Steel members come in a wide variety of shapes; the properties of the cross section are needed for analysis and design. (Bob Scott/Getty Images)

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1 Seel memers come in a wide varie of sapes; e properies of e cross secion are needed for analsis and design. (Bo Sco/Ge Images) 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

2 10 Review of enroids and Momens of Ineria HAPTER VERVIEW Topics covered in aper 10 include cenroids and ow o locae em (Secions 10. and 10.3), momens of ineria (Secion 10.4), parallelais eorems (Secion 10.5), polar momens of ineria (Secion 10.6), producs of ineria (Secion 10.7), roaion of aes (Secion 10.8), and principal aes (Secion 10.9). nl plane areas are considered. Tere are numerous eamples wiin e caper and prolems a e end of e caper availale for review. A ale of cenroids and momens of ineria for a varie of common geomeric sapes is given in Appendi D, availale online, for convenien reference. aper 10 is organized as follows: 10.1 Inroducion enroids of Plane Areas enroids of omposie Areas Momens of Ineria of Plane Areas Parallel-Ais Teorem for Momens of Ineria Polar Momens of Ineria Producs of Ineria Roaion of Aes Principal Aes and Principal Momens of Ineria 5 Prolems engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

3 4 aper 10 Review of enroids and Momens of Ineria 10.1 INTRDUTIN Tis caper is a review of e definiions and formulas peraining o cenroids and momens of ineria of plane areas. Te word review is appropriae ecause ese opics are usuall covered in earlier courses, suc as maemaics and engineering saics, and erefore mos readers will alread ave een eposed o e maerial. However, since cenroids and momens of ineria are used repeaedl rougou e preceding capers, e mus e clearl undersood e reader and e essenial definiions and formulas mus e readil accessile. Te erminolog used in is and earlier capers ma appear puzzling o some readers. For insance, e erm momen of ineria is clearl a misnomer wen referring o properies of an area, since no mass is involved. Even e word area is used inappropriael. Wen we sa plane area, we reall mean plane surface. Sricl speaking, area is a measure of e size of a surface and is no e same ing as e surface iself. In spie of is deficiencies, e erminolog used in is ook is so enrenced in e engineering lieraure a i rarel causes confusion. 10. ENTRIDS F PLANE AREAS Te posiion of e cenroid of a plane area is an imporan geomeric proper. To oain formulas for locaing cenroids, we will refer o Fig. 10-1, wic sows a plane area of irregular sape wi is cenroid a poin. Te coordinae ssem is oriened ariraril wi is origin a an poin. Te area of e geomeric figure is defined e following inegral: A da (10-1) in wic da is a differenial elemen of area aving coordinaes and (Fig. 10-1) and A is e oal area of e figure. Te firs momens of e area wi respec o e and aes are defined, respecivel, as follows: Q da Q da (10-a,) da FIG Plane area of arirar sape wi cenroid Tus, e firs momens represen e sums of e producs of e differenial areas and eir coordinaes. Firs momens ma e posiive or negaive, depending upon e posiion of e aes. Also, firs momens ave unis of leng raised o e ird power; for insance, mm 3. Te coordinaes and of e cenroid (Fig. 10-1) are equal o e firs momens divided e area: 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

4 Secion 10. enroids of Plane Areas 5 da Q A da Q da A da (10-3a,) FIG. 10- Area wi one ais of smmer FIG Area wi wo aes of smmer FIG Area a is smmeric aou a poin If e oundaries of e area are defined simple maemaical epressions, we can evaluae e inegrals appearing in Eqs. (10-3a) and (10-3) in closed form and ere oain formulas for and. Te formulas lised in Appendi D were oained in is manner. In general, e coordinaes and ma e posiive or negaive, depending upon e posiion of e cenroid wi respec o e reference aes. If an area is smmeric aou an ais, e cenroid mus lie on a ais ecause e firs momen aou an ais of smmer equals zero. For eample, e cenroid of e singl smmeric area sown in Fig. 10- mus lie on e ais, wic is e ais of smmer. Terefore, onl one coordinae mus e calculaed in order o locae e cenroid. If an area as wo aes of smmer, as illusraed in Fig. 10-3, e posiion of e cenroid can e deermined inspecion ecause i lies a e inersecion of e aes of smmer. An area of e pe sown in Fig is smmeric aou a poin. I as no aes of smmer, u ere is a poin (called e cener of smmer) suc a ever line drawn roug a poin conacs e area in a smmerical manner. Te cenroid of suc an area coincides wi e cener of smmer, and erefore e cenroid can e locaed inspecion. If an area as irregular oundaries no defined simple maemaical epressions, we can locae e cenroid numericall evaluaing e inegrals in Eqs. (10-3a) and (10-3). Te simples procedure is o divide e geomeric figure ino small finie elemens and replace e inegraions wi summaions. If we denoe e area of e i elemen A i, en e epressions for e summaions are A n i1 A i Q n i1 ia i Q n ia i i1 (10-4a,,c) in wic n is e oal numer of elemens, i is e coordinae of e cenroid of e i elemen, and i is e coordinae of e cenroid of e i elemen. Replacing e inegrals in Eqs. (10-3a) and (10-3) e corresponding summaions, we oain e following formulas for e coordinaes of e cenroid: Q A n i1 i A i n A i i1 Q A n i1 i A i n A i i1 (10-5a,) Te cenroid of wide-flange seel secions lies a e inersecion of e aes of smmer (Poo coures of Louis Gescwinder) Te accurac of e calculaions for and depends upon ow closel e seleced elemens fi e acual area. If e fi eacl, e resuls are eac. Man compuer programs for locaing cenroids use a numerical sceme similar o e one epressed Eqs. (10-5a) and (10-5). 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

