Aschheim - 2002 - Seismic Design Based on the Yield Displacement

7/28/2019 Aschheim - 2002 - Seismic Design Based on the Yield Displacement

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Seismic Design Based on the YieldDisplacement

Mark Aschheim,a) M.EERI

Although seismic design traditionally has focused on period as a primary

design parameter, relatively simple arguments, examples, and observationsdiscussed herein suggest that the yield displacement is a more stable andmore useful parameter for seismic design. The stability of the yield displace-ment is illustrated with four detailed examples, consisting of moment-resistant frame buildings. Each frame is designed to limit roof drift for a spe-cific ground motion using an equivalent SDOF model in conjunction with

Yield Point Spectra. The effectiveness of the simple design method is estab-lished by nonlinear dynamic analysis. Yield displacements were stable andconsistent while the fundamental periods of vibration (and lateral stiffness)

required to meet the performance objective differed substantially.[DOI: 10.1193/1.1516754]

INTRODUCTION

Since the introduction of the response spectrum by Benioff (1934) and Biot (1941) ithas become second nature to think of the seismic design task in terms of the fundamen-tal period of vibration of a structure. Contemporary and traditional design approachesrely upon the fundamental period of vibration of the structure to determine the requiredstrength (or stiffness) for the design level seismic actions. This use of the fundamental

period of vibration is based on the premise that it may be estimated given the initialstructural concept and dimensions of the structure. This paper develops the idea that theyield displacement is a more natural and more useful parameter to use in the seismic

design of structures responding nonlinearly.The use of period as a fundamental design parameter draws directly from the equa-

tion of motion for linear elastic response, for which the peak displacement, Sd, is a func-tion of the period of vibration, T. Because T2(m/k)0.5, structures of varied heights(Figure 1) having the same ratio of mass, m, and stiffness, k, all have the same peakdisplacement. The period of vibration is useful for characterizing the peak response ofan infinite variety of systems (such as those in Figure 1) whose response is linear elastic.

Although many formulas and design procedures are based on the idea that the period

of vibration of the structure can be estimated early in the design process, the reality isthat the period of vibration may vary significantly as the initial design concept is refinedinto the final product, particularly if the lateral strength and stiffness must be adjusted to

a) Mid-America Earthquake Center, University of Illinois, 2118 Newmark Laboratory, 205 N. Mathews, Urbana,

IL 61801

581

Earthquake Spectra, Volume 18, No. 4, pages 581600, November 2002; 2002, Earthquake Engineering Research Institute


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satisfy precisely defined performance objectives. The difficulty of accurately estimatingthe period of vibration of the final design is apparent in the work of Chopra and Goel(2000), who examined a large number of buildings in California.

Alternatives to the period of vibration are available for proportioning structures for

seismic actions. This paper describes a technique in which the yield displacement of thestructure is used to determine the base shear strength required to satisfy one or more

performance objectives. In this design approach, the period of the structure is seen to bea consequence of decisions made in the design process to achieve a desired seismic per-formance; its value is unknown at the beginning of the design process and is not needed.

In principle, either period or yield displacement may be used for design. Demands

can be represented as a function of either parameter, using traditional (period-based) re-sponse spectra or Yield Point Spectra. The yield displacement can be estimated accu-rately early in the design process and is relatively stable. While period-based approacheswere adequate in the past, the stability of the yield displacement is noteworthy now thatseismic performance expectations are being defined with greater precision (e.g., FEMA273, ATC 1997; SEAOC 1999).

DESIGN AND THE KINEMATICS OF YIELD

While analysis often is necessary for design, the design process is fundamentally dif-ferent from analysis. A structural system must be conceived before its properties may beassessed. In the usual case, the geometry of the structure is established by other design

professionals and is not easily modified. The structural engineer may have one or several

structural systems in mind. Materials to be used (e.g., steel and concrete) are largelydetermined by market forces, which dictate what may be obtained economically. Ap-

proximate member depths can be established early in the design process. Most of thesubsequent effort is directed at proportioning the members of the structural system tohave sufficient strength to satisfy the load combinations specified in the building code,sufficient stiffness to satisfy code provisions for interstory drifts, and appropriate details

Figure 1. The systems above all have the same period. Their response histories and peak dis-

placement responses are identical, provided that response is linear elastic.

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for structural integrity and ductile behavior. Member depths and spans usually deviate

little throughout the design effort from the values initially assumed, even as memberstrengths and stiffnesses are adjusted to satisfy code requirements. The design context issuch that the layout of the structure, member depths, and material properties envisionedat the start of the design process typically will change little in the development of thefinal design. The choices available to designers operating in this context are illustrated inthe following examples.

