CENTROID (AĞIRLIK MERKEZİ )
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2 CENTOD (ĞLK MEKEZİ ) centrod s a geometrcal concept arsng from parallel forces. Tus, onl parallel forces possess a centrod. Centrod s tougt of as te pont were te wole wegt of a pscal od or sstem of partcles s lumped. f proper geometrcal odes possess an as of smmetr, te centrod wll le on ts as. f te od possesses two or tree smmetr aes, ten te centrod wll e located at te ntersecton of tese aes. f one, two or tree dmensonal odes are defned as analtcal functons, te locatons of ter centrods can e calculated usng ntegrals.
3 composte od s one wc s comprsed of te comnaton of several smple odes. n suc odes, te centrod s calculated as follows: Lnea tn rod (Çg) reaa flat plate wt constant tcness (lan) Volumea spere or a cone (Hacm) 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 d d r d=rdqdr 1 d cosθ ρ d sn ρ d d sn ρ ρsnθρdρdθ d π/ π/ π/ q q q r q d d =rsnq =rcosq r q dq dr π π π d d
5 G r G
6 t s often necessar to calculate te moments of unforml dstruted loads aout an as lng wtn te plane te are appled to or perpendcular to ts plane. Generall, te magntudes of tese forces per unt area (pressure or stress) are proportonal to dstance of te lne acton of te force from te moment as. Te elemental force actng on an element of area, ten s proportonal to dstance tmes dfferental area, and te elemental moment s proportonal to dstance squared tmes dfferental area. Elemantar moment s proportonal to dstance dm=d d dfferental area: Tus, te total moment: dm=m=d d. Ts ntegral s named as rea Moment of nerta or Second Moment of rea.
7 Moment of nerta s not a pscal quantt suc as veloct, acceleraton or force, ut t enales ease of calculaton; t s a functon of te geometr of te area. Snce n Dnamcs tere s no suc concept as te nerta of an area, te moment of nerta as no pscal meanng. But n mecancs, moment of nerta s used n te calculaton of endng of a ar, torson of a saft and determnaton of te stresses n an cross secton of a macne element or an engneerng structure.
8 ectangular Moments of nerta = d = d nerta moment of area wt respect to as nerta moment of area wt respect to as Polar Moments of nerta o = =r d r = + o = = +
9 Product of nerta (Çarpım lan talet Moment) n certan prolems nvolvng unsmmetrcal cross sectons and n calculaton of moments of nerta aout rotated aes, an epresson d =d occurs, wc as te ntegrated form =d
10 Propertes of moments of nerta : 1. rea moments of nerta o,, are alwas postve.. ma e (), (+) or ero wenever eter of te reference aes s an as of smmetr, suc as te as n te fgure.. Te unt for all area moments of nerta s te. power of tat taen for lengt (L ).
11 . Te smallest value of an area moment of nerta tat an area can ave s realed wt respect to an as tat passes from te centrod of ts area. Te area moment of nerta of an area ncreases as te area goes furter from ts as. Te area moment of nerta wll get smaller wen te dstruton of an area gets closer to te as as possle.
12 Jrason (talet Elemsl) Yarıçapı Consder an area, wc as rectangular moment of nerta. We now vsuale ts area as concentrated nto a long narrow strp of area a dstance from te as. B defnton, te moment of nerta of te strp aout te as wll e te same as tat of te orgnal area f = Te dstance s called te radus of graton of te area aout te as. O O
13 ad of graton aout te and aes are otaned n te same manner. O O o o lso snce,
14 Te moment of nerta of an area aout a noncentrodal as ma e easl epressed n terms of te moment of nerta aout a parallel centrodal as. d e e G r de O r d Two ponts tat sould e noted n partcular aout te transfer of aes are: Te two transfer aes must e parallel to eac oter One of te aes must pass troug te centrod of te area
15 Te Parallels Teorems also old for rad of graton as: r de were s te radus of graton aout a centrodal as parallel to te as aout wc apples and r s te perpendcular dstance etween te two aes. For product of nerta: de
16 1) ECTNGLE G / / d d=d d d d d 1 d d d
17 1) ECTNGLE d G / / d=d d d d 1 e e e d
18 . TNGLE G / / d n n From smlart of te trangles, 1 d nd d 6 1 d n
19 . TNGLE / G / d m n a smlar manner, d 1
20 . SOLD CCLE G G r dr π r π dr r π rdr r rdr d d r o Due to smmetr; π
21 . SEM CCLE G O / d 8
22 5. QUTE CCLE G / / d e
23 n Mecancs t s often necessar to calculate te moments of nerta aout rotated aes. Te product of nerta s useful wen we need to calculate te moment of nerta of an area aout nclned aes. Ts consderaton leads drectl to te mportant prolem of determnng te aes aout wc te moment of nerta s a mamum and a mnmum.
24 sn q cos q cos q cos q sn q sn q Gven d, d, d We ws to determne moments and product of nerta wt respect to new aes and. Note: cosq snq cosq snq
25 tan q m 1 ma mn Te equaton for q m defnes two angles 9 o apart wc correspond to te prncpal aes of te area aout O. ma and mn are te prncpal moments of nerta. Te product of nerta s ero for te prncple aes of nerta.
CENTROID (AĞIRLIK MERKEZİ )
CENTOD (ĞLK MEKEZİ ) centrod s a geometrcal concept arsng from parallel forces. Tus, onl parallel forces possess a centrod. Centrod s tougt of as te pont were te wole wegt of a pscal od or sstem of partcles
CENTROID (AĞIRLIK MERKEZİ )
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.
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