5 6 aper 10 Review of enroids and Momens of Ineria Eample 10-1 A = f() da d FIG Eample enroid of a paraolic semisegmen B A paraolic semisegmen AB is ounded e ais, e ais, and a paraolic curve aving is vere a A (Fig. 10-5). Te equaion of e curve is f () 1 in wic is e ase and is e eig of e semisegmen. Locae e cenroid of e semisegmen. Soluion To deermine e coordinaes and of e cenroid (Fig. 10-5), we will use Eqs. (10-3a) and (10-3). We egin selecing an elemen of area da in e form of a in verical srip of wid d and eig. Te area of is differenial elemen is da d 1 d (a) () Terefore, e area of e paraolic semisegmen is A da 1 0 d 3 Noe a is area is /3 of e area of e surrounding recangle. Te firs momen of an elemen of area da wi respec o an ais is oained mulipling e area of e elemen e disance from is cenroid o e ais. Since e and coordinaes of e cenroid of e elemen sown in Fig are and /, respecivel, e firs momens of e elemen wi respec o e and aes are Q da 0 1 d 4 (d) 15 Q da 0 1 d (e) 4 in wic we ave susiued for da from Eq. (). We can now deermine e coordinaes of e cenroid : (c) Q 3 A 8 Q A 5 (f,g) Tese resuls agree wi e formulas lised in Appendi D, ase 17. Noes: Te cenroid of e paraolic semisegmen ma also e locaed aking e elemen of area da as a orizonal srip of eig d and wid 1 () Tis epression is oained solving Eq. (a) for in erms of. 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

6 Secion 10.3 enroids of omposie Areas ENTRIDS F MPSITE AREAS In engineering work we rarel need o locae cenroids inegraion, ecause e cenroids of common geomeric figures are alread known and aulaed. However, we frequenl need o locae e cenroids of areas composed of several pars, eac par aving a familiar geomeric sape, suc as a recangle or a circle. Eamples of suc composie areas are e cross secions of eams and columns, wic usuall consis of recangular elemens (for insance, see Figs. 10-, 10-3, and 10-4). Te areas and firs momens of composie areas ma e calculaed summing e corresponding properies of e componen pars. Le us assume a a composie area is divided ino a oal of n pars, and le us denoe e area of e i par as A i. Ten we can oain e area and firs momens e following summaions: A c (a) () A FIG enroid of a composie area consising of wo pars A n A i i1 Q n i1 i A i Q n i A i in wic i and i are e coordinaes of e cenroid of e i par. Te coordinaes of e cenroid of e composie area are Q A n i1 i A i n A i i1 Q A i1 n i1 (10-6a,,c) (10-7a,) Since e composie area is represened eacl e n pars, e preceding equaions give eac resuls for e coordinaes of e cenroid. To illusrae e use of Eqs. (10-7a) and (10-7), consider e L-saped area (or angle secion) sown in Fig. 10-6a. Tis area as side dimensions and c and ickness. Te area can e divided ino wo recangles of areas A 1 and A wi cenroids 1 and, respecivel (Fig. 10-6). Te areas and cenroidal coordinaes of ese wo pars are A ia i n A i i1 A (c ) c Terefore, e area and firs momens of e composie area (from Eqs. 10-6a,, and c) are A A 1 A ( c ) 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

7 8 aper 10 Review of enroids and Momens of Ineria Q 1A 1 A ( c ) Q 1A 1 A ( c ) Finall, we can oain e coordinaes and of e cenroid of e composie area (Fig. 10-6) from Eqs. (10-7a) and (10-7): Q c A ( c ) Q c (10-8a,) A ( c ) a f e (a) () FIG omposie areas wi a cuou and a ole g c d A similar procedure can e used for more comple areas, as illusraed in Eample 10-. Noe 1: Wen a composie area is divided ino onl wo pars, e cenroid of e enire area lies on e line joining e cenroids 1 and of e wo pars (as sown in Fig for e L-saped area). Noe : Wen using e formulas for composie areas (Eqs and 10-7), we can andle e asence of an area suracion. Tis procedure is useful wen ere are cuous or oles in e figure. For insance, consider e area sown in Fig. 10-7a. We can analze is figure as a composie area suracing e properies of e inner recangle efg from e corresponding properies of e ouer recangle acd. (From anoer viewpoin, we can ink of e ouer recangle as a posiive area and e inner recangle as a negaive area. ) Similarl, if an area as a ole (Fig. 10-7), we can surac e properies of e area of e ole from ose of e ouer recangle. (Again, e same effec is acieved if we rea e ouer recangle as a posiive area and e ole as a negaive area. ) uous in eams mus e considered in cenroid and momen of ineria calculaions (Don Farrall/Ge Images) 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

8 Secion 10.3 enroids of omposie Areas 9 Eample 10- Te cross secion of a seel eam is consruced of a HE 450 A wide-flange secion wi a 5 cm 1.5 cm cover plae welded o e op flange and a UPN 30 cannel secion welded o e oom flange (Fig. 10-8). Locae e cenroid of e cross-secional area. Plae 5 cm 1.5 cm 1 HE 450A 1 c 3 FIG Eample 10-. enroid of a composie area UPN 30 3 Soluion Le us denoe e areas of e cover plae, e wide-flange secion, and e cannel secion as areas A 1, A, and A 3, respecivel. Te cenroids of ese ree areas are laeled 1,, and 3, respecivel, in Fig Noe a e composie area as an ais of smmer, and erefore all cenroids lie on a ais. Te ree parial areas are A 1 (5 cm)(1.5 cm) 37.5 cm A 178 cm A cm in wic e areas A and A 3 are oained from Tales E-1 and E-3 of Appendi E. Le us place e origin of e and aes a e cenroid of e wide-flange secion. Ten e disances from e ais o e cenroids of e ree areas are as follows: 440 mm 15 mm mm 440 mm mm 46 mm coninued 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

9 10 aper 10 Review of enroids and Momens of Ineria in wic e perinen dimensions of e wide-flange and cannel secions are oained from Tales E-1 and E-3. Te area A and firs momen Q of e enire cross secion are oained from Eqs. (10-6a) and (10-6) as follows: A n A i A 1 A A 3 i cm 178 cm 75.8 cm 91.3 cm Q n i A i 1 A 1 A 3 A 3 i1 (.75 cm)(37.5 cm ) 0 (4.6 cm)(75.8 cm ) 101 cm 3 Now we can oain e coordinae o e cenroid of e composie area from Eq. (10-7): Q A 101 cm cm mm Since is posiive in e posiive direcion of e ais, e minus sign means a e cenroid of e composie area is locaed elow e ais, as sown in Fig Tus, e disance c eween e ais and e cenroid is c mm Noe a e posiion of e reference ais (e ais) is arirar; owever, in is eample we placed i roug e cenroid of e wide-flange secion ecause i sligl simplifies e calculaions. 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