EXAMPLE 1: AXIAL RESPONSE OF A BAR

The first example illustrates the essence of the seismic design problem in a verysimple form, although the physical form may be unfamiliar. A member having a prede-termined length is to be designed to resist an earthquake, with the earthquake causing

only axial loads to develop over the length of the member (Figure 2). Mass is locatedonly at the top of the structure. Grade 50 steel (fy345 MPa) is to be used. The strainsthat result from inertial forces are uniform over the length of the member, resulting inthe displacements shown. This displacement profile is the only mode of deformation that

may develop and thus is the predominant mode of response. A capacity curve may bedetermined by nonlinear static analysis. Forces are applied proportional to the mass andthe displacement at the top of the member, resulting in the capacity curves shown inFigure 2 for different cross-sectional areas. The cross-sectional area of the member de-termines its strength and stiffness, but the yield displacement (y) is constant becausethe steel yield strain and predominant mode shape are independent of the strength pro-vided to the system. Note that the yield strain, yield displacement, and predominantmode shape are easily determined at the start of the design. The engineers latitude is in

selecting the cross-sectional area of the member to achieve acceptable performance. Ifthe engineer should choose to limit the peak ductility () and peak displacement (uy) responses, the required strength is easily determined using the Yield Point Spec-tra representation of demand, described later in this paper.

Figure 2. An axially excited bar has the force, strain, and displacement distributions shown.

Increasing the cross-sectional area of the bar increases its strength and stiffness; the yield dis-

placement of the bar depend on the yield strain of the material.

SEISMIC DESIGN BASED ON THE YIELD DISPLACEMENT 583


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EXAMPLE 2: FLEXURAL RESPONSE OF CANTILEVER BEAMS AND WALLS

The second example considers the flexural response of cantilevered steel and rein-forced concrete beams. Mass is assumed to be located only at the tips of the cantilevers.

Based on the moment diagram, overall yielding of the beams will commence when thesections closest to the support yield. For the steel beam, the yield curvature (Figure 3a)is given by yy /(d/2), where y=the yield strain of the steel and d=the depth of thesection. For the concrete beam (Figure 3b), yy /(dkd), where d=the depth to thetension steel and kd=the depth to the neutral axis. For both beams, the yield displace-ment is governed by yielding of the steel, and its value is given by yyL

2/3, whereL=cantilever span. It is apparent that the yield displacement is a function of the beamgeometry and the yield strain of the steel. For a steel beam of a given depth and grade of

steel, the yield displacement is independent of the strength of the beam. For the concretebeam, changes in longitudinal reinforcement have a small effect on the depth of the

compression zone, and this has a minor influence on the yield curvature and yield dis-placement. These observations are fundamental and have been reported by others (e.g.,Priestley 2000).

The previous discussion relating to the reinforced concrete beam also applies to acantilevered structural wall, shown in Figure 4. A lateral force applied to the top of the

Figure 3. Idealized moment-curvature response of (a) steel and (b) reinforced concrete sec-

tions, illustrating the relative stability of the yield curvature with changes in steel area, for con-

stant section depth.

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wall is associated with inertial forces that develop when the mass at the top of the wall

is excited. To calculate the displacement at the top of the wall when the reinforcement at

the base of the wall yields requires knowledge of the geometry of the wall and grade ofreinforcement. As for the reinforced concrete beam, a cantilever wall loaded at its tip hasyield displacement given by yyH

2/3, where Hthe height of the wall. The curvatureof the wall at first yielding of the extreme longitudinal reinforcement is given by yy /(Lwkd), where Lwthe length of the wall and kdthe depth of the neutral axis.For relatively light axial loads and low longitudinal steel ratios, the denominator oftenmay be estimated as about 0.8Lw . As the curvature increases, yielding spreads to inter-mediate reinforcing bars. A curvature representing this state may be estimated as y(1.8 to 2)y /Lw according to Priestley and Kowalsky (1998) and Paulay (2002) for

walls of rectangular cross section with axial loads less than about 0.15Ag fc , where Agthe gross cross-sectional area of the wall and fcthe concrete compressive strength.

Both expressions fory indicate that the yield curvature is a function of the overall di-

mensions of the wall and the yield strain of the steel, and is largely independent of theflexural strength provided to the wall.

