Active Displacement Feedback Control of a Smart Beam: Analytic and Numerical Solutions

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Procdings of th Intrnationa MutiConfrnc of Enginrs and Computr Scintists 9 Vo II IMECS 9, March 8 -, 9, Hong Kong Activ Dispacmnt Fdback Contro of a Smart Bam: Anaytic and Numrica Soutions C.

1 Procdings of th Intrnationa MutiConfrnc of Enginrs and Computr Scintists 9 Vo II IMECS 9, March 8 -, 9, Hong Kong Activ Dispacmnt Fdback Contro of a Smart Bam: Anaytic and Numrica Soutions C. SPIER, I.S. SADEK, J. C. BRUCH, JR. 3, J. M. SLOSS 4, S. ADALI 5 Abstract: Activ vibration contro of a smart bam with intgratd pizocramic actuator and snsor patchs is considrd. An anaytica soution of this probm is workd out for th cas of th controd bam incuding th mass and stiffnss of th pizocramic patchs. Th quation of motion for th controd bam incuds Havisid functions and drivativs of th Havisid function du to finit patch ngths. This maks th probm difficut to sov using convntiona mthods. An intgra quation is introducd, whr th ignsoutions of th intgra quation ar ignsoutions of th diffrntia quation of motion for th controd bam. A finit mnt mod of a controd bam is aso formuatd. Th mod contains modifid bam mnt mass and stiffnss matrics to account for th pizo patchs and contro ffct. Two cas studis ar prsntd and th first thr natura frquncis and mod shaps ar found using th finit mnt and intgra quation soutions. Th rsuts from th intgra quation soution match vry cosy th rsuts from th finit mnt soution. Kywords: Dispacmnt fdback contro, pizocramic actuators and snsors, smart structurs. INTRODUCTION In th prsnt work, an anaytica soution is proposd to find th natura frquncis and mod shaps for a controd bam with pizocramic snsor and actuator patchs. Activ vibration contro is impmntd using cosd-oop dispacmnt fdback with pizocramic patchs. Prvious soutions ngct th mass and stiffnss of th pizo patchs, which can b significant if pizocramic matrias ar usd. Th spcific controd structur to b studid is a cantivr bam with a pizocramic snsor bondd to th top surfac of th bam and a pizocramic actuator bondd to th bottom surfac of th bam. Th snsor on th top dtcts a strain whn th bam is vibrating and th output votag signa of th snsor is ampifid and snt to th actuator. Th signa snt to th actuator attmpts to countract th vibration of th bam and contro its motion. Vibration contro using pizoctric patchs is a widy rsarchd topic. Howvr, thr is no anaytica soution avaiab for an activy controd bam with Dpartmnt of Mchanica Enginring, Univrsity of Caifornia, Santa Barbara, CA 936, USA(spir@nginring.ucsb.du) Dpartmnt of Mathmatics and Statistics, Amrican Univrsity of Sharjah, Unitd Arab Emirats(sadk@aus.du) 3 Dpartmnt of Mchanica Enginring and Dpartmnt of Mathmatics, Univrsity of Caifornia, Santa Barbara, CA 936, USA (jcb@nginring.ucsb.du) 4 Dpartmnt of Mathmatics, Univrsity of Caifornia, Santa Barbara, CA 936 USA (JMSLOSS@SILCOM.COM) 5 Schoo of Mchanica Enginring, Univrsity of KwaZuu-Nata, Durban, South Africa (adai@ukzn.ac.za) finit ngth patchs whr th patch stiffnss and mass ar incudd in th quation of motion. In [], Tzou drivs th govrning quations usd in th anaysis of an activy controd structur. Th govrning quations incud th snsor votag cratd du to a vibrating bam and th trnay appid momnt th pizo actuator crats on th bam. Th snsor votag and th appid momnt ar th basis for activ contro and show up in th quation of motion. An quation of motion incuding th stiffnss and mass of th pizo patchs is drivd