Engineering Structures 69 (2014) 1–11

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Engineering Structures

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An equivalent accidental eccentricity to account for the effects oftorsional ground motion on structures

http://dx.doi.org/10.1016/j.engstruct.2014.02.0380141-0296/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Department of Civil Engineering, Indian Institute ofTechnology, Gandhinagar, India. Tel.: +91 9925433861.

E-mail address: dbasu@iitgn.ac.in (D. Basu).

Dhiman Basu a,b,⇑, Michael C. Constantinou b, Andrew S. Whittaker b

a Department of Civil Engineering, Indian Institute of Technology, Gandhinagar, Indiab Department of Civil, Structural and Environmental Engineering, University at Buffalo, State University of New York, Buffalo, New York, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 April 2013Revised 26 February 2014Accepted 28 February 2014

Keywords:TorsionEccentricityEarthquake ground motionSeismic designSeismic isolation

The seismic design of buildings and other structures should include provisions for inherent and acciden-tal torsion effects. Procedures developed decades ago for use with equivalent lateral force (static) analysishave been often used for response-history analysis with no investigation of whether the proceduresachieve the desired result; namely, robust framing systems with limited susceptibility to excessive tor-sional displacement. The utility of procedures presented in ASCE 7 for treating accidental eccentricity asmeans for accounting for the effects of torsional ground motion is examined by analysis of simple linearand nonlinear systems. Results indicate that these standards-based procedures do not achieve the desiredtrends when torsional ground motion effects are considered, namely, increased component demandswith increasing accidental eccentricity. An alternate approach for using accidental eccentricity conceptsin accounting for torsional ground effects is then proposed and verified in representative examples forsimple linear and nonlinear systems. In each case, component demand increases monotonically as theaccidental eccentricity increases.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Calculation of the seismic response of buildings and other struc-tures requires consideration of torsion. Standards of design prac-tice recognize the importance of torsional contributions tohorizontal displacement response and simplified procedures havebeen proposed to estimate these contributions. Two types of tor-sion are considered: natural (or inherent) and accidental. Naturaltorsion is the product of non-coincident centers of mass (CM)and rigidity (CR) at one or more floor levels in a structure.Accidental torsion is used to indirectly account for: (a) plandistributions of reactive mass that differ from those assumed indesign, (b) variations in the mechanical properties of structuralcomponents in the seismic force-resisting system, (c) non-uniformyielding of components in the seismic force-resisting system, and(d) torsional ground motion.

Seismic analysis and design of buildings require considerationof natural and accidental torsion. Rules are presented in ASCEStandard 7, Minimum Design Loads for Buildings and Other Structures[1] for use with Equivalent Lateral Force (ELF) or static analysis,and dynamic analysis, as summarized below.

1.1. ASCE 7 standard

Section 12.8.4 of ASCE 7 presents rules for addressing torsion ifthe ELF procedure is used to analyze a building. Section 16.1presents rules for use with dynamic analysis. Specifically,Section 16.1.5, Horizontal Shear Distribution, states ‘‘The distribu-tion of horizontal shear shall be in accordance with Section 12.8.4except that amplification of torsion in accordance withSection 12.8.4.3 is not required where the accidental torsion effectsare included in the dynamic analysis model.’’ That is, ASCE 7 allowsthe analyst to include accidental torsion in the models for dynamicanalysis but does not provide guidance as to how to do so. It hasbeen common practice to include these effects: (a) by ignoringthem in the dynamic analysis and then considering those in accor-dance with Section 12.8.4.3 of ASCE 7, or (b) by explicit consider-ation of the effects through the use of accidental eccentricity in amanner similar to that used in the ELF procedure but in dynamicanalysis.

A number of studies on accidental torsion have been reported inthe archival literature. De-La-Llera and Chopra [8–10] calculated avalue of the accidental eccentricity for use with the ELF procedureby studying the dynamic response of single and multistory build-ings subjected to torsional ground motion. The ground motionswere calculated from records of horizontal acceleration at thefoundation level of instrumented buildings by dividing the

2 D. Basu et al. / Engineering Structures 69 (2014) 1–11

difference in accelerations recorded at adjacent accelerographs bythe distance between them. These studies validated the ASCE-7procedure (see above) for including accidental eccentricity in theELF procedure. The studies also concluded that results obtainedfrom the study of single-story buildings provide approximatelycorrect results for multistory buildings. A limitation of thesestudies is that the torsional ground motions used for analysis arespecific to the instrumented buildings and do not represent free-field rotational motions that could be used for the analysis of anystructure. De-La-Llera and Chopra [11] also analyzed one-storysystems using (a) the ELF, and (b) response- history analysis proce-dures with only horizontal ground excitation. Five percent acciden-tal eccentricity was added to the actual eccentricity for analysis. Acomparison of results showed significant differences. De-La-Lleraand Chopra [12] studied the variation of the torsional amplification(see above) as a function of the ratio of the uncoupled torsional totranslational frequencies in one-story systems and proposed anenveloping procedure to compute member forces for design usingthe ELF procedure. De-La-Llera and Chopra [13] also developed amulti-step procedure to determine the increase in edge displace-ment due to accidental torsion by utilizing data recorded at thefoundation and floor levels of an instrumented building. The proce-dure was then extended and evaluated from data recorded in 12buildings [17]. Chandler [5] studied the effect of accidental torsionon inelastic response of buidings. Torsional component of groundmotion can also be contributed from the spatially varying horizon-tal components and its effect on structural response was investi-gated by Hao [15]. Experimental investigations have also beenreported studying the effect of torsional ground motion (De-La-Colina et al. [7] and Wolff et al. [22]). Recently, Sheikhabadi [19]investigated the adequeacy of code specified accidental eecntricity.

