Nonlinear dynamic stability of damped Beck s column with variable cross-section

Transcription

1 Noninear dynamic stabiity of damped Beck s coumn with variabe cross-section J.T. Katsikadeis G.C. Tsiatas To cite this version: J.T. Katsikadeis G.C. Tsiatas. Noninear dynamic stabiity of damped Beck s coumn with variabe cross-section. Internationa Journa of Non-Linear Mechanics Esevier () pp.64. <0.06/j.ijnoninmec >. <ha > HAL Id: ha Submitted on Ju 00 HAL is a muti-discipinary open access archive for the deposit and dissemination of scientific research documents whether they are pubished or not. The documents may come from teaching and research institutions in France or abroad or from pubic or private research centers. L archive ouverte puridiscipinaire HAL est destinée au dépôt et à a diffusion de documents scientifiques de niveau recherche pubiés ou non émanant des étabissements d enseignement et de recherche français ou étrangers des aboratoires pubics ou privés.

2 Author s Accepted Manuscript Noninear dynamic stabiity of damped Beck s coumn with variabe cross-section J.T. Katsikadeis G.C. Tsiatas PII: S (07) DOI: doi:0.06/j.ijnoninmec Reference: NLM 37 To appear in: Internationa Journa of Non- Linear Mechanics Received date: 4 Juy 006 Revised date: 6 September 006 Accepted date: October 006 Cite this artice as: J.T. Katsikadeis and G.C. Tsiatas Noninear dynamic stabiity of damped Beck s coumn with variabe cross-section Internationa Journa of Non-Linear Mechanics (007) doi:0.06/j.ijnoninmec This is a PDF fie of an unedited manuscript that has been accepted for pubication. As a service to our customers we are providing this eary version of the manuscript. The manuscript wi undergo copyediting typesetting and review of the resuting gaey proof before it is pubished in its fina citabe form. Pease note that during the production process errors may be discovered which coud affect the content and a ega discaimers that appy to the journa pertain.

3 Noninear dynamic stabiity of damped Beck s coumn with variabe cross-section J.T. Katsikadeis * G.C. Tsiatas Schoo of Civi Engineering Nationa Technica University of Athens Zografou Campus GR Athens Greece Abstract In this paper the noninear dynamic stabiity of Beck s coumn with variabe mass and stiffness properties in the presence of damping (both interna and eterna) is investigated using a compete noninear dynamic anaysis. This approach permits the eamination of the goba stabiity of the system in contrast to the static noninear one which though more economica in computationa cost is associated ony with the oss of oca stabiity via futter or divergence. The governing equations describing the dynamic response are derived in terms of the dispacements taking aso into account the aia deformation which has a striking infuence on the critica oad. Since the cross-sectiona properties of the beam vary aong its ais the resuting couped noninear differentia equations have variabe coefficients. Their soution is achieved using the anaog equation method (AEM) of Katsikadeis. Besides its accuracy and effectiveness this method overcomes the shortcoming of a FEM soution which may eperience ack of convergence. Interesting concusions are drawn. The important however finding is that the incusion of the aia deformation affects highy the critica oad of Beck s coumn with varying cross sectiona properties whie it eaves it unatered for Beck s coumn with uniform cross section. Keywords: Beck s coumn; noninear dynamic stabiity; noninear dynamic anaysis; anaog equation method; variabe cross-section * Corresponding author. Te.: ; fa: E-mai addresses: jkats@centra.ntua.gr (J.T. Katsikadeis) gtsiatas@centra.ntua.gr (G.C. Tsiatas).

