eeri paper chao goel lee 2007 8

8/17/2019 EERI Paper Chao Goel Lee 2007 8

1/23

A Seismic Design Lateral ForceDistribution Based on Inelastic Stateof Structures

Shih-Ho Chao,a)

M.EERI, Subhash C. Goel,b)

M.EERI, and Soon-Sik Leec)

It is well recognized that structures designed by current codes undergolarge inelastic deformations during major earthquakes. However, lateral forcedistributions given in the seismic design codes are typically based on results of elastic-response studies. In this paper, lateral force distributions used in thecurrent seismic codes are reviewed and the results obtained from nonlinear dynamic analyses of a number of example structures are presented and discussed. It is concluded that code lateral force distributions do not representthe maximum force distributions that may be induced during nonlinear

response, which may lead to inaccurate predictions of deformation and forcedemands, causing structures to behave in a rather unpredictable and undesirable manner.A new lateral force distribution based on study of inelastic

behavior is developed by using relative distribution of maximum story shearsof the example structures subjected to a wide variety of earthquake ground motions. The results show that the suggested lateral force distribution,especially for the types of framed structures investigated in this study, is morerational and gives a much better prediction of inelastic seismic demands atglobal as well as at element levels. DOI: 10.1193/1.2753549

INTRODUCTION

The current lateral seismic-force distributions in building codes are generally based

on first-mode dynamic solution of lumped multiple-degree-of-freedom (MDOF) elasticsystems (ATC 1978, Clough and Penzien 1993, Chopra 2000, BSSC 2003), which can

be expressed as:

f i1 = V 1 wi i1 j =1

n w j j 1 1

where f i1 is the lateral force at level i; V 1 is the first-mode base shear; w j is lumped seis-mic weight at j th level; and j is amplitude of the first mode at j th level. The code ex-

pression assumes that the lateral force distribution can be expressed using the first-mode

a) Postdoctoral Research Fellow, Dept. of Civil and Environmental Engineering, University of Michigan, Ann

Arbor, MI; E-mail: [email protected] b) Professor, Dept. of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI; E-mail:

[email protected]) Senior Bridge Engineer, URS Corporation, Roseville, CA; E-mail: [email protected]

547

Earthquake Spectra, Volume 23, No. 3, pages 547–569, August 2007; © 2007, Earthquake Engineering Research Institute


2/23

deflected shape of an elastic lumped-mass system subjected to dynamic loading, whichcan be written as:

i1 = hik / L 2

where L is the total height of the structure; h ik is the height of level i above the base with

exponent k being related to the building’s fundamental period T . Observations of theelastic response of buildings suggest that the first-mode shape is close to a straight linek = 1 when the fundamental period is 0.5 s or less; and is close to a parabola k = 2when the fundamental period is 2.5 s or longer. This leads to the following code lateralforce distribution (BSSC 2003, ICC 2006):

F i = C viV 3

where

C vi =wihi

k

j =1n w j h j

k 4

and F i is the lateral force applied at level i; C vi is the vertical distribution factor; V is thetotal design base shear, which replaces V 1 in Equation 1 since V 1 is the dominant part of the total force V ; wi and w j are the total effective seismic weights at levels i and j , re-spectively; h i and h j are the heights of levels i and j from the ground, respectively; and n is the number of stories. For structures with natural period between 0.5 and 2.5 s, k isdetermined by linear interpolation between 1 and 2.

Equations 3 and 4 were originally adopted by ATC 3-06 Provisions (ATC 1978),which served as the basis of the NEHRP Provisions (BSSC 2003). These expressions are

also used by the current International Building Code (ICC 2006). Similarly, the UBC 97(ICBO 1997) used distribution of design lateral forces based on a linear mode shape,together with an additional concentrated lateral force at the top level to account for higher mode effects (SEAOC 1999):

F i = V − F t wihi

j =1n w j h j

5

The force at the top level computed from Equation 5 is increased by an additionalforce:

F t = 0.07TV if T 0.7 s 6a

F t = 0 if T 0.7 s 6b

where F t is the additional concentrated force applied at the top level of the structure, inorder to account for higher mode effects. It is noted that the IBC/NEHRP and UBC dis-tributions are essentially the same when the fundamental period of the structure is lessthan 0.5 s.

