FBD Vs DBD

Engineering Structures 51 (2013) 128–140

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Engineering Structures

journal homepage: www.elsevier .com/locate /engstruct

Drift, strain limits and ductility demands for RC moment framesdesigned with displacement-based and force-based design methods

0141-0296/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2013.01.004

⇑ Corresponding author.E-mail addresses: [email protected], [email protected] (A.L. Vidot-Vega).

Aidcer L. Vidot-Vega a,⇑, Mervyn J. Kowalsky b

a Department of Engineering Science and Materials, University of Puerto Rico, Mayagüez (UPRM), Puerto Ricob Department of Civil, Construction, and Environmental Engineering, North Carolina State University, Raleigh, NC, United States

a r t i c l e i n f o

Article history:Received 25 October 2010Revised 16 February 2011Accepted 14 January 2013Available online 26 February 2013

Keywords:Displacement-based designDriftsReinforced concrete framesForce-based designMaterial strainsDuctility

a b s t r a c t

This paper presents the results of the non-linear time history analysis of six different reinforced concretemoment frames. The frames were designed using direct displacement-based design (DDBD) and tradi-tional force-based design methods. Frames of 4–12 storeys tall and with two and three bays were studied.The interstorey drifts, displacements, and material strains obtained from the analyses of the framesdesigned using both design methods are compared. The implications of code implied ductility and allow-able drifts were also studied. Target steel tensile strains and interstorey drifts for the frames designedusing DDBD correlated well with the values obtained from the analyses.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Traditionally, seismic structural design has been based primar-ily on forces (force-based seismic design). In general, this type ofdesign has worked well for providing safe structures, but has notensured uniform damage among them. In response to this situa-tion, methods in the family of performance-based design haveevolved over the last twenty years in an attempt to design struc-tures in which their behavior will be controllable under definedlevels of seismic hazard.

Performance-based seismic engineering (PBSE) consists of a ‘‘setof engineering procedures for the design and construction of struc-tures to achieve predictable levels of performance in response tospecified level of hazards, within definable levels of reliability’’[14]. Performance-based seismic design explicitly evaluates howa building is likely to perform given the potential hazard it is likelyto experience. It permits the design of new buildings or upgrades ofexisting buildings with a realistic understanding of the risk of casu-alties, occupancy interruption, and economic loss that may occuras a result of future earthquakes [9]. This is accomplished by thedefinition of performance objectives that are selected by the ownerand engineer prior to the design. The direct displacement-baseddesign (DDBD) method has evolved as a way to implement PBSEin a direct manner [10]. The basic idea of DDBD is to design a

structure whereby it achieves a predefined level of deformation(and hence damage) under a predefined level of seismic intensity.This method is based on the substitute structure approach pio-neered by Gulkan and Sozen [3].

The first objective of this paper is to compare important designparameters such as: drifts, displacements, and material strains ob-tained from the non-linear time history analysis of RC frames de-signed using DDBD and force-based design (FBD). The secondobjective is to explore inconsistencies between forced reductionfactors, displacement amplification factors, and drift limits (suchas those defined in ASCE 7-05 [1] for frames designed accordingto FBD. Reinforced concrete frame buildings of 4, 6, 8, and 12 sto-reys with two and three bays were designed using both methods.The ASCE7-05 [1] was used to design the frames using the force-based approach. The program OpenSees [7] was used to performthe non-linear time history analyses.

2. Force-based vs. displacement-based design methods

The main difference between the traditional force-based design(FBD) and displacement-based design (DBD) methods is the start-ing point of the design. The starting point of the FBD method is theestimation of the member sizes and the elastic stiffness of themembers (usually cracked) to determine the approximate funda-mental period of the structure. Then, the period is used to estimatelateral forces and base shear. The base shear is reduced using force-reduction factors which are based on the type of structural system.

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A.L. Vidot-Vega, M.J. Kowalsky / Engineering Structures 51 (2013) 128–140 129

The displacements or drifts are compared at the end of the designwith the code-based limits. Usually a linear elastic analysis of thestructure is performed under the lateral forces calculated fromthe procedure and the displacements obtained are multiplied bya deflection amplification factor and then compared to allowabledrift limits.

Displacement-based methods use displacement or drifts as thestarting point of the design. Strength and stiffness are not variablesin the procedure -they are the end results. Performance-basedmethods such as DDBD allow the assessment of the actual perfor-mance capability of individual building designs, which cannot beaccomplished by FBD methods. As a result, the performance capa-bility of buildings designed using traditional methods can be vari-able and, for a given building, may not be specifically known. Theperformance of some buildings designed using FBD can be betterthan the minimum standards anticipated by the code, while theperformance of others could be worse [9].

3. DDBD for frame buildings

The DDBD procedure was developed with the aim of providing agreater emphasis on displacements throughout a variety of perfor-mance limit states. Using this method, a structure is designed toachieve a predefined level of lateral displacement when subjectedto a predefined level of seismic intensity [11]. DDBD uses the con-cept of an equivalent linear system (ELS) defined by an equivalentdamping and the secant stiffness evaluated at maximum responseto represent the response of a non-linear system (Fig. 1). The basicsteps of the DDBD method for moment frame buildings are de-scribed briefly. Additional information can be found elsewhere[11].