4 . Introduction The infuence of damping on the stabiity of inear eastic nonconservative uniform coumns has been etensivey investigated. The reated main concusions are the stabiizing effect of the eterna damping [] and to the parado of the destabiizing effect of the interna damping [ 3]. However when nonuniform coumns are eamined [4 5] the infuence of interna damping can have a stabiizing character under certain conditions whie the eterna damping has aways a stabiizing effect [4]. Some recent resuts for inear viscoeastic modes in which creep reaation and hysteresis effects are taken into account can be found in a survey paper by Gau [6]. A specia case of the infuence of damping on Hamitonian autonomous systems is very recenty reported by Kounadis [7]. The stabiity behavior of nonconservative coumns can be eamined ony via noninear anaysis static or dynamic [8-]. The atter one permits the eamination of the goba stabiity of the system instead of the static noninear anaysis which is associated with the oss of oca stabiity via futter or divergence [ 3]. The work that has been done on the noninear dynamic anaysis is imited ony to uniform damped Beck s coumns. The bifurcation may be subcritica or supercritica. The subcritica can ehibit instabiity beow the critica oad (from the inear theory) whie the supercritica is the stabe soution to which a soutions tend independenty of the initia conditions. The initia studies of Kokka [4] and Chen [4] have shown that the bifurcation is supercritica for the uniform Beck s coumn. Recenty Andersen and Thomsen [6] studying a uniform Beck s coumn with a tip mass at its free end observed that the rotary inertia of the mass can change the type of bifurcation from supercritica to subcritica. To the authors knowedge pubications on the soution of the probem of Beck s coumn with variabe mass and stiffness properties are not avaiabe in iterature.

5 In this paper the noninear dynamic stabiity of Beck s coumn with variabe mass and stiffness properties in the presence of damping (both interna and eterna) using a compete noninear dynamic anaysis is investigated. For homogeneous materia variabe mass and stiffness is due to the variation of the beam cross-section. The noninearity resuts from retaining the square of the sope in the strain-dispacement reations (intermediate noninear theory). In this case the transverse defection affects the aia force and the resuting equations in terms of the dispacements are couped noninear with variabe coefficients. The anaysis is performed with and without considering the aia deformation. This requires the soution of two different initia boundary vaue probems. The deviations of the two approaches are studied and compared with the inear theory. The soution of the probems was achieved using the anaog equation method (AEM) of Katsikadeis [7] as it was deveoped for the noninear dynamic anaysis of beams [8]. According to this method the two couped noninear hyperboic partia differentia equations with variabe coefficients are repaced by two uncouped inear ones pertaining to the aia and transverse deformation of a substitute beam with unit aia and bending stiffness respectivey under fictitious time dependent oad distributions. Besides its accuracy this method overcomes the shortcoming of a FEM soution which eperiences ack of convergence [6] and depends on discretization. Eampe probems of uniform Beck coumn and Beck coumn with ineary varying height are presented which iustrate the effectiveness of the empoyed method to hande this probem. Moreover usefu concusions are drawn concerning the infuence of the aia deformation. Thus in case of uniform Beck coumn this infuence is negigibe whie in case of Beck s coumn with ineary varying height is dominant and shoud be aways incuded in the anaysis. 3

6 . Governing equations Consider an initiay straight beam of ength of viscoeastic materia vibrating in a viscous medium. The beam has variabe cross-section A= A() and moment of inertia I = I( ). The ais coincides with the neutra ais of the beam which is bent in its pane of symmetry z under the action of a tangentia compressive foower tip oad P (see Fig. a). We assume that there is no abrupt variation in cross-section of the beam so that the Euer-Bernoui theory is vaid [9]. In the foowing the equations are derived (a) for noninear anaysis incuding the aia deformation (b) for noninear anaysis ecuding the aia deformation and (c) for inear anaysis.. Noninear theory incuding aia deformation Moderate arge defections are considered. In this case the noninear kinematic reation retains the square of the sope of the defection whie the strain component remains sti sma compared with the unity. Thus we have ε ( z) = u + w + z κ () where u = u() t and w = w() t are dispacements aong the and z ais respectivey and κ is the curvature of the defected ais given as w κ = + w ( ) 3/ The equations of motion are derived by considering the equiibrium of the deformed () eement. Thus referring to Fig. b and taking into account the inertia and eterna damping forces we obtain mu cu + ( N cos θ Q sin θ ) = 0 (3) mw cw + ( N sin θ + Q cos θ ) = 0 (4) 4

