International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Performance-based seismic analysis of light SDoF secondary substructures Kasinos, Stavros; Palmeri, Alessandro; Lombardo, Mariateresa Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Performance-Based Seismic Analysis of Light SDoF SecondarySubstructuresStavros KasinosPhD Candidate, School of Civil & Building Engineering, Loughborough University, UKAlessandro PalmeriSenior Lecturer, School of Civil & Building Engineering, Loughborough University, UKMariateresa LombardoLecturer, School of Civil & Building Engineering, Loughborough University, UKABSTRACT: A novel procedure is presented for the application of the PBE (performance-based engi-neering) methodology to the seismic analysis and design of light secondary substructures. In the proposedtechnique, uncertainty is conveniently represented in the reduced modal subspace rather than geometricdomain, which significantly reduces the number of uncertain parameters. The random response of aprimary structure under earthquake excitation is investigated, various cases of linear and nonlinear sec-ondary subsystems are examined and the propagation of uncertainty from the dynamic properties of theprimary structure to the seismic performance of the secondary subsystems is quantified.1. INTRODUCTIONSecondary subsystems are components or contentsof buildings that do not form part of the primaryload-bearing structure. Examples include architec-tural, mechanical and electrical components, build-ing equipment or furniture, which can be modelledas single-degree-of-freedom (SDoF) oscillators ormulti-degree-of-freedom (MDoF) structures, eitherlinear or nonlinear, singly or multiply connected tothe primary structure. Their seismic analysis anddesign is a topic of key engineering interest becausetheir damage can cause injuries or deaths, as wellas interruption of services, which in turn can leadto further human and economic losses (e.g. Taghaviand Miranda, 2003) .The primary focus in earthquake engineering hashistorically been on structural resistance, provid-ing designers with guidance to ensure life safety.This has mainly been addressed by specifying pre-scriptive and inexplicit requirements, e.g. limitingstresses and deformations determined from nomi-nal design loads. Aimed at enabling a more pre-dictable performance, as well as allowing di↵erenttargets to be achieved for building structures in dia-logue with the relevant stakeholders, performance-based engineering (PBE) philosophy has recentlyemerged as a broad spectrum of design solutionsunderpinned by well-defined case-specific perfor-mance objectives.Inherent uncertainties in the specification ofground shaking and structural properties (e.g.strength and sti↵ness of members and connec-tions) induce variation in the seismic performanceof structures. While these uncertainties are im-plicitly considered within a prescriptive design (i.e.through partial safety factors and characteristic val-ues), PBE allows a rational estimation of their ef-fects in a probabilistic manner.The implementation of PBE for secondary sub-structures is limited within the technical literature.Goulet et al. (2007) demonstrated the application ofthe methodology to reinforced-concrete moment-resisting frames, highlighting the impact of keymodelling assumptions on the accurate calculationof damage and associated repair costs. Yang et al.(2007) adopted a full probabilistic implementationof PBE for the performance evaluation of facilities,in which nonstructural components were classified112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015into performance groups based on their sensitivityto engineering demand parameters (EDPs).In this paper, a simulation-based procedure ispresented for the application of the PBE method-ology to the seismic analysis of light subsystems.Rather than being directly defined in the geomet-ric (physical) domain, in the proposed approach,uncertainty is characterised in the modal subspace,with modal shapes, modal frequencies and damp-ing ratios constituting the random quantities, whichnoticeably reduces the number of uncertain param-eters and the size of the dynamic problem. Bothlinear and nonlinear SDoF oscillators (attached to alinear MDoF system) are considered, as represen-tatives of a wider spectrum of nonstructural com-ponents, and the propagation of uncertainty fromthe primary structure to the secondary subsystemsis quantified.2. GOVERNING EQUATIONSIf the secondary system is assumed to