CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE

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1 CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng methods for ntegraton of dfferental equatons. Tme-Steppng Method The equaton to be solved s or ( ) && & for lnearly elastc system mu + cu + ku = p t S (, ) ( ) && & & for nelastc system mu + cu + f u u = p t wth ntal condton u ( 0) = u0 u& ( 0) = u& 0 The appled force s gven by a set of dscrete values p = p( t ) where =0 to N. The tme nterval Δ t = t t + s usually constant, although ths s not necessary. The response s determned at dscrete tme nstant t. The dsplacement, velocty and acceleraton at tme t, denoted by u, u&, and u&&, respectvely, are assumed to be known and satsfy the equaton mu&& + cu& + ku = p 5 -

2 The numercal procedure to be presented wll enable us to determne the response quanttes u +, u & +, and u && + at tme + whch satsfy the equaton mu&& + cu& + ku = p We frst apply the procedure to tme = 0 to determne response at tme = and repeat the procedure agan to determne response at tme = 2 and so on. Therefore, ths progressve calculaton s called tme-steppng method. The response at tme + determned from response at tme s usually not exact. Many approxmate procedures mplemented numercally are possble. The requrements for a numercal procedure are () Convergence the numercal soluton should approach the exact soluton as the tme step decreases (2) Stablty the numercal soluton should be stable even f there s some round-off error or approxmaton. (3) Accuracy the numercal soluton should provde results that are close enough to the exact soluton. 5-2

3 These ssues are very mportant n numercal methods of solvng equatons. They wll govern the lmtaton of tmesteppng procedures. Three types of methods wll be dscussed: ) Method based on nterpolaton of exctaton 2) Method based on fnte dfference expresson of velocty and acceleraton 3) Method based on assumed varaton of acceleraton. 5-3

4 Method Based on Interpolaton of Exctaton Ths method s hghly effcent by nterpolaton exctaton durng a tme step as a lnearly varyng functon. p Δp ( τ ) p = + Δ t where Δ p = p + p and the tme varable τ vares from 0 to τ Δ t. For smplcty, we wll show dervaton of ths procedure for a system wthout dampng, although ths procedure can be extended to damped systems. The equaton to be solved s mu&& ku p Δp + = + t τ Δ The response u( τ ) over tme 0 τ Δ t s the sum of three parts: ) Free vbraton due to ntal dsplacement u and velocty u& at τ = 0 2) Response to step force p wth zero ntal condton Δp Δt τ 3) Response to ramp force wth zero ntal condton 5-4

5 Analytcal soluton derved n Chapter 3 can be used to determned the above three parts of responses and we wll get These formulae are derved from exact soluton of the equaton of moton. Therefore the result s exact f the exctaton s actually vares lnearly durng each tme step as usually assumed for earthquake ground exctaton whch s recorded at closely spaced tme ntervals. The exact soluton used n dervng ths procedure s avalable only f the system s lnear. The only restrcton on the sze of tme step s that t permts a close approxmaton to the exctaton functon and t provdes response results at closely spaced tme ntervals so that the response peaks are not mssed. 5-5

6 If the tme step Δ t s constant, the coeffcents A, B, D ' n ths procedure need to be computed only once. 5-6

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9 Central Dfference Method Ths method s based on a fnte dfference approxmaton of the tme dervatves of dsplacements, whch are velocty and acceleraton. Suppose Δ t s constant Δ t. The central dfference expresson for velocty and acceleraton at tme are u u 2Δt + & = and u u&& u 2u + u = + 2 ( Δt) Substtutng these n the equaton of moton at tme, we get u+ 2u + u u+ u m + c + ku 2 = p 2Δt ( Δt) We assume that u and u are known from prevous steps. Transferrng known quanttes to the rght hand sde, we get or where m c m c 2m + u = p u k u ( Δt) 2Δt + ( Δt) 2Δt ( Δt) kˆ ( Δt) 2 ku ˆ m c = + 2 Δ t = + pˆ m c 2m pˆ = p u k u ( Δt) 2Δt ( Δt) 2 2 The unknown u + s then gven by u + = pˆ k ˆ 5-9

10 Note that u + s obtaned wthout usng equaton of moton at tme + but from equaton of moton at tme. And u + can be computed explctly from the known dsplacement u and u. Such method s called an explct method. When = 0, u s needed for computng u, so we consder u u 2Δt & 0 = and u u&& u 2u + u = ( Δt) Usng the frst equaton to elmnate u n the second equaton, we then have ( Δt ) 2 u = uo Δ t( u& 0) + u&& 0 2 And consder equaton of moton at tme = 0 we get mu&& + cu& + ku = p u&& = p cu& ku m to be used for determnng u The procedure s summarzed next 5-0

