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MODAL ANALYSIS OF LINEAR DYNAMIC SYSTEMS:PHYSICAL INTERPRETATION

By Anil K. Chopra,l Member, ASCE

ABSTRACT: The modal analysis procedure for calculating the dynamic response of classically damped linearstructural systems is reformulated with the objective of developing the physical significance of the elements ofthe solution and defining modal contribution factors. The resulting concepts are used to develop an approachfor predicting how the relative contributions of various modes to the response and the number of modes thatshould be included in the solution depend on the vibration properties of the system, on the response quantityof interest, and on (1) the spatial distribution and time variation of applied dynamic forces in the case of forceexcited systems, or (2) the response (or design) spectrum for an earthquake excitation. To illustrate this approach,numerical results are presented for a five-story frame for a wide range of parameters and a base-isolated building.

where

(3)

(4)

(5)

(2b)

(2a)

(6a-c)

p(t) =sp(t)

N N

u(t) =2: un(t) =2: <\>nqn(t)n-l n-l

Eq. (1) also governs the vector u of nodal displacementsrelative to the moving ground undergoing acceleration iig(t) if

mu + cti + ku = p(t) (1)

where m, C, and k = mass, damping, and stiffness matrices ofthe system. We will consider a common loading case in whichthe applied forces pj(t)-j = 1,2, ... ,N-have the same timevariation p(t), and their spatial distribution is defined by s,independent of time. Thus

RESPONSE HISTORY ANALYSIS: REFORMULATIONAND INTERPRETATION

Equations of Motion

A viscously damped system with classical damping and having N degrees of freedom is considered. The response of thesystem described by the vector u of nodal displacements dueto external forces p(t) is governed by the N differential equations

Modal Expansion of Displacements and Forces

The natural frequencies W nand natural modes <Pn of vibration of the system satisfy the following matrix eigenvalueproblem:

The displacement u of the system can be expressed as thesuperposition of modal contributions un(t):

where t = influence vector representing the displacements ofthe masses resulting from static application of a unit-grounddisplacement, ug = 1. Thus, the spatial distribution of the effective earthquake forces, (2b) is s = mt.

where qn(t) = modal coordinates.A central idea of this formulation is to expand the spatial

distribution s of applied forces or s = mt of the effective earthquake forces as

INTRODUCTION

The classical mode superposition method or classical modalanalysis method is widely recognized as a powerful methodfor calculating the dynamic response of viscously dampedlinear structural systems with classical damping. The methodis attractive because the response of a multi-degree-of-freedom (MDF) system is expressed as the superposition ofmodal responses, each modal response is determined from thedynamic analysis of a single-degree-of-freedom (SDF)system, and these dynamic analyses need to be implementedonly for those modes with significant contribution to the response.

For analysis of earthquake-excited systems, two versions ofthe procedure are in use: (1) the response history version, inwhich the modal responses are computed as a function of timeand then superposed to obtain the response history of the system; and (2) the response spectrum version, in which the peakvalues of modal responses are determined from the responsespectrum or design spectrum that characterizes the excitation,and the peak response of the system is then estimated by appropriately combining the modal peaks.

While classical modal analysis is described in many bookson structural dynamics and is widely applied in structural engineering practice, it has usually not been formulated in a manner that emphasizes the physical significance of the elementsof the solution. The present paper is intended to be responsiveto this need.

Its objectives are: (1) to reformulate classical modal analysisin order to provide a useful physical interpretation of themethod; and (2) to develop a methodology for predictinghow the relative contributions of various modes to theresponse and the number of modes necessary in the solutiondepend on the vibration properties of the system, on the response quantity of interest, and on either the spatial distribution and time variation of the applied dynamic forces in thecase of force-excited systems, or on the response (or design)spectrum for an earthquake excitation. The physical interpretation of modal analysis presented in this paper is the centralconcept utilized extensively in a recent book (Chopra 1995)to analyze and understand the dynamic response of linear systems.

I Johnson Professor of Civil Engineering, 721 Davis Hall, Univ. ofCalifornia, Berkeley, CA 94720.

Note. Associate Editor: James M. Nau. Discussion open until OctoberI, 1996. To extend the closing date one month, a written request mustbe filed with the ASCE Manager of Journals. The manuscript for thispaper was submitted for review and possible publication on July 5, 1995.This paper is part of the ]ounuJl of Structural Engineering, Vol. 122,No.5, May, 1996. ©ASCE, ISSN 0733-9445/96/0005-0517-0527/$4.00+ $.50 per page. Paper No. 11065.

JOURNAL OF STRUCTURAL ENGINEERING / MAY 1996/517

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MODAL CONTRIBUTION FACTORS

Eq. (14), specialized for the displacement response, is equivalent to (11) [Chopra (1995) page 470].

Total Response

Combining the response contributions of all the modes givesthe total response:

(14)

(15)

N N

r(t) =2: r.(t) =2: r:A.(t)n-l "_1

Interpretation of Modal Analysis

In the first phase of the modal analysis procedure, the vibration properties-natural frequencies and modes-of thestructure are computed and the force distribution vector S orrna. is expanded into its modal components s•. Fig. 1 showsschematically the rest of the analysis procedure for earthquakeexcitation to emphasize the underlying concepts; with appropriate modifications it also applies to force-excited systems.The contribution of the nth mode to the dynamic response isobtained by multiplying the results of two analyses: (1) staticanalysis of the structure subjected to external forces, s., and(2) dynamic analysis of the nth-mode SDF system excited byground acceleration iig(t). Thus, modal analysis requires staticanalysis of the structure for N sets of external forces: s., n =1,2, ... ,N, and dynamic analysis ofN different SDF systems.Combining the modal responses gives the dynamic responseof the structure.

