chapter 04-dynamic response of buildings

183

Chapter 4

Dynamic Response of Structures

James C. Anderson, Ph.D.Professor of Civil Engineering, University of Southern California, Los Angeles, California

Key words: Dynamic, Buildings, Harmonic, Impulse, Single-Degree-of –Freedom, Earthquake, Generalized Coordinate,Response Spectrum, Numerical Integration, Time History, Multiple-Degree-of-Freedom, Nonlinear,Pushover, Instrumentation

Abstract: Basic principles of structural dynamics are presented with emphasis on applications to the earthquakeresistant design of building structures. Dynamic characteristics of single degree of freedom systems arediscussed along with their application to single story buildings. The response of these systems to harmonicand impulse loading is described and illustrated by application to simple structures. Consideration of theearthquake response of these systems leads to the concept of the elastic response spectrum and thedevelopment of design spectra. The use of procedures based on a single degree of freedom is extended tomultiple degree of freedom systems through the use of the generalized coordinate approach. Thedetermination of generalized dynamic properties is discussed and illustrated. A simple numerical integrationprocedure for determining the nonlinear dynamic response is presented. The application of matrix methodsfor the analysis of multiple degree of freedom systems is discussed and illustrated along with earthquakeresponse analysis. A response spectrum procedure suitable for hand calculation is presented for elasticresponse analyses. The nonlinear static analysis for proportional loading and the nonlinear dynamic analysisfor earthquake loading are discussed and illustrated with application to building structures. Finally, the useof the recorded response from buildings containing strong motion instrumentation for verification ofanalytical models is discussed.

184 Chapter 4

4. Dynamic Response of Structures 185

4.1 Introduction

The main cause of damage to structuresduring an earthquake is their response to groundmotions which are input at the base. In order toevaluate the behavior of the structure under thistype of loading condition, the principles ofstructural dynamics must be applied todetermine the stresses and deflections, whichare developed in the structure. Structuralengineers are familiar with the analysis ofstructures for static loads in which a load isapplied to the structure and a single solution isobtained for the resulting displacements andmember forces. When considering the analysisof structures for dynamic motions, the termdynamic simply means “time-varying”. Hencethe loading and all aspects of the response varywith time. This results in possible solutions ateach instant during the time interval underconsideration. From an engineering standpoint,the maximum values of the structural responseare usually the ones of particular interest,specially in the case of structural design.

The purpose of this chapter is to introducethe principles of structural dynamics withemphasis on earthquake response analysis.Attention will initially be focused on theresponse of simple structural systems, whichcan be represented in terms of a single degreeof freedom. The concepts developed for thesesystems will then be extended to includegeneralized single-degree-of-freedom (SDOF)systems using the generalized-coordinateapproach. This development in turn leads to theconsideration of the response of structureshaving multiple degrees of freedom. Finally,concepts and techniques used in nonlineardynamic-response analysis will be introduced.

4.2 Dynamic Equilibrium

The basic equation of static equilibriumused in the displacement method of analysis hasthe form,

kvp = (4-1)

where p is the applied force, k is the stiffnessresistance, and v is the resulting displacement.If the statically applied force is now replaced bya dynamic or time-varying force p(t), theequation of static equilibrium becomes one ofdynamic equilibrium and has the form

)()()()( tkvtvctvmtp ++= &&& (4-2)

where a dot represents differentiation withrespect to time.

A direct comparison of these two equationsindicates that two significant changes, whichdistinguish the static problem from the dynamicproblem, were made to Equation 4-1 in order toobtain Equation 4-2. First, the applied load andthe resulting response are now functions oftime, and hence Equation 4-2 must be satisfiedat each instant of time during the time intervalunder consideration. For this reason it is usuallyreferred to as an equation of motion. Secondly,the time dependence of the displacements givesrise to two additional forces which resist theapplied force and have been added to the right-hand side.

The equation of motion represents anexpression of Newton’s second law of motion,which states that a particle acted on by a force(torque) moves so that the time rate of changeof its linear (angular) momentum is equal to theforce (torque):

)()(dt

dvm

dt

dtp = (4-3)

where the rate of change of the displacementwith respect to time, dv/dt, is the velocity, andthe momentum is given by the product of themass and the velocity. Recall that the mass isequal to the weight divided by the accelerationof gravity. If the mass is constant, Equation 4-3becomes

)()()( tvmdt

dv

dt

dmtp &&== (4-4)

186 Chapter 4

which states that the force is equal to theproduct of mass and acceleration. According tod’Alembert’s principle, mass develops aninertia force, which is proportional to itsacceleration and opposing it. Hence the firstterm on the right-hand side of Equation 4-2 iscalled the inertia force; it resists theacceleration of the mass.

Dissipative or damping forces are inferredfrom the observed fact that oscillations in astructure tend to diminish with time once thetime-dependent applied force is removed. Theseforces are represented by viscous dampingforces, that are proportional to the velocity withthe constant proportionality referred to as thedamping coefficient. The second term on theright-hand side of Equation 4-2 is called thedamping force.

Inertia forces are the more significant of thetwo and are a primary distinction between staticand dynamic analyses.

It must also be recognized that all structuresare subjected to gravity loads such as self-weight (dead load) and occupancy load (liveload) in addition to dynamic base motions. Inan elastic system, the principle of superpositioncan be applied, so that the responses to staticand dynamic loadings can be consideredseparately and then combined to obtain the totalstructural response. However, if the structuralbehavior becomes nonlinear, the responsebecomes load-path-dependent and the gravityloads must be considered concurrently with thedynamic base motions.

Under strong earthquake motions, thestructure will most likely display nonlinearbehavior, which can be caused by materialnonlinearity and/or geometric nonlinearity.Material nonlinearity occurs when stresses atcertain critical regions in the structure exceedthe elastic limit of the material. The equation ofdynamic equilibrium for this case has thegeneral form

)()()()()( tvtktvctvmtp ++= &&& (4-5)

in which the stiffness or resistance k is afunction of the yield condition in the structure,

which in turn is a function of time. Geometricnonlinearity is caused by the gravity loadsacting on the deformed position of the structure.If the lateral displacements are small, thiseffect, which is often referred to as P-delta, canbe neglected. However, if the lateraldisplacements become large, this effect must beconsidered. In order to define the inertia forcescompletely, it would be necessary to considerthe accelerations of every mass particle in thestructure and the corresponding displacements.Such a solution would be prohibitively time-consuming. The analysis procedure can begreatly simplified if the mass of the structurecan be concentrated (lumped) at a finite numberof discrete points and the dynamic response ofthe structure can be represented in terms of thislimited number of displacement components.The number of displacement componentsrequired to specify the position of the masspoints is called the number of dynamic degreesof freedom. The number of degrees of freedomrequired to obtain an adequate solution willdepend upon the complexity of the structuralsystem. For some structures a single degree offreedom may be sufficient, whereas for othersseveral hundred degrees of freedom may berequired.

4.3 SINGLE-DEGREE-OF-FREEDOM SYSTEMS

4.3.1 Time-Dependent Force

The simplest structure that can beconsidered for dynamic analysis is an idealized,one-story structure in which the single degreeof freedom is the lateral translation at the rooflevel as shown in Figure 4-1. In thisidealization, three important assumptions aremade. First, the mass is assumed to beconcentrated (lumped) at the roof level. Second,the roof system is assumed to be rigid, andthird, the axial deformation in the columns isneglected. From these assumptions it followsthat all lateral resistance is in the resistingelements such as columns, walls, and diagonal


braces located between the roof and the base.Application of these assumptions results in adiscretized structure that can be represented asshown in either Figure 4-lb or 4-1c with a time-dependent force applied at the roof level. Thetotal stiffness k is simply the sum of thestiffnesses of the resisting elements in the storylevel.

The forces acting on the mass of thestructure are shown in Figure 4-1d. Summingthe forces acting on the free body results in thefollowing equation of equilibrium, which mustbe satisfied at each instant of time:

)(tpfff sdi =++ (4-6)

where

fi = inertia force = um &&

fd = damping (dissipative) force= vc &fs = elastic restoring force = kv

)(tp = time-dependent applied force

u&& is the total acceleration of the mass, andvv,& are the velocity and displacement of the

mass relative to the base. Writing Equation 4-6in terms of the physical response parametersresults in

)(tpkvvcum =++ &&& (4-7)

It should be noted that the forces in thedamping element and in the resisting elementsdepend upon the relative velocity and relativedisplacement, respectively, across the ends ofthese elements, whereas the inertia forcedepends upon the total acceleration of the mass.The total acceleration of the mass can be

Figure 4-1. single-degree-of-freedom system subjected to time-dependent force.

188 Chapter 4

expressed as

)()()( tvtgtu &&&&&& += (4-8)

where

)(tv&& = acceleration of the mass relative tothe base

)(tg&& = acceleration of the base

In this case, the base is assumed to be fixedwith no motion, and hence 0)( =tg&& and

)()( tvtu &&&& = . Making this substitution forthe acceleration, Equation 4-7 for a time-dependent force becomes

)(tpkvvcvm =++ &&& (4-9)

4.3.2 Earthquake Ground Motion

When a single-story structure, shown inFigure 4-2a, is subjected to earthquake groundmotions, no external dynamic force is applied atthe roof level. Instead, the system experiencesan acceleration of the base. The effect of this onthe idealized structure is shown in Figure 4-2band 4-2c. Summing the forces shown in Figure4-2d results in the following equation ofdynamic equilibrium:

0=++ sdi fff (4-10)

Substituting the physical parameters for fi, fd

and fs in Equation 4-10 results in an equilibriumequation of the form

0=++ kvvcum &&& (4-11)

Figure 4-2. Single-degree-of-freedom system subjected to base motion.


This equation can be written in the form ofEquation 4-9 by substituting Equation 4-8 intoEquation 4-11 and rearranging terms to obtain

)(tpkvvcvm e=++ &&& (4-12)

where

)(tpe = effective time-dependent force = )(tgm &&−Hence the equation of motion for a structure

subjected to a base motion is similar to that fora structure subjected to a time-dependent forceif the base motion is represented as an effectivetime-dependent force which is equal to theproduct of the mass and the groundacceleration.

4.3.3 Mass and Stiffness Properties

Most SDOF models consider structures,which experience a transactional displacementof the roof relative to the base. In this case thetranslational mass is simply the concentratedweight divided by the acceleration of gravity(32.2 ft/sec2 or 386.4 in./sec2). However, casesdo arise in which the rotational motion of thesystem is significant. An example of this mightbe the rotational motion of a roof slab whichhas unsymmetrical lateral supports. Newton’ssecond law of motion states that the time rate ofchange of the angular momentum (moment ofmomentum) equals the torque. Considering aparticle of mass rotating about an axis o, asshown in Figure 4-3, the moment of momentumcan be expressed as

dt

dmrtvrmL

θ== 2)(& (4-13)

The torque N is then obtained by taking thetime derivative:

θ== &&Idt

dLN (4-14)

Figure 4-3. Rotating particle of mass.

where

2mrI = = mass moment of inertia

For a rigid body, the mass moment of inertiacan be obtained by summing over all the massparticles making up the rigid body. This can beexpressed in integral form as

∫ ρ= dmI 2 (4-15)

where ρ is the distance from the axis ofrotation to the incremental mass dm. Fordynamic analysis it is convenient to treat therigid-body inertia forces as though thetranslational mass and the mass moment ofinertia were concentrated at the center of mass.The mass and mass moment of inertiaof several common rigid bodies are summarizedin Figure 4-4.

Example 4-1 (Determination of MassProperties)

Compute the mass and mass moment ofinertia for the rectangular plate shown in Figure4-5.

• Translational mass:

190 Chapter 4

abtm µµν ==

whereµ = mass density = mass per unit volumeV = total volume

• Rotational mass moment of inertia:

∫ ρ= dmI 2 , Where 222 yx +=ρ

tdxdydVdm µ=µ=

∫ ∫ ∫ +µ=ρ=2/

0

2/

0

222 )(4a b

dxdyyxtdmI

12

12484

22

2233

bamI

ababt

baabtI

+=

+µ=+µ=

Figure 4-4. Rigid-body mass and mass moment of inertia.


Figure 4-5. Rectangular plate of example 4-1.

In order to develop dynamic models ofSDOF systems, it is necessary to review the

force—displacement (stiffness) relationships ofseveral of the more common lateral forcemembers used in building structures. Asindicated previously, the assumptions used indeveloping the SDOF model restrict lateralresistance to structural members between theroof and base. These might include suchmembers as columns, diagonal braces, andwalls. Stiffness properties for these elementsare summarized in Figure 4-6.

4.3.4 Free Vibration

Free vibration occurs when a structureoscillates under the action of forces that areinherent in the structure without any externally

Figure 4-6. Stiffness properties of lateral force resisting elements.

192 Chapter 4

applied time-dependent loads or groundmotions. These inherent forces arise from theinitial velocity and displacement the structurehas at the beginning of the free-vibration phase.

Undamped Structures The equation ofmotion for an undamped SDOF system in freevibration has the form

0)()( =+ tkvtvm && (4-16)

which can be written as

0)()( 2 =ω+ tvtv&& (4-17)

where mk /2 =ω . This equation has thegeneral solution

tBtAtv ω+ω= cossin)( (4-18)

in which the constants of integration A and Bdepend upon the initial velocity )0(v& and initialdisplacement v(0). Applying the initialconditions, the solution has the form

tvtv

tv ω+ωω

= cos)0(sin)0(

)(&

(4-19)

This solution in time is representedgraphically in Figure 4-7.