Since most building structures (especially those with large R values) designed ac-

548 S. H. CHAO, S. C. GOEL, AND S. S. LEE


3/23

cording to current code procedures are expected to undergo large deformations in theinelastic range when subjected to major earthquakes, lateral force distributions can be

quite different from those given by the code formulas such as Equation 4. In order toachieve the main goal of performance-based seismic design, i.e., a desirable and predict-able structural response, it is important to consider inelastic behavior of structures di-rectly in the design process. The commonly used elastic analysis and design proceduresin current practice, together with elastic-design lateral force distributions, may not bewell suited to fulfill this goal in a realistic manner.

One of the essential elements of performance-based seismic design of structuresshould be to use more realistic design lateral force distribution, which represents peak lateral force distribution in a structure in the inelastic state and includes the higher modeeffects. This paper presents a new lateral force distribution based on the study of inelas-tic responses of various types of structural systems, using extensive nonlinear dynamicanalysis results.

NEW LATERAL FORCE DISTRIBUTION

The format for this new design lateral force distribution based on inelastic state of astructure was originally proposed by Lee and Goel (2001) by using shear distributionfactor derived from the relative distribution of maximum story shears of a large number of steel moment frames subjected to selected earthquake records. The suggested expres-sion has a format similar to that of the IBC/NEHRP expression:

F i = C vi V 7

where

C vi = i − i+1 wnhn

j =1n w j h j

T −0.2

when i = n, n+1 = 0 8a

i =V i

V n= j =in w j h j

wnhn T

−0.2

8b

where i is the shear distribution factor at level i; V i and V n, respectively, are the storyshear forces at level i and at the top (nth) level; w j is the seismic weight at level j ; h j isthe height of level j from the base; w n is the weight at the top level; hn is the height of roof level from the base; T is the fundamental period; F i is the lateral force at level i; and V is the total design base shear. The value of parameter was originally proposed as 0.5

by Lee and Goel (2001), which was later revised to 0.75 based on more extensive non-

linear dynamic analyses on eccentrically braced frames (EBFs) and special truss mo-ment frames (STMFs) by Chao and Goel (2005 and 2006a) and is briefly presented inthis paper. It is interesting to note that, by using a similar approach (story shear distri-

bution), Park and Medina (2006) proposed a design lateral force distribution for momentframe structures exposed to pulse-type near-fault ground motions.

A SEISMIC DESIGN LATERAL FORCE DISTRIBUTION BASED ON INELASTIC STATE OF STRUCTURES 549


4/23

It should be mentioned that the lateral force distribution as presented in this study is based on story shear distribution V i / V n, because it facilitates calculation of lateral forcesstarting from the top (level n).

JUSTIFICATION OF THE NEW LATERAL FORCE DISTRIBUTION

Four types of steel frames were evaluated in this study: moment frames (MFs),EBFs, STMFs, and concentrically braced frames (CBFs). The floor plan and elevation

views are shown in Figures 1–5. The nine-story MF is the same that was studied byGupta and Krawinkler (1999), and was designed based on UBC lateral force distribu-

Figure 1. (a) Floor plan of nine-story building and (b) nine-story moment frame.

Figure 2. (a) Floor plan of three-story building and (b) three-story eccentrically braced frame.



5/23

tion. The three- and 10-story EBFs were designed according to both the IBC and thesuggested lateral force distributions (Chao and Goel 2005). The nine-story STMF asshown in Figure 4 was designed by using the suggested expression (Chao and Goel2006a). The six-story CBF (Sabelli 2000), which was originally designed using NEHRPlateral force distribution and 1997 AISC Seismic Provisions (AISC 1997), was rede-

Figure 3. (a) Floor plan of 10-story building and (b) 10-story eccentrically braced frame.

Figure 4. (a) Floor plan of nine-story building and (b) nine-story special truss moment frame.



6/23

signed for this study according to the 2005 AISC Seismic Provisions (AISC 2005). Themember sizes and other details for the study frames can be found in the related refer-ences.