(7)

where tn is the damping ratio for the nth mode. The factor f nthat multiplies the force p(t) or ground acceleration ug(t) isoften called a modal participation factor, implying that it is ameasure of the degree to which the nth mode participates inthe response. This is not a useful definition, however, becausef n is not independent of how the mode is normalized, nor ameasure of the contribution of the mode to a response quantity.Both these drawbacks are overcome by modal contributionfactors that will be defined later.

The solution for the modal coordinate qn (t) is

which is independent of how the modes are normalized.Eq. (5) may be viewed as an expansion of the applied force

distribution s or rna. in terms of inertia force distributions Snassociated with natural modes (Crandall and McCalley 1961;Veletsos, unpublished notes, 1977). The expansion of (5) hasthe useful property that the force vector snp(t) or -SnUg(t)produces response only in the nth mode but no response inany other mode. This implies that the response in the nth modeis due entirely to the partial force vector snp(t) or -SnUg(t).

Modal Equations

By using (4), (1) can be transformed to a system of uncoupled equations in modal coordinates:

iin + 2t.wnt/. + w~q. = f.p(t) or -f.uit), n =1,2, ... , N(8)

Eq. (6) for the coefficient f ncan be derived by premultiplyingboth sides of (5) by <1>; and using the orthogonality propertyof modes. The contribution of the nth mode to the vector S orrna. is

(9)

(16)

(17)

(18)

The contribution r. of the nth mode to response quantity r,(14), can be expressed as

These modal contribution factors have three important properties. First, by definition they are dimensionless. Second, theyare independent of how the modes are normalized. This property becomes obvious by noting that r: is the static effect ofSn' which does not depend on the normalization, and the modalproperties do not enter into r st

• Third, the sum of the modalcontribution factors over all modes is unity, that is

where rst = static value of r due to external forces s, and f.,the modal contribution factor for the nth mode, is defined as

(11)

q.(t) = f.D.(t)

where D.(t) is governed by

D. + 2t.w.D. + w~Dn =p(t) or -ug(t) (10)

This is the equation of motion for the nth-mode SDF system(an SDF system with vibration properties-natural frequencyW n and damping ratio tn-of the nth mode of the MDF system) excited by ground acceleration ug(t). For force-excitedsystems the nth-mode SDF system is defined as having unitmass, and its response to the force p(t) is also governed by(10).

Modal Responses

The contribution of the nth mode to nodal displacementsu(t) is

To determine the forces in various structural elementsbeams, columns, walls, etc.-from the displacements un(t),we define the equivalent static forces associated with the nthmode response: fn = kun(t). Substituting (11) and using (3)gives

This important result can be proven by recognizing that S =2 s. [(5) and (7)], which implies that rsr = 2 r~'. Dividing byr st gives the desired result.

PEAK MODAL RESPONSES

f.(t) =s.A.(t)

where the pseudo-acceleration is defined as

A.(t) =w~Dn(t)

(12)

(13)

Consider the displacement Dn(t) and pseudo-accelerationAn(t) of the nth-mode SDF system and define their peak values(denoted by subscript "0") as the maximum of their absolutevalues:

The nth-mode contribution r.(t) to any response quantity r(t)is determined by static analysis of the structure subjected toexternal forces fn(t). If r: denotes the modal static response,the static value (indicated by superscript "st") of r due toexternal forces s., then

Dno =maxID.(t)I; A.o =max\A.(t) I

The corresponding value of rn(t), (16), is

(19a,b)

(20)

518/ JOURNAL OF STRUCTURAL ENGINEERING / MAY 1996


Static Analysis of Dynamic Analysis of Modal Contribution toMode

Structure SDFSystem Dynamic Response

Forcess,

fA""I ,,(I) = ,;' A,(I)

" OJ" 1;,(,

~

::'l~ ::'l~ ::'l~ _ii,<l)

Forces

S2

fA""2 '2(1) = ,~l Az(1)

" ~,1;2

Il2

~

7,'7,:z ::'l7,:z 7,'7,:z _ii.(t)

. · · . · ·. · · . · ·

. . · · · ·Forces

SN

fAJ"N 'I'll) = ,~l AI'II)

" OJN,1;NIlN

~

7,'7,7,; ~~ ~7,7,; _ii.(I)

NTotal response r(1) = ::r(1)

FIG. 1. Conceptual Explanation of Modal Analyele

For force-excited systems we shall rewrite this equation interms of a dimensionless dynamic response factor. For the nthmode SDF system this factor is Rdn = Dno/(Dn.s,)o, where(Dn.s,)o is the peak value of Dn.s,(t), the static response, Obtained by dropping the Dn and Dn terms in (10), Dn,,,(t) =p(t)/w~ and its peak value is (Dn.s,)o = Po/w~. Therefore, (20)becomes

(21)

The quantities r" and Fn depend on the spatial distribution sof the applied forces but are independent of their time variation p(t); on the other hand, Rdn depends on p(t), but is independent of s.

For earthquake-excited systems (20) can be rewritten as

(22)

where An == A no is the ordinate A(Tn, 'n) of the pseudo-acceleration response (or design) spectrum.