� Step 1: Selection of target drift—based on the desired perfor-mance. This is selected by the designer and is often specifiedon the basis of drift or strain. The structure is represented byan SDOF oscillator, as illustrated in Fig. 1a, having an effectivemass (me) and an effective height (He).� Step 2: The displacement profile (Di) is obtained based on the

first inelastic mode shape (di) of the structure using Eq. (1),where Dc is the critical or maximum drift; which usually occursbetween the ground base and first floor in a RC frame. Theinelastic mode shape approximates the displaced shape of aninelastic system at peak response. The critical displacement(Dc) is then obtained by multiplying the height of the criticalfloor to the target drift (Step 1). According to past researchers[10–12] for frame buildings the normalized inelastic modeshapes are given by (Eq. (2)):

Di ¼ diDc

dc

� �ð1Þ

(a) (bFig. 1. Basic step

di ¼Hi

Hnn 6 4

di ¼43

Hi

Hn

� �1� Hi

4Hn

� �� n > 4

ð2Þ

where n is the total number of storeys, Hi is the height of storey ifrom the base, Hn is the total height of the building, and dc is themode shape value at the critical level.� Step 3: Calculation of system displacement (Dsys)—this is

obtained using the displacement profile from step 2 and themasses (mi) at each storey. It represents the equivalent systemdisplacement of the SDOF substitute structure and is given asfollows:

Dsys ¼Pn

i¼1miD2iPn

i¼1miDið3Þ

� Step 4: Selection of seismic demand and calculation of effectivedamping—for DDBD this may be expressed as a displacementresponse spectra (DRS) generated for several levels of damping,as illustrated in Fig. 1b. Expressions for equivalent viscousdamping include components due to hysteretic and viscousdamping and can be defined as a function of ductility for differ-ent materials and systems, which are found in the literature[2,12].� Step 5: Calculation of structure effective period— the period (Te)

is found by entering with the equivalent system displacementfound in Step 3 in a design spectra corresponding to the calcu-lated equivalent viscous damping from Step 4. Fig. 1b illustratesthis step.� Step 6: Calculation of design base shear—once the effective per-

iod of the substitute structure is obtained in Step 5, the effectivestiffness (Ke) is obtained using Eq. (4). The effective stiffness ofthe substitute structure is defined as the secant stiffness tomaximum response, as shown in Fig. 1c. The design base shearforce (Vbase) at the design limit state is then obtained by multi-plying the effective stiffness (Eq. (4)) by the system displace-ment (Eq. (3)).

Ke ¼4p2me

T2e

ð4Þ

Vbase ¼ KeDsys ð5Þ

� Step 7: Structural analysis and design—A frame analysis is per-formed under the action of the lateral force vector. Capacityprotected members are defined using cracked section proper-ties, while plastic hinge members are defined using effectivestiffness properties. The beam and column elements aredesigned using the forces obtained from the analysis and capac-ity design principles.

) (c)s for DDBD.

Table 1Properties of the frame buildings designed using DDBD and FBD (revised).

No. storeys No. bays Floor height (m) Beam length (m) DDBD FBD DDBD FBD DDBD FBD

Beam section (mm �mm) Column section (mm �mm) Target drift (%) Drift limit (%)

4 2 3.35 4.88 406 � 508 457 � 609 508 � 508 660 � 660 2.0 2.04 3 3.35 4.88 406 � 508 406 � 558 508 � 508 660 � 660 2.0 2.06 2 3.66 6.10 406 � 508 457 � 609 508 � 508 609 � 609 2.0 2.06 3 3.66 6.10 406 � 508 406 � 558 508 � 508 609 � 609 2.0 2.08 2 3.66 6.10 406 � 508 457 � 609 508 � 508 609 � 609 2.0 2.0

12 2 3.35 5.49 457 � 609 609 � 762 609 � 609 864 � 864 2.0 2.0

Table 2Nominal moment capacities of the beams for FBD and DDBD frames.

No. bays No. Storey FBD initial FBD revised DDBD

Nominal moment capacity (Mn): KN-mTwo bay 4 473 581 325

6 456 683 4118 514 746 495

12 911 1174 875

Three bay 4 395 533 3546 549 724 514

130 A.L. Vidot-Vega, M.J. Kowalsky / Engineering Structures 51 (2013) 128–140

4. Application of FBD and DDBD methods To RC frame buildings

The DDBD method is illustrated with the design of several RCframes with 4, 6, 8, and 12-storeys. The FBD method described inASCE7-05 [1] was also used to design the same RC frames. Thebuildings were symmetrical with two and three bays (Table 1).These frames were designed to deform according to the beam–sway mechanism, in which flexural plastic hinges at the ends ofthe beams and at the column base of the first floor are expected.All RC sections have distributed reinforcing steel around the perim-eter. Sample sections for the beams and columns used in all theexamples are shown in Fig. 2. Longitudinal reinforcing ratios variedfrom 1% to 2.5% in beams and columns.

For the force-based designed frames, the sections were modi-fied to satisfy the allowable drift limit of 2% imposed in the ASCE7-05 [1] code. The allowable drift of 2% corresponds to occupancycategories of I or II [1]. Reduction factors (R) and deflection ampli-fication factors (Cd) of 8 and 5.5 were used for design (special RCmoment-frames), respectively. The sections presented on Table 1for FBD frames represent the final revised designs that satisfy thedrift limit of 2%. All the DDBD and FBD frames started with thesame sections; however for the DDBD frames, no iteration wasneeded. Table 2 shows the nominal moment capacities for thebeam sections used for FBD initial, FBD revised and DDBD frames.

In addition to the self-weight of the beams and columns, thefloor weights per length without the top floor (slab + superimposeddead load) were 75 kN/m for the 4-storey frames, 56 kN/m for the6 and 8-storeys frames, and 95 kN/m for the 10 and 12-storeysframes. The floor weights per length for the top floors were61 kN/m for the 4-storey frames, 45 kN/m for the 6 and 8-storeysframes, and 79 kN/m for the 10 and 12-storeys frames.

