DYNAMIC ANALYSIS BY NUMERICAL INTEGRATION

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1 DYNAMIC ANALYSIS BY NUMERICAL INTEGRATION Normally, For Earhquake Loading Direc Numerical Inegraion Is Very Slow 0. INTRODUCTION The mos general approach for he soluion of he dynamic response of srucural sysems is he direc numerical inegraion of he dynamic equilibrium equaions. This involves, afer he soluion is defined a ime zero, he aemp o saisfy dynamic equilibrium a discree poins in ime. Mos mehods use equal ime inervals a,, 3... N. Many differen numerical echniques have previously been presened; however, all approaches can fundamenally be classified as eiher explici or implici inegraion mehods. Explici mehods do no involve he soluion of a se of linear equaions a each sep. Basically, hese mehods use he differenial equaion a ime o predic a soluion a ime +. For mos real srucures, which conain siff elemens, a very small ime sep is required in order o obain a sable soluion. Therefore, all explici mehods are condiionally sable wih respec o he size of he ime sep. Implici mehods aemp o saisfy he differenial equaion a ime afer he soluion a ime is found. These mehods require he soluion of a se of linear equaions a each ime sep; however, larger ime seps may be used. Implici mehods can be condiionally or uncondiionally sable.

2 STATIC AND DYNAMIC ANALYSIS There exis a large number of accurae, higher-order, muli-sep mehods ha have been developed for he numerical soluion of differenial equaions. These mulisep mehods assume ha he soluion is a smooh funcion in which he higher derivaives are coninuous. The exac soluion of many nonlinear srucures requires ha he acceleraions, he second derivaive of he displacemens, are no smooh funcions. This disconinuiy of he acceleraion is caused by he nonlinear hyseresis of mos srucural maerials, conac beween pars of he srucure, and buckling of elemens. Therefore, only single-sep mehods will be presened in his chaper. Based on a significan amoun of experience, i is he conclusion of he auhor ha only single-sep, implici, uncondiional sable mehods be used for he sep-by-sep seismic analysis of pracical srucures. 0. NEWMARK FAMILY OF METHODS In 959 Newmark [] presened a family of single-sep inegraion mehods for he soluion of srucural dynamic problems for boh blas and seismic loading. During he pas 40 years Newmark s mehod has been applied o he dynamic analysis of many pracical engineering srucures. In addiion, i has been modified and improved by many oher researchers. In order o illusrae he use of his family of numerical inegraion mehods consider he soluion of he linear dynamic equilibrium equaions wrien in he following form: Mu && + Cu& + Ku = F (0.) The direc use of Taylor s series provides a rigorous approach o obain he following wo addiional equaions: 3 u = u- + u- + & u&& - + &&& u (0.a) 6 = + + u& u& - u&& - &&& u (0.b)

3 DIRECT INTEGRATION METHODS 3 Newmark runcaed hese equaions and expressed hem in he following form: 3 u = u- + u- + & u&& - + β &&& u (0.a) u = u + u + γ u & & - && - &&& (0.b) If he acceleraion is assumed o be linear wihin he ime sep, he following equaion can be wrien: &&& u = ( u&& u&& ) (0.3) The subsiuion of Equaion (0.3) ino Equaions (0.a and b) produces Newmark s equaions in sandard form u = u- + u& - + ( ) u&& - u&& β + β (0.4a) u& = u& + ( γ) u&& + γ u&& (0.4b) - - Newmark used Equaions (0.4a, 0.4b and 0.) ieraively, for each ime sep, for each displacemen DOF of he srucural sysem. The erm &&u was obained from Equaion (0.) by dividing he equaion by he mass associaed wih he DOF. In 96 Wilson [] formulaed Newmark s mehod in marix noaion, added siffness and mass proporional damping, and eliminaed he need for ieraion by inroducing he direc soluion of equaions a each ime sep. This requires ha Equaions (0.4a and 0.4b) be rewrien in he following form: u&& = b( u u ) + bu& + b3u&& (0.5a) u& = b4( u u ) + b5u& + b6u&& (0.5b) where he consans b o b 6 are defined in Table 0.. The subsiuion of Equaions (0.5a and0.5b) ino Equaion (0.) allows he dynamic equilibrium of he sysem a ime o be wrien in erms of he unknown node displacemens u. Or,

4 4 STATIC AND DYNAMIC ANALYSIS ( b M + b4c + K) u = F + M( bu bu& b 3u&& ) + C u u u ( b4 b5 & b6 && ) (0.6) The Newmark direc inegraion algorihm is summarized in Table 0.. Noe ha he consans b i need be calculaed only once. Also, for linear sysems, he effecive dynamic siffness marix K is formed and riangularized only once. 0.3 STABILITY OF NEWMARK S METHOD For zero damping Newmark s mehod is condiionally sable if γ, β and ω MAX γ - β (0.7) where ω MAX is he maximum frequency in he srucural sysem []. Newmark s mehod is uncondiionally sable if β γ (0.8) However, if γ is greaer han ½, errors are inroduced. These errors are associaed wih numerical damping and period elongaion. For large muli degree-of-freedom srucural sysems he ime sep limi, given by Equaion (0.7), can be wrien in a more useable form as T MIN π γ - β (0.9) Compuer models of large real srucures normally conain a large number of periods which are smaller han he inegraion ime sep; herefore, i is essenial ha one selec a numerical inegraion mehod ha is uncondiional for all ime seps.