10 Secion 10.4 Momens of Ineria of Plane Areas MMENTS F INERTIA F PLANE AREAS Te momens of ineria of a plane area (Fig. 10-9) wi respec o e and aes, respecivel, are defined e inegrals da I da I da (10-9a,) FIG Plane area of arirar sape in wic and are e coordinaes of e differenial elemen of area da. Because e elemen da is muliplied e square of e disance from e reference ais, momens of ineria are also called second momens of area. Also, we see a momens of ineria of areas (unlike firs momens) are alwas posiive quaniies. To illusrae ow momens of ineria are oained inegraion, we will consider a recangle aving wid and eig (Fig ). Te and aes ave eir origin a e cenroid. For convenience, we use a differenial elemen of area da in e form of a in orizonal srip of wid and eig d (erefore, da d). Since all pars of e elemenal srip are e same disance from e ais, we can epress e momen of ineria I wi respec o e ais as follows: I da / d / 1 3 (a) In a similar manner, we can use an elemen of area in e form of a verical srip wi area da d and oain e momen of ineria wi respec o e ais: I da / / 3 d 1 () B da d B If a differen se of aes is seleced, e momens of ineria will ave differen values. For insance, consider ais BB a e ase of e recangle (Fig ). If is ais is seleced as e reference, we mus define as e coordinae disance from a ais o e elemen of area da. Ten e calculaions for e momen of ineria ecome I BB da d (c) FIG Momens of ineria of a recangle Noe a e momen of ineria wi respec o ais BB is larger an e momen of ineria wi respec o e cenroidal ais. In general, e 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

11 1 aper 10 Review of enroids and Momens of Ineria 1 1 momen of ineria increases as e reference ais is moved parallel o iself farer from e cenroid. Te momen of ineria of a composie area wi respec o an paricular ais is e sum of e momens of ineria of is pars wi respec o a same ais. An eample is e ollow o secion sown in Fig a, were e and aes are aes of smmer roug e cenroid. Te momen of ineria I wi respec o e ais is equal o e algeraic sum of e momens of ineria of e ouer and inner recangles. (As eplained earlier, we can ink of e inner recangle as a negaive area and e ouer recangle as a posiive area. ) Terefore, (a) 3 I (d) 1 1 () FIG omposie areas Tis same formula applies o e cannel secion sown in Fig , were we ma consider e cuou as a negaive area. For e ollow o secion, we can use a similar ecnique o oain e momen of ineria I wi respec o e verical ais. However, in e case of e cannel secion, e deerminaion of e momen of ineria I requires e use of e parallel-ais eorem, wic is descried in e ne secion (Secion 10.5). Formulas for momens of ineria are lised in Appendi D. For sapes no sown, e momens of ineria can usuall e oained using e lised formulas in conjuncion wi e parallel-ais eorem. If an area is of suc irregular sape a is momens of ineria canno e oained in is manner, en we can use numerical meods. Te procedure is o divide e area ino small elemens of area A i, mulipl eac suc area e square of is disance from e reference ais, and en sum e producs. Radius of Graion A disance known as e radius of graion is occasionall encounered in mecanics. Radius of graion of a plane area is defined as e square roo of e momen of ineria of e area divided e area iself; us, r I A r I A (10-10a,) in wic r and r denoe e radii of graion wi respec o e and aes, respecivel. Since momen of ineria as unis of leng o e four power and area as unis of leng o e second power, radius of graion as unis of leng. Aloug e radius of graion of an area does no ave an ovious psical meaning, we ma consider i o e e disance (from e reference ais) a wic e enire area could e concenraed and sill ave e same momen of ineria as e original area. 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

12 Secion 10.4 Momens of Ineria of Plane Areas 13 Eample 10-3 A = f () Deermine e momens of ineria I and I for e paraolic semisegmen AB sown in Fig Te equaion of e paraolic oundar is da d FIG Eample Momens of ineria of a paraolic semisegmen B f() 1 (Tis same area was considered previousl in Eample 10-1.) Soluion To deermine e momens of ineria inegraion, we will use Eqs. (10-9a) and (10-9). Te differenial elemen of area da is seleced as a verical srip of wid d and eig, as sown in Fig Te area of is elemen is da d 1 d (e) (f) Since ever poin in is elemen is a e same disance from e ais, e momen of ineria of e elemen wi respec o e ais is da. Terefore, e momen of ineria of e enire area wi respec o e ais is oained as follows: I da 0 1 d = 1 5 To oain e momen of ineria wi respec o e ais, we noe a e differenial elemen of area da as a momen of ineria di wi respec o e ais equal o 3 (g) di 1 3 (d)3 d 3 3 as oained from Eq. (c). Hence, e momen of ineria of e enire area wi respec o e ais is I 0 3 d d Tese same resuls for I and I can e oained using an elemen in e form of a orizonal srip of area da d or using a recangular elemen of area da d d and performing a doule inegraion. Also, noe a e preceding formulas for I and I agree wi ose given in ase 17 of Appendi D. 3 () 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

13 14 aper 10 Review of enroids and Momens of Ineria 10.5 PARALLEL-AXIS THEREM FR MMENTS F INERTIA In is secion we will derive a ver useful eorem peraining o momens of ineria of plane areas. Known as e parallel-ais eorem, i gives e relaionsip eween e momen of ineria wi respec o a cenroidal ais and e momen of ineria wi respec o an parallel ais. To derive e eorem, we consider an area of arirar sape wi cenroid (Fig ). We also consider wo ses of coordinae aes: (1) e c c aes wi origin a e cenroid, and () a se of parallel aes wi origin a an poin. Te disances eween e wo ses of parallel aes are denoed d 1 and d. Also, we idenif an elemen of area da aving coordinaes and wi respec o e cenroidal aes. From e definiion of momen of ineria, we can wrie e following equaion for e momen of ineria I wi respec o e ais: I ( d 1 ) da da d 1dA d 1 da (a) Te firs inegral on e rig-and side is e momen of ineria I c wi respec o e c ais. Te second inegral is e firs momen of e area wi respec o e c ais (is inegral equals zero ecause e c ais passes roug e cenroid). Te ird inegral is e area A iself. Terefore, e preceding equaion reduces o I I c Ad 1 (10-11a) Proceeding in e same manner for e momen of ineria wi respec o e ais, we oain I I c Ad (10-11) c d da c d d 1 FIG Derivaion of parallel-ais eorem 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

