NUMERICAL SCHEME INFLUENCE ON PSEUDO DYNAMIC TESTS RESULTS

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1 October 1-17, 8, Bejng, Chna NUMERICAL SCHEME INFLUENCE ON PSEUDO DYNAMIC TESTS RESULTS ABSTRACT : G. Amato 1, L. Cavaler 1 PhD Student, Dpartmento d Ingegnera Strutturale e Geotecnca, Palermo. Italy Assocate Professor, Dpartmento d Ingegnera Strutturale e Geotecnca, Palermo. Italy Emal: cavaler@dseg.unpa.t The pseudo-dynamc test method s a well establshed and wdely spreadng expermental technque for structural analyss but the convenence of the chosen tme-steppng scheme s not often adequately stressed and reported n expermental results. In fact to obtan results relable such as those granted by shakng table tests t s necessary to stand a notceable computatonal effort to ntegrate step by step the dynamc equlbrum equaton. In ths survey a comparson between a number of numercal ntegraton technques and some numercal examples s presented. The ntegraton methods have been chosen among the most establshed and newest ones wth the am to nvestgate ther advantages and downsdes partcularly wth regard to computatonal effort and error propagaton for lnear and non-lnear structural systems. Furthermore some general nformaton and defntons concernng numercal ntegraton tecnques and ther stablty and accuracy propertes are gven; Central Dfference Method, Newmark Explct Method, the α-method, as conceved by Hlber, Hughes and Taylor n 1977, and the Tme Dscontnuous Galerkn Method, appled by means of predctor-corrector tecnque by Burs, Bonell n 1 and, are dscussed. KEYWORDS: Pseudo-dynamc tests, errors, numercal algorthms, non-lnear systems. 1. INTRODUCTION A pseudo-dynamc test can be consdered a smulaton of a shakng table test by means of expermental equpment usually used to perform quas-statc tests. A greater computatonal effort though s demanded by pseudo-dynamc test because the equpment must be drected by an electronc controller to perform a closed loop test. In fact at each step the structural dfferental equaton of moton has to be ntegrate to obtan the dsplacements to mpose on the specmen n a quas-statc way. Ths computaton employs specmen restorng forces measurements and thus t takes place durng the test. The ntegraton of the equaton of moton and the mnmzaton of the delay between actual and command dsplacements, are the man tasks performed by the controller. Because of the ntegraton of the dynamc equaton of moton s the core of the pseudo-dynamc test method, t consttutes the current focus of the research. Two are the man research felds: the numercal analyss of stablty and error propagaton propertes of the algorthms and the exploraton of numercal schemes wth hgher performances. The ultmate target s performng pseudo-dynamc test on structural non-lnear systems excted by random forces. In fact there are many crcumstances nfluencng the results relablty, for example the choose of the temporal ntervals t and T whch represent, respectvely, the numercal ntegraton step and the tme span effectvely runnng durng a test between two ensung loadng steps. The choose of t s obvously lnked to many expermental aspects as t has been dscussed by Mahn and Shng (1985) whle the numercal temporal step s strctly lnked to the scheme propertes. The level of vscous dampng, the knd of external exctaton and the structural system characterstcs are others determnant elements n the choose of the proper numercal ntegraton scheme.

