Stress concentration in castellated I-beams under transverse bending

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Stress concentration in castellated I-beams under transverse bending"

Transcription

1 466 ISSN MECHANIKA olume 22(6): Stress onentrtion in stellte I-ems uner trnsverse ening A. Pritykin Kliningr Stte Tehnil University (KGTU), Sovetsky v. 1, , Kliningr, Russi, E-mil: Immnuel Knt Blti Feerl University, A. Nevskogo str. 14, , Kliningr, Russi 1. Introution Although stellte ems re lrey pplie out 100 yers, the theory of their stress stte still is not still finlly elorte. It n e onfirme, for exmple, with sene of reommentions on esigning of suh ems in Eurooe 3. This sitution n e expline, first of ll, with omplexity of prolem. To hoie the optiml imensions of stellte em it is nee to ppreite the imum level of stresses in it, euse this is one of importnt prmeters in struturl norms. Stress istriution in stellte ems ws investigte in ny works [1-13] minly using FEM n experiments. Anlytil reltions were nlyze in works [14-15]. However relile formul for stress level ws not otine. ery suite instrument for ompre of stress stte of ems with ifferent we-utting pttern is the stress onentrtion ftor α σ (SCF), representing y itself non-imension mgnitue. It is possile etermine SCF s rtio of imum equivlent stresses in zone of opening to imum stress in flnge of em with soli we uner given externl lo. In this work the etermintion of oeffiient α σ ws performe for se of trnsverse flexure. 2. Equivlent stresses in em For estimtion of the stress onentrtion level uner trnsverse ening it ws initilly onsiere simply supporte stellte I-em, performe on unwste tehnology from rolle profile #50 (GOST ), loe with onentrte fore, pplie in mile of spn. Depth of holes ws opte equl to h H, s more useful in struturl prtie. We-post with ws equl to sie of hexgonl hole, i. e. lssi sheme of the em perfortion ws onsiere (Fig. 1). In this se imensions of ems were l сm In ommon se it n e interprete s l H t t m h / H /, i. e. length totl w f f height of em we thikness flnge with - thikness of flnge reltive epth of opening reltive with of wepost. Conentrte fore P = kn ws onstnt in ll ses of loing. Aury of lultions y FEM is minly etermine with the finite element sizes: the less FE, the more urte is lultion. But pplition of ll refine elements is not suitle euse of the restrite omputer memory n essentil inresing of ompute time. For exmple, solution of the eqution system with unknowns in omputer with 4 G RAМ emn more the one minute. Reuing the size of the eqution system n respetively time of lultion n e hieve using ifferent pprohes: pplition the super element metho; tking into ount the struture symmetry n onsiering only hlf of em; using no uniform mesh of finite elements n others. Lst two pprohes re simplest n rther effetive, ut pper question, wht size of elements will e suffiient for getting of emne ury of lultion? Theoretilly to etermine optimum sizes of FE is rther omplex, tht is why in most ses they re hosen on se of lultions with suessive reuing of element sizes, until ifferene in results will e negligile. Fig. 1 Clultion sheme of stellte em In this work the refine mesh of FE ws opte only in viinity of one opening, s it gives the lest system of equtions. Then this refine mesh ws isple in turn to eh opening. It is ler the size of element is to e onnete with rius of urvture of hexgonl opening. The smller fillet rius the lesser elements re to e, otherwise ontour of opening tkes form not smooth ut roken line, n ury of lultion will e reue. Although formlly openings in wes performe on unwste tehnology re not roune in relity they hve some fillets. In orne with reommentions of AISC (Amerin Institute of Steel Construtions) in lultions for strength of stellte ems the fillet rius of hexgonl openings is neessry opt equl to r 2t w, or to r 5/ 8", epening wht is igger. In lultions performing elow the rius of fillet ws tken r 0. 04h (h epth of opening). This orrespons pproximtely to onition r 2t. w Fulfille nlysis show the stisftory ury is rehe uner sizes of finite elements equl to 0.05r, tht is why in lultions sizes of FE ner the ontour of opening they were equl FE 1 mm, n in other prts of em their imensions were 20 mm. Depth of openings ws h = 500 mm-600 mm. Uner trnsverse ening the importnt role in vlue of α σ ply s ening moment M so n sher fore. The first one etermines level of norml stress σ x, n seon is onnete with the sher stress τ xy in we. In tehnil