by Banks t a. []. In [], pizocramic patchs ar usd for th damag dtction of a structur whr th stiffnss and mass hav bn changd du to damag. Th motiv bhind Banks t a. [] is diffrnt than what is dsird for activ vibration supprssion, but th quation of motion for a cantivr bam with pizo patchs is th sam for both cass. Thr hav bn svra rsuts (s.g. [3]) for controd bams using pizocramic actuators. Howvr, ths rsuts ony us pizo patchs as actuators and do not us thm as snsors. A constant votag is appid to th actuators in ths cass and cosd-oop contro is imposd whr th snsor is continuousy snding a changing signa to th actuator. Tani t a. [4] prformd an primnt using a gap snsor to dtct th dispacmnt of th bam and th information is convrtd to votag and fd to pizocramic actuators. Anaytica soutions hav bn obtaind using an intgra quation approach for an activy controd bam [5-6]. Soss t a. [5] driv an anaytic soution to find th natura frquncis for finit ngth pizo patchs, but dos not incud th mass and stiffnss of th patch. Th soution for using mutip pizo snsor and actuator patchs, aso not incuding th stiffnss and mass of th patchs, is found in [6].. PROBLEM FORMULATION Th cantivr bam is uniform with rctanguar cross sction and is campd at and has a fr nd at. Pizocramic patchs ar bondd to th top and bottom surfacs of th bam as shown in Figur. Fig. : Schmatic of snsor and actuator patch ocations ISBN: IMECS 9

2 Procdings of th Intrnationa MutiConfrnc of Enginrs and Computr Scintists 9 Vo II IMECS 9, March 8 -, 9, Hong Kong Th pizocramic ayrs ar considrd to b prfcty bondd to th top and bottom surfacs of th astic bam and th physica proprtis of th bonding matria ar not considrd. Aso, th ffctiv ais of th pizoctric ayrs is aignd with th -ais to nsur th maimum pizoctric ffcts in snsor and actuator appications. Th ocation of th snsor is givn by s s and th ocation of th actuator is givn by a a.th transvrs dfction of th bam is rprsntd by w(,t) and th quation of motion of th activy controd bam with coocatd snsor and actuator patchs is w ρ( ) + L [ w] t whr th oprator L is dfind by s w w α ξ s L[ w] ( ) b S ( ) (.) (.) whr th prims rfr to diffrntiation with rspct to and α(), ρ(), S () andξ ar givn by (.3) (.4) (.5) ρ( ) ρbab + ρpap H ( a) H ( a) α ( ) EbIb+ EpIp H( a ) H( a) S( ) H ( a) H ( a) GEahsrsr ad3, ah3, s ξ s s (.6) for < < and t >, whr th subscripts b, a and s rfr to th proprtis of th bam, th pizocramic actuator matria and th pizocramic snsor matria, rspctivy, and ρ is th dnsity, A is th cross sction ara, E is th astic moduus, I is th momnt of inrtia, b is th bam width, and H() is th Havisid function. In quations (.3)-(.4), th patch proprtis ar now givn by E p E a E s, I p I a I s, ρ p ρ a ρ s, and A p A a A s bcaus th snsor and actuator hav th sam siz and ar of th sam matria. In quation (.6), r a (h a +h)/ is th ffctiv momnt arm, d 3,a is a pizoctric constant of th actuator, r s (h s +h)/, h 3,s is a pizoctric constant of th snsor, h is th bam thicknss, and h s is th snsor thicknss. Th output votag of a pizoctric snsor is givn by Tzou [] s h s w ϕs() t h3, srs (.7) In s s s ordr to achiv mor ffctiv vibration contro, th snsor signa is ampifid by a gain G and th rsuting signa snt to th actuator is 3. FREE VIBRATION ANALYSIS Standard fr vibration anaysis is prformd on th bam. Th transvrs motion of th bam is a function of both position and tim and is givn by wt (, ) ψ ( ) λt (3.) whr λiω. Taking th partia drivativs of w(,t) and substituting thm into th quation of motion givs λρ ψ ψ [ ] ( ) ( ) + L ( ) (3.) 