2 October 1-17, 8, Bejng, Chna The algorthms reported n ths paper are the Central Dfference Method, the Explct Newark Algorthm, the α-method (Hlber, Hughes, Taylor) and the Tme Dscontnuous Galerkn Method. Ther descrpton has the am to provde a useful tool to approach the pseudo-dynamc test method and to show through some numercal comparsons the nfluence of the used algorthm on the results obtaned. Central Dfference Method and Explct Newmark Algorthm (1959) are the frst numercal schemes whch have been used to carry out a pseudo-dynamc test. Referrng to Newmark algorthms, nterestng analyses have been done by dfferent authors. Xe and Steven (199) have analyzed ts stablty, n the case of non-lnear systems whle Shng and Mahn (1987 b) have studed the possblty of modfyng the scheme to make t more effectve n connecton wth hgher spurous modes. The so called α-method (Hlber, Hughes and Taylor 1977) too has been subjected to several mplementatons and comparsons (e.g. Shng and Mannvan 199, Shng, Vannan and Cater 1991, Bonell and Burs 199, Burs and Shng 1996, Bonell and Burs ). A weak formulaton of the Galerkn method has been used and tested va numercal and symbolc applcatons not long ago (Bonell, Burs and Mancuso 1 and ) and s here descrbed snce t uses a very dfferent approach to the problem.. EQUATION OF MOTION OF A M.D.O.F. SYSTEM AND SOME DEFINITIONS FOR ITS NUMERICAL INTEGRATION The dynamc equlbrum equaton of a mult degrees of freedom structure can be expressed by Eqn. Errore. L'orgne rfermento non è stata trovata. where d, v, a are respectvely the dsplacements, veloctes and acceleratons vectors of the system, M e C are the mass and dampng matrces, R(d) s the restorng forces vector and F the external forces vector. Ma+Cv+R(d) = F (.1) Eqn. Errore. L'orgne rfermento non è stata trovata. together wth the correspondng ntal data d( t ) = d, v( t ) = vconsttutes a Cauchy problem whose closed form soluton s known only for specal cases. To approach a numercal soluton of ths problem the frst step s the dscretzaton of the temporal doman by choosng a number N of tme ntervals lastng t and ths way Eqn. (1) assumes at the tme nstant t= t the followng form: Ma +Cv +R(d) = F (.) In order to evaluate the numercal ntegraton methods, used to smulate and carry out pseudo-dynamc tests, and to check ther apttude to smulate the dynamc response of a structural system, some defntons concernng the numercal propertes and error propagaton of the schemes can be useful (Quarteron et al. ). The accuracy acheved by a numercal scheme s strctly lnked to the local truncaton error τ ( t). Ths s the error exstng at each step between the exact soluton and the approxmated one under the hypothess that at the prevous step numercal and exact solutons were concdent. The maxmum value of the local truncaton error s the global truncaton error τ ( t) and an ntegraton algorthm s of k-th order consstency f ths error s an nfntesmal of k-th order respect to t. To acheve convergence though, a global truncaton error nfntesmal respect to t s not enough, n fact, the stablty of the numercal scheme (the ablty of workng despte perturbatons and errors accumulaton) s a necessary condton. Ths defnton of stablty s related to the behavor of the algorthm when t s nfntesmal but stablty concerns the asymptotc behavor of the scheme too. In ths sense an algorthm s called stable f the numercal soluton s bounded when tme goes to nfnty. Numercal schemes can be stable dependng or not on the ntegraton nterval, ther stablty wll be thus condtoned or uncondtoned. Usually explct schemes are condtonally stable and the mplct ones uncondtonally stable, but ths s only a general rule. For a structural system dynamcally forced (whch s a set of damped oscllators) a numercal scheme s stable f ts free vbratons don t grow wthout boundares regardless of the ntal condtons. To analyze the stablty of a numercal scheme, the free vbratons can be studed consderng that the response can be expressed n the