Idealized capacity curves (plotting the shear force developed at the base of the wallversus the displacement at the top of the wall) are presented in Figure 4 for differentlongitudinal reinforcement contents. As in the previous examples, changing the strength

of the wall by changing the steel content has a significant effect on the strength andstiffness of the wall. Given the wall geometry and grade of reinforcing steel, the onlyoption available to the engineer is to determine the amount of longitudinal steel neces-sary to provide sufficient strength to the wall so that its seismic performance is accept-able. Yield Point Spectra are useful for this task if performance is indexed by the duc-tility and drift responses of the wall. The wall reinforcement and wall thickness may then

be determined. Minor changes in yield displacement associated with the final detailing

of the wall usually are not of significance.

The example of Figure 4 may be extended to include multiple floors. Now, the pre-dominant mode shape should reflect the development of lateral forces at multiple floorlevels. A Rayleigh approach may be used to identify the first mode shape. This modeshape depends on the distribution of strength (and stiffness) throughout the wall height,

Figure 4. Idealized response of a slender reinforced concrete structural wall.



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but is independent of the absolute strength of the wall. Experience suggests that approxi-mate mode shapes are adequate for design purposes, and the elastic deflected shape ob-tained by applying a typical code distribution of lateral force over the height of the

building often will suffice. Applying this approach with a triangular distribution of lat-eral forces (following the Uniform Building Code [ICBO 1997]) for cases in which massis uniformly distributed results in an estimate of the roof displacement at yield of

y11

40yH

2 (1)

where Hthe height of the wall. Using established procedures (e.g., ATC 40, ATC1996), the yield displacement of an equivalent single-degree-of-freedom (ESDOF)structure may be determined based on the deflected shape and mass distribution. Theyield curvature of the wall, the deflected shape, and the distribution of mass in the struc-ture can be estimated accurately at the start of the design and are insensitive to the

strength provided to the wall. This assumes that the distribution of strength and stiffness

provided to the structure results in a predominant mode of response that is consistentwith the deflected shape assumed in design. This requirement is not onerous; rather, it isnaturally satisfied if the engineer prescribes the way the structure will respond to anearthquake by making intelligent choices in the design of the structure.

As in the previous examples, the engineer may seek to determine the lateral strength

required to limit system ductility and drift responses to acceptable values. The requiredstrength may be determined using Yield Point Spectra based on an estimate of the yielddisplacement, as described by Tjhin et al. (2002). This estimate is based on quantitiesestablished at the start of the design (the height of the structure, the length (Lw) of thewall, the yield strain of the material, the distribution of mass, and the deflected shapeassociated with the predominant mode of response); the yield displacement will varylittle as the initial structural concept is refined into the final design.

EXAMPLE 3: MOMENT-RESISTANT FRAMES

A typical perimeter moment-resistant frame building will have relatively small tribu-tary gravity loads if located in a relatively high seismic region. This results in antisym-metric beam moment diagrams that place the largest flexural demands on the beamsnearest the columns, similar to the cantilever beams of Example 2. The elastic modeshape depends on the distribution of stiffness and mass throughout the structure. If thecolumn strengths are proportioned such that a predominantly beam-hinging mechanism

develops, then the displacement response usually is represented adequately by the firstelastic mode shape. The elastic mode shape is independent of the absolute values ofstrength and stiffness provided to the structure; only the relative distributions of strengthand stiffness matter, and these are the province of the engineer.

The yield displacement of the system is largely determined by yielding of the beams.The yield displacement of the beams depends on their spans, member depths, and ma-

terial properties. The displacement of the roof when the beams yield depends on the pre-dominant mode shape and is affected by the stiffness of the columns. As for the case ofstructural walls, sufficient precision may be obtained by assuming the predominant

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mode to be equal to the first elastic mode shape and then estimating this shape. Basedon fairly simple assumptions, one may derive an approximate expression for estimatingthe yield displacement of regular steel moment-resistant frames:

y

H

y

6 h

dcolCOF

2L

dbm(2)

where y=the estimated roof displacement at yield, H=the height of the building, y=theyield strain of the steel, L=the beam span, h=story height, dbm=the beam depth, dcol=thecolumn depth, and COF=the column overstrength factor. In general, the member andstory properties to be used in Equation 2 should be representative of the frame. The col-umn overstrength factor (COF) should be selected to ensure that a first-mode response is

dominant; that is, weak story mechanisms must be precluded. For typical configurations,the term associated with the beam flexibility is dominant, and variations of COF withinreasonable limits (perhaps 1.2 to 1.5) have little effect on the yield displacement esti-mate. Even if experience should suggest refinements to Equation 2, its position here

serves to emphasize that the yield displacement may be estimated based only on geom-etry, material properties, and simple parameters relating the strengths or stiffnesses ofthe beams and columns.