4. INTEGRAL EQUATION SOLUTION An anaytic soution to quation (3.) for a bam with coocatd pizo snsor and actuator patchs is proposd. An intgra quation is introducd and it wi b shown that th ignsoutions of th intgra quation ar ignsoutions to th quation of motion. Considr th foowing intgra quation σψ ( ) ρ( sks ) (, ) ψ ( sds ) (4.) whr σ ω - and th krn is givn by K( s, ) Gs (, ) + pqs ( ) ( ) (4.) whr G(,s), p(), and q(s) ar auiiary functions. Th auiiary functions ar chosn such that thy satisfy th foowing conditions [ ] L K(, s) δ ( s) (4.3a) Gs (, ) α( ) δ( s) d d p( ) ( ) S ( ) α d d 4. Equivanc (4.3b) (4.3c) Using th rationship (4.3a), it can b shown that th ignsoutions to th intgra quation ar ignsoutions of th diffrntia quation of motion givn by quation (3.). This is dmonstratd in th foowing mannr: Assum ψ() to b a soution of th intgra quation (4.) and appy th L oprator to both sids of th intgra quation (4.), thn on obtains ϕ () t Gϕ () t (.8) a s [ ] [ ] σl ψ( ) L K(, s) ρ( s) ψ( s) ds (4.4) Th cantivr bam is campd at rsuting in zro dispacmnt and sop. Th bam is fr at giving zro shar and zro rsutant momnt. This rsuts in th foowing boundary conditions for t > w(, t ), w α ( ) w, w α( ) (.9) (.) Using th condition (4.3a) and th fundamnta proprty of th dta function, th quation (4.4) can b rducd to [ ] σl ψ( ) ρ( ) ψ( ) (4.5) which taks th sam form as th diffrntia quation givn by (3.). Hnc with an appropriaty chosn krn and auiiary functions G(,s), p(), and q(s), th ignsoutions of th intgra quation ar ignsoutions of th quation of motion. ISBN: IMECS 9

3 Procdings of th Intrnationa MutiConfrnc of Enginrs and Computr Scintists 9 Vo II IMECS 9, March 8 -, 9, Hong Kong 4. Th q(s) Function To dtrmin q(s), th rationships dfind in quations (.), (4.) and (4.3b-c), ar usd to writ L[K(,s)] as s s G p G p L[ K(, s) ] α( ) q( s) bξ q( s) S ( ) + + s s (4.6) Soving quations (4.3a)- (4.6) for q(s) yids th foowing [ ( s, ) ( s, )] bξ [ p p ] bξ G s G s (4.7) qs () ( ) ( ) s s 4.3 Th p() Function To dtrmin th patch function p(), th foowing rationship is usd d d d p α( ) H a H a d ( ) ( ) Upon intgrating quation (4.8) twic rsuts in d p (4.8) α ( ) H( a ) H( a) (4.9) d with th constants of intgration bing qua to zro. Th soution of quation (4.9) is a picwis function comprisd of th foowing < p ( ) p( ) p( ) a < a a a (4.) whr p () is a quadratic function and p () is a inar function. Th functions p () and p () ar givn by ( /)( ) a p ( ) EI b b+ EI p p ( )( ) ( )( ) a a a + / a a p( ) EI + EI b b p p (4.a) (4.b) which ar ony vaid for aignd snsor and actuator patchs ( s a and s a ). 4.4 Th G(,s) Function Th ast auiiary functions to b dtrmind ar th G(,s) and g(,s) functions. Lt g(,s) b dfind as ( ) ( ) g( s, ) gsh?(, ) s + gsh (, ) s (4.3) Th function ĝ(,s) is chosn such that g (,s) δ( s). For a cantivr bam, a function for ĝ(,s) which satisfis this condition is 3 gˆ( s, ) + s 6 < s (4.4a) 3 gˆ( s, ) s + s 6 > s (4.4b) Using th foowing rationship, G(,s) can b dtrmind Gs (, ) α( ) g (, ) s Intgrating quation (4.5) twic rsuts in th foowing (, ) ηη ( η, ) α ( η) (, ) (, ) ( ) (, ) ( ) ( ) (, ) ηη (4.5) G s g s (4.6a) G s gηη η s α η dη Gs η α η g η sdη whr th intgration constants ar qua to zro. 5. METHOD OF SOLUTION (4.6b) (4.6c) a. Choos a compt orthonorma st of functions. b. Epand th krn in trms of th Fourir sris of th orthonorma functions. c. Rduc th intgra quation to an infinit st of inar quations. d. Dtrmin th numbr of trms N such that succssiv soutions diffr by a ngigib amount.. Considr th intgra quation in