The ELF definition of accidental eccentricity has been used forresponse-history analysis to account for multiple effects, includinguncertainty in mass distribution and torsional ground effects butits technical basis has not been demonstrated. The effect of tor-sional ground motion on the torsional response and its relationto accidental eccentricity is studied in this paper. The effects ofuncertainty in mass distribution are not studied in the paper asthey are appropriately accounted for by the use of accidentaleccentricity. It is shown in this paper that the shifting of the centerof mass results in torsional response that does not necessarilyincrease as the accidental eccentricity increases, therefore it doesnot properly account for the effects of torsional ground motion.

Consider the simple three degree-of-freedom system shown inFig. 1. The plan dimensions of the single story structure are a � b.The floor plate is supported by six columns that have lateral stiff-ness K1, K2 and K3 as shown in the figure. The CR is located a dis-tance e from the CM. The calculation of CR is simple for thisstructure. The offset of the CR from the CM produces the naturalor inherent torsional moment, Mt. The total torsional moment forthis structure, including dynamic amplification, could be calcu-

Fig. 1. Analytical model.

lated as the product of the translational inertial force and a designeccentricity, ed, as follows:

ed ¼ aðe� bbÞ ð1Þ

where a is a dynamic amplification factor, b is a decimal fraction(set equal to 0.05 in ASCE 7), and b is the building plan dimensionperpendicular to the applied translational force. The product bb isthe accidental eccentricity. In ASCE 7, the dynamic amplificationfactor is applied to the accidental torsion only, and a = Ax, whereAx is the torsional amplification factor and has values in the rangebetween 1 and 3. The approach adopted in ASCE 7 is straightfor-ward and does not require explicit calculation of the CR at each floorlevel, since the natural torsion is directly taken into account in theanalysis of a mathematical model by applying the code-specifiedlateral load profile through the CM at each floor level. The calcula-tion of the CR at each floor level is not straightforward for a multi-story building and is dependent on the lateral force profile used forthe ELF procedure as discussed in Hejal and Chopra [16] and Basuand Jain [3]. Although procedures have been developed to accountfor the dynamic amplification of natural torsion when using anELF procedure (e.g., [20,14,3] they have not been adopted in ASCE 7.

The ASCE 7 rules for imposing accidental torsion in multi-storystructures do not provide guidance on whether a shift in the CM ofeach floor is to be ordered or random, and if the same shift is to beused at each floor level. If the dominant contributor to accidentaltorsion is torsional ground motion, an ordered shift in the CM ofeach floor plate is reasonable. If this is not the case, random shiftsare reasonable, with the net effect of accidental torsion likely beingsmall in the lower stories of a medium-to-high-rise structure.

Shifting the CM at each floor level to consider the effects of acci-dental torsion alters the modal properties of a structure and itsmodal damping ratios if Rayleigh damping is used with a standardsoftware [6] to describe the inherent damping in the structure.Note that actual torsional ground motion does not produce thesechanges. The impact of shifting the CM on the modal propertiesand structural response has not been discussed in the literatureand is studied below.

Single-story three degree-of-freedom systems are subjected totranslational and torsional components of seismic excitation tostudy the conventional treatment of accidental torsion. The studyshows the limitations of the conventional approach when usedwith response-history analysis. An alternative definition of acci-dental eccentricity is proposed and verified by a series of analysesof single story elastic systems and nonlinear seismic isolationsystems.

2. Mathematical model for dynamic analysis

The one-story singly symmetric system shown in Fig. 1, com-posed of a rigid deck of mass m supported on six massless lat-eral-load-resisting elements, is used for analysis. The CM of thedeck is located at its geometric center and its radius of gyrationabout a vertical axis passing through the CM is r. Each lateral-load-resisting element has identical translational stiffness in thetwo orthogonal directions but no torsional stiffness. The systemis symmetric about the x axis but has an eccentricity e about they axis. This system could represent a seismic isolation system sup-porting a rigid superstructure or a single story singly symmetricbuilding. The system is subjected to both translational seismicexcitation along the y axis and torsional ground excitation.