Nonlinear dynamic analyses were carried out to evaluate the validity of the sug-gested design lateral force distribution. A total of 21 SAC Los Angeles–region ground motions records (Somerville et al. 1997; fifteen at 10% and six at 2% probability of ex-ceedance in 50 years) were used. The characteristics of the selected SAC ground mo-tions are given in Table 1. Analyses for MFs and CBFs were carried out by using theSNAP-2DX program (Rai et al. 1996), while analyses for EBFs and STMFs were con-ducted by using Perform-3D (RAM 2003) and Perform-2D (RAM 2003), respectively.Details regarding modeling work can be found elsewhere (Lee and Goel 2001, Chao and Goel 2005, 2006a, 2006b). Table 2 summarizes the fundamental periods of all the studyframes. The merits of the suggested expression over the code distributions are discussed

below.

RELATIVE STORY SHEAR DISTRIBUTIONS

The suggested new lateral force distribution was justified by investigating the rela-tive story shear distributions (defined as the ratio of maximum story shear force at level

i to that at top level n; i.e.,V i / V n) as obtained from the nonlinear dynamic analyses. Notethat story shear distribution and lateral force distribution have a direct relationship. Thestory shear, V x, in any story is the sum of the lateral forces above that story:

Figure 5. (a) Floor plan of six-story building and (b) six-story concentrically braced frame.



7/23


8/23

smaller member sizes and therefore larger story drifts and member deformations at thoselevels. This trend was noticed in all types of study frames.

On the other hand, the suggested lateral force distribution (thus, the relative storyshear distribution) is closer to the results obtained from nonlinear dynamic analyses. It isnoted, as shown in Figure 6, that relative story shear distribution using =0.5 generallyrepresents a lower bound of the nonlinear dynamic analysis results. This would normallylead to larger design forces at upper floors, which may result in concentration of inelas-tic deformation at the lower levels. Further analyses by Chao and Goel (2005 and 2006a)show that relative story shear distribution using =0.75 represents an upper bound of the nonlinear dynamic analysis results (Figure 6) and generally leads to more uniformdeformations of elements as well as stories over the height of the structure, which will

be discussed later.

It is also noticed that, by applying an additional lateral force at the top level, theUBC equation gives better prediction than the IBC/NEHRP equation. The reason thatthe ATC 3-06 (thus IBC and NEHRP) does not include the additional top force can beattributed to the concern that an underestimation of story shears in the lower storiesmight be more serious than those in the top stories (Newmark and Rosenblueth 1971).However, based on nonlinear dynamic analysis results, it is obvious that the IBC/

NEHRP equation can result in significant underdesign of the top stories with consequentadverse results.

Figures 7–10 show the nonlinear dynamic analysis results for three- and 10-storyEBFs designed by using both the IBC and the suggested expressions. It is seen that, gen-erally, the difference between the analysis results and the code predictions becomeslarger when the building height (or fundamental period) increases (see Table 2). It is also

Figure 6. Relative story shear distributions from nonlinear dynamic analyses, code expressions,and suggested expression for nine-story moment frame designed based on NEHRP expression.



9/23

noticed from Figures 9 and 10 that the relative story shear distributions obtained fromnonlinear dynamic analyses have the same trend (i.e., are closer to the suggested distri-

bution) regardless of whether the frame was designed according to the code or the sug-gested formula. These observations are further confirmed through nonlinear dynamic

Figure 7. Relative story shear distributions from nonlinear dynamic analyses, code expressions,and suggested expression for three-story eccentrically braced frame designed based on IBCexpression.

Figure 8. Relative story shear distributions from nonlinear dynamic analyses, code expressions,and suggested expression for three-story eccentrically braced frame designed based on sug-gested expression.



10/23


11/23

The suggested design lateral force distribution was further verified by using struc-tural wall systems from the analytical results carried out by other researchers (Iqbal and Derecho 1980). By using the DRAIN-2D program, Iqbal and Derecho conducted non-linear dynamic analyses of a series of isolated reinforced-concrete structural walls hav-

Figure 11. Relative story shear distributions from nonlinear dynamic analyses, code expres-sions, and suggested expression for nine-story special truss moment frame designed based onsuggested expression.