which th krn is rpacd by th N-trm Fourir sris pansion of th krn. f. Th nw intgra quation is rducd to soving a inar systm of N+ quations with N+ unknowns. For th soution of th intgra quation, a st of orthonorma ignfunctions φ n () is chosn which ar thos for an uncontrod bam with no contro momnt trm. Th mod shaps of a standard cantivr bam with boundary conditions for a fid nd at and a fr nd at ar U ( Ωn ) Wn( ) V Ω n/ + T Ωn/ T ( Ω ) n ( ) ( ) whr th ignvaus ar dfind by Ω 4 (ρa/ei) 4 ω (5.) in which (5.) T( ) [ sinh( ) sin( ) ] (5.3a) U( ) [ cosh( ) + cos( ) ] (5.3b) V( ) [ cosh( ) cos( ) ] (5.3c) Th orthonorma ignfunctions ar givn by Wn ( ) ϕ n( ) W ( ) n L (5.4) Th function G(,s) can b pandd in trms of th ignfunctions φ n () of th uncontrod bam Gs (, ) Gφ ( ) φ ( s) j k jk j k (5.5) ISBN: IMECS 9

4 Procdings of th Intrnationa MutiConfrnc of Enginrs and Computr Scintists 9 Vo II IMECS 9, March 8 -, 9, Hong Kong whr jk φ k φj G G(, s) ( ) ( s) dds (5.6) Equation (4.) in trms of th auiiary functions and quations (4.) and (5.6) givs σψ ( ) G φ ( ) C + p( ) C whr j k jk j k Ck φk() sψ() s ρ() s ds, C q() sψ() s ρ() s ds (5.7) (5.8) To convrt quation (5.7) into a st of inar quations capab of bing sovd, mutipy by ρ()φ n () and intgrat from to. Aso, quation (5.7) can b mutipid by ρ()q() and intgratd from to rsuting in n jk j n k + m j k σc ρ( ) G φ ( ) φ ( ) C d ρ( ) p( ) φ ( ) C d jk j k j k (5.9) σc ρ( ) G φ ( ) C q( ) d+ ρ( ) p( ) q( ) C d for n,,,. (5.) Th st of quations can now b sovd to dtrmin th ignvaus (natura frquncis). Howvr, thr ar an infinit numbr of trms for th systm of quations. An finit numbr of trms must b sctd rsuting in a finit st of inar quations. Now th intgra quation has bn rducd to soving a inar systm of N+ quations with N+ unknowns. Onc th ignvaus ar obtaind, th mod shaps can b found using quation (5.7). 6. FINITE ELEMENT SOLUTION Th finit mnt mthod is usd as an atrnativ to th anaytic soution obtaind from th intgra quation approach. A finit mnt mod of a controd bam is constructd and th corrsponding mass and stiffnss matrics of th controd structur ar drivd. Standard bam mnts ar usd as a foundation for th mod and ar modifid to incud th mass and stiffnss of th pizo patchs and aso account for th contro momnt inducd by th pizo actuator patch. Th natura frquncis and norma mods of th controd bam ar obtaind by soving th matri quation [ M]{ U} + [ K]{ U} { } (6.) A harmonic soution is assumd for {U} in th form { U} { U} iωt (6.) rsuting in [ K] ω [ M] { U} { } (6.3) For thr to b a nontrivia soution, th dtrminant of th cofficint matri of {Ū} must b zro [ K] ω [ M] (6.4) Th poynomia of quation (6.4) can now b sovd to dtrmin th natura frquncis of th systm and th mod shaps can thn b dtrmind from quation (6.3). 6. Finit Emnt Mod Th finit mnt mod uss a combination of bam mnts and patch mnts. Patch mnts ar modifid bam mnts, which ar ncssary to account for th addd stiffnss and mass of th pizo patchs aong with th contro ffct cratd by th pizo actuator. Standard bam mnts ar usd for th sction of th bam whr thr ar no pizo patchs. Th approimating function for w(,t) is givn by w ( ) wn ( ) + θn( ) + wn 3( ) + θn4( ) (6.5) whr w and w ar th noda transvrs dispacmnts and θ and θ ar th noda rotations. Th shap functions ar th standard bam mnt shap functions 3 3 ( ) 3 +, 3 N N ( ) +, whr is th mnt ngth. 6. Mass Matrics 3 N ( ) (6.6a) N ( ) + (6.6b) Th consistnt mass matri for a uniform two-nod bam mnt is givn by ( ) T [ ] ρ [ ] [ ] m A N N d (6.7) whr [N][N () N () N 3 () N 4 ()] and is a row matri containing th shap functions. (i) Bam Emnt Mass Matri with no Pizo Patchs Th intgra in quation (6.7) can b carrid out using ρ b A b for th bam dnsity and ara and th foowing matri is obtaind ( ) ρ ba b b [ m ] (ii)patch Emnt Mass Matri with Pizo Patchs (6.8) This is simiar to th mass matri obtaind in quation (6.8) but th masss of th snsor and actuator patchs ar addd to th mass of th bam rsuting in ISBN: IMECS 9