2 467 theory of flexure the stress stte of em with soli we is onsiering tking into ount only two stress omponents σ x n τ xy. In perforte ems ner openings the norml stress σ y is lso hieve ig vlues. Tht is why in stellte ems uner trnsverse ening tkes ple omplex stress stte, the integrl prmeter of whih for evlution of SCF uner joint tion of σ x, σ y n τ xy n e equivlent stress von Mises. In ommon form it n e expresse vi stress omponents s: tion of n M. For tht simply supporte stellte I-em of ritrry length, loe with onentrte fore in mile of spn ws onsiere. In this se ner ene opening uner ny length of em the sher fore will e onstnt n ening moment. Ner ny other opening the joint tion of n M tkes ple. M 0 P Const (1) x x y x xy Appreite vlue of with formul: /, (2) where is imum stress in flnge of em with soli we, etermine on tehnil theory of flexure s: М / W, (3) where W is moulus of inerti of em s ross setion with soli we: 2 f f w W t H H t / 6. (4) Of ourse suh pproh to lultion of SCF on Eq. (2) hs some peulirity, euse the level of imum stress is mesuring in one ross setion, n se vlue is tken in nother, ut it hs no importnt sense, euse α σ is non-imension mgnitue. Bsi vntge here is ommoity of lultion, n orresponene of otine vlue α σ to physil piture of stress stte of we in the onentrtion zone. With the im to istinguish influene of n M on vlue α σ it ws performe lultions uner onstnt vlue n prtilly sent moment М n uner joint Fig. 2 e versus 0 of ene we-post: - - ; с - e - с 0. 4Н с0 0. 2Н 0 с Н ; - с Н с Н First of ll efine originl lotion of ene opening, i.e. istne 0 from the opening ege to the support setion. Otine y FEM lultions of versus with of ene we-post re shown in Fig. 2, from whih it n e seen the pik of stress tkes ple in zone ( )Н, fter tht level of stress is stilizing, lthough flexure moment is growing. Differene in vlues (Fig. 2) oes not exee 3%, i. е. this mgnitue is rther stle uner hnging of with of ene we-post. All this llow in further lultions opt с H. Evlute now influene of em length l t imum level of equivlent stress ner first opening lote ner support. As show lultions y FEM (Fig. 3), uner onstnt trnsverse fore n sent flexure moment М imum equivlent stress ner ontour of ene hole will e prtilly onstnt. Moreover relte length of em ose not ply ny role (ompre Fig. 3, а n 3, ). From this n e onlue the stress stte in viinity of ene hole of simply supporte em ompletely etermine with vlue of trnsverse fore. ; ; Fig. 3 Stress stte ner ene hole uner onstnt fore in ems l см of ifferent length: - l = 15H; - l 20 Н lue of stress, proue with tion fore, n e represente with reltion:, (5) Ht w where α is numeril oeffiient, etermine from FE nlysis. Clultion with FEM shows the oeffiient 41. For exmple, for stellte I-em with imensions m uner tion of onentrte fore Р = kn level of imum equivlent stress in viinity of 1-st hole oring to Eq. (5) will e: / MP. Now perform with help of lultions y FEM nlysis the influene on mgnitue of flexure moment in the sme em with imensions сm , ut in zone of seon n following openings (Fig. 4). As it n e seen, level of imum stress grows proportionlly to moment М. On se of these results the stress in viinity of ny opening n e represente

3 468 s sum of two items: one use y sher fore in orne with Eq. (5) n seon use y moment М: М, (6) W М Htw where α M is numeril oeffiient, etermine y FEM nlysis. Flexure moment М for n-th opening n e pproximte s: М x n 1 s, (7) where s is step of openings; n is orinl numer of opening, in viinity of whih the stress vlue is etermining. In ommon se the step of openings uner ny perfortion n e written s: 2 2 3, (8) s а H / where is reltive with of we-post; reltive epth of opening. с / h / H is Sustitution of Eq. (4), Eq. (7) n Eq. (8) in Eq. (6) ring to expression: М n t / Ht 1 Ht f f w w, n 2 In Eq. (9) vlue (n - 2) is written inste of (n - 1), euse proportionl growth of stresses is ppering only eginning from 2n opening. Tking into ount the oeffiient is onstnt for ll lulte elow ems Eq. (9) n e rewritten s: М экв 6.4 n t / Ht 1 Ht f f w w (9). (10) erify Eq. (10) for stellte I-em сm loe with onentrte fore Р kn, pplie in mi-spn. For this vrint it ws opte vlue. Then for 6-th hole uner n in orne with Eq. (10) it n e otine МP Fig. 4 Stress stte of I-em with imensions m uner tion of onentrte fore Р = kn: - 2n; - 3r; - 4th; - 6th holes Aoring to result of FEM, s shown in Fig. 4, for 6th opening vlue 433 МP. It inites on prtilly full oiniene with result of Eq. (11). For other openings of this em the vlues otine nlytilly y Eq. (10) re shown in Tle 1. It is nee to note tht Eq. (10) remins pplile for ny length of em uner unhngele prmeters of perfortion. So for rtio of em length l / Н 20 in orne with Eq. (10) vlue 495 МP, n lultion y FEM gives МPа, tht inite on ivergene pproximtely in 1%. For other em with imensions сm uner the sme lo the stress stte of em otine y FEM is represente in Fig. 5. / Fig. 5 Stress stte of I-em with imensions сm uner trnsverse ening: - 2n; - 4th; - 5th; - 6th holes

4 469 Clultion oring to Eq. (10) uner ξ = 1, β = n α = 41 for 6th opening gives МP n oring to FEM (Fig. 5, ) result. is МP (the ivergene oes not exee 2.6%). Suh result is stisfying to engineering ury of lultions. But it is possile to otin prtilly full oiniene of results y FEM n y Eq. (10) for this em, if perform orretion of oeffiient α, reuing it from 41 to 39.5 n remining seon oeffiient α M unhngele n equl to 6.4. In this se results otine y FEM n y Eq. (10) prtilly oinie. lues of stresses for other openings of this em n for ems of other sizes re shown in Tle 1. In Tle 1 it were onsiere stellte ems frite on unwste tehnology from rolle profiles #45, 50, 55 n 60. They re quite similr in proportions of height n we thikness ut min ifferene is in reltions of were to flnge-re. This ifferene n e seen in vrition of oeffiient α whih is hnging in very nrrow rnge: from 38.8 to 41. Stress Tle 1 in zone of hexgonl openings with rius of fillet in I-ems of ifferent profile uner onentrte lo Р = kn pplie in mi-spn r 0. 04h Numer of hole Prmeters of em ; ; сm Stress y FEM Stress y Eq. (10), MP Divergene, % Prmeters of em ; ; 15Н m Stress y FEM Stress y Eq. (10), MP Divergene, % Prmeters of em ; ; 15Н m Stress y FEM Stress y Eq. (10), MP Divergene, % Prmeters of em ; М 6. 4 ; 15Н m Stress y FEM Stress y Eq. (10), MP Divergene, % М М М Influene of epth of opening t the stress level Evlute now influene of epth of opening t the stress. Consier ems with reltive epth of opening 07. n Results of lultion of em with imensions сm-0.7-1, performe y Eq. (10) with vlue of α = 46.5 n ompute y FEM re shown in Tle 2. Stress stte of we ner the ifferent openings of this em uner onentrte lo Р = kn is lso shown in Fig. 6. It n e seen from Tle 2 n Tle 1 the inreseing of the epth of opening les to growth of oeffiient of fore α, i. e. role of sher fore in vlue of equivlent stresses is inresing. Depenene α on mgnitue β is proportionl n n e pproximte with expression: (11) Similr linel epenene there is n for ems with other imensions. Stresses экв in viinity of hexgonl opening of ifferent epth in I-ems uner onentrte lo Р = kn Numer of hole Prmeters of em ; M 6. 4 ; сm Stress y FEM Stress y Eq. (10), MP Divergene, % Prmeters of em ; 6. 4 ; сm Stress y FEM Stress y Eq. (10), MP Divergene, % M Tle 2