be "light"(e.g. Muscolino and Palmeri, 2007), i.e. the sec-ondary attachment’s mass mS is much less thanthe mass of the primary structure MP (mS ⌧ MP),a cascade-type approach is admissible, with thetwo systems being decoupled and sequentially anal-ysed. Initially, the seismic response of the primarysystem is evaluated neglecting the feedback of thesecondary, with the response of the secondary suc-cessively being computed at the points of attach-ment. In this approach no primary-secondary inter-action is taken into account.2.1. Linear primary systemLet us consider the case of aMDoF primary system.Within the linear-elastic range, its seismic motion isruled by:M · u¨(t) +C · u˙(t) +K ·u(t) = M·⌧ · u¨g (t) , (1)where M, C and K are matrices of mass, equivalentviscous damping and elastic sti↵ness, respectively;u(t) is the array collecting the degrees of freedom(DoFs) of the system; ⌧ is a vector of seismic inci-dence; u¨g (t) is the ground acceleration.The equations of motion can be projected to themodal space, reducing the size of the dynamic prob-lem from n (system’s DoFs) to m (the number ofmodes retained within the analysis). This requiressolving the real-valued eigenproblem:M · ·⌦2 = K · , (2)where is the normalized modal matrix and⌦ thediagonal spectral matrix.It has been shown (Palmeri and Lombardo, 2011)that the truncation error introduced by the reducedmodes can be corrected via a dynamic mode ac-celeration method (DyMAM). Accordingly, the dy-namic response can be expressed as the sum ofmodal contributions and a corrective term:u(t) = ·q(t) +b!2f✓(t) , (3)where q(t) is the array collecting the modal coordi-nates, ruled by the equation of motion in the modalspace:q¨(t) +2 ⇣⌦ · q˙(t) +⌦2 ·q(t) = p · u¨g(t) , (4)in which:p = > ·M ·⌧ , (5)while b is the static correction vector and ✓(t) isthe response of the oscillator satisfying:✓¨(t) +2 ⇣ f ! f ✓˙(t) +!2f ✓(t) = u¨g (t) , (6)in which ! f and ⇣ f are chosen as:! f = 2min {⌦} ; ⇣f =1p2. (7)2.2. Uncertainty in the modal subspaceIn the preceding subsection, the analysis procedureto calculate the deterministic response of the pri-mary structure was described, in which the equa-tions of motion are conveniently projected to thereduced modal subspace, reducing the size of thedynamic problem (Falsone and Muscolino, 2004).Likewise, ecient analysis methods of combined212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015primary-secondary systems (Biondi and Muscol-ino, 2000) utilise modal analysis to predict the dy-namic interaction between the two components. Inaccordance with the PBE philosophy, the proba-bilistic response is of interest, whereby uncertaintyconsideration needs to be explicit (FIB, 2012).Contrary to the existing methods where uncertaintyis treated in the full nodal space, the present study,is motivated by the need to characterise uncertaintyin the reduced modal domain significantly reducingthe number of uncertain parameters. This can beachieved by considering some random fluctuationsin the eigenvectors, such that the stochastic modalmatrix will read:ˆ =2666666666664>1 +↵1,2>2 +↵1,3>3 + · · ·+↵1,m>m↵2,1>1 +>2 +↵2,3>3 + · · ·+↵2,m>m...↵m,1>1 +↵m,2>2 +↵m,3>3 + · · ·+>m3777777777775>,(8) being a modal shape, and ↵ an independent ran-dom coecient for each mode.It can be shown that by modifying the determin-istic Eq. (4) in light of Eq. (8), and applying similarconsiderations to the modal frequencies and the en-ergy dissipation, the stochastic solution of the sys-tem will be governed by a di↵erential equation ofthe form:mˆ · ¨ˆq(t) + cˆ · ˙ˆq(t) + kˆ · qˆ(t) = pˆ · u¨g (t) , (9)where mˆ, cˆ and kˆ represent the stochastic mass,damping and sti↵ness matrices in the reducedmodal space respectively, such that:mˆ = Im+↵ +↵> ; (10)cˆ = 2 ⇣⇣ ⇥Im+↵ + ⇤⌦+⌦⇥↵ +⇤>⌘ ; (11)kˆ = ⇥Im+↵ + ⇤⌦2 +⌦2⇥↵ + ⇤> , (12)while qˆ(t) is the array collecting the random re-sponse and pˆ the seismic incidence vector:pˆ =fIm+↵ +↵>gp . (13)It is worth emphasising here that, in the pro-posed formulation, three sources of uncertainty areconsidered, namely the mass (through the modalshapes), modal frequencies and viscous dampingratios, respectively, each associated with a zeromean randommatrix, namely,↵ ,  and  , such that:↵ =26666666640 ↵1,2 . . . ↵1,m↵2,1 0 . . . ↵2,m...... . . ....↵m,1 ↵m,2 . . . 03777777775; (14) =266666641. . .m37777775; =266666641. . .m37777775,(15)giving rise to a total of m2+m statistically indepen-dent random coecients.2.2.1. VerificationIt is possible to confirm the validity of the afore-mentioned formulation for a SDoF case (m = 1) byevaluating the stochastic quantities !