Physical insight into Timoshenko beam theory and its modification with extension

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1 Structural Egieerig ad Mechaics, Vol. 48, No. 4 (013) DOI: Physical isight ito Timosheko beam theory ad its modificatio with extesio Ivo Sejaović ad Nikola Vladimir Faculty of Mechaical Egieerig ad Naval Architecture, Uiversity of Zagreb, Ivaa Lučića 5, Zagreb, Croatia (Received July 4, 013, Revised October 9, 013, Accepted November 1, 013) Abstract. A outlie of the Timosheko beam theory is preseted. Two differetial equatios of motio i terms of deflectio ad rotatio are comprised ito sigle equatio with deflectio ad aalytical solutios of atural vibratios for differet boudary coditios are give. Double frequecy pheomeo for simply supported beam is ivestigated. The Timosheko beam theory is modified by decompositio of total deflectio ito pure bedig deflectio ad shear deflectio, ad total rotatio ito bedig rotatio ad axial shear agle. The goverig equatios are codesed ito two idepedet equatios of motio, oe for flexural ad aother for axial shear vibratios. Flexural vibratios of a simply supported, clamped ad free beam are aalysed by both theories ad the same atural frequecies are obtaied. That fact is proved i a aalytical way. Axial shear vibratios are aalogous to stretchig vibratios o a axial elastic support, resultig i a additioal respose spectrum, as a ovelty. Relatioship betwee parameters i beam respose fuctios of all type of vibratios is aalysed. Keywords: Timosheko beam theory; flexural vibratio; axial shear vibratio; vibratio parameter; aalytical solutio; double frequecy pheomeo 1. Itroductio Beam is used as a structural elemet i may egieerig structures like frame ad grillage oes (Pilkey 00, Pavazza 007, Carrera et al. 011). Moreover, the whole complex structure ca be modelled as a beam to some exted like ship hulls, floatig airports, etc (Sejaović et al. 009). The Euler-Beroulli theory is widely used for simulatio of a sleder beam behaviour. For thick beam Timosheko theory has bee developed by takig shear ifluece ad rotary iertia ito accout (Timosheko 191, 19). Shear effect is extremely large i higher vibratio modes due to reduced mode half wave legth. The Timosheko beam theory deals with two differetial equatios of motio with deflectio ad cross-sectio rotatio as the basic variables (Timosheko 191, 19). The system is reduced ito a sigle four order partial differetial equatio by Timosheko (1937), where oly approximate solutios are give as commeted i (Ima 1994) ad (va Resburg ad va der Merve 006). I the most papers the first approach with two differetial equatios is used i order to esure cotrol of exact ad complete beam behaviour, (Geist ad McLaughli 1997, va Correspodig author, Ph.D., ikola.vladimir@fsb.hr Copyright 013 Techo-Press, Ltd. ISSN: (Prit), (Olie)

2 50 Ivo Sejaović ad Nikola Vladimir Resburg ad va der Merve 006). Possibility to operate with sigle equatio of motio i terms of pure bedig deflectio is oticed ad recetly used, due to reaso of simplicity, as a approximated but reliable eough solutio (Sejaović et al. 1989, Li 008). The Timosheko beam theory is applied as a base for more complex problems, like beam vibratios o elastic foudatio (De Rosa 1995), beam vibratios ad bucklig o elastic foudatio (Matsuaga 1999), vibratios of double-beam system with trasverse ad axial load (Stojaović ad Kozić 01), vibratio ad stability of multiple beam systems (Stojaović et al. 013), beam respose movig to load (Siady 008), etc. Recetly, the Timosheko beam theory is used i aotechology for vibratio aalysis of aotubes, as for istace (Simsek 011). Timosheko idea of shear ad rotary iertia ifluece o deflectio is ot oly limited to beams. These effects are also icorporated i the Midli thick plate theory as a D problem (Midli 1951). Timosheko beam static fuctios are ofte used as coordiate fuctios for thick plate vibratio aalysis by the Rayleigh-Ritz method (Zhou 001). Furthermore, differetial equatio of beam torsio, with shear ifluece is based o aalogy with that for beam bedig (Pavazza 005). Hece, i case of coupled flexural ad torsioal vibratios of a girder with ope crosssectio the same mathematical model is used for aalysis of both resposes (Sejaović et al. 009). The Timosheko beam theory plays a importat role i developmet of sophisticated beam fiite elemets. Various fiite elemets have bee worked out i the last decades. They are distiguished i the choice of iterpolatio fuctios for mathematical descriptio of deflectio ad rotatio. Applicatio of the same order polyomials leads to so-called shear lockig, sice bedig strai eergy for a sleder beam vaishes before shear strai eergy. If static solutio of Timosheko beam is used for deflectio ad rotatio fuctios this problem is overcome (Reddy 1997, Sejaović et al. 009). I spite of the fact that may papers have bee published o Timosheko beam theory durig log period of time, it seams that all pheomea hidde i that theory are ot yet ivestigated. Motivated by the state-of-the art, some additioal ivestigatio has bee udertake ad the obtaied results preseted i this paper shed more light o the cosidered subject. I Sectio a outlie of the Timosheko beam theory is preseted, where basic equatios i terms of deflectio ad cross-sectio rotatio are listed, ad geeral solutio for atural vibratios is give. I Sectio 3 the Timosheko beam theory is modified i such a way that deflectio is split ito pure bedig deflectio ad shear deflectio, while rotatio is decomposed ito cross-sectio rotatio due to pure bedig ad axial shear agle, as a ovelty. Applicatio of both theories is illustrated i Sectio 4 withi umerical examples for simply supported, clamped ad free beam. I Sectio 5 compariso of the theories is doe. It is foud that flexural part of the modified beam theory, used i the literature as a approximate alterative, is actually rigorous as that based o the origial theory. Axial shear vibratios extracted from the Timosheko beam theory, gives a additioal atural frequecy spectrum. I Appedix A frequecy equatios for clamped ad free Timosheko beam are specified, ad i Appedix B the same is doe for the modified beam theory. Liear relatio betwee the above frequecy equatios is preseted i Sectio 5. A detail aalysis of vibratio parameters i argumets of hyperbolic ad trigoometric fuctios i solutios of beam respose is performed i Appedix C. Their exact asymptotic values as fuctio of frequecy are specified, that is a improvemets comparig to the kow approximate values. It is cofirmed that double frequecy spectrum is pheomeo related oly to the simply supported beam. I that way dilemma cocerig this subject is overcome. I Sectio 6 valuable coclusios based o the performed detail aalysis are draw.