5 Procdings of th Intrnationa MutiConfrnc of Enginrs and Computr Scintists 9 Vo II IMECS 9, March 8 -, 9, Hong Kong ( ) ( ρbab + ρpap) m p (6.9) For a on mnt patch N () N () N (), N ( ) N ( ) N ( ), N (), N ( ) 4 (6.5) 6.3 Stiffnss Matrics Th stiffnss matri for a bam mnt is obtaind using th intgra ( ) [ ] [ ] [ ] T k B EI B d (6.) whr [B] is th scond drivativ of [N] with rspct to. (i) Bam Emnt Stiffnss Matri with no Pizo Patchs Th intgra in quation (6.) can b vauatd using E b I b as th stiffnss of th bam to dtrmin th stiffnss matri for a bam mnt with no pizo patchs [ k] ( ) bnding Eb I 3 b (ii) Patch Emnt Stiffnss Matri with Pizo patchs [ ] [ ] ( ) ( ) ( ) p bnding contro k k + k whr [ k ] () is givn by (6.) and [ k] () bnding foowing sction. 6.4 Contro Matri contro (6.) (6.) wi b drivd in th To dtrmin th contro componnt of th stiffnss matri, th contro trm is mutipid by th shap functions and intgratd s w bξ [ H ( a) H ( a) ] Ni( ) d s (6.3) for i,,..,n n whr N n is th numbr of nods and Ni( ) ar th intrpoating functions dfind ovr th ntir domain. Substituting th drivativs of th shap functions into th approimating function and vauating at s and s which in trms of oca mnt coordinats is η and η whr η ( s ) Substitut quations (6.4)-(6.5) back into th origina intgra givn in quation (6.3) and intgrat th rsut by parts to obtain bξ( θ θ) N ( ) ( ) ( ) ( ) i S d Ni S (6.6) Using quations (6.6) and th fact that S ()S (), S ()[δ( a ) δ( a )], and η and η with η( a ), th contro componnt of th stiffnss matri bcoms th foowing for a sing mnt pizo patch [ ] ( k ) contro bξ bξ bξ bξ (6.7) Th patch mnt stiffnss matri with stiffnss (6.) and contro componnts (6.7) is thn dtrmind by quation (6.). 7. NUMERICAL RESULTS To vaidat th intgra quation soution, th rsuts obtaind from th intgra quation soution wi b compard with th finit mnt rsuts. An amp with coocatd pizocramic (PZT) snsor and actuator patchs wi b studid. Th first thr natura frquncis, first thr mod shaps, and tip dispacmnt givn initia dispacmnt and vocity wi b shown. Computr simuations using th Map 9.5 softwar packag ar prformd to obtain th rsuts. Th st of quations from quations (5.9) and (5.) must b truncatd to a finit st of quations in ordr to achiv a numrica soution. It is found that a si trm soution (N5) producs accurat rsuts. A finit mnt cod is aso writtn in Map 9.5 to find th natura frquncis of th controd structur. A tn mnt mod is usd whr th bam is dividd up into. mnt ngths. It was found that whn tn mnts ar usd, th frquncis wr convrging and not changing much whn mor mnts ar addd. Th mod shaps for th first two cas studis ar aso found using a Map 9.5 program. Th structur consists of an auminum bam with PZT-5H snsor and actuator patchs. Th matria proprtis and dimnsions of th bam and pizo patchs ar simiar to th primnt prformd by Xu and Koko [7] and ar istd in Tab I. For th anaysis prformd, ony coocatd snsor and actuator positions ar considrd. Th snsor and actuator aso hav th sam physica proprtis, i.. E p E a E s, ρ p ρ a ρ s, and h p h a h s. dw d [ wn ( ) θ N ( ) wn ( ) θ N ( )] [ wn () θ N () wn () θ N ()] (6.4) ISBN: IMECS 9