7 M = Q (5) where m = m( ) = ρa( ) being the mass density per unit ength and c is the coefficient of the eterna viscous damping. For the case of moderate arge defections the foowing reations are vaid [8] ds = d (6) cos θ sin θ w θ (7ab) κ = w (8) The stress resutants are evauated by integrating appropriatey the norma stress ε σ = Eε+ E = E + E ε t t Thus the aia force and the bending moment are obtained as N = EA+ E A u + w t ( ) ( ) ( ) = EAu + w + EAu + w w M = EI + E I w t = EIw E Iw (9) (0) () where E is the coefficient of dynamic visco-eastic resistance [0]. Substituting Eqs. (7) into Eqs. (3) and (4) and using Eq. (5) to eiminate Q we obtain equations of motion in the form mu cu + N ( M w ) = 0 () mw cw + M + ( Nw ) = 0 (3) which by virtue of Eqs. (0) and () become ( ) ( ) mu cu + EA u + w + E A u + w w + ( + ) EIw E Iw w = 0 (4) 5

8 mw cw ( EIw + E Iw ) + { ( EA u w ) E A( u w w ) w } = 0 (5) The pertinent boundary conditions are u ( 0) = 0 and N() = P = EA+ E A u + w t = (6ab) w ( 0) = 0 and Q () = 0 = EI+ EI w t = (7ab) w ( 0) 0 () 0 M = = EI + E I w t = (8ab) and the initia conditions are u( 0 ) = u ( ) u ( 0 ) = u ( ) (9ab) w( 0 ) = w ( ) w ( 0 ) = w ( ) (0ab) where u ( ) u ( ) w ( ) w ( ) are prescribed spatia functions. Without restricting the generaity in our anaysis we negect the aia inertia and damping forces whose infuence wi be the subject of further investigation. Thus the equations of motion are reduced to ( ) ( ) EA u + w + EIw w = 0 () { } ( ) ( ) mw cw EIw + E Iw + EA u + w w = 0 () Moreover the boundary conditions after dropping the time dependent terms [4 5 ] become u ( 0) = 0 and EA ( u + w ) = P (3ab) = w ( 0) = 0 and ( ) = 0 (4ab) EIw = w ( 0) = 0 and EIw = 0 (5ab) = whie the initia conditions are imited ony to Eqs. (0). 6

9 . Noninear theory ecuding aia deformation We start with Eq. () in which we drop the aia inertia and damping forces. This resuts in equation N ( M w ) = 0 (6) which can be readiy integrated independenty to yied N = M w + c (7) Then using the boundary conditions (6b) and (8b) for the beam end = we obtain c = P (8) Thus the aia force is given as N () = P M w (9) Introducing the bending moment M from Eq. () without the contribution of the time dependent term into Eq. (9) we obtain N ( ) = P ( EIw ) w (30) which is substituted into Eq. (3) to yied the counterpart of Eq. (5) when the aia deformation is negected. Thus we obtain the noninear equation of motion ecuding the aia deformation as mw + cw + ( EIw + E Iw ) + Pw + ( EIw ) w = 0 (3) The boundary conditions (4) and (5) hod aso in this case together with the initia conditions (0)..3 Linear theory In this case Eqs. (30) and (3) are simpified to [] 7

10 N ()= P (3) mw + cw + ( EIw + E Iw ) + Pw = 0 (33) under the boundary conditions (4) and (5) and the initia conditions (0). 3. The AEM soution for the noninear dynamic anaysis of Beck s coumn Eqs. () and () are soved using the AEM which for the probem at hand is appied as foows. Let u = u() t and w = w() t be the sought soutions which are two and four times differentiabe in ( 0 ) respectivey. Noting that Eqs. () and () are of the second order with respect to u and of fourth order with respect to w respectivey we obtain by differentiating u = b ( t) (34) w = b ( t) (35) Eqs. (34) and (35) describe the aia and bending inear response of a beam with constant unit aia and feura stiffness subjected to the fictitious time dependent aia b and transverse b respectivey. They indicate that the soution of Eqs. () and () can be estabished by soving Eqs. (34) and (35) under the boundary conditions (3)-(5) provided that the fictitious oad distributions b b are first determined. Eqs. (34) and (35) are quasistatic that is the time is considered as a parameter. Note that Eqs. (34) and (35) are referred to as the anaog equations to Eqs. () and (). The fictitious oads are estabished by deveoping a procedure based on the integra equation method for one-dimensiona probems. Thus the integra representations of the soutions of Eqs. (34) and (35) are written as ut () = c+ c + G( ξ) b( ξ t) dξ (36) 0 3 wt () = c + c + c+ c + G ( ξ) b( ξ td ) ξ (37)