3 Physical isight ito Timosheko beam theory ad its modificatio with extesio 51. Timosheko beam theory.1 Basic equatios Timosheko beam theory deals with beam deflectio ad agle of rotatio of cross-sectio, w ad ψ, respectively (Timosheko 191, 19). The sectioal forces, i.e., bedig momet ad shear force read w M D, Q S x x where D=EI is flexural rigidity ad S=kGA is shear rigidity, A is cross-sectio area ad I is its momet of iertia, k is shear coefficiet, ad E ad G=E/((1+ ν)) is Youg's modulus ad shear modulus, respectively. Value of shear coefficiet depeds o beam cross-sectio profile (Cowper 1966, Sejaović ad Fa 1990). Stiffess properties for complex thi-walled girder are determied by the strip elemet method (Sejaović ad Fa 1993). Beam is loaded with trasverse iertia load per uit legth, ad distributed bedig momet w, x qx m m J t where m=ρa is specific mass per uit legth ad J=ρI is its momet of iertia. Equilibrium of momets ad forces leads to two coupled differetial equatios From (5) yields t M Q Q mx, q x x w D S J x x t 0 w w S x x t m 0 w m w x x S t ad by substitutig (6) ito (4) derived per x, oe arrives at the sigle beam differetial equatio of motio w J m w m J w x D S x t D t S t 4 4 w 0 4 Oce (7) is solved agle of rotatio is obtaied from (6) as x (1) () (3) (4) (5) (6). (7) w m w d x f t x S (8) t

4 5 Ivo Sejaović ad Nikola Vladimir where f(t) is rigid body motio. If w is extracted from (4) ad substituted i (5) the same type of differetial equatio as (7) is obtaied for ψ ad (8) for w.. Geeral solutio of atural vibratios I atural vibratios w=w si ωt ad ψ=ψ si ωt, ad Eqs. (7) ad (8) are reduced to the vibratio amplitudes where 4 d W J m d W m J 1 W 0 4 dx D S dx D S dw dx Ψ Wdx C (9) m S. (10) Solutio of (9) ca be assumed i the form W=Ae γx that leads to biquadratic equatio Roots of (11) read where i 1 ad a 4 a b 0 (11) J m, m J 1 D S D S. (1) b,, i, i (13) m J 4m m J S D D S D (14) m J 4m m J S D D S D. (15) Deflectio fuctio with its derivatives ad the first itegral ca be preseted i the matrix form W sh x ch x si x cos x W ch x sh x cos x si x A 1 sh ch si cos A W x x x x W ch x sh x cos x si x A A 4 Wx d ch x sh x cos x si x. (16) Accordig to the solutio of Eq. (9), Eq. (10) ad Eq. (1), beam displacemets ad forces read

5 Physical isight ito Timosheko beam theory ad its modificatio with extesio 53 W Ash x A ch x A si x A cos x (17) m m m m Ψ 1 A 1ch x 1 A sh x 1 A 3 cos x 1 A 4 si x S S S S m m M D A1 sh x Ach x A3 si x A4 cos x S S (18) (19) m Q A1 ch x Ash x A3 cos x A4 si x. (0) Relative values of costats A i, i=1,,3,4, are determied by satisfyig four boudary coditios. Sice there is o additioal coditio costat C i (10) is igored. Coefficiet α, Eq. (14), ca be zero, i which case 0 S/ J ad S D m J 0 / /. Deflectio fuctio accordig to (17) takes the form W A x A A si x A cos x (1) where the first two terms describe rigid body motio. If 0, the i, where ad deflectio fuctio reads m J m J 4m S D S D D () W A si x A cos x A si x A cos x. (3) Expressios for displacemets ad forces Eqs. (17)-(0) have to be trasformed accordigly. Hece, chx cosx, shx isix, where imagiary uit is icluded i costat A 1,, istead of sigle factor α it is ecessary to write, ad fially all fuctios associated with A 1 ad A must have the same sig as those with A 3 ad A 4. The above aalysis shows that beam has a lower ad higher spectral respose, ad trasitio oe. Frequecy spectra are shifted for threshold frequecy ω 0. This problem is also ivestigated i (Geist ad McLaughli 1997, va Resburg ad va der Merve 006, Li 008). The basic differetial Eqs. (4) ad (5) are solved i (va Resburg ad va der Merve 006) by assumig solutio i the form w=ae γx ad ψ=be γx ad the same expressios for displacemets (17) ad (18) are obtaied..3 Simply supported beam Origi of the coordiate system is located i the middle of beam legth due to reaso of simplicity. Symmetric atural modes for lower frequecy spectrum are cosidered for which costat A 1 =A 3 =0. Boudary coditios read W(l/)=0 ad M(l/)=0, ad oe obtais from (18) ad (19) system of equatios