3 October 1-17, 8, Bejng, Chna followng recursve form x A x (.3) + 1= where A s the amplfcaton matrx of the numercal scheme, that s ndependent on, but s strctly dependent on the adopted scheme, and x s the vector contanng some or all state varables of the system such as dsplacements, veloctes and acceleratons. Some condtons on the engenvalues of the matrx A gve proof of stablty of the algorthm. Snce the amplfcaton matrx A depends on the dynamcal characterstcs of the system such as mass, stffness and dampng and on those attanng the scheme, these condtons convert on nequaltes correlatng the own pulsatons ω k of the system and the temporal step t. 3. CENTRAL DIFFERENCE METHOD For the case of sngle degree of freedom systems, ths numercal scheme assumes that velocty and acceleraton at the -th step can be descrbed by the followng equatons (Mahn and Shng 1985) ( ) ( ) v = d d / t; a = d + d d / t ; (3.1) Equlbrum Eqn.(.1) specalzed n the case of a s.d.o.f. system, among wth Eqs. (3.1) can be rearranged to express the dsplacement n the form ( ) (.5 ) (.5 ) 1 d 1 t F R tc m d 1 md tc m + = (3.) For a lnear s.d.o.f. system under an external force and wth zero vscous dampng, a drect step-by-step ntegraton algorthm can be expressed n a recursve form as x = Ax + l f (3.3) n+1 n n where l s the load operator and f n s the value of the external exctaton at the n-th step but ths formulaton does not take nto account the expermental and numercal errors whch are always present n a numercal smulaton or n an actual test. These errors are due for example to floatng-pont computaton or they are nherent to the numercal scheme, but the man source s the restorng force feedback, further the latter s due both to measurement mprecson and to the dfference between the actual dsplacements and the command ones whch affects the restorng forces. By takng nto account only the expermental errors Eqn. (3.3) assumes a modfed form whch depends not only on A and l but also on the measured restorng force, affected by the restorng force feedback r error e n ntroduced at the n-th step. For Central Dfference Method a formulaton of the cumulatve error on the response system due to ths feedback error has been gven by Shng and Mahn (1987 a) and Shng and Mahn (199). Furthermore Mahn and Shng (198) showed how the natural frequency of the system s dstorted by the error presence, n fact the apparent perod of the oscllator undergoes a shrnkage whch depends on the tme step t. As t s shown n Fg. 1-a the numercal pulsaton approaches the real one when the tme step t goes to zero. In Fg. 1-b one can see the dfferent response obtaned for a sngle degree of freedom system, undamped and unforced, featured by an ntal velocty v = 1 for dfferent ntegraton tme steps. The rato between the maxmum cumulatve error e + and the standard devaton of the expermental random feedback errors at each nstants n 1 max S e, estmated as a functon of the number of steps n and the product ω t s shown n Fg. 1-c. An example of numercal errors due to the scheme s gven by a comparson wth an alternatve formulaton of the z = d d t nstead of d (Eqn.(3.)) s used Central Dfference Method. In ths case the summed varable ( ) / 1

4 October 1-17, 8, Bejng, Chna to avod round-off errors caused.e. by floatng pont computatons or n the feedback process (Mahn and Shng 1985). A smple comparson of the dfference between the two formulatons s shown n Fg. -b where the dsplacement response of a s.d.o.f. system wth natural angular frequency ω=π (rad/s) and vscous dampng ζ=.1 frst and ζ=.1 after, s corrupted, step after step, by addng a random varable wth zero mean (µ=) and standard devaton σ =.1 (Fg. -a). The system sn t externally excted and null ntal condtons are mposed. The ntegraton step used s t=.1 (s). By usng both formulaton a response dfferent from the attended null one s obtaned because of the error nfluence but the second formulaton acheves a smaller system response for ts greater capacty of workng n spte of perturbatons. Observe that a better effect s obtaned for a greater value of vscous dampng. (T-T') / T t / T dsplacement t =.5 (s) t =.5 (s) t =.1 (s) 6 8 e n+1 max / S e 3 1 ω t = π 1/1 ω t = π 5/1 ω t = π 1/1 8 Number of steps (a) (b) (c) Fgure 1. Perod shrnkage by Central Dfference Method (a); Free vbraton response of a s.d.o.f. system (ω = π) for dfferent tme steps (b). Normalzed cumulatve bounds for random errors ( e / n+ 1 S max e) for dfferent values of t (c)... µ =, σ =.1 ζ.1 ζ.1 I formulaton II formulaton nose -. d(t)/ - d(t)/ (a) (b) (c) Fgure. Nose wth zero mean and standard devaton σ =.1 (a); Normalzed dsplacements of a not excted s.d.o.f. system calculated by usng the two dfferent formulatons of the Central Dfference Method n presence of the nose and for two dfferent value of vscous dampng: ζ =.1 (b) and ζ =.1 (c).. NEWMARK ALGORITHMS AND α-method The α-method famly of algorthms (Hlber, Hughes and Taylor 1977) s based on the followng set of equatons: Ma + (1 + α ) Cv αcv + (1 + α ) R αr = (1 + α ) F αf (.1) (.5 ) d + = d + t v + t β a + βa + (.) 1 1 ( 1 ) v + = v + t γ a + γa + (.3) 1 1 The algorthms of ths famly are a functon of three parameters α, β, γ and they nclude the Newmark algorthms