5 470 e Fig. 6 Stresses in simply supporte I-em сm uner trnsverse ening: - 1st; - 2n; - 3r; - 4th; e - 5th; f - 6th holes f Fig. 7 Stresses in simply supporte I-em сm uner trnsverse ening: - 2n; - 3r; - 4th; - 6th holes For relte epth of openings β = 0.73 oeffiient α tkes vlue 55.3 ut α M remins unhngele. lues of for I-em with imensions сm re shown in Tle 2, from whih it is seen the ivergene in ifferent lultions oes not exee 2.5%. Results of lultion y FEM re shown in Fig. 7. Clultions performe for ems m n m show tht if opt oeffiients α = 47.3 n α = 56.2 respetively ivergene oes not exee 1% for thir n following openings (see Tle 3). It is importnt the high ury tkes ple ner the most loe openings, lote in the mile prt of em. Stress istriution in em m is shown in Fig. 8. Stresses ner hexgonl openings of ifferent epth in I-ems Tle 3 Numer of hole Prmeters of em ; 6. 4 ; m Stress y FEM Stress y Eq. (10), MP Divergene, % Prmeters of em ; 6. 4 ; m Stress y FEM Stress y Eq. (10), MP Divergene, % M M Fig. 8 Stresses in simply supporte I-em m uner trnsverse ening: - 2n; - 6th holes

6 Influene of reltive with of we-post t stress level As it is known, eveloping of perforte ems is irete on lightening of we with ifferent wys: inresing the epth of openings; inresing the length of opening, y performing them with elongte form suh s ovl, retngulr or sinusoil; reuing the with of we-posts. Author propose tehnology of frition of stellte ems with regulr hexgonl openings uner ny with of weposts [16]. Tht is why influene of reltive with of wepost t stress is onsiering elow. Results of lultion y FEM on progrm ANSYS of I-em сm with reltive with of we-post β = 0.5 with sequent isplement of smll mesh re shown in Fig. 9. Due to reuing of with of we-posts the numer of openings t hlf of the em length inrese to 7. Anlytil lultion of equivlent stress llows onfirm, tht oring to Eq. (10) reuing of β le to less level of. In Eq. (10), s in previous vrints with β = 1 ftor of influene of moment remins the sme α M = 6.4, ut ftor of sher fore tkes vlue α = Results of lultion of inite em y Eq. (10) re shown in Tle 4. Reue the reltive we-post with till β = 0.3 n lulte gin the sme simply supporte em with Н = 90 m uner tion of onentrte lo Р = kn. The vlues of stresses otine y FEM re shown in Fig 10. Due to reuing the we-post with the numer of openings t hlf of em length inrese to 8. Clultions in oring to Eq. (10) re performe uner the sme vlues of α Q = 37.6 n α M = 6.4. e f Fig. 9 Stress stte of I-em with imensions сm uner trnsverse ening: - 2n; - 3r; - 4th ; - 5th ; e - 6th ; f - 7th holes Stress ner hexgonl openings with rius of fillet r 0. 04h n ifferent with of we-posts in simply supporte I-em uner trnsverse ening Numer of hole Bem s prmeters α = 37.6; α M = 6.4; сm Stress y FEM, MP Stress y Eq. (10), MP Divergene, % Bem s prmeters α = 37.6; α M = 6.4; сm Stress y FEM, MP Stress y Eq. (10), MP Divergene, % Tle 4 Fig. 10 Stress stte of I-em with imensions сm uner trnsverse ening: - 2n; - 5th; - 6th; - 7th holes

7 Experimentl investigtion In orer to verify Eq. (10) it ws put n experiment on steel moel in form of oule ntilevere I-em with imensions m , loe y two onentrte fores = 10 kn pplie t the ene setions vi ynmometers DR-20 (Fig. 11). Mteril of em ws steel S345. Instlltion h two rigi posts lote t istne 1 m from eh other. Length of eh ntilever ws 1.5 m. During loing the level of stresses in viinity of openings ws mesure y strin guges with se 1mm lote on we in form of strin rosettes. Reings of guges were registere with Dt Aquisition Controller of English firm Shlumerger. Guges were glue ner the fillet orner openings in ples, etermine with lultion y FEM. Bem ws simply supporte n loe symmetrilly t ens. Fig. 11 Test set-up with stellte I-em moel m equivlent stress Results of tests show the mesure imum ner ontour of 2-n opening ws equl to 181 MP n in viinity of 3-r opening it ws 190 MP. Clultion of em with finite element metho (Fig. 12, n 11, ) inite vlues of in the sme lotions equl to 184 MP n 195 MP respetively. Differene in vlues reh 2.5%. Determintion of equivlent stress ppering in viinity of thir opening in orne with Eq. (10) for teste em gives: / МP (12) Otine results inite the stresses lulte y Eq. (12), y FEM n registere in experiment uner trnsverse ening re in goo orreltion. As it n e seen oeffiient α M = 6.4 is onstnt for ll imensions of ems n ifferent perfortion. Fig. 12 Stress istriution in oule ntilevere simply supporte I-em moel m uner trnsverse ening: - 1st; - 2n; - 3r openings 6. Stress onentrtion ftor Evlute now stress onentrtion ppering in we uner tion of flexure moment M n trnsverse fore. For this purpose it will e using Eq. (2) in whih we sustitute vlue of imum equivlent stress in ritrry se- ТТ tion Eq. (10) n stress, etermine on tehnil theory of flexure s: ТТ l 2 t H H t / 6 2 f f w. (13) Sustituting Eq. (10) n Eq. (13) in Eq. (2), the stress onentrtion oeffiient α σ is etermine s follows * , (14). n / where ω * = 6 f t f / Ht w + 1 n η = l / H. As it n e seen from Eq. (14) SCF oes not epen on lo ftors ut is etermine only with geometry of em, reltive length η = l / H n prmeters of perfortion in non-imension form ξ n β. Clulte y Eq. (14) oeffiients α σ for ems with imensions сm n сm will e equl α σ = 433 / 110 = 3.93 n α σ = 298 / 81.4 = 3.67 respetively. The less vlue α σ for em with height Н = 90 сm ompre with em with Н = 75 сm n e expline y reution of relte re of flnge: if in em with Н = 75 сm