6 54 Ivo Sejaović ad Nikola Vladimir l l ch cos A 0. (4) m l m l A4 0 ch cos S S Its determiat has to be equal to zero for o-trivial solutio l l. (5) Det ch cos 0 The above frequecy equatio is satisfied if βl/=( 1)π/. By employig expressio (15) for β, yields where Two positive solutios of (6) read a b 0 (6) a 4 S D S J J m (7) DS 4 1 b,, 1,... (8) Jm l S D J D J DJ J S m S m Sm 1, 4. (9) They characterize the first ad the secod frequecy spectrum, respectively. Relative values of itegratio costats ca be determied from the first of Eq. (4) A l l cos 0, A ch. (30) 4 Sice oe costat is zero, aother is arbitrary ad atural modes read W =A cos(( 1)πx/l), =1,,... If 0 tha frequecy equatio (5) is trasformed ito Now l l l cos cos 0. (31) / 1 / ad by employig () for the same expressio for atural frequecies as i the previous case is obtaied, i.e., Eq. (9). Ratio of the itegratio costats is A A 4 l cos 0 (3) l cos 0

7 Physical isight ito Timosheko beam theory ad its modificatio with extesio 55 ad both costats are arbitrary, that results i the commo atural modes W =A cos(( 1)πx/l), where > 0, 0 = β 0 l/π. Hece, for a iteger two frequecy spectra exist, oe due to β ad aother due to, which are shifted for ω 0. Sice l/ l/ their atural modes are idetical. I similar way eigepairs for atisymmetric modes takig A A4 0 ito accout, ca be determied. I that case siβ l/=0 ad si / 0, that requires l / l /, 1,... l Formula (9) for atural frequecies is valid with / l. Itegratio costats are expressed with sh ad si fuctios i a previously aalogous way. Natural frequecies ca be also directly determied from differetial Eq. (9) by assumig atural modes i the form W =A si(πx/l). Formula (9) is obtaied with / l, 1,... that icludes both symmetric ( 1,3...) ad atisymmetric (,4...) modes. Double frequecy pheomeo is aalysed i (va Resburg ad va der Merve 006), startig from basic Eqs. (4) ad (5) with two variables, ad the same results as preseted above are obtaied..4 Clamped beam Symmetric atural modes are cosidered, takig A 1 =A 3 =0. Boudary coditios read W(l/)=0 ad Ψ(l/)=0 ad oe obtais by employig Eqs. (18) ad (19) frequecy equatio for lower spectrum (A1) show i Appedix A. The itegratio costats are represeted with Eq. (30). Frequecy equatio for atisymmetric modes is obtaied by takig costats A =A 4 =0, Eq. (A). I similar way frequecy equatios for symmetric ad atisymmetric modes for higher spectrum are specified, Eqs. (A3) ad (A4), respectively..5 Free beam I this case boudary coditios read M(l/)=0 ad Q(l/)=0. Frequecy equatios for lower ad higher spectrum, ad symmetric ad atisymmetric modes, are also give i Appedix A, Eqs. (A5), (A6), (A7) ad (A8), respectively. 3. Modified beam theory 3.1 Differetial equatios of motio Beam deflectio w ad agle of rotatio ψ are split ito their costitutive parts, Fig. 1, i.e. wb w wb ws,,, (33) x where w b ad w s is beam deflectio due to pure bedig ad trasverse shear, respectively, ad φ is agle of cross-sectio rotatio due to bedig, while ϑ is cross-sectio slope due to axial shear. Equilibrium Eqs. (4) ad (5) ca be preseted i the form with the separated variables w b ad w s, ad ϑ

8 56 Ivo Sejaović ad Nikola Vladimir Fig. 1 Thick beam displacemets (a) total deflectio ad rotatio w,ψ, (b) pure bedig deflectio ad rotatio w b,φ, (c) trasverse shear deflectio w s, (d) axial shear agle ϑ 3 wb wb ws D J S D S J 3 x t x x x t (34) w S m w w S s b s x t x. (35) Sice oly two equatios are available for three variables oe ca assume that flexural ad axial shear displacemet fields are ot coupled. I that case, by settig both left ad right had side of (34) zero, yields from the former D wb J wb ws. (36) S x S t By substitutig (36) ito (35) differetial equatio for flexural vibratios is obtaied, which is expressed with pure bedig deflectio 4 4 wb J m wb m J w b S w 4 b x D S. (37) x t D t S t D x Disturbig fuctio o the right had side i (37) ca be igored due to assumed ucouplig. Oce w b is determied, the total beam deflectio, accordig to (33), reads D w J w w wb S x S t b b. (38) The right had side of (34) represets differetial equatio of axial shear vibratios

9 Physical isight ito Timosheko beam theory ad its modificatio with extesio 57 S J x D D t 0. (39) 3. Geeral solutio of flexural atural vibratios Natural vibratios are harmoic, i.e., w b =W b siωt ad ϑ=θsiωt, so that equatios of motio (37) ad (39) are related to the vibratio amplitudes 4 d Wb J m d Wb m J 1 W 0 4 b dx D S dx D S (40) d Θ S J dx D S Θ Amplitude of total deflectio, accordig to (38), reads 1 0. (41) J d b W D W b W 1 S S dx. (4) Eq. (40) is kow i literature as a reliable alterative of Timosheko differetial equatios, (Sejaović ad Fa 1989, Sejaović et al. 009, Li 008). By comparig (40) with (9) it is obvious that differetial equatio of flexural vibratios of the modified beam theory is of the same structure as that of Timosheko beam theory, but they are related to differet variables, i.e., W ad W b deflectio, respectively. Therefore, geeral solutio for W preseted i Sectio. is valid for W b with all derivatives. I that case flexural displacemets ad sectioal forces read J D J D W B11 sh x B 1 ch x S S S S J D J D B31 si x B4 1 cos x S S S S (43) dw Φ b B1 ch x Bsh x B3 cos x B4 si x dx (44) d W M D b D B1 sh x B ch x B3 si x B4 cos x dx (45) 3 d Wb dwb J J Q D J D B 3 1 ch x B sh x dx dx D D J J B3 cos x B4 si x. D D (46)

