PROPOSING INPUT-DEPENDENT MODE CONTRIBUTION FACTORS FOR SIMPLIFIED SEISMIC RESPONSE ANALYSIS OF BUILDING SYSTEMS

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1 he 4 th World Coferece o Earthquake Egieerig October -7, 008, Beiig, Chia PROPOSING INPU-DEPENDEN ODE CONRIBUION FACORS FOR SIPLIFIED SEISIC RESPONSE ANALYSIS OF BUILDING SYSES ahmood Hosseii ad Laya Abbasi Associate Professor, Structural Egieerig Research Ceter, Iteratioal Istitute of Earthquake Egieerig ad Seismology (IIEES), ehra, Ira, hosseii@iiees.ac.ir Graduate Studet, Graduate Program o Earthquake Egieerig, Iteratioal Istitute of Earthquake Egieerig ad Seismology (IIEES), ehra, Ira, l.abbasi@iiees.ac.ir ABSRAC : Whe the excitatio frequecy at the base of a buildig is close to oe of its modal frequecies other tha its first mode, the cotributio of that mode i compariso with the first mode is much more comparig with other cases of excitatio, however, this fact is ot take ito accout by the commo defiitios of ode Participatio / Cotributio Factors i which always the first mode is domiat. If the effect of iput frequecy is icorporated by some way i the defiitio of these factors, it will be possible to calculate the approximate structural resposes to various dyamic loads, icludig seismic forces, with o eed to time history aalysis. For this purpose, i this paper at first various defiitios, proposed by differet scholars, for modal participatio factors have bee itroduced ad discussed briefly. he a simple DOF system has bee cosidered subected to siusoidal base excitatios with various frequecies. For each iput, the ratio of the correspodig maximum respose of every mode i each degree of freedom of the system to the total respose of that degree of freedom has bee calculated ad compared to those of other modes. he the resposes of the cosidered system to some seismic iputs with various frequecy cotets have bee calculated ad the same ratios have bee obtaied agai to fid out the possibility of defiig iput-frequecy depedet mode participatio factors. Numerical results show that this defiitio is possible ad these factors ca be used for Simplified Seismic Respose Aalysis of buildig systems. EYWORDS: Iput-frequecy Depedet ode Participatio Factors, Simplified Dyamic Aalysis. INRODUCION I almost all resources of structural dyamics the mode participatio factors are idepedet of the iput characteristics, ad are oly fuctios of specificatios of the structure. his basically results i domiacy of the first atural mode of the system i its dyamic respose to ay base excitatio, regardless of the type ad frequecy cotet of the excitatio. However, it is clear that i case of a buildig, excited at its base by a excitatio havig a frequecy close to oe of its modal frequecies other tha its first mode, the cotributio of that mode i its dyamic respose i compariso with the cotributio of the first mode is much more tha other cases of excitatio. herefore, if the effect of the iput frequecy ca be icorporated i the defiitio of mode cotributio factors by some way, it will be possible to calculate the approximate values of maximum resposes of structures to various dyamic loads, icludig earthquake iduced forces, by some easy calculatios with o eed to ay time history aalysis. his ca be called a Simplified Dyamic Aalysis Approach. For this purpose, i this paper at first various defiitio proposed by differet scholar for modal participatio factors have bee itroduced ad discussed briefly. he a simple DOF system has bee cosidered subected to siusoidal base excitatios with various frequecies. For each iput the ratio of the correspodig maximum respose of every mode i each degree of freedom of the system to the total respose of that degree of freedom has bee calculated ad compared to those of other modes. he the resposes of the cosidered system to a set of seismic iputs with various frequecy cotets, from low to high, have bee calculated ad the same ratios have bee obtaied agai to fid out the possibility of defiig iput-frequecy depedet mode participatio factors. he details of the study are explaied i the ext sectios of the paper.