Several important concepts of oscillatorymotion can be illustrated with this result. Theamplitude of vibration is constant, so that thevibration would, theoretically, continueindefinitely with time. This cannot physicallybe true, because free oscillations tend todiminish with time, leading to the concept ofdamping. The time it takes a point on the curveto make one complete cycle and return to itsoriginal position is called the period ofvibration, T. The quantity ω is the circularfrequency of vibration and is measured inradians per second. The cyclic frequency f isdefined as the reciprocal of the period and ismeasured in cycles per second, or hertz. Thesethree vibration properties depend only on themass and stiffness of the structure and arerelated as follows:

fk

mT

12

2 =π=ωπ= (4-20a)

The amplitude of motion is given as:

( ) ( )[ ]2

2.

00

vw

vp +

= (4-20b)

Figure 4-7. Free-vibration response of an undampedSDOF system.

It can be seen from these expressions that iftwo structures have the same stiffness, the onehaving the larger mass will have the longerperiod of vibration and the lower frequency. Onthe other hand, if two structures have the samemass, the one having the higher stiffness willhave the shorter period of vibration and thehigher frequency.

Example 4-2 (Period of undamped freevibration)

Construct an idealized SDOF model for theindustrial building shown in Figure 4-8, andestimate the period of vibration in the twoprincipal directions. Note that vertical cross


bracings are made of 1-inch-diameter rods,horizontal cross bracing is at the bottom chordof trusses, and all columns are W8 × 24.

•Weight determination: Roof level: Composition roof 9.0 psf Lights, ceiling, mechanical 6.0 psf Trusses 2.6 psf Roof purlins, struts 2.0 psf Bottom chord bracing 2.1 psf Columns (10 ft, 9 in.) 0.5 psf Total 22.2 psf Walls: Framing, girts, windows 4.0 psf Metal lath and plaster 6.0 psf Total 10.0 psf

Total weight and mass:

W = (22.2)(100)(75) + (10)(6)(200 + 150)W = 187,500 lb = 187.5 kips

485.04.386

5.187 ===g

Wm kips-sec2/in.

•Stiffness determination: North—south (moment frames):

kips/in.6.231)6.9(24

kips/in.6.9

)144(

)8.82)(29000)(12(12

24

1

33

===

=

==

∑=i

i

i

i

kk

k

L

EIk

East—west (braced frames):

kips/in.7.358)7.59(6

kips/in.7.59280

)858.0)(29000)(785.0(

585.0)31(,31)20/12(tan

in.280ft3.232012

785.04/

cos

6

1

2

1

22

2

2

===

==

===θ

==+=

=π=

θ=

∑=

−

ii

i

i

kk

k

Cos

L

dA

L

AEk

oo

• Period determination:

North—south:

Hz48.31

sec.287.08.21

22

rad/sec8.21485.0

6.231

==

===

===

Tf

T

m

k

πωπ

ω

Figure 4-8. Building of Example 4-2.

East—west:

HZ3.41

sec.23.02.27

22

rad/sec2.27485.0

7.358

==

=π=ωπ=

===ω

Tf

T

m

k

Damped Structures In an actual structurewhich is in free vibration under the action ofinternal forces, the amplitude of the vibration

194 Chapter 4

tends to diminish with time and eventually themotion will cease. This decrease with time isdue to the action of viscous damping forceswhich are proportional to the velocity. Theequation of motion for this condition has theform

0)()()( =++ tkvtvctvm &&& (4-21)

This equation has the general solution

ω+

ωω

λω+= λω− tvt

vvetv dd

dt cos)0(sin

])0()0([)( &

(4-22a)

where

ω==λ

m

C

C

C

cr 2 = percentage of critical

damping

21 λ−ω=ωd = damped circular

frequency

Figure 4-9. Free vibration response of a damped SDOFsystem.

The solution to this equation with time isshown in Figure 4-9. The damping in theoscillator is expressed in terms of a percentageof critical damping, where critical damping is

defined as 2mω and is the least amount ofdamping that will allow a displaced oscillator toreturn to its original position withoutoscillation. For most structures, the amount ofviscous damping in the system will varybetween 3% and 10% of critical. Substitutingan upper value of 20% into the aboveexpression for the damped circular frequencygives the result that ω d = 0.98ω . Since the twovalues are approximately the same for values ofdamping found in structural systems, theundamped circular frequency is used in place ofthe damped circular frequency. In this case theamplitude of motion is given as:

( ) ( ) ( )[ ]2

2.

000

vw

wvvp

D

+

+= λ(4-22b)

One of the more useful results of the free-vibration response is the estimation of thedamping characteristics of a structure. If astructure is set in motion by some externalforce, which is then removed, the amplitudewill decay exponentially with time as shown inFigure 4-9. It can further be shown that the ratiobetween any two successive amplitude peakscan be approximated by the expression

πλ=+

2

)1(

)(e

iv

iv(4-23)

Taking the natural logarithm of both sidesresults in

πλ=+

=δ 2)1(

)(ln

iv

iv(4-24)

where the parameter δ is called thelogarithinic decrement. Solving for thepercentage of critical damping, λ, gives

πδ≈λ

2(4-25)


The above equation provides one of themore useful means of experimentally estimatingthe damping characteristics of a structure.

4.4 Response to Basic DynamicLoading

4.4.1 Introduction

Time histories of earthquake accelerationsare in general random functions of time.However, considerable insight into the responseof structures can be gained by considering theresponse characteristics of structures to twobasic dynamic loadings; harmonic loading andimpulse loading. Harmonic loading idealizesthe earthquake acceleration time history as atrain of sinusoidal waves having a givenamplitude. These might be representative of theaccelerations generated by a large, distantearthquake in which the random wavesgenerated at the source have been filtered bythe soil conditions along the travel path.

Impulse loading idealizes the earthquakeaccelerations as a short duration impulseusually having a sinusoidal or symmetrical(isosceles) triangular shape. The idealizationmay be a single pulse or it may be a pulse traincontaining a limited number of pulses. Thisloading is representative of that which occurs inthe near fault region.

This section will present a brief overview ofthe effects of harmonic loading and impulseloading on the response of building structures.

4.4.2 Harmonic Loading

For an undamped system subjected tosimple harmonic loading, the equation ofmotion has the form

ptpkvvm sin0=+&& (3-26a)

where P0 is the amplitude and p is thecircular frequency of the harmonic load.For a ground acceleration, the acceleration

can be represented as ptg sin0&& , the

equivalent force amplitude as ooe gmp &&=and the frequency ratio β = p/ω. Thesolution for the time dependentdisplacement has the form

)sin(sin)1(

1)(

2tpt

k

gmtv o ωβ−

β−×=

&&

(3-26b)

where

ntdisplaceme static the// == kpkgm oeo&&

21

1

β− = dynamic amplification factor

sin pt = steady state response

βsin ω t = transient response induced by the initial conditions

From equation (4-26b) it can be seen that forlightly damped systems, the peak steady stateresponse occurs at a frequency ratio near unitywhen the exciting frequency of the applied loadequals the natural frequency of the system. Thisis the condition that is called resonance. Theresult given in Equation (4-26b) implies that theresponse of the undamped system goes toinfinity at resonance, however, a closerexamination in the region of β equal to unity,Clough and Penzien (4-4) , shows that it onlytends toward infinity and that several cycles arerequired for the response to build up. A similaranalysis for a damped system shows that atresonance, the dynamic amplificationapproaches a limit that is inversely proportionalto the damping ratio

λ=

2

1DA (4-26c)

For both the undamped and the dampedcases, the response builds up with the numberof cycles as shown in Figure 4-10a.

196 Chapter 4

Figure 4-10a. Resonance response.

The required number of cycles for the dampedcase can be estimated as 1/λ. The condition ofresonance can occur in buildings which aresubjected to base accelerations having afrequency that is close to that of the buildingand having a long duration. The duration of theground shaking is an important factor in thistype of response for the reasons just discussed.The Mexico City earthquakes (1957, 1979,1985) have produced good examples ofharmonic type ground motions which have astrong resonance effect on buildings. Groundmotions having a period of approximately 2seconds were recorded during the 1985earthquake and caused several buildings tocollapse in the upper floors.

It must be recognized that as the responsetends to build up, the effective damping willincrease and as cracking and local yieldingoccur the period of the structure will shift. Bothof these actions in the building will tend toreduce the maximum response. Since thedynamic amplification and number of cycles toreach the maximum response are both inverselyproportional to the damping, the use ofsupplemental damping in the building tocounter this type of ground motion is attractive.

4.4.3 Impulse Motion

Much of the initial work on impulse loadswas done during the period of 1950-1965 and isdiscussed by Norris et al.(4-15). The force onstructures generated by a blast or explosion canbe idealized as a single pulse of relatively shortduration. More recently it has becomerecognized that some earthquake motions,particularly those in the near fault region, canbe idealized as either a single pulse or as asimple pulse train consisting of one to threepulses. The accelerations recorded in Bucharest,Romania during the Vrancea, Romaniaearthquake (1977), shown in Figure 4-10b, are agood example of this type of motion. It is ofinterest to note that this site is more than 100miles from the epicenter, indicating that thistype of motion is not limited to the near faultregion.

Figure 4-10b. Bucharest (1977) ground acceleration.

The maximum response to an impulse loadwill generally be attained on the first cycle. Forthis reason, the damping forces do not havetime to absorb much energy from the structure.Therefore, damping has a limited effect incontrolling the maximum response and isusually neglected when considering themaximum response to impulse type loads.

The rectangular pulse is a basic pulse shape.This pulse has a zero (instantaneous) rise time


and a constant amplitude, po, which is appliedto the structure for a finite duration td. Duringthe time period when the load is on the structure(t < td) the equation of motion has the form

opkvvm =+&& (4-26d)

which has the general solution

)cos1()( 0 tk

ptv ω−= (4-26e)

When the impulse load is no longer actingon the structure, the system is responding infree vibration and the equation of motionbecomes

ttvttv

tv dd ω+ω

ω= cos)(sin

)()(

&(4-26f)

where

dttt −=

The displacement, v(td) and the velocity)( dtv& at the end of the loading phase

become the initial conditions for the freevibration phase. It can be shown that thedynamic amplification, DA, which isdefined as the ratio of the maximumdynamic displacement to the staticdisplacement, will equal 2 if td ≥ T/2 andwill equal 2sin(π td /T) if 2/Ttd ≤ . For

elastic response, the dynamic amplificationis a function of the shape of the impulseload and the duration of the load relative tothe natural period of the structure as shownin Figure 4-10c.

For nonlinear behavior, the equation ofmotion becomes more complex, requiring theuse of numerical methods for solution. Resultsof initial studies for basic pulse shapes werepresented in the form of response charts(4-15)

such as the one shown in Figure 4-10d whichcan be thought of as a constant strengthresponse spectra. For nonlinear response, thedynamic amplification factor is replaced by the

displacement ductility ratio which is defined asthe ratio of the maximum displacement to thedisplacement at yield.

yieldv

vmax=µ (4-26g)

Figure 4-10c. Maximum elastic response, rectangular andtriangular load pulses.[4-16]

Figure 4-10d. Maximum elasto-plastic response,rectangular load pulse.[4-16]

It can also be seen that the single curverepresenting the elastic response becomes afamily of curves for the inelastic response.

198 Chapter 4

These curves depend upon the ratio of themaximum system resistance, Rm, to themaximum amplitude of the impulse load. Notethat the bottom curve in Figure 4-10d which hasa resistance ratio of 2 represents the elasticresponse curve with the ductility equal to orless than unity for all values of td /T. It can alsobe seen that as the resistance ratio decreases,the ductility demand increases.

4.4.4 Example 4-3 (Analysis for ImpulseBase Acceleration)

The three bay frame shown in Figure 4-10eis assumed to be pinned at the base. It issubjected to a ground acceleration pulse whichhas an amplitude of 0.5g and a duration of 0.4seconds. It should be noted that thisacceleration pulse is similar to one recorded atthe Newhall Fire Station during the Northridgeearthquake (1994). The lateral resistance atultimate load is assumed to be elasto-plastic.The columns are W10× 54 with a clear heightof 15 feet and the steel is A36 having a nominalyield stress of 36 ksi. Estimate the following:

Figure 4-10e. Building elevation, resistance and loading,Example 4-3.

(a) the displacement ductility demand, (b) themaximum displacement and (c) the residualdisplacement.

For a W10× 54 column, I = 303 in4 and Z= 66.6 in3 The lateral stiffness of an individualcolumn is calculated as

in

kip

L

EIki 5.4

)1215(

303)29000(3333

=×

×==

and the total stiffness becomes

in

kip0.185.44 =×== ∑ ikK

The mass is the weight divided by theacceleration of gravity,

in

seckips26.0

4.386

100 2

2sec

in

kips −===g

Wm

The period of vibration of the structure cannow be calculated as

.sec75.00.18

26.022 =π=π=

k

mT

and the duration ratio becomes

53.075.0

4.0 ==T

td

The effective applied force, Pe is given as

kips505.05.0 ==×== WgmgmP oe &&

The ultimate lateral resistance of thestructure occurs when plastic hinges form atthe tops of the columns and a sway mechanismis formed. The nominal plastic momentcapacity of a single column is

kips-in 240066636 =×== .. Z F M yP

and the shear resistance is

kips. 13.33 180

2400P ===h

MVi

The total lateral resistance is

KipsVR i 33.534 ==

The resistance to load ratio, is then given as

1.150

3.53 ==evP

R


Using this ratio and the duration ratio, td

/T and entering the response spectrum givenin Figure 4-10d, the displacement ductilitydemand is found to be 2.7. Thedisplacement at yield can be obtained as

.0.318

3.53in

K

Rvy ===

and the maximum displacement is

in.1.80.37.2max =×=×= ylcv µ

The residual or plastic deformation is thedifference between the maximum displacementand the displacement at yield.

inchesvv presidual 1.50.31.8)( =−==

More recently, these calculations have beenprogrammed for interactive computation onpersonal computers. The program NONLIN (4-

14) can be used to do this type of calculation andto gain additional insight through the graphicsthat are available. Using the program, themaximum displacement ductility is calculatedto be 2.85, the maximum displacement is 8.4inches, and the plastic displacement is 5.6 in. Aplot of the calculated time history of thedisplacement, shown in Figure 4-10f, indicates

that structure reaches the maximumdisplacement on the first cycle and that fromthis time onward, it oscillates about a deformedposition of 5.6 inches which is the plasticdisplacement. This can also be seen in a plot ofthe force versus displacement, shown in Figure4-10g which indicates a single yield excursionfollowed by elastic oscillations about theresidual displacement of 5.6 inches.