6 Procdings of th Intrnationa MutiConfrnc of Enginrs and Computr Scintists 9 Vo II IMECS 9, March 8 -, 9, Hong Kong Tab I: Proprtis of bam and pizo patchs Bam Proprtis Eastic Moduus E b 68 GPa Dnsity ρ b 8 kg/m 3 Width b.5 m Hight h b.965 mm Lngth.6 m PZT Proprtis Eastic Moduus E p 6 GPa Dnsity ρ p 75 kg/m 3 Width b.5 m Hight h p.75 mm Pizo Constants d V/m h V/m CASE I. Th Pizo Patchs at Coocatd Locations s a. and s a.3. Th frquncis found from th intgra quation and th finit mnt soutions for this cas ar prsntd in Tab II. Th first thr natura frquncis for various gain vaus ar shown. Tab II: Natura frquncy rsuts Gain Intgra Equation (rad/sc) Finit Emnt (rad/sc) ω.3. ω ω ω.7.6 ω ω ω ω ω ω ω ω Th first thr mod shaps for this cas using dispacmnt fdback contro dtrmind from th intgra quation soution and th finit mnt soution ar dispayd in Fig.. Th finit mnt mod is rprsntd by dashd in whi a soid in rprsnts th mod shap obtaind from th intgra quation. Fig. First Thr Mod shaps CASE II: Various Patch Configurations Th first thr natura frquncis using th intgra quation soution for coocatd patchs starting nar th bas of th bam with a s 6 and tnding in. incrmnts ar istd in Tab III. Tab III: Natura frquncis for various patch ocations (rad/sc) a s Patch Locations Gain ω ω ω ω ω ω ω ω ω ω ω ω CONCLUSION Anaytic and finit mnt soutions ar obtaind for two cas studis using dispacmnt fdback contro. Th first thr natura frquncis and mod shaps ar found for th intgra quation and finit mnt soutions. No anaytic soution istd prviousy for a controd bam that incuds th mass and stiffnss of th pizo patchs or no primnt has bn prformd to find th natura frquncy of a controd structur with pizocramic patchs. Sinc thr was no primnta data to compar th anaytic soution to, a finit mnt mod is usd to vrify th intgra quation soution. Th natura frquncy rsuts for th two cas studis obtaind from th intgra quation soution matchd vry cosy with th finit mnt soution, thrfor vrifying th anaytic soution. Th mod shaps found for Cass I and II using th intgra quation soution aso matchd vry cosy with th finit mnt soution. REFERENCES [] S. Tzou, Pizoctric Shs, Kuwr Acadmic Pubishrs, 993. [] H. T. Banks, D.J. Inman, D. J. Lo, and Y. Wang, An primntay vaidatd dtction thory in smart structurs, Journa of Sound and Vibration, vo. 9, 996, pp [3] N. D. Maw, and S. F. Asokanthan, Moda charactristics of a fib bam with mutip distributd actuators, Journa of Sound and Vibration, vo. 69, 4, pp [4] J. Tani, S. Chonan, Y. Liu, F. Takahashi, K. Ohtomo, and Y. Fuda, Y., Digita activ vibration contro of a cantivr bam with pizoctric actuators, JSME Intrnationa Journa, vo. 34, 99, pp [5] J. M. Soss, J.C. Bruch, Jr., S. Adai, and I. S. Sadk, Pizoctric patch contro using an intgra quation approach, Thin-Wad Structurs, vo. 39,, pp [6] J. M. Soss, J.C. Bruch, Jr., S. Adai, and I. S. Sadk, Intgra quation approach for bams with muti-patch pizo snsors and actuators, Journa of Vibration and Contro, vo. 8,, pp [7] S. X. Xu, and T.S. Koko, Finit mnt anaysis and dsign of activy controd pizoctric smart structurs, Finit Emnts in Anaysis and Dsign, vo. 4, 4, pp. 4-6 ISBN: IMECS 9

SME 3033 FINITE ELEMENT METHOD. Bending of Prismatic Beams (Initial notes designed by Dr. Nazri Kamsah)

SME 3033 FINITE ELEMENT METHOD. Bending of Prismatic Beams (Initial notes designed by Dr. Nazri Kamsah) Bnding of Prismatic Bams (Initia nots dsignd by Dr. Nazri Kamsah) St I-bams usd in a roof construction. 5- Gnra Loading Conditions For our anaysis, w wi considr thr typs of oading, as iustratd bow. Not: