=tion /an s/t pn

A B

( L'

Pen4elesaian  Torsi 4an+ .eker1a pa/a setiap se+-ent  Torsi pa/a 1n+!1n+ C &, sehin++a ti/ak a/a te+an+an /an ti/ak a/a pntiran< Torsi pa/a se+-ent (D C !T'  T% C ! %83  23& C  '83 NTorsi pa/a se+-ent B( C ! T' C ! %83 NMo-en inersia polar  I C /2 C 8:332 --2 p 0% Maka te+an+an +eser 4an+ .eker1a pa/a se+-ent  S/t pntiran 4an+ ter1a/i pa/a se+-ent 

QB( C Q(D C φB( C φ(D C

3'<9:: /0 00<&%8 TL !&<&%% φ C #Ip &<&'' φBD C !&<&'' ra/ C QC

'5T

Mpa Mpa ra/ ra/ !&<5&9

o

0<3 Te+an+an /an re+an+an pa/a .e.an +eser -rni QW σW Q Jika a/a se.ah kepin+ lo+a- se+i!2 /en+an sat sisi searah * W /en+an s-. , s-. aial, /an sisi lainn4a /ala- arah 4 x ke atas, -aka 1ika .en/a ts. -en+ala-i .e.an torsi -rni, Q ele-en se+i!2 ts. han4a akan -eneri-a .e.an +eser Q<


22 / 56

Jika .en/a silin/ris ts. /ipan=n+ se=ara -irin+ /en+an s/t W, -aka pa/a per-kaan -iri 4an+ ter1a/i akan .eker1a te+an+an!te+an+an nor-al σW /an +eser QW< Den+an perhitn+an -ate-atika /an kesei-.an+an +a4a )te+an+an*, -aka akan /iperoleh  σW C Q sin %W QW C Q =os %W Re+an+an pa/a +eser -rni< Q σ)in =  Q 

σ)ax = Q

23o



Jika .e.ann4a -asih -ask elastis linier, -aka .erlak Hk- Hooke ntk +eseran QC# ata  C Q  # ).an/in+kan /en+an tarikan  C G E *

σ)ax = Q σ)in =  Q Q Karena σ -a, -in ter1a/i pa/a s/t -irin+ 23o , /iperoleh ε)ax =   %

(ontoh soal 0<5 Se.ah ta.n+ Al-ini- /en+an /o C 9& -- /an /i C 5& -- /ikenai torsi T C 2 KN-< a* Tentkan /an +a-.arkan letak te+an+an +eser, tarik, tekan -a< .* Tentkan re+an+an -a< Pen4elesaian  # al-ini- C %8 #Pa 2 Mo-en inersia polar I p C 0% )/o ! /i2 * C %<83E!&5 -2 a* Q-a C TR  I p C 39<'9% MPa Q-a C 39,% Mpa σ=C ! 39,%MPa Q-a σt * *

x



Q-a

c



23o

x

G

σ=

Q -a

σt C  39,%MPa

Gt .* Re+an+an -aksi--  Hk- Hooke Q-a C -a # -a C Q-a # C &<&&%%  εt -a C &<&&'' )tarik* Re+an+an -ai-G-a C -a ε= -a C !&<&&'' )=o-presi ata tekan* % 0<5 H.n+an antara Mo/ls elastisitas E /en+an -o/ls +eser #< S/ah /i.ahas /i -ka, .ahwa a/a keterh.n+an antara E /an # -elali Poisson Ratio< E #C %)'  *

0<8 H.n+an antara Da4a P /en+an Torsi T< PCT[ /i -ana [ a/alah o-e+a, ke=epatan s/t )ra//etik*


23 / 56

Jika ; -enn1kkan ;rekwensi, ptaran per/etik, -aka .erlak h.n+an  [ C %  ; Satan ;rekwensi a/alah Hert7 )H7* ata /etik !' < Satan /a4a a/alah kw ata Hp< ' Hp /ikonersikan -en1a/i 825 watt< ' Hp C 33& ;t!l.s

0<9 Torsi pa/a statis tak tent< Ta.n+ lar Jika a/a .atan+ ko-posit 4an+ terssn /ari % -aterial /ikenai torsi A B se=ara .ersa-a!sa-a, -aka .erlak h.n+an  ,n- abn T Torsi T ter.a+i pa/a -asin+!-asin+ -aterial T'  T% C T φ1 = φ2 Selain it, s/t pntirn4a sa-a T' L T% L Karena a/a r-s TL φ' C φ% C φ C -aka # I p #' I p' #% I p% φ1 = φ2 T' T% T' T% Karena -aka ata C ! C & #' I p' #% I p% #' I p' #% I p% T'  T% C T /iperoleh  #' Ip' #% Ip% ' C T % C T #' Ip'  #% Ip% #' Ip'  #% Ip% Den+an /iketahin4a T' /an T% , -aka .esaran!.esaran lain .isa /ihitn+< (ontoh 0<: A (

B

Diketahi .atan+ A(B /en+an /A, LA,I pA /an /B, LB, I pB Bahan ke/a .a+ian sa-a< Torsi To /i.erikan /i (< a* Hitn+ Torsi TA /an TB< .* Te+an+an +eser -a /i setiap se+-ent, QA( /an Q(B =* S/t rotasi )pntir* /i (, φ(

Pen4elesaian  a* TA  TB C To Torsi To ini .eker1a /i ten+ah .atan+, -aka akan -eni-.lkan s/t pntir /i B C φ' Tetapi, To ini -eni-.lkan reaksi /i B /en+an TB, -aka TB ini -eni-.lkan φ% Karena 1n+ B terikat, -aka To LA TL φ' C φ C # I p # I pA

φ1 + φ2 = 0 φ% C

To LA # I pA TB LA TB LB To LA  C I pA I pB I pA I pA LB C

TB LA ! # I pA TB LA ! ! # I pA I L TB pB A !

C

TB LB # I pB TB LB C & # I pB To LA  I pA LB C I pA I pB I pA I pB LA


A

24 / 56

o

B o I pB LA  I pA LB I pB LA  I pA LB TA /A TB /B Te+an+an +eser -a T/ Q C Q C QC A( (B % I p % I pA % I pB S/t rotasi φ( akan sa-a .aik /i pena-pan+ A( ata (B< TA LA T L φ( C  B B # I pA # I pB

Latihan soal  0<9!' B 3o A L2

Batan+ pa/at AD /i.eri torsi 0To /an To seperti

( o

L2

D L%

/i +a-.ar< Hitn+ s/t pntir -ai--< Pen4elesaian  Kea/aann4a statis tak tent< Jika reaksi /i A /an D a/alah TA /an TD, -aka TA  TD C 0 To  To C 2 To

Statis tak tent a/alah pen1-lahan al1a.ar % ata le.ih statis tertent< Kea/aan /i atas a/alah seperti pen1-lahan .atan+ AD /en+an kea/aan 1epit /i A .e.as /i D 4an+ -en+ala-i torsi TB, T( /an TD ke-/ian pen1-lahan -asin+!-asin+ s/t pntirn4a C & ata φD' + φD% + φ"3 = & 0 To L2 0 To L2 T L% TB L 3 To Karena TB -aka φD' = TB C  o C #I #I #I #I 2 To L% '' To Karena T( -aka φD% = TA C #I 2 ! TB L 3 To T Karena TD -aka φD0 = T(D C 2 To ! TA C TB( C o #I 2 2 ! '' To L To L 3 To L TL φAB = φBC = φC" = φ = #I '5 #I 2 #I 9 #I φAB =

0<9!% A

B

Hitn+ Torsi TA /an TD

(

o

L0

φ)ax

o

L0

Karena TB -aka φD' = Karena T( -aka φD% = Karena TB -aka φD0 =

D

L0 ! To L 0#I !% To L 0#I TB L #I

Hitn+ s/t pntir -a /an s/t pntir /i ten+ah TA  TB C & Jawa.  T(D C ! TA TB( C To ! TA φD' + φD% + φ"3 = &

TB C To

TA C ! To

AB =  o

BC = 0

C" = o

φAB =

! To L 0 #I

φBC =

&

φC" =

To L 0 #I


0<9!3 A

25 / 56

Batan+ A( /ia-eter &,5%3 in, (B &,83 in< B Berapa T 1ika te+an+an +eser i7in :&&& psi # C '&<9&& ! ''<9&& psi A-.il # C ''&&& psi 9 '5 Jawa.  Kea/aan statis tak tent Persa-aan torsi  TA ! T  TB C & 

(

Kea/aan statis tak tent C pen1-lahan .e.erapa kea/aan tertent   A B C 

TB

I A( C  / 2 0% C I (B C

&<&'3 in2 &<&0'' in2 Karena T, 1n+ B akan ter1a/i pntiran se.esar Karena TB, 1n+ B akan ter1a/i pntiran