The design spectral acceleration and displacement spectrumsare presented in Fig. 3, which corresponds to ASCE7-05 [1] siteclass C with S1 and Ss equal to 0.47 and 1.37, respectively. TheDDBD method can be used to account for different performancelimit states such as serviceability, damage control and survival,among others. As the purpose of the examples is to compare the re-sults with the buildings designed using FBD, several limits werenot considered as part of the initial comparison. In order tocompare both methods, the frames were also designed for 2%

BEAM COLUMN

Fig. 2. Sample RC sections used for FBD and DDBD examples.

target drift using the DDBD method. Section 6.4 of the paper ex-plores the design of the frames using DDBD for the serviceabilitylimit state.

5. Non-linear time history analysis of the RC frames

5.1. Ground motions used in the NLTHA

Non-linear time history analysis (NLTHA) of the frames de-signed with DDBD and FBD methods were performed using spec-trum compatible records [15]. A combined frequency–timedomain scaling type was used to match the target spectrum. Toavoid the known shortcomings of fully artificial records, the re-cords used in this work were obtained from the modification ofhistoric records. The continuous wavelet transform (CWT) wasused to de-compose the original acceleration time-series into anumber of time series with energy in non-overlapping frequency-bands. An iterative procedure [15] is used to scale each time his-tory so that when they are added together they produce a spec-trum-compatible ground motion. Seven earthquakes were usedin the analysis (Fig. 4). Table 3 shows the name, year, and stationof the earthquakes used in the NLTHA. These ground motions wereselected in order to have a good range of different characteristicsincluding strong ground motion duration, and number of cycles.

5.2. Modeling approach and hysteric material models

The frame models were developed using the program OpenSees[7]. The elements were modeled using a fiber based lumped plas-ticity approach (Fig. 5). Within OpenSees, this was accomplishedby using the ‘‘beam with hinges’’ element [13]. This element con-sists of flexural plastic hinges that are modeled with fiber ele-ments, while regions outside the plastic hinge are modeled usingtraditional frame element properties. The confined and unconfinedconcrete in the fiber sections was modeled using the Kent and Park[5] concrete model with degraded linear unloading/reloading stiff-ness [4]. This model (Concrete01 in OpenSees) assumes no tensilestrength for concrete. The concrete 28 day compressive strength(fpc), concrete strain at maximum strength (epsc0), and concretecrushing strain (epsU) and strength (fpcu) are the required inputfor this concrete model. These parameters were calculated as

Spec

tral A

ccel

erat

ion

(g)

0

0.2

0.4

0.6

0.8

1.0

0 5Period (sec)

10 15

Sd [m

]

0.25

0.50

0.75

1.00

1.25

1.50

00

5Period [sec]

10 15

(a) (b)Fig. 3. (a) Spectral acceleration and (b) displacement spectrums.

Fig. 4. Acceleration response spectrum of the earthquakes used in the NLTHA.

Table 3Earthquakes used in the NLTHA.

Number Earthquake Name Year Station name

EQ1 Northridge-01 1994 Bell Gardens – JaboneriaEQ2 Northridge-01 1994 Canoga Park – Topanga CanEQ3 Chi-Chi, Taiwan-05 1999 CHY017EQ4 Imperial Valley-06 1979 El Centro Array #5EQ5 Chi-Chi, Taiwan-04 1999 CHY046EQ6 Chi-Chi, Taiwan-06 1999 TCU109EQ7 Superstition Hills-02 1987 El Centro Imp. Co. Cent

Linear Elastic

Lpi Lpj

Node i Node j

Elastic properties: E, Iz, A

Fig. 5. Beam with hinges element.


proposed by Mander et al. [6]. The concrete strength remains con-stant beyond the crushing point. The steel fibers were modeledusing a reinforcing steel model developed by Moehle and Kunnath[8]. Linear elastic properties are used for the rest of the element.The bond slip (yield penetration) was not directly modeled in themembers. However, the plastic hinge length does account for thisindirectly through the length of strain penetration, thus addressing

the impact of bar slip on the member deformation. The elasticdamping in all the dynamic simulations was represented by 2.0%tangent-stiffness proportional damping.

Steel tensile and concrete compression strains at maximumdrifts were obtained at each analysis. The strains were obtainedat the critical section in the beam plastic hinges since the desiredmechanism of inelastic deformation for these frames involves theformation of the flexural plastic hinges at the ends of the beams.Displacements at each storey were also obtained.

6. NLTHA results and comparison

6.1. Drift and displacement profiles for the FBD frames before driftlimit check

This section presents the results obtained from the NLTHA ofthe frames designed using ASCE7-05 [1] design code that did notsatisfy the drift limit at the initial iteration where the drift is eval-uated by taking the elastic drift and multiplying by Cd. Figs. 6–8show the average interstorey drift profile of the frames obtainedfrom the NLTHA, the code allowable drift limit and the drifts ob-tained from elastic analyses times the deflection amplification fac-tor (Cd). Figs. 9–11 show the displacement profiles obtained fromthe NLTHA and the elastic displacements times Cd. The y-axis ofthese plots represents the height of each storey expressed as per-centage of the total height (Hn) of the frame. It is noted that thedrifts obtained from NLTHA are very different from the drifts ob-tained from elastic analyses times Cd for all cases. It is also ob-served that the elastic analyses multiplied by Cd over predict thedisplacements in the frames that were analyzed. It was also ob-served that the initially designed frames have satisfied the driftlimits imposed on the ASCE7-05 [1] in the cases studied in this pa-per (when examining the non-linear time history analysis). This re-sult shows that in the majority of the cases the drift limits imposedby codes force the designer to increase section size when it is actu-ally not needed and that elastic analysis clearly deviates from thenon-linear response of the structure.