10 58 Ivo Sejaović ad Nikola Vladimir Fig. Aalogy betwee axial shear model ad stretchig model Parameters α ad β are specified i Sectio., Eqs. (14) ad (15), respectively. I this case also parameter α ca be zero that gives 0 S/ J ad S D m J 0 / /. By takig this fact ito accout, bedig deflectio W b is of the form (1), while total deflectio accordig to (43), reads D W B1 x B 0 B3 si 0x B4 cos 0x S (47) where B 1 ad B are ew itegratio costats istead of B 1 ad B, which are ifiite due to zero coefficiets. Cocerig the higher order frequecy spectrum the goverig expressios for displacemets ad forces, Eqs. (43)-(46), have to be trasformed i the same maer as explaied i Sectio Geeral solutio of axial shear atural vibratios Differetial Eq. (41) for atural axial shear vibratios of beam reads It is similar to the equatio for rod stretchig vibratios d Θ J S Θ 0 dx D D. (48) d u m 0 R u. (49) dx EA Differece is additioal momet SΘ, which is associated to iertia momet ω JΘ, ad represets reactio of a imagied rotatioal elastic foudatio with stiffess equal to the shear stiffess S, as show i Fig.. du Solutio of (49) ad correspodig axial force N EA read d x where R m / EA agle ad momet u C si x C cos x (50) 1 1 N EA C cos x C si x (51). Based o aalogy betwee (48) ad (49) oe ca write for shear slope

11 Physical isight ito Timosheko beam theory ad its modificatio with extesio 59 where Θ C six C cosx (5) 1 1 M D C cos x C six (53) J S. (54) D D Betwee atural frequecies of axial shear beam vibratios ad stretchig vibratios there is relatio, where 0 S/ J belogs to the axial shear mode obtaied from (44), 0 R Θ A0 A1x, (which associates o sheared set of playig cards). It is iterestig that 0 is at the same time threshold frequecy of flexural vibratios, as explaied i Sectio Simply supported beam Let us cosider symmetric modes for which A 1 =A 3 =0, ad boudary coditios W(l/)=0 ad M(l/)=0. By employig formulae (43) ad (45) oe obtais frequecy equatio i the form J l l S Det 1 ch cos 0. (55) It icludes additioal factor comparig to (5) based o Timosheko beam theory, from which threshold frequecy 0 S/ J is determied. Sice the remaied part of (55) is idetical to (5), everythig what is writte i Sectio.3 is valid i this case icludig formula (9) for atural frequecies. If double frequecy pheomeo is aalysed i the same way as that for Timosheko beam, the same results are obtaied. 3.5 Clamped beam Boudary coditios read W=0 ad Φ=0 at x=±l/. By employig (43) ad (44) oe obtais frequecy equatios for the first respose spectrum ad symmetric ad atisymmetric modes listed i Appedix B, Eqs. (B1) ad (B). I a similar way, after modificatio of Eqs. (43) ad (44) for higher spectrum, the obtaied frequecy equatios are preseted by Eqs. (B3) ad (B4). 3.6 Free beam For a free beam M=0 ad Q=0 at x=±l/. By employig (45) ad (46) oe obtais frequecy equatios for the first respose spectrum ad symmetric ad atisymmetric modes show i Appedix B, Eqs. (B5) ad (B6). Frequecy equatios for higher spectrum, after modificatio of Eqs. (45) ad (46), are represeted by Eqs. (B7) ad (B8). 3.7 Axial shear vibratios A beam performig axial shear vibratios ca be fixed or free at both eds, or oe ed ca be fixed ad aother free. Mode fuctio Θ =Csiη x, η =π/l, =1,... satisfies boudary coditios

12 530 Ivo Sejaović ad Nikola Vladimir for fixed beam Θ(0)= Θ(l)=0. By takig ito accout Eq. (54) for η oe obtais expressio for atural frequecies S D J J l. (56) If beam is free M(0)=M(l)=0, ad frequecy equatio reads η siηl=0. The first coditio η=0 gives accordig to (54) threshold frequecy 0 S/ J, while the secod coditio siηl=0 requires η l=π/l. Hece, atural frequecies are represeted by Eq. (56) ad atural mode is Θ =Csiη x. For combied fixed-free boudary coditios, Θ(0)=0 ad M(l)=0, frequecy equatio reads ηcosηl=0. Agai, η=0 gives ω 0 ad cosηl=0 requires η =( 1)π/l, =1,... Expressio for atural frequecies reads Natural mode is Θ C si x. S D 1 J J l. (57) 4. Illustrative umerical examples 4.1 Simply supported beam A beam of I-profile with height-to-legth ratio h/l=0. ad shear coefficiet k=5/6 is aalysed. Due to reaso of simplicity dimesioless frequecy parameter λ=ω/ω 0 is itroduced. Natural frequecies of flexural vibratios are give by (9) ad frequecy parameter ca be preseted i the form where f 1, c c d (58) I c 1, e, d e e. (59) k k Al Its values for the first ad secod frequecy spectrum are listed i Table 1. They are the same for both Timosheko beam theory (TBT) ad modified theory (MBT). I flexural vibratios of a simply supported beam, agle of rotatio is free. Therefore, let us cosider axial shear vibratios of free beam. Natural frequecies are give by (56) ad frequecy parameter ca be preseted i the form s 1 1 e. (60) k