!TL #I T L φB2 = B ' # IA( se.esar TB C Persa-aan s/t pntir  φB1 + φB2 = 0 -aka TA C Pa/a .a+ian A( ter1a/i torsi TA , .a+ian (B torsi TB< φB1 =

C !&<&2: T 

TB L% C &<&:3 TB # I(B

&<3&: T &<2:' T

Pa/a .a+ian A( ter1a/i te+an+an

Q-a C

T /% C '&<%09 T I p

T C 98:<&:

Pa/a .a+ian (D ter1a/i te+an+an

Q-a C

T /% C 5<'20 T I p

T C '253<%

Maka T i7in C 98:<&: 0<9!5 A

l.
Batan+ A( /ia-eter %& -- , /an (B 2& -B Berapa T 1ika te+an+an +eser i7in 33 MPa # C 83 ! 9& #Pa A-.il # C 9& #Pa %&& 3&& Jawa.  Kea/aan statis tak tent Persa-aan torsi  TA ! T  TB C & I A( C  / 2 0% C '38'2 --2 I (B C %3'2%: --2 

(

Karena T, 1n+ B akan ter1a/i pntiran se.esar Karena TB, 1n+ B akan ter1a/i pntiran se.esar Persa-aan s/t pntir 

φB1 + φB2 = 0

!TL C !&<'3: T #I T L TB L% φB2 = B '  C &<'92 TB # IA( # I(B TB C &<952: T -aka TA C &<'03' T φB1 =


26 / 56

Pa/a .a+ian A( ter1a/i torsi TA , .a+ian (B torsi TB< Pa/a .a+ian A( ter1a/i te+an+an

Q-a C

T /% C &<&95& <'& !0 T I p

T C 50:<38 N-

Pa/a .a+ian (D ter1a/i te+an+an

Q-a C

T /% C &<&599 <'& !0 T I p

T C 8::<25 N-

Maka T i7in C 50:<38 Latihan 

Ker1akan /i r-ah 0<9!8 0<9!'& 0<9!''

N-

DIA#RAM #A$A #ESER DAN M>MEN LENTUR 2<' Pen/ahlan< Dia+ra- +a4a +eser /an -o-en lentr s/ah /i.erikan pa/a -ata kliah Statika Strktr, -aka pa/a -ata kliah Mekanika Kekatan Bahan ini ti/ak akan /i.erikan teorin4a la+i< 2<% (ontoh!=ontoh soal< (ontoh 2<' P Mo A L2 L2 L%

Diketahi .atan+ se/erhana AB pan1an+ L /i.e.ani B +a4a P /an -o-en lentr Mo seperti pa/a +a-.ar #a-.ar /ia+ra- +a4a +eser /an -o-en lentrn4a<

Pen4elesaian  Mla!-la /i+a-.ar /l DBB!n4a ntk -en+hitn+ +a4a!+a4a reaksi t-pa P Mo MB C & R A )L* ! P )0L2*  Mo C & A B R A C 0 P 2 ! Mo  L L2 L2 L% R A R B R B C P 2  Mo  L Dia+ra- #a4a +eser  Han4a a/a % wila4ah &    L2 /an L2    L " C R A )konstan, +aris lrs -en/atar* &    L2 "  A R A 0 P 2 ! Mo  L L2 R A L2    L

P " A

L2

" C R A ! P C R B )konstan, +aris lrs -en/atar*



R A

R A

0 P 2 ! Mo  L Dia+ra- +a4a +eser P 2  Mo  L

R B

Dia+ra- -o-en lentr  A/a 0 wila4ah &    L2 , L2    L%, /an L%  x   &    L2 M M C R A )* C 0P2 ! MoL )linier, lrs -irin+*  A 0PL'5 ! Mo2 L2  C & Mo C & R A  C L2 ML2 C 0PL'5 ! Mo2 A L2 L2    L% A R A

L2  L%

P M 

R A M C R A )* ! P ) ! L2* C 0P2 ! MoL ! P  PL2 C PL2 ! MoL ! P2 )linier, lrs -irin+* ML2 C PL2 ! Mo2 ! PL'5 C 0PL'5 ! Mo2

2$/56

DIA#RAM #A$A #ESER DAN M>MEN LENTUR ML%! C PL2 ! Mo% ! PL9 C PL9 ! Mo% ML% C PL9 ! Mo%  Mo C PL9  Mo% 0PL'5 ! Mo2

L%  x   P A

2&/56

Mo

L2

M

A

L% R A

)kiri* )kanan*

L2 R A



M C R A )* ! P ) ! L2* ! Mo C PL2 ! MoL ! P2  Mo ML% C PL2 ! Mo% ! PL9  Mo C & C PL9  Mo% A ML C PL2 ! Mo ! PL2  Mo C & R A

PL9 ! Mo%

L% 0PL'5 ! Mo2

PL9  Mo% & B

L2 PL9 ! Mo%

L%

L Dia+ra- -o-en lentr Kesi-plan, 1ika .e.ann4a .erpa sesat 4an+ tetap seperti +a4a terpsat, -o-en terpsat, -aka w1/ /ia+ra- +a4a +esern4a akan .er.entk kotak!kotak );n+si konstan*, /an w1/ /ia+ra- -o-en lentrn4a akan .er.entk se+i!0 se+i!0 );n+si linier, +aris lrs -irin+*< (ontoh 2<%

)*

o

A

Batan+ =antileer AB pan1an+ L /i.eri .e.an ter/istri.si  .er! .entk se+i!0 /en+an tin++i -a o< B



L ? )*

o

#a-.ar /ia+ra- +a4a +eser /an -o-en lentrn4a< Jawa.  n+si .e.ann4a a/alah,  C o )L* Be.an ? C o L% , sehin++a R B C ? C  o L% Jarak ? ke pin++ir a/alah L0, -aka MB C ?L0 C o L%5

o /2

MB Dia+ra- +a4a +eser 

6

L

R B Be.an 6 C  % C o %%L C " "o C & "L C o L%%L C o L% C R B

 );n+si pan+kat %, para.ola*

 " M RB  L

Dia+ra- -o-en lentr M C 6 0 C  % )0* C o )L*)%*)0* C o 05L C o 05L )kra pan+kat 0* Mo C & ML C o L%5 C MB

o /2



MB

DIA#RAM #A$A #ESER DAN M>MEN LENTUR

2'/56

Kesi-plan, 1ika .e.ann4a ter/istri.si, L -aka /ia+ra- +a4a +esern4a akan -erpakan +aris len+kn+< De-ikian pla /ia+ra- -o-en lentrn4a akan ' /era1at /i atas kra +a4a +eser< )kala kra +a4a +eser pan+kat %, -aka kra -o-en lentr pan+kat 0, /st* (ontoh 2<0 P C '2k  C %&&l.;t #a-.arlah /ia+raA

: ;t

B %2 ;t P

(

5 ;t 6 C 5k

: ;t A

'3 ;t

B %2 ;t

( 5 ;t

o 2/6

Jawa., -la!-la +a-.ar DBB!n4a< 6 C  L C %&& )0&* C 5&&& l. C 5k 6 .eker1a /i ten+ah!ten+ah .atan+, se1arak '3 ;t /ari A MB C & R A C P )'3*  6 ):* C ''&&& l. %2 R B C P  6 ! R A C :&&& l.

Dia+ra- +a4a +eser, a/a 0 interal &  x  ' '  x  24 an 24  x  30 " &  x  ' 6 4 C & R A ! 6 ! " C & " C R A ! 6 C ''&&& ! %&&  )linier, -irin+*  R A "o C ''&&& : ''&&& ": C ''&&& ! %&& ):* :%&& C :%&& 6 P "

'  x  24

: R A



4 C & R A ! 6 ! P ! " C & " C R A ! P ! 6

: R A

C ''&&& ! '2&&& ! %&&  C ! 0&&& ! %&&  ": C ! 0&&& ! '9&& ''&&&

C !29&& "%2 C ! 0&&& ! 29&& C !89&& P 6

%2 \  \ 0&

)linier, -irin+*

:%&&

" !29&&

:

%2

R B

!89&&

R A  4 C & R A  R B ! P ! 6 ! " C & " C R A  R B ! P ! 6 C ''&&&  :&&& ! '2&&& ! %&&  C ''&&&  :&&& ! '2&&& ! %&&  C 5&&& ! %&&  "%2 C 5&&& ! 29&& C '%&&