6.2. Interstorey drift vs. target or allowable values for FBD revised andDDBD frames

From the results of the previous section, it is apparent that thereis a discrepancy between the drifts obtained from non-linear timehistory analysis and the multiplication of the elastic drift times Cd.Assuming that the non-linear time history analysis is correct, thestructures as designed in Section 6.1 would largely be acceptable.However, it is more likely that the performance of those structureswould have been evaluated on the basis of the amplified elastic

0 1 2 3 4 50

20

40

60

80

100

Interstorey Drift (%)

H/H

n (%)

NLTHACd*elasticDrift Limit

0 1 2 3 4 50

20

40

60

80

100


H/H

n (%)


(a) (b)

Fig. 6. Average drift profiles for the FBD four storey frames with (a) two, (b) three bays before drift check.

0 1 2 3 4 50

20

40

60

80

100


H/H

n (%)

NLTHACd*elastic

Drift Limit

0 1 2 3 4 50

20

40

60

80

100


H/H

n (%)


(a) (b)

Fig. 7. Average drift profiles for the FBD six storey frames with (a) two, (b) three bays before drift check.

0 1 2 3 4 50

20

40

60

80

100


H/H

n (%)


0 2 4 60

20

40

60

80

100


H/H

n (%)


(a) (b)

Fig. 8. Average drift profiles for the FBD frames with (a) 8 and (b) 12 storeys with two bays before drift check.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

Displacement (m)

H/H

n (%)

EQ1EQ2EQ3EQ4EQ5EQ6EQ7Average THelastic*Cd

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

Displacement (m)

H/H

n(%)


(a) (b)

Fig. 9. Displacement profiles for the FBD four storey frames with (a) two bays and (b) three bays before drift check.


0 0.2 0.4 0.6 0.80

20

40

60

80

100

Displacement (m)

H/H

n (%)


0 0.2 0.4 0.6 0.80

20

40

60

80

100

Displacement (m)

H/H

n (%)


(a) (b)

Fig. 10. Displacement profiles for the FBD six storey frames with (a) two bays nd (b) three bays before drift check.

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

Displacement (m)

H/H

n (%)


0 0.5 1 1.5 20

20

40

60

80

100

Displacement (m)

H/H

n (%)


(a) (b)

Fig. 11. Displacement profiles for the FBD frames with (a) 8 and (b) 12 storeys with two bays before drift check.


drifts, which indicate unacceptable results. As a consequence, inthis section, the same structures are re-designed until the drifts ob-tained by multiplying the elastic drift by Cd equal to allowable driftlimit. This will of course result in structures that are much strongerthan they need to be. However, unless the engineer employs non-linear time history analysis of the structures designed in Sec-tion 6.1, that information would not be known. Also presented inthis section are the results of structures designed according toDDBD where the target displacement is based on the code allow-able drift limit. Table 2 shows that the moment capacity of thebeam sections for the revised FBD frames (that satisfy the allow-able drift limit) are much larger than the initial FBD (that did notsatisfy the allowable drift limit) and DDBD sections (designed for2% target drift).

Then, NLTHA of the revised FBD frames and DDBD were per-formed to obtain drift and displacement profiles in order to com-pare both approaches. The drift profiles of the revised FBD andDDBD are shown in Figs. 12–15. The interstorey drifts of the framesare shown on the x-axis and the height (y-axis) of each level is ex-pressed as percentage of the total height (Hn). Also shown in thesefigures is the average drift from all seven earthquakes and the tar-get drift in the case of DDBD or the allowable drift for the revisedFBD frames. The target drift for DDBD and allowable drift for FBDare both 2%. Tables 4 and 5 present the maximum interstorey driftsobtained for the NLTHA with the seven earthquakes, average val-ues and target or allowable drifts for the frames designed withDDBD and FBD, respectively. From Figs. 12 and 14 it can be ob-served that the target interstorey drift at the critical storey wasreached in all the DDBD cases. The average interstorey drift ob-tained from the seven simulations was in close agreement withthe target drift for the frames with two and three bays. It can beobserved from Figs. 13 and 15 that the allowable drift for theforce-based designed frames was met when compared with the

average drift obtained from the seven simulations, but in generalthere is an underestimation of the drifts. However, it is interestingto compare the drifts obtained with the NLTHA and the ones ob-tained using elastic analyses and the deflection amplification factorCd established by the code (Figs. 13 and 15). Notice that the driftsobtained by the elastic analyses times the Cd are very different(usually lower) at the first floors until the critical storey, then thedrift profile shape becomes similar to the shape obtained fromNLTHA. The elastic analyses overestimate the drift at the higherstoreys in the frames. Next, the displacement profiles will becompared.