Figure 12. Relative story shear distributions from nonlinear dynamic analyses, code expres-sions, and suggested expression for six-story concentrically braced frame designed based on

NEHRP (IBC) expression.



12/23

ing 10, 20, 30, and 40 stories. These individual cantilever walls represent interior ele-ments of a building and provide the entire lateral load resistance for their tributary floor area. The corresponding parameters and the six ground motions used are given in Table3. Based on the analysis results, they concluded that the UBC and ATC 3-06 formulastend to underestimate the shears at the top levels for all the study structural walls. Amodified UBC force distribution was proposed based on their observations (Iqbal and Derecho 1980):

(a) Normalized shear forces in the top 25% of the wall height should be increased by a factor of:

1 = 2 −T

3 where 1.0 1 1.5 10

(b) Use of 1 should not result in a normalized shear force greater than that allo-cated to any portion in the lower 75% of the wall height using UBCdistribution.

Relative story shear distributions for the 10- and 20-story structural walls based ontheir proposed formula, along with the code distributions and expression suggested inthis study, are plotted and shown in Figure 13. As can be seen, the distribution proposed

by Iqbal and Derecho for reinforced-concrete structural walls generally agrees with thefindings in this study for frame systems, in that the current code formulas can signifi-cantly underestimate the story shears at the upper levels when a structural wall deformsinto the inelastic range. Note that the straight lines in Iqbal and Derecho’s distribution atupper levels in Figure 13 are due to the constraint (b) in their proposed method. Based on Figure 13, it appears that the suggested formula in this study can also be applied tostructural wall systems. In any case, further research, such as using more ground mo-tions as well as coupled wall systems, should be undertaken to determine if a slightly

smaller value of would be more appropriate for structural wall systems.

In order to examine the effects of inelastic and elastic behavior on the lateral forcedistributions, elastic dynamic analyses were carried out for the nine-story STMF. As in-

Table 3. Structural parameters and ground motions used for analyses of the 10-, 20-, 30-, and

40-story isolated structural walls (Iqbal and Derecho 1980)

WallFirst-StoryHeight (ft.)

Other StoryHeight (ft.)

FundamentalPeriod, T (s)

Stiffness and Mass

Earthquake Ground Motions

10-Story 12 8.75 0.5 Uniform over wall height

1971 Pacoima Dam, S16E


1971 Holiday Orion, E-W1952 Taft, S69E


1940 El Centro, E-WS1 (Artificial Acc.)


1940 El Centro, N-S



13/23

dicated in Figure 14, the elastic relative story shear distributions have a tendency to shifttoward the code distribution a bit more as opposed to the inelastic analysis results asshown in Figure 11. However, the code expression still does not predict the elastic re-sponses well enough, even though it was developed based on elastic behavior. Figure 15shows the relative story shear distributions for a 10-story EBF, as obtained from elasticand inelastic dynamic response analyses. As can be seen, the elastic relative story shear distributions are not unique and can vary quite a bit with the ground motion used. On theother hand, the inelastic relative story shear distributions were closer to the values given

by the suggested expression (Equation 8; =0.75). This is quite understandable because

Figure 13. Comparison between relative story shear distributions for 10- and 20-story struc-tural walls.

Figure 14. Relative story shear distributions from elastic dynamic analyses, code expressions,and suggested expression for nine-story special truss moment frame.



14/23

the story shears in the inelastic state are limited by the strength of the system. It is in-teresting to note that a similar trend was also observed by Goel (1967) in his early studyof inelastic seismic behavior of multistory steel moment frames, as shown in Figure 16.

MAXIMUM INTERSTORY DRIFT DISTRIBUTIONS

Since the suggested lateral force distribution is based on inelastic response, thestructures designed by using such distribution tend to be better proportioned. In other words, the possibility of overdesign or underdesign in certain regions is greatly reduced.Figure 17 shows that an EBF designed by using the suggested lateral force distribution

Figure 15. Selected results of relative story shear distributions from elastic and inelastic dy-namic analyses for 10-story eccentrically braced frame.