The remainder of the paper draws on empirical data focusing on the response ofmoment-resistant frames, although the underlying logic is fundamental and is more gen-erally applicable, as suggested by the previous discussion.

THE STABILITY OF FRAME YIELD DISPLACEMENTS

Figure 5 shows the capacity curves obtained for two moment-resistant frames, each

four stories in height with three bays and having the same nominal geometry and mem-ber depths. The weights of the member cross sections were varied between the two

frames in order to change the lateral strength. The nonlinear static analysis was con-ducted by applying lateral forces proportional to the mode shape amplitude and mass ateach floor level. Bilinear curves were fitted to the resulting capacity curves to identifythe yield displacement of each structure. The yield displacements are seen to be nearly

invariant (about 0.75% of the height), although the strengths and stiffnesses of the twoframes differ considerably. Given that A36 steel was used for the frames, the yield dis-

placement estimated with Equation 2 is 0.74% of the height. (Had Grade 50 steel beenused, a yield drift of 0.75% (50/36)=1.04% would be expected for this frame. This valuelies in the range of 1 to 1.2% that often brackets the yield drifts observed in nonlinear

static analyses of steel moment-resistant frames.)

The preceding example illustrates the stability of the yield displacement as the lat-

eral strength was changed for a frame having a fixed geometry. Frames of varied heights

are now considered. Gupta and Kunnath (2000) report the capacity curves obtained us-ing three pushover techniques applied to 4-, 8-, 12-, and 20-story buildings having re-inforced concrete moment-resistant frames along each column line. Each building hasthe same nominal floor plan, with 12-ft. (3.7 m) story heights and 24-ft (7.3 m) column

spacings. The designs were based on the 1988 Uniform Building Code, with Rw12 andZ0.4, but using the effective stiffnesses (corresponding to first yield) recommended by



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FEMA 273 (ATC 1997). The fundamental periods of vibration computed for the build-ings, after design, were 0.90, 1.75, 2.21, and 3.31 sec, respectively. The capacity curvesobtained using different pushover techniques are plotted in Figure 6. The thick dashedline represents a bilinear approximation to the capacity curves obtained using the FEMA273 pushover technique, in which lateral forces are determined as a function of the

period-dependent exponent, k, given by Equation 3-8 of that document. The yield driftscorresponding to the bilinear curves of Figure 6 are approximately 0.6, 0.6, 0.5, and0.5%, respectively, for the four frames. It would be feasible to design any of these struc-

tures assuming the yield displacement to be about 0.5 or 0.6% of the building height.

Given the apparent stability of the yield drift (with changes in strength and changes

in the number of stories) and the claim that it may be estimated accurately early in thedesign process, one may wonder if it even is necessary to determine the period of vi-

bration in order to carry out a seismic design. The design of four moment-resistantframes to limit peak drift and ductility responses is described later in the paper. The pro-cedure used for their design and the representation of ground motion demands using

Yield Point Spectra are described next.

YIELD POINT SPECTRA

A designer wishing to determine the strength required to limit the peak ductility anddrift responses of the structure may do so using Yield Point Spectra (YPS). In essence,YPS are constant ductility spectra plotted on the axes of yield strength coefficient andyield displacement, for a range of oscillator periods and for a specified load-deformation

relationship. The yield strength of the SDOF system, Vy , is normalized by its weight, W,to obtain the yield strength coefficient, Cy . Figure 7 plots values of Cy versus the yield

Figure 5. Capacity curves determined by nonlinear static (pushover) analysis for two four-story

moment-resistant steel frames, each having three bays. The frames have the same nominal sec-

tion depths; the weights of the sections were changed to change the lateral strength. The yield

displacement is nearly constant even though the strengths, stiffnesses, and periods of vibration

of the frames differed substantially. (From Black and Aschheim, 2000.)

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displacement y for displacement ductilities =1, 2, 4, and 8, for a bilinear load-

deformation response having post-yield stiffness equal to 5% of the initial stiffnessand viscous damping equal to 5% of critical damping. The classic 1940 NS El Centrorecord is used in this example. When YPS are plotted using linear scales, periods areconstant along radial lines that emanate from the origin; values are indicated (in sec-onds). Peak displacements u of the SDOF systems relative to the ground are given by

y , where the value ofmay be estimated by interpolating between the curves of con-stant ductility.