6.3. Displacement profiles vs. target or allowable values for FBDrevised and DDBD frames

Displacement profiles for frames with 4, 6, 8 and 12 storeys areshown in Figs. 16–21. For the DDBD frames it is shown in the fig-ures the target displacement profiles (dashed line); meanwhile forthe FBD frames it is shown the elastic displacements times thedeflection amplification factor (dashed line). Figs. 16 and 17 showsthe results for the 4-storey frames with two and three bays de-signed using DBDD and FBD, respectively. It is noted that the targetdisplacement shape from DDBD is in close agreement with theNLTHA results (Fig. 16).The displacements are usually overesti-mated at the top floors due to the use of the same sections in allthe storeys. Nevertheless the results obtained show confidence inthe prediction of storey displacements using direct-displacementbased design. From Fig. 17 it is observed that for some of the earth-quakes, the displacements obtained exceed the elastic displace-ments times the deflection amplification factor for the FBDframes. In general, the displacement shape obtained from the elas-tic analyses underestimate the average displacements obtainedfrom NLTHA at the floors located at approximately 45% of the total

0 1 2 30

20

40

60

80

100


H/H

n (%)

EQ1EQ2EQ3EQ4EQ5EQ6EQ7Average THDDBD Target

0 1 2 30

20

40

60

80

100


H/H

n (%)


0 1 2 30

20

40

60

80

100


H/H

n (%)


0 1 2 30

20

40

60

80

100


H/H

n (%)


n = 6

n = 8

n = 4

n = 12

Fig. 12. Interstorey drift profiles for the 4, 6, 8, 12-storey frames with two bays designed using DDBD.

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100


H/H

n (%

)

EQ1EQ2EQ3EQ4EQ5EQ6EQ7Average THDrift Limitelastic*C d

0 1 2 30

20

40

60

80

100


H/H

n (%

)

EQ1EQ2EQ3EQ4EQ5EQ6EQ7Average THDrift Limitelastic*Cd

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100


H/H

n (%)


0 0.5 1 1.5 2 2.5 30

20

40

60

80

100


H/H

n (%

)


n = 4

n = 8

n = 6

n = 12

Fig. 13. Interstorey drift profiles for the 4, 6, 8, 12-storey frames with two bays designed using FBD.

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100


H/H

n (%)


0 0.5 1 1.5 2 2.5 30

20

40

60

80

100


H/H

n (%)


n=6n=4

Fig. 14. Interstorey drift profiles for the 4 and 6-storey (n) frames with three bays designed using DDBD.


height, meanwhile at the top floors (after 45% of height) the dis-placements are overestimated by small amount in the FBD frames.Similar results are noted for the 6, 8 and 12-story buildings inFigs. 18–21. From this section, one can conclude that it is possibleto achieve designs with FBD that will meet the allowable drift if

used in an iterative manner. In many case, 2–3 iterations wereneeded to converge. It is important to remember that such itera-tion in the design is actually not needed if one examines thenon-linear time history results of Section 6.1 However, assumingthat the engineer does not perform a non-linear time history

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100


H/H

n (%)

H/H

n (%)


0 0.5 1 1.5 2 2.5 30

20

40

60

80

100



n=6n=4

Fig. 15. Interstorey drift profiles for the 4 and 6-storey (n) frames with three bays designed using FBD.

Table 4Interstorey drift values (%) for frames designed using DDBD.

Storey-bay Eq1 Eq2 Eq3 Eq4 Eq5 Eq6 Eq7 Average Target

4-2bay 1.96 2.41 1.71 1.96 1.85 1.85 1.97 1.91 2.006-2bay 1.94 2.20 1.83 2.34 2.00 1.90 1.95 1.95 2.008-2bay 2.06 1.85 2.04 2.01 2.06 2.00 2.35 2.05 2.0012-2bay 1.99 1.78 1.94 2.31 1.71 2.00 2.21 2.00 2.004-3bay 1.56 2.15 1.64 1.77 1.69 1.99 2.4 1.90 2.006-3bay 1.71 2.30 1.67 1.71 1.99 1.97 1.81 1.88 2.00

Table 5Interstorey drift values (%) for frames designed using FBD.

Storey-bay

Eq1 Eq2 Eq3 Eq4 Eq5 Eq6 Eq7 Average Limit

4-2bay 1.61 1.79 1.93 1.83 1.42 1.67 2.34 1.79 2.006-2bay 1.52 1.86 1.45 1.46 1.54 1.44 1.52 1.54 2.008-2bay 1.53 1.84 1.31 1.72 1.90 1.63 1.65 1.59 2.0012-2bay 1.80 1.34 1.72 1.49 1.81 1.37 1.70 1.60 2.004-3bay 1.60 2.48 2.04 1.80 1.67 1.63 2.04 1.89 2.006-3bay 1.48 1.54 1.74 1.23 1.55 1.58 1.69 1.54 2.00


analysis, one is left with the impression that the structures of Sec-tion 6.1 will greatly exceed the code drift limits and that the FBDrevised designs shown in this section represent the appropriate de-signs. Through the process of iteration, the strength of the FBDstructures changed as shown in Table 2.The revised FBD sectionshave larger required moment capacities. Lastly, it is important torealize that through the use of DDBD, one can arrive directly, with-out iteration, to a suitable and efficient design that meet the driftcriteria with significantly lower levels of required strength (see Ta-ble 2). In the next section, material strains (steel and concrete) willbe compared for the serviceability limit.