Figure 4-10g. Computed force versus displacement.

Figure 4-10f. Computed displacement time history

200 Chapter 4

4.4.5 Approximate resopnse to impulseloading

In order to develop a method for evaluatingthe response of a structural system to a generaldynamic loading, it is convenient to firstconsider the response of a structure to a short-duration impulse load as shown in Figure 4-10h, If the duration of the applied impulse load,t, is short relative to the fundamental period ofvibration of the structure, T, then the effect ofthe impulse can be considered as an incrementalchange in velocity. Using the impulse-momentum relationship, which states that theimpulse is equal to the change in momentum,the following equation is obtained:

∫=t

dttpm

tv0

)(1

)(& (4-26)

Following the application of the short-duration impulse load, the system is in freevibration and the response is given by Equation4-19. Applying the initial conditions at thebeginning of the free vibration phase,

∫= 1

0 11 negligible)(,)(1

)(t

tvdttpm

tv&

Figure 4-10h. Short duration rectangular impulse.

Equation 4-19 becomes

∫ −ωω

=− 1

0 11 )(sin)(1

)(t

ttdttpm

ttv (4-27)

For a damped structural system, the free-vibration response is given by Equation 4-22Applying the above initial conditions toEquation 4-22 results in the following equationfor the damped response:

)(sin

)(1

)(

1

0

)(1

11

tt

edttpm

ttv

d

t tt

d

−ω×ω

=− ∫ −ωλ−

(4-28)

4.4.6 Response to General DynamicLoading

The above discussion of the dynamicresponse to a short-duration impulse load canreadily be expanded to produce an analysisprocedure for systems subjected to an arbitraryloading time history. Any arbitrary time historycan be represented by a series of short-durationimpulses as shown in Figure 4-11. Consider oneof these impulses which begins at time ℑ afterthe beginning of the time history and has aduration dτ. The magnitude of this differentialimpulse is p(τ) dτ, and it produces a differentialresponse which is given as

ωτ′ωτ=τ

m

dtpdv

sin)()( (4-29)

The time variable t′ represents the free-vibration phase following the differentialimpulse loading and can be expressed as

τ−=′ tt (4-30)

Substituting this expression into Equation 4-29 results in

ωττ−ωτ=τ

m

dtpdv

)(sin)()( (4-31)

The total response can now be obtained bysuperimposing the incremental responses of allthe differential impulses making up the time


history. Integrating Equation 4-31, the totaldisplacement response becomes

∫ ττ−ωτω

=t

dtpm

tv0

)(sin)(1

)( (4-32)

which is known as the Duhamel integral. Whenconsidering a damped structural system, thedifferential response is given by Equation 4-28and the Duhamel integral solution becomes

∫−

=−−t

o d

dt

m

dteptv

ωττωτ τλω )(sin)(

)()(

(4-33)

Figure 4-11. Differential impulse response.

Since the principle of superposition wasused in the derivation of Equations 4-32 and 4-33, the results are only applicable to linearstructural systems. Furthermore, evaluation of

the integral will require the use of numericalmethods. For these two reasons, the use of adirect numerical integration procedure may bepreferable for solving for the response of adynamic system subjected to general dynamicload. This will be addressed in a later section onnonlinear response analysis. However, theDuhamel-integral result can be applied in aconvenient and systematic manner to obtain asolution for the linear elastic structural responsefor earthquake load.

4.4.7 Earthquake Response of ElasticStructures

Time-History Response The response toearthquake loading can be obtained directlyfrom the Duhamel integral if the time-dependent force p(t) is replaced with theeffective time-dependent force Pe(t), which isthe product of the mass and the groundacceleration. Making this substitution inEquation 4-33 results in the followingexpression for the displacement:

ω= )(

)(tV

tv (4-34)

where the response parameter V(t) representsthe velocity and is defined as

∫ ττ−ωτ= τ−ωλ−t

dt dtegtV

0

)( )(sin)()( && (4-35)

The displacement of the structure at anyinstant of time during the entire time history ofthe earthquake under consideration can now beobtained using Equation 4-34. It is convenientto express the forces developed in the structureduring the earthquake in terms of the effectiveinertia forces. The inertia force is the product ofthe mass and the total acceleration. UsingEquation 4-11, the total acceleration can beexpressed as

)()()( tvm

ktv

m

ctu −−= &&& (4-36)

202 Chapter 4

If the damping term can be neglected ascontributing little to the equilibrium equation,the total acceleration can be approximated as

)()( 2 tvtu ω−=&& (4-37)

The effective earthquake force is then given as

)()( 2 tvmtQ ω= (4-38)

The above expression gives the value of thebase shear in a single-story structure at everyinstant of time during the earthquake timehistory under consideration. The overturningmoment acting on the base of the structure canbe determined by multiplying the inertia forceby the story height:

)()( 2 tvhmtM ω= (4-39)

Response Spectra Consideration of thedisplacements and forces at every instant oftime during an earthquake time history canrequire considerable computational effort, evenfor simple structural systems. As mentionedpreviously, for many practical problems andespecially for structural design, only themaximum response quantities are required. Themaximum value of the displacement, asdetermined by Equation 4-34, will be defined asthe spectral displacement

max)(tvSd = (4-40)

Substituting this result into Equations 4-38and 4-39 results in the following expressionsfor the maximum base shear and maximumoverturning moment in a SDOF system:

dSmQ 2max ω= (4-41)

dShmM 2max ω= (4-42)

An examination of Equation 4-34 indicatesthat the maximum velocity response can beapproximated by multiplying the spectraldisplacement by the circular frequency. Thisresponse parameter is defined as the spectralpseudovelocity and is expressed as

dpv SS ω= (4-43)

In a similar manner, Equation 4-37 indicatesthat the maximum total acceleration can beapproximated as the spectral displacementmultiplied by the square of the circularfrequency. This product is defined as thespectral pseudoacceleration and is expressed as

dpa SS 2ω= (4-44)

A plot of the spectral response parameteragainst frequency or period constitutes theresponse spectrum for that parameter. Aschematic representation of the computation ofthe displacement spectrum for the north-southcomponent of the motion recorded at El Centroon May 18, 1940 has been presented byChopra(4-1) and is shown in Figure 4-12.Because the three response quantities arerelated to the circular frequency, it isconvenient to plot them on a single graph withlog scales on each axis. This special type of plotis called a tripartite log plot. The three responseparameters for the El Centro motion are shownplotted in this manner in Figure 4-13. For aSDOF system having a given frequency(period) and given damping, the three spectralresponse parameters for this earthquake can beread directly from the graph.

Two types of tripartite log paper are used forplotting response spectra. Note that on thehorizontal axis at the bottom of the graph inFigure 4-13, the period is increasing from left toright. For this reason, this type of tripartite logpaper is often referred to as period paper. Asimilar plot of the response spectra for the ElCentro N-S ground motion is shown in Figure4-14. Here it can be seen that frequency, plottedon the horizontal axis, is increasing from left toright. This type of tripartite paper is referred toas frequency paper.


Figure 4-12. Computation of deformation (or displacement) response spectrum. [After Chopra (4-1)].

204 Chapter 4

Figure 4-13. Typical tripartite response-spectra curves.


Figure 4-14. Response spectra, El Centro earthquake, May 18,1940, north-south direction.

Figure 4-15. Site-specific response spectra.

206 Chapter 4

4.4.8 Design Response Spectra

Use of the elastic response spectra for a singlecomponent of a single earthquake record

(Figure 4-13), while suitable for purposes ofanalysis, is not suitable for purposes of design.The design response spectra for a particular siteshould not be developed from a single

Figure 4-16. Smoothed site-specific design spectra.


acceleration time history, but rather should beobtained from the ensemble of possibleearthquake motions that could be experiencedat the site. This should include the effect ofboth near and distant earthquakes. Furthermore,a single earthquake record has a particularfrequency content which gives rise to thejagged, sawtooth appearance of peaks andvalleys shown in Figure 4-13. This feature isalso not suitable for design, since for a givenperiod, the structure may fall in a valley of theresponse spectrum and hence be underdesignedfor an earthquake with slightly differentresponse characteristics. Conversely, for a smallchange in period, the structure might fall on apeak and be overdesigned. To alleviate thisproblem the concept of the smoothed responsespectrum has been introduced for design.Statistics are used to create a smoothedspectrum at some suitable design level. Themean value or median spectrum can generallybe used for earthquake-resistant design ofnormal building structures. Use of this spectrumimplies there is a 50% probability that thedesign level will be exceeded.

Structures that are particularly sensitive toearthquakes or that have a high risk may bedesigned to a higher level such as the mean plusone standard deviation, which implies that theprobability of exceedance is only 15.9%.Structures having a very high risk are oftendesigned for an enveloping spectrum whichenvelopes the spectra of the entire ensemble ofpossible site motions. Response spectra whichare representative of a magnitude-6.5earthquake at a distance of 15 miles, developedby the Applied Technology Council (4-2), areshown in Figure 4-15. The correspondingsmoothed design spectra are shown in Figure 4-16.

Newmark and Hall (4-3) have proposed amethod for constructing an elastic designresponse spectrum in which the primary inputdatum is the anticipated maximum groundacceleration. The corresponding values for themaximum ground velocity and the maximumground displacement are proportioned relativeto the maximum ground acceleration, which is

normalized to 1.0g. The maximum groundvelocity is taken as 48 in./sec, and themaximum ground displacement is taken as 36in. It should be noted that these values representmotions which are more intense than thosenormally considered for earthquake-resistantdesign; however, they are approximately in thecorrect proportion for earthquakes occurring oncompetent soils and can be scaled forearthquakes having lower ground acceleration.

Table 4-1. Relative values of spectrum amplificationfactors (4-3).

Amplification factor forPercentageof criticalDamping Displacement Velocity Acceleration

0 2.5 4.0 6.40.5 2.2 3.6 5.81 2.0 3.2 5.22 1.8 2.8 4.35 1.4 1.9 2.6

10 1.1 1.3 1.520 1.0 1.1 1.2

Three principal regions of the responsespectrum are identified, in which the structuralresponse can be approximated as a constant,amplified value. Amplification factors areapplied to the ground motions in these threeregions to obtain the design spectrum for aSDOF elastic system. Based on a large database of recorded earthquake motions,amplification factors which give a probabilityof exceedance of about 10% or less are given inTable 4-1 for various values of the structuraldamping. The basic shape of the Newmark—Hall design spectrum using the normalizedground motions and the amplification factorsgiven in Table 4-1 for 5% damping is shown inFigure 4-17. The displacement region is thelow-frequency region with frequencies less than0.33 Hz (periods greater than 3.0 sec). Themaximum displacement of the SDOF system isobtained by multiplying the maximum grounddisplacement by the displacement amplificationfactor given in Table 4-1. The velocity region isin the mid-frequency region between 0.33 Hz(3.0 sec) and 2.0 Hz (0.5 sec). Maximumvelocities in this region are obtained by

208 Chapter 4

multiplying the maximum ground velocity bythe amplification factor for the velocity (Table4-1). An amplified acceleration region liesbetween 2.0 Hz (0.5 sec) and 6.0 Hz (0.17 sec).The amplified response is obtained in the samemanner as in the previous two cases. Structureshaving a frequency greater than 30 Hz (periodless than 0.033 sec) are considered to be rigidand have an acceleration which is equal to theground acceleration. In the frequency rangebetween 6 Hz (0.17 sec) and 30 Hz (0.033 sec)there is a transition region between the ground

acceleration and the amplified accelerationregion.

Similar design spectra corresponding to thepostulated ground motion presented in Figures4-15 and 4-16 are shown in Figure 4-18. Inorder to further define which response spectrumshould be used for design, it is necessary toestimate the percentage of critical damping inthe structure. A summary of recommendeddamping values for different types of structuresand different stress conditions is given in Table4-2 as a guideline.

Figure 4-17. Basic New mark-Hall design spectrum normalized to 1.0g for 5% damping (4-3).


Example 4-4 (Construction of a Newmark-Hall Design Spectrum)

Construct a Newmark-Hall design spectrumfor a maximum ground acceleration of 0.2g,and use it to estimate the maximum base shearfor the industrial building of Example 4-1.Assume the damping is 5 percent of critical.

•Determine ground motion parameters:ground acceleration = (1.0)(0.2) = 0.2gground velocity = (48.0)(0.2)=9.6in./sec.ground displacement=(36.0)(0.2)=7.2 in.

•Amplified response parameters:acceleration = (0.2)(2.6) = 0.52g

velocity = (9.6)(1.9) = 18.2 in./secdisplacement = (7.2)(1.4) = 10.0 in.

The constructed design spectrum is shownin Figure 4-19.

From Example 4-1:N-S:T = 0.287 sec.ω = 21.8 rad/sec,f = 3.48 HZ

From the design spectrum for f = 3.48 Hz:Sd=v(t)max =0.42 in.

Figure 4-18. A New mark-Hall design spectra.

210 Chapter 4

From Equation 4-42:

Qmax = (0.485)(21.8)2(0.42) = 96.8 kipsE-W:T = 0.23 sec,ω = 27.2 rad/sec,From the design spectrum for f = 4.3 Hz:

Sd = 0.28 in.

From Equation 4-42:

Qmax=(0.485)(21.8)2(0.28) = 64.5 kips

Figure 4-19. Response spectrum of Example 4-3.