14 Secion 10.5 Parallel-Ais Teorem for Momens of Ineria 15 Equaions (10-11a) and (10-11) represen e parallel-ais eorem for momens of ineria: B da d FIG Momens of ineria of a recangle (Repeaed) B Te momen of ineria of an area wi respec o an ais in is plane is equal o e momen of ineria wi respec o a parallel cenroidal ais plus e produc of e area and e square of e disance eween e wo aes. To illusrae e use of e eorem, consider again e recangle sown in Fig Knowing a e momen of ineria aou e ais, wic is roug e cenroid, is equal o 3 /1 (see Eq. a of Secion 10.4), we can deermine e momen of ineria I BB aou e ase of e recangle using e parallel-ais eorem: I BB I Ad Tis resul agrees wi e momen of ineria oained previousl inegraion (Eq. c of Secion 10.4). From e parallel-ais eorem, we see a e momen of ineria increases as e ais is moved parallel o iself farer from e cenroid. Terefore, e momen of ineria aou a cenroidal ais is e leas momen of ineria of an area (for a given direcion of e ais). Wen using e parallel-ais eorem, i is essenial o rememer a one of e wo parallel aes mus e a cenroidal ais. If i is necessar o find e momen of ineria I aou a noncenroidal ais - (Fig ) wen e momen of ineria I 1 aou anoer noncenroidal (and parallel) ais 1-1 is known, we mus appl e parallel-ais eorem wice. Firs, we find e cenroidal momen of ineria I c from e known momen of ineria I 1 : I c I 1 Ad 1 () Ten we find e momen of ineria I from e cenroidal momen of ineria: I I c Ad I 1 A(d d 1 ) (10-1) Tis equaion sows again a e momen of ineria increases wi increasing disance from e cenroid of e area. c c d d FIG Plane area wi wo parallel noncenroidal aes (aes 1-1 and -) 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

15 16 aper 10 Review of enroids and Momens of Ineria Eample 10-4 Te paraolic semisegmen AB sown in Fig as ase and eig. Using e parallel-ais eorem, deermine e momens of ineria I c and I c wi respec o e cenroidal aes c and c. c A FIG Eample Parallel-ais eorem B c Soluion We can use e parallel-ais eorem (raer an inegraion) o find e cenroidal momens of ineria ecause we alread know e area A, e cenroidal coordinaes and, and e momens of ineria I and I wi respec o e and aes. Tese quaniies were oained earlier in Eamples 10-1 and Te also are lised in ase 17 of Appendi D and are repeaed ere: A I I To oain e momen of ineria wi respec o e c ais, we use Eq. () and wrie e parallel-ais eorem as follows: I c I A (10-13a) In a similar manner, we oain e momen of ineria wi respec o e c ais: I c I A (10-13) Tus, we ave found e cenroidal momens of ineria of e semisegmen. 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

16 Secion 10.5 Parallel-Ais Teorem for Momens of Ineria 17 Eample 10-5 Plae 5 cm 1.5 cm HE 450A 1 Deermine e momen of ineria I c wi respec o e orizonal ais - roug e cenroid of e eam cross secion sown in Fig (Te posiion of e cenroid was deermined previousl in Eample 10- of Secion 10.3.) Noe: From eam eor (aper 5), we know a ais - is e neural ais for ending of is eam, and erefore e momen of ineria I c mus e deermined in order o calculae e sresses and deflecions of is eam. 1 UPN 30 c 3 Soluion We will deermine e momen of ineria I c wi respec o ais - appling e parallel-ais eorem o eac individual par of e composie area. Te area divides naurall ino ree pars: (1) e cover plae, () e wide-flange secion, and (3) e cannel secion. Te following areas and cenroidal disances were oained previousl in Eample 10-: 3 A cm A 178 cm A cm FIG Eample Momen of ineria of a composie area mm mm c mm Te momens of ineria of e ree pars wi respec o orizonal aes roug eir own cenroids 1,, and 3 are as follows: 3 I (5 cm)(1.5 cm) cm 4 I 6370 cm 4 I cm 4 Te momens of ineria I and I 3 are oained from Tales E-1 and E-3, respecivel, of Appendi E. Now we can use e parallel-ais eorem o calculae e momens of ineria aou ais - for eac of e ree pars of e composie area: (I c ) 1 I 1 A 1 ( 1 c) cm 4 (37.5 cm )(6. cm) 5790 cm 4 (I c ) I A c 6370 cm 4 (178 cm )(34.73 cm) cm 4 (I c ) 3 I 3 A 3 ( 3 c) 597 cm 4 (75.8 cm )(1.13 cm) cm 4 Te sum of ese individual momens of ineria gives e momen of ineria of e enire cross-secional area aou is cenroidal ais -: I c (I c ) 1 (I c ) (I c ) cm 4 Tis eample sows ow o calculae momens of ineria of composie areas using e parallel-ais eorem. 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

17 18 aper 10 Review of enroids and Momens of Ineria 10.6 PLAR MMENTS F INERTIA r da Te momens of ineria discussed in e preceding secions are defined wi respec o aes ling in e plane of e area iself, suc as e and aes in Fig Now we will consider an ais perpendicular o e plane of e area and inersecing e plane a e origin. Te momen of ineria wi respec o is perpendicular ais is called e polar momen of ineria and is denoed e smol I P. Te polar momen of ineria wi respec o an ais roug perpendicular o e plane of e figure is defined e inegral FIG Plane area of arirar sape I P r da (10-14) in wic r is e disance from poin o e differenial elemen of area da (Fig ). Tis inegral is similar in form o ose for momens of ineria I and I (see Eqs. 10-9a and 10-9). Inasmuc as r, were and are e recangular coordinaes of e elemen da, we oain e following epression for I P : I P r da ( )da da da Tus, we oain e imporan relaionsip I P I I (10-15) d d c da d 1 FIG Derivaion of parallel-ais eorem (Repeaed) c Tis equaion sows a e polar momen of ineria wi respec o an ais perpendicular o e plane of e figure a an poin is equal o e sum of e momens of ineria wi respec o an wo perpendicular aes and passing roug a same poin and ling in e plane of e figure. For convenience, we usuall refer o I P simpl as e polar momen of ineria wi respec o poin, wiou menioning a e ais is perpendicular o e plane of e figure. Also, o disinguis em from polar momens of ineria, we someimes refer o I and I as recangular momens of ineria. Polar momens of ineria wi respec o various poins in e plane of an area are relaed e parallel-ais eorem for polar momens of ineria. We can derive is eorem referring again o Fig Le us denoe e polar momens of ineria wi respec o e origin and e cenroid (I P ) and (I P ), respecivel. Ten, using Eq. (10-15), we can wrie e following equaions: (I P ) I I (I P ) I c I c (a) Now refer o e parallel-ais eorems derived in Secion 10.5 for recangular momens of ineria (Eqs a and 10-11). Adding ose wo equaions, we ge I I I c I c A(d 1 d ) 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