8 473 vlue ω f / ω w = 0.345, then in em with Н = 90 сm it will e ω f / ω w = The flnge effet n e ompre with inresing the plte imensions uner evlution of stress onentrtion in viinity of lone opening uner plne stress stte. It is nee to rememer the otine results re pplile to stellte ems with fillet rius of opening r = 0.04h. 7. Conlusions 1. Anlytil expression for SCF for se of trnsverse ening is otine s sum of two omponents refleting influene of sher fore n flexure moment М respetively. 2. Otine reltions for α σ n for equivlent stresses re pplile for reltive epth of openings in ipson n for reltive with of weposts in ipson uner fillet rius r 0. 04h 3. Ftor of influene of moment α M = 6.4 oes not epen on the reltive vlues ξ n β. 4. Ftor of influene of sher fore α grows with inresing of epth openings n is lmost proportionl to vlue β. 5. Stress onentrtion ftor ner hexgonl openings uner trnsverse ening n reh vlue α σ = 4. Referenes 1. esrghvhry, K Stress istriution in stellte em, Pro of ASCE, Strut Div 95(2): Cheng, W.K.; Hosin, M.U.; Neis, Anlysis of stellte steel ems y the finite elements metho, Pro of Speil Conf on FEM in Civil Eng, Moutree, Cn, Liu, T.C.H.; Chung, K.F Steel em with lrge we openings of vrious shpes n sizes: finite element investigtion, J Constr Steel Res 59(9): erissimo, G.S.; Fkury, R.H.; Riero, J.C Design is for unreinfore we openings in steel n omposite ems with W-shpes, Eng J. 20(3): Dionisio, M. t l Determintion of ritil lotion for servie lo ening stresses in non-omposite ellulr ems, Reserh Report 8. illnov University. 6. Lgros, N.D. t l Optimum esign of steel strutures with we opening, J of Eng Strut 30(4): Devinis, B.; Kvers A.K Investigtion of rtionl epth of stellte steel I-em, J of Civil Eng. n Mngement 149(3): Hoffmn, R. t l Anlysis of stress istriution n filure ehvior of ellulr ems, Reserh Report 7. illnov University. 9. Tsvriis, K.D Filure moes of omposite n non-omposite perforte ems setions with vrious shpes n sizes of we openings, PhD thesis, City University, Lonon. 10. Сhhpkhne, N.K.; Sshiknt, R.K Anlysis of stress istriution in stellte em using finite element metho n experimentl tehniques, Int. J. of Meh Eng Appl Res 3(3): Wkhure, M.R.; Sge, A Finite element. nlysis of stellte steel em, Int. J. of Eng. n Innovtive Tehnology (IJEIT), 2(1): Wng, P.; Wng, X.; M, N ertil sher ukling pity of we-posts in stellte steel ems with fillet orner hexgonl we openings, Engineering Strutures. 75: Durif, S.; Bouhir, A.; ssrt, O Experimentl n numeril investigtion on we-post memer from ellulr ems with sinusoil openings, Eng. Strut. 59: Dorhev,.М.; Litvinov, Е Anlytil etermintion of stress-strin stte of we-post of perforte em, Izvesti vuzov, Constrution 5: (in Russin). 15. Pritykin, A Stress onentrtion in ems with one row of hexgonl openings, estnik of Mosow Stte Strut. University 1: (in Russin). 16. Ptent Russin Feertion Perforte metlli em. Pulishe Bulletin 30(3). А. Pritykin STRESS CONCENTRATION IN CASTELLATED I-BEAMS UNDER TRANSERSE BENDING S u m m r y In the work on se of lultions y FEM of stellte I-ems the pproximte reltions for evlution of stress level n stress onentrtion ftor in viinity of hexgonl fillet openings uner trnsverse ening re erive. Clultion of simply supporte stellte I-ems uner tion of one onentrte fore pplie in mi-spn n two symmetrilly pplie fores ws performe. Propose reltion for equivlent stress ner openings ifferentite role of eh fore ftor n М n llow etermine level of stresses in stellte ems in wie ipson of the opening prmeters uner ifferent length rtio with engineering ury. Otine results were verifie with experiment test on steel stellte em with 4 m length. Keywors: stress onentrtion ftor, stellte I-ems, hexgonl openings, von Mises stress, FEM, experiment. Reeive Novemer 04, 2015 Aepte Novemer 25, 2016

MCH T 111 Handout Triangle Review Page 1 of 3

MCH T 111 Handout Triangle Review Page 1 of 3 Hnout Tringle Review Pge of 3 In the stuy of sttis, it is importnt tht you e le to solve lgeri equtions n tringle prolems using trigonometry. The following is review of trigonometry sis. Right Tringle:

More information

I 3 2 = I I 4 = 2A

I 3 2 = I I 4 = 2A ECE 210 Eletril Ciruit Anlysis University of llinois t Chigo 2.13 We re ske to use KCL to fin urrents 1 4. The key point in pplying KCL in this prolem is to strt with noe where only one of the urrents

More information

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion

More information

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!

Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite! Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:

More information

The DOACROSS statement

The DOACROSS statement The DOACROSS sttement Is prllel loop similr to DOALL, ut it llows prouer-onsumer type of synhroniztion. Synhroniztion is llowe from lower to higher itertions sine it is ssume tht lower itertions re selete

More information

Lecture 11 Binary Decision Diagrams (BDDs)

Lecture 11 Binary Decision Diagrams (BDDs) C 474A/57A Computer-Aie Logi Design Leture Binry Deision Digrms (BDDs) C 474/575 Susn Lyseky o 3 Boolen Logi untions Representtions untion n e represente in ierent wys ruth tle, eqution, K-mp, iruit, et

More information

SIDESWAY MAGNIFICATION FACTORS FOR STEEL MOMENT FRAMES WITH VARIOUS TYPES OF COLUMN BASES

SIDESWAY MAGNIFICATION FACTORS FOR STEEL MOMENT FRAMES WITH VARIOUS TYPES OF COLUMN BASES Advned Steel Constrution Vol., No., pp. 7-88 () 7 SIDESWAY MAGNIFICATION FACTORS FOR STEEL MOMENT FRAMES WIT VARIOUS TYPES OF COLUMN BASES J. ent sio Assoite Professor, Deprtment of Civil nd Environmentl

More information

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1) Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion

More information

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

Project 6: Minigoals Towards Simplifying and Rewriting Expressions MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

More information

6.5 Improper integrals

6.5 Improper integrals Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

More information

Proportions: A ratio is the quotient of two numbers. For example, 2 3

Proportions: A ratio is the quotient of two numbers. For example, 2 3 Proportions: rtio is the quotient of two numers. For exmple, 2 3 is rtio of 2 n 3. n equlity of two rtios is proportion. For exmple, 3 7 = 15 is proportion. 45 If two sets of numers (none of whih is 0)

More information

Ranking Generalized Fuzzy Numbers using centroid of centroids

Ranking Generalized Fuzzy Numbers using centroid of centroids Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July ning Generlize Fuzzy Numers using entroi of entrois S.Suresh u Y.L.P. Thorni N.vi Shnr Dept. of pplie Mthemtis GIS GITM University Vishptnm

More information

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,

More information

APPROXIMATION AND ESTIMATION MATHEMATICAL LANGUAGE THE FUNDAMENTAL THEOREM OF ARITHMETIC LAWS OF ALGEBRA ORDER OF OPERATIONS

APPROXIMATION AND ESTIMATION MATHEMATICAL LANGUAGE THE FUNDAMENTAL THEOREM OF ARITHMETIC LAWS OF ALGEBRA ORDER OF OPERATIONS TOPIC 2: MATHEMATICAL LANGUAGE NUMBER AND ALGEBRA You shoul unerstn these mthemtil terms, n e le to use them ppropritely: ² ition, sutrtion, multiplition, ivision ² sum, ifferene, prout, quotient ² inex

More information

Symmetrical Components 1

Symmetrical Components 1 Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent

More information

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points.

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points. Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You

More information

Algebra 2 Semester 1 Practice Final

Algebra 2 Semester 1 Practice Final Alger 2 Semester Prtie Finl Multiple Choie Ientify the hoie tht est ompletes the sttement or nswers the question. To whih set of numers oes the numer elong?. 2 5 integers rtionl numers irrtionl numers

More information

Introduction to Olympiad Inequalities

Introduction to Olympiad Inequalities Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................

More information

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem. 27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

More information

QUADRATIC EQUATION. Contents

QUADRATIC EQUATION. Contents QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

More information

(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.

(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P. Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time

More information

The Stirling Engine: The Heat Engine

The Stirling Engine: The Heat Engine Memoril University of Newfounln Deprtment of Physis n Physil Oenogrphy Physis 2053 Lortory he Stirling Engine: he Het Engine Do not ttempt to operte the engine without supervision. Introution Het engines

More information

MAC-solutions of the nonexistent solutions of mathematical physics

MAC-solutions of the nonexistent solutions of mathematical physics Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE

More information

U Q W The First Law of Thermodynamics. Efficiency. Closed cycle steam power plant. First page of S. Carnot s paper. Sadi Carnot ( )

U Q W The First Law of Thermodynamics. Efficiency. Closed cycle steam power plant. First page of S. Carnot s paper. Sadi Carnot ( ) 0-9-0 he First Lw of hermoynmis Effiieny When severl lterntive proesses involving het n work re ville to hnge system from n initil stte hrterize y given vlues of the mrosopi prmeters (pressure p i, temperture

More information

Generalized Kronecker Product and Its Application

Generalized Kronecker Product and Its Application Vol. 1, No. 1 ISSN: 1916-9795 Generlize Kroneker Prout n Its Applition Xingxing Liu Shool of mthemtis n omputer Siene Ynn University Shnxi 716000, Chin E-mil: lxx6407@163.om Astrt In this pper, we promote

More information

Section 4: Integration ECO4112F 2011

Section 4: Integration ECO4112F 2011 Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

More information

Materials Analysis MATSCI 162/172 Laboratory Exercise No. 1 Crystal Structure Determination Pattern Indexing

Materials Analysis MATSCI 162/172 Laboratory Exercise No. 1 Crystal Structure Determination Pattern Indexing Mterils Anlysis MATSCI 16/17 Lbortory Exercise No. 1 Crystl Structure Determintion Pttern Inexing Objectives: To inex the x-ry iffrction pttern, ientify the Brvis lttice, n clculte the precise lttice prmeters.