4.4 GENERALIZED-COORDINATE APPROACH

Up to this point, the only structures whichhave been considered are single-story buildingswhich can be idealized as SDOF systems. Theanalysis of most structural systems requires amore complicated idealization even if theresponse can be represented in terms of a singledegree of freedom. The generalized-coordinateapproach provides a means of representing theresponse of more complex structural systems interms of a single, time-dependent coordinate,known as the generalized coordinate.

Displacements in the structure are related tothe generalized coordinate as

)()(),( tYxtxv φ= (4-45)

Where Y(t) is the time-dependentgeneralized coordinate and )(xφ is a spatialshape function which relates the structuraldegrees of freedom, v(x, t), to the generalizedcoordinate. For a generalized SDOF system, itis necessary to represent the restoring forces inthe damping elements and the stiffnesselements in terms of the relative velocity andrelative displacement between the ends of theelement:

)()(),( tYxtxv && φ∆=∆ (4-46)

Table 4-2 Recommended Damping Values (4-3)Stress level Type and condition of

structurePercentageof criticaldamping

Stress level Type and condition ofstructure

Percentageof criticaldamping

Workingstress,<1/2yield point

Vital pipingWelded steel, prestressedconcrete, well-reinforcedconcrete(only slight cracking)

1-22-3

At or justbelow yieldpoint

Vital pipingWelded steel, prestressedconcrete(without completeloss in prestress)

2-35-7

Reinforced concrete withconsiderable cracking

3-5 Prestressed concrete withno prestress left

7-10

Bolted and / or riveted steel,wood structures with nailedor bolted joints.

5-7 Bolted and / or rivetedsteel, wood structures withnailed or bolted joints.

10-15

Wood structures withnailed joints

15-20


)()(),( tYxtxv φ∆=∆ (4-47)

Most structures can be idealized as a verticalcantilever, which limits the number ofdisplacement functions that can be used torepresent the horizontal displacement. Once thedisplacement function is selected, the structureis constrained to deform in that prescribedmanner. This implies that the displacementfunctions must be selected carefully if a goodapproximation of the dynamic properties andresponse of the system are to be obtained. Thissection will develop the equations fordetermining the generalized responseparameters in terms of the spatial displacementfunction and the physical response parameters.Methods for determining the shape functionwill be discussed, and techniques fordetermining the more correct displacementfunction for a particular structure will bepresented.

4.4.1 Displacement Functions andGeneralized Properties

Formulation of the equation of motion interms of a generalized coordinate will berestricted to systems which consist of anassemblage of lumped masses and discreteelements. Lateral resistance is provided bydiscrete elements whose restoring force isproportional to the relative displacementbetween the ends of the element. Dampingforces are proportional to the relative velocitybetween the ends of the discrete dampingelement. Formulation of the equation of motionfor systems having distributed elasticity isdescribed by Clough and Penzien. (4-4) Thegeneral equation of dynamic equilibrium isgiven in Equation 4-6, which represents asystem of forces which are in equilibrium atany instant of time. The principle of virtualwork in the form of virtual displacements statesthat

If a system of forces which are inequilibrium is given a virtual displacementwhich is consistent with the boundaryconditions, the work done is zero.

Applying this principle to Equation 4-6results in an equation of virtual work in theform

0)( =−∆+∆+ vtpvfvfvf sdi δδδδ (4-48)

where it is understood that ),( txvv = and thatthe virtual displacements applied to thedamping force and the elastic restoring forceare virtual relative displacements. The virtualdisplacement can be expressed as

)()(),( tYxtxv δφ=δ (4-49)

and the virtual relative displacement can bewritten as

)()(),( tYxtxv δφ∆=∆δ (4-50)

where

)()()()()()(),( tYxtYxtYxtxv ji φ∆=φ−φ=∆

The inertia, damping and elastic restoringforces can be expressed as

Ykvkf

Ycvcf

Ymvmf

s

d

i

φ∆=∆=φ∆=∆=

φ==&&&

&&&&

(4-51)

Substituting Equations 4-49, 4-50, and 4-51into Equation 4-48 results in the followingequation of motion in terms of the generalizedcoordinate:

)(**** tpYkYcYm =++ &&& (4-52)

where m*, c*, k*, and p* are referred to as thegeneralized parameters and are defined as

212 Chapter 4

force dgeneralize

stiffness dgeneralize

damping dgeneralize

mass dgeneralize

*

2*

2*

2*

=φ=

=φ∆=

=φ∆=

=φ=

∑∑∑∑

iii

iii

iii

iii

pp

kk

cc

mm

(4-53)

For a time-dependent base acceleration thegeneralized force becomes

Lgp &&=* (4-54)

where

factor ionparticipat earthquake=

φ= ∑i

iimL(4-55)

It is also convenient to express thegeneralized damping in terms of the percent ofcritical damping in the following manner:

ωλ=φ∆= ∑ *2* 2)( micci

i (4-56)

Where ω represents the circular frequencyof the generalized system and is given as

*

*

m

k=ω (4-57)

The effect of the generalized-coordinateapproach is to transform a multiple-degree-of-freedom dynamic system into an equivalentsingle-degree-of-freedom system in terms of thegeneralized coordinate. This transformation isshown schematically in Figure 4-20. The degreeto which the response of the transformedsystem represents the actual system will dependupon how well the assumed displacement shaperepresents the dynamic displacement of theactual structure. The displacement shapedepends on the aspect ratio of the structure,which is defined as the ratio of the height to thebase dimension. Possible shape functions forhigh-rise, mid-rise, and low-rise structures aresummarized in Figure 4-21. It should be notedthat most building codes use the straight-lineshape function which is shown for the mid-risesystem. Once the dynamic response is obtainedin terms of the generalized coordinate, Equation4-45 must be used to determine thedisplacements in the structure, and these in turn

Figure 4-20. Generalized single-degree-of-freedom system.


can be used to determine the forces in theindividual structural elements.

In principle, any function which representsthe general deflection characteristics of thestructure and satisfies the support conditionscould be used. However, any shape other thanthe true vibration shape requires the addition ofexternal constraints to maintain equilibrium.These extra constraints tend to stiffen thesystem and thereby increase the computedfrequency. The true vibration shape will haveno external constraints and therefore will havethe lowest frequency of vibration. Whenchoosing between several approximatedeflected shapes, the one producing the lowestfrequency is always the best approximation. Agood approximation to the true vibration shapecan be obtained by applying forces representingthe inertia forces and letting the staticdeformation of the structure determine thespatial shape function.

Example 4-5 (Determination of generalizedparameters)

Considering the four-story, reinforced-concrete moment frame building shown inFigure 4-22, determine the generalized mass,generalized stiffness, and fundamental period ofvibration in the transverse direction using thefollowing shape functions:

(a) )2/sin()( Lxx π=φ and (b)

Lxx /)( =φ .All beams are 12in.× 20 in.,

and all columns are 14 in× 14 in. cf ′ =4000

psi, and the modulus of elasticity of

concrete is 6106.3 × psi. Reinforcing steelis made of grade-60 bars. Floor weights(total dead load) are assumed to be 390 kipsat the roof, 445 kips at the fourth and thirdlevels, and 448 kips at the first level. Liveloads are 30 psf at the roof and 80 psf pertypical floor level.

Figure 4-22. Building of Example 4-5.

Assuming beams are rigid relative tocolumns (Figure 4-23),

Figure 4-21. Possible shape functions based on aspect ratio.

214 Chapter 4

Figure 4-23. Assumed shape of column deformation.

in.

kips140

)144(

)3201)(106.3)(12)(3(

in.

kips209

)126(

)3201)(106.3)(12)(3(

frame) one(3

in.800012

)20(12

in.320112

)14(14

12

12

3

3

1

3

3

2,3,4

3

1

43

43

3

3

=×=

=×=

==

==

==

=∆

=

∆=

∑=

K

K

KKK

I

I

L

EIVK

L

EIV

ii

istory

beam

col

i

Calculating generalized properties (seeFigure 4-24):

(a) Assuming )2/sin()( Lxx π=φ :

Level K M φi ∆φi 2iM φ 2

iK φ∆

4 0.252 1.000 0.252209 0.071 1.054

3 0.288 0.929 0.249209 0.203 8.613

2 0.288 0.726 0.152209 0.306 19.570

1 0.290 0.420 0.051140 0.420 24.696

M* = 0.704 K* = 53.933

sec72.0

secrad/75.8704.0

93.53*

*

=

===

aTandm

kω

(b) Assuming Lxx /)( =φ

Level K M φi ∆φi 2iM φ 2

iK φ∆

4 0.252 1.000 0.252209 0.241 12.139

3 0.288 0.759 0.166209 0.242 12.240

2 0.288 0.517 0.077209 0.241 12.139

1 0.290 0.276 0.022140 0.276 10.665

M* = 0.517 K* = 47.183

.sec66.0 andrad/sec55.9517.0

183.47*

*

==

==ω

bTm

k

Since )2/sin()(, LxxTT ba π=φ> is a

better approximation to the deflected shapethan Lxx /)( =φ

4.4.2 Rayleigh’s Method

Rayleigh’s method is a procedure developedby Lord Rayleigh (4-5) for analyzing vibratingsystems using the law of conservation ofenergy. Its principal use is fordetermining an accurate approximation of thenatural frequency of a structure. The success of


the technique in accomplishing this has beenrecognized by most building codes, which haveadopted the procedure as an alternative forestimating the fundamental period of vibration.In addition to providing an estimate of thefundamental period, the procedure can also beused to estimate the shape function φ (x).

In an undamped elastic system, themaximum potential energy can be expressed interms of the external work done by the appliedforces. In terms of a generalized coordinate thisexpression can be written as

22)(

*

max

Ypp

YPE ii =φ= ∑ (4-58)

Similarly, the maximum kinetic energy canbe expressed in terms of the generalizedcoordinate as

22)(

*222

22

max

mYm

YKE

iii

ω=φω= ∑ (4-59)

According to the principle of conservationof energy for an undamped elastic system, thesetwo quantities must be equal to each other andto the total energy of the system. Equating

Equation 4-58 to Equation 4-59 results in thefollowing expression for the circular frequency:

Ym

p*

*

=ω (4-60)

Substituting this result into Equation 4-20for the period results in

*

*

2p

YmT π= (4-61)

Multiplying the numerator and denominatorof the radical by Y and using Equation 4-45results in the expression for the fundamentalperiod:

∑∑π=

ii

i ii

vpg

vwT

2

2 (4-62)

which is the expression found in most buildingcodes.

The forces which must be applied laterallyto obtain either the shape function φ (x) or thedisplacement v(x) represent the inertia forces,which are the product of the mass and theacceleration. If the acceleration is assumed tovary linearly over the height of a building with

Figure 4-24. Development of a generalized SDOF model for building of Example 4-4.

216 Chapter 4

uniform weight distribution, a distribution ofinertia force in the form of an inverted trianglewill be obtained, being maximum at the top andzero at the bottom. This is similar to thedistribution of base shear used in most buildingcodes and can be a reasonable one to use whenapplying the Rayleigh method. The resultingdeflections can be used directly in Equation 4-62 to estimate the period of vibration or theycan be normalized in terms of the generalizedcoordinate (maximum displacement) to obtainthe spatial shape function to be used in thegeneralized-coordinate method.

Example 4-6 (Application of Rayleigh’sMethod)

Use Rayleigh’s method to determine thespatial shape function and estimate thefundamental period of vibration in thetransverse direction for the reinforced-concretebuilding given in Example 4-4.

We want to apply static lateral loads that arerepresentative of the inertial loads on thebuilding. Since the story weights areapproximately equal, it is assumed that theaccelerations and hence the inertial loads varylinearly from the base to the roof (see Figure 4-25).

Note that the magnitude of loads isirrelevant and is chosen for ease ofcomputation. The following computations (onthe bottom of this page) are a tabular solution ofEquation 4-61.

*

*

2p

YmT π= , or

sec712.0912.16

)3343.0)(666.0(2 =π=T

Note that since T = 0.721 is greater thaneither of the periods calculated in Example 4-5,

the deflected shape given by applying the staticloads is a better approximation than either ofthe two previous deflected shapes.

Figure 4-25.Frame of Example 4-5.

4.4.3 Earthquake Response of ElasticStructures

Time-History Analysis Substituting thegeneralized parameters of Equations 4-53 and4-54 into the Duhamel-integral solution,Equation 4-33, results in the following solutionfor the displacement:

ωφ=

*

)()(),(

m

tVxtxv

L(4-63)

Using Equation 4-37, the inertia force at anyposition x above the base can be obtained from

Level K m P V ∆=V/k v φ 2iim φ Pi φi

4 0.252 8.0 0.3343 1.000 0.252 8.000209 8 0.0383

3 0.288 6.0 0.2960 0.886 0.226 5.316209 14 0.0670

2 0.288 4.0 0.2290 0.685 0.135 2.740209 18 0.0861

1 0.288 2.0 0.1429 0.428 0.053 0.856140 20 0.1429 0.000 0.000 0.666 16.912


),()(),()(),( 2 txvxmtxvxmtxq ω== && (4-64)

which, using Equation 4-63, becomes

*

)()()(),(

m

tVxxmtxq

ωφ= L(4-65)

The base shear is obtained by summing thedistributed inertia forces over the height H ofthe structure:

∫ ω== )(),()(*

2

tVm

dxtxqtQL

(4-66)

The above relationships can be used todetermine the displacements and forces in ageneralized SDOF system at any time duringthe time history under consideration.