The algebraic sign of rno is the same as that of r',,' = r"Fn

because Rdn and An are positive by definition. Although it hasan algebraic sign, rno will be referred to as the peak value ofthe contribution of the nth mode to response r or, for brevity,the peak modal response because it corresponds to the peakvalue of Dn(t) or An(t).

EARTHQUAKE RESPONSE SPECTRUM ANALYSIS

The peak value r o of the total response r(t) is estimated bycombining the peak modal responses r no (n = 1, 2, ... , N)according to the well-known modal combination rules: squareroot-of-sum-of-squares (SRSS) rule or complete quadraticcombination (CQc) rule, as appropriate. The algebraic sign ofrno is relevant in the CQC rule, but inconsequential in theSRSS rule.

The response spectrum analysis (RSA) is a procedure fordynamic analysis of a structure subjected to earthquake excitation, but it reduces to a series of static analyses. For each



mode considered, static analysis of the structure subjected toforces S. provides the modal static response, r~, which is multiplied by the spectral ordinate A. to obtain the peak modelresponse rtlO , (22).

HOW MANY MODES TO INCLUDE

The response contributions of all the natural modes must beincluded if the"exact" value of the structural response to dynamic excitation is desired, but the first few modes can usuallyprovide sufficiently accurate results. The number of modes tobe included depends on two factors, modal contribution factori. and dynamic response factor Rdn or spectral ordinate A., thatenter into the modal response equations (21) and (22).

If only the first J modes are included, the error in the staticresponse is

mFlexural rigidity

m ColumnsElc

Beams Elhm

-I::V)

II-I:: m@iV)

m

For a fixed J the error e, depends on the spatial distribution S

of the applied forces. For any s the error e, will be zero whenall the modes are included (J = N) because of (18), and theerror will be unity when no modes are included (J = 0). Thus,modal analysis can be truncated when le,l, the absolute valueof e" becomes sufficiently small for the response quantity rof interest.

The dynamic response factors Rdn or spectral ordinates A.also influence the relative values of the modal responses andhence the number of modes that should be included in theanalysis. Rdn and A. depend on the shape of the dynamic response factor plot or on the shape of the earthquake response(or design) spectrum respectively, and on the vibration properties T. and ~. of the nth mode.

,e, = 1 - 2: i.

.-1

(23)2h

FIG. 2. Properties of Uniform Flve-Story Frames

Modal Expansion of Forces

Consider two different sets of applied forces: p(t) = sap(t)and p(t) = SbP(t), where s~ = (0 0 0 0 1) and sr = (0 0 0 -12); note that the resultant force is unity in both cases (Fig. 3).Substituting for m, <1>., and S = Sa in (6) and (7) gives themodal contributions S. shown in Fig. 3. The contributions ofthe higher modes, especially the second and third modes, to S

are larger for Sb than for Sa, suggesting that these modes maycontribute more to the response if the force distribution is Sb

than if it is Sa' We will return to this observation later.

EXAMPLE 1: FORCE-EXCITED, FIVE-STORY SHEARBUILDING

System Considered

To illustrate the concepts developed in the preceding sections, consider the structure of Fig. 2: a five-story shear building (Elb = 00, i.e., flexurally rigid floor beams and slabs) withlumped mass m at each floor, and same story stiffness k =24Elclh3 for all stories.

Modal Contribution Factors

We first study how the modal contribution factors i. dependon the spatial distribution S of the applied forces. Static analysis of the building subjected to forces S and S. gives r" andr~, respectively, and (17) leads to i.; for each S = Sa and Sb

the S. vectors are displayed in Fig. 3. For the base shear androof displacement of the building, the modal contribution factors and their cumulative values considering the first J modesare presented in Table 1. Consistent with (18), the sum of

0.356 0.301 0.106 0.029

0.327 0.195 0.077

0.250 0.055 0.101

= + + + +0.327 0.149 0.093

0.101 0.250 0.272 0.179 0.055

Sa St ~ S3 S4 S5

2 0.385 0.508 0.564 0.135

0.354 0.403 0.746

0.679 0.475

= + + + +0.553 0.569 0.646

0.110 0.739 0.685

Sb SI ~ S3 S4 Ss

FIG. 3. Modal Expansion of Excitation Vectors s. and Sb

520 I JOURNAL OF STRUCTURAL ENGINEERING I MAY 1996


TABLE 1. Modal and Cumulative Contribution Factors

FORCE DISTRIBUTION, s. FORCE DISTRIBUTION, Sb

Mode Roof Displacement Base Shear Roof Displacement Base Shearnor J J J J

number 2: aSn 2: IIbn 2: aSn 2: IIbnof modes, J asn n.1 IIbn n.1 asn n.1 IIbn n.1

(1 ) (2) (3) (4) (5) (6) (7) (8) (9)

1 0.880 0.880 1.252 1.252 0.792 0.792 1.353 1.3532 0.087 0.967 -0.362 0.890 0.123 0.915 -0.612 0.7413 0.024 0.991 0.159 1.048 0.055 0.970 0.431 1.1724 0.008 0.998 -0.063 0.985 0.024 0.994 -0.242 0.9305 0.002 1.000 0.015 1.000 0.006 1.000 0.070 1.000

modal contribution factors over all modes is unity, althoughthe convergence mayor may not be monotonic. For the structure and force distributions considered, the convergence ismonotonic for roof displacement but not for base shear.

The data of Table 1 permit two useful observations pertaining to relative values of the modal responses:

1. For a particular spatial distribution of forces, the modalcontribution factors for higher modes are larger for baseshear than for roof displacement, suggesting that thehigher modes contribute more to base shear (and otherelement forces) than to roof displacement (and otherfloor displacements).