Figure 16. Elastic and inelastic story shear distributions of a 10-story moment frame (Goel1967).



15/23

exhibited more uniform distribution of maximum interstory drift than the EBF designed by IBC lateral force distribution. A similar trend was also observed in the nine-storySTMF designed according to the suggested distribution, as shown in Figure 18. A uni-form interstory drift distribution also implies that intended “fuse” elements (such as

beams in moment frames, shear links in EBFs, and chord members in STMFs) at all

Figure 17. Comparison of maximum interstory drift distribution between EBFs designed by

using the suggested and code design lateral force distributions (all ground motions have a prob-ability of exceedence of 10% in 50 years): (a) suggested, (b) IBC 2006.

Figure 18. Maximum interstory drift distribution of nine-story STMF designed by the sug-gested lateral force distribution (all ground motions have a probability of exceedence of 10% in50 years).



16/23

levels can be mobilized to dissipate earthquake energy, thus minimizing the possibilityof concentration of excessive inelastic deformation and damage in some stories or lo-calized regions.

COLUMN DESIGN MOMENTS

It has been pointed out by several investigators in the past that when a structure issubjected to seismic loading, especially in the inelastic state, large moments can occur inthe columns, which can be quite different from those calculated by elastic analysis (eg,Paulay and Priestley 1992). Conventional design approaches usually do not accuratelyestimate the maximum column moments and their location (Bondy 1996, Medina and Krawinkler 2005). In fact, the column moments are quite often underestimated because

the columns are subjected to moments not only from those delivered from the beams or other members framing into the columns (conventional capacity design approach), butalso from their own deformation (Bondy 1996).

In view of the above-mentioned shortcoming in conventional design approach, an al-ternative design method was proposed by Leelataviwat et al. (1999) in considering theequilibrium of an entire “column tree” in the extreme limit state. Figure 19 shows thefree-body diagram of an exterior “column tree” of a moment frame for calculating thedesign column moments in each story. In order to ensure the formation of strong-columnweak-beam mechanism, columns should be designed for maximum expected forces byconsidering a reasonable extent of strain-hardening in the beam plastic hinges and ma-terial overstrength. The moment at a strain-hardened beam plastic hinge can be obtained

by multiplying its nominal plastic moment M pb by an overstrength factor , which

accounts for the effect of strain hardening and material overstrength. The column mo-ments and shear forces can be calculated by applying the expected beam end momentsand equivalent lateral (inertia) forces applied at each level F iu necessary to keep the“column tree” in equilibrium. Through this approach the moments due to column elasticdeformation are also accounted for. It is worth mentioning that conventional capacitydesign approaches only consider individual joints, in which the column moment capacity

Figure 19. Free body diagram of an exterior “column tree.”



17/23

must be greater than the moment demands coming from beam ends. The approach used herein and illustrated in Figure 19 considers the entire column tree under all applicableforces needed for equilibrium. An appropriate moment at the column base must also beapplied. It is in the calculation of column design moments and shears that the impor-tance of using a realistic lateral force distribution becomes more critical.

Figure 20 shows a comparison of the column moments calculated by the above-mentioned procedure with the new lateral force distribution, the UBC lateral force dis-

tribution, and maximum column moment envelopes obtained from nonlinear dynamicanalyses for the exterior and interior columns of the nine-story moment frame (Lee and Goel 2001). The column design moments calculated by using the new lateral force dis-tribution agree very well with the maximum column moment envelopes at most levels.On the other hand, the design column moments calculated by using the UBC lateralforce distribution significantly deviate from the nonlinear dynamic analysis results. Theoverestimation becomes even greater for taller frames (Figure 21). It should be men-tioned that the overestimated column design moments due to code lateral force distribu-tion occur only when the proposed column design method (i.e., “column tree” approach)is used. As mentioned earlier, the column design moments can be underestimated whenconventional elastic analysis and design methods are employed along with the code dis-tribution of lateral forces.