Response-Spectrum AnalysisThe maximum value of the velocity given

by Equation 4-35 is defined as the spectralpseudovelocity (Spv), which is related to thespectral displacement (Sd) by Equation 4-43.Substituting this value into Equation 4-63results in an expression for the maximumdisplacement in terms of the spectraldisplacement:

*max

)()(

m

Sxxv dLφ= (4-67)

The forces in the system can readily bedetermined from the inertia forces, which canbe expressed as

max2

maxmax )()()()()( xvxmxvxmxq ω== &&

(4-68)

Rewriting this result in terms of the spectralpseudo-acceleration (Spa) results in thefollowing:

*max

)()()(

m

Sxmxxq paLφ

= (4-69)

Of considerable interest to structuralengineers is the determination of the base shear.This is a key parameter in determining seismicdesign forces in most building codes. The baseshear Q can be obtained from the aboveexpression by simply summing the inertiaforces and using Equation 4-55:

*

2

max m

SQ paL

= (4-70)

It is also of interest to express the base shearin terms of the effective weight, which isdefined as

∑∑

φ

φ=

i ii

ii i

w

wW

2

2

*)(

(4-71)

The expression for the maximum base shearbecomes

gSWQ pa /*max = (4-72)

This form is similar to the basic base-shearequation used in the building codes. In the codeequation, the effective weight is taken to beequal to the total dead weight W, plus apercentage of the live load for specialoccupancies. The seismic coefficient C isdetermined by a formula but is equivalent to thespectral pseudoacceleration in terms of g. Thebasic code equation for base shear has the form

CWQ =max (4-73)

The effective earthquake force can also bedetermined by distributing the base shear overthe story height. This distribution depends uponthe displacement shape function and has theform

Lii

i

mQq

φ= max (4-74)

If the shape function is taken as a straightline, the code force distribution is obtained. Theoverturning moment at the base of the structure

218 Chapter 4

can be determined by multiplying the inertiaforce by the corresponding story height abovethe base and summing over all story levels:

∑=i

iiO qhM (4-75)

Example 4-7 (Spectrum Analysis ofGeneralized SDOF System)

Using the design spectrum given in Figure4-26, the shape function determined in Example4-6, and the reinforced-concrete moment frameof Example 4-5, determine the base shear in thetransverse direction, the correspondingdistribution of inertia forces over the height ofthe structure, and the resulting overturningmoment about the base of the structure.

rad/sec.8.715

Hz,39.1/1sec.,721.0

=ω=== TfT

From the design spectrum Spa = 0.185g.

Level mi φi 2iim φ miφi miφi/L qmax Vmax

4 0.252 1.000 0.252 0.252 0.305 27.10

27.10

3 0.288 0.866 0.226 0.255 0.308 27.36

54.46

2 0.288 0.685 0.135 0.197 0.238 21.14

75.60

1 0.288 0.428 0.053 0.123 0.149 13.24

0.666 0.827 88.84

Figure 4-26. Design spectrum for Example 4-6.

From Equation 4-66,

kips84.88666.0

)4.386)(185.0()827.0( 2

max ==Q

The overturning moment is: (see Fig, 4-27)

Figure 4-27. Story shears and overturning moment(Example 4-6)

kipsft2716

)12(24.13)5.22(14.21

)33(36.27)5.43(10.27

−=+

++=oM

The displacement is

dd SSmv αφ=ϕφ= *)/(max

where


in.50.0in.80.0

in.035.1in.168.1

168.1)941.0)(242.1(

242.1666.0

827.0

941.0)715.8(

)4.386)(185.0(

/and/

12

34

2

*2

====

φ=φ=

==α

==

ϕ=αω=

vv

vv

v

S

mSS

iii

d

pad

4.5 RESPONSE OFNONLINEAR SDOFSYSTEMS

In an earlier section it was shown that theresponse of a linear structural system could beevaluated using the Duhamel integral. Theapproach was limited to linear systems becausethe Duhamel-integral approach makes use ofthe principle of superposition in developing themethod. In addition, evaluation of the Duhamelintegral for earthquake input motions willrequire the use of numerical methods inevaluating the integral. For these reasons it maybe more expedient to use numerical integrationprocedures directly for evaluating the responseof linear systems to general dynamic loading.These methods have the additional advantagethat with only a slight modification they can beused to evaluate the dynamic response ofnonlinear systems. Many structural systems willexperience nonlinear response sometime duringtheir life. Any moderate to strong earthquakewill drive a structure designed by conventionalmethods into the inelastic range, particularly incertain critical regions. A very useful numericalintegration technique for problems of structuraldynamics is the so called step-by-stepintegration procedure. In this procedure thetime history under consideration is divided intoa number of small time increments ∆ t. Duringa small time step, the behavior of the structureis assumed to be linear. As nonlinear behavioroccurs, the incremental stiffness is modified. Inthis manner, the response of the nonlinearsystem is approximated by a series of linearsystems having a changing stiffness. The

velocity and displacement computed at the endof one time interval become the initialconditions for the next time interval, and hencethe process may be continued step by step.

4.5.1 Numerical Formulation ofEquation of Motion

This section considers SDOF systems withproperties m, c, k(t) and p(t), of which theapplied force and the stiffness are functions oftime. The stiffness is actually a function of theyield condition of the restoring force, and thisin turn is a function of time. The dampingcoefficient may also be considered to be afunction of time; however, general practice is todetermine the damping characteristics for theelastic system and to keep these constantthroughout the complete time history. In theinelastic range the principle mechanism forenergy dissipation is through inelasticdeformation, and this is taken into accountthrough the hysteretic behavior of the restoringforce.

The numerical equation required to evaluatethe nonlinear response can be developed by firstconsidering the equation of dynamicequilibrium given previously by Equation 4-6.It has been stated previously that this equationmust be satisfied at every increment of time.Considering the time at the end of a short timestep, Equation 4-6 can be written as

)()()()( ttpttfttfttf sdi ∆+=∆++∆++∆+(4-76)

where the forces are defined as

∑

∑

−

=

=

∆=

−∆+=∆

∆+=∆=

∆+=∆+=

1

1

1

)()(

)()()(

)()()()(

)(

)(

n

iiit

n

itiis

d

i

tvtkr

tvttvtv

tvtkrtvtkf

ttvcf

ttvmf

&

&&

(4-77)

and in the case of ground accelerations

220 Chapter 4

)()()( ttgmttpttp e ∆+−=∆+=∆+ && (4-78)

Substituting Equations 4-77 and 4-78 intoEquation 4-76 results in an equation of motionof the form

)()()( ttgmvkttvcttvm ii ∆+−=∆+∆++∆+ ∑ &&&&&

(4-79)

It should be noted that the incrementalstiffness is generally defined by the tangentstiffness at the beginning of the time interval

dv

dfk s

i = (4-80)

In addition, the dynamic properties given inEquations 4-77 and 4-78 can readily beexchanged for the generalized properties whenconsidering a generalized SDOF system.

4.5.2 Numerical Integration

Many numerical integration schemes areavailable in the literature. The techniqueconsidered here is a step-by-step procedure inwhich the acceleration during a small timeincrement is assumed to be constant. A slightvariation of this procedure, in which theacceleration is assumed to vary linearly duringa small time increment, is described in detail byClough and Penzien.(4-4). Both procedures havebeen widely used and have been found to yieldgood results with minimal computational effort.

If the acceleration is assumed to be constantduring the time interval, the equations for theconstant variation of the acceleration, the linearvariation of the velocity and the quadraticvariation of the displacement are indicated inFigure 4-28. Evaluating the expression forvelocity and displacement at the end of the timeinterval leads to the following two expressionsfor velocity and displacement:

Figure 4-28. Increment motion (constant acceleration).

2)(

2)()()(

ttv

tttvtvttv

∆+∆∆++=∆+ &&&&&& (4-81)

4)(

4)(

)()()(22 t

tvt

ttv

ttvtvttv

∆+∆∆++

∆+=∆+

&&&&

&

(4-82)

Solving Equation 4-82 for the acceleration)( ttv ∆+&& gives

)()(44

)(2

tvtvt

vt

ttv &&&&& −∆

−∆∆

=∆+ (4-83)



)(4

)(2

tAvt

ttv +∆∆

=∆+&& (4-84)

where

)()(4

)(

)()(

tvtvt

tA

tvttvv

&&& −∆

−=

−∆+=∆

Note that this equation expresses theacceleration at the end of the time interval as afunction of the incremental displacement andthe acceleration and velocity at the beginning ofthe time interval. Substituting Equation 4-83into Equation 4-81 gives the followingexpression for the velocity at the end of thetime increment:

)(2

)( tvvt

ttv && −∆∆

=∆+ (4-85)


)(2

)( tBvt

ttv +∆∆

=∆+& (4-86)

where

)()( tvtB &−=

It is convenient to express the damping as alinear function of the mass:

ωλ=λ=α= mCmc cr 2 (4-87)

Use of this equation allows theproportionality factor α to be expressed as

λω=α 2 (4-88)

Substituting Equations 4-85, 4-86, and 4-88into Equation 4-79 results in the following formof the equation for dynamic equilibrium:

vktRtBvt

mtAvt

m ∆++

+∆

∆α+

+∆

∆)()(

2)(

24

)( ttgm ∆+= && (4-89)

Moving terms containing the responseconditions at the beginning of the time intervalto the right-hand side of the equation results inthe following so-called pseudo-static form ofthe equation of motion:

)()( ttpvk t ∆+=∆ (4-90)

where

)]()([

)()()(

242

tBtAm

tRttgmttp

kt

m

t

mk tt

α−−−∆+−=∆+

+∆α+

∆=

&&

The solution procedure for a typical timestep is as follows:

1. Given the initial conditions at the beginningof the time interval, calculate thecoefficients A(t) and B(t).

2. Calculate the effective stiffness.3. Determine the effective force.4. Solve for the incremental displacement

tkpv /= (4-91)

5. Determine the displacement, velocity andacceleration at the end of the time interval:

)(4

)(

)(2

)(

)()(

2tA

tttv

tBt

ttv

vtvttv

+∆

=∆+

+∆

=∆+

∆+=∆+

&&

& (4-92)

6. The values given in Equation 4-92 becomethe initial conditions for the next timeincrement, and the procedure is repeated.

222 Chapter 4

The above algorithm can be easilyprogrammed on any microcomputer. If it iscombined with a data base of recordedearthquake data such as EQINFOS,(4-6) it can beused to gain considerable insight into the linearand nonlinear response of structures that can bemodeled as either a SDOF system or as ageneralized SDOF system. It also forms thebackground material for later developments formultiple-degree-of-freedom systems.

An important response parameter that isunique to nonlinear systems is the ductilityratio. For a SDOF system, this parameter can bedefined in terms of the displacement as

yield)(

plastic)(0.1

yield)(

(max)

v

v

v

v +==µ (4-93)

As can be seen from the above equation, theductility ratio is an indication of the amount ofinelastic deformation that has occurred in thesystem. In the case of a SDOF system orgeneralized SDOF system the ductility obtainedfrom Equation 4-93 usually represents theaverage ductility in the system. The ductilitydemand at certain critical regions, such asplastic hinges in critical members, may beconsiderably higher.

4.6 MULTIPLE-DEGREE-OF-FREEDOM SYSTEMS

In many structural systems it is impossibleto model the dynamic response accurately interms of a single displacement coordinate.These systems require a number of independentdisplacement coordinates to describe thedisplacement of the mass of the structure at anyinstant of time.

4.6.1 Mass and Stiffness Properties

In order to simplify the solution it is usuallyassumed for building structures that the mass ofthe structure is lumped at the center of mass ofthe individual story levels. This results in a

diagonal matrix of mass properties in whicheither the translational mass or the massmoment of inertia is located on the maindiagonal.

=

nn

i

v

v

v

v

m

m

m

m

f

.

.

.

.

.

.}{3

2

1

3

2

1

(4-94)

It is also convenient for building structuresto develop the structural stiffness matrix interms of the stiffness matrices of the individualstory levels. The simplest idealization for amultistory building is based on the followingthree assumptions: (i) the floor diaphragm isrigid in its own plane; (ii) the girders are rigidrelative to the columns and (iii) the columns areflexible in the horizontal directions but rigid inthe vertical. If these assumptions are used, thebuilding structure is idealized as having threedynamic degrees of freedom at each story level:a translational degree of freedom in each of twoorthogonal directions, and a rotation about avertical axis through the center of mass. If theabove system is reduced to a plane frame, itwill have one horizontal translational degree offreedom at each story level. The stiffnessmatrix for this type of structure has thetridiagonal form shown below:

For the simplest idealization, in which eachstory level has one translational degree offreedom, the stiffness terms ki in the aboveequations represent the translational storystiffness of the ith story level. As theassumptions given above are relaxed to includeaxial deformations in the columns and flexuraldeformations in the girders, the stiffness term ki

in Equation 4-95 becomes a submatrix ofstiffness terms, and the story displacement vi


becomes a subvector containing the variousdisplacement components in the particular storylevel. The calculation of the stiffnesscoefficients for more complex structures is astandard problem of static structural analysis.For the purposes of this chapter it will beassumed that the structural stiffness matrix isknown.

4.6.2 Mode Shapes and Frequencies

The equations of motion for undamped freevibration of a multiple-degree-of-freedom(MDOF) system can be written in matrix formas

}0{}]{[}]{[ =+ vKvM && (4-96)

Since the motions of a system in freevibration are simple harmonic, the displacementvector can be represented as

tvv ω= sin}{}{ (4-97)

Differentiating twice with respect to timeresults in

}{}{ 2 vv ω−=&& (4-98)

Substituting Equation 4-98 into Equation 4-96 results in a form of the eigenvalue equation,

( ) }0{}{][][ 2 =ω− vMK (4-99)

The classical solution to the above equationderives from the fact that in order for a set ofhomogeneous equilibrium equations to have anontrivial solution, the determinant of thecoefficient matrix must be zero:

}0{])[]det([ 2 =ω− MK (4-100)

Expanding the determinant by minorsresults in a polynomial of degree N, which iscalled the frequency equation. The N roots ofthe polynomial represent the frequencies of theN modes of vibration. The mode having thelowest frequency (longest period) is called thefirst or fundamental mode. Once thefrequencies are known, they can be substitutedone at a time into the equilibrium Equation 4-99, which can then be solved for the relativeamplitudes of motion for each of thedisplacement components in the particularmode of vibration. It should be noted that sincethe absolute amplitude of motion isindeterminate, N-1 of the displacementcomponents are determined in terms of onearbitrary component.