13 Physical isight ito Timosheko beam theory ad its modificatio with extesio 531 Table 1 Frequecy parameter λ=ω/ω 0 of simply supported beam, h/l=0. Flexural, TBT ad MBT 1 st 1 spectrum, f d spectrum, Stretchig, f t Axial Shear, * 1.000* * 1.000* *Threshold s The secod term i (60) belogs to the stretchig vibratios ad values for both parameters are 1 listed i Table 1. Values of are larger tha f due to higher tesioal tha flexural stiffess. t f s Both the secod flexural spectrum,, ad axial shear spectrum,, start with threshold parameter 0 1, ad it is iterestig that they are very close i spite of differet umber of modal odes, Table Clamped beam Values of atural frequecies for TBT i the lower ad higher spectrum are determied by frequecy equatios (A1), (A), (A3) ad (A4) for symmetric ad atisymmetric modes. Eqs. (B1), (B), (B3) ad (B4) are used for determiig frequecies of MBT. Values of frequecy parameters are equal for both TBT ad MBT ad are listed i Table. Frequecy parameter for axial shear vibratios of fixed beam, which is equal to that of free beam is also listed i Table. I spite of the fact that ad start with the threshold value λ 0 =1, they diverge for higher modes. 4.3 Free beam fh j s Values of atural frequecies accordig to TBT ad MBT are determied by Eqs. (A5), (A6), (A7) ad (A8), ad Eqs. (B5), (B6), (B7) ad (B8), respectively. Values of frequecy parameters are equal ad are show i Table 3, together with those for axial shear vibratios, which are the same as i the previous cases.

14 53 Ivo Sejaović ad Nikola Vladimir Table Frequecy parameter λ=ω/ω 0 of clamped beam, h/l=0., k=5/6 Mode o. j Lower spectrum, Flexural TBT ad MBT fl j Higher spectrum, fh j Axial shear, * * 1.000* 1.000* *Threshold Table 3 Frequecy parameter λ=ω/ω 0 of free beam, h/l=0., k=5/6 Mode o. j Lower spectrum, Flexural TBT ad MBT fl j Higher spectrum, Axial shear, * * 1.000* 1.000* *Threshold fh j s j s j 5. Compariso of Timosheko beam theory ad modified beam theory 5.1 Natural frequecies Timosheko beam theory deals with two differetial equatios of motio with two basic

15 Physical isight ito Timosheko beam theory ad its modificatio with extesio 533 variables, i.e., deflectio ad agle of rotatio. That system is reduced to oe equatio i terms of deflectio ad all physical quatities deped o its solutio. O the other side, i the modified beam theory total deflectio is split ito pure bedig deflectio ad shear deflectio, while total agle of rotatio cosists of pure bedig rotatio ad axial shear agle. The goverig equatios are codesed ito sigle oe for flexural vibratios with bedig deflectio as the mai variable, ad aother for axial shear vibratios. Differetial equatios for flexural vibratios i both theories are of the same structure so that expressios for atural frequecies of simply supported beam are idetical. Numerical examples show that values of atural frequecies for other boudary coditios are also the same, i spite of the fact that frequecy equatios are differet. Such a result is ot expected sice for clamped Timosheko beam boudary agle Ψ Φ Θ 0, while i the modified theory oly Φ 0. Hece, oe could coclude that the Timosheko beam theory will give somewhat higher frequecy values tha the modified theory due to fixatio of the complete agle. Similar situatio occurs i case of free beam, where total momet for Timosheko beam MΨ MΦ M Θ 0, ad i the modified theory M Φ 0. Equal atural frequecies of flexural vibratios determied umerically ca ot be accepted as a rule. That fact should be cofirmed i a aalytical way. By comparig, for istace, frequecy equatios for clamped beam i lower spectrum ad symmetric modes, Eqs. (A1) ad (B1), they have the same fuctios but differet coefficiets. Sice the equatios give the same atural frequecies, their coefficiets should be proportioal. These equatios ca be writte i matrix otatio J D J D 1 1 l l ch si S S S S 0. (61) m m l l sh cos S S To meet the above coditio of equal frequecies, determiat of the system (61) has to be zero. After some algebra determiat ca be preseted i the form 4 Det DS Sm Jm. (6) S By substitutig Eqs. (14) ad (15) for α ad β ito (6) yields that the term i the brackets is zero. I similar way oe ca prove that determiats of all pairs of frequecy equatios (A i ) ad (B i ), i=1,...8 i Appedix A ad B respectively, are zero. That is also valid for a beam with mixed boudary coditios, i which case complete expressios for displacemets ad forces with all four itegratio costats are take ito accout. 5. Natural modes Formulas for displacemets ad forces i TBT ad MBT, Eqs. (17)-(0) ad (39)-(4), respectively, are expressed with the same hyperbolic ad trigoometric fuctios but their coefficiets are differet. Hece, it is ecessary to compare mode shapes determied by TBT ad MBT. For that purpose the previous example of clamped beam with symmetric modes i the lower frequecy spectrum is take ito cosideratio. Natural modes are characterized by shape, while their amplitude is arbitrary. From the first