HIGHER MODE EFFECTS

The new lateral force distribution increases the forces in the upper stories, thereby better representing the higher mode effects. Figure 22 shows the shapes of IBC (or NE-HRP) and the suggested lateral force distributions for the example nine-story momentframe; the corresponding parameters are summarized in Tables 4 and 5. It can be seen

Figure 20. Design column moments and maximum column moment envelopes from nonlinear dynamic analyses for a nine-story moment frame (Lee and Goel 2001).



18/23

that the new lateral force distribution results in larger force at the upper level than thatgiven by the IBC lateral force distribution. This implies that the additional design forceat the top level as given by the IBC (intended to reflect the higher mode effects) may not

be adequate, especially when the structures are taller and respond inelastically.

Dynamic analyses conducted on moment frames with various beam-to-column stiff-ness ratios have shown that, as the beam-to-column stiffness decreases, the higher mode

Figure 21. Design column moments and maximum column moment envelopes from nonlinear dynamic analyses for a 20-story moment frame (Lee and Goel 2001).

Figure 22. The suggested lateral force distribution and IBC 2006 lateral force distribution for the nine-story moment frame.



19/23

response becomes an increasing percentage of the total response (Chopra and Cruz1986, Chopra 2005). For the extreme case when the beam stiffness approaches zero, thelateral forces at the upper levels of a moment frame become much larger than those cal-culated by the code expression, as shown in Figure 23. A moment frame with negligible

beam stiffness resembles a moment frame in which plastic hinges have occurred at the beam ends. Therefore, the response as shown in Figure 23b supports the observations inthis study that the influence of higher mode effects increases in the inelastic state whenthe plastic hinges form at the beam ends during strong shaking.

Nonlinear dynamic analyses carried out by Villaverde (1991,1997) also showed thatusing the linear elastic first-mode distribution of lateral forces (without accounting for the fact that a structure would enter inelastic state during a major earthquake) could be

the primary reason for numerous upper story collapses during the 1985 Mexico Cityearthquake. The modal periods of a structure elongate as plastic deformation occurs.That leads to increased contribution from the higher modes, hence columns designed

Table 4. IBC/NEHRP design lateral forces and design story shears for the nine-story moment

frame

Floor hi (ft.) wi (kips) wihik (kip-ft) F i (kips) Story Shear (kips)

9 122 1177.0 945870 0.239V 0.239V

8 109 1092.5 750484 0.190V 0.429V

7 96 1092.5 628844 0.159V 0.588V

6 83 1092.5 513514 0.130V 0.718V

5 70 1092.5 405082 0.102V 0.821V

4 57 1092.5 304304 0.077V 0.898V

3 44 1092.5 212212 0.054V 0.951V

2 31 1092.5 130316 0.033V 0.984V

1 18 1111.5 62195 0.016V 1.000V

Table 5. Suggested design lateral forces and design story shears for the nine-story moment

frame

Floor h j (ft.) w j (kips) w j h j (kip-ft) w j h j i F i (kips) Story Shear (kips)

9 122 1177.0 143594 143594 1.000 0.323V 0.323V

8 109 1092.5 119083 262677 1.538 0.174V 0.498V

7 96 1092.5 104880 367557 1.955 0.135V 0.632V

6 83 1092.5 90678 458234 2.288 0.108V 0.740V 5 70 1092.5 76475 534709 2.554 0.086V 0.826V

4 57 1092.5 62273 596982 2.763 0.068V 0.894V

3 44 1092.5 48070 645052 2.920 0.051V 0.944V

2 31 1092.5 33868 678919 3.029 0.035V 0.979V

1 18 1111.5 20007 698926 3.092 0.021V 1.000V



20/23

based on elastic first-mode distribution of story shears are subjected to significantlyhigher moments when a structure deforms in the inelastic range, especially those in theupper stories (Villaverde 1991). Villaverde’s conclusions further justify the importanceof using a lateral force distribution, such as the one suggested herein.