2. For a particular response quantity, the modal contributionfactors for higher modes are larger for force distributionSb than for Sa, suggesting that the higher modes contribute more to a response in the Sb case. Recall that themodal expansions of Sa and Sb had suggested the sameconclusion.

Rdn in (21). In Fig. 4 Rd for harmonic force of period T isplotted against TnfT for SDF systems with natural period Tn

and two damping ratios: , = 5 and 70%; Rd for a half-cyclesine pulse force of duration td is plotted against Tnftd in Fig. 5for undamped SDF systems.

How Rdn , for a given excitation p(t), varies with n dependson where the natural periods Tn fallon the period scale. In thecase of impulsive excitation, Fig. 5 shows that Rdn varies overa narrow range for a wide range of Tn and could have similarvalues for several modes. Thus, several modes would generally have to be included in modal analysis with their relativeresponse contributions, (21), determined primarily by the relative values of fn' the modal contribution factors. The sameconclusion also applies to highly damped systems subjected toharmonic force because as seen in Fig. 4, several modes couldhave similar values of Rdn • However, for lightly damped systems subjected to harmonic excitation, Fig. 4 indicates that Rdn

is especially large for modes with natural period Tn close to

10r--------........----------,

..-!<

.!C>::sII 1+---==:..-....::--+---'r-------1~

..-!<1,~ 1~::::::::::::::....----.:~--__+----~--_lII

~

2r---------,-----------,

0.1 '-,-_-'----'---''--'-....L..J...LJ...l-_-'-_..Iol._J.....JL..L..L.l...LJ0.1 1 10

Tn _ Undamped natural periodT - Forcing period

FIG. 4. Dynamic Respons. Factors for Harmonic Force; t =5% and 70%

How many modes should be included in modal analysis?We first examine how the number of modes required to keepthe error in static response below some selected value is influenced by the spatial distribution S of the applied forces. If theobjective is to keep lell < 0.05 (5%) for the base shear, thedata of Table 1 indicate that three modes suffice for the forcedistribution Sa, whereas all five modes need to be included inthe case of Sb' For the same accuracy in the roof displacement,two modes suffice for the force distribution Sa, but three modesare needed in the case of Sb' More modes need to be includedfor the force distribution Sb than for Sa because, as mentionedearlier, the modal contribution factors for higher modes arelarger for Sb than for Sa'

We next examine how the number of modes required isinfluenced by the response quantity of interest. If the objectiveis to keep lell < 0.05 (5%), three modes need to be includedto determine the base shear for force distribution Sa, whereastwo modes would suffice for roof displacement. To achievethe same accuracy for force distribution Sb, all five modes areneeded for base shear, whereas three modes would suffice forroof displacement. More modes need to be included for baseshear than for roof displacement because the modal contribution factors for higher modes are larger for base shear than forroof displacement.

It is not necessary to repeat the preceding analysis for allresponse quantities. Instead, some of the key response quantities, especially those that are likely to be sensitive to highermodes, should be identified for deciding the number of modesto be included in modal analysis.

Dynamic Response Factor

We now study how the modal response contributions depend on the time variation of the excitation. The dyamic response to p(t) is characterized by the dynamic response factor

o,:---'---'---'-.....................~---'----'---'--'- .........~0.1 1 10

'[g = Undamped natural periodtd Force duration

FIG. 5. Dynamic Response Factor for Half-Cycle Sine PulseForce; t=O



TABLE 2. Modal Contribution Factors for Vb and V.

TABLE 3. Modal Contribution Factors for Mb and Us

OIl

..(c0.~...0'0u~6"0::l 0.10

'"p..

(a) (b) (c)

FIG. 6. Deflected Shapes: (a) p = OJ (b) P = 118; and (c) p =00

5..-----------------...,

Base Shear Vb Top-Story Shear V.

Mode p=o p = 1/8 p=oo p=o p = 1/8 p=oo(1 ) (2) (3) (4) (5) (6) (7)

1 0.679 0.796 0.879 1.38 1.30 1.252 0.206 0.117 0.087 -0.528 -0.441 -0.3623 0.070 0.051 0.024 0.204 0.211 0.1594 0.033 0.026 0.007 -0.080 -0.089 -0.0635 0.012 0.009 0.002 0.020 0.023 0.015

Base OverturningMoment Mb Top-Floor Displacement u.

Mode p=o p = 1/8 p=oo p=o p = 1/8 p = 00

(1 ) (2) (3) (4) (5) (6) (7)

1 0.898 0.985 1.030 1.009 1.027 1.0302 0.Q78 -0.003 -0,035 -0.009 -0.030 -0,0353 0.016 0.014 0.006 0.0005 0.003 0.0064 0.006 0.003 -0.001 -0.00005 -0.0005 -0.0015 0.002 -0.001 0.0003 0.000005 0.00007 0.0003

0.1 1 10Natural vibration period Tn- sec

FIG. 7. Elastic Pseudo-Acceleration Design Spectrum forGround Motions with UI/O =1 g, ulIO =48lnJsec, and ul/O =36 In.; t=5% (1 in. =25.4 mm)

quantity. For four response quantities-base shear Vb, topstory shear Vs, base overturning moment M b , and top-floordisplacement us-the modal contribution factors, are presentedin Tables 2 and 3 for p = 0, 1/8, and 00; these results areindependent of T!. Consistent with (18), for each responsequantity and each p the sum of modal contribution factors overail modes is unity, although the convergence mayor may not

the forcing period T. These modes would contribute most tothe response and are perhaps the only modes that need to beincluded in modal analysis unless the modal contribution factors f n for these modes are much smaller than for some othermodes.