11 where c = c() t ( i =...6) are arbitrary time dependent integration constants to be i i determined from the boundary conditions and G = ξ (38) G = ξ ( ξ) (39) are the fundamenta soutions (free space Green s functions) of Eqs. (34) and (35) respectivey. The derivatives of u and w are obtained by direct differentiation of Eqs. (36) and (37). This yieds u ( t) c G ( ξ) b ( ξ t) dξ = + u ( t) b ( t) 0 = (40ab) w ( t) = 3c + c + c + G ( ξ) b ( ξ t) dξ (4a) w ( t) = 6c + c + G ( ξ) b ( ξ t) dξ (4b) w ( t) 6 c G ( ξ) b ( ξ t) dξ 3 0 = + w ( t) b ( t) = (4cd) Substituting the above derivatives into Eqs. () and () yieds the equations from which the fictitious sources b and b can be determined. This can be impemented ony numericay as foows. The interva ( 0 ) is divided into N equa eements (see Fig. ) on which b and b are assumed to vary according to a certain aw (constant inear paraboic etc). The constant eement assumption is empoyed here because the numerica impementation becomes very simpe and the obtained numerica resuts are very good. After discretization of Eqs. (36) and (37) we obtain N j () = j() + ( j) ( ) j j= j= (4) ut c t b ξ G ξ dξ 9

12 4 N 4 j () = j+ () + ( j) ( ) j j= j= (43) wt c t b ξ G ξ dξ or ut () = H () c + G () b (44) wt () = H () c + G () b (45) where G ( ) and G () are N known matrices originating from the integration of the kernes G( ξ ) and G( ξ ) on the eements respectivey; H ( ) = and H ( ) 3 = ; c = { c c } T ; c = { c3 c4 c5 c6} T ; b b are the vectors containing the vaues of the fictitious oads at the noda points respectivey. Simiary we obtain for Eqs. (40) and (4) u() t = H () c + G () b u ( t ) = b (46ab) w ( t) = H( ) c + G( ) b w () t = H() c + G() b (47ab) w () t = H() c + G() b w ( t ) = b (47cd) where G () G () G ( ) are N known matrices originating from the integration of the derivatives of the kernes G ( ξ ) G ( ξ ) on the eements; H () is a known matri resuting from the differentiation of H () whereas H ( ) H () H () are 4 known matrices resuting from the differentiation of H (). Finay coocating Eqs. () and () at the N noda points and substituting the reevant derivatives from Eqs. (46) and (47) yieds the foowing equations of motion K( b b c c)= 0 (48) Mb + Cb K ( b b c c ) = 0 (49) 0

13 where M and C are known N N generaized mass and damping matrices and K ( i b b ) c c ( i = ) generaized stiffness vectors. The boundary conditions (3)-(5) are in genera noninear and can be written as fb ( b c c)= 0 (50) Eq. (49) is the semi-discretized equation of motion of the Beck s coumn. The associated initia conditions resut from Eq. (45) when appied to a noda points and combined with Eqs. (0). Thus we have b ( 0) = G ( w H c ) (5) b ( 0) = G w (5) The time step integration method for noninear equations of motion can be empoyed to sove Eq. (49). In each iteration for b within a time step the current vaue of b is utiized to update the vectors b and c c on the basis of Eqs. (48) and (50). This demands the soution of a noninear system of agebraic equations which is performed using the modified Newton- Raphson method. In this paper the average acceeration time step integration method was empoyed to sove Eq. (49) and the resuts were cross-checked by a time step integration method based on the anaog equation method [3]. Once the vectors b b c c are computed the dispacements u = u() t and w = w() t and their derivatives at any instant t are evauated from Eqs. (44) through (47). 4. Numerica eampes On the base of the procedure described in previous section a FORTRAN program has been written for estabishing the dynamic noninear response of damped both eternay and internay beam coumns with variabe mass and stiffness properties subjected to foower