11 Ths central dfference method wll gve meanngless results, called unstable, f the tme step s not short enough. The requrement for stablty of ths procedure s that Δ t < T π n However, ths requrement s never a constrant because the tme step needs to be much shorter, typcally Δt/ T n 0., to obtan acceptable accuracy of results. In analyss of earthquake response, a tme step about sec up to 0.02 sec s chosen to defne ground exctaton. 5 -

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13 Newmark s Method Ths method s developed by Nathan M. Newmark n 959 based on the followng equatons: ( γ) ( γ ) u & = u & + Δ t u && + Δt u && + + ( ) ( 0.5 β)( ) β( ) = + Δ & + Δ && + Δ && + u u t u t u t u The parameters β and γ defne the varaton of acceleraton over a tme step and determne the stablty and accuracy characterstcs of the method. Typcal selecton s γ =0.5 and β

14 Specal Cases. Average acceleraton If 2 u + γ = and β = 4 are chosen, the above equatons for u + and & corresponds to the specal case that acceleraton durng the tme step s constant and equal to the average of u&& and u && + as can be shown below. 2. Lnear acceleraton If 2 u + γ = and β = 6 are chosen, the above equatons for u + and & corresponds to the specal case that acceleraton durng the tme step vares lnearly between u&& and u && + as can be shown below. 5-4

15 Tme-Steppng Formula Ths method uses equlbrum equaton at tme and tme +, whch nvolves response quanttes at tme +,.e. u +, u & +, and u && +. Such method s called an mplct method. Let us defne the ncremental form Δ u = u+ u Δ u = u+ u & & & Δ u&& = u&& + u&& Δ p = p + p From the bass equatons of Newmark and Solve for ( γ) ( γ ) ( ) ( γ ) Δ u & = Δ t u && + Δ t u && = Δ t u && + Δt Δu && + ( ) ( 0.5 β)( ) β( ) 2 2 & && && + Δ u = Δ t u + Δ t u + Δt u Δ&& u and substtute ( ) ( Δt) 2 ( ) = Δ t u& + u&& + β Δt Δu&& 2 Δ u&& = Δu u& u&& β t βδt 2β ( Δ ) 2 Δ&& u n equaton for Δ u& γ γ γ Δ u & = Δu u +Δt u βδt β & 2 β && Then, substtute Δ u& and equaton of moton 2 Δ&& u n the ncremental form of mδ u&& + cδ u& + kδ u =Δp It can be wrtten as kˆ Δ u =Δ pˆ We obtan Δpˆ Δ u = k ˆ 5-5

16 where and Once γ ˆk = k + c+ m βδt β ( Δt) 2 γ γ Δ pˆ =Δ p + m c u m t c u + β t β & + +Δ Δ 2β 2β && Δ u s known, u Δ &, Δ&& u, u +, u & +, and u && + can be computed u = u +Δ + u u & = u & +Δ + u & u&& = u&& +Δ + u&& Alternatvely, u && + can be computed from u&& + = p cu& ku m

17 Newmark s method s stable f Δt T π 2 γ 2β For Average acceleraton method γ = and β = 2 4 n Δ t < T Ths mples that average acceleraton method s stable for any Δ t, although results would not be accurate for large Δ t. n For Lnear acceleraton method Δ t γ = 2 and β = 6 < 0.55 T Ths requrement s not sgnfcant because a much smaller tme step s requred for accurate representaton of exctaton and response. n 5-7

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20 Stablty Numercal procedures that gve bounded results f tme step s shorter than a certan lmt are called condtonally stable. Numercal procedures that gve bounded results regardless of tme step sze, no matter how large, are called uncondtonally stable. Stablty of the method s mportant for mult-degree-offreedom system where a uncondtonally stable method s sometmes necessary. Computatonal Error Error s nherent n any numercal method both from round-off error and approxmaton of soluton. Let us consder solutons of free vbraton usng dfferent procedures dscussed earler; Δ t = 0.T n ; and compare to the exact analytcal soluton. 5-20

21 All numercal methods gve results that have ampltude decay, mplyng that these procedures ntroduce numercal dampng. Most methods make the perod of vbraton longer except the central dfference method, whch gves result that has shorter perod than the exact result. Perod shortenng n the central dfference method s hghly sgnfcant when Δ t Ts close to ts stablty lmt /π. / n 5-2

22 The lnear acceleraton Newmark s method seems to be most accurate n the sense of least perod elongaton error for these methods consdered for lnear SDF system. The choce of methods would be dfferent for MDF system or nonlnear response analyss. The choce of tme step also depends on the tme varaton of the dynamc exctaton and natural perod of the system. Δ t = 0.T n gves reasonably accurate results, but t also has to be small enough to avod dstorton of the exctaton functon. For earthquake exctaton Δ t s usually less than 0.02 sec. A useful technque for selectng the tme step s to solve the problem wth a tme step that seems reasonable and resolve the problem wth a small tme step. The tme step s deemed small enough f results from two analyses are essentally the same, otherwse reduce the tme step and repeat such comparson untl two successve solutons are close enough. 5-22