More information

Chem Homework 11 due Monday, Apr. 28, 2014, 2 PM

Chem Homework 11 due Monday, Apr. 28, 2014, 2 PM Chem 44 - Homework due ondy, pr. 8, 4, P.. . Put this in eq 8.4 terms: E m = m h /m e L for L=d The degenery in the ring system nd the inresed sping per level (4x bigger) mkes the sping between the HOO

More information

Data Structures LECTURE 10. Huffman coding. Example. Coding: problem definition

Data Structures LECTURE 10. Huffman coding. Example. Coding: problem definition Dt Strutures, Spring 24 L. Joskowiz Dt Strutures LEURE Humn oing Motivtion Uniquel eipherle oes Prei oes Humn oe onstrution Etensions n pplitions hpter 6.3 pp 385 392 in tetook Motivtion Suppose we wnt

More information

Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and

Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and Leture 7: iffusion of Ions: Prt : oupled diffusion of tions nd nions s desried y Nernst-Plnk Eqution Tody s topis Continue to understnd the fundmentl kinetis prmeters of diffusion of ions within n eletrilly

More information

Chapter 4 State-Space Planning

Chapter 4 State-Space Planning Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 Stte-Spe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different

More information

Forces on curved surfaces Buoyant force Stability of floating and submerged bodies

Forces on curved surfaces Buoyant force Stability of floating and submerged bodies Stti Surfe ores Stti Surfe ores 8m wter hinge? 4 m ores on plne res ores on urved surfes Buont fore Stbilit of floting nd submerged bodies ores on Plne res Two tpes of problems Horizontl surfes (pressure

More information

Sturm-Liouville Theory

Sturm-Liouville Theory LECTURE 1 Sturm-Liouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory

More information

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri

More information

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

More information

Particle Lifetime. Subatomic Physics: Particle Physics Lecture 3. Measuring Decays, Scatterings and Collisions. N(t) = N 0 exp( t/τ) = N 0 exp( Γt/)

Particle Lifetime. Subatomic Physics: Particle Physics Lecture 3. Measuring Decays, Scatterings and Collisions. N(t) = N 0 exp( t/τ) = N 0 exp( Γt/) Sutomic Physics: Prticle Physics Lecture 3 Mesuring Decys, Sctterings n Collisions Prticle lifetime n with Prticle ecy moes Prticle ecy kinemtics Scttering cross sections Collision centre of mss energy

More information

ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS

ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX 6310 30 Armstrong Hll Morgntown, WV 6506 millerd@mth.wvu.edu

More information

8.3 THE HYPERBOLA OBJECTIVES

8.3 THE HYPERBOLA OBJECTIVES 8.3 THE HYPERBOLA OBJECTIVES 1. Define Hperol. Find the Stndrd Form of the Eqution of Hperol 3. Find the Trnsverse Ais 4. Find the Eentriit of Hperol 5. Find the Asmptotes of Hperol 6. Grph Hperol HPERBOLAS

More information

HARMONIC BALANCE SOLUTION OF COUPLED NONLINEAR NON-CONSERVATIVE DIFFERENTIAL EQUATION

HARMONIC BALANCE SOLUTION OF COUPLED NONLINEAR NON-CONSERVATIVE DIFFERENTIAL EQUATION GNIT J. nglesh Mth. So. ISSN - HRMONIC LNCE SOLUTION OF COUPLED NONLINER NON-CONSERVTIVE DIFFERENTIL EQUTION M. Sifur Rhmn*, M. Mjeur Rhmn M. Sjeur Rhmn n M. Shmsul lm Dertment of Mthemtis Rjshhi University

More information

New Expansion and Infinite Series

New Expansion and Infinite Series Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

More information

Due to gravity and wind load, the post supporting the sign shown is subjected simultaneously to compression, bending, and torsion.

Due to gravity and wind load, the post supporting the sign shown is subjected simultaneously to compression, bending, and torsion. ue to grvit nd wind lod, the post supporting the sign shown is sujeted simultneousl to ompression, ending, nd torsion. In this hpter ou will lern to determine the stresses reted suh omined lodings in strutures

More information

y = c 2 MULTIPLE CHOICE QUESTIONS (MCQ's) (Each question carries one mark) is...

y = c 2 MULTIPLE CHOICE QUESTIONS (MCQ's) (Each question carries one mark) is... . Liner Equtions in Two Vriles C h p t e r t G l n e. Generl form of liner eqution in two vriles is x + y + 0, where 0. When we onsier system of two liner equtions in two vriles, then suh equtions re lle

More information

NON-DETERMINISTIC FSA

NON-DETERMINISTIC FSA Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

More information

Chapter Gauss Quadrature Rule of Integration

Chapter Gauss Quadrature Rule of Integration Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to

More information

mesuring ro referene points Trget point Pile Figure 1. Conventionl mesurement metho of the pile position In the pst reserh the uthors hve evelope Bum

mesuring ro referene points Trget point Pile Figure 1. Conventionl mesurement metho of the pile position In the pst reserh the uthors hve evelope Bum Development of New Metho for Mesurement of Centrl Ais of Clinril Strutures Using Totl Sttion Kzuhie Nkniw 1 Nouoshi Yuki Disuke Nishi 3 Proshhnk Dzinis 4 1 CEO KUMONOS Corportion Osk Jpn. Emil: nkniw09@knkou.o.jp

More information

This enables us to also express rational numbers other than natural numbers, for example:

This enables us to also express rational numbers other than natural numbers, for example: Overview Study Mteril Business Mthemtis 05-06 Alger The Rel Numers The si numers re,,3,4, these numers re nturl numers nd lso lled positive integers. The positive integers, together with the negtive integers

More information

Thermal energy 2 U Q W. 23 April The First Law of Thermodynamics. Or, if we want to obtain external work: The trick of using steam

Thermal energy 2 U Q W. 23 April The First Law of Thermodynamics. Or, if we want to obtain external work: The trick of using steam April 08 Therml energy Soures of het Trnsport of het How to use het The First Lw of Thermoynmis U W Or, if we wnt to otin externl work: U W 009 vrije Universiteit msterm Close yle stem power plnt The trik

