CENTROID (AĞIRLIK MERKEZİ )
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2 CENTOD (ĞLK MEKEZİ ) centrod s geometrcl concept rsng from prllel forces. Tus, onl prllel forces possess centrod. Centrod s tougt of s te pont were te wole wegt of pscl od or sstem of prtcles s lumped. f proper geometrcl odes possess n s of smmetr, te centrod wll le on ts s. f te od possesses two or tree smmetr es, ten te centrod wll e locted t te ntersecton of tese es. f one, two or tree dmensonl odes re defned s nltcl functons, te loctons of ter centrods cn e clculted usng ntegrls.
3 composte od s one wc s comprsed of te comnton of severl smple odes. n suc odes, te centrod s clculted s follows: Lne tn rod (Çg) re flt plte wt constnt tckness (ln) Volume spere or cone (Hcm) Composte Composte Composte dl dl dl dl dl dl l l l l l l d d d d d d dv dv dv d dv dv V V V V V V
4
5 =rsnq dr dq d d r r d d=rdqdr ρ d ρsnθρdρdθ ρ snqdrdq snqdq q =rcosq ρ π/ cosθ 1 π π π/ π π/
6 G r
7 G
8 w= d / d d w= d
9 upper lower rgt left d d d d ld d d d / / / / / / lower upper d d / / / / d l = = 1/ 1/
10 upper lower rgt left d d wd d left rgt d w d d 6 = = /
11 t s often necessr to clculte te moments of unforml dstruted lods out n s lng wtn te plne te re ppled to or perpendculr to ts plne. Generll, te mgntudes of tese forces per unt re (pressure or stress) re proportonl to dstnce of te lne cton of te force from te moment s. Te elementl force ctng on n element of re, ten s proportonl to dstnce tmes dfferentl re, nd te elementl moment s proportonl to dstnce squred tmes dfferentl re. Elemntr moment s proportonl to dstnce dfferentl re: dm=d d Tus, te totl moment: dm=m=d d. Ts ntegrl s nmed s re Moment of nert or Second Moment of re.
12 Moment of nert s not pscl quntt suc s veloct, ccelerton or force, ut t enles ese of clculton; t s functon of te geometr of te re. Snce n Dnmcs tere s no suc concept s te nert of n re, te moment of nert s no pscl menng. But n mecncs, moment of nert s used n te clculton of endng of r, torson of sft nd determnton of te stresses n n cross secton of mcne element or n engneerng structure.
13 ectngulr Moments of nert = d = d nert moment of re wt respect to s nert moment of re wt respect to s Polr Moments of nert o = =r d r = + o = = +
14 Product of nert (Çrpım ln tlet Moment) n certn prolems nvolvng unsmmetrcl cross sectons nd n clculton of moments of nert out rotted es, n epresson d =d occurs, wc s te ntegrted form =d
15 Propertes of moments of nert : 1. re moments of nert o,, re lws postve.. m e (), (+) or ero wenever eter of te reference es s n s of smmetr, suc s te s n te fgure.. Te unt for ll re moments of nert s te. power of tt tken for lengt (L ).
16 Propertes of moments of nert :. Te smllest vlue of n re moment of nert tt n re cn ve s reled wt respect to n s tt psses from te centrod of ts re. Te re moment of nert of n re ncreses s te re goes furter from ts s. Te re moment of nert wll get smller wen te dstruton of n re gets closer to te s s possle.
17 Jrson (tlet Elemslk) Yrıçpı Consder n re, wc s rectngulr moment of nert. We now vsule ts re s concentrted nto long nrrow strp of re dstnce k from te s. B defnton, te moment of nert of te strp out te s wll e te sme s tt of te orgnl re f =k Te dstnce k s clled te rdus of grton of te re out te s. O O k k
18 d of grton out te nd es re otned n te sme mnner. k O O k k k k k k o o k k k lso snce,
19 G O d e r Te moment of nert of n re out noncentrodl s m e esl epressed n terms of te moment of nert out prllel centrodl s. de r e d
20 e d Te sum of tese two equtons gves r, nd snce G O d e r de For product of nert
21 Te Prllels teorems lso old for rd of grton s: k k r k k de were k s te rdus of grton out centrodl s prllel to te s out wc k pples nd r s te perpendculr dstnce etween te two es. For product of nert: k k de
22 Two ponts tt sould e noted n prtculr out te trnsfer of es re: Te two trnsfer es must e prllel to ec oter One of te es must pss troug te centrod of te re
23 1) ECTNGLE G / / d d=d d d d d 1 d d d
24 1) ECTNGLE d G / / d=d d d d 1 e e e d
25 . TNGLE G / / d n n From smlrt of te trngles, 1 d nd d 6 1 d n
26 . TNGLE / G / d m n smlr mnner, d 1
27 . SOLD CCLE G G r dr π r π dr r π rdr r rdr d d r o Due to smmetr; π
28 . SEM CCLE G O / d 8
29 5. QUTE CCLE G / / d e
30 n Mecncs t s often necessr to clculte te moments of nert out rotted es. Te product of nert s useful wen we need to clculte te moment of nert of n re out nclned es. Ts consderton leds drectl to te mportnt prolem of determnng te es out wc te moment of nert s mmum nd mnmum.
31 sn q cos q cosq cos q sn q sn q Gven d, d, d We ws to determne moments nd product of nert wt respect to new es nd. Note: cosq snq cosq snq
32 tn q m 1 m mn Te equton for q m defnes two ngles 9 o prt wc correspond to te prncpl es of te re out O. m nd mn re te prncpl moments of nert. Te product of nert s ero for te prncple es of nert.
33 5 mm 5 mm Smple Prolem: Determne te orentton of te prncpl es of te nert troug te centrod te ngle secton nd determne te correspondng mmum nd mnmum moments of nert. 1 mm.5 mm 1 mm.5 mm d G mm 7.5 mm 1 mm d G G d 1 G 1 d 1 mm 7.5 mm 1 mm For te complete secton; = mm Te product of nert out te es for prt s; mm (1) d 1 Te product of nert out te es for prt s; mm d (1) d d 1
34 5 mm d 1 mm.5 mm G d G d 1 G 1 d 1 mm 7.5 mm 1 mm Te moments of nert out te nd es for prt s; d e mm mm Te moments of nert out te nd es for prt s; d e mm mm For te complete secton; = mm = mm = mm tn q m ( 7.5) q m 1 o m mn 1 m.7 1 mm mn mm
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