18 Secion 10.6 Polar Momens of Ineria 19 Susiuing from Eqs. (a), and also noing a d d 1 d (Fig ), we oain (I P ) (I P ) Ad (10-16) Tis equaion represens e parallel-ais eorem for polar momens of ineria: Te polar momen of ineria of an area wi respec o an poin in is plane is equal o e polar momen of ineria wi respec o e cenroid plus e produc of e area and e square of e disance eween poins and. To illusrae e deerminaion of polar momens of ineria and e use of e parallel-ais eorem, consider a circle of radius r (Fig ). Le us ake a differenial elemen of area da in e form of a in ring of radius r and ickness dr (us, da pr dr). Since ever poin in e elemen is a e same disance r from e cener of e circle, e polar momen of ineria of e enire circle wi respec o e cener is r (I P ) r da pr 3 dr p r 4 (10-17) 0 Tis resul is lised in ase 9 of Appendi E. Te polar momen of ineria of e circle wi respec o an poin B on is circumference (Fig ) can e oained from e parallel-ais eorem: (I P ) B (I P ) Ad p r 4 p r (r ) 3p (10-18) As an incidenal maer, noe a e polar momen of ineria as is smalles value wen e reference poin is e cenroid of e area. A circle is a special case in wic e polar momen of ineria can e deermined inegraion. However, mos of e sapes encounered in engineering work do no lend emselves o is ecnique. Insead, polar momens of ineria are usuall oained summing e recangular momens of ineria for wo perpendicular aes (Eq ). r 4 r r dr FIG Polar momen of ineria of a circle da B 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

19 0 aper 10 Review of enroids and Momens of Ineria 10.7 PRDUTS F INERTIA Te produc of ineria of a plane area is defined wi respec o a se of perpendicular aes ling in e plane of e area. Tus, referring o e area sown in Fig , we define e produc of ineria wi respec o e and aes as follows: da I da (10-19) FIG Plane area of arirar sape da da FIG Te produc of ineria equals zero wen one ais is an ais of smmer d c da d 1 FIG Plane area of arirar sape c From is definiion we see a eac differenial elemen of area da is muliplied e produc of is coordinaes. As a consequence, producs of ineria ma e posiive, negaive, or zero, depending upon e posiion of e aes wi respec o e area. If e area lies enirel in e firs quadran of e aes (as in Fig ), en e produc of ineria is posiive ecause ever elemen da as posiive coordinaes and. If e area lies enirel in e second quadran, e produc of ineria is negaive ecause ever elemen as a posiive coordinae and a negaive coordinae. Similarl, areas enirel wiin e ird and four quadrans ave posiive and negaive producs of ineria, respecivel. Wen e area is locaed in more an one quadran, e sign of e produc of ineria depends upon e disriuion of e area wiin e quadrans. A special case arises wen one of e aes is an ais of smmer of e area. For insance, consider e area sown in Fig. 10-0, wic is smmeric aou e ais. For ever elemen da aving coordinaes and, ere eiss an equal and smmericall locaed elemen da aving e same coordinae u an coordinae of opposie sign. Terefore, e producs da cancel eac oer and e inegral in Eq. (10-19) vanises. Tus, e produc of ineria of an area is zero wi respec o an pair of aes in wic a leas one ais is an ais of smmer of e area. As eamples of e preceding rule, e produc of ineria I equals zero for e areas sown in Figs , 10-11, 10-16, and In conras, e produc of ineria I as a posiive nonzero value for e area sown in Fig (Tese oservaions are valid for producs of ineria wi respec o e paricular aes sown in e figures. If e aes are sifed o anoer posiion, e produc of ineria ma cange.) Producs of ineria of an area wi respec o parallel ses of aes are relaed a parallel-ais eorem a is analogous o e corresponding eorems for recangular momens of ineria and polar momens of ineria. To oain is eorem, consider e area sown in Fig. 10-1, wic as cenroid and cenroidal c c aes. Te 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

20 Secion 10.7 Producs of Ineria 1 produc of ineria I wi respec o an oer se of aes, parallel o e c c aes, is I ( d )( d 1 )da da d 1 da d da d 1 d da in wic d 1 and d are e coordinaes of e cenroid wi respec o e aes (us, d 1 and d ma ave posiive or negaive values). Te firs inegral in e las epression is e produc of ineria I c c wi respec o e cenroidal aes; e second and ird inegrals equal zero ecause e are e firs momens of e area wi respec o e cenroidal aes; and e las inegral is e area A. Terefore, e preceding equaion reduces o I I c c Ad 1 d (10-0) Tis equaion represens e parallel-ais eorem for producs of ineria: Te produc of ineria of an area wi respec o an pair of aes in is plane is equal o e produc of ineria wi respec o parallel cenroidal aes plus e produc of e area and e coordinaes of e cenroid wi respec o e pair of aes. To demonsrae e use of is parallel-ais eorem, le us deermine e produc of ineria of a recangle wi respec o aes aving eir origin a poin a e lower lef-and corner of e recangle (Fig. 10-). Te produc of ineria wi respec o e cenroidal c c aes is zero ecause of smmer. Also, e coordinaes of e cenroid wi respec o e aes are d 1 d Susiuing ino Eq. (10-0), we oain c c FIG. 10- Parallel-ais eorem for producs of ineria B I I c c Ad 1 d 0 (10-1) 4 Tis produc of ineria is posiive ecause e enire area lies in e firs quadran. If e aes are ranslaed orizonall so a e origin moves o poin B a e lower rig-and corner of e recangle (Fig. 10-), e enire area lies in e second quadran and e produc of ineria ecomes /4. Te following eample also illusraes e use of e parallel-ais eorem for producs of ineria. 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