''&&& :%&&

'%&&

DIA#RAM #A$A #ESER DAN M>MEN LENTUR

30/56

"0& C 5&&& ! 5&&& C &

&

Dia+ra- +a4a +eser )ti/ak /iskala* !29&& Kra +a4a +eser -e-oton+ s-.  /i !89&&  C : /an  C %2< & : %2 0& Mo-en lentr -ai-- saat +a4a +eser C &< Dia+ra- -o-en lentr a/a 0 interal &  x  ' '  x  24 an 24  x  30 M C & &  x  ' 6 % Dia+ra- -o-en lentr M M  R A  ! 6 % C & & M C 6 % ! R A   R A : C '&&  % ! ''&&&   Mo C & : M: C 9'&& ! ::&&& C !:&:&& '0'00 6 P '  x  24 M M C & M  R A  ! 6 % ! P)!:* C & : R A M C 6 %  P)!:* ! R A   M C '&&  % ! ''&&&   '2&&&  ! '%5&&& Dia+ra- -o-en lentr 05&& C '&&  %  0&&&  ! '%5&&& & & M: C !:&:&& ).enar* M%2 C 05&&  %2 '0'00 %2 \  \ 0& P 6 M R B : %2 R A  M C & M  R A  ! 6 % ! P)!:*  R B )!%2* C & M C 6 %  P)!:* ! R A  ! R B )!%2* Dia+ra- -o-en lentr C '&&  %  0&&&  ! '%5&&& ! :&&&   %'5&&& C '&&  % ! 5&&&   :&&&& & M%2 C 05&& ).enar* M0& C & ).enar*  :

05&& &

'0'00

%2 Kesi-plan  0& Lan+kah perta-a a/alah -en++a-.ar DBB ntk -en=ari +a4a!+a4a reaksi t-pan< Beriktn4a -en=ari +a4a +eser ntk setiap interal< Batas!.atas interal a/alah setiap a/a .e.an +a4a lar ata a/a t-pan<

DIA#RAM #A$A #ESER DAN M>MEN LENTUR Jika a/a .e.an -o-en lar, ti/ak akan -e-pen+arhi +a4a +eser, ti/ak -ena-.ah interal< Setiap -elewati .atas interal, akan ter1a/i lon=atan +a4a +eser< Besarn4a lon=atan C +a4a +eser atas ! +a4a +eser .awah C nilai .e.an lar ata nilai reaksi t-pan< Nilai!nilai +a4a +eser pa/a .atas interal akan -erpakan alat pe-eriksaan ke.enaran hasil perhitn+an< Setelah it .eralih ke -o-en lentr< Tentkan 1-lah interal, ke-/ian =ari r-s -o-en ntk tiap interal< (ari har+a -o-en pa/a .atas interal, 4an+ akan -erpakan alat pe-eriksaan ke.enaran hasil perhitn+an< R-s -o-en lentr selal ' /era1at /iatas r-s +a4a +eser, karena Mo-en C #a4a  1arak< '< Bo.ot 0&< Diketahi .atan+ DE /iletakkan /i atas .atan+ A( P P -asin+!-asin+ sepan1an+ 0a )lihat +a-.ar*< D E Pa Batan+ DE /i.eri .e.an % +a4a terpsat P /an B A ( -o-en lentr Pa< a a a a #a-.ar /ia+ra- +a4a +eser /an -o-en lentr .atan+ AB( Pen4elesaian MD C & Pa  P)a* ! R( )%a* C & R( C P P P R ( P R D P R D C P D E Pa B R ( ( B A R D MA C & R B C %P a a a a a a R A R B R A C & Dia+ra- +a4a +eser .atan+ AB P B A a a a a a a B !P A Dia+ra- -o-en lentr .atan+ AB !Pa

31/56

TE#AN#AN TARIK  TEKAN DAN #ESER

3<' Pen/ahlan< Perhitn+an +a4a +eser /an -o-en lentr /i+nakan ntk -en+hitn+ te+an+an tarik /an te+an+an +eser -aksi--< Dala- peren=anaan, te+an+an!te+an+an ini ti/ak .oleh -ele.ihi te+an+an i7in 4an+ /iper.olehkan< 3<% Te+an+an tariktekan< Untk ti/ak -ena-.ah .e.an pikiran, lan+sn+ sa1a /i.erikan r-sn4a /i -ana σ4 a/alah te+an+an tariktekan pa/a 1arak 4 /ari +aris netral 4 C M4 I te+an+an tekan 1ika ke=en/ern+ann4a -e-en/ek, te+an+an tarik 1ika ke=en/ern+ann4a -e-an1an+< 4 a/alah 1arak /i atas ata /i .awah +aris netral +aris netral a/alah .a+ian 4an+ ti/ak -en+ala-i per.ahan P I a/alah -o-en inersia pena-pan+< A B (ontoh, 1ika .atan+ AB /i.eri .e.an P /i ten+ah, -aka ke=en/e! rn+ann4a .atan+ akan -elen+kn+ ke .awah< 4 tekan ! Maka .a+ian atas akan -en+ala-i tekan, .awah kena tarikan< +aris #aris netral a/alah +aris .i/an+ 4an+ ti/ak -en+ala-i per.ahan<  netral #aris netral a/alah +aris 4an+ -elali titik .erat area pena-pan+< tarik  Terlihat .ahwa te+an+an ter.esar terletak /i 4 ter1ah, 4ait /i 4 klit per-kaan atas ata .awah< Untk pena-pan+ se+i!2, -o-en inersia I C ''% .h 0 h Untk pena-pan+ lin+karan, I C I44 C '52 7/2 ]In+at, Ip C '0% 7/2 ^   Jika +aris netraln4a .er+eser, -aka I .ar C I+aris .erat  r % A . /i -ana r C 1arak per+eseran s-. A C las pena-pan+ Jika .en/an4a ko-posit, .erlak Iko-posit C I-asin+!-asin+ Posisi +aris netral .en/a ko-posit 1+a hars /i=ari /l< (ontoh 3<0 : ;t P C '% kp  C ',3 kp;t A B %% ;t P 6 C 00 kp : ;t '' ;t

4+aris netral C 4A A /i -ana A C las pena-pan+

%8 Diketahi .atan+ se/erhana AB /i.e.ani seperti pa/a +a-.ar< Pena-pan+ .atan+ 9<83 a/alah tin++i h C %8 in /an le.ar . C 9,83 in Hitn+ te+an+an tarik tekan -ai--< Jawa. Perta-a kali /i=ari reaksi t-pan /en+an DBB Diperoleh R A C %0<3:' kp

0%35


R A

R B %% ;t Dia+ra- -o-en lentr &  x  '  C ',3 kp;t R A



R B C M C R A  ! &,83  %

 C ',3 kp;t P C '% kp 

%'<2&: kp Mo C M: C

&

kp!;t '3'<38 kp!;t

M :  x  22

R A

0035

M

M C R A  ! &,83  % ! P )!:*

C ! &,83 %  ''<3:'   '&9 M: C '3'<38 kp!;t M%% C & kp!;t Diperoleh M-a C M: C '3'<38 kp!;t

).enar* ).enar*

Karena pena-pan+n4a .er.entk 2 perse+i!pan1an+, -aka +aris netral a/a /i ten+ah< I C ''% .h0 C '203%<'99 in2 4 C h% C '0<3 in -aka tekan C tarik C -a C M4 C '8'&<9 psi I (ontoh 3<2  C 0 kNBatan+ se/erhana AB( /it-p /i en+sel A /an rol B seperti A ( pa/a +a-.ar /en+an .e.an -erata  C 0 kNR A R B 0Jika pena-pan+n4a .er.entk kanal ter.alik, hitn+ .esar ',3 /an lokasi te+an+an tarik tekan -a< 4 Dari DBB /iperoleh R A C 0<083 kN R B C '&<'%3 kN '% 9& M C R A  ! &,3  % Mo C & .enar &x3 '% M0 C !0<083  M-a ter1a/i /i /M/ C & 0&& /M/ C R A !  C &  C RA   C Pena-pan+ AB( )--* '<'%3 M'<'%3 C '<9:92 M C R A  ! &,3  %  R B )!0* M0 C !0<083 kN!0  x  45 .enar M2,3 C & '<9:92 .enar Mo-en -a ',9:9 kN!- ter1a/i /i  C ','%3 & B & Karena tan/a -o-en positip, -aka .atan+ -elen+kn+ A ( ke .awah, .a+ian atas kena tekan, .awah kena tarik< !0<083 Mo-en -in 0,09 kN!- ter1a/i /i  C 0 Karena tan/a -o-en ne+atip, -aka .a+ian atas kena tarik, .awah kena tekan< 2 8 4 Ben/a . h A 4 4A '% ' C 0 '% 59 9'5 02 %8822 1


'%

05&& 82 %552&& 9  3%0% 0%'999 61.523 Ben/a I r I  r % A 4 M C '9:9<2 !0083 N!' C 0 0'220% %8<3%0 :0%35'<8: '9<288  C !'2<%' %3<%5 Mpa % 20%&& '%<288 5&0508<58 5'<3%0  C 28<0' !92<'' MPa 9 %25985'<% σtarik -a C 28<0' Mpa ter1a/i pa/a posisi  C ','%3 - /i .a+ian .awah .atan+< σtekan -a C !92<'' Mpa ter1a/i pa/a posisi  C 0 - /i .a+ian .awah .atan+< Latihan

%

3

P C '&&& l.