2 he 4 th World Coferece o Earthquake Egieerig October -7, 008, Beiig, Chia. ODE PARICIPAION CONCEPS AND DEFINIIONS I calculatio of dyamic respose of structures to seismic excitatios by mode combiatio techique the modal equatios of motio of the form give by Eq () are used: y + Cy + y = P(t) () where is the mode umber,, C ad are respectively the modal mass, modal dampig, ad modal stiffess, ad y ad its time derivatives are modal displacemet, velocity, ad acceleratio respectively, ad P (t) is the modal load. I cases of ordiary multi-story buildigs subected to ust a horizotal seismic excitatio the modal load is give by: t { } { } P(t)= φ P (t) =L x (t) () eff g i which {φ } t is the traspose of th modal vector or mode shape of the structure, ad {P eff (t)} is the earthquake effective load vector give by: eff { } {P (t)}=-[] x (t) (3) where [] is the mass matrix, {} is the x earthquake ifluece vector havig elemets of uity, ad x g (t) is the time history of groud displacemet. O this basis the modal respose to base excitatios are obtaied by: g { } [ ]{} { φ} [ ]{ φ} φ L y (t)= V (t)= V (t) ωd (4) I Eq (4) L is called the modal ifluece factor ad V (t) is the modal itegral of seismic respose, which is a fuctio of groud acceleratio as well as the values of modal frequecy, ω, ad modal dampig, ξ. Up to this poit there is o differece i the formulatio proposed by various scholars for calculatio of seismic respose of buildig systems, however, cosiderig the algebraic term beside V (t) i Eq (4) as a factor which is oly depedet o the modal characteristics of the system, while V (t) is a fuctio of both earthquake ad system characteristics, it has bee tried to itroduce some kid of modal participatio or cotributio factor for simplificatio of seismic respose calculatios, ad several defiitios have bee proposed for these factors so far. Clough ad Pezie (975) have suggested the followig formulas, callig γ s as the modal effective masses. L γ = γ = = {} [ ]{} (5) = hey have called the total mass of the buildig. Amii (983) have itroduced the followig formula: { φ } [ ]{} r { φ } [ ]{ φ } α = α = () = callig α s the mode participatio factors ad {r} has the same defiitio of {} i Eq (3). Adeli (98) has proposed the followig formula, callig L s the modal participatio factors, which their summatio is uity, implyig that the whole modes together makes the whole respose of the structure:

3 he 4 th World Coferece o Earthquake Egieerig October -7, 008, Beiig, Chia { φ} [ ]{} { φ } [ ]{ φ } L = L = (7) Gaylord ad Gaylord (989) ad also Paz (994) have used aother form of Eq (7) by usig γ istead of L, which ca be used whe the mass matrix is diagoal, such as the case of multistory buildigs with m i as mass of the i th story. m iφi γ = i= γ = (8) m i φi i= Naeim (993) have suggested the followig formulas. L N e= e=otal ass (9) = where e is called the effective modal mass. It ca be see that Eqs (), (7) ad (8) are all based o the same cocept. Eqs (5) ad (9) are i fact very similar i the cocept, ad they have ust used differet otatios. By payig attetio to Eqs () to (8) it ca be easily see that if the modal vectors are calculated by differet approaches, surprisigly, differet values will be obtaied for each of the so-called modal participatio factors. Chopra (995) has metioed implicitly this shortcomig ad has itroduced some other form of modal cotributio factors which are differet kids of resposes such as displacemets, story shear forces ad story momets. Eqs (5) ad (9), o the other had, do ot have this shortcomig, but their use is limited to oly the case of shear-beam-type structures subected to ust horizotal groud excitatio, ad they ca ot be used for other types of structures, ad eve for the case of shear-beam-type buildigs they ca ot be used for rotatioal or torsioal excitatios. Hosseii ad Yaghoobi Veyegha (998, 999) have proposed, by usig the defiitio of Clough ad Pezie, a somehow ew defiitio for modal participatio factors of a structure subected to multicompoet groud motio by itroducig Eq (0). L k γ k = k γ = (0) ek = i which γ k ad ek are respectively the modal participatio factor of the structure ad its effective mass for the k th compoet of groud motio. his effective mass is defied as: { r } [ ]{ r } ek = k k () where {r k }is the earthquake ifluece vector for the k th compoet of groud motio, of which the elemet r k is defied as the geeralized displacemet created i the i th degree of freedom of the structure due to the uite positive geeralized displacemet of the k th compoet of groud motio. By Eqs (0) ad (), i fact, istead of oe participatio factor for each mode as a sigle valued parameter which is ot depedet o how the structure is excited by the groud motio, up to six differet compoetial factors ca be defied for each mode of the structure, depedig o the umber of cosidered compoet of groud motio. Although by this approach the modal participatio factors are defied much deliberately ad are depedet to the excitatio compoet(s), still they are quite idepedet from the excitatio frequecy. his is while, obviously, excitig a structure by a excitatio havig a domiat frequecy very close to oe of the modal frequecies of the structure will cause the participatio of that mode icrease remarkably. his fact is ot take ito cosideratio i the defiitio of modal participatio factors, eve by usig modal-compoetial factors. I the ext sectio