21 aper 10 Review of enroids and Momens of Ineria Eample 10-6 Deermine e produc of ineria I of e Z-secion sown in Fig Te secion as wid, eig, and consan ickness. A 1 A A 3 FIG Eample Produc of ineria of a Z-secion Soluion To oain e produc of ineria wi respec o e aes roug e cenroid, we divide e area ino ree pars and use e parallel-ais eorem. Te pars are as follows: (1) a recangle of wid and ickness in e upper flange, () a similar recangle in e lower flange, and (3) a we recangle wi eig and ickness. Te produc of ineria of e we recangle wi respec o e aes is zero (from smmer). Te produc of ineria (I ) 1 of e upper flange recangle (wi respec o e aes) is deermined using e parallel-ais eorem: (I ) 1 I c c Ad 1 d (a) in wic I c c is e produc of ineria of e recangle wi respec o is own cenroid, A is e area of e recangle, d 1 is e coordinae of e cenroid of e recangle, and d is e coordinae of e cenroid of e recangle. Tus, I c c 0 A ( )() d 1 d Susiuing ino Eq. (a), we oain e produc of ineria of e recangle in e upper flange: (I ) 1 I c c Ad 1 d 0 ( )() ( )( ) 4 Te produc of ineria of e recangle in e lower flange is e same. Terefore, e produc of ineria of e enire Z-secion is wice (I ) 1, or I ( )( ) (10-) Noe a is produc of ineria is posiive ecause e flanges lie in e firs and ird quadrans. 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

22 Secion 10.8 Roaion of Aes RTATIN F AXES Te momens of ineria of a plane area depend upon e posiion of e origin and e orienaion of e reference aes. For a given origin, e momens and produc of ineria var as e aes are roaed aou a origin. Te manner in wic e var, and e magniudes of e maimum and minimum values, are discussed in is and e following secion. Le us consider e plane area sown in Fig. 10-4, and le us assume a e aes are a pair of ariraril locaed reference aes. Te momens and producs of ineria wi respec o ose aes are I da I da I da (a,,c) in wic and are e coordinaes of a differenial elemen of area da. Te 1 1 aes ave e same origin as e aes u are roaed roug a counerclockwise angle u wi respec o ose aes. Te momens and produc of ineria wi respec o e 1 1 aes are denoed I 1, I 1, and I 1 1, respecivel. To oain ese quaniies, we need e coordinaes of e elemen of area da wi respec o e 1 1 aes. Tese coordinaes ma e epressed in erms of e coordinaes and e angle u geomer, as follows: 1 cos u sin u 1 cos u sin u (10-3a,) Ten e momen of ineria wi respec o e 1 ais is I 1 1 da ( cos u sin u) da cos u da sin u da sin u cos u da or, using Eqs. (a), (), and (c), I 1 I cos u I sin u I sin u cosu (10-4) Now we inroduce e following rigonomeric ideniies: cos u 1 (1 cos u) sin u 1 (1 cos u) sinucos u sin u 1 u da FIG Roaion of aes 1 1 u 1 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

23 4 aper 10 Review of enroids and Momens of Ineria Ten Eq. (10-4) ecomes I 1 I I I I cos u I sin u (10-5) In a similar manner, we can oain e produc of ineria wi respec o e 1 1 aes: I da 5( cos u sin u)( cos u sin u)da (I I )sin u cos u I (cos u sin u) (10-6) Again using e rigonomeric ideniies, we oain I 1 1 I I sin u I cos u (10-7) Equaions (10-5) and (10-7) give e momen of ineria I 1 and e produc of ineria I 1 1 wi respec o e roaed aes in erms of e momens and produc of ineria for e original aes. Tese equaions are called e ransformaion equaions for momens and producs of ineria. Noe a ese ransformaion equaions ave e same form as e ransformaion equaions for plane sress (Eqs. 6-4a and 6-4 of Secion 6.). Upon comparing e wo ses of equaions, we see a I 1 corresponds o s 1, I 1 1 corresponds o 1 1, I corresponds o s, I corresponds o s, and I corresponds o. Terefore, we can also analze momens and producs of ineria e use of Mor s circle (see Secion 6.4). Te momen of ineria I 1 ma e oained e same procedure a we used for finding I 1 and I 1 1. However, a simpler procedure is o replace u wi u 90 in Eq. (10-5). Te resul is I 1 I I I I cos u I sin u (10-8) Tis equaion sows ow e momen of ineria I 1 varies as e aes are roaed aou e origin. A useful equaion relaed o momens of ineria is oained aking e sum of I 1 and I 1 (Eqs and 10-8). Te resul is I 1 I 1 I I (10-9) Tis equaion sows a e sum of e momens of ineria wi respec o a pair of aes remains consan as e aes are roaed aou e origin. Tis sum is e polar momen of ineria of e area wi respec o e origin. Noe a Eq. (10-9) is analogous o Eq. (6-6) for sresses. 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