0&&

'%

&,3

(ari .esar /an lokasi te+an+an tarik /an tekan -ai--<

 C '%9 l.;t 2 ;t

0 9 ;t

3 ;t

&,3 %,3 Pena-pan+ )in=h*

Latihan  C 0,3 kN%L

L

%. .

Batan+ se/erhana /en+an pan1an+ oerstek L C ',3 /i.e.ani +a4a persatan pan1an+  C 0,3 kN-< Jika .erat 1enis .ahan  C 88 kN-0 /an pena-pan+n4a .er.entk 2!perse+i!pan1an+ tin++i %. /an le.ar ., /an te+an+an nor-al i7in  C 5& Mpa, a* Hitn+ le.ar . /en+an -en+a.aikan .erat /iri< .* Hitn+ le.ar . /en+an -e-perhitn+kan .erat /iri<

3<0 Te+an+an +eser< Jika sat .atan+ /i.e.ani sehin++a ter1a/i lentran, -aka a/a .a+ian 4an+ -en+ala-i pe-en! /ekan /an /i.a+ian se.elahn4a -en+ala-i pe-an1an+an< Aki.atn4a, pa/a setiap lapisan akan ter1a/i per.e/aan re+an+an 4an+ -enn1kkan a/an4a te+an+an +eser< 4 Lihat +a-.ar /i sa-pin+< Te+an+an +eser 4an+ ter1a/i /i lapisan se1arak 4' /i atas h%

4 4'

h .

+aris netral +aris 7 netral

/i -ana

"6 I. " C +a4a lintan+ )+eser* I C -o-en inersia pena-pan+ . C le.ar pena-pan+ 6 C -o-en las C 4 /A QC

0235


h%

0335

h% 6C 4 /A C .4 /4 C .% 4 % C .% ])h%* % ! 4'% ^ 4' 4' 4'

Untk /aerah knin+

h%

6 .isa 1+a /ihitn+ /ari perkalian las /aerah knin+  1arak titik .erat area ke +aris netral 6 C ].  )h% ! 4'*^  ]4'  )h%! 4'*%^ C .% ]h% ! 4'^]% 4'  h% ! 4'^ C C .% ]h% ! 4'^]h%  4'^ C .% ])h%*% ! 4'% ^ Distri.si te+an+an +eser< Pa/a per-kaan atas.awah, te+an+an +eser C & karena ti/ak a/a h% lapisan /i atas.awahn4a< Ata 1ika /ilihat /ari r-s, 4 ' C h% , Q-a -aka 6 C & Te+an+an +eser -erpakan ;n+si linier /ari 6, /an 6 -erpakan ;n+si kwa/rat /ari 4' h%

Q-a C

(ontoh soal<

" . )h%* % I. %

;<2

C

9I

 C '5& l.in Titik (

A

( 0

2

B 9

05

%

4(

h% . C '

C

Batan+ AB /en+an pena-pan+ 2  ' /i.e.ani  C '5& l.in< Tentkan te+an+an nor-al σc /an te+an+an ' +eser :=< #a-.arlah arah te+an+ann4a< R A C R B C %99& l. Jawa.  "( C R A ! '5&_%9 C !'5&& l. M( C R A )%9* ! '5&%_%9% C '8:%& l. i

I C ''% .h0 C 3<0000 in2 A( C )'*)'* C ' in% 8 C ',3 in ( C :( C

M4=

I "6=

I.

(ontoh< P a C &<3 A R A

C !005& psi )tekan* C

23& psi

0" %A

6( C 8A C '<3 in0 : σ :

23&

Arah te+an+an +eser σ 005& psi -en+ikti arah +a4a +eser< Di se.elah kiri ele-en, +a4a +eser ke .awah<

P &<3 -

Balok AB -e-iliki pena-pan+ .  h C '&&  '3& -Jika te+an+an lentr i7in σi7in C '' MPa /an te+an+an +eser B i7in :i7in C ',% MPa, hitn+ .e.an Pi7in< R B Jawa. M-a C Pa C &,3P ter1a/i /i antara % .e.an P "-a C R A C P ter1a/i antara t-pan /an .e.an

I C ''% .h0 C %9'%3&&& --2 M4 I I C MC PC CPa C 9<%3 I 4 a4

kN C


"6 : C I.

"C

:I.

6

PC

=

%< Bo.ot 0&<

:I.

6

,

Diketahi .atan+ se/erhana AB( /i.eri .e.an terpsat P C '&&& l. /an .e.an ter! /istri.si  C '%9 l.;t seperti /i +a-.ar< Pan1an+ /an 1arak!1arak /ala- ;eet<

'&&& l.

A

2 ;t

B

hC0 (

9 ;t

3 ;t

-a

C '%<&& kN

&,3 '%9 l.;t

0535

&,3

. C %,3 Pena-pan+ AB( .er.entk T ter.alik Pena-pan+ AB( seperti /i +a-.ar, /i-ensi /ala- in=h< Periksa, /i -ana /an .erapa .esar te+an+an tarik ter.esar /an te+an+an tekan ter.esar< Mo-en -a ter1a/i /i trnan /M/ C & 1ika ;n+si M)* kontin< '&&& A

2 ;t

MA C &

52& B

R B C R A C

(

'02& 0&&

MB C

&

.enar

9 ;t

3 ;t R A R B M C R A  C 0&& M2 C '%&& &\`2 M C R A  ! '&&&)!2* C ! 8&&  2&&& M9 C !'5&& 2\`9 Mo-en -a ter1a/i /i  C 2 se.esar '%&& l.<;t, .a+ian atas tekan, .awah tarik< Mo-en -in ter1a/i /i  C 9 se.esar '5&& l.<;t, .a+ian atas tarik, .awah tekan< &,3

'

hC0

% &,3 . C %,3

Pena-pan+ ko-posit ter/iri atas .entk ' &,3  %,3 in, /an .entk % %,3  &,3 in< Perhitn+an s-. .erat /an -o-en inersian4a 8 , > ,+>2A Ben/a . h Las A 4 4A ' &<3 %<3 '<%3 '<83 %<'983 &<53' &<83 '<032% % %<3 &<3 '<%3 &<%3 &<0'%3 &<&%5 &<83 &<8%:% %<3

9

%<3

4A 8 C A 4atas C 4 .awah C

2 1

M2 C '%&& M9 C !'5&&

σ4 ?kan

=

σ& ?kan

=

σ?kan )ax =

Jawa.an .k 0< Bo.ot 2&< 0P

σ?kan )ax =

'

rC!4 '09%2 :%'5 '09%2 '09&&

psi psi psi psi

%<&9000 , C

12

σ4 a>ik

=

σ& a>ik

=

σa>ik )ax = σa>ik )ax =

b<3

5:'% '920% '920% '92&&

I C I'  I% psi psi psi psi

Batan+ se/erhana AB( /i.eri .e.an terpsat 0P /an P seperti /i +a-.ar< Jika P C 3 kN, L C 0,5 - /an h C %9& -P /an .erat 1enis .ahan  C 3,3 kN-0


A

L%

B L

R A

(

h

L0 R B

0835

a* Tentkan le.ar . 1ika te+an+an +eser i7in Q-a C &,8 MPa

. .* Tentkan le.ar . 1ika te+an+an lentr i7in σ-a C 9,% MPa Pena-pan+ AB( i* Den+an -en+a.aikan .erat /iri ii* Den+an -e-perhitn+kan .erat /iri R A C '<'58 P C 3<900 kN Jawa., /en+an DBB /iperoleh R B C %<900 P C '2<'8 kN '3 kN 3 kN A '<9 B ( Dia+ra- +a4a +eser " C R A "L% C 3<900 kN 0<5 '<% &   ` L% R A R B " C R A! 0P "L C !:<'58 kN L%   ` L " C R A! 0P  R B "2L0 C 3<&&& kN L   ` 2L0 6 C ).h%*)h2* C .h%9 6 .h%9 0 3<038' C C C %9& I C .h0'% I. .%h0'% %.h . "6 2<85': 2<85': QC . C C " " C 5%<039 -: . I. . Jika .erat /iri /iperhitn+kan, -aka a/a .e.an -erata se.esar w C '3 6 3 ? C 2L0 C 80:% . NR A C 3900  %252 . N A '<9 B ( R B C '2'58  2:%9 . N 0<5 '<% R A R B %<2 Dia+ra- +a4a +eser " C R A ! w "& C 3900  &   ` '<9 "L%! C 3900  C " C R A ! w ! 0P "L% C !:'58  '<9   ` 0<5 "L! C !:'58  " C R A ! w ! 0P  R B "L C 3&&&  0,5   ` 2,9 "2L0 C 3&&&  6 3<038' C I. . "6 3<038' 2:'&8 QC C :'58  0&9& . C  '53&& C I. . . . C .* Dia+ra- -o-en lentr tanpa .erat /iri< R A C '3&&& 3&&& R B C A '<9 B (

3900<0 N '2'58 N

h

C '32&

%252 !0&9 !0&9 !0&9& !0&9& !2:%9

. N-

. &<&8'9 . . . -a . .