5 October 1-17, 8, Bejng, Chna when α =. In ts general form a Newmark algorthm s mplct because at the temporal step t +1 the dsplacement depends on the acceleraton at the same nstant. The two parameters β and γ control the accuracy and stablty propertes of the scheme,.e. by assumng β =.5 and γ =.5 we are usng the trapezodal rule or constant average acceleraton method, whereas the choose β = and γ =.5 provdes an explct formulaton whch s the one that has been frst used to carry out pseudo-dynamc tests. Ths Explct Newmark Method and the Central Dfference Method possess the same mathematcal propertes that s they are both condtonally stable and ther stablty lmt, n the case of a lnear system, s ω k t, where ω k s the generc natural pulsaton of the system. For example referrng to errors nfluence, the results nserted n Fg. 1 are stll vald for the Newmark algorthms famly and f the smulaton to whch Fg. refers s repeated, the same curve of the second formulaton of the Central Dfferent Method s obtaned. The am of α-method famly of algorthms s to overcome some lmts of the Newmark algorthms regardng n specfc the numercal dsspaton, that s the ablty of the numercal scheme to damp the so-called spurous hgher modes of the amplfcaton matrx, those produced by the dscretzaton of the moton equatons. Newmark algorthms possess ths property whch s partally controlled by the γ parameter. What the method wants to elmnate s the dsspaton of the basc modes of the system when spurous hgher modes are dsspated by means of the parameter γ. In ths sense the ntroducton of the parameter α nverts the trend of the basc modes to be dsspated when the spurous hgher modes are dsspated. The α parameter (α < ), ntroduces a negatve dampng apttude whch s smlar to the vscous dampng and, what s more mportant, s not effectve on the hgher modes. These propertes led the authors of the method n object to arrange the γ-dsspaton and a negatve α-dsspaton so that t s possble to get rd of drawbacks lke the dsspaton of the lowest modes and gan the benefts (the dsspaton of spurous modes) of the γ-dsspaton. The new algorthm whle been functon of the three afore mentoned parameters, s thus ruled by only one of them because of the expressons statng the uncondtonal stablty (β =.5 (γ +.5) ) and a postve global numercal dsspaton (α + γ = 1/). dsplacement dof 8-8 α-method α= t =.1 (s) t =.5 (s) dsplacement dof 8-8 α-method α=-.1 C= C=.1 M C=.5 M 8 1 dsplacement dof α-method α= dsplacement dof Implct Newmark Method γ= t =.1 (s) t =.5 (s) dsplacement dof Implct Newmark Method γ= dsplacement dof Implct Newmark Method γ=.6 C= C=.1 M C=.5 M Fgure. Dsplacements of a d.o.f. non-lnear system calculated for two dfferent values of t by means of Implct Newmark Method and α-method; In the followng numercal example the α Method s tested n comparson wth the Newmark scheme. The dynamcal system studed s a d.o.f. system whose non-lnear restorng forces are expressed by

6 October 1-17, 8, Bejng, Chna 3 3 R1 = k11 d1 + k1 d + k13 d1 and R = k1 d1 + k d + k3 d. The system has been excted by a snusodal force F 1 (t)= m 1 sn(π t) appled at the frst degree of freedom and the structural response has been obtaned for two dfferent values of the tme step, t=.1 s and t=.5 s. The parameters characterzng the system stffness are k 11 =1 N/m, k = N/m, k 1 = k 1 =- N/m, k 13 =5 N/m 3, k 3 =5 N/m 3 whle the masses are m 1 =m =1 kg. The smulaton has been repeated for two dfferent values of the vscous dsspaton matrx: C=.1 M and C=.5 M. In ths second case the maxmum restorng force value s about forty tmes the maxmum vscous force value. The value of numercal dsspaton chosen for the α-method s α=-.1 whch mples, γ=.6 for the mplct Newmark scheme. In Fg. the results obtaned for dfferent values of the matrx C are plotted. Each fgure shows the dsplacements of the nodes 1 and computed by usng the mplct Newmark method and the α-method. One can see how dsplacements calculated by usng α-method don t change for the dfferent ntegraton step values, both for the undamped system and for the damped one, meanng convergence has been acheved for the chosen t. Ths doesn t happen wth the mplct Newmark method when the system s not damped by vscous forces. Further observe as the ncreasng of dsspaton produces the agreement of the responses obtaned wth the two methods. 