0 0.1 0.2 0.3 0.40

20

40

60

80

100

Displacement (m)

H/H

n(%) EQ1

EQ2EQ3EQ4EQ5EQ6EQ7Average THDDBD Target

(a)Fig. 16. Displacement profiles for the DDBD four stor

6.4. Serviceability limit for DDBD frames

The validity of the DDBD method to design for any limit state isexplored in this section. The serviceability limit state based onmaterial strains was selected to design the same frames previouslydiscussed. The serviceability concrete compression strain is de-fined as the strain at which crushing is expected to begin (usually0.004), while the serviceability steel tensile strain is defined as thestrain (0.015) at which residual crack widths would exceed 1 mm[11]. The serviceability limit is controlled by the strain limit thatis reached first. Target drifts according these values of strain werefound using Eqs. (6) and (7) [16]. These equations can be used todirectly correlate material strains to interstorey drifts in RC framebuildings. The steel tensile strain controls the serviceability limit inthese examples. Therefore a steel strain of 0.015 was used to com-pute target drifts using Eq. (6) in this section. In these equations,hby is the yield beam drift (Eq. (8)), Lp is the plastic hinge length(Eq. (10)), dbl is the assumed diameter of the reinforcing bar, qlong

is the assumed longitudinal reinforcing ratio, LB is the beam length,HB is the beam depth, es is the steel strain, ec is the concrete strain,concrete compression strength (fc0), and Fy and Fu are the yield andultimate strength, respectively.

0 0.1 0.2 0.3 0.420

40

60

80

100

Displacement (m)

H/H

n (%)


(b)ey frames with (a) two bays and (b) three bays.

0 0.1 0.2 0.3 0.40

20

40

60

80

100

Displacement (m)

H/H

n (%) EQ1

EQ2EQ3EQ4EQ5EQ6EQ7Average THelastic*Cd

0 0.1 0.2 0.3 0.40

20

40

60

80

100

Displacement (m)

H/H

n (%

) EQ1EQ2EQ3EQ4EQ5EQ6EQ7Average THelastic*Cd

(a) (b)Fig. 17. (a) Displacement profiles for the FBD four storey frames with two bays and (b) with three bays.

0 0.1 0.2 0.3 0.4 0.50

20

40

60

80

100

Displacement (m)

H/H

n (%) EQ1

EQ2EQ3EQ4EQ5EQ6EQ7Average THDDBD Target

0 0.1 0.2 0.3 0.4 0.50

20

40

60

80

100

Displacement (m)

H/H

n (%)


(a) (b)Fig. 18. Displacement profiles for the DDBD six storey frames with (a) two bays and (b) three bays.

0 0.1 0.2 0.3 0.4 0.50

20

40

60

80

100

Displacement (m)

H/H

n (%)


0 0.1 0.2 0.3 0.4 0.50

20

40

60

80

100

Displacement (m)

H/H

n (%)


(a) (b)Fig. 19. Displacement profiles for the FBD six storey frames with (a) two bays and (b) three bays.


Using steel tensile strains (es):

ht arg et ¼ hby þ 1:75esx0:15s � 2:1ey

� � Lp

HB

� �� 0:70e�0:080

s ð6Þ

Using concrete compression strain (ec):

ht arget ¼ hbyþðð�1:6�xsþ4Þecþ0:004�2:1eyÞLp

HB

� �� 0:85e�0:065

c ð7Þ

hby ¼ 2:1eYLB

6HB

� �ð8Þ

xs ¼ qlongFy

f 0cð9Þ

Lp ¼ maxkLþ 0:022Fydbl

0:044Fydbl

� ; k ¼ 0:2

Fu

FY� 1

� �6 0:08;

L ¼ LB=2 ðdouble bendingÞ ð10Þ

The frames designed for the serviceability limit were analyzedalso using OpenSees. NLTHA of the frames using the same earth-quake records (Table 3) previously discussed were also performed.The focus of these analyses was to determine if the strains obtainedfrom the analysis correlated well with the target strains. Steel ten-sile strains were obtained in the beams at critical storeys (usuallyfirst or second storey). The strains for the FBD frames were also ob-tained for the initial and revised designs in order to see tendenciesin the level of strains at the allowable drift limit of 2%. The strainresults for the frames designed for serviceability limit using DDBD

0 0.2 0.4 0.6 0.80

20

40

60

80

100

Displacement (m)

H/H

n (%)


0 0.2 0.4 0.6 0.80

20

40

60

80

100

Displacement (m)

H/H

n (%)


(a) (b)Fig. 20. Displacement profiles for the DDBD (a) eight storey frame and (b) 12 storey frame with two bays.

0 0.2 0.4 0.6 0.80

20

40

60

80

100

Displacement (m)

H/H

n (%)


0 0.2 0.4 0.6 0.80

20

40

60

80

100

Displacement (m)

H/H

n (%)


(a) (b)Fig. 21. Displacement profiles for the FBD (a) eight storey frame and (b) 12 storey frame with two bays.


and FBD (initial and revised) were plotted in the same figures.Fig. 22a and b shows the average maximum steel tensile strainsat the beams for the frames designed both methods with twoand three bays, respectively. Again, the strains were obtained atthe beams since the frames were designed to follow the beam-sway mechanism (weak beams–strong columns). The steel tensilestrains are pretty close to the serviceability limit of 0.015 (dashedline) for which the frames were designed using DDBD. For the FBDframes, the maximum tensile strains exceed the serviceabilitystrains in some cases, especially for the two-bay frames (steelstrain as high as 0.020) for the revised frames. Nevertheless, FBDmethods are not formulated to control strain levels. Similar steeltensile strains were obtained for DDBD and the initial FBD frames.There is no clear tendency in the strains obtained for the FBDframes, although these results show that at the allowable driftlimit of 2% the FBD frames will be close to the serviceability limitbased on steel and concrete strains of 0.015 and 0.004,respectively.