Superimposed on the YPS of Figure 7 is a capacity curve for a SDOF system. Theyield displacement is equal to about 4 cm and the yield coefficient is about 0.18, result-ing in the yield point being located on the =2 curve. This indicates that the peak dis-

placement will be twice the yield displacement, or about 8 cm. If either the drift or duc-tility responses exceed limiting values associated with a performance objective, the

lateral strength may be increased. If the yield displacement is constant, additionalstrength will reduce the ductility demand and the associated peak displacement responseof the structure.

The curves shown in Figure 7 are computed for a specific ground motion record.Alternately, smoothed YPS may be used for design purposes. For example, a smoothed

Figure 6. Capacity curves presented by Gupta and Kunnath (2000) for 4-, 8-, 12-, and 20-story

reinforced concrete frames designed to satisfy the 1988 Uniform Building Code. Dashed lines

indicate response computed using the FEMA 273/274 procedure for nonlinear static analysis.

Yield drifts were inferred, and range between 0.5 and 0.6% regardless of the number of stories.

Fundamental periods of vibration vary between 0.90 and 3.31 sec.



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elastic design spectrum may be coupled with a smoothed R--T relationship to deter-mine the strengths required for constant ductility responses. This follows the same ideaexpressed by Chopra and Goel (1999) and Fajfar (1999) for improving the seminal Ca-

pacity Spectrum Method (Freeman 1978), from which all of these representations are

derived. In the case of Yield Point Spectra, the yield displacement is plotted on the ab-scissa, while the other representations explicitly plot the peak displacement. The authors

preference for YPS derives in part from the directness with which the strength requiredto limit drift and ductility responses to acceptable values may be determined. In short,the yield displacement is a property of the structure; the ductility demands (and peak

displacement response) associated with a yield strength coefficient are properties of thehazard. The computation of YPS for specific ground motion records can be made using

the computer program USEE 2001 (Inel et al. 2001). A more complete description ofthis representation is given by Aschheim and Black (2000).

SEISMIC DESIGN BASED ON YIELD DISPLACEMENT

Black and Aschheim (2000) developed and implemented a simple seismic design

procedure for determining the strength required to limit peak ductility and drift de-mands. One begins with an estimated value of the yield displacement. This estimate may

be made based on experience, a previous nonlinear static analysis, or explicit formulasthat consider the framing system, geometry, yield strain, and an approximate modeshape. The design base shear strength may be determined using this procedure in place

of the code base shear strength provisions, and the remainder of the design can followthe provisions of a building code. In the present implementation, design and response

are computed at design strength levels; any service level gravity loads present in thenonlinear static and dynamic analyses are therefore used for determining the momentsused for the design of the beams. The sequence of steps is as follows:

Figure 7. Yield Point Spectra computed for the 1940 NS El Centro record. The response of a

system with yield displacement equal to approximately 4 cm, yield strength coefficient equal to

about 0.18, and period equal to 1 s is shown. The yield point falls on the 2 curve, indicating

that the peak displacement is twice the yield displacement.

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Step 1: Estimate the yield displacement that would be observed in a nonlinear static(pushover) analysis of the building when responding in its predominant mode. In thefour examples that follow, the yield displacement is estimated to be 0.75% of the height,

based on a yield strength fy36 ksi (250 MPa).

Step 2: Based on the performance objectives, determine the allowable ductility forthe system. This is the minimum of (1) the system ductility limit associated with limitingstructural damage at the performance level, and (2) the ratio of the drift limit associated

with limiting nonstructural damage at the performance level and the yield drift estimatedin Step 1.

Step 3: Estimate values of the terms 1 (modal participation factor) and 1 (massparticipation factor) based on an assumed mode shape and distribution of mass. The

mode shape is normalized to have unit amplitude at the roof. Estimate the yield displace-ment of the equivalent SDOF system as y /1 . Experience indicates that reasonable ap-

proximations of the mode shape are adequate for design purposes. Values of these modal

parameters can be calculated for various mode shapes; tabulated results are available forthe case of uniform mass in Abrams (1985) and Black and Aschheim (2000).

Step 4: Enter the YPS with the estimated ESDOF yield displacement and the allow-able ductility (determined in Steps 2 and 3); read off the required yield strength coeffi-cient, Cy . The required base shear coefficient for the building is 1Cy .