5. TIME DISCONTINUOUS GALERKIN METHOD The applcaton of the Galerkn technque to ntegrate dfferental equaton as requested n pseudo-dynamc test arses from the wll to extend ths expermental technque to hghly non-lnear problems. Nevertheless the frst formulaton of the method s addressed to lnear problems. Observe that the numercal propertes such as uncondtonal stablty or accuracy order, verfed for a numercal scheme used to solve a lnear problem, don t apply mmedatly for non-lnear systems for whch stablty and accuracy have to be proved for each case. The formulaton of the Galerkn method used here s the one mplemented n a predctor-corrector form by Bonell and Burs () for non-lnear cases. The procedure has been checked on classcal stff problems such as Duffng oscllator and stff strng pendulum. The formulaton s based on two ndependent felds, dsplacement d(t) and momentum p(t) = M v(t). The second order dfferental equaton (.1) and the correspondent ntal condtons are rewrtten n the followng frst order scheme where the superposed dots ndcate tme dervatve p t CM p t R d t F t M p t d t t I t N 1 1 () + () + ( ()) = (); () () =, = (, ); d() = d ; p() = p = Mv ; (5.1) The numercal propertes of the algorthm both n the lnear verson and n non-lnear one are extensvely dscussed n Bonell, Burs and Mancuso (1) and () and the scheme n ts non-lnear verson has been evaluated (Bonell, Burs and Mancuso ) on a Duffng system whose closed-form soluton s known for ntal condtons d() = d ; p() = p and no external exctaton. The results appear to be satsfyng n respect of accuracy and energy-decayng propertes. Nevertheless non-lnear problems, far from be solved, are underlned n the followng numercal example showng the nablty of the current schemes when non-lnear systems are excted by random forces to reach a relable soluton. The consdered system s a s.d.o.f. oscllator whose restorng force s modeled by the Duffng expresson R( d) = S1d( t) ( 1 + Sd( t) ). The system characterstcs are m=1, S 1 =1, S =. The ntal condtons of the system are null whle exctaton s gven by a Gaussan whte nose wth zero mean (µ=) and standard devaton σ = 79.. Dsplacements have been calculated by means of α-method (α=-1/3) and Tme Dscontnuous Galerkn Method (a=b=1/6, number of teratons of the corrector k=) and for two values of the vscous dampng coeffcent c=.1s 1 and c=.1s 1. The exctaton tme step s t F =.1 s and two dfferent values of the ntegraton step have been chosen, frst t = t F =.1 s and then t =.1 t F =.1 s. For each method dsplacements obtaned wth

7 October 1-17, 8, Bejng, Chna the two tme steps are shown n Fg. 7. These evdence how both methods are relable only for few seconds, but n the case of the Galerkn Method the response for the two chosen t remans concdent for a longer tme. For both schemes dfferences between response obtaned wth dfferent tme step reduce when c ncreases. In spte of the convergence of the two methods for a suffcently hgh value of c, the relablty of the response s stll to be evaluated. In fact the balance of the energes of the system, the one gven by the external forces, the one dsspated and that stored n terms of knetc and potental energy, could gve an ndcaton but not a defntve response. dsplacement dsplacement - - α-method c=.1*s Galerkn Method c=.1*s dsplacement dsplacement - - α-method c=.1*s1 Galerkn Method t =.1 (s) t =.1 (s) c=.1*s Fgure 7. Duffng oscllator wth vscous dampng coeffcent c=.1*s 1 and c=.1*s 1 ; dsplacements calculated wth dfferent values of ntegraton step 6. CONCLUSIONS In ths paper Central Dfference Method, Newmark Explct Method, α-method (Hlber, Hughes and Taylor 1977), Tme Dscontnuous Galerkn Method (Burs, Bonell 1 and ) have been reported and some smulatons of pseudodynamc tests have been carred out on s.d.o.f. and m.d.o.f. systems to assess the numercal scheme propertes and ther relablty when appled to lnear and non-lnear cases. In spte of the dfferences among the analyzed schemes, from the above tests t manly emerges that the ncreasng of the physcal dsspaton produces a reducton of the dfferences between those schemes. Alternatvely, low levels of the dsspaton coeffcents pose the problem of the dentfcaton of a suffcently relable algorthm. In the case of lnear systems some rules whch make t possble to evaluate the algorthm stablty, are avalable. On the contrary, n the case of non lnear systems, t s not possble to obtan rules vald n each case. Certanly some verfcaton, for example by computng the energes nvolved n the test may be carred out at the end of a pseudo-dynamc test n order to verfy ts relablty. What above makes some questons rse about pseudo-dynamc tests each tme the algorthm used s not declared and a dscusson about t s not offered before the results are proposed. Further detals about the dfferences nvolvng the numercal schemes analyzed can be found n (Amato and Cavaler, 7). REFERENCES Amato G., Cavaler L., (7). About Numercal Integraton Methods for Pseudo-Dynamc Tests: State of the Art. XII Congresso Nazonale ANIDIS "L'ngegnera ssmca n Itala". Psa, 1-1 gugno 7.