It is not necessary to compute all the natural periods of asystem having a large number of degrees of freedom in ascertaining which of the Rdn values are significant. Only thefirst few natural periods need to be calculated and located onthe plot showing the dynamic response factor. Then the approximate locations of the higher natural periods become readily apparent, thus providing sufficient information to estimatethe range of Rdn values and to make a preliminary decision onthe modes that may contribute significant response. Precisevalues of Rdn can then be calculated for these modes to beincluded in modal analysis.

The reduction in computational effort achieved by considering only the first few modes may not be significant in dynamic analysis of systems with a small number of dynamicdegrees of freedom, such as the five-story shear frame considered here. However, substantial reduction in computation canbe achieved for practical complex structures that may requirehundreds or thousands of degrees of freedom for their idealization.

Design Spectrum

The earthquake excitation is characterized by the designspectrum of Fig. 7, multiplied by 0.5, so that it applies toground motions with a peak ground acceleration UgO =0.5 g,peak ground velocity ugo =24 in.lsec, and peak ground displacement ugo = 18 in. In the design spectrum shown for 5%damping, the acceleration-sensitive, velocity-sensitive, anddisplacement-sensitive regions are identified.

EXAMPLE 2: EARTHQUAKE-EXCITED, FIVE-STORYBUILDINGS

Systems Analyzed

To illustrate the concepts developed in the preceding sections, consider the structure of Fig. 2, a single-bay, five-storyframe with constant story height =h and bay width =2h. Allthe beams have the same flexural rigidity, Elb , and the columnrigidity, EIe. does not vary with height. The building is idealized as a lumped-mass system with the same mass m at allthe floor levels. The damping ratio for all five natural vibrationmodes is assumed to be 5%.

Only two additional parameters are needed to define thesystem completely: the fundamental natural vibration periodT1 and the beam-to-column stiffness ratio p, where p = h/4/e •

The latter parameter indicates how much the system may beexpected to behave as a frame. For p =0 the beams imposeno restraint on joint rotations, and the frame behaves as aflexural beam [Fig. 6(a)]. For p = 00 the beams restrain completely the joint rotations, and the structure behaves as a shearbeam with double-curvature bending of the columns in eachstory [Fig. 6(c)). An intermediate value of p represents a framein which beams and columns undergo bending deformationwith joint rotation [Fig. 6(b)]. As an example for the frame ofFig. 2, P = 1/8 represents Ib =le12, which implies a frame withcolumns stiffer than the beams, typical of earthquake-resistantconstruction. The parameter p controls the vibration propertiesof the frame [Roehl (1971) and Chopra (1995) pages 642644].

Modal Contribution Factors

We first study how the modal contribution factors dependon the beam-to-column stiffness ratio p and on the response



100.1

Natural vibration period Tn. sec

aL- ~.........J.......J..L-...J...... .......L..~o.L_~__L~.........=

0.01

0.5

OIl

~ ac:0 (b) Tl = 3 sec'.0e p=oo~(,)(,)<U

0"0::3

0.5Q.)

'"p..

a

Ts

0.51----/

frames (p =00) with fundamental natural periods TI =0.5 and3.0 s, respectively, are identified. For the building with T I = 3s, the An values for the higher modes are larger than A 1 forthe fundamental mode, whereas for the building with T1 = 0.5s, the An (n 2: 2) values are either equal to or smaller thanAI' Thus, the higher-mode response, expressed as a percentageof the total response, should be larger for the building with T1

= 3 s than for the T1 = 0.5 s building. In general, for thespectrum selected, as T1 increases within the velocity- and displacement-sensitive regions of the spectrum, the higher-mode

1.5....-----------r-.-r----r--..,---------,

FIG. 8. Natural Perloda and Spectral Ordinatea for ThreeCaaea: (a) 7; =0.5 a, p =""; (b) 7; =3 a, p =""; and (c) T, =3 a,p=O

1. For a fixed value of p and each of the four responsequantities, the modal contribution factor '1 for the firstmode is larger than the factors 'n for the higher modes,suggesting that the fundamental mode should have thelargest contribution to each of these responses.

2. For a fixed value of p, the absolute values of T. for thesecond and higher modes are larger for V~ than for Vb'and the values for Vb in turn are larger than those for Mband u~. This observation suggests that the second- andhigher-mode response contributions should be more significant for base shear Vb than for the base overturningmoment Mb or top-floor displacement u~. Among thestory shears the higher-mode responses should be moresignificant for the fifth-story shear than for the baseshear.

3. As p decreases, the absolute values of the higher-modecontribution factors Tn for V~, Vb' and Mb increase (butfor minor exceptions), especially in the second mode(Tables 2 and 3). This observation suggests that thehigher-mode contributions to any of these forces shouldbecome a larger fraction of the total response as p decreases and should be largest for a flexural beam with p=O.