This method can be used satisfactorily forsystems having a limited number of degrees offreedom. Programmable calculators haveprograms for solving the polynomial equationand for doing the matrix operations required todetermine the mode shapes. However, for

)953(

.

.

.

..

...

...

...}{

1

3

2

1

1

4323

3212

21

−

+−

−

−+−

−+

−

=

−

− n

n

nnn

n

s

v

v

v

v

v

kkk

k

kkkk

kkkk

kk

f

224 Chapter 4

problems of any size, digital computerprograms which use numerical techniques tosolve large eigenvalue systems(4-7) must beused.

Example 4-8 (Mode Shapes and Frequencies)It is assumed that the response in the

transverse direction for the reinforced-concretemoment frame of Example 4-4 can berepresented in terms of four displacementdegrees of freedom which represent thehorizontal displacements of the four storylevels. Determine the stiffness matrix and themass matrix, assuming that the mass is lumpedat the story levels. Use these properties tocalculate the frequencies and mode shapes ofthe four-degree-of-freedom system.

•Stiffness and mass matrices: The stiffnesscoefficient ijk is defined as the force atcoordinate i due to a unit displacement atcoordinate j, all other displacements being zero(see Figure 4-29):

where B = ω2/800•Characteristic equation:

0486.0430.62476.113183.64

0][][ 2

=+−+−

=ω−

BBBB

MK

Solution:

sec124.04,800.504,800

24225.34

sec155.03,388.403,800

23039.23

sec244.01,768.252,800

22830.02

sec744.01,438.81,800

21089.01

==ωω

==

==ωω

==

==ωω

==

==ωω

==

TB

TB

TB

TB

•Mode shapes (see Figure 4-29) areobtained by substituting the values of Bi, one ata time, into the equations

}0{}]){[]([ 2 =ω− vMK

and determining N-1 components of thedisplacement vector in terms of the firstcomponent, which is set equal to unity. Thisresults in the modal matrix

−−−−

−−=Φ

92.024.105.147.0

75.175.078.074.0

78.107.120.091.0

00.100.100.100.1

][

Solution of the above problem using thecomputer program ETABS (4-12) gives thefollowing results:

−−−

−−−

=

34920900

2094182090

0209418209

00209209

][K

=

16.1000

015.100

0015.10

00001.1

4

1][M

−−−−−

−−−−−

=ω−

B

B

B

B

MK

16.174.105.100

05.115.109.205.10

005.115.109.205.1

0005.101.105.1

200][][ 2


=

107.0

152.0

268.0

838.0

}{T

−−−

−−=Φ

92.024.105.147.0

75.175.078.074.0

78.107.120.091.0

00.100.100.100.1

][

This program assumes the floor diaphragmis rigid in its own plane but allows axialdeformation in the columns and flexuraldeformations in the beams. Hence, with theseadded degrees of freedom (fewer constraints)the fundamental period increases. However,comparing the results of this example withthose of Example 4-5, it can be seen that forthis structure a good approximation for the first-mode response was obtained using thegeneralized SDOF model and the staticdeflected shape.

Figure 4-29. Stiffness determination and modeshape(Example 4-8).

4.6.3 Equations of Motion in NormalCoordinates

Betti’s reciprocal work theorem can be usedto develop two orthogonality properties ofvibration mode shapes which make it possibleto greatly simplify the equations of motion. Thefirst of these states that the mode shapes areorthogonal to the mass matrix and is expressedin matrix form as

)(}0{}]{[}{ nmM mT

n ≠=φφ (4-101)

Using Equations 4-99 and 4-101, the secondproperty can be expressed in terms of thestiffness matrix as

)(}0{}]{[}{ nmK mT

n ≠=φφ (4-102)

which states that the mode shapes areorthogonal to the stiffness matrix. It is furtherassumed that the mode shapes are alsoorthogonal to the damping matrix:

)(}0{}]{[}{ nmC mT

n ≠=φφ (4-103)

Sufficient conditions for this assumptionhave been discussed elsewhere.(4-8) Since anyMDOF system having N degrees of freedomalso has N independent vibration mode shapes,it is possible to express the displaced shape ofthe structure in terms of the amplitudes of theseshapes by treating them as generalizedcoordinates (sometimes called normalcoordinates). Hence the displacement at aparticular location, vi, can be obtained bysumming the contributions from each mode as

∑=

φ=N

nnini Yv

1

(4-104)

In a similar manner, the completedisplacement vector can be expressed as

226 Chapter 4

}]{[}{}{1

YYvN

nnn Φ=φ= ∑

=

(4-105)

It is convenient to write the equations ofmotion for a MDOF system in matrix form as

)}({}]{[}]{[}]{[ tPvKvCvM =++ &&& (4-106)

which is similar to the equation for a SDOFsystem, Equation 4-9. The differences arisebecause the mass, damping, and stiffness arenow represented by matrices of coefficientsrepresenting the added degrees of freedom, andthe acceleration, velocity, displacement, andapplied load are represented by vectorscontaining the additional degrees of freedom.The equations of motion can be expressed interms of the normal coordinates by substitutingEquation 4-105 and its appropriate derivativesinto Equation 4-106 to give

)}({}]{][[}]{][[}]{][[ tPYKYCYM =Φ+Φ+Φ &&&

(4-107)

Multiplying the above equation by thetranspose of any modal vector {φn} results inthe following:

)}({}{}]{][[}{

}]{][[}{}]{][[}{

tPTnYKT

n

YCTnYMT

n

φ=Φφ+

Φφ+Φφ &&&(4-108)

Using the orthogonality conditions ofEquations 4-101, 4-102, and 4-103 reduces thisset of equations to the equation of motion for ageneralized SDOF system in terms of thegeneralized properties for the n th mode shapeand the normal coordinate Yn:

)(**** tPYKYCYM nnnnnn =++ &&& (4-109)

where the generalized properties for the nthmode are given as

)}({}{loading dgeneralize)(

}]{[}{

stiffness dgeneralize

2}]{[}{

damping dgeneralize

}]{[}{mass dgeneralize

n*

*2

*

*n

*

n*

tPtP

MK

K

MC

C

MM

Tn

nnnT

n

n

nnnnT

n

nT

n

φ==

ω=φφ=

=

ωλ=φφ=

=

φφ==

(4-110)

The above relations can be used to furthersimplify the equation of motion for the nthmode to the form

*

*2 )(

2n

nnnnnnn M

tPYYY =ω+ωλ+&& (4-111)

The importance of the abovetransformations to normal coordinates has beensummarized by Clough and Penzien,(4-4) whostate that

The use of normal coordinates serves totransform the equations of motion from a set ofN simultaneous differential equations which arecoupled by off diagonal terms in the mass andstiffness matrices to a set of N independentnormal coordinate equations.

It should further be noted that theexpressions for the generalized properties ofany mode are equivalent to those definedpreviously for a generalized SDOF system.Hence the use of the normal modes transformsthe MDOF system having N degrees of freedominto a system of N independent generalizedSDOF systems. The complete solution for thesystem is then obtained by superimposing theindependent modal solutions. For this reasonthis method is often referred to as the modal-superposition method. Use of this method alsoleads to a significant saving in computationaleffort, since in most cases it will not benecessary to use all N modal responses toaccurately represent the response of thestructure. For most structural systems the lowermodes make the primary contribution to thetotal response. Therefore, the response can


usually be represented to sufficient accuracy interms of a limited number of modal responsesin the lower modes.

4.6.4 Earthquake-Response Analysis

Time-History Analysis As in the case ofSDOF systems, for earthquake analysis thetime-dependent force must be replaced with theeffective loads, which are given by the productof the mass at any level, M, and the groundacceleration g(t). The vector of effective loadsis obtained as the product of the mass matrixand the ground acceleration:

)(}]{[)( tgMtPe &&Γ= (4-112)

where {Γ} is a vector of influence coefficientsof which component i represents theacceleration at displacement coordinate i due toa unit ground acceleration at the base. For thesimple structural model in which the degrees offreedom are represented by the horizontaldisplacements of the story levels, the vector{Γ} becomes a unity vector, {1}, since for aunit ground acceleration in the horizontaldirection all degrees of freedom have a unithorizontal acceleration. Using Equation 4-108,the generalized effective load for the nth modeis given as

)()(* tgtP nen L= (4-113)

Where }]{[}{ Γφ= MTnnL

Substituting Equation 4-113 into Equation4-111 results in the following expression for theearthquake response of the nth mode of aMDOF system:

*2 /)(2 nnnnnnnn MtgYYY &&&&& ϕ=ω+ωλ+ (4-114)

In a manner similar to that used for theSDOF system, the response of this mode at any

time t can be obtained by the Duhamel integralexpression

nn

nnn M

tVtY

ωϕ=

*

)()( (4-115)

where Vn(t) represents the integral

∫ ττ−ωτ= τ−ωλ−t

nt

n dtegtV nn

0

)( )(sin)()( && (4-116)

The complete displacement of the structureat any time is then obtained by superimposingthe contributions of the individual modes usingEquation 4-105

∑=

Φ=φ=N

nnn tYtYtv

1

)}(]{[)(}{)}({ (4-117)

The resulting earthquake forces can bedetermined in terms of the effectiveaccelerations, which for each mode are givenby the product of the circular frequency and thedisplacement amplitude of the generalizedcoordinate:

*2 )(

)()(n

nnnnnne M

tVtYtY

ωϕ=ω=&& (4-118)

The corresponding acceleration in thestructure due to the n th mode is given as

)(}{)}({ tYtv nenne&&&& φ= (4-119)

and the corresponding effective earthquakeforce is given as

*/)(}]{[

)}(]{[)}({

nnnnn

nn

MtVM

tvMtq

ϕωφ=

= &&(4-120)

The total earthquake force is obtained bysuperimposing the individual modal forces toobtain

228 Chapter 4

∑=

ωΦ==N

nn tYMtqtq

1

2 )(]][[)()( (4-121)

The base shear can be obtained by summingthe effective earthquake forces over the heightof the structure:

)(

)}({}1{)()(1

tVM

tqtqtQ

nnen

H

in

Tinn

ω=

== ∑= (4-122)

where *2 / nnen MM L= is the effective mass for

the nth mode.The sum of the effective masses for all of

the modes is equal to the total mass of thestructure. This results in a means ofdetermining the number of modal responsesnecessary to accurately represent the overallstructural response. If the total response is to berepresented in terms of a finite number ofmodes and if the sum of the correspondingmodal masses is greater than a predefinedpercentage of the total mass, the number ofmodes considered in the analysis is adequate. Ifthis is not the case, additional modes need to beconsidered. The base shear for the nth mode,Equation 4-122, can also be expressed in termsof the effective weight,Wen, as

)()( tVg

WtQ nn

enn ω= (4-123)

where

( )∑∑

=

=

φ

φ=

H

i ini

H

i ini

enW

WW

1

2

2

1 (4-124)

The base shear can be distributed over theheight of the building in a manner similar toEquation 4-74, with the modal earthquakeforces expressed as

n

nnn

tQMtq

L

)(}]{[)}({

φ= (4-125)

4.6.5 Response-Spectrum Analysis

The above equations for the response of anymode of vibration are exactly equivalent to theexpressions developed for the generalizedSDOF system. Therefore, the maximumresponse of any mode can be obtained in amanner similar to that used for the generalizedSDOF system. By analogy to Equations 4-34and 4-43 the maximum modal displacement canbe written as

dnn

nn S

tVtY =

ω= max

max

)()( (4-126)

Making this substitution in Equation 4-115results in

*max / ndnnn MSY ϕ= (4-127)

The distribution of the modal displacementsin the structure can be obtained by multiplyingthis expression by the modal vector

*maxmax

}{}{}{

n

dnnnnnn M

SYv

Lφ=φ= (4-128)

The maximum effective earthquake forcescan be obtained from the modal accelerations asgiven by Equation 4-120:

*max

}]{[}{

n

pannnn M

SMq

ϕφ= (4-129)

Summing these forces over the height of thestructure gives the following expression for themaximum base shear due to the nth mode:

*2max / npannn MSQ ϕ= (4-130)


which can also be expressed in terms of theeffective weight as

gSWQ panenn /max = (4-131)

where Wen is defined by Equation 4-124.Finally, the overturning moment at the base of

the building for the nth mode can bedetermined as

[ ]{ } */ npannno MSMhM Lφ= (4-132)

where h is a row vector of the story heights

above the base.

4.6.6 Modal Combinations

Using the response-spectrum method forMDOF systems, the maximum modal responseis obtained for each mode of a set of modes,which are used to represent the response. Thequestion then arises as to how these modalmaxima should be combined in order to get thebest estimate of the maximum total response.The modal-response equations such asEquations 4-117 and 4-121 provide accurateresults only as long as they are evaluatedconcurrently in time. In going to the response-spectrum approach, time is taken out of theseequations and replaced with the modal maxima.These maximum response values for theindividual modes cannot possibly occur at thesame time; therefore, a means must be found tocombine the modal maxima in such a way as toapproximate the maximum total response. Onesuch combination that has been used is to takethe sum of the absolute values (SAV) of themodal responses. This combination can beexpressed as

∑=

≤N

nnrr

1

(4-133)

Since this combination assumes that themaxima occur at the same time and that theyalso have the same sign, it produces an upper-bound estimate for the response, which is tooconservative for design application. A morereasonable estimate, which is based onprobability theory, can be obtained by using thesquare-root-of-the-sum-of-the-squares (SRSS)method, which is expressed as

∑=

≈N

nnrr

1

2 (4-134)

This method of combination has been shownto give a good approximation of the responsefor two-dimensional structural systems. Forthree-dimensional systems, it has been shownthat the complete-quadratic-combination (CQC)method (4-9) may offer a significantimprovement in estimating the response ofcertain structural systems. The completequadratic combination is expressed as

∑∑= =

≈N

i

N

jjiji rprr

1 1

(4-135)

where for constant modal damping

2222

2/32

)1(4)1(

)1(8

ζ+ζλ+ζ−ζζ+λ=ijp (4-136)

and

cr

ij

cc /

/

=λ

ωω=ζ

Using the SRSS method for two-dimensional systems and the CQC method foreither two- or three-dimensional systems willgive a good approximation to the maximumearthquake response of an elastic systemwithout requiring a complete time-historyanalysis. This is particularly important forpurposes of design.