Step 5: Distribute the base shear over the height of the building and design the build-

ing according to the equivalent static lateral force procedure of a modern building code.Black and Aschheim (2000) used the UBC (ICBO 1997) lateral force distribution fordesign of the beams and employed additional criteria for sizing the columns to ensurethat weak story mechanisms did not result.

If multiple performance objectives are to be considered, Steps 2 and 4 are repeatedfor each performance objective, and design of the members (Step 5) continues with the

largest of the base shear coefficients determined in Step 4.

The most direct approach to verify the adequacy of a design is to compute the elastic

mode shapes and the capacity curve (by nonlinear static analysis) to verify that the yielddisplacement and yield coefficient of the equivalent SDOF structure are consistent withthose assumed in design. If they should differ and it appears that the performance ob-

jective will not be satisfied, a more accurate value of the required base shear coefficientcan be determined by repeating Step 4, with the computed yield displacement and modal

parameters used in place of the estimated values.

In the work by Black and Aschheim (2000) an alternate approach was used, which

did away with the need for nonlinear static analysis. The determination of an AdmissibleDesign Region (Aschheim and Black 2000) identifies families of equivalent SDOF sys-

tems (having different periods of vibration) that just satisfy the performance objectives.The yield point of an equivalent SDOF system is defined by three quantities (of whichonly two are independent): yield displacement, yield strength, and elastic stiffness. The

procedure outlined in Steps 1 through 5 identifies one of the many equivalent SDOFsystems that satisfies the performance objective. The structure may be designed to con-form to the properties of this particular equivalent SDOF system (identified in Steps 1



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through 5) using elastic analysis alone, by focusing on the period of the MDOF systemdesigned using the design base shear obtained from Step 5. If the fundamental periodshould deviate from the period of the equivalent SDOF system, member sizes are then

refined to shift the period to match that of the equivalent SDOF system, without reduc-ing member strengths. As a final check, the mass participation factor (1) is recomputedfor the current elastic mode shape to verify that the beam strengths are adequate for the

base shear computed using the current value of1 .

This alternate procedure was employed in the design of the four frames reported inBlack and Aschheim (2000). Nonlinear static (pushover) analyses were done only afterthe design of the frames was complete, as part of the documentation and validation of

the design method. Nonlinear dynamic analyses were then conducted to determine thepeak roof drifts so that the adequacy of design based on an equivalent SDOF systemcould be assessed.

DESIGN EXAMPLES

Four regular moment-resistant steel frame buildings were designed based on an es-timated yield displacement using the design method described above. Steel was chosen

for simplicity to permit bilinear load-deformation models to be used, although the find-ings are equally applicable to reinforced concrete frames (for which stiffness-degradingresponses ideally should be embodied in the Yield Point Spectra). The frames were de-signed and analyzed only for lateral load to exercise the design method without intro-ducing complications resulting from gravity loads. Details are provided in Black and

Aschheim (2000).

Pairs of 4- and 12-story frames were designed. Each frame was designed to limit

roof drift to 1.5% of the building height when subjected to a specific earthquake groundmotion, with the understanding that this relatively severe drift limit may result in inter-story drifts on the order of 1.7 (1.5%)=2.5%. The designs were made for specific groundmotions so that nonlinear dynamic analyses could be used to validate the design proce-dure; it is understood that real designs would be based on smoothed design spectra or a

larger number of individual ground motions. Each building in a pair was designed foreither a relatively weak or a relatively strong earthquake ground motion. The buildingsdesigned for the weaker records were more flexible, and are designated by the identifiersFlexible-4 and Flexible-12. The buildings designed for the stronger motions werestiffer, and hence were designated as Rigid-4 or Rigid-12. A diverse set of motionswas used, including near field and nearly harmonic motions (Table 1), to illustrate the

robustness of the design approach.

For design purposes, yield displacements were estimated to be 0.75% of the heightbased on the assumption that fy36 ksi (250 MPa). This corresponds to an estimatedyield displacement of 0.128 and 0.368 m for the 4- and 12-story frames, respectively.

Because peak roof displacements were limited to 1.5% of the height, control of drift re-quired that the ductility response of the frames (in the first mode) not exceed a targetvalue of 1.5/0.75=2. The relatively low ductility associated with this drift limit indicatesthat structural damage will be relatively light, provided that the structure is proportionedto develop beam hinging over its height.