ˆ and ⇣ˆ . Par-ticularising Eqs. (10) to (12), imposing a Taylor ex-pansion about ↵, , = 0, and dropping high orderterms, one gets:!ˆ =skˆmˆ= ! ·r1+2↵+2 1+2↵ ! · 1+ 2↵ 22+ . . .! !1+ (16)and:⇣ˆ =cˆ2!ˆ mˆ= ⇣1+2↵+ +1+2↵+ +2↵ = ⇣1+2↵ 2↵  + . . . ⇣1+.(17)312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015It follows that the standard deviation of the dimen-sionless random variables ↵,  and  is the coe-cient of variation (CoV) of the associated physicalquantity.2.3. Nonlinear secondary oscillatorBoth linear and nonlinear SDoF oscillators havebeen considered as secondary subsystems. In thefirst case the equation of motion reads:u¨s(t) +2 ⇣s!s u˙s(t) +!2s us(t) = u¨(a)p (t) , (18)where u¨(a)p (t) = u¨p(t)+ u¨g(t) is the absolute acceler-ation of the primary structure at the position wherethe secondary system is attached.For the nonlinear case, the governing equationcan be posed in the form:u¨s(t) +1msfs,nonlin(t) = u¨(a)p (t) , (19)in which fs,nonlin(t) is the nonlinear restoring forcein the secondary SDoF oscillator, whose mathe-matical definition depends on the particular type ofnonlinear behaviour.For an elastic-perfectly plastic secondary subsys-tem, the evolution in time of the restoring force isruled by:f˙s,nonlin(t) =8>>>>>>>>>>>><>>>>>>>>>>>>:Ks u˙s(t)if  fs,EPP(t) < fs,yor  fs,EPP(t) = fs,y andfs,EPP(t) u˙s(t) < 0;0otherwise,(20)in which fs,y is the yielding force in the secondaryoscillator.The case of a rigid-perfectly plastic SDoF systemcan be considered as the limiting case of the aboverestoring force, when Ks ! +1. As a result, theequation of motion becomes:u¨s(t) =8>>>>>>>><>>>>>>>>:u¨(a)p (t) + µs g sgn⇣u¨(a)p (t)⌘if u¨(a)p (t) > µs g ;0otherwise,(21)Figure 1: Structural frame which µs is the friction coecient for the sec-ondary system and g is the acceleration due to grav-ity.3. NUMERICAL APPLICATIONThe proposed formulation has been applied for theanalysis of two subsystems in cascade.Fig. 1 shows aMDoF primary system comprisingof a 5-storey single-bay moment-resisting frame,being irregular in plan, with position S denotingthe point of attachment of a light secondary SDoFoscillator of unit mass, modelled as (i) linear, (ii)elastoplastic and (iii) rigid-plastic. Floors are rigidin plane, while the self-weight and super-dead loadconstitute the mass source of the structure. The fun-damental period of vibration is Tp=0.498s for theprimary, and 0.9Tp for the secondary (cases (i) and(ii)). The total number of DoFs is n=15, with onlym=6 retained in the analysis, chosen such that atleast 90% of the modal mass participates in the seis-mic motion, a criterion set by current codes of prac-tice (e.g. Eurocode 8, 2004).A range of recorded accelerograms are appliedin the x direction, chosen as representative of var-ious scenarios (Cecini and Palmeri, 2015), namelyEl Centro 1940, Erzincan 1992 and Irpinia 1980,with peak ground accelerations (PGAs) of 0.313g,0.515g and 0.177g, respectively.3.1. Uncertainty characterisationUncertainty in the mass, modal frequencies andviscous damping ratios of the primary system isrepresented and propagated to the secondary sys-tem, with CoVs chosen as 0.025, 0.05 and 0.15 re-spectively, for two separate assumed distributionsnamely, uniform and Gaussian.412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Although extendible to other methods, the re-sults are presented only for a Monte-Carlo simu-lation (MCS), which is used to generate the ran-domised matrices, with a number of realisationsnsym = 1,000.3.2. Performance measuresTo quantify the response statistics and assess thepropagation of uncertainty from the primary to thesecondary subsystem, di↵erent performance mea-sures (PMs) are defined. It is acknowledged that,various components are sensitive to di↵erent struc-tural response parameters and thus PMs are chosenas: the maximum absolute acceleration and relativedisplacement respectively, for both linear primaryand secondary systems; the total accumulated plas-tic deformation for the elastoplastic nonlinear; andmaximum sliding distance for the rigid plastic.3.3. Primary system responseA selection of results for the primary system is pre-sented in this section. In a first stage, the frequencyresponse function (FRF) has been evaluated for theprimary system. Fig. 2(a) shows the exact FRF inthe geometric space, by retaining all modes (m=n);by using the mode displacement method (MDM),with m=6, where no correction is applied; by