8&&&&& Pa &<&8'9 -

R A  R B C %&&&& N ).enar*


0<5

'<% R B M C R A  M'<9 C '&3&& N&   ` '<9 M C R A  ! '3&&&)!'<9* M0<5 C !5&&& NM C R A  ! '3&&&)!'<9*  R B)!0<5* M2<9 C & NKarena M ;n+si linier -aka M-a C '&3&& N-

R A '<9   ` 0<5 0<5   ` 2<9 σ C

M4 5M C I .h%

0935

. C

5M C &<&:9 σh%

-

Jika .erat /iri /iperhitn+kan, -aka a/a .e.an -erata se.esar w C '3 ? 3 ? C 2L0 C 80:% . NR A C 3900  %252 . N A '<9 B ( R B C '2'58  2:%9 . N 0<5 '<% R A R B %<2

h

C '32&

. N-

Karena ;n+si M ti/ak linier, -aka le.ih -/ah pakai =o.a!=o.a< ε C . C &<&::9&2 &<&&& C ;n+si pe-eriksaan )pro+ra- e=el* Be.an -erata w C '30<8 N? C 808<83 N R A C 5&8:<0 N R B C '253: N R A  R B ! %&&&& C 808<83 N M C R A ! w%% /M/ C RA ! w C & &   ` '<9 '<9   ` 0<5

0<5   ` 2<9

.enar  C RAw C 0:<330 M'<9 C '&5:2 N-

M C R A ! w%% ! '3&&&)!'<9* /M/ C R A ! w ! '3&&& C &

 C )RA ! '3&&&*w C !39<&2 M0<5 C !5''' NM C R A ! w%% ! '3&&&)!'<9*  R B)!0<5* /M/ C R A ! w ! '3&&&  R B C &  C )RA!'3&&&R *w C 08<00' B M2<9 C & N-

M-a C '&5:2 N .h0 IC C &<&&&% '%

σ C

M4 C 9,%&&,&&& I

σ)ax C

9,%&&,&&&

.enar

(atatan Soal /i atas 1ika s/ah pernah /iker1akan /i r-ah, -aka ti/ak akan ter1a/i keslitan saat hars -en+er1akan ke-.ali /i kelas /en+an .oleh -e-.ka .k =atatan< Jika /ihitn+ /en+an . C '&& --, hasiln4a s/ah =kp -e-a/ai<

ANALISA TE#AN#AN /an RE#AN#AN

5<' Pen/ahlan< Sela-a ini te+an+an han4a /ihitn+ pa/a .i/an+ 4an+ searah /en+an s-. ta-a< Pa/ahal, karena a/an4a .e.erapa .e.an 4an+ arahn4a .erariasi, -n+kin sa1a arah te+an+an -aksi-- ti/ak pa/a arah s-. ta-a< Pe-aha-an ini pentin+ .a+i teknisi 4an+ akan -e-asan+ .alok ka4 -isaln4a, karena ka4 -e-iliki serat 4an+ san+at -e-pen+arhi kekatan .ahan pa/a .er.a+ai arah< 5<% Te+an+an .i/an+< Tan/a arah te+an+an positip 4

4'

Arah s/t ptar W positip

Q4

' Q'4'





Q'4'

1ika s-. !4 /iptar ke kiri se.esar W -e-.entk s-. ' ! 4'

' Q4

4

4'

4'

4 '

W 4  W  Karena sisti- .era/a /ala- kea/aan seti-.an+, -aka Q4 C Q4 /e-ikian pla :'4' Q'4' ' Pan/an+ pena-pan+ .er.entk se+i!0 /i sa-pin+ W Ao Jika las .i/an+ te+ak a/alah Ao -aka las .i/an+ 4an+ lain Aose= W .isa /ihitn+< Q4Ao Kesei-.an+an +a4a /ala- arah '  Q4Aotan W ' Ao se= W !  Ao =os W ! Q4 Ao sin W ! 4 Ao tan W sin W ! 4Aotan W Q4 tan W =os W C & Q'4' Ao se= W   Ao sin W ! Q4 Ao =os W ! 4 Ao tan W =os W  Den+an =ara 4an+ sa-a Q4 tanW sin W C & Karena Q4 C Q4 , /an /en+an pen+atran penlisan, /iperoleh σx1 = σx cos2W  σ* sin2W  %:x* sin W =os W :x1*1 =  (σx  σ*@ sinW =os W  :x* (cos2W ! sin%W*

Den+an pen+atran .ahwa =os%W C '% )'  =os %W* sin W =os W C '% sin %W     !  ' C  4   4 =os %W  :4 sin %W % %  ! 4 :' C !  sin %W  :4 =os %W %

sin%W C '% )' ! =os %W* -aka /iperoleh r-s Mohr R-s Lin+karan Mohr ntk te+an+an R-s 1-lah te+an+an

3'/56


'  4' C   4 (ontoh soal 8<' Se.ah ele-en -en+ala-i te+an+an  C '5&&& psi, 4 C 5&&& psi /an Q4 C 2&&& psi< 4 4' Hitn+ te+an+an!te+an+an Q4 ' 4an+ .eker1a pa/a .i/an+ Q'4' -irin+ W C 23o W C 23o   sin %W C sin :&o C ' Q'4' =os %W C =os :&o C & ' Q4 4'   4 C ''&&& psi 4 %  ! 4 C 3&&& psi ' C ''&&&  3&&& )&*  2&&& C '3&&& psi % :' C !3&&& )'*  2&&& )&* C !3&&& psi 4' /apat /iperoleh /ari r-s σ' /en+an -en++anti s/t W -en1a/i W  :&o 4' C

  4

% 4' C ''&&& !

!

 ! 4

=os %W

% 3&&& )&* !

! :4 sin %W 2&&& )'* C

8&&& psi

(ontoh 8<% Diketahi Lihat +a-.ar< 4 σ4' Ditan4akan Te+an+an!te+an+an pa/a s/t '3o< Q4  C !25 MPa σ4' '3o Pen4elesaian  Q'4' 4 C '%MPa σ' o 15  Q4 C !': MPa 25 MPa   4 σ' C !'8 MPa ': MPa % Q'4'  ! 4 σ4' '% MPa C !%: MPa % ' C

  4

% C !'8

4' C

  4

% C !'8

C

4 ! 

 

! !

 ! 4 % !%:  ! 4 % !%:

=os %W  :4 sin %W  &<955&0 

=os %W

!': 

!&<3 C !0%<5'3 Mpa )tekan*

! :4 sin %W

 &<955&0 !

!': 

!&<3 C !'<0930 Mpa )tekan*

40/56


'

41/56

4

%  !&<3  C %: Lin+karan Mohr ntk te+an+an< 4 Q4 W 

!': 

&<955 C !0&<:32 MPa Q! Q-a C R 4 4 Q4

 Q4

4'



& :'4'

4

'

Q4

' σ



41

-in

%W %φ %

 P 

'

Q

-a

Perhatikan, s-. : positip ke .awah (ara -en++a-.ar lin+karan Mohr ti/ak hars .erskala, 4an+ pentin+ lo+is<

:'4'

4'

Untk -en=ari te+an+an!te+an+an pa/a s-. ' ! 4' /apat -en++nakan lin+karan Mohr te+an+an< R-s!r-s 4an+ /i+nakan -en+a-.il r-s se.el- ini< ' C 4' C :' C

  4 %   4 !

 !