230 Chapter 4

Example 4-9 (Response Spectrum Analysis)Use the design response spectrum given in

Example 4-7 and the results of Example 4-8 toperform a response-spectrum analysis of thereinforced concrete frame. Determine the modalresponses of the four modes of vibration, andestimate the total response using the SAV,SRSS, and CQC methods of modalcombination. Present the data in a tabular formsuitable for hand calculation. Finally, comparethe results with those obtained in Example 4-6for a generalized SDOF model.

From Example 4-7,

=

16.1000

015.100

0015.10

00001.1

4

1][M

sec

80.50

39.40

77.25

44.8

}{r

=ω

Table 3-3. Computation of response for model of Example 4-8Modal Modal ResponseParam

etern = 1 2 3 4

ω = 8.44 25.77 40.39 50.80αn = 1.212 -0.289 0.075 0.010Sd = 1.190 0.155 0.062 0.039

1.00 1.00 1.00 1.000.91 0.20 -1.07 -1.78

Response 0.74 -0.78 -0.75 1.75 Combined ResponseQuantity

φ =

0.47 -1.05 1.24 -0.92 SAV SRSS CQCDisplacement n = 4 1.44 -0.045 0.019 -0.002 1.506 1.441 1.441vn=φnαnSdn 3 1.31 -0.009 -0.020 0.003 1.342 1.310 1.310(Eq.3.128) 2 1.07 0.035 -0.014 -0.003 1.122 1.071 1.071

1 0.68 0.047 0.023 0.001 0.751 0.682 0.682

Acceleration n= 4 102.6 -29.9 31.0 -5.1 168.6 111.4 110.7

nvnnv 2ω=&& 3 93.3 -6.0 -32.6 7.7 139.6 99.3 98.9

2 76.2 23.2 -22.8 -7.7 129.9 83.2 83.31 48.4 31.2 37.5 2.6 119.7 68.8 70.0

Inertia force n = 4 25.91 -7.54 7.83 -1.30 42.6 28.1 27.9

nvMnq &&= 3 26.82 -1.72 -9.38 2.23 40.2 28.6 28.4

2 21.91 6.68 -6.56 -2.23 37.4 23.9 23.91 14.03 9.05 11.35 0.75 35.2 20.2 20.6

Shear n= 4 25.91 -7.54 7.83 -1.30 42.6 28.1 28.0Qn=Σqn 3 52.73 -9.26 -1.55 0.93 64.5 53.6 53.5

2 74.64 -2.58 -8.11 -1.30 86.6 75.1 75.11 88.67 6.47 3.24 -0.55 98.9 89.0 89.0

Overturning n= 4 272.1 -79.2 82.2 -13.7 447.2 295.4 293.6Moment 3 825.7 -176.4 65.9 -3.9 1071.9 846.9 845.3(ft-kips) 2 1609.4 -203.5 -19.2 -17.5 1849.6 1622.4 1621.3

1 2673.4 -125.9 19.7 -24.1 2843.1 2676.5 2675.7


nvndn

v

SS

S

f

ω=

=

=π

ω=

−−−−

−−=Φ

/

in./sec

0.2

5.2

0.4

0.10

Hz

09.8

43.6

10.4

34.1

2}{

92.024.105.147.0

75.175.078.074.0

78.107.120.091.0

00.100.100.100.1

][

From Equation 4-128,

∑=

=

ω==

αφ=ϕφ=

N

inin

nnn

dndnnnnn

qQ

vMvMq

SSMv

1

2

*max

}{][}]{[}{

}{)/}({}{

&&

For CQC combination,05.0=λ = constant for all modes

=

0000.11582.00193.00017.0

1582.00000.10452.00025.0

0193.00452.00000.10062.0

0017.0025.0062.00000.1

ijp

The computation of the modal and thecombined response is tabulated in Table 4-3. The results are compared with thoseobtained for the SDOF model in Table 4-4.

Table 4-4. Comparison of results obtained from MDOFand SDOF models.

Responseparameter

MDOF(Example 3-9)

SDOF(Example 3-7)

Period (sec) 0.744 0.721Displacements(in)Roof3rd

2nd

1.441.311.07

1.171.040.80

Responseparameter

MDOF(Example 3-9)

SDOF(Example 3-7)

1st 0.68 0.50Inertia force (kips)Roof3rd

2nd

1st

28.128.623.920.2

27.127.421.113.2

Base shear (kips) 89.0 88.8Overturningmoment (ft-kips)

2678 2716

4.7 NONLINEAR RESPONSEOF MDOF SYSTEMS

The nonlinear analysis of buildings modeledas multiple degree of freedom systems (MDOF)closely parallels the development for singledegree of freedom systems presented earlier.However, the nonlinear dynamic time historyanalysis of MDOF systems is currentlyconsidered to be too complex for general use.Therefore, recent developments in the seismicevaluation of buildings have suggested aperformance-based procedure which requiresthe determination of the demand and capacity.Demand is represented by the earthquakeground motion and its effect on a particularstructural system. Capacity is the structure'sability to resist the seismic demand. In order toestimate the structure's capacity beyond theelastic limit, a static nonlinear (pushover)analysis is recommended (4-17). For moredemanding investigations of building response,nonlinear dynamic analyses can be conducted.

For dynamic analysis the loading timehistory is divided into a number of small timeincrements, whereas, in the static analysis, thelateral force is divided into a number of smallforce increments. During a small time or forceincrement, the behavior of the structure isassumed to be linear elastic. As nonlinearbehavior occurs, the incremental stiffness ismodified for the next time (load) increment.Hence, the response of the nonlinear system isapproximated by the response of a sequentialseries of linear systems having varyingstiffnesses.

232 Chapter 4

Static Nonlinear AnalysisNonlinear static analyses are a subset of

nonlinear dynamic analyses and can use thesame solution procedure without the timerelated inertia forces and damping forces. Theequations of equilibrium are similar to Equation4-1 with the exception that they are written inmatrix form for a small load increment duringwhich the behavior is assumed to be linearelastic.

}{}]{[ PvK ∆=∆ (4-136a)

For computational purposes it is convenientto rewrite this equation in the following form

}{}{}]{[ PRvK tt =+∆ (4-136b)

where Kt is the tangent stiffness matrix forthe current load increment and Rt is therestoring force at the beginning of the loadincrement which is defined as

∑−

=

∆=1

1

}]{[}{n

iitit vKR

Figure 4-29a. Pushover Curve, Six Story Steel Building.

The lateral force distribution is generallybased on the static equivalent lateral forcesspecified in building codes which tend to

approximate the first mode of vibration. Theseforces are increased in a proportional mannerby a specified load factor. The lateral loading isincreased until either the structure becomesunstable or a specified limit condition isattained. The results from this type of analysisare usually presented in the form of a graphplotting base shear versus roof displacement.The pushover curve for a six-story steelbuilding (4-18) is shown in Figure 4-29a and thesequence of plastic hinging is shown in Figure4-29b.

Figure 4-29b Sequence of Plastic Hinge Formation, SixStory Steel Building.

The equations of equilibrium for a multipledegree of freedom system subjected to baseexcitation can be written in matrix form as

)(}]{[

}]{[}]{[}]{[

tgM

vKvCvM

&&

&&&

Γ−=++

(Eq.4-137)

This equation is of the same form as that ofEq. 4-76 for the single degree of freedomsystem. The acceleration, velocity anddisplacement have been replaced by vectorscontaining the additional degrees of freedom.The mass has been replaced by the mass matrixwhich for a lumped mass system is a diagonalmatrix with the translational mass androtational mass terms on the main diagonal. Theincremental stiffness has been replaced by theincremental stiffness matrix and the dampinghas been replaced by the damping matrix. This


latter term requires some additional discussion.In the mode superposition method, the dampingratio was defined for each mode of vibration.However, this is not possible for a nonlinearsystem because it has no true vibration modes.A useful way to define the damping matrix for anonlinear system is to assume that it can berepresented as a linear combination of the massand stiffness matrices of the initial elasticsystem

][][][ KMC β+α= (Eq 4-138)

Where α and β are scaler multipliers whichmay be selected so as to provide a givenpercentage of critical damping in any twomodes of vibration of the initial elastic system.These two multipliers can be evaluated from theexpression

λλ

ω−ωωω

ωω−

ω−ω=

βα

j

i

ij

ji

ij

ij

22112 (Eq.4-139)

where ωi and ωj are the percent of criticaldamping in the two specified modes. Once thecoefficients α and β are determined, thedamping in the other elastic modes is obtainedfrom the expression

22k

kk

βω+ωα=λ (Eq. 4-140)

A typical damping function which was usedfor the nonlinear analysis of a reinforcedconcrete frame (4-10) is shown in Figure 4-30.Although the representation for the damping isonly approximate it is justified for these typesof analyses on the basis that it gives a goodapproximation of the damping for a range ofmodes of vibration and these modes can beselected to be the ones that make the majorcontribution to the response. Also in nonlineardynamic analyses the dissipation of energythrough inelastic deformation tends ofovershadow the dissipation of energy through

viscous damping. Therefore, an exactexpresentation of damping is not as importantin a nonlinear system as it is in a linear system.One should be aware of the characteristics ofthe damping function to insure that importantcomponents of the response are not lost. Forinstance, if the coefficients are selected to givea desired percentage of critical damping in thelower modes and the response of the highermodes is important, the higher mode responsemay be over damped and its contribution to thetotal response diminished.

Figure 4-30. Damping functions for a framed tube.

Substituting Eq. 4-138 into Eq. 4-137 resultsin

)(}]{[

}]{[}]{[}]{[}]{[

tgM

vKvKvMvM i

&&

&&&&

Γ−=+β+α+

(Eq. 4-141)

where Ki refers to the initial stiffness.

234 Chapter 4

Representing the incremental stiffness interms of the tangent stiffness, Kt, andrearranging some terms, results in

}]{[}{}]{[}]{[ vKRvKvK ttt ∆+=∆= (Eq. 4-142)

where

∑−

=

∆=1

1

}]{[}{n

iitit vKR

Using the step-by-step integration procedurein which the acceleration is assumed to beconstant during a time increment, equationssimilar to Eqs. 4.84 and 4-86 can be developedfor the multiple degree of freedom systemwhich express the acceleration and velocityvectors at the end of the time increment interms of the incremental displacement vectorand the vectors of initial conditions at thebeginning of the time increment:

}{}){4

()}({2 tAv

ttv +∆

∆=&& (Eq. 4-143)

}{}){2

()}({ tBvt

tv +∆∆

=& (Eq. 4-144)

}{)}({)}({ vttvtv ∆+∆−= (Eq. 4-144a)

where

)}({)}({4

}{ ttvttvt

At ∆−−∆−∆

−= &&& (Eq. 4-145)

)}({}{ ttvBt ∆−−= & (Eq. 4-146)

Substituting Eqs. 4-142 through 4-146 intoEq. 4-141 and rearranging some terms leads tothe pseudo-static form

}~

{}]{~

[ PvK =∆ (Eq. 4-147)

where

[ ]

{ }

tC

ttC

BKBAM

RtPP

KKCMCK

titt

t

ti

∆=

∆+

∆=

−+−−=

++=

β

αβα

2

24

}]{[}{}{][

}{)}({}~

{

][][][]~

[

1

20

10

The incremental displacement vector can beobtained by solving Eq. 4-147 for {∆v} Thisresult can then be used in Eqs. 4-143, 4-144 and4-144a to obtain the acceleration vector, thevelocity vector and the displacement vector atthe end of the time interval. These vectors thenbecome the initial conditions for the next timeinterval and the process is repeated.

Output from a nonlinear response analysisof a MDOF system generally includes responseparameters such as the following: an envelopeof the maximum story displacements, anenvelope of the maximum relative storydisplacement divided by the story height(sometimes referred to as the interstory driftindex (IDI), an envelope of maximum ductilitydemand on structural members such as beams,columns, walls and bracing, an envelope ofmaximum rotation demand at the ends ofmembers, an envelope of the maximum storyshear, time history of base shear, momentversus rotation hysteresis plots for criticalplastic hinges, time history plots of storydisplacements and time history plots of energydemands (input energy, hysteretic energy,kinetic energy and dissapative energy).

For multiple degree of freedom systems, thedefinition of ductility is not as straight-forwardas it was for the single degree of freedomsystems. Ductility may be expressed in terms ofsuch parameters as displacement, relativedisplacement, rotation, curvature or strain.Example 4-10.Seismic Response Analyses

The following is a representative responseanalysis for a six story building in whichthe lateral resistance is provided by momentresistant steel frames on the perimeter. The


structure has a rectangular plan with typicaldimensions of 48822 ′×′ as shown inFigure 4-31. The building was designed forthe requirements of the 1979 Edition of theUniform Building Code (UBC) with theseismic load based on the use of staticequivalent lateral forces.

Elastic AnalysesAs a first step in performing the analyses,

the members of the perimeter frame will bestress checked for the design loading conditionsand the dynamic properties of the building willbe determined. This will help to insure that theanalytical model of the building is correct andthat the gravity loading which will be used forthe nonlinear response analysis is alsoreasonable. This will be done using a threedimensional model of the lateral force systemand the ETABS (4-11) computer program. Thisprogram is widely used on the west coast forseismic analysis and design of buildingsystems. An isometric view of the perimeterframe including the gravity load is shown inFigure 4-32. The location of the concentratedand distributed loads depends upon the framingsystem shown in Figure 4-31.