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The design method was tested using a relatively imprecise estimate of the predomi-nant mode shape, given by a simple straight-line deflected shape. Lumped masses wereuniformly distributed over the height of the frames. For this mode shape and mass dis-tribution, 11.33 and 1.44 and10.86 and 0.79 for the 4- and 12-story frames, re-spectively. These values may be computed from standard formulas for these quantities orcan be obtained from tables provided by Abrams (1985) and reproduced by Black andAschheim (2000). These values are based on mode shapes normalized to unit amplitudeat the roof. The estimated yield displacements of the equivalent SDOF system are given

by 0.128/1.33=0.096 m and 0.368/1.44=0.26 m (Table 2). The required yield strength

coefficient was determined to obtain a displacement ductility of 2 for the estimatedSDOF yield displacements using Yield Point Spectra prepared for each ground motion

(Figure 8), resulting in the yield strength coefficients given in Table 2. The required baseshear coefficients of Table 1 were obtained as the product of the yield strength coeffi-cients and the mass participation factors of Table 1. The frames were designed for these

base shear coefficients using the lateral force distribution of the Uniform Building Code(ICBO 1997) to obtain the frames shown in Figure 9. The periods of the equivalentSDOF systems were not required for design, but were employed to ensure that the stiff-

Table 1. Frame design parameters

Designation

Design

Ground

Motion

Estimated

Yield

Displacement,

m

Allowable

System

Ductility

Modal

Participation

Factor,

1

Mass

Participation

Factor,

1

Required

Base Shear

Coefficient

Flexible-4 1992 Landers-

Lucerne 250

0.128 2 1.33 0.86 0.258

Rigid-4 1994

Northridge-

Newhall 360

0.128 2 1.33 0.86 0.688

Flexible-12 1985

Michoacan-

SCT 270

0.368 2 1.44 0.79 0.174

Rigid-12 1995 Kobe-

Takatori 360

0.368 2 1.44 0.79 0.474

Table 2. Equivalent SDOF parameters

Designation Estimated SDOF YieldDisplacement, m AllowableDuctility Required YieldStrength Coefficient ESDOFPeriod, sec

Flexible-4 0.096 2 0.30 1.13

Rigid-4 0.096 2 0.80 0.70

Flexible-12 0.26 2 0.22 2.18

Rigid-12 0.26 2 0.60 1.32



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ness and strength of each frame were consistent with the estimated properties of theequivalent SDOF systems. All columns were nominally 14 in. (0.36 m) in depth; nomi-

nal beam depths ranged from 18 to 27 in. (0.46 to 0.69 m).

Nonlinear static (pushover) analyses were done using Drain-2DX (Prakash et al.

1993) with forces applied proportional to the amplitude of the actual elastic mode shapeand the mass at each floor (Figure 10). Bilinear curves were fitted to the resulting ca-

pacity curves, leading to the values of base shear coefficient and yield displacementshown in Table 3. The yield drifts of the 4-story frames were slightly larger than the

estimate of 0.75%, and those of the 12-story frames were slightly lower than the estimateof 0.75%; in the worst case the actual value differed by just 9% from the estimate. Thus,the yield drifts were stable even as the base shear coefficient changed from 0.265 to0.680 for the 4-story frames and from 0.173 to 0.469 for the 12-story frames, as the

Figure 8. Yield Point Spectra used for the design of the four frames: (a) Lucerne, (b) Newhall,

(c) SCT, and (d) Takatori design ground motions. The YPS shown were computed for=1, 2, 4,

and 8, for a bilinear load-deformation model with post-yield stiffness equal to 10% of the initial

stiffness and viscous damping equal to 5% of critical damping. Lines of constant period are

parallel to one another when YPS are plotted on logarithmic axes.

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periods changed from 0.71 to 1.16 s for the 4-story frames and from 1.25 to 2.17 sec-onds for the 12-story frames, and even as the number of stories changed from four totwelve.

Nonlinear dynamic analyses were conducted to determine the peak roof displace-

ment response of the frames under the design ground motions (Figure 11). Peak roof

Figure 9. Framing designed for the (a) Flexible-4, (b) Rigid-4, (c) Flexible-12, and (d) Rigid-

12 frames.



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displacements (Table 3) were within 87 to 95% of the 1.5% drift target (given by 0.255and 0.735 m for the 4- and 12-story frames, respectively). Cuesta and Aschheim (2001)observed a tendency for peak roof displacements to be slightly smaller than the esti-mates made using ESDOF systems, for a varied set of ground motions. This observation

suggests that this design method has a slightly conservative bias.