Influence of T, on Higher-Mode Response

be monotonic. For the class of structures considered the convergence is monotonic for base shear, but not for the otherthree response quantities. The data of Tables 2 and 3 permitthree useful observations that have a bearing on relative valuesof the modal responses:

In this section we use the preceding concepts and data topredict how the modal response contributions depend on thefundamental natural period T1 of the structure. For this purposewe examine the three factors that enter into (22) for the peakmodal response: (I) The static value r" of r is a common factorin all modal responses and therefore does not influence therelative values of the modal responses; (2) as mentioned earlier, for a fixed p the modal contribution factors '. are independent of T1; and (3) the pseudo-acceleration spectrum ordinate An is the only factor in (22) that depends on T1 andperiod ratios T1IT.; for a fixed p, T1ITn do not depend on T1 •

This is illustrated for the selected design spectrum in parts (a)and (b) of Fig. 8, wherein the natural periods T. of two shear

20

40

80

60

10

(c)p=oo

0.110

(b) P = 1/8

0.110

(a) p = 0

...,-----------...-------------,----------~loo100

80-c.,~0.

oS 60'"c8-'".,.....,'8 40e.:..,..cOIl

:E20

00.1

Fundamental natural period Tb sec

FIG. 9. Higher Mode Reaponseln Vb' V., Mb, and u. for Uniform Five-Story Framea for Three Valuea of p



response is expected to become an increasing percentage ofthe total response.

This prediction is confirmed by the results of dynamic analyses. The peak values of the response of a frame with specifiedTI and p is determined by considering (1) all five modes, and(2) only the first mode. The difference between the two resultsfor the peak value is the higher mode response, i.e, the combined response due to all modes higher than the first mode.Such analyses were repeated for three values of p - 0, 1/8,and oo-and many values of TI • The higher mode response,expressed as a percentage of the total response, is presentedas a function of TI in Fig. 9 for the four response quantities.The higher-mode response is negligible for TI in the acceleration-sensitive region of the spectrum and increases with increasing TI in the velocity- and displacement-sensitive regions.Such results are useful in evaluating the lateral force provisions in building codes [Chopra (1995) Chapter 21].

Influence of p on Higher-Mode Response

In this section we predict how the modal response contributions depend on the beam-to-column stiffness ratio p. Forthis purpose we examine the three factors that enter into (22)for the peak modal response: (1) The static value ,Sf of , is acommon factor in all modal responses and therefore does notinfluence the relative values of the modal response; (2) as pdecreases, the absolute values of the higher-mode contributionfactors Pn for the base shear and top-story shear increase, especially in the second mode (Table 2); and (3) the pseudoacceleration spectrum ordinates depend on TI and on TlfTn ;

the latter becomes larger as p decreases [Roehl (1971) andChopra (1995) pages 642-644] and the Tn values are spreadout over a wider period range of the design spectrum. This isillustrated in parts (b) and (c) of Fig. 8. Both frames have thesame T1 = 3 s, but they differ in p-one is a shear beam (p= 00), and the other a flexural beam (p =0). As a result, theratio A 2 for the second mode-generally the most significantof the higher modes-to Al for the first mode is larger forbuildings with p = 0 than for the p = 00 case. Thus, puttingthese two reasons together, both the modal contribution factorPn and the spectral ordinate An for the second mode are largerfor the p = 0 frame; therefore, the higher-mode response isexpected to be more significant in this case than for the framewith p = 00, In general, for the design spectrum selected andfor TI within the velocity- and displacement-sensitive regions

of the spectrum, the ratio AnfAI increases (or more precisely,does not decrease) with decreasing p, and this trend shouldlead to increased higher-mode response.

This prediction is confirmed by the results of dynamic analyses in Fig. 9, which demonstrate that for each response quantity the higher-mode response is least significant for systemsbehaving like shear beams (p = 00), becomes increasingly significant as p decreases, and is largest for systems deforminglike flexural beams (p = 0).

The influence of TI and p on the higher mode response hasbeen studied previously (Cruz and Chopra 1986), but it wasnot as elegant without the concept of modal contribution factors introduced in this paper.

How Many Modes to Include

We first examine the error e2 in static response if two modesare included. Table 4 shows this error computed from (23) andnumerical values for the modal contribution factors in Tables2 and 3. The error e2 is below 0.15 or 15% for the four response quantities when the first two modes are included. Fora fixed p the error varies with the response quantity. It issmaller in the base overturning moment M b relative to the baseshear Vb, and in Vb compared to the top-story shear V~. Theerror is much smaller for the top-floor displacement u~, and itis less than 3% if the first mode alone is considered. For aparticular response quantity the error e2 varies with p, beingsmallest for p = 00 (i.e., shear beams) and largest for p = 0(i.e., flexural beams). The top-floor displacement displaystrends opposite to the forces in the sense that e2 increases withincreasing p, but e2 is so small that higher modes are of littleconsequence. These data suggest that the first one or twomodes may provide a good approximation to the total response, with the accuracy depending on the response quantityand on p.

We next examine how the spectral ordinates An influence

TABLE 4. ..=1 - ~~.1 f n

Response p=o p = 1f8 p=oo(1 ) (2) (3) (4)

V, 0.144 0.144 0.110Vb 0,115 0.086 0.033Mb 0,024 0.18 0.005u, 0.0004 0,003 0.005

10 10

(a) p = 0 (b) P = 1/8 (c)p=oa

No of modes5

.~-

......;:,.""