24 Secion 10.9 Principal Aes and Principal Momens of Ineria PRINIPAL AXES AND PRINIPAL MMENTS F INERTIA Te ransformaion equaions for momens and producs of ineria (Eqs. 10-5, 10-7, and 10-8) sow ow e momens and producs of ineria var as e angle of roaion u varies. f special ineres are e maimum and minimum values of e momen of ineria. Tese values are known as e principal momens of ineria, and e corresponding aes are known as principal aes. Principal Aes To find e values of e angle u a make e momen of ineria I 1 a maimum or a minimum, we ake e derivaive wi respec o u of e epression on e rig-and side of Eq. (10-5) and se i equal o zero: (I I )sin u I cos u 0 (a) Solving for u from is equaion, we ge I an u p I I (10-30) in wic u p denoes e angle defining a principal ais. Tis same resul is oained if we ake e derivaive of I 1 (Eq. 10-8). Equaion (10-30) ields wo values of e angle u p in e range from 0 o 360 ; ese values differ 180. Te corresponding values of u p differ 90 and define e wo perpendicular principal aes. ne of ese aes corresponds o e maimum momen of ineria and e oer corresponds o e minimum momen of ineria. Now le us eamine e variaion in e produc of ineria I 1 1 as u canges (see Eq. 10-7). If u 0, we ge I 1 1 I, as epeced. If u 90, we oain I 1 1 I. Tus, during a 90 roaion e produc of ineria canges sign, wic means a for an inermediae orienaion of e aes, e produc of ineria mus equal zero. To deermine is orienaion, we se I 1 1 (Eq. 10-7) equal o zero: (I I )sin u I cos u 0 Tis equaion is e same as Eq. (a), wic defines e angle u p o e principal aes. Terefore, we conclude a e produc of ineria is zero for e principal aes. In Secion 10.7 we sowed a e produc of ineria of an area wi respec o a pair of aes equals zero if a leas one of e aes is an ais of smmer. I follows a if an area as an ais of smmer, a ais and an ais perpendicular o i consiue a se of principal aes. Te preceding oservaions ma e summarized as follows: (1) principal aes roug an origin are a pair of orogonal aes for wic e momens of ineria are a maimum and a minimum; () e orienaion of e principal aes is given e angle u p oained from Eq. (10-30); (3) e produc of ineria is zero for principal aes; and (4) an ais of smmer is alwas a principal ais. 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

25 6 aper 10 Review of enroids and Momens of Ineria FIG Recangle for wic ever ais (in e plane of e area) roug poin is a principal ais (a) () FIG Eamples of areas for wic ever cenroidal ais is a principal ais and e cenroid is a principal poin Principal Poins Now consider a pair of principal aes wi origin a a given poin. If ere eiss a differen pair of principal aes roug a same poin, en ever pair of aes roug a poin is a se of principal aes. Furermore, e momen of ineria mus e consan as e angle u is varied. Te preceding conclusions follow from e naure of e ransformaion equaion for I 1 (Eq. 10-5). Because is equaion conains rigonomeric funcions of e angle u, ere is one maimum value and one minimum value of I 1 as u varies roug a range of 360 (or as u varies roug a range of 180 ). If a second maimum eiss, en e onl possiili is a I 1 remains consan, wic means a ever pair of aes is a se of principal aes and all momens of ineria are e same. A poin so locaed a ever ais roug e poin is a principal ais, and ence e momens of ineria are e same for all aes roug e poin, is called a principal poin. An illusraion of is siuaion is e recangle of wid and eig sown in Fig Te aes, wi origin a poin, are principal aes of e recangle ecause e ais is an ais of smmer. Te aes, wi e same origin, are also principal aes ecause e produc of ineria I equals zero (ecause e riangles are smmericall locaed wi respec o e and aes). I follows a ever pair of aes roug is a se of principal aes and ever momen of ineria is e same (and equal o 4 /3). Terefore, poin is a principal poin for e recangle. (A second principal poin is locaed were e ais inersecs e upper side of e recangle.) A useful corollar of e conceps descried in e preceding four paragraps applies o aes roug e cenroid of an area. onsider an area aving wo differen pairs of cenroidal aes suc a a leas one ais in eac pair is an ais of smmer. In oer words, ere eis wo differen aes of smmer a are no perpendicular o eac oer. Ten i follows a e cenroid is a principal poin. Two eamples, a square and an equilaeral riangle, are sown in Fig In eac case e aes are principal cenroidal aes ecause eir origin is a e cenroid and a leas one of e wo aes is an ais of smmer. In addiion, a second pair of cenroidal aes (e aes) as a leas one ais of smmer. I follows a o e and aes are principal aes. Terefore, ever ais roug e cenroid is a principal ais, and ever suc ais as e same momen of ineria. If an area as ree differen aes of smmer, even if wo of em are perpendicular, e condiions descried in e preceding paragrap are auomaicall fulfilled. Terefore, if an area as ree or more aes of smmer, e cenroid is a principal poin and ever ais roug e cenroid is a principal ais and as e same momen of ineria. Tese condiions are fulfilled for a circle, for all regular polgons (equilaeral riangle, square, regular penagon, regular eagon, and so on), and for man oer smmeric sapes. In general, ever plane area as wo principal poins. Tese poins lie equidisan from e cenroid on e principal cenroidal ais aving e 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

26 Secion 10.9 Principal Aes and Principal Momens of Ineria 7 larger principal momen of ineria. A special case occurs wen e wo principal cenroidal momens of ineria are equal; en e wo principal poins merge a e cenroid, wic ecomes e sole principal poin. R = I I I u p I I I FIG Geomeric represenaion of Eq. (10-30) Principal Momens of Ineria Le us now deermine e principal momens of ineria, assuming a I, I, and I are known. ne meod is o deermine e wo values of u p (differing 90 ) from Eq. (10-30) and en susiue ese values ino Eq. (10-5) for I 1. Te resuling wo values are e principal momens of ineria, denoed I 1 and I. Te advanage of is meod is a we know wic of e wo principal angles u p corresponds o eac principal momen of ineria. I is also possile o oain general formulas for e principal momens of ineria. We noe from Eq. (10-30) and Fig (wic is a geomeric represenaion of Eq ) a in wic cos u p I I R R I sin u p I R (10-31a,) I I (10-3) is e poenuse of e riangle. Wen evaluaing R, we alwas ake e posiive square roo. Now we susiue e epressions for cos u p and sin u p (from Eqs a and ) ino Eq. (10-5) for I 1 and oain e algeraicall larger of e wo principal momens of ineria, denoed e smol I 1 : I 1 I I I I I (10-33a) Te smaller principal momen of ineria, denoed as I, ma e oained from e equaion I 1 I I I (see Eq. 10-9). Susiuing e epression for I 1 ino is equaion and solving for I, we ge I I I I I I (10-33) Equaions (10-33a) and (10-33) provide a convenien wa o calculae e principal momens of ineria. Te following eample illusraes e meod for locaing e principal aes and deermining e principal momens of ineria. 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