Figure 10. Capacity curves obtained by nonlinear static analysis of the four frames, for lateral

forces applied proportional to the first mode amplitude and mass at each floor: (a) Flexible-4;(b) Rigid-4; (c) Flexible-12; and (d) Rigid-12. Bilinear curves fitted to the capacity curves are

shown with yield displacements and yield strengths indicated.

Table 3. Frame characteristics and performance from nonlinear static and dynamic analysis.

Designation

Yield

Displacement,m YieldDrift, %

Base Shear

Coefficient(at Yield) FundamentalPeriod, s

Peak Roof

Displacement,m

Flexible-4 0.129 0.759 0.265 1.16 0.241

Rigid-4 0.133 0.782 0.680 0.71 0.223

Flexible-12 0.353 0.720 0.173 2.17 0.666

Rigid-12 0.335 0.684 0.469 1.25 0.650

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The present examples demonstrate that drift and ductility demands can be controlledfor a range of building heights and earthquake intensities using a simple design methodthat is based on the stability of the yield displacement. Extremely simple assumptions

were employed and no nonlinear analyses were done to refine the designs. The strengthsrequired to satisfy the performance objective, and the resulting periods, differed substan-tially, but the yield displacements of the buildings were nearly constant. Assuming that a

period-based design approach would result in buildings of similar proportions, such anapproach would have to find the periods of the resulting designs. This implies that

period-based procedures would require some number of iterations. Convergence using a

yield displacement estimate was rapid enough that no iteration was necessary.

CONCLUSIONS

This paper focuses on the yield displacement of a structure responding in its pre-dominant mode to a seismic excitation. The yield displacement is defined with respect to

Figure 11. Roof displacement histories of the frames subjected to the design ground motions,

determined by nonlinear dynamic analysis: (a) Flexible-4; (b) Rigid-4; (c) Flexible-12; and (d)

Rigid-12. Peak displacements are indicated.



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a capacity curve which is determined by nonlinear static analysis for applied forces thatare consistent with the predominant mode of response. Logical arguments and empiricaldata were employed to demonstrate that:

1. The yield displacement of a structure dominated by flexural response is a func-tion of the yield strain of the material, the height of the structure, the depth of

the yielding members, the shape of the predominant mode of response, and thedistribution of mass and stiffness throughout the structure. (The depth of theyielding members does not affect the yield displacement of a structure domi-nated by axial response.)

2. The parameters (above) that determine the yield displacement are known early

in the design process, allowing accurate estimates of the yield displacement tobe made prior to the detailed design of the structure.

3. The yield displacement is nearly independent of the strength of the system, al-lowing the strength to be selected to achieve a desired performance, based on aninitial estimate of the yield displacement.

4. Seismic design approaches are available that rely on an estimate of the yielddisplacement. The period of vibration is not an essential ingredient of a seismicdesign procedure.

5. Estimates of the yield displacement may be made assuming that the drift atyield (y /H) depends on the structural configuration and materials, and islargely independent of the number of stories and lateral strength of the struc-ture.

Given a structural configuration, the strength required to limit the ductility and peakdisplacement responses of a system determines the stiffness and vibration characteristics

of the system. The fundamental period of vibration is seen to be a consequence ofchoices made in design to control performance. While the influence of strength on the

period(s) of vibration has been overlooked in the past, it is significant now thatperformance-based design objectives are being specified with greater precision. Conven-tional design approaches can be modified to consider the influence of strength on the

period of vibration, but such modifications introduce iterations into the design process.The examples presented herein show that design approaches based on the stability of theyield displacement can converge rapidly upon a solution, entirely eliminating the needfor iteration if reasonable assumptions are made in design.

These conclusions, drawn from fundamentals and illustrated for structures having asharply defined yield point, are also applicable to structures in which the progression ofyielding results in a less sharply defined yield displacement.

ACKNOWLEDGMENTS

Interaction with a number of people aided the development of this paper. Discus-sions with Craig Comartin, Steve Mahin, and Jack Moehle were particularly valuable.Support provided by a CAREER Award from the National Science Foundation (Grant

No. CMS-9984830) is gratefully acknowledged. This work made use of Earthquake En-

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gineering Research Centers Shared Facilities supported by the National Science Foun-dation under Award Number EEC-9701785. Figure 6 was reproduced with the permis-sion of the Earthquake Engineering Research Institute.

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(Received 2 January 2001; accepted 14 August 2002)

600 M. ASCHHEIM

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Aschheim - 2002 - Seismic Design Based on the Yield Displacement

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