0,1 0,1

0.1 10 0,1 10 0,1 10

Fundamental natural period TI • sec

FIG. 10. Normalized Base Shear In Uniform Flve-5tory Frames for Three Values of p



and

(a) (b)

FIG. 11. N-Story Building: (a) Fixed-Base; (b) Isolated

Analysis Procedure

With ground motion characterized by the design spectrumof Fig. 7, scaled to agO = 0.5 g, the RSA procedure summarizedearlier will be used to analyze two systems: (1) the buildingon a fixed base; and (2) the same structure supported on anisolation system. In applying the RSA procedure to the isolated

(24b)

(24a)

Isolationsystem

mi

ml

mb

~!'fi. ~"

No

m·

1 ml

7J.~ 7,:~

}

t - Cbb - 2(M + mb)wb

in which M = 5m is the mass of the building excluding thebase slab. The five-story building on a base isolation systemis a six-degree-of-freedom system with nonclassical dampingbecause damping in the isolation system is typically muchmore than in the building. In this example the base slab massmb = m and the stiffness and damping of the isolation systemare such that Tb = 2.0 s and ~b = 10%.

The natural periods and modal damping ratios of both systems are presented in Table 5.

the number of modes that should be included in the analysis.For a fixed p and T) in the velocity- or displacement-sensitiveregions of the spectrum, the ratio An/A) is larger for frameswith longer fundamental period T) [Fig. 8(a) and (b)]. Thusfor the same desired accuracy, more modes should be includedin the analysis of buildings with longer T) than the number ofmodes necessary for shorter T) buildings. For a fixed T) in thevelocity-sensitive or displacement-sensitive regions of thespectrum, the ratio An/A) is larger for frames with smaller p[Fig. 8(b) and (c)]. Thus, for the same desired accuracy, moremodes should be included in the analysis of buildings withsmaller p compared to the number of modes necessary forbuildings with larger p.

These expectations regarding how T) and p influence thenumber of modes that should be included in earthquake response analysis are confirmed by the results of Fig. 10, where,for each p value, five response curves for base shear are identified by indicating the number of modes included in the response spectrum analysis. It is clear that the first two modesprovide a reasonably accurate value for the base shear inframes with T1 in the velocity-sensitive region of the spectrum,and one mode is sufficient in the acceleration-sensitive region.This conclusion is also valid for shears in all the stories andoverturning moments at all floors. The first mode alone provides accurate results for Us over the entire range of Th andfor all p values, as indicated in Fig. 9.

In light of the preceding observations, it is instructive toexamine the 90% rule for participating mass specified in theUniform Building Code. Because the effective modal mass isequal to the modal static response V:n for base shear [Chopra(1995) pages 476-484], the prior rule implies that enoughsay i-modes should be included so that eJ for base shear isless than 10%. However, as noted earlier, eJ varies with theresponse quantity, and therefore this error may exceed 10%for other response quantities such as shears in upper storiesand bending moments and shears in some structural elements.

EXAMPLE 3: EARTHQUAKE-EXCITED, BASEISOLATED BUILDING

System Considered and Parameters

The building to be isolated [Fig. II(a)] is a five-story shearframe (i.e., beam-to-column stiffness ratio p = 00) with massand stiffness properties uniform over its height: lumped massmJ = m = 100 kips/g at each floor, and stiffnesses k for eachstory; k is chosen so that the fundamental natural vibrationperiod TIf = 0.4 s. The damping matrix is proportional to thestiffness matrix with 2% damping in the fundamental mode.

As shown in Fig. l1(b), this five-story building is mountedon a base slab of mass mb' supported in tum on a base-isolation system with lateral stiffness kb and linear viscous damping Cb' Two parameters, Tb and ~b' are introduced to characterize the isolation system:

TABLE 5. Natural Periods and Modal Damping Ratios

Fixed-Base Building Isolated Building

Mode Tnl (s) tnl (%) Mode Tn (s) tn (%)(1) (2) (3) (4) (5) (6)

- - - 1 2.030 9.581 0.400 2.00 2 0.217 5.642 0.137 5.84 3 0.114 7.873 0.087 9.20 4 0.080 10.34 0.068 11.8 5 0.066 12.35 0.059 13.5 6 0.059 13.6

TABLE 6. Calculation of Base Shear In Fixed-Base and Isolated Buildings

Fixed-Base Building Isolated Building

Mode An/g V~~/m Vb/W Mode An/g V:'/m VblW(1 ) (2) (3) (4) (5) (6) (7) (8)

- - - - 1 0.359 5.028 0.3611 1.830 4.398 1.609 2 1.291 -0.021 -0.0052 1.272 0.436 0.111 3 1.058 -0.005 -0.0013 0.859 0.121 0.021 4 0.792 -0.002 -0.0004 0.700 0.038 0.005 5 0.682 -0.0005 -0.0005 0.638 0.008 0.001 6 0.635 -0.0001 -0.000

SRSS - - 1.613 SRSS - - 0.361

JOURNAL OF STRUCTURAL ENGINEERING 1MAY 1996/525


structure we are ignoring the coupling of modal equations dueto nonclassical damping. The peak response due to the nthmode of vibration is given by (22), which is specialized forthe base shear Vb in the building and the base displacement orisolator deformation Ub

(25a,b)

where Dn = Anlw; is the deformation spectrum ordinate. Thepeak modal responses are combined by the SRSS rule.

Modal Static Responses

The modal static responses V~n and u~n for Vb and Ub, respectively, are presented in Tables 6 and 7. It is clear thatV~n has significant values in the first two modes of the fixedbase structure, and these modes are expected to contribute significantly to the response. However, for the isolated building,V~n is small in all the higher modes and the response in thesemodes should be negligible, i.e., the first mode with the dominant value V~I should provide most of the response.