%  ! 4

 ! 4 %  ! 4 % sin %W

=os %W  :4 sin %W =os %W

! :4 sin %W

 :4 =os %W

% Pen+e-.an+ann4a a/alah )lihat +a-.ar*    P C  4 % -aka -a C P  R R C %

 ! 4 %

%

< : % 4

-in C P ! R :-a C R

Posisi -a /apat /iperoleh /en+an -en+hitn+ s/t %<

R-sn4a R sin % C Q4 Ata % C sin !' :4  R

S-. te+an+an -a!-in /iperoleh /en+an -en++eser s-. 4 ke kiri se.esar . S/t %φ /apat /iperoleh /en+an -en+ran+i s/t %W ! %


42/56

Maka ' C P  R =os %φ 4' C P ! R =os %φ :'4' C R sin %φ (ontoh!=ontoh soal< 8<' Se.ah ele-en -en+ala-i te+an+an  C '5&&& psi, 4 C 5&&& psi /an Q4 C 2&&& psi< 4 4' '    Q4 P C  4 C Q'4' % W C 23o   '5&&&  5&&& Q'4' ' % Q4 4' C ''&&& psi 4  ! 4 '5&&& ! 5&&& C C 3&&& psi Q! % % Q-a C R 4  !  % < R % C  4  :4% Q4 ' % R C 52&0<' 

& Q4

-in



41

P  Q

%W %φ %

-a

σ

% C sin !' Q4R C &<5828 ra/ C %W C :&o %φ C 3' o C &<9:5' ra/ ' C P  R =os %φ C 4' C P ! R =os %φ C :'4' C R sin %φ C

0:

o

'3&&& psi 8&&& psi !3&&& psi

(ontoh 8<% Diketahi Lihat +a-.ar< 4 σ4' Ditan4akan Te+an+an!te+an+an pa/a s/t '3o< Q4    σ4' '3o Pen4elesaian  Q'4' P C  4 C ! 25  '% σ' % % o 15  C !'8 25 MPa  ! 4 σ' ! 25 ! '% C ': MPa % % Q'4' σ4' C !%: '% MPa :! R C 02<58 % C sin !' Q4R C &<39& ra/ C 00 o 1 %W C 0&o  %φ 




%

%φ C ' C 4' C :'4' C

σ

4 4'

43/56

50 o C '<'&2 ra/ P ! R =os %φ C !0%<5' MPa P  R =os %φ C !'<0: MPa R sin %φ C !0&<:3 MPa

:

(ontoh 8!0 4 Q4

(ari /an +a-.arkan posisi te+an+an ta-a<    :! P C  4 C arah s-. ta-a %   !2&3& psi    ! 4  σ '%0&& psi % C 4 % 28&& psi !9%3& psi 2%&& psi R C :2:2<: % C sin !' Q4R C &<3'9 ra/ C %:<58 o :  C '2<92 o σ)ax = 3222<: psi σ)in = !'0323 psi :)ax = :2:2<: psi Hasil /i .k .er.e/a karena kelir soalV

Lin+karan Mohr ntk re+an+an< G4 '% ! ! '% -a ! '% 4 4 4'

GP C Grata!rata R % C ε



G-in

%W



%

'

%

Grata!rata G G-a G' ' 2

G  G4 %

 < 4 %

%

%

<

G' C GP  R =os )%  2W* % C sin !'

 '% 4 '% 

G ! G4

C

4

%R G-a C GP  R G-in C GP ! R -a C %R G  G G ! G  C  4   4 =os %W  4 sin %W % % % G ! G  C !  4 sin %W  4 =os %W % %

(ontoh soal 8!8 G4 C Diketahi G C 02&  '& !5 ''&  '& !5 Hitn+ a* re+an+an pa/a ele-en 4an+ .erorientasi 0&o

4 C

'9&  '& !5


44/56

.* re+an+an ta-a =* re+an+an +eser -ai-- , /an +a-.ar -asin+!-asin+ posisin4a G  G Jawa. GP C  4 C 02&  ''&  '& !5 C %%3  '& !5 % % G ! G4 02& ! ''& C  '& !5 C ''3  '& !5 % % G ! G4 %< 4 %< %  R C % % R C '25<&0  '& !5 '% ! ! '% -a ! '% 4

G4 % C sin !'

4

4 %R

C &<552 ra/ C 09<&3

o

' %φ



G-in

%W



ε

%

o

C &<090% ra/

G' C GP  R =os %φ C 05&<22  '& !5 G4' C GP ! R =os %φ C 9:<339  '& !5 '4' C %R sin %φ C !'&:<%  '& !5

 Grata!rata '% 4 '% 

%φ C %W ! % = %%

G G-a Re+an+an G-a /apat /iperoleh /en+an % =ara  ' s-.  /iptar ke kanan ata ke kiri< Besarn4a sa-a, han4a s/tn4a .e/a

4' ε4'

4 ε4

4

'4'

ε'

0&o

 ε

(ontoh 8!9 (

Se.ah Roset Re+an+an 23o ter/iri /ari 0 loa/ =ell 4an+ /iatr seperti pa/a +a-.ar< Teran+kan =ara ker1a alat ts.< 4' 0 alat kr ts. -en+kr Go, G23 /an G:& ' /i -ana Go C G /an G:& C G4

B

23o

A

> Maka

W G  G4

G ! G4

GW C 4

G  G4 %



G ! G4 %

=os %W 

4 %

sin %W


45/56

23

sehin++a 4 /apat /i=ari % % % 4 = %G23 ! Go ! G:& Jika 4 s/ah /i/apat, -aka lin+karan Mohr re+an+an /apat /i+a-.ar< Re+an+an pa/a setiap posisi /apat /i=ari< H.n+an re+an+an /en+an te+an+an< R-s Hooke ntk ' /era1at ke.e.asan /apat /iperlas sa-pai 0 /era1at ke.e.asan< Untk ' /era1at,  C GE /an Q C # Untk % /era1at ke.e.asan .erlak  ! 4  !  )  4* G C  G4 C 4 G7 C ! E E E σ C

/i -ana

E (ε  ε *  4 ' ! %

σ4 C

E (ε  ε * 4  ' ! %

E C -o/ls elastisitas tarik tekan # C -o/ls elastisitas +eser  C Poisson ratio H.n+an # /en+an E a/alah 

 C

Se/an+kan ntk 0 /era1at ke.e.asan   )   * G C ! 4 7 E E 7  )   * G7 C !  4 E E σ C

E %)'  * G4 C

4 E

!

 )   *  7 E

E )' ! *G  )G4  G7* )'  *)' ! %* E )' ! *G4  )G  G7* σ4 C )'  *)' ! %* E )' ! *G7  )G  G4* σ7 C )'  *)' ! %*

Latihan soal Ker1akan soal!soal no-or 8<%!3, 8<%!5 /an 8<%!8 8<%!'%, 8<%!'0 8<%!'2, 8<%!'3, 8<%!'5 8<8!'3, 8<8!'5 8<8!%&, 8<8!%', 8<8!%%

b

Ker1akan /en+an r-s /an /en+an lin+karan Mohr

LENDUTAN ! DELEKSI

8<' Pen/ahlan< Dala- peren=anaan -esin, te+an+an 4an+ ter1a/i -e-an+ ti/ak .oleh -ele.ihi te+an+an i7in, tetapi serin+ a/a ketentan lain 4an+ -alah le.ih pentin+, 4ait len/tan ata /e;leksi< Dala- peren=anaan alat!alat an+kat a/a ketentan .ahwa len/tan ran+ka ti/ak .oleh le.ih /ari ''&&& sa-pai '8&& pan1an+ ran+ka< De-ikian pla .atan+ lan+sin+ 4an+ -en+ala-i pe-.e.anan aksial, akan +a+al 1ika ter1a/i tekkan ).=klin+* pa/ahal .e.ann4a 1ah /i .awah .e.an i7in patah< 8<% Persa-aan /e;leksi< 4 P

M)* 

R A

R B

R A

R-s len/tann4a a/alah

Dia+ra- -o-en EI 4  C M)*

Se.a+ai a=an, lihat pe-.e.anan /i sa-pin+< Jika /ia+ra- M)* a/alah positip se.a+ai /i +a-.ar, R B -aka nilai len/tan ne+atip akan -elen/t ke .awah<

/i -ana E C Mo/ls elastisitas tarik I C -o-en inersia pena-pan+ 4  C trnan kle/a /ari len/tan 4 M)* C persa-aan -o-en lentr

Jika /inte+rasi sekali akan -en+hasilkan EI 4  C M)* /

/i -ana 4  C W C s/t ke-irin+an len/tan

Diinte+rasi la+i -en+hasilkan EI 4 C

M)* /%

/i -ana 4 C len/tan

(atatan R-s!r-s /i atas .erlak 1ika I /an E a/alah konstan ntk sepan1an+ .atan+< Jika I /an E ti/ak konstan, -aka hars -ask /ala- inte+ran< Karena /iinte+rasi % kali, -aka akan -n=l /a konstanta .e.as, sehin++a ntk -enentkan .esarn4a konstanta .e.as ts., /iperlkan % )/a* s4arat .atas< S4arat .atas ts. serin+ ata ha-pir selal ti/ak /ise.tkan se=ara eksplisit )teran+!teran+an*, tetapi se=ara i-plisit a/a ter=ant- /ala- soal ata +a-.ar< Misaln4a len/tan pa/a titik t-pan -en/atar C &, s/t W ke-irin+an len/tan C & pa/a t-pan 1epit -en/atar< Disa-pin+ it, 1ika a/a per.ahan r-s -o-en karena -elintasi )-en4e.eran+i* sat .e.an ata +a4a reaksi t-pan, -aka .erlak ketentan W' C W% , ata 4  kiri C 4  kanan ).atan+ ti/ak patah*, /an 4 kiri C 4 kanan ).atan+ ti/ak pts*<