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

Maximum size of a minimum watching system and the graphs achieving the bound

Maximum size of a minimum watching system and the graphs achieving the bound Mximum size of minimum wthing system n the grphs hieving the oun Tille mximum un système e ontrôle minimum et les grphes tteignnt l orne Dvi Auger Irène Chron Olivier Hury Antoine Lostein 00D0 Mrs 00 Déprtement

More information

Theme 8 Stability and buckling of members

Theme 8 Stability and buckling of members Elsticity nd plsticity Theme 8 Stility nd uckling o memers Euler s solution o stility o n xilly compressed stright elstic memer Deprtment o Structurl Mechnics culty o Civil Engineering, VSB - Technicl

More information

Solutions to Problem Set #1

Solutions to Problem Set #1 CSE 233 Spring, 2016 Solutions to Prolem Set #1 1. The movie tse onsists of the following two reltions movie: title, iretor, tor sheule: theter, title The first reltion provies titles, iretors, n tors

More information

Proving the Pythagorean Theorem

Proving the Pythagorean Theorem Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or

More information

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique? XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

More information

Algorithm Design and Analysis

Algorithm Design and Analysis Algorithm Design nd Anlysis LECTURE 8 Mx. lteness ont d Optiml Ching Adm Smith 9/12/2008 A. Smith; sed on slides y E. Demine, C. Leiserson, S. Rskhodnikov, K. Wyne Sheduling to Minimizing Lteness Minimizing

More information

Section 6.1 Definite Integral

Section 6.1 Definite Integral Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

More information

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS 9 The cgrw-hill Compnies, Inc. All rights reserved. Fifth SI Edition CHAPTER 5 ECHANICS OF ATERIALS Ferdinnd P. Beer E. Russell Johnston, Jr. John T. DeWolf Dvid F. zurek Lecture Notes: J. Wlt Oler Texs

More information

ENERGY AND PACKING. Outline: MATERIALS AND PACKING. Crystal Structure

ENERGY AND PACKING. Outline: MATERIALS AND PACKING. Crystal Structure EERGY AD PACKIG Outline: Crstlline versus morphous strutures Crstl struture - Unit ell - Coordintion numer - Atomi pking ftor Crstl sstems on dense, rndom pking Dense, regulr pking tpil neighor ond energ

More information

Identifying and Classifying 2-D Shapes

Identifying and Classifying 2-D Shapes Ientifying n Clssifying -D Shpes Wht is your sign? The shpe n olour of trffi signs let motorists know importnt informtion suh s: when to stop onstrution res. Some si shpes use in trffi signs re illustrte

More information

Monochromatic Plane Matchings in Bicolored Point Set

Monochromatic Plane Matchings in Bicolored Point Set CCCG 2017, Ottw, Ontrio, July 26 28, 2017 Monohromti Plne Mthings in Biolore Point Set A. Krim Au-Affsh Sujoy Bhore Pz Crmi Astrt Motivte y networks interply, we stuy the prolem of omputing monohromti

More information

CIT 596 Theory of Computation 1. Graphs and Digraphs

CIT 596 Theory of Computation 1. Graphs and Digraphs CIT 596 Theory of Computtion 1 A grph G = (V (G), E(G)) onsists of two finite sets: V (G), the vertex set of the grph, often enote y just V, whih is nonempty set of elements lle verties, n E(G), the ege

More information

Prefix-Free Regular-Expression Matching

Prefix-Free Regular-Expression Matching Prefix-Free Regulr-Expression Mthing Yo-Su Hn, Yjun Wng nd Derik Wood Deprtment of Computer Siene HKUST Prefix-Free Regulr-Expression Mthing p.1/15 Pttern Mthing Given pttern P nd text T, find ll sustrings

More information

New centroid index for ordering fuzzy numbers

New centroid index for ordering fuzzy numbers Interntionl Sientifi Journl Journl of Mmtis http://mmtissientifi-journlom New entroi inex for orering numers Tyee Hjjri Deprtment of Mmtis, Firoozkooh Brnh, Islmi z University, Firoozkooh, Irn Emil: tyeehjjri@yhooom

More information

5: The Definite Integral

5: The Definite Integral 5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

More information

Automata and Regular Languages

Automata and Regular Languages Chpter 9 Automt n Regulr Lnguges 9. Introution This hpter looks t mthemtil moels of omputtion n lnguges tht esrie them. The moel-lnguge reltionship hs multiple levels. We shll explore the simplest level,

More information

Computing all-terminal reliability of stochastic networks with Binary Decision Diagrams

Computing all-terminal reliability of stochastic networks with Binary Decision Diagrams Computing ll-terminl reliility of stohsti networks with Binry Deision Digrms Gry Hry 1, Corinne Luet 1, n Nikolos Limnios 2 1 LRIA, FRE 2733, 5 rue u Moulin Neuf 80000 AMIENS emil:(orinne.luet, gry.hry)@u-pirie.fr

More information

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2 1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

More information

10. AREAS BETWEEN CURVES

10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

More information

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR 6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht

More information

CS 360 Exam 2 Fall 2014 Name

CS 360 Exam 2 Fall 2014 Name CS 360 Exm 2 Fll 2014 Nme 1. The lsses shown elow efine singly-linke list n stk. Write three ifferent O(n)-time versions of the reverse_print metho s speifie elow. Eh version of the metho shoul output

More information

1.3 SCALARS AND VECTORS

1.3 SCALARS AND VECTORS Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd

More information

Line Integrals and Entire Functions

Line Integrals and Entire Functions Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series

More information

MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES

MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES MA10207B: ANALYSIS SECOND SEMESTER OUTLINE NOTES CHARLIE COLLIER UNIVERSITY OF BATH These notes hve been typeset by Chrlie Collier nd re bsed on the leture notes by Adrin Hill nd Thoms Cottrell. These

More information

Available online at Procedia Computer Science 5 (2011)

Available online at  Procedia Computer Science 5 (2011) Aville online t www.sieneiret.om Proei Computer Siene 5 (20) 505 52 The 2n Interntionl Conferene on Amient Systems, Networks n Tehnologies Spe Vetor Moultion Diret Torque Spee Control Of Inution Motor