16 534 Ivo Sejaović ad Nikola Vladimir equatio i (8) oe fids l ch A 4 A (63) l cos where costat A is chose as referet oe. I order to esure the same total beam deflectio withi TBT ad MBT, Eqs. (17) ad (39), respectively, the followig relatios for itegratio costats i (39) have to be applied B 1 J 1 S A D S B 1 J 1 S 4 4 (64) A. (65) D S Numerical calculatio of displacemets ad forces withi TBT ad MBT is performed for the followig iput data: h= m, l=10 m, E= N/m, ν =0.3, ρ =7850 kg/m 3. Natural frequecy is calculated from kow frequecy parameter λ i i Table, as ω j = ω 0 λ j. Diagrams for total deflectio, W T ad W M, agle of rotatio, Ψ ad Φ, bedig momet, M T ad M M, ad shear force, Q T ad Q M, for the first mode are show i Fig. 3. Exactly the same values for TBT ad MBT are obtaied. Bedig deflectio W b ad shear deflectio W s, determied withi MBT, are also icluded i Fig. 3. Their boudary values are cacelled, resultig i zero edge total deflectio. Shape of shear deflectio mode is similar to that of bedig momet, as result of their structure, Eqs. (38) ad (41), respectively. Diagrams of displacemets ad forces for the fifth mode determied by TBT ad MBT, are show i Fig. 4, ad also are idetical. Boudary values of bedig deflectio ad shear deflectio are quite large, but their sum is zero. Equal displacemet ad force modes determied by TBT ad MBT idicate that coefficiets i correspodig equatios are idetical ad this ca be proved aalytically. Let us compare, for istace, the secod coefficiet i the TBT shear force ad that of MBT, Eqs. (0) ad (4), respectively m J A D B D. (66) By takig ito accout (64) the above relatio ca be preseted i the form J J D m 1 0 D S S. (67) By substitutig Eq. (14) for α ito (67) all terms are cacelled. Beam deflectio W is expressed with hyperbolic ad trigoometric fuctios, Eq. (17). The latter are related to simply supported beam ad the former compesate boudary ifluece, which is reduced to local effect for higher modes, as ca be see by comparig the first ad fifth modes

17 Physical isight ito Timosheko beam theory ad its modificatio with extesio 535 Fig. 3 The first flexural mode of clamped beam, A = 1 m Fig. 4 The fifth flexural mode of clamped beam, A = 0.1m show i Figs. 3 ad 4. After threshold frequecy ω 0 boudary iterferece almost disappears ad modes are expressed oly with trigoometric fuctios as i the case of simply supported beam. Therefore, atural frequecies ω j > ω 0 of clamped ad free beam are very close, Tables ad 3,

18 536 Ivo Sejaović ad Nikola Vladimir ad are of the same order of magitude as those i the first frequecy spectrum of simply supported beam, Table 1. Vibratio parameters i argumets of trigoometric fuctios coverge to the asymptotic value J / D ad m/ S with frequecy icreased, as elaborated i Appedix C. Axial shear vibratios are aalysed withi MBT assumig zero deflectio. Their first mode occurs at threshold frequecy 0, which correspods to trasitio flexural mode, Eq. (1), with a larger umber of modal odes where deflectio is zero. Hece, assumptio of ucoupled flexural ad axial shear vibratio is realistic. The same differetial equatio for axial vibratio as (37) ca be obtaied i TBT from Eq. (4), by igorig deflectio. 5.3 Static solutio Compariso of TBT ad MBT for static aalysis is also iterestig. Oe expects that expressios for static displacemets ca be obtaied directly by deductio of dyamic expressios. I case of TBT static term of Eq. (9) leads to W=A 0 +A 1 x+a x +A 3 x 3, ad Eq. (10) gives Ψ = (A 1 +A x+3a 3 x ). That results i zero shear force Q, Eq. (1), ad is also obvious from (0) if ω=0 is take ito accout. Therefore, i order to overcome this problem, it is ecessary to retur back to Eqs. (4) ad (5) with static terms. By substitutig (5) ito (4), yields Dd 3 Ψ /dx 3 =0, i.e., Ψ = (A 1 +A x+3a 3 x ). Based o kow Ψ, oe obtais from (4) D dψ D W Ψ x A A A x A x A x A A x S dx S 3 d (68) O the other side, static part of Eq. (36) of MBT gives W b =B 0 +B 1 x+b x +B 3 x 3, ad from (38) directly yields D d Wb 3 D W Wb B 0 B1 x Bx B3x B 3B3x (69) S dx S which is the same as (68). Agle of rotatio is Φ = dw b /dx= (B 1 +B x+3b 3 x ) that is the same as the above Ψ i TBT. If static solutio for W ad Φ, which are strogly depedet, is used for developmet of beam fiite elemet shear lockig, as metioed i the Itroductio, does ot occur. 6. Coclusios The research is motivated by the fact that a overall physical isight ito Timosheko beam theory has ot bee doe after more tha 90 years of its wide ad successful applicatio. The modified Timosheko beam theory is result of such ivestigatio. Based o the performed comparative aalysis betwee the Timosheko beam theory (TBT) ad the modified beam theory (MBT), the followig coclusios are draw: TBT deals with two differetial equatios of motio with total deflectio ad rotatio, which are codesed ito sigle equatio i terms of deflectio. I MBT total deflectio is split ito pure bedig deflectio ad trasverse shear deflectio, ad total rotatio is decomposed ito bedig agle ad axial shear agle. MBT operates with two ucoupled differetial equatios of motio, oe for flexural ad