27 8 aper 10 Review of enroids and Momens of Ineria Eample Deermine e orienaions of e principal cenroidal aes and e magniudes of e principal cenroidal momens of ineria for e cross-secional area of e Z-secion sown in Fig Use e following numerical daa: eig 00 mm, wid 90 mm, and consan ickness 15 mm. FIG Eample Principal aes and principal momens of ineria for a Z-secion u 1 p Soluion Le us use e aes (Fig. 10-8) as e reference aes roug e cenroid. Te momens and produc of ineria wi respec o ese aes can e oained dividing e area ino ree recangles and using e parallel-ais eorems. Te resuls of suc calculaions are as follows: I mm 4 I mm 4 I mm 4 Susiuing ese values ino e equaion for e angle u p (Eq ), we ge I an u p u p 38.4 and 18.4 I I Tus, e wo values of u p are u p 19. and 109. Using ese values of u p in e ransformaion equaion for I 1 (Eq. 10-5), we find I mm 4 and mm 4, respecivel. Tese same values are oained if we susiue ino Eqs. (10-33a) and (10-33). Tus, e principal momens of ineria and e angles o e corresponding principal aes are I mm 4 u p1 19. I mm 4 u p 109. Te principal aes are sown in Fig as e l 1 aes. 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

28 HAPTER 10 Prolems 9 PRBLEMS HAPTER 10 enroids of Areas Te prolems for Secion 10. are o e solved inegraion Deermine e disances and o e cenroid of a rig riangle aving ase and aliude (see ase 6, Appendi D, availale online) Deermine e disance o e cenroid of a rapezoid aving ases a and and aliude (see ase 8, Appendi D, availale online) Deermine e disance o e cenroid of a semicircle of radius r (see ase 10, Appendi D, availale online) Deermine e disances and o e cenroid of a paraolic spandrel of ase and eig (see ase 18, Appendi D, availale online) Deermine e disances and o e cenroid of a semisegmen of n degree aving ase and eig (see ase 19, Appendi D, availale online). enroids of omposie Areas Te prolems for Secion 10.3 are o e solved using e formulas for composie areas Deermine e disance o e cenroid of a rapezoid aving ases a and and aliude (see ase 8, Appendi D, availale online) dividing e rapezoid ino wo riangles ne quarer of a square of side a is removed (see figure). Wa are e coordinaes and of e cenroid of e remaining area? alculae e disance o e cenroid of e cannel secion sown in e figure if a 150 mm, 5 mm, and c 50 mm. B PRBS , , and c Wa mus e e relaionsip eween e dimensions a,, and c of e cannel secion sown in e figure in order a e cenroid will lie on line BB? Te cross secion of a eam consruced of a HE 600 B wide-flange secion wi an 00 mm 0 mm cover plae welded o e op flange is sown in e figure. Deermine e disance from e ase of e eam o e cenroid of e cross-secional area. a a B Plae 00 mm 0 mm a a HE 600B a a PRBS and PRBS and engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

29 30 aper 10 Review of enroids and Momens of Ineria Deermine e disance o e cenroid of e composie area sown in e figure. 170 mm 30 mm 30 mm 180 mm 180 mm 105 mm 15 mm 30 mm 30 mm 90 mm 90 mm PRB mm 80 mm 50 mm 50 mm 80 mm 300 mm 80 mm 150 mm 80 mm 10 mm PRBS , , and Deermine e coordinaes and of e cenroid of e L-saped area sown in e figure. Momens of Ineria Prolems roug are o e solved inegraion Deermine e momen of ineria I of a riangle of ase and aliude wi respec o is ase (see ase 4, Appendi D, availale online) Deermine e momen of ineria I BB of a rapezoid aving ases a and and aliude wi respec o is ase (see ase 8, Appendi D, availale online). 1 mm 150 mm 1 mm 100 mm PRBS , , , and Deermine e momen of ineria I of a paraolic spandrel of ase and eig wi respec o is ase (see ase 18, Appendi D, availale online) Deermine e momen of ineria I of a circle of radius r wi respec o a diameer (see ase 9, Appendi D, availale online). Prolems roug are o e solved considering e area o e a composie area Deermine e coordinaes and of e cenroid of e area sown in e figure Deermine e momen of ineria I BB of a recangle aving sides of lengs and wi respec o a diagonal of e recangle (see ase, Appendi D, availale online). 01 engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.

30 HAPTER 10 Prolems alculae e momen of ineria I for e composie circular area sown in e figure. Te origin of e aes is a e cener of e concenric circles, and e ree diameers are 0, 40, and 60 mm. Parallel-Ais Teorem alculae e momen of ineria I of a HE 30 B wide-flange secion wi respec o is ase. (Use daa from Tale E-l, Appendi E, availale online.) Deermine e momen of ineria I c wi respec o an ais roug e cenroid and parallel o e ais for e geomeric figure descried in Pro For e cannel secion descried in Pro , calculae e momen of ineria I c wi respec o an ais roug e cenroid and parallel o e ais. PRB alculae e momens of ineria I and I wi respec o e and aes for e L-saped area sown in e figure for Pro Te momen of ineria wi respec o ais 1-1 of e scalene riangle sown in e figure is mm 4. alculae is momen of ineria I wi respec o ais A semicircular area of radius 150 mm as a recangular cuou of dimensions 50 mm 100 mm (see figure). alculae e momens of ineria I and I wi respec o e and aes. Also, calculae e corresponding radii of graion r and r. PRB mm 15 mm 50 mm 50 mm 50 mm 150 mm 150 mm For e eam cross secion descried in Pro , calculae e cenroidal momens of ineria I c and I c wi respec o aes roug e cenroid suc a e c ais is parallel o e ais and e c ais coincides wi e ais. PRB alculae e momens of ineria I 1 and I of a HE 450 wide-flange secion using e cross-secional dimensions given in Tale E-l, Appendi E, availale online. (Disregard e cross-secional areas of e filles.) Also, calculae e corresponding radii of graion r 1 and r, respecivel alculae e momen of ineria I c wi respec o an ais roug e cenroid and parallel o e ais for e composie area sown in e figure for Pro alculae e cenroidal momens of ineria I c and I c wi respec o aes roug e cenroid and parallel o e and aes, respecivel, for e L-saped area sown in e figure for Pro engage Learning. All Rigs Reserved. Ma no e scanned, copied or duplicaed, or posed o a pulicl accessile wesie, in wole or in par.