4 he 4 th World Coferece o Earthquake Egieerig October -7, 008, Beiig, Chia it is tried to defie a ew cocept of iput-frequecy depedet mode participatio factor by which it is believed that the seismic respose aalysis of buildig systems ca be doe with very little calculatios. 3. INPU-FREQUENCY DEPENDEN ODE PARICIPAION FACOR he modal participatio or cotributio factors by the defiitios give i previous sectio all results i the domiacy of the first mode of ay structure i calculatio of its seismic respose, regardless of the domiat frequecy of the excitatio. o fid out if it is possible to defie mode participatio factors which deped i some way o the iput frequecy a 5-story buildig show i Figure is cosidered. m 5 m 4 m 3 m m k 5 k 4 k 3 k k m = m = 5 0 = 4 0 = 3 0 =.7 0 =... = m = N / m N / m N / m N / m N / m = 50 to Figure. he 5-story buildig model used i the study I this model the height of all stories has bee assumed to be 3.30 m, ad stories mass ad stiffess values are as give i Figure. For seismic desig of this buildig model the soil type has bee assumed to be of type II, ad the seismicity of the regio has bee cosidered to be moderate to high. he buildig has bee desiged based o drift cotrol. he modal frequecies ad modal shapes are obtaied as: 4.7. {} = 8.3 rad / Sec ω [] φ.993 = Now, by itroducig the followig formulas: v i ( t ) η i = v i ( t ) v i ( t ) = k ϕ i y ( t ) i = i k ϕ i = k il ϕ l l = v i ( t ) = k ϕ i y ( t ) = i = i ()

5 he 4 th World Coferece o Earthquake Egieerig October -7, 008, Beiig, Chia where η i is the modal story shear force ratio, v i is the cotributio of mode i the shear force of story i, v i, it is possible to relate the modal cotributio factors to the iput frequecy as described hereiafter. wo sets of base excitatios have bee cosidered, a siusoidal ad a seismic. I case of siusoidal excitatios two frequecy values, correspodig to the secod ad the forth modes of the buildig, respectively. r/s ad 4.97 r/s have bee used. I case of seismic excitatios the accelerograms of three earthquakes of Emeryville, Chichi, ad Corralitos, have bee used, respectively as low frequecy, mid frequecy, ad high frequecy excitatio. he results of dyamic respose calculatios for siusoidal excitatios are show i Figures to, where the horizotal axes shows the time i sec ad vertical axes show the modal shear values i kn. Figure. he modal ad total shear force values at the st story of the buildig for the case of siusoidal base excitatio with the secod modal frequecy Figure 3. he modal ad total shear force values at the d story of the buildig for the case of siusoidal base excitatio with the secod modal frequecy Figure 4. he modal ad total shear force values at the 3 rd story of the buildig for the case of siusoidal base excitatio with the secod modal frequecy Figure 5. he modal ad total shear force values at the 4 th story of the buildig for the case of siusoidal base excitatio with the secod modal frequecy Figure. he modal ad total shear force values at the 5 th story of the buildig for the case of siusoidal base excitatio with the secod modal frequecy

6 he 4 th World Coferece o Earthquake Egieerig October -7, 008, Beiig, Chia It is see i Figures to that the first mode does ot have much cotributio, particularly i higher stories. he same is true for the case of excitatio of the buildig by the frequecy of its forth mode of vibratio, of which the results ca ot be show here because of lack of space (the complete results ca be foud i the mai report of the study (Abbasi 008)). he results of seismic excitatios are show i Figures 7 to. Figure 7. he modal ad total shear force values at the st story of the buildig for the case of seismic excitatio by Chichi (mid Figure 8. he modal ad total shear force values at the d story of the buildig for the case of seismic excitatio by Chichi (mid Figure 9. he modal ad total shear force values at the 3 rd story of the buildig for the case of seismic excitatio by Chichi (mid Figure 0. he modal ad total shear force values at the 4 th story of the buildig for the case of seismic excitatio by Chichi (mid Figure. he modal ad total shear force values at the 5 th story of the buildig for the case of seismic excitatio by Chichi (mid It ca be see i Figures 7 to that cotributio of the secod mode is remarkable i the respose of the system, ad eve i some cases, more tha the first mode, particularly i the higher stories.