19 Physical isight ito Timosheko beam theory ad its modificatio with extesio 537 aother for axial shear vibratios i terms of pure bedig deflectio ad axial shear agle, respectively. TBT ad MBT flexural differetial equatios are of the same structure ad give the same values of atural frequecies ad mode shapes, ot oly for simply supported beam, but also for ay combiatio of boudary coditios. Two flexural respose spectra are obtaied for simply supported beam by both theories, shifted for threshold frequecy. For a beam with mixed boudary coditios lower frequecy spectrum is obtaied up to threshold frequecy, ad the higher spectrum is cotiued. Double frequecy spectrum does t occur i this case. Natural modes of higher spectrum are siusoidal as i case of simply supported beam, ad ifluece of boudary coditios is cosiderably reduced. Threshold frequecy depeds o shear stiffess ad mass momet of iertia, ad its value is icreased for more sleder beams. Axial shear vibratios result with a additioal frequecy spectrum, which starts with threshold frequecy. Differetial equatio for axial shear vibratios ca also be extracted from Timosheko equatios by assumig zero deflectio. MBT with its differetial equatio is already kow i literature, as a approximate alterative of TBT developed uder some assumptio. The performed comparative aalysis shows that itroduced assumptio actually represets the reality, ad therefore MBT is rigorous theory as well as TBT. Moreover, MBT holds mathematical model of axial shear vibratios, extracted from TBT, which is ot maifested i flexural respose of Timosheko beam sice flexural ad axial displacemet fields are ot coupled. The obtaied results withi this ivestigatio could have some impact o the other aspects of applicatio of the Timosheko beam theory, as referred i the Itroductio, like beam o elastic foudatio, beam stability, elastically coected multiple beams, thick plate, beam ad plate fiite elemets, etc. Timosheko beam theory ad its modificatio are the first order shear deformatio theories. I future work it would be iterestig to ivestigate possibility to exted the modified beam theory to the secod order, as it is doe for Timosheko beam theory by Leviso (1981a, 1981b). High order shear deformatio beam theory is importat for istace for logitudial stregth aalysis of multideck ships like Cruise Vessels. They are characterized with quite stiff hull up to mai deck, ad high ad light superstructure, that maifests o-uiform profile of axial displacemet of ship cross-sectio, (Sejaović ad Tomašević 1999). Ackowledgmets This paper is dedicated to the memory o Stephe Prokofievitch Timosheko, distiguished scietist, who was Professor of Techical Mechaics at the Uiversity of Zagreb from 190 to 19, before he moved to the USA ad fially joied to the Staford Uiversity, ad his famous beam theory published i 191 ad 19, which is still challegig topic of ivestigatio ad subject of applicatio i theory ad practice. The ivestigatio was supported by the Natioal Research Foudatio of Korea (NRF) grat fuded by the Korea govermet (MSIP) through GCRC-SOP (No ).

20 538 Ivo Sejaović ad Nikola Vladimir Refereces Carrera, E., Giuto, G. ad Petrolo M. (011), Beam Structures, Classical ad advaced Theories, Joh Wiley & Sos Ic., New York, NY, USA. Cowper, G.R. (1966), The shear coefficiet i Timosheko s beam theory, J. Appl. Mech., 33, De Rosa, M.A. (1995), Free vibratios of Timosheko beams o two-parametric elastic foudatio, Comput. Struct., 57, Geist, B. ad McLaughli, J.R. (1997), Double eigevalues for the uiform Timosheko beam, Appl. Math. Letters, 10(3), Ima, D.J. (1994), Egieerig Vibratio, Pretice Hall, Ic., Eglewood Cliffs, New Jersey. Leviso, M. (1981a), A ew rectagular beam theory, J. Soud Vib., 74(1), Leviso, M. (1981b), Further results of a ew beam theory, J. Soud Vib., 77(3), Li, X.F. (008), A uified approach for aalysig static ad dyamic behaviours of fuctioally graded Timosheko ad Euler-Beroulli beams, J. Soud Vib., 3(5), Matsuaga, H. (1999), Vibratio ad bucklig of deep beam-colums o two-parameter elastic foudatios, J. Soud Vib., 8, Midli, R.D. (1951), Ifluece of rotary iertia ad shear o flexural motios of isotropic elastic plates, J. Appl. Mech., 18(1), Pavazza, R. (005), Torsio of thi-walled beams of ope cross-sectios with ifluece of shear, It. J. Mech. Sci., 47, Pavazza, R. (007), Itroductio to the Aalysis of Thi-Walled Beams, Kige, Zagreb, Croatia. (i Croatia) Pilkey, W.D. (00), Aalysis ad Desig of Elastic Beams, Joh Wiley & Sos Ic., New York, NY, USA. Reddy, J.N. (1997), O lockig free shear deformable beam elemets, Comput. Meth. Appl. Mech. Eg., 149, Sejaović, I. ad Fa, Y. (1989), A higher-order flexural beam theory, Comput. Struct., 10, Sejaović, I. ad Fa, Y. (1990), The bedig ad shear coefficiets of thi-walled girders, Thi-Wall. Struct., 10, Sejaović, I. ad Fa, Y. (1993), A fiite elemet formulatio of ship cross-sectioal stiffess parameters, Brodogradja, 41(1), Sejaović, I. ad Tomašević, S. (1999), Logitudial stregth aalysis of a Cruise Vessel i early desig stage, Brodogradja, 47(4), Sejaović, I., Tomašević, S. ad Vladimir, N. (009), A advaced theory of thi-walled structures with applicatio to ship vibratios, Mar. Struct., (3), Simsek, M. (011), Forced vibratio of a embedded sigle-walled carbo aotube traversed by a movig load usig olocal Timosheko beam theory, Steel. Comp. Struct., 11(1), Siady, P. (008), Dyamic respose of a Timosheko beam to a movig force, J. Appl. Mech., 75(), Stojaović, V. ad Kozić, P. (01), Forced trasverse vibratio of Rayleigh ad Timosheko double-beam system with effect of compressive axial load, It. J. Mech. Sci., 60, Stojaović, V., Kozić, P. ad Jaevski, G. (013), Exact closed-form solutios for the atural frequecies ad stability of elastically coected multiple beam system usig Timosheko ad higher-order shear deformatio theory, J. Soud Vib., 33, Timosheko, S.P. (191), O the correctio for shear of the differetial equatio for trasverse vibratio of prismatic bars, Phylosoph. Magazie, 41(6), Timosheko, S.P. (19), O the trasverse vibratios of bars of uiform cross sectio, Phylosoph. Magazie, 43, Timosheko, S.P. (1937), Vibratio Problems i Egieerig, d Editio, D. va Nostrad Compay, Ic. New York, NY, USA. va Resburg, N.F.J. ad va der Merve, A.J. (006), Natural frequecies ad modes of a Timosheko