SUMMARY AND CONCLUSIONS

This paper presents a study aimed at formulating a more realistic design lateral forcedistribution that accounts for inelastic behavior of structures when subjected to major earthquakes. Use of a realistic force distribution based on inelastic response is one of theimportant steps in a comprehensive seismic design methodology if accurate representa-tion of expected structural response is to be realized. In this study, extensive nonlinear dynamic analyses were carried out on different types of frames to verify the suggested distribution, and the results showed that the suggested lateral force distribution, espe-cially for the types of framed structures investigated in this study, is more rational and gives a much better prediction of inelastic seismic demands at global as well as at ele-ment levels. It should be mentioned that the suggested lateral force distribution can beeasily further refined and modified by using more ground motions, such as near-field

Figure 23. (a) Moment frame with small beam-to-column stiffness ratio, and (b) equivalentstatic lateral force distribution from dynamic analysis and IBC expression (Chopra 2005).



21/23

records and other structural systems, than those used in this study. It is also noted thatthe current codes generally use one distribution (derived based on first-mode elastic dy-

namic response) for all types of structures. The suggested lateral force distribution can be applied to most of the commonly used frame types, and it can also be modified for other structures by changing the value of the factor.

The following conclusions can be drawn from the results of this study:

1. Steel frames (MFs, EBFs, STMFs, and CBFs) designed by using the suggested lateral force distribution resulted in maximum story shears that agreed well withthose obtained from nonlinear dynamic analyses. Maximum story shear distri-

butions as given in the codes, which are based on first-mode elastic behavior,deviate significantly from the time-history dynamic analysis results regardlessof whether the structures respond in the elastic or inelastic range.

2. Frames designed by using the suggested lateral force distribution experienced more uniform maximum interstory drifts along the heights than the frames de-signed based on current code distributions.

3. The proposed column design procedure, using the “column tree” concept and the new design lateral force distribution, gives a very good estimation of maxi-mum column moment demands when the structures respond to severe ground motions and deform into their inelastic state.

Higher mode effects are also well reflected in the suggested design lateral force distribution.

ACKNOWLEDGMENTS

The financial support for this study was provided by NUCOR Research & Develop-ment and the G. S. Agarwal Fellowship Fund in the Department of Civil & Environmen-

tal Engineering at the University of Michigan. The opinions and views expressed in this paper are solely those of the authors and do not necessarily reflect those of the sponsors.

REFERENCES

AISC, 1997. Seismic Provisions for Structural Steel Buildings, American Institute of Steel Con-struction, Chicago, Ill.

AISC, 2005. Seismic Provisions for Structural Steel Buildings, ANSI/AISC 341-05, AmericanInstitute of Steel Construction, Chicago, Ill.

Applied Technology Council, 1978. Tentative Provisions for the Development of Seismic Regu-lations for Buildings, Report ATC 3-06 , NBS Special Publication 510, NSF Publication 78-08, National Science Foundation, Washington, D.C.

Bondy, K. D., 1996. A more rational approach to capacity design of seismic moment frame

columns, Earthquake Spectra 12, 395–406.Building Seismic Safety Council (BSSC), 2003. National Earthquake Hazard Reduction Pro-

gram (NEHRP) Recommended Provisions for Seismic Regulations for New Buildings and

Other Structures (Part 1: Provisions, FEMA 450-1, and Part 2: Commentary, FEMA 450-2) ,Federal Emergency Management Agency, Washington, D.C.

Chao, S.-H., and Goel, S. C., 2005. Performance-Based Seismic Design of EBF Using Target



22/23

Drift and Yield Mechanism as Performance Criteria, Report No. UMCEE 05-05, Departmentof Civil and Environmental Engineering, University of Michigan, Ann Arbor.

––—, 2006a. Performance-Based Plastic Design of Seismic Resistant Special Truss Moment Frames, Report No. UMCEE 06-03, Department of Civil and Environmental Engineering,University of Michigan, Ann Arbor.

––—, 2006b, A seismic design method for steel concentric braced frames for enhanced perfor-mance, Paper No. 227, Proceedings, Fourth International Conference on Earthquake Engi-neering, October 12–13, Taipei, Taiwan.

Chopra, A. K., 2000. Dynamics of Structures—Theory and Applications to Earthquake Engi-neering , Second Edition, Prentice Hall, Englewood Cliffs, NJ.

––—, 2005. Earthquake Dynamics of Structures—A Primer , Second Edition, Earthquake Engi-neering Research Institute, Oakland, Calif.