46/56


Sk!sk /ala- r-s M)* hars sa-a satann4a, C satan +a4a  satan 1arak< (ontoh!=ontoh soal (ontoh :<' Batan+ AB pan1an+ L /i.e.ani .e.an -erata w )lihat +a-.ar* w (ari persa-aan len/tann4a, /an hitn+ len/tan -aksi--< Jawa. Den+an DBB /i=ari reaksi!reaksi t-pan< Diperoleh R A C R B C wL% A L B M)* C R A<)* ! w<)%* C wL%  ! w% % w w EI 4  C wL%  ! w% % EI 4  C wL2 % ! w5 0  =' % M)* R A R B R A EI 4 C wL'% 0 ! w%2 2  ='  =%  S4arat .atas  C&,4C& CL,4C&

=% C & EI 4 C wL'% 0 ! w%2 2  =' w'% L2 ! w%2 L2  ='L C & =' C ! w%2 L0

EI 4 C wL'% 0 ! w%2 2 ! w%2 L0  Karena si-etris, len/tan -ai-- ter1a/i /i  C L% EI 4-a C w:5 L2 ! w092 L2 ! w29 L2 C ! 3w092 L2 )tan/a -ins .erarti arahn4a ke .awah* Karena si-etris, -aka s4arat .atasn4a .isa .erta-.ah  C L% -aka 4  C & EI 4 L% C )wL2 % ! w5 0 ! w%2 L0*  C L% C w'5 L0 ! w29 L0 ! w%2 L0 C & .enar Wo C ! WL , s/t ke-irin+an /i  C & sa-a .esarn4a /en+an /i  C L, han4a .e/a tan/a< EI 4 & C )wL2 % ! w5 0 ! w%2 L0*  C & C ! w%2 L0 EI 4 L C )wL2 % ! w5 0 ! w%2 L0*  C L C w2 L0 ! w5 L0 ! w%2 L0 C w%2 L0 .enar (ontoh :<% w A

L wL

MA R A

Batan+ =antileer AB pan1an+ L /i.e.ani .e.an -erata w< Lihat +a-.ar< (ari persa-aan len/tann4a, /an hitn+ len/tan -ai--< R A C wL B Jawa. Den+an DBB /iperoleh MA C w% L% M)* C R A< ! MA ! w% % C wL ! w% L% ! w% % EI 4  C wL ! w% L% ! w% % EI 4  C wL% % ! w% L%  ! w5 0  ='

L%

w M

EI 4 C wL5 0 ! w2 L% % ! w%2 2  ='  =% S4arat .atas

4$/56


MA R A

C&,4C& C&,4C&

% 

4&/56

=% C & =' C &

EI 4 C wL5 0 ! w2 L% % ! w%2 2 (ontoh :<0 P a

.

A

L

&\\a R A



a\\L

Diketahi .atan+ se/erhana AB pan1an+ L /i.eri .e.an terpsat P Lihat +a-.ar< (ari persa-aan len/tann4a, /an =ari len/tan -a< R A C P.L B Jawa., /en+an DBB /iperoleh R B C PaL M)* C R A< C P.L  M EI 4  C P.L  EI 4  C P.%L %  =' EI 4 C P.5L 0  ='  =% M)* C R A< ! P)!a* C P.L  ! P) ! a*

P a

R A



S4arat .atas

M !a C& CL Ca Ca

EI 4  C P.L  ! P)!a* EI 4  C P.%L % ! P% )!a*%  =0 EI 4 C P.5L 0 ! P5 )!a*0  =0  =2 4C& 4C& 4 kiri C 4 kanan 4  kiri C 4  kanan

Ca

EI 4 a C )P.%L %  ='* C a C P.a%%L  =' EI 4 a C )P.%L % ! P% )!a*%  =0*Ca C P.a%%L  =0

Ca

EI 4a C )P.5L 0  ='  =%*Ca C P.a05L  ='a  =%

=' C =0 =% C =2

EI 4a C )P.5L 0 ! P5 )!a*0  ='  =2*Ca C P.a05L  ='a  =2 C& &\\a a\\L CL

EI 4o C )P.5L 0  ='  =%*C& C =% C & EI 4 C P.5L 0  =' EI 4 C P.5L 0 ! P5 )!a*0  ='

=% C =2 C &

EI 4L C )P.5L 0 ! P5 )!a*0  ='*CL C P.L%5 ! P5 )L!a*0  ='L C & C P.L%5 ! P.05  ='L C & =' C =0 C ! P.L5  P.05L

&\\a

EI 4 C P.5L 0 ! P.L5   P.05L  C

P.

%

%

%


4'/56

!

5L a\\L EI 4 C P.5L 0 ! P5 )!a*0 ! P.L5   P.05L  P. P)!a*0 C )% ! L%  .%* ! 5L 5 Persa-aan s/t ke-irin+an W )ra/* &\\a P. EI 4  C P.%L % ! P.L5  P.05L C )0% ! L%  .%* 5L EI 4  C P.%L % ! P% )!a*% ! P.L5  P.05L C a\\L P. P)!a*% % % % C )0 ! L  . * ! 5L % Len/tan -a< Jika a c . , -aka len/tan akan ter1a/i /i & \  \ a , pa/a EI 4  C & L% ! .% P. % % % )0 ! L  . * C &  C 0 5L (atatan, 1ika a ` . , -aka 1arak  .isa /ipan/an+ .er+erak /ari kanan, posisi a /an . /i.alik< Maskkan har+a ini /ala- persa-aan len/tan< Jika a C . ata .e.an P .eker1a /i ten+ah!ten+ah, -aka .erlak

EI 4 C

P )2% ! 0L%* 29

Meto/a sperposisi< Jika sat .atan+ -en+ala-i .e.erapa pe-.e.anan, -aka len/tan totaln4a akan sa-a /en+an hasil pen1-lahan /ari len/tan tiap .e.an< (ontoh P w w P C



w P )%L% ! 0 ! L0*  )2% ! 0L%* %2 29 Meto/a sperposisi .isa 1+a /iaplikasikan -en1a/i kass pen+ran+an< Den+an a/an4a -eto/a sperposisi ini -aka ta.el len/tan ntk kass!kass -- /apat /iaplikasikan< Jika a/a .atan+ oerhan+ ata =antileer /en+an .e.an ti/ak /i 1n+, -aka .erlak  P len/tan /i ( C len/tan /i B  L WB L F( C FB  L WB A B ( EI 4

C


Soal!soal latihan Ker1akan

:<0!'3, :<0!'8 , :<0!'9 , :<0!': :<3!' , :<3!% , :<3!5 , :<3!'0 , :<3!%'

50/56

STATIS TAK TENTU

Kea/aan statis tak tent a/alah kea/aan 4an+ .erle.ihan sehin++a -eni-.lkan .an4akn4a aria.el 4an+ ti/ak /iketahi -en1a/i le.ih .an4ak /aripa/a .an4akn4a persa-aan 4an+ terse/ia< (ontoh, .atan+ AB 4an+ seharsn4a =kp /it-p se=ara se/erhana /i A /an B, ata =kp /i1epit /i A sa1a, tern4ata -alah /i 1epit /i A /an /it-p 1+a /i B< P P P A B A B A B Maka 1ika se-la han4a a/a % aria.el 4an+ ti/ak /iketahi, 4ait R A /an R B, ata R A /an MA, -aka karena kon/isin4a -en1a/i 1epit /an t-p aria.eln4a -en1a/i R A, R B /an MA se-entara .an4akn4a persa-aan han4a %, 4ait kesei-.an+an +a4a /an kesei-.an+an -o-en< Maka perl /i=ari akal )trik* a+ar -asalah ts. /apat /ipe=ahkan< (ontoh '&<'

Se.ah .atan+ AB pan1an+ L /i1epit /i A /an /it-p /i B /i.eri .e.an -erata w persatan pan1an+< Den+an DBB /apat /iperoleh 0 reaksi R A, R B /an MA

w A

L

B wL

MA R A

L%

R A C wL ! R B

L%

R B

Karena han4a terse/ia % persa-aan, kesei-.an+an +a4a /an kesei-.an+an -o-en, -aka /ari 0 reaksi ts. hars a/a sat 4an+ /i1a/ikan re/n/an ata reaksi ta-.ahan, 4ait R B

MA C '% wL% ! R B L

M C R A  ! MA ! '% w% Mo-en lentr pa/a posisi  /ari 1n+ 1epit % % EI 4  C M C wL ! RB  ! wL  R B L ! w % % % % % EI 4  C '% wL ! '% R B  ! '% wL   R B L ! '5 w0  =' EI 4 C '5 wL0 ! '5 R B 0 ! '2 wL% %  '% R B L% ! '%2 w2  ='  =% Di-askkan s4arat .atas =% C & C& 4C& =' C & C& 4C& EI 4 C '5 wL0 ! '5 R B 0 ! '2 wL% %  '% R B L% ! '%2 w2 CL 4C&