21 Physical isight ito Timosheko beam theory ad its modificatio with extesio 539 beam, Wave Motio, 44, Zhou, D. (001), Vibratios of Midli rectagular plates with elastically restraied edges usig static Timosheko beam fuctios with Rayleigh-Ritz method, Itl. J. Solids Struct., 38,

22 540 Ivo Sejaović ad Nikola Vladimir Appedix A. frequecy equatios for Timosheko beam theory Clamped beam: Lower spectrum, symmetric modes m l l m l l 1 ch si 1 sh cos 0 S S (A1) Lower spectrum, atisymmetric modes m l l m l l 1 sh cos 1 ch si 0 S S (A) Higher spectrum, symmetric modes m l l m l l 1 cos si 1 si cos 0 S S (A3) Higher spectrum, atisymmetric modes m l l m l l 1 si cos 1 cos si 0 S S (A4) Free beam: Lower spectrum, symmetric modes 3 m l l 3 m l l 1 ch si 1 sh cos 0 S S (A5) Lower spectrum, atisymmetric modes 3 m l l 3 m l l 1 sh cos 1 ch si 0 S S (A6) Higher spectrum, symmetric modes 3 m l l 3 m l l 1 cos si 1 si cos 0 S S (A7) Higher spectrum, atisymmetric modes 3 m l l 3 m l l 1 si cos 1 cos si 0 S S (A8)

23 Physical isight ito Timosheko beam theory ad its modificatio with extesio 541 Appedix B. frequecy equatios for modified beam theory Clamped beam: Lower spectrum, symmetric modes J D l l J D l l S S S S 1 ch si 1 sh cos 0 (B1) Lower spectrum, atisymmetric modes J D l l J D l l S S S S 1 sh cos 1 ch si 0 (B) Higher spectrum, symmetric modes J D l l J D l l S S S S 1 cos si 1 si cos 0 (B3) Higher spectrum, atisymmetric modes J D l l J D l l S S S S 1 si cos 1 cos si 0 (B4) Free beam: Lower spectrum, symmetric modes J l l J l l 1 ch si 1 sh cos 0 D D (B5) Lower spectrum, atisymmetric modes J l l J l l 1 sh cos 1 ch si 0 D D (B6) Higher spectrum, symmetric modes J l l J l l 1 cos si 1 si cos 0 D D (B7) Higher spectrum, atisymmetric modes J l l J l l 1 si cos 1 cos si 0 D D (B8)

24 54 Ivo Sejaović ad Nikola Vladimir Appedix C. aalysis of vibratio parameters Vibratio parameters,, ad i argumets of hyperbolic ad trigoometric fuctios of beam respose ca be ormalized i dimesioless form ad preseted as fuctio of threshold frequecy 0 i order to aalyse their relatioship. Beam parameters are the followig: EA 1 m A, J I, D EI, S kga,. (C1) k By employig (C1) threshold frequecy reads S EA 0. (C) J I Ay frequecy ca be expressed as fractio of threshold frequecy, i.e. 0. Terms i Eq. (14) for take the followig form m J 1 (C3) S D E By substitutig the above formulas ito Eqs. (14) ad (15) yields 4m 4 D E. (C4) r r (C5) where r I / A is the radius of gyratio. I the case of threshold frequecy 0, 0 1 ad 0r 0, while For very high frequecies 1, both 1 0r 1. (C6) r ad r coverge to the asymptotic values ar, ar. (C7) I similar way parameter of axial shear vibratios ca be preseted i the form By takig ito accout (C1) oe obtais J S S D D D 1. (C8)

25 Physical isight ito Timosheko beam theory ad its modificatio with extesio 543 Fig. C1 Diagrams of beam vibratio parameters Asymptotic value of r is idetical to that of r, Eq. (C7). 1 r. (C9) Vibratio parameters for rod stretchig vibratio reads R m / EA. By takig ito accout R 0 ad 0 S/ J, oe obtais that r is idetical to asymptotic value r, Eq. a (C7). Diagrams of dimesioless beam vibratio parameters r, r, r, r ad r as fuctio of are show i istructive Fig. C1. Parameter r is trasformed ito r at the threshold frequecy, where 0r 0, while r 0 is preseted with (C6). Both r ad r coverge to asymptotic values which are differet. Parameter of axial shear vibratios r follows r, givig a close higher frequecy spectrum. A similar parametric aalysis is performed by va Resburg ad va der Merve (006), where flexural parameters,, ad as fuctios of are show. However, oly slopes of their asymptotes are determied ad idicated i correspodig figure of (va Resburg ad va der Merve 006) i ituitive positios, which does t provide realistic isight ito parameter covergece. I geeral case atural frequecies are determied from frequecy equatio Det F, 0, which is formulated by satisfyig boudary coditios. Step-by-step umerical procedure is used util such values of coupled vibratio parameters ad meet the above coditio. These values are distict poits i correspodig diagrams show i Fig. C for clamped beam. If beam is simply supported values of parameter pairs are kow a priori r. (C10) r r l

26 544 Ivo Sejaović ad Nikola Vladimir Fig. C Relatios betwee vibratio parameters ad atural frequecies for clamped beam Fig. C3 Relatios betwee vibratio parameters ad atural frequecies for simply supported beam Fig. C4 Relatios betwee vibratio parameters ad atural frequecies for axial vibratios

27 Physical isight ito Timosheko beam theory ad its modificatio with extesio 545 By eterig i parameter diagrams, atural frequecies for the first ad secod spectrum ca be determied as show i Fig. C3. Hece, for oe value of there are two differet frequecies but oe mode shape. I the above way it is proved i a physically trasparet way that double frequecy pheomeo is a characteristic of simply supported beam oly. Axial vibratios have also two spectra for ay boudary coditios, oe for stretchig motio ad aother for shear motio. For give, pairs of frequecies are obtaied, also show i Fig. C4. Correspodig atural modes are of the same shape, but of differet physical meaig.