Chopra, A. K., and Cruz, E. F., 1986. Evaluation of building code formulas for earthquakeforces, J. Struct. Eng. 112, 1881–1899.

Clough, R. W., and Penzien, J., 1993. Dynamics of Structures, Second Edition, McGraw-Hill,Inc., New York.

Goel, S. C., 1967. Inelastic Behavior of Multistory Building Frames Subjected to EarthquakeMotion, Ph.D. Thesis, University of Michigan, Ann Arbor.

Gupta, A., and Krawinkler, H., 1999. Seismic Demands for Performance Evaluation of Steel Moment Resisting Frame Structures, John A. Blume Earthquake Engineering Center Report

No. 132, Department of Civil Engineering, Stanford University, Stanford, Calif.

International Code Council (ICC), 2006. International Building Code, ICC, Birmingham, Ala.

International Conference of Building Officials (ICBO), 1997. Uniform Building Code, ICBO,Whittier, Calif.

Iqbal, M., and Derecho, A. T., 1980. Inertial forces over height of reinforced concrete structuralwalls during earthquakes, in Reinforced Concrete Structures Subjected to Wind and Earth-quake Forces, ACI Publication SP-63, American Concrete Institute, Detroit, Mich., 173–196.

Lee, S.-S., and Goel, S. C., 2001. Performance-Based Design of Steel Moment Frames Using Target Drift and Yield Mechanism, Report No. UMCEE 01-17 , Department of Civil and En-vironmental Engineering, University of Michigan, Ann Arbor.

Leelataviwat, S., Goel, S. C., and Stojadinović, B., 1999. Toward performance-based seismicdesign of structures, Earthquake Spectra 15, 435–461.

Medina, R. A., and Krawinkler, H., 2005. Strength demand issues relevant for the seismic de-sign of moment-resisting frames, Earthquake Spectra 21, 415–439.

Newmark, N. M, and Rosenblueth, E., 1971. Fundamentals of Earthquake Engineering , Pren-tice Hall, Inc., Englewood Cliffs, NJ.

Park, K., and Medina, R. A., 2006. Lateral load patterns for conceptual seismic design of frames exposed to near-fault ground motions, Paper No. 949, Proceedings, 8th U.S. National Conference on Earthquake Engineering , April 18–22, San Francisco, Calif.

Paulay, T., and Priestley, M. J. N., 1992. Seismic Design of Reinforced Concrete and Masonry Buildings, John Wiley & Sons, Inc., New York.

Rai, D. C., Goel, S. C., and Firmansjah, J., 1996. SNAP-2DX: General Purpose Computer Pro- gram for Static and Dynamic Nonlinear Analysis of Two Dimensional Structures, Report No.

UMCEE 96-21, Department of Civil and Environmental Engineering, University of Michi-gan, Ann Arbor.



23/23

RAM International, 2003. Perform-2D User Guide, RAM International, Carlsbad, Calif.

––—, 2003. Perform-3D User Guide, RAM International, Carlsbad, Calif.

Sabelli, R., 2000. Research on Improving the Design and Analysis of Earthquake Resistant Steel Braced Frames, FEMA/EERI Report , Earthquake Engineering Research Institute, Oakland,Calif.

Seismology Committee of Structural Engineers Association of California (SEAOC), 1999. Rec-ommended Lateral Force Requirements and Commentary, Seventh Edition, SEAOC, Sacra-mento, Calif.

Somerville, P. G., Smith, M., Punyamurthula, S., and Sun, J., 1997. Development of Ground Motion Time Histories for Phase 2 of the FEMA/SAC Steel Project, Report No. SAC/BD-97/

04, SAC Joint Venture, Sacramento, Calif.

Villaverde, R., 1991. Explanation for the numerous upper floor collapses during the 1985Mexico City earthquake, Earthquake Eng. Struct. Dyn. 20, 223–241.

––—, 1997. Discussion of “A more rational approach to capacity design of seismic moment

frame columns,” Earthquake Spectra 13, 321–322.(Received 23 August 2006; accepted 13 February 2007


eeri paper chao goel lee 2007 8

Documents