5 DIRECT INTEGRATION METHODS 5 Table 0.. Summary of he Newmark Mehod for Direc Inegraion I. INITIAL CALCULATION A. Form saic siffness marix K, mass marix M and damping marix C B. Specify inegraion parameers β and γ C. Calculae inegraion consans b = β b b = +γ b b = (+γ b γ) 5 = b 3 = β b =γ b β 6 3 D. Form effecive siffness marix K = K+b M + b C 4 E. Triangularize effecive siffness marix K = LDL T F. Specify iniial condiions u0, u& 0, u&& 0 II. FOR EACH TIME STEP =,, A. Calculae effecive load vecor 4 F = F + M( bu b u& b u&& ) + C( b u bu& b && u ) B. Solve for node displacemen vecor a ime T LDL u = F forward and back-subsiuion only C. Calculae node velociies and acceleraions a ime u& = b ( u u ) + bu& + b u&& u&& = b ( u u ) + b u& + b u&& 3 D. Go o Sep II.A wih =+

6 6 STATIC AND DYNAMIC ANALYSIS 0.4 THE AVERAGE ACCELERATION METHOD The average acceleraion mehod is idenical o he rapezoidal rule ha has been used o numerically evaluae second order differenial equaions for approximaely 00 years. I can easily be derived from he following runcaed Taylor s series expansion: 3 τ τ uτ = u-+ τu-+ u-+ u- + 6 u τ u- u + τu& + ( && + && ) - - & && &&&... (0.0) where τ is a variable poin wihin he ime sep. The consisen velociy can be obained by differeniaion of Equaion (0.0). Or, u & & ( && u u u && τ = + τ ) (0.) If τ = u= u- + u- + & u&& -+ && u (0.a) 4 4 u& = u& - + u && - + u&& (0.b) These equaions are idenical o Newmark s Equaions (0.4a and b) wih γ = / and β = / 4. I can easily be shown ha he average acceleraion mehod conserves energy for he free vibraion problem, Mu&& + Ku = 0, for all possible ime seps [4].. Therefore, he sum of he kineic and srain energy is consan. Or, T T T T E = u& Mu& + u Ku = u& -Mu& - + u- Ku- (0.3)

7 DIRECT INTEGRATION METHODS WILSON S θ FACTOR In 973, he general Newmark mehod was made uncondiionally sable by he inroducion of a θ facor [3]. The inroducion of he θ facor is moivaed by he observaion ha an unsable soluion ends o oscillae abou he rue soluion. Therefore, if he numerical soluion is evaluaed wihin he ime incremen he spurious oscillaions are minimized. This can be accomplished by a simple modificaion o he Newmark mehod by using a ime sep defined by = θ (0.4a) and a load defined by R = R-+ θ ( R R-) (0.4b) where θ 0.. Afer he acceleraion &&u vecor is evaluaed by Newmark s mehod a he inegraion ime sep θ, values of node acceleraions, velociies and displacemens are calculaed from he following fundamenal equaions: u&& = u&& -+ ( u&& u&& -) θ (0.5a) u& = u& + ( γ ) u&& + γ u&& (0.5b) - - u = u- + u- + ( β) & u&& - + β u&& (0.5c) The use of he θ facor ends o numerically damp ou he high modes of he sysem. If θ equals.0 Newmark s mehod is no modified. However, for problems where he higher mode response is imporan, he errors ha are inroduced can be large. In addiion, he dynamic equilibrium equaions are no exacly saisfied a ime. Therefore, he auhor no longer recommends he use of he θ facor. A he ime of he inroducion of he mehod, i solved all problems associaed wih sabiliy of he Newmark family of mehods. However, during he pas weny years new and more accurae numerical mehods have been developed.

8 8 STATIC AND DYNAMIC ANALYSIS 0.6 THE USE OF STIFFNESS PROPORTIONAL DAMPING Because of he uncondiional sabiliy of he average acceleraion mehod, i is he mos robus mehod o be used for he sep-by-sep dynamic analysis of large complex srucural sysems in which a large number of high frequencies, shor periods, are presen. The only problem wih he mehod is ha he shor periods, which are smaller han he ime sep, oscillae indefiniely afer hey are excied. The higher mode oscillaion can be reduced by he addiion of siffness proporional damping. The addiional damping ha is added o he sysem is of he form C D =δ K (0.6) where he modal damping raio, given by Equaion (8.8) is defined by ξ n π = δ ωn = δ T n (0.7) One noes ha he damping is large for shor periods and small for he long periods or low frequencies. I is apparen ha periods which are greaer han he ime sep canno be inegraed accuraely by any direc inegraion mehod. Therefore, i is logical o damp hese shor periods o preven hem from oscillaing during he soluion procedure. For a ime sep equal o he period, Equaion (0.7) can be rewrien as T δ = ξ n π (0.8) Hence, if he inegraion ime sep is 0.0 seconds and we wish o assign a minimum of.0 o all periods shorer han he ime sep, a value of δ = should be used. The damping raio in all modes is now predicable for his example from Equaion (0.7). Therefore, he damping raio for a.0 second period is 0.0 and for a 0.0 second period is 0..