usingthe MAM and DyMAM, which introduce a staticand dynamic correction, respectively. It is evidentthat, while MAM produces an error in the high fre-quency range, this is seen to be corrected by Dy-MAM, which gives an improved approximation andis thus exploited in the subsequent stages.Fig. 2(b) compares the exact deterministic re-sponse (black line) with the 1,000 realisations ob-tained with the proposed randomisation of themodal information, also corrected by DyMAM, forthe case of uniform distribution (light grey), andGaussian distribution (grey). For both cases, theproposed randomisation seems to be satisfactoryin the frequency domain, with a higher fluctuationcaused by the Gaussian distribution attributed to itsunbounded nature.Carrying out the dynamic analysis in the time do-main (Fig. 3), one can observe that the oscillationsof the randomised response tend to show significantfluctuations around the deterministic ones, with theuncertainty propagating with time, in all accelera-tion and displacement time histories.Fig. 4 quantifies the statistics of the two PMsof the primary system under Irpinia ground mo-tion record and Gaussian distribution, with µ,  andCoV denoting mean, standard deviation and coe-cient of variation of the stochastic output, and x be-ing the deterministic (reported as a reference value).Notably, the output CoV shows a 47% in-crease from acceleration (CoV=0.09) to displace-ment (CoV=0.132), suggesting that the choice ofthe PM is significant. Furthermore, both CoVsexceed the assumed input CoVs for mass and fre-quency (0.025, 0.05 respectively), while displace-ment PM lies close to the chosen input CoV fordamping.3.4. Secondary system responseFollowing the seismic response of the primarystructure, our analyses proceed with the cascade re-sponse of three secondary oscillators. Fig. 5 com-pares the stochastic and deterministic force (top)and displacement (bottom) time histories for thelinear (left), elastoplastic (middle) and rigid plastic(right) secondary systems, respectively.In all three cases, as expected, the proposed ran-domisation seems satisfactory, with the time histo-ries of the PMs fluctuating around the deterministicresponse.Fig. 6 illustrates the normalised frequency distri-bution diagrams of the PM for the linear elastic,elastoplastic and rigid-plastic subsystems, respec-tively.For the elastic case, the randomisation pre-dicts a mean response of µ=0.179m, which is ingood agreement with the deterministic value ofx=0.182m. When compared to the correspondingrelative displacement of the primary structure, anamplification of about 8 times is seen, which is at-tributed to resonance (the two systems being almostin-tune). The CoV=0.199 is higher than that ofthe primary system (0.132) as well as of the threesources of uncertainty of the input variables, sug-gesting that uncertainty has been amplified in thesubsystem.In the elastoplastic oscillator a value ofx=0.723m in the corresponding PM shows that the512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015101 102 103107106105104103102101! [rad/s]|H( !) |ExactMDMMAMDyMAM(a)101 102 103107106105104103102101! [rad/s]|H( !) |ExactGaussianUniform(b)Figure 2: FRF for modal correction methods (a) and stochastic response (b).2 4 6 81050510t [s]u¨(t)h ms2i(a)2 4 6 810010t [s]u¨(t)h ms2i(b)2 4 6 810505·102t [s]u(t)[ m](c)Figure 3: Acceleration time histories of Irpinia (left) and El Centro (middle); displacement time history of ElCentro (right), drawn from Gaussian distributions.4 4.5 5 5.5 6|u¨max |hms2if( |u¨max|)µ = 4.97,  = 0.44CoV = 0.09, x = 5.15(a)1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4·10220406080100|umax | [m]f( |umax|)µ = 0.023,  = 0.003CoV = 0.132, x = 0.022(b)Figure 4: Frequency diagrams for acceleration (a) and displacement (b) primary system PMs for Irpinia earth-quake, drawn from Gaussian distributions.mean total plastic deformation is slightly underes-timated (µ=0.746m), while a CoV=0.395, beingmuch higher than the input parameters, suggests anincreased dispersion in the results.When the rigid-plastic oscillator is considered,a decrease of an order of magnitude in the corre-612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015sponding PM is observed (when compared to theprevious cases). Again, the resulting CoV of 0.279is significantly higher than in the input, showing ev-idence of dispersion, which needs to be accountedfor.It therefore appears that, depending on the dy-namic behaviour of the secondary system and itsmechanical