14 forces. The uniform and nonuniform Beck s coumn has been studied as an iustrative eampe of the deveoped anaysis and soution method. Eampe : Uniform Beck s coumn The noninear dynamic stabiity of Beck s coumn with uniform rectanguar crosssection b h 0 and ength =.0 m has been studied. The empoyed data are: E = kn/m 0.0m b = h 0 = 0.5 m - ρ = knm sec c = 0. and E 4 3 = 0.0E. The empoyed initia conditions are w ( ) = 8w( ξ 4ξ + 6 ξ )/4 and w () = 0with ξ = / and w is the initia tip defection of the beam. In Tabe the computed critica oads are presented from (i) inear theory (ii) noninear theory ecuding the aia deformation and (iii) noninear theory incuding the aia deformation. The convergence of the method is shown by increasing the number of eements N. In a three theories the computed vaues of the critica oad are identica and coincide with those obtained by Andersen and Thomsen [6] who used a perturbation anaysis of the noninear equations of motion ignoring however the aia deformation of the beam. The FEM soution using geometric noninear dynamic finite eement mode for case (ii) gives resuts which are quaitative and quantitative the same which however ehibit ack of convergence in some cases [6]. In Tabe the finite tip ampitude is presented from the noninear theory (i) ecuding and (ii) incuding the aia deformation for various vaues of the aia oad P. The two noninear theories give the same resuts. However as compared with those obtained in [6] ony the case for P = 3.53 are found in good agreement. From the obtained resuts it becomes apparent that the infuence of the aia deformation on the critica oad is negigibe for uniform Beck s coumn. This is an interesting finding.

15 Moreover the time histories of the tip defection for two different initia tip defections w are shown in Fig. 3 vaidating the supercritica bifurcation character of uniform Beck s coumn whie in Fig. 4 and Fig. 5 the variation of the critica oad in regard to the eterna and interna damping is depicted respectivey. From Fig. 4 it can be pointed out that the critica oad increases monotonicay with the coefficient of the eterna viscous damping. On the contrary the curve of Fig. 5 shows that the vaue of the critica oad decreases to a minimum vaue ( P cr =.03 at E = 0.03E ) and thereafter it increases with increasing vaue of the coefficient of dynamic visco-eastic resistance (interna damping). Eampe : Beck s coumn with variabe mass and stiffness properties The noninear dynamic stabiity of Beck s coumn with variabe cross section has been studied. The empoyed data are the same with those in previous eampe. The height of the beam varies according to the inear aw h( ) = h0 + a( /) (see Fig. 6) with a = tanφ being the taper ratio and h 0 the height at the haf ength. In order to compare the resuts with those of the previous eampe the voume of the beam i.e. V = bh0 was kept constant. The resuting beam shoud have no abrupt change of the cross-section so that the Euer-Bernoui theory remains vaid [9]. Boey [4] has shown that for a beam with unit constant width a rate of change of the cross-section a 0.35 yieds an error of 7.5% whie for a 0.7 the error is.8%. This was aso verified by the authors who treated the beam as a D easticity probem and used the BEM to obtain the soution [5]. The computed critica oads from (i) inear theory (ii) noninear theory ecuding and (iii) incuding the aia deformation for various vaues of the ratio a are shown in Tabe 3. The resuts obtained on the basis of the first two theories are coincident but differ consideraby from those of the third one. This significant finding demands that the aia deformation on the noninear dynamic stabiity of Beck s coumn shoud be aways incuded 3