9 DIRECT INTEGRATION METHODS THE HILBER, HUGHES AND TAYLOR α METHOD The α mehod [4] uses he Newmark mehod o solve he following modified equaions of moion: Mu && +(+ α) Cu& +(+ α) Ku = (+ α) F αf + αcu& + αku (0.9) Wih α equals zero he mehod reduces o he consan acceleraion mehod. I produces numerical energy dissipaion in he higher modes; however, i canno be prediced as a damping raio as in he use of siffness proporional damping. Also, i does no solve he fundamenal equilibrium equaion a ime. However, i is currenly being used in many compuer programs. The performance of he mehod appears o be very similar o he use of siffness proporional damping. 0.8 SELECTION OF A DIRECT INTEGRATION METHOD I is apparen ha a large number of differen direc numerical inegraion mehods are possible by specifying differen inegraion parameers. A few of he mos commonly used mehods are summarized in 0.. Table 0.. Summary of Newmark Mehods Modified by he δ Facor METHOD γ β δ T MIN ACCURACY Cenral Difference / Excellen for small Unsable for large Linear Acceleraion / / Very good for small Unsable for large Average Acceleraion / /4 0 Good for small No energy dissipaion Modified Average Acceleraion / /4 T π Good for small Energy dissipaion for large

10 0 STATIC AND DYNAMIC ANALYSIS For single degree-of-freedom sysems he cenral difference mehod is mos accurae; and he linear acceleraion mehod is more accurae han he average acceleraion mehod. However, if only single degree-of-freedom sysems are o be inegraed he piece-wise exac mehod, previously presened, should be used since here is no need o use an approximae mehod. I appears ha he modified average acceleraion mehod, wih a minimum addiion of siffness proporional damping, is a general procedure ha can be used for he dynamic analysis of all srucural sysems. Using δ =T/ π will damp ou periods shorer han he ime sep and inroduces a minimum error in he long period response. 0.9 NONLINEAR ANALYSIS The basic Newmark Consan acceleraion mehod can be exended o nonlinear dynamic analysis. This requires ha ieraion mus be performed a each ime sep in order o saisfy equilibrium. Also, he incremenal siffness marix mus be formed and riangularized a each ieraion or a selecive poins in ime. Many differen numerical ricks, including elemen by elemen mehods, have been developed in order o minimize he compuaional requiremens. Also, he riangularizaion of he effecive incremenal siffness marix may be avoided by he inroducion of ieraive soluion mehods. 0.0 SUMMARY For earhquake analysis of linear srucures, i should be noed ha he direc inegraion of he dynamic equilibrium equaions is normally no numerically efficien as compared o he mode superposiion mehod using LDR vecors. If he riangularized siffness and mass marices and oher vecors canno be sored in high-speed sorage, he compuer execuion ime can be large. Afer using direc inegraion mehods for approximaely fory years, he auhor can no longer recommend he Wilson mehod for he direc inegraion of he dynamic equilibrium equaions. The Newmark consan acceleraion mehod, wih he addiion of very small amoun of siffness proporional damping, is recommended for dynamic analysis nonlinear srucural sysems. For all mehods of direc

11 DIRECT INTEGRATION METHODS inegraion grea care should be aken o make cerain ha he siffness proporional damping does no eliminae imporan high-frequency response. Mass proporional damping canno be jusified because i causes exernal forces o be applied o he srucure ha reduce he base shear for seismic loading. In he area of nonlinear dynamic analysis one canno prove ha any one mehod will always converge. One should always check he error in he conservaion of energy for every soluion obained. In fuure ediions of his book i is hoped ha numerical examples will be presened in order ha he appropriae mehod can be recommended for differen classes of problems in srucural analysis. 0. REFERENCES. Newmark, N. M., A Mehod of Compuaion for Srucural Dynamics, ASCE Journal of he Engineering Mechanics Division, Vol. 85 No. EM3, (959).. Wilson, E. L., Dynamic Response by Sep-By-Sep Marix Analysis, Proceedings, Symposium On The Use of Compuers in Civil Engineering, Labororio Nacional de Engenharia Civil, Lisbon, Porugal, Ocober -5, (96). 3. Wilson, E. L., I. Farhoomand and K. J. Bahe, Nonlinear Dynamic Analysis of Complex Srucures, Earhquake Engineering and Srucural Dynamics,, 4-5, (973). 4. Hughes, Thomas, The Finie Elemen Mehod - Linear Saic and Dynamic Finie Elemen Analysis, Prenice Hall, Inc., (987).

12 STATIC AND DYNAMIC ANALYSIS