'5 wL2 ! '5 R B L0 ! '2 wL2  '% R B L0 ! '%2 wL2 C & ! '5 wL2  '2 wL2  '%2 wL2 0wL R B C C ! '5 R B L0  '% R B L0 9 R A C wL ! R B C 39 wL

51/56

STATIS TAK TENTU

MA C '% wL% ! R B L C '9 wL% Setelah R A, R B /an MA /iketahi, -aka r-s +a4a +eser /an -o-en lentr /apat /ihitn+< EI 4 C '5 wL0 ! ''5 wL 0 ! '2 wL% %  0'5 wL%% ! '%2 w2 w% )3L ! 0L% ! %%* 29 Len/tan -ai-- /apat /i=ari /ari trnan perta-a, 4  C & 90 ! '3L%  5L% C & ata 9% ! '3L  5L% C & '3L d LZ)%%3 ! ':%* C '5  C )'3 ! Z00*'5 L C &<3893 L EI 4-a C !&<&&32'5'% wL C

329 wL0 ! ''5 wL% % ! '%2 w2 C

Jika /ipan/an+ se.a+ai sperposisi, -aka kea/aann4a a/alah seperti =antileer /en+an .e.an -erata w  =antileer /en+an .e.an terpsat R B /i 1n+ .e.as< R B w Pers4aratan ta-.ahan 4an+ ter1a/i  a/alah, len/tan /i B C &, ata A L B A L B len/tan /i B karena .e.an -erata  len/tan /i B karena .e.an terpsat hars salin+ -en+hilan+kan )sa-a*< Dari ta.el len/tan -aka '0 R BL0 ! '9 wL2 C & Untk .e.an ter/istri.si EI4-a C !'9 wL2 EI4-a C '0 R BL0 ata R B C 09 wL Untk .e.an terpsat Untk selan1tn4a /i.olehkan -en+a-.il /ata, -en4alin /ari .k, ke=ali /in4atakan lain< (ontoh '&<%

Batan+ AB pan1an+ L /i1epit /i 1n+!1n+ A /an B, /i.eri .e.an P /i ten+ah!ten+ah< (ari persa-aan len/tann4a< Karena kea/aann4a si-etris, -aka R A C R B C P%, MA C ! MB

P A

L

B P

MA R A

L%

L%

Kassn4a -erpakan pen1-lahan 0 -a=a- kea/aan Be.an terpsat P /i ten+ah  .e.an terpsat P% /i 1n+  -o-en MB /i 1n+<

MB Dari ta.el /iperoleh, len/tan /i 1n+ C R B EI4B C 329 PL0 !029 PL0  '% MBL% C & Be.an P EI4B C ! '5 PL0 MB C '9 PL Be.an P% Mo-en MB EI4B C '% MBL%

Maka persa-aan len/tann4a, ntk & `  ` L%  EI4 C ''% P% )% ! 0L*  ''% P% )0L ! *  ''5 PL % C '29 P% )2  0L* Latihan Ker1akan soal!soal /i atas 1ika +a4a terpsat P ti/ak .eker1a /i ten+ah!ten+ah, -elainkan

52/56

STATIS TAK TENTU

53/56

se1arak a /ari 1n+ A, /i -ana a  . C L Meto/a 0 Mo-en< w A IA R A LA

Misalkan a/a .atan+ pan1an+, /i -ana .a+ian ten+ahn4a /i.eri na-a A, B /an ( seperti pa/a +a-.ar, /en+an pe-.e.anan /an -o-en inersia -asin+!-asin+< Jika /ipoton+ /i A, B /an (, -aka akan ti-.l Mo-en MA, MB /an M(

P B

( IB LB

R (

R B MA

MB

A

MB

M(

B IA

R A

IB LA

R B R B

Misalkan karena pe-.e.anan ts. .atan+n4a -elen+kn+ seperti /i +a-.ar sehin++a -e-.entk s/t ke-irin+an WBA /an WB(

LB

R ( S/t ke-irin+an len/tan ts. /apat /i.a+i % WBA' a/alah /ise.a.kan oleh pe-.e.anan /an WBA WBC WBA% 4an+ /ise.a.kan oleh -o-en pe-oton+an< De-ikian pla WB( 4an+ /apat /i.a+i %, WB(' aki.at pe-.e.anan /an WB(% aki.at -o-en< Karena .atan+n4a ti/ak patah ata pts, -aka .erlak ketentan WBA C ! WB( Besarn4a s/t ke-irin+an len/tan aki.at -o-en ts. a/alah MALA MBLA MBLB M(LB WBA% C WB(% C   5EIA 0EIA 0EIB 5EIB

(atatan, -o/ls E sa-a<

Maka, s/t ke-irin+an totaln4a MALA MBLA WBA C WBA'  WBA% C WBA'   5EIA 0EIA WB( C WB('  WB(% C WB(' 

MBLB M(LB  0EIB 5EIB

Karena WBA C ! WB( -aka MA Jika IA C IB -aka

LA L L L  % MB A  B  M( B C ! 5E ) WBA' ! WB(' * IA IA IB IB

MA LA  % MB )LA  LB*  M( LB C ! 5EI )WBA'  WB('*

Ke-/ian 1ika LA C LB

MA  2 MB  M( C 5EI )! WBA' ! WB('* L

(ontoh '&<8 w A

0L2 B

(

P C wL D

Hitn+ reaksi!reaksi t-pan< Karena L, I /an E sa-a -aka .erlak r-s

STATIS TAK TENTU

L R A

L R (

Pa/a .entan+ AB( .erlak

Maka hasiln4a

MA  2 MB  M( C 5EI )! WBA' ! WB('* L

L

R B

54/56

R D

MA C & karena 1n+ A .e.as EI WBA' C '%5 wL0 lihat ta.el ntk .e.an ter/istri.si EI WB(' C & karena ti/ak a/a pe-.e.anan /i B(

% 0 2 MB  M( C ! 5EI wL C ! wL 2 L %2EI

Pa/a .entan+ B(D .erlak

EI W(B' C &

EI W(D' C

Pa. )L.* 5L

)_*

MD C & C

P)0L2*)L2*)LL2* 5L

0 % MB  2 M( C ! 5EI 3wL C ! '3 wL 52 L '%9 EI Dari persa-aan )_* /an )__* /iperoleh 

C

3PL% '%9

C

)__* MB C

2: wL% ! :5&

M( C

'' wL% ! %2&

Untk -en+hitn+ reaksi!reaksi t-pan /i+nakan DBB< R-sn4a a/alah R B C R BA'  R BA%  R B('  R B(% R BA' C reaksi /i B karena .e.an pa/a ras BA /i -ana R BA% C reaksi /i B karena -o-en /i t-pan A

C

MA M ! B LA LA

C

M( M ! B LB LB

R B(' C reaksi /i B karena .e.an pa/a ras B( R B(% C reaksi /i B karena -o-en /i t-pan ( R B C R BA' 

MA M M M ! B  R B('  ( ! B LA LA LB LB

Untk -en+hitn+ reaksi t-pan A, pra!pra a/a .atan+ AA /i se.elah kiri AB< w 2: wL% MA C & MA C & MB C ! A A B :5& R AA' C & R AB' C wL% % L L Maka R A C &  &  &  wL ! 2: wL C 20' wL R A R B % :5& L :5& w

C

3wL0 '%9

C

2: wL%

C

'' wL%

STATIS TAK TENTU

A

B

(

A

B

!

55/56

(

!

:5& %2& % % % R B C wL  &  2: wL  &  2: wL ! '' wL L L % :5& L :5& L %2& R A R B R ( R B C 302 wL :5& % % 0L2 wL MB C ! 2: wL M( C ! '' wL MDC & B ( D :5& %2& R (D' C wL2 L L 2: wL% '' wL% wL '' wL% R ( C & !   & R B R ( R D :5& L %2& L 2 %2& L R ( C %8: wL :5& % 0L2 wL M( C ! '' wL MDC & R D(' C 0wL ( D D %2& 2 % R D C 0wL ! '' wL  &  &  & C '5: wL L L 2 %2& L %2& R ( R D Pe-eriksaan R A  R B  R (  R D C 20'  302  %8:  '5: C % .enar :5& :5& :5& %2& Latihan Ker1akan /en+an =ara inte+rasi Ker1akan /en+an =ara sperposisi Ker1akan /en+an persa-aan 0 -o-en

'&<0!% '&<0!% '&<2!' '&<3!' '&<3!8

'&<0!3 '&<0!3 '&<2!% '&<3!% '&<3!'&

'&<0!9 '&<0!9 '&<2!0 '&<3!2 '&<3!'%

'&<0!'& '&<0!'& '&<2!5 '&<3!5 '&<3!'0

Tegangan Tarik, Tekan Dan Geser

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