(a)Fig. 22. Average steel tension strains from the NLT

6.5. Concrete compression strains

Although the DDBD frames in Section 6.4 were designed for asteel tensile strain of 0.015 since this limit controls the designand not a serviceability concrete strain of 0.004, it was deemedimportant to also obtain the concrete compression strains in thecritical beams (where maximum strain occurs). Fig. 23a and bshows average values of maximum concrete compression strainsobtained from the NLTHA as a function of the number of storeysfor frames with two and three bays, respectively. In these figuresit is also included the serviceability strain limit of 0.004 for com-parison purpose. Higher strains were obtained in frames designedusing DDBD than in the revised FBD frames. Similar concrete com-pression strains were obtained for DDBD and the initial FBDframes. The concrete compression strains for all DBDB frames donot exceed the serviceability limit of 0.004. This outcome was ex-pected since the frames were designed to reach first the steel strainof 0.015 and not the concrete strain of 0.004.

(b)HA for frames with (a) two and (b) three bays.

(a) (b)Fig. 23. Average compression concrete strains from the NLTHA for frames with (a) two and (b) three bays.


7. Trends at the ASCE 07-05 implied ductility and drift limits

This section explores the implications of the reduction factors,deflection amplification factor (Cd) and drift limits imposed inthe ASCE 07-05 [1] design code. Past experience and observationof building behavior following earthquakes has shown that a struc-ture can be economically designed for a fraction of the estimatedelastic seismic design forces, while maintaining the basic life safetyperformance objective. This design philosophy implies that struc-tural inelastic behavior (and damage) is expected. This reductionin design seismic force is effected through the use of a Seismic Re-sponse Modification Factor, R. A Deflection amplification factor isintroduced to predict expected maximum deformations from thatproduced by the design seismic forces. To control primarily non-structural damage, several drift limits are adopted.

Using Eqs. (6) and (7) trends for steel strains, concrete strainsand drifts were explored for frames with beam aspect ratios vary-ing from 4 to 16 and with the implied ductility by the ASCE7-05 Cd

factor, which is equal to 5.5 for special RC moment frames. Beamaspect ratios are defined by the ratio between the length and depthof the beam (LB/HB). Different longitudinal steel ratios on beamswere also considered (1–3.0%). A reinforcing steel bar diameter of25.4 mm was used in all the calculations. Eqs. (6) and (7) were di-vided by the yield drift of the frames (Eq. (11)) to obtain expres-sions to compute displacement ductilities as function of differentstrain limits (concrete and steel). The yield drift (Eq. (11)) includesthe contribution of several components according to Priestley [12]such as: (1) the joint rotation due to shear and flexure, and (2) col-umn deformation due to shear and flexure.

hy ¼ 0:5eYLB

HB

� �ð11Þ

Fig. 24a and b shows the steel tensile strains as function ofbeam aspect ratios for different longitudinal steel ratios and forbeam depths of 508 mm and 750 mm, respectively. These plotswere obtained by substituting the implied ductility of 5.5 on the

4 6 8 10 12 14 160

0.02

0.04

0.06

0.08

stee

l stra

in

aspect ratio

1.0%1.5%2.0%2.5%3.0%

(a)Fig. 24. Steel tensile strains for frames with different beam aspect ratios

expressions of displacement ductilities (htarget/hy) based on strainlimits. Fig. 25a and b shows the concrete compression strains asfunction of beam aspect ratios for different longitudinal steel ratiosand for beam depths of 508 mm and 750 mm, respectively. It isnoted that the steel tensile and concrete compression strains in-crease as the beam aspect ratio increases. There is a change of slopein the curve that occurs due to the plastic hinge length value. Asthe beam becomes larger, the plastic hinge length becomes domi-nated by the length of the beam itself and not for the strain pene-tration part (Eq. (10) – first one). For shorter beams, the strainpenetration part becomes more important (Eq. (10) – secondone), thus the value is constant until the change of slope occurs.The longitudinal steel ratios have less impact on the strain limits.The steel tensile and concrete compression strains for the impliedductility of 5.5 exceed the serviceability values (0.015 and 0.004,respectively) in all cases. This result indicates that for the impliedductility level of 5.5 based on Cd the damage level, as defined bystrains in beam plastic hinges will vary greatly. The use of constantforce reduction and deflection amplification factors does not implyconstant level of damage across a variety of building geometries fora given structure type. The determination of the difference in thelevel of damage for the same type of structure subjected to thesame earthquake however cannot be quantified using these analy-ses. To determine the difference in damage it will be necessary toperform detailed analyses and support them by experimentaltesting.

The interstorey drifts for the steel and concrete strains at theimplied ductility were obtained and are shown in Fig. 26. It is ob-served that the drifts exceed in all cases the allowable drift of 2%imposed by the ASCE 7-05 code. The allowable drift of 2% was usedto compute displacement ductilities at the strain levels implied by5.5 (Fig. 27) for frames with different aspect ratios. It is also shownthat the drift of 2% does not match with the implied ductility levelof the code. As a result, in many cases, the code drift limit willgovern the design with ductility levels far smaller than thatimplied by the force reduction factor given by the same code.

4 6 8 10 12 14 160.02

0.03

0.04

0.05

0.06

0.07

0.08

stee

l stra

in

aspect ratio

1.0%1.5%2.0%2.5%3.0%

(b)and implied ductility of 5.5 for (a) H = 508 mm and (b) H = 750 mm.

4 6 8 10 12 14 160

0.005

0.010

0.015

0.020

0.025

conc

rete

stra

in

aspect ratio

1.0%1.5%2.0%2.5%3.0%

4 6 8 10 12 14 160.005

0.010

0.015

0.020

0.025

conc

rete

stra

in

aspect ratio

1.0%1.5%2.0%2.5%3.0%

(a) (b)Fig. 25. Concrete compression strains for frames with different beam aspect ratios and implied ductility of 5.5 for (a) H = 508 mm and (b) H = 750 mm.