8 October 1-17, 8, Bejng, Chna Bonell A., Burs O., (). Generalzed α-methods for sesmc structural testng. Earthquake Engneerng & Structural Dynamcs, 33, Bonell A., Burs O., Mancuso M, (1). Explct predctor-multcorrector tme dscontnuous Galerkn methods for lnear dynamcs. Journal of Sound and Vbraton, 6:, Bonell A., Burs O., Mancuso M., (). Explct predctor-multcorrector tme dscontnuous Galerkn methods for non-lnear dynamcs. Journal of Sound and Vbraton, 56:, Burs O., Shng P. B., (1996). Evaluaton of some mplct tme-steppng algorthms for pseudo-dynamc tests. Earthquake Engneerng & Structural Dynamcs, 5, Burs O., Shng P. B., Radakovc-Guzna Z., (199). Pseudo-dynamc testng of stran-softenng systems wth adaptatve tme steps. Earthquake Engneerng & Structural Dynamcs, 3, Chang S. Y., (). Uncondtonal stablty for explct pseudodynamc testng. Structural Engneerng & Mechancs, 18, 118. Hlber H. M., Hughes T. J. R., Taylor R. L., (1977). Improved numercal dsspaton for tme ntegraton algorthms n structural dynamcs. Earthquake Engneerng & Structural Dynamcs, 5, Mahn S. A., Shng P. B., (1985). Pseudodynamc method for sesmc testng. Journal of Structural Engneerng ASCE, 111, Newmark N.M., (1959). A method of computaton for structural dynamcs. Journal of Engneerng Mechancs ASCE, 85:3, Quarteron A., Sacco R., Saler F., (). Matematca numerca, Sprnter-Verlag Itala. Shng P. B., Mahn S. A., (1983). Expermental error propagaton n pseudodynamc testng. Report UCB/EERC-83/1, Earthquake Engneerng Research Center, Unversty of Calforna, Berkeley, Calf. Shng P. B., Mahn S. A., (198). Pseudodynamc test method for sesmc performance evaluaton: theory, mplementaton. Report UCB/EERC-8/1, Earthquake Engneerng Research Center, Unversty of Calforna, Berkeley, Calf. Shng P. B., Mahn S. A., (1987) (a). Cumulatve expermental errors n pseudodynamc tests. Earthquake Engneerng & Structural Dynamcs, 15, 9. Shng P. B., Mahn S. A., (1987) (b). Elmnaton of spurous hgher mode response n pseudodynamc tests. Earthquake Engneerng & Structural Dynamcs, 15, 55. Shng P. B., Mahn S. A., (199). Expermental error effects n pseudodynamc testng. Journal of Engneerng Mechancs ASCE, 116, Shng P. B., Vannan M. T., Cater E., (1991). Implct tme ntegraton for pseudodynamc tests. Earthquake Engneerng & Structural Dynamcs,, Xe Y. M., Steven G. P., (199). Instablty, chaos, growth and decay of energy tme-steppng schemes for non-lnear dynamc equatons. Communcaton n Numercal Methods n Engneerng, 1, 3931.