16 in the anaysis when the beam has variabe mass and stiffness properties. It shoud be aso observed in the noninear theory (iii) that the critica oad increases monotonicay with the taper ratio which means that the materia must be shifted towards the free end in order to obtain the maimum critica oad. Moreover in Fig. 7 are shown the bifurcation diagrams from noninear theory (iii) for various vaues of the taper ratio. The shape of the bifurcation branches becomes steeper as the taper ratio decreases. For this reason the case of taper ratio a = 0.5 is investigated for possibe subcritica bifurcation. Fig. 8 and Fig. 9 show the variation of the critica oad in regard to the eterna and interna damping respectivey. It is apparent from these figures that the same quaitativey concusions with those of uniform Beck s coumn can be drawn. Namey the critica oad increases monotonicay with the coefficient of the eterna viscous damping whie it is decreasing to a minimum vaue ( P cr = 5.6 at E = 0.05E ) and thereafter increases for further increase of the interna damping. Finay in order to estabish the bifurcation character two noninear dynamic anaysis are performed one beow ( P = 5.00 ) and another above ( P = 5.0 ) the critica oad. The time histories of the tip defection are shown in Fig. 0 vaidating the supercritica bifurcation character of nonuniform Beck coumn. 5. Concusions In this paper the noninear dynamic stabiity of Beck s coumn with variabe mass and stiffness properties in the presence of damping (both interna and eterna) has been investigated using a compete noninear dynamic anaysis which incudes the aia deformation. The soution of the derived couped noninear equations of motion was achieved effectivey using the anaog equation method. This investigation has reached to certain striking effects concerning the infuence of the aia deformation on the critica oad of Beck s coumn with variabe cross section. The main concusions can be summarized as 4

17 The bifurcation is aways supercritica in uniform and nonuniform Beck s coumn. The aia deformation affects consideraby the critica oad when the beam has variabe cross section. It may give remarkaby ower or arger critica oads. Therefore it shoud be aways incuded in the anaysis. On the contrary it has negigibe infuence for uniform cross section. The ecusion of the aia deformation in the noninear dynamic anaysis of Beck s coumn with variabe mass and stiffness properties may yied arger oads reducing thus the safety of the structure. The critica oad increases monotonicay in uniform and nonuniform Beck s coumn with the coefficient of the eterna viscous damping. The critica oad decreases to a minimum vaue and thereafter increases for further increase of the vaue of the coefficient of dynamic visco-eastic resistance (interna damping) in uniform and nonuniform Beck s coumn. In the noninear theory incuding the aia deformation the critica oad increases monotonicay with the taper ratio. This suggests shifting of the materia towards the free end in order to maimize the critica oad The shape of the bifurcation branches becomes steeper as the taper ratio decreases. References [] R.H. Paut E.F. Infante The effect of eterna damping on the stabiity of Beck s coumn Int. J. Soids Struct. 6 (970) [] V.V. Bootin Nonconservative Probems of the Theory of Eastic Stabiity (Moscow Engish transation) Pergamon Press Ltd. Oford 963. [3] G. Herrmann I.C. Jong On the destabiizing effect of damping in nonconservative eastic systems J. App. Mech. 3 (965)

18 [4] R.C. Kar Stabiity of a nonuniform cantiever subjected to dissipative and nonconservative force Comput. Struct. (980) [5] B.N. Rao G.V. Rao Stabiity of tapered cantiever coumns subjected to a tipconcentrated foower force with or without damping Comput. Struct. 37 (990) [6] L. Gau The infuence of damping on waves and vibrations Mechanica Systems and Signa Processing 3 (99) -30. [7] A.N. Kounadis Hamitonian weaky damped autonomous systems ehibiting periodic attractors Z. angew. Math. Phys. 57 (006) [8] R.V. Vitaiani A.M. Gasparini A.V. Saetta Finite eement soution of the stabiity probem for noninear undamped and damped systems under nonconservative oading Int. J. Soids Struct. 34 (997) [9] A.N. Kounadis On the parado of the destabiizing effect of damping in non-conservative systems Int. J. Non-Linear Mech. 7 (99) [0] A.N. Kounadis Non-potentia dissipative systems ehibiting periodic attractors in region of divergence Chaos Soitons & Fractas 8 (997) [] V.V. Bootin A.A. Grishko M.Yu. Panov Effect of damping on the postcritica behaviour of autonomous non-conservative systems Int. J. Non-Linear Mech. 37 (00) [] A.N. Kounadis Some new instabiity aspects for nonconservative systems under foower oads Int. J. Mech. Sci. 33 (99) [3] A.N. Kounadis On the faiure of static stabiity anayses of nonconservative systems in regions of divergence instabiity Int. J. Soids Struct. 3 (994) [4] R.W. Kokka On the non-inear Beck s probem with eterna damping Int. J. Non- Linear Mech. 4 (984)