More information

Composite Pattern Matching in Time Series

Composite Pattern Matching in Time Series Composite Pttern Mthing in Time Series Asif Slekin, 1 M. Mustfizur Rhmn, 1 n Rihnul Islm 1 1 Deprtment of Computer Siene n Engineering, Bnglesh University of Engineering n Tehnology Dhk-1000, Bnglesh slekin@gmil.om

More information

In Search of Lost Time

In Search of Lost Time In Serh of Lost Time Bernette Chrron-Bost 1, Mrtin Hutle 2, n Josef Wier 3 1 CNRS / Eole polytehnique, Pliseu, Frne 2 EPFL, Lusnne, Switzerln 3 TU Wien, Vienn, Austri Astrt Dwork, Lynh, n Stokmeyer (1988)

More information

Outline Data Structures and Algorithms. Data compression. Data compression. Lossy vs. Lossless. Data Compression

Outline Data Structures and Algorithms. Data compression. Data compression. Lossy vs. Lossless. Data Compression 5-2 Dt Strutures n Algorithms Dt Compression n Huffmn s Algorithm th Fe 2003 Rjshekr Rey Outline Dt ompression Lossy n lossless Exmples Forml view Coes Definition Fixe length vs. vrile length Huffmn s

More information

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero

More information

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

More information

MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further

More information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

More information

Lesson 55 - Inverse of Matrices & Determinants

Lesson 55 - Inverse of Matrices & Determinants // () Review Lesson - nverse of Mtries & Determinnts Mth Honors - Sntowski - t this stge of stuying mtries, we know how to, subtrt n multiply mtries i.e. if Then evlute: () + B (b) - () B () B (e) B n

More information

Equations of Motion. Figure 1.1.1: a differential element under the action of surface and body forces

Equations of Motion. Figure 1.1.1: a differential element under the action of surface and body forces Equtions of Motion In Prt I, lnce of forces nd moments cting on n component ws enforced in order to ensure tht the component ws in equilirium. Here, llownce is mde for stresses which vr continuousl throughout

More information

8 Laplace s Method and Local Limit Theorems

8 Laplace s Method and Local Limit Theorems 8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved

More information

m A 1 1 A ! and AC 6

m A 1 1 A ! and AC 6 REVIEW SET A Using sle of m represents units, sketh vetor to represent: NON-CALCULATOR n eroplne tking off t n ngle of 8 ± to runw with speed of 6 ms displement of m in north-esterl diretion. Simplif:

More information

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR 6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht

More information

Arrow s Impossibility Theorem

Arrow s Impossibility Theorem Rep Voting Prdoxes Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Voting Prdoxes Properties Arrow s Theorem Leture Overview 1 Rep

More information

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a. Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining

More information

Functions. mjarrar Watch this lecture and download the slides

Functions. mjarrar Watch this lecture and download the slides 9/6/7 Mustf Jrrr: Leture Notes in Disrete Mthemtis. Birzeit University Plestine 05 Funtions 7.. Introdution to Funtions 7. One-to-One Onto Inverse funtions mjrrr 05 Wth this leture nd downlod the slides

More information

REINFORCED CONCRETE. Reinforced Concrete Design. A Fundamental Approach - Fifth Edition

REINFORCED CONCRETE. Reinforced Concrete Design. A Fundamental Approach - Fifth Edition CHAPTER REINFORCED CONCRETE Reinored Conrete Design A Fundmentl Approh - Fith Edition Fith Edition FLEXURE IN BEAMS A. J. Clrk Shool o Engineering Deprtment o Civil nd Environmentl Engineering 5 SPRING

More information

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions. Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene

More information

Example 3 Find the tangent line to the following function at.

Example 3 Find the tangent line to the following function at. Emple Given n fin eh of the following. () () () () Emple Given fin. Emple Fin the tngent line to the following funtion t. Now ll tht we nee is the funtion vlue n erivtive (for the slope) t. The tngent

More information

Planar Conformal Mappings of Piecewise Flat Surfaces

Planar Conformal Mappings of Piecewise Flat Surfaces Plnr Conforml Mppings of Pieewise Flt Surfes Philip L. Bowers n Moni K. Hurl Deprtment of Mthemtis, The Flori Stte University, Tllhssee, FL 32306, USA. owers@mth.fsu.eu mhurl@mth.fsu.eu Introution y There

More information

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ). AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

More information

Bivariate drought analysis using entropy theory

Bivariate drought analysis using entropy theory Purue University Purue e-pus Symposium on Dt-Driven Approhes to Droughts Drought Reserh Inititive Network -3- Bivrite rought nlysis using entropy theory Zengho Ho exs A & M University - College Sttion,

More information

= state, a = reading and q j

= state, a = reading and q j 4 Finite Automt CHAPTER 2 Finite Automt (FA) (i) Derterministi Finite Automt (DFA) A DFA, M Q, q,, F, Where, Q = set of sttes (finite) q Q = the strt/initil stte = input lphet (finite) (use only those

More information

( ) as a fraction. Determine location of the highest

( ) as a fraction. Determine location of the highest AB/ Clulus Exm Review Sheet Solutions A Prelulus Type prolems A1 A A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f( x) Set funtion equl to Ftor or use qudrti eqution if qudrti Grph to

More information

THERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION

THERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION XX IMEKO World Congress Metrology for Green Growth September 9,, Busn, Republic of Kore THERMAL EXPANSION COEFFICIENT OF WATER FOR OLUMETRIC CALIBRATION Nieves Medin Hed of Mss Division, CEM, Spin, mnmedin@mityc.es

More information

Naming the sides of a right-angled triangle

Naming the sides of a right-angled triangle 6.2 Wht is trigonometry? The word trigonometry is derived from the Greek words trigonon (tringle) nd metron (mesurement). Thus, it literlly mens to mesure tringle. Trigonometry dels with the reltionship

More information