Using the post-processor programSTEELER (4-12), the lateral force system is stresschecked using the AISC-ASD criteria. Thestress ratio is calculated as the ratio of theactual stress in the member to the allowablestress. Applying the gravity loads incombination with the static equivalent lateralforces in the transverse direction produces thestress ratios shown in Figure 4-33. This resultincludes the effect of an accidental eccentricitywhich is 5% of the plan dimension. Themaximum stress ratio in the columns is 0.71and the maximum in the beams is 0.92. Thesevalues are reasonable based on standardpractice at the time the building was designed.Ideally, the stress ratio should be just less thanone, however, this is not always possible due tothe finite number of steel sections that areavailable.Modal analyses indicate that the first threelateral modes of vibration in each direction

represent more than 90% of the participatingmass. In the transverse direction, these modeshave periods of vibration of 1.6, 0.6 and 0.35seconds. In the longitudinal direction, theperiods are slightly shorter.

Dynamic analyses are conducted using thesame analytical model and considering anensemble of five earthquake ground motionsrecorded during the Northridge earthquake. Arepresentative time history of one of thesemotions is shown in Figure 4-34. Thecorresponding stress ratios in the perimeterframe are shown in Figure 4-35 for earthquakemotion applied in the transverse direction.Stress ratios in the beams of the transverseframes range from 2.67 to 4.11 indicatingsubstantial inelastic behavior. Stress ratios inexcess of 1.12 are obtained in all of thecolumns of the transverse frames, however, itshould be recalled that there is a factor of safetyof approximately 1.4 on allowable stress andplastic hinging.

Nonlinear AnalysesIn order to estimate the lateral resistance of

the building at ultimate load, a static, nonlinearanalysis (pushover) is conducted forproportional loading. The reference lateral loaddistribution is that specified in the 1979 UBC.This load distribution is then multiplied by aload factor to obtain the ultimate load. Thenonlinear model is a two dimensional model inwhich the plasticity is assumed to beconcentrated in plastic hinges at the ends of themembers.

The results of the pushover analysis areusually represented in terms of a plot of the roofdisplacement versus the base shear as shown inFigure 4-36. This figure indicates that firstyielding occurs at a base shear of approximately670 kips and a roof displacement ofapproximately 7.25 inches. The UBC 1979static equivalent lateral forces for this frameresults in a base shear of 439 kips whichimplies a load factor of 1.52 on first yield. At aroof displacement of 17.5 inches, a swaymechanism forms with all girders hinged and

Figure 4-31 Typical floor framing plan ~ Fourth & fifth floors

Figure 4-32. Gravity Loading Pattern, ETABS

Figure 4-33. Calculated Stress Ratios, Design Loads, ETABS

Figure 4-34. Recorded Base Acceleration, Sta. 322, N-S

240 Chapter 4

Figure 4-35. Calculated Stress Ratios, Sta. 322 Ground Motion

Figure 4-36. Static Pushover Curve

hinges at the base of the columns. At thisdisplacement, the pushover curve is becomingalmost horizontal indicating a loss of most ofthe lateral stiffness. This behavior ischaracterized by a large increase indisplacement for a small increase in lateral loadsince lateral resistance is only due to strainhardening in the plastic hinges. The ultimateload is taken as 840 kips which divided by thecode base shear for the frame (439 kips) resultsin a load factor of 1.91 on ultimate.

Note that the elastic dynamic analysis forthe acceleration shown in Figure 4-34 results ina displacement at the roof of 16.7 inches.Comparing this to the pushover curve (Figure4-36) indicates that the structure should be wellinto the inelastic range based on thedisplacement response.

Figure 4-37. Calculated Nonlinear Dynamic Response.

242 Chapter 4

The nonlinear dynamic response of astructure is often presented in terms of thefollowing response parameters: (1) envelope ofmaximum total displacement, (2) envelopeof maximum story to story displacementdivided by the story height (interstory driftindex), (3) maximum ductility demand for thebeams and columns, (4) envelopes of maximumplastic hinge rotation, (5) moment versusrotation hysteresis curves for critical membersand (6) envelopes of maximum story shear.Representative plots of four of these parametersare shown in Figure 4-37. The lateraldisplacement envelope (Figure 4-37a) indicatesthat the maximum displacement at the rooflevel is 12.3 inches which is less than the 16.7inches obtained from the elastic dynamicanalysis. The interstory drift and total beamrotation curves are shown in Figure 4- 37bwhich indicates that the interstory drift rangesfrom 0.01 (1%) to 0.024 (2.4%). The beamrotation can be seen to range between 0.016 and0.025. The curvature ductility demands of thebeams and columns is shown in Figure 4-37c.

The maximum ductility demand for thecolumns is 1.8 and for the beams it is 3.3. Thehysteretic behavior of a plastic hinge in acritical beam is shown in a plot of momentversus rotation in Figure 4- 37d.

A final plot, Figure 4-38, shows thenonlinear displacement time history of the roof.This figure illustrates the displacement of apulse type of input. After some lessor cyclesduring the first 7 seconds of the time history,the structure sustains a strong displacement atapproximately 8 seconds which drives the roofto a displacement of 12 inches relative to thebase. Note the acceleration pulse at this time inthe acceleration time history (Figure 4-34).Following this action, the structure begins tooscillate about a new, deformed position at fourinches displacement. This is a residualdisplacement, which the structure will havefollowing the earthquake and is characteristic ofinelastic behavior. Additional details of thisanalysis example can be found in the literature(4-13).

Figure4-38. Nonlinear Displacement, Roof Level


4.8 VERIFICATION OFCALCULATED RESPONSE

The dynamic response procedures discussedin the previous sections must have the ability toreliably predict the dynamic behavior ofstructures when they are subjected to criticalseismic excitations. Hence, it is necessary tocompare the results of analytical calculationswith the results of large-scale experiments. Thebest large-scale experiment is when anearthquake occurs and properly placedinstruments record the response of the buildingto ground motions recorded at the base. Theinstrumentation (accelerometers) placed in asix-story reinforced concrete building by theCalifornia Strong Motion InstrumentationProgram (CSMIP) is indicated in Figure 4-39.The lateral force framing system for the

building, shown in Figure 4-40, indicates thatthere are three moment frames in the transverse(E-W) direction and two moment frames in thelongitudinal (N-S) direction. Note that thetransverse frames at the ends of the building arenot continuous with the longitudinal frames. Itis assumed that the floor diaphragms are rigidin their own plane. During the Loma Prietaearthquake the instrumentation recordedthirteen excellent records of building responsehaving a duration of more than sixty seconds (4-

19). Since the response was only weaklynonlinear, the calculations can be made usingthe ETABS program, however, similar analysescan also be conducted with a nonlinear responseprogram (4-20).

Figure 4-39. Location of Strong Motion Instrumentation

244 Chapter 4

Figure 4-40. ETABS Building Model

To improve the evaluation of the recordedresponse, spectral analyses are conducted inboth the time domain (response spectra) andfrequency domain (Fourier spectra). A furtherrefinement of the Fourier analysis can beattained by calculating a Fourier amplitudespectra for a segment (window) of the recordedtime history. The fixed duration window is thenshifted along the time axis and the process is

repeated until the end of the time history record.This results in a “moving window Fourieramplitude spectra” (MWFAS) which indicatesthe changes in period of the building responseduring the time history as shown in Figure 4-41.In this example a ten-second window was usedwith a five-second shift for the first sixtyseconds of the recorded response. In general,the length of the “window” should be at least2.5 times the fundamental period of thestructure.

If the connections (offsets) are assumed to berigid, the initial stiffness of the building prior toany cracking of the concrete can be estimatedusing the analytical model with memberproperties of the gross sections. This results in aperiod of 0.71 seconds in the E-W direction and0.58 seconds in the N-S direction. Thiscondition can also be evaluated by the resultsobtained from the initial window of theMWFAS. An examination of Figure 4-41indicates an initial period of 0.71 in the E-Wdirection and 0.58 seconds in the N-S direction.Identical results were also obtained fromambient vibration tests conducted by Marshall,et al. (4-21).

Figure 41. Variation of Building Period with Time


During the strong motion portion of theresponse, cracking in the concrete and limitedyielding of the tension steel will cause theperiod of vibration to lengthen. In order torepresent this increased flexibility in the elasticanalytical model, the flexibility of theindividual members can be reduced to aneffective value or the rigid offsets at theconnections (4-13) can be reduced in length. Forthis example, the rigid offsets were reduced byfifty percent. This results in a period of 1.03seconds in the E-W direction and 0.89 secondsin the N-S direction which are in the range ofvalues obtained from the MWFAS. Consideringthe entire duration of the recorded response, theFourier amplitude spectra indicates a period of1.05 seconds in the E-W direction and 0.85 inthe N-S direction. Corresponding valuesobtained from a response spectrum analysis

indicate 1.0 E-W and 0.90 N-S. It can beconcluded that for this building, all of thesevalues are in good agreement. The MWFASalso indicate an increase in period ofapproximately fifty percent in both principaldirections during the earthquake. This amountof change is not unusual for a reinforcedconcrete building (4-22), however, it does indicatecracking and possible limited yielding of thereinforcement. The time histories of theacceleration and displacement at the roof levelare shown in Figure 4-42. This also shows agood correlation between the measured and thecalculated response.

Figure 4-42. Time History Comparisons of Acceleration, Displacement

246 Chapter 4

REFERENCES

4-1 Chopra, A, K., Dynamics of Structures, A Primer,Earthquake Engineering Research institute,Berkeley, CA, 1981.

4-2 Applied Technology Council, An Evaluation of aResponse Spectrum Approach to Seismic Design ofBuildings, ATC-2, Applied Technology Council,Palo Alto, CA, 1974.

4-3 Newmark, N. M., and Hall, W. J., "Procedures andCriteria for Earthquake Resistant Design", BuildingPractices for Disaster Mitigation, U. S. Departmentof Commerce, Building Science Series 46, 1973.

4-4 Clough, R. W., and Penzien, J., StructuralDynamics, McGraw-Hill, Inc., New York, NY,1975.

4-5 Rayleigh, Lord, Theory of Sound, DoverPublications, New York, NY, 1945.

4-6 Lee, V. W., and Trifunac, M. D., "StrongEarthquake Ground Motion Data in EQINFOS, PartI," Report No.87-01, Department of CivilEngineering, USC, Los Angeles, CA, 1987.

4-7 Bathe, K. J., and Wilson, E. L., Numerical Methodsin Finite Element Analysis, Prentice Hall, Inc.,Englewood Cliffs, New Jersey, 1976.

4-8 Caughy,T. K., "Classical Normal Modes in DampedLinear Dynamic systems," Journal of AppliedMechanics, ASME, Paper No. 59-A-62, June, 1960.

4-9 Wilson, E. L., Der Kiureghian, A.Rand Bayo, E. P.,"A Replacement for the SRSS Method in SeismicAnalysis", Earthquake Engineering and StructuralDynamics, Vol. 9, 1981.

4-10 Anderson, J. C., and Gurfinkel, G., "SeismicBehavior of Framed Tubes," International Journal ofEarthquake Engineering and Structural Dynamics,Vol. 4, No. 2, October-December, 1975.

4-11 Habibullah, A., "ETABS, Three DimensionalAnalysis of Building Systems," User's Manual,Version 6.0, Computers & Structures, Inc. Berkeley,California, 1994.

4-12 Habibullah, A., "ETABS Design Postprocessors,"Version 6.0, Computers & Structures, Inc. Berkeley,California, 1994.

4-13 Anderson, J. C., "Moment Frame Building",Buildings Case Study Project, SSC 94-06, SeismicSafety Commission, State of California,Sacramento, California, 1996.

4-14 Charney, F.A., NONLIN, Nonlinear Dynamic TimeHistory Analysis of Single Degree of FreedomSystems, Advanced Structural Concepts, Golden,Colorado, 1996.

4-15 Norris, C.H., Hansen, R.J., Holley, M.J., Biggs,J.M., Namyet, S., and Minami, J.K., StructuralDesign for Dynamic Loads, McGraw-Hill BookCompany, New York, New York, 1959.

4-16 U.S. Army Corps of Engineers, Design of Structuresto Resist the Effects of Atomic Weapons, EM 1110-345-415, 1957.

4-17 Applied Technology Council, “Seismic Evaluationand Retrofit of Concrete Buildings”, ATC-40,Applied Technology Council, Redwood City,California, 1991.

4-18 Anderson, J.C. and Bertero, V.V., “SeismicPerformance of an Instrumented Six Story SteelBuilding”, Report No. UCB/EERC-91/111,Earthquake Engineering Research Center,University of California at Berkeley, Berkeley,California, 1991.

4-19 California Division of Mines and Geology, “CSMIPStrong-Motion Records from the Santa CruzMountains (Loma Prieta) California Earthquake ofOctober 17, 1989”, Report OSMS 89-06.

4-20 Anderson, J.C. and Bertero, V.V., “SeismicPerformance of an Instrumented Six Story SteelBuilding”, Report No. UCB/EERC-93/01,Earthquake Engineering Research Center,University of California at Berkeley.

4-21 Marshall, R.D., Phan, L.T. and Celebi, M., “FullScale Measurement of Building Response toAmbient Vibration and the Loma PrietaEarthquake”, Proceedings, Fifth NationalConference on Earthquake Engineering, Vol. II,Earthquake Engineering Research Institute,Oakland, California, 1994.

4-22 Anderson, J.C., Miranda, E. and Bertero, V.V.,“Evaluation of the Seismic Performance of a Thirty-Story RC Building”, Report No. UCB-EERC-93/01,Earthquake Engineering Research Center,University of California, Berkeley.

chapter 04-dynamic response of buildings

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