19 [5] M. Chen Hopf bifurcation in Beck s probem Noninear Ana. Theory Methods App. (987) [6] S.B. Andersen J.J. Thomsen Post-critica behaviour of Beck s coumn with a tip mass Int. J. Non-Linear Mech. 37 (00) [7] J.T. Katsikadeis The anaog equation method. A boundary-ony integra equation method for noninear static and dynamic probems in genera bodies Theor. App. Mech. 7 (00) [8] J.T. Katsikadeis G.C. Tsiatas Non-inear dynamic anaysis of beams with variabe stiffness J. Sound Vib. 70 (004) [9] J.T. Katsikadeis G.C. Tsiatas Bucking oad optimization of beams Arch. App. Mech. 74 (005) [0] J.A. Hudson The Ecitation and Propagation of Eastic Waves Cambridge University Press Cambridge 980. [] J.L. CLaudon Détermination et maimisation de a charge critique d une coonne de Hauger en présence d amortisement J. App. Math. Physic. 9 (978) [] M. Beck Knickast des einseitig eingespannten tangentia gedrücten Stabes Zeitscrift für Angewandte Mathematik und Physik 3 (95) 5-8. [3] J.T. Katsikadeis A new time step integration scheme for structura dynamics based on the anaog equation method in Coection of papers dedicated to Prof. P.S. Theocaris Nationa Technica University of Athens (994) [4] B.A. Boey On the accuracy of the Bernoui-Euer theory for beams of variabe section J. App. Mech. ASME 30 (963) [5] J.T. Katsikadeis Boundary Eements: Theory and Appications. Amsterdam-London: Esevier 00. 7

20 Tabe. Eampe : Critica oad of Beck s coumn ( c = 0. E = 0.0E) from (i) inear theory (ii) noninear theory ecuding and (iii) noninear theory incuding the aia deformation for various vaues of the boundary eements N. N = 0 N = 5 N = 30 N = 35 N = 65 [6] (i) (ii) (iii) Tabe. Eampe : Finite tip ampitude of Beck s coumn ( c = 0. E = 0.0E) from noninear theory (i) ecuding and (ii) incuding the aia deformation for various vaues of boundary eements N. P N = 0 N = 5 N = 30 N = 35 [6] 3.8 (i) (ii) (i) (ii) (i) (ii) Tabe 3. Eampe : Critica oad of Beck s coumn with ineary varying height ( c = 0. E = 0.0E) from (i) inear theory (ii) noninear theory ecuding and (iii) noninear theory incuding the aia deformation for various vaues of the taper ratio a ( N = 30 ). a (i) (ii) (iii)

21 z EI() EA() m() wt () () a () b P N M Q ds θ Q M + dq + dm θ + dθ N + dn Fig.. (a) Beck coumn with variabe mass and stiffness properties and (b) forces and moments acting on the deformed eement. Noda points 0 N N + Fig.. Descritization of the interva and distribution of the noda points. (i) (ii) Fig. 3. Eampe : Time history of the tip defection of uniform Beck coumn ( c = 0. E = 0.0E P =.94 ) for P = 3.33 (i) w = 0.05 and (ii) w = 0.0. cr 9

22 Fig. 4. Eampe : Critica oad of uniform Beck coumn versus eterna damping ( E = 0.0E N = 30 ). Fig. 5. Eampe : Critica oad of uniform Beck coumn versus interna damping ( c = 0. N = 30 ). h ( ) a = tanφ φ h 0 / / Fig. 6. Linear variation of the height of the Beck coumn in Eampe. 0

23 Fig. 7. Eampe : Bifurcation diagrams of Beck coumn with ineary varying height from noninear theory incuding the aia deformation for various vaues of the taper ratio a ( c = 0. E = 0.0E N = 30 ). Fig. 8. Eampe : Critica oad of Beck coumn with ineary varying height ( a = 0.5 ) versus eterna damping ( E = 0.05E N = 30 ).

24 Fig. 9. Eampe : Critica oad of Beck coumn with ineary varying height ( a = 0.5 ) versus interna damping ( c = 0. N = 30 ). (i) Fig. 0. Eampe : Time history of the tip defection of Beck coumn with ineary varying height ( c = 0. E = 0.05E a = 0.5 P cr = 5.6 w = 0.5 ) for (i) P = 5.00 and (ii) P = 5.0. (ii)