parameters, the seismic response canbe significantly amplified, and the dispersion of re-sults increased, meaning that a simple prescriptiveapproach to the design of such component can beunder- or over- conservative.4. CONCLUSIONSA simulation-based procedure was presented forthe application of the PBE methodology to the seis-mic analysis of light SDoF subsystems attached toa primary MDoF structure. A novel feature of theproposed approach is the characterisation of un-certainty in the reduced modal space, rather thanin the full geometric domain, and its applicationin conjunction with a dynamic mode accelerationmethod (DyMAM). As demonstrated with MonteCarlo simulations, the proposed approach is capa-ble of accurately representing the random dynamicresponse, despite the fact that the number of un-certain parameters is reduced to m2+m statisticallyindependent coecients (m being the number ofmodes retained in analysis). The resulting modelappears to be adequate for the purpose of assess-ing how uncertainty in the primary structure prop-agates to the seismic performance of the secondarysubsystems.In a first stage, the response of a primarystructure subjected to di↵erent accelerograms wasquantified through relevant performance measures(PMs), with uncertainty comprising the mass dis-tribution, modal frequencies and damping. In a sec-ond stage, uncertainty in the primary system waspropagated to i) linear, ii) elastoplastic and iii)rigid-plastic secondary systems, and the responsestatistics were quantified.Future investigations will be carried out to assessthe sensitivity of the output to the chosen randomvariation of the input parameters as well as theirprobabilistic distribution. The case of the rockingmotion of a secondary subsystem will also be ex-amined.5. REFERENCESBiondi, B. and Muscolino, G. (2000). “Component-mode synthesis method variants in the dynamics ofcoupled structures.” Meccanica, 35(1), 17–38.Cecini, D. and Palmeri, A. (2015). “Spectrum-compatible accelerograms with harmonic wavelets.”Computers and Structures, 147, 26–35.Eurocode 8 (2004). European Commettee for Standardi-sation. Design of structures for earthquake resistance.Falsone, G. and Muscolino, G. (2004). “New real-valuemodal combination rules for non-classically dampedstructures.” Earthquake Engineering and StructuralDynamics, 33, 1187–1209.FIB (2012). Probabilistic performance-based seismicdesign. International Federation for Structural Con-crete (fib), technical report, bulletin 68 Report, Lau-sanne, Switzerland.Goulet, C. A., Haselton, C. B., Mitrani-Reiser, J., Beck,J. L., Deierlein, G. G., Porter, K. A., and Stewart, J. P.(2007). “Evaluation of the seismic performance of acode-conforming reinforced-concrete frame building– From seismic hazard to collapse safety and eco-nomic losses.” Earthquake Engineering and Struc-tural Dynamics, 36(13), 1973–1997.Muscolino, G. and Palmeri, A. (2007). “An earthquakeresponse spectrum method for linear light secondarysubstructures.” ISET Journal of Earthquake Technol-ogy, 44(1), 193–211.Palmeri, A. and Lombardo, M. (2011). “A new modalcorrection method for linear structures subjected todeterministic and random loadings.” Computers andStructures, 89, 844–854.Taghavi, S. and Miranda, E. (2003). Response assess-ment of nonstructural building elements. PEER Re-port 2003/05, University of California Berkeley.Yang, T. Y., Moehle, J., Stojadinovic, B., and Ki-ureghian, A. D. (2007). “Seismic performance evalu-ation of facilities: Methodology and implementation.”Journal of Structural Engineering, 135(10), 1146–1154.712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20156 8 10 1250050t [s]F(t)[ kN]6 8 10 12201001020t [s]F(t)[ kN]6 8 10 12202t [s]F(t)[ kN]6 8 10 [s]u(t)[ m]6 8 10 120.100.1t [s]u(t)[ m]6 8 10 1210123·102t [s]u(t)[ m]Figure 5: Force (top) and displacement (bottom) time histories a linear (left), elastoplastic (middle) and rigid-plastic (right) secondary system, for Irpinia earthquake, drawn from Gaussian distributions.0.1 0.15 0.2 0.25 0.3 0.35246810|umax | [m]f( |umax|)µ = 0.179,  = 0.035CoV = 0.199, x = 0.182(a)0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 [m]f⇣ up⌘µ = 0.746,  = 0.294CoV = 0.395, x = 0.723(b)0.5 1 1.5 2 2.5·10220406080100120umax [m]f( umax)µ = 0.012,  = 0.003CoV = 0.279, x = 0.013(c)Figure 6: PM frequency diagram for a linear (a) elastoplastic (b) and rigid-plastic (c) secondary system due toIrpinia earthquake, drawn from Gaussian distribution.8


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