7 he 4 th World Coferece o Earthquake Egieerig October -7, 008, Beiig, Chia Figure. he modal ad total shear force values at the st story of the buildig for the case of seismic excitatio by Corralitos (high Figure 3. he modal ad total shear force values at the d story of the buildig for the case of seismic excitatio by Corralitos (high Figure 4. he modal ad total shear force values at the 3 rd story of the buildig for the case of seismic excitatio by Corralitos (high Figure 5. he modal ad total shear force values at the 4 th story of the buildig for the case of seismic excitatio by Corralitos (high Figure. he modal ad total shear force values at the 5 th story of the buildig for the case of seismic excitatio by Corralitos (high It ca be see i Figures to that i the case of a high frequecy earthquake the cotributio of oe of the higher modes (i this case the third mode) is quite domiat comparig with the cotributio of the first mode. Now, goig back to Eqs () it is see that defiig the story shear modal cotributio factors is reasoable, but, as it ca be see i the show results that excitig the buildig with a excitemet whose frequecy is very close to oe specific mode of the buildig does ot ecessarily leads to the domiacy of that mode i the shear values i all stories. A very good sample of such case ca be see i Figure 4, which shows that the cotributio of the secod mode i the shear force of the 3 rd story of the buildig is very little i spite of that the

8 he 4 th World Coferece o Earthquake Egieerig October -7, 008, Beiig, Chia excitatio frequecy is exactly equal to the secod modal frequecy of the buildig. he reaso behid this fact is described here. he story shear force vector ca be writte as: 0 { V(t) } = [ I ][ ][ φ]{ y(t) } [] I = 0 0 (i the case of 5-story buildig) (3) By defiig [ λ ]=[ I][ ][ φ], ay elemet of this matrix, λ i, ca be called the coefficiet of the th modal respose i the shear force of the i th story. For the cosidered 5-soty buildig [ λ ] is obtaied as: [ λ ] = I fact, [ λ ] shows the structure depedet part of story shear modal cotributio factors. It is see i the above matrix that all mode has the same cotributio i the shear force of the st story, while for other stories the modal cotributios are quite differet, ad sometimes they have opposite effects. he reaso of little cotributio of the secod mode i the shear force of third story, which was metioed before, ca also be foud i this matrix by comparig -0.3 with other values i the third row of the matrix. 4. CONCLUSIONS Based o the umerical results it ca be said that the iput-frequecy depedet mode participatio factors are meaigful factors that ca be defied for ay structure, ad their umber is equal to (for a -degree of freedom system) istead of. Furthermore, by usig these factors the calculatio of seismic respose of structures ca be doe by little effort, with o eed to time history aalysis. herefore, they are very useful tools for Simplified Seismic Respose Aalysis of buildig systems. REFERENCES - Abbasi, L. (008). Studyig the Possibility of Simplified Seismic Aalysis of Buildigs by Usig the Iput-frequecy Depedet ode Participatio Factors, aster hesis uder Supervisio of Dr. ahmood Hosseii, Submitted to the Graduate Program of IIEES, ehra, Ira. - Adeli, Hoatollah (983). Earthquake Egieerig (i Persia), Dehkhoda Publicatios, ehra. - Amii, Ali (983). Ivestigatio of Respose to Earthquake Excitatio: A Statistical Basis for Spectrum Superpositio, USC Report No. CE 83-0, Los Ageles, Califoria. - Chopra, A.. (995). Dyamics of Structures, Pretice Hall. - Clough, R. W. ad Pezie, J. (975). Dyamics of Structures, d Editio, c GrawHill. - Gaylord, E. H. ad Gaylord, C. N. (990). Structural Egieerig Hadbook, 3 rd Ed., c GrawHill. - Hosseii,. ad Yaghoobi-e-Vayegha, F. (998). A Study o Earthquake Ifluece Factors i Buildig Structures, Peouheshameh (he Persia Joural ad Bulleti of IIEES), Vol., No Hosseii,. ad Yaghoobi-e-Vayegha, F. (999). A ore Precise Isight ito the Cocept ad Defiitios of the ass Participatio Factor i Seismic Aalysis of Structures, Peouheshameh (he Persia Joural ad Bulleti of IIEES), Vol., No Paz,. (994). Iteratioal Hadbook of Earthquake Egieerig, Chapma ad Hall.