4 6 8 10 12 14 160

0.02

0.04

0.06

0.08

0.10

inte

rsto

rey

drift

aspect ratio

ASCE 7 -05 drift limit

Fig. 26. Maximum interstorey drifts for frames with different beam aspect ratiosand implied ductility of 5.5.

4 6 8 10 12 140

1

2

3

4

5

6

disp

lace

men

t duc

tility

aspect ratio

ASCE 7-05 implied ductility

Fig. 27. Displacement ductility for frames with different beam aspect ratios forallowable drift of 2% at strain limits for 5.5, H = 750 mm.


While such an outcome may acceptable, the problem arises withthe incompatibilities between drift limits and force reduction fac-tors used in force-based design. It should be noted that this doesnot suggest that structures designed by such an approach are un-safe. To the contrary, they are likely to have greater than requiredstrength and required ductility capacity, both of which are fine forlife safety considerations. However, such an approach is of limitedvalue if one desires to implement some form of performance basedseismic design and to guarantee uniform risk among designedstructures. The design of a RC frame with an implied ductility of5.5 will generally will be dominated by gravity loads instead ofseismic loads.

8. Summary and conclusions

This paper had two main objectives: (1) Compare importantparameters such as: drifts, displacements, and material strains

obtained from DDBD and FBD frames. (2) Investigate the impactsof the ASCE7-05 design code allowable drifts and ductility limitson the design of RC frames. To accomplish these objectives, sixdifferent buildings were designed using force-based and directdisplacement-based methods. Results from this research indi-cates the following: (1) Smaller RC sections were obtained usingDDBD, resulting in some savings in material, (2) both the initialand revised FBD sections have larger required moment capacitiesthan those from DDBD, (3) maximum interstorey drifts werecontrolled with good accuracy using both methods, although inthe case of FBD, there is a large discrepancy between expectedand actual drifts based on non-linear time history analysis unlessan iterative approach is utilized, (4) elastic analyses underesti-mates the displacement at the first storeys (45% of total frameheight) and overestimate displacement at the top storeys, (5)constant values of drift ratios and ductility demands imposedin building codes are in general not in agreement with eachother, and that in neither case do they imply uniform levels ofdamage, (6) elastic drift and displacements with the deflectionapplication factor in FBD are substantially different from theNLTHA, (7) unless a NLTHA is conducted on FBD designed struc-tures, the engineer may be sent into an unnecessary loop of stiff-ening the structure as was the case for the FBD revisedstructures discussed in this paper, and (8) the DDBD methodcontrols with good accuracy the strains at the beams for the ser-viceability limit state.

The direct displacement-based design allows the engineer to re-late drifts to material strains at the beams, which are better indica-tors of damage in the sections. This allows the designer to have abetter idea of the level of damage that is accepting during the de-sign process. The steel tensile strains were not exceeded for theDDB designed buildings.

References

[1] ASCE. Minimum design loads for buildings and other structures. ASCE7-05;2005.

[2] Dwairi HM. Equivalent damping in support of direct displacement-baseddesign with applications to multi-span bridges. Ph.D. Dissertation, Departmentof Civil, Construction and Environmental Engineering, North Carolina StateUniversity, Raleigh, NC; 2004.

[3] Gulkan P, Sozen M. Inelastic response of reinforced concrete structures toearthquake motions. ACI J 1974;71(12):604–10.

[4] Karsan I, Jirsa J. Behavior of concrete under compressive loadings. ASCE J StructDiv 1969;95(12):2543–63.

[5] Kent DC, Park R. Flexural members with confined concrete. ASCE J Struct Eng1971;97(7):1969–90.

[6] Mander JP, Priestley MJN, Park R. Theoretical stress–strain model for confinedconcrete. ASCE J Struct Eng 1988;114(8):1804–26.

[7] McKenna F, Fenves GL, Scott MH, Jeremic B. Open system for earthquakeengineering simulation-opensees; 2000 <http://opensees.berkeley.edu>.

[8] Mohle J, Kunnath S. Reinforcing steel: opensees user’s manual; 2006 <http://opensees.berkeley.edu>.

http://opensees.berkeley.edu




[9] FEMA 445. Next-generation performance-based seismic design guidelines(NEHRP). Washington, DC: ATC; 2006.

[10] Pettinga JD, Priestley M. Dynamic behavior of reinforced concrete framesdesigned with direct displacement-based design. J Earthq Eng 2005;9(2):309–30.

[11] Priestley MJN. Brief comments on elastic flexibility of RC frames, andsignificance to seismic design. Bullet New Zeal Nat Soc Earthq Eng 1998;31(4):246–59.

[12] Priestley M, Calvi GM, Kowalsky M. Displacement-based seismic design ofstructures. Italy: IUSS Press; 2007.

[13] Scott M, Fenves G. Plastic hinge integration methods for force-based beam–column elements. ASCE J Struct Eng 2006;132(2):244–52.

[14] SEAOC. Recommended lateral force requirements and commentary, 7th ed.;1995.

[15] Suarez LE, Montejo LA. Generation of artificial earthquakes via the wavelettransform. Int J Solids Struct 2005;42(21–22):5905–19.

[16] Vidot-Vega AL, Kowalsky MJ. Relationships between strain, curvature, anddrift in RC moment frames in support of performance-based seismic design.ACI Struct J 2010;107(3):291–9.

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FBD Vs DBD