Peak Modal Responses

The preceding predictions from the modal static responsesare confirmed by the results of dynamic analysis in Tables 6and 7. The peak value of the earthquake response due to eachnatural mode of both systems is determined from (25), wherethe spectral ordinates Anlg are shown in Figs. 12 and 13. Thedynamic response of the isolated building due to all highermodes is negligible. The first mode alone produces essentiallythe entire response: isolator deformation of 14.045 in. and baseshear equal to 36.1% of W, the 500-kip weight of the buildingexcluding the base slab. The first two natural modes providesignificant response of the fixed based building; however,the second mode contributes little to the combined SRSSvalue.

2.0

Effects of Base IsolatIon

In this section we use the preceding concepts to predictthe influence of base isolation on the base shear in the building, and to identify precisely the underlying reason for thiseffect. The peak modal response is the product of two parts:the modal static response V~n and the pseudo-accelerationAn. Each part is examined for the first mode of the baseisolated building and of the fixed-base building; this is themode that provides most of the response in each case. Observe that V~I =5.028m for the isolated building is somewhatlarger than V~I = 4.398m for the fixed-base building. However, the pseudo-acceleration Al = 0.359 g for the isolatedbuilding is only one-fifth of Al =1.830 g for the fixed-basebuilding; as a result, the first-mode base shear coefficient of36.1 % for the isolated building is much smaller than the160.9% for the fixed-base building. The isolation system reduces the base shear primarily because the natural period ofthe first mode, providing most of the response, is muchlonger than the fundamental period of the fixed-base structure, leading to a smaller spectral ordinate, as seen by comparing Figs. 12 and 13. This is typical of design spectra onfirm ground and fixed-base structures with fundamental natural period in the flat portion of the acceleration-senstive region of the spectrum.

This prediction is confirmed by the results of dynamicanalysis, considering all modes, in Table 6. These results

TABLE 7. Calculation of Isolator Deformation

Mode Dn w~u:' u"" (in.)(1 ) (2) (3) (4)

1 14.470 0.971 14.0452 0.597 0.022 0.0133 0.133 0.005 0.0014 0.050 0.002 0.0005 0.029 0.001 0.0006 0.022 0.0001 0.000

SRSS - - 14.045

00\0oo

~=2% 1.830g

0.0

0.02 0.05 0.1 0.2 0.5 2 3

Natural vibration period Tn. sec

FIG. 12. Design Spectrum and Spectral Ordinates for FIXed-Base Flve-5tory Building



2.0

~=2%

320.50.20.10.05

0.0

0.02

1.55%

1.291gOll

..:0::

10%0'll 1.0ju1;j6"0

""'"Po.0.5

Natural vibration period Tn' sec

FIG. 13. Design Spectrum and Spectral Ordinates for Isolated Flve-8tory Building

demonstrate that base isolation reduces the base shear coefficient from 161.3% to 36.1 %.

CONCLUSIONS

With the reformulation of classic.al modal analysis and physical insight contributed in this paper, the following conclusionsresult:

1. The contribution of the nth natural vibration mode to thedynamic response of a multi-degree-of-freedom structure toapplied forces p(t) = sp(t) or ground acceleration iig(t) can beinterpreted as the product of the results of two analyses: (1)static analysis of the structure subjected to external forces Sm

the contribution of the nth mode to spatial distribution s ofp(t) or ml of the effective earthquake forces; and (2) dynamicanalysis of the nth-mode SDF system excited by the forcep(t) or ground acceleration iig(t).

2. The modal contribution factors f n, which depend on s,are dimensionless and independent of how the modes are normalized; their sum over all modes is unity.

3. The relative contributions of various modes to the dynamic response and the number of modes that should be included in modal analysis to achieve the desired accuracy canbe determined from numerical values of the modal contribution factors and of the dynamic response factor for p(t) or theearthquake response (or design) spectrum for iig(t).

4. Examples have been presented to illustrate that the physical interpretation of modal analysis and the concept of modalcontribution factors, developed in this paper, are useful in predicting how the relative contributions of the various modes to

the response and the number of modes necessary in the solution depend on the vibration properties of the system, on theresponse quantity of interest, and on: (1) the spatial distribution s and time variation p(t) of the applied dynamic forces;or (2) the response (or design) spectrum for an earthquakeexcitation.

5. Another example has been presented to demonstrate thatthe physical interpretation of modal analysis is useful in predicting that the base shear in a building is reduced due to baseisolation and in identifying precisely the underlying reasonsfor this reduction.

ACKNOWLEDGMENTSThis paper is based on Chapters 12, 13, 18, and 20 of a recent book

(Chopra 1995). The author is grateful to Dr. Rakesh K. Goel for implementing the calculations and preparing the figures, and to Dr. Gregory L.Fenves for valuable discussions.

APPENDIX. REFERENCESChopra, A. K. (1995). Dynamics of structures: theory and applications

to earthquake engineering, Prentice Hall, Englewood Cliffs, N.J.Crandall, S. H., and McCalley, R. B. Jr. (1961). "Numerical methods of

analysis." Shock and vibration handbook, Chapter 28, C. M. Harrisand C. E. Crede, eds., McGraw-Hili, New York, N.Y.

Cruz, E. E, and Chopra, A. K. (1986), "Elastic earthquake response ofbuilding frames." J. Struct. Engrg., ASCE, 112,443-459.

Roehl, J. L. (1971). "Dynamic response of ground-excited buildingframes," PhD thesis, Rice Univ., Houston, Tex.

Uniform building code. (1994). Int. Conf. of Build. Officials, Whittier,Calif.

Veletsos, A.S. (1977), Structural Dynamics, class notes, Univ. of Calif.,Berkeley.



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