The parameters used to characterize the model are: (1) Ky =total lateral stiffness in the y direction (equal to that in the xdirection), (2) xy = (Ky/m)0.5 = uncoupled lateral frequency, (3)KhR = torsional stiffness about the CR, (4) xh = (KhR/mr2)0.5 = uncou-pled torsional frequency, and (5) X = (KhR/r2Ky)0.5 = ratio ofuncoupled torsional frequency to translational frequency. For a

Fig. 2. Station layout in Lotung array [18].

D. Basu et al. / Engineering Structures 69 (2014) 1–11 3

given aspect ratio and location of the elements with respect to theCM, the lateral stiffness of each of the elements may be expressedas

K1;K3 ¼ Ky �1

2sx

eb

� �� 1

4sysa

sx

� �2

þX2R2a þ

1s2

x

eb

� �2" #

;

K2 ¼ Ky12

1þ sysa

sx

� �2( )

� 2X2R2a �

2s2

x

eb

� �2" #

ð2Þ

In Eq. (2), sx = b*/b, sy = a*/a, sa = a/b, Ra ¼ ½1þ s2a �

0:5=ð2

ffiffiffi3p

sxÞ. Assum-ing a unit mass, Ky in Eq. (2) may be replaced by x2

y . Given thedimensions of the deck and location of the elements, this elasticsystem is uniquely described by three normalized parameters:xy, X and e/b.

Specific to this six-element model, the arbitrary selection of thethree normalized parameters does not lead to a physical or realsystem (which requires K1, K2 and K3 to be positive). Note thatxy can be chosen regardless of X and e/b for a physical system.Further, if X is specified, e/b cannot be arbitrary. For example,when X = 1.0, the range of e/b for a physical system is 0 6 e/b 6 0.35. Similarly, 0 6 e/b 6 0.30 and 0 6 e/b 6 0.10 are forX = 1.25 and 1.50, respectively.

Note that X P 1.0 implies a system with uncoupled transla-tional frequency less than the uncoupled torsional frequency. Sucha system is denoted herein as a torsionally stiff system. Conversely,a torsionally flexible system is characterized by X < 1.0. Physically,a torsionally stiff system has stiffer members located towards theperiphery of the structure, whereas in a torsionally flexible systemthese members are located towards its center. In practice, torsion-ally flexible systems are uncommon and may be considered unre-alistic, particularly for seismically isolated structures. From themathematical model considered here, it may be noted that 0 6 e/b 6 0.03 for X = 0.8.

The range of e/b discussed above is specific to the mathematicalmodel considered here. A different range can be obtained if thenumber and location of the elements are different. Nevertheless,the mathematical model considered here covers a wide range oftorsionally stiff systems and is sufficient for the purposes of thisstudy.

3. Conventional calculation of accidental eccentricity

It is common practice to shift the CM at each floor level by a dis-tance equal to the accidental eccentricity to amplify the maximumtranslational response when performing response-history analysis.This approach is studied herein and its effect on the displacementdemand is examined. For convenience, the two sides with respectto the CR of the model are denoted as Side A and Side B as shown inFig. 1. In a torsionally stiff system (X P 1.0), elements located onSide A are expected to sustain more displacement than those onSide B. In a torsionally flexible system (X < 1.0), elements locatedon Side B are more critical than those on Side A. Since torsionallyflexible systems are not very common in practice, the proceduresfollowed in this paper are first formulated for torsionally stiff sys-tems. Torsionally flexible systems are then also analyzed withappropriate modifications.

The CM is first shifted away from the CR (increasing the actualeccentricity) and denoted here as Shift 1. The CM is then shifted toeach side in turn (increasing and then decreasing the actual eccen-tricity) and denoted as Shift 2.

3.1. Ground motion considered

This study of accidental torsion includes an explicit consider-ation of torsional ground motion. Time series of torsional motions

cannot be directly recorded at this time and need to be extractedby analysis of translational time series (e.g., [4]). Herein, a proce-dure based on earthquake acceleration time series (M6.1 event ofJanuary 16, 1986, source-to-site distance of 20 km) recorded atthe Large Scale Seismic Testing (LSST) array in Lotung, Taiwan, isutilized to extract the rotational ground motions. The Lotung-LSST(LLSST) site is a part of the SMART-1 array. The fifteen free-surfaceaccelerometers at the LLSST are positioned along three arms atapproximately 120-degree intervals (Fig. 2). Each arm extends forabout 50 m and the spacing between the surface stations variesfrom 3 m to 90 m. Each arm contains five stations that are desig-nated as 1–5 starting from the center of the array. That is the threestations numbered 1 in the array represent tier 1; the three sta-tions numbered 2 in the array represent tier 2; and so on.Furthermore, each station is identified herein as FAi_j, where i isthe arm (1–3) and j is the station (1–5). Further details can be ob-tained from [18].

The Surface Distribution Method (SDM) [4] is used to computethe torsional ground motion. The following adjustments weremade to the procedure to develop an upper bound torsional spec-trum: (i) The recorded EW (y) and NS (x) components are consid-ered as the horizontal acceleration field instead of the SH wavecomponent, (ii) The shear wave velocity at the surface layer(=140 m/s per [21]) is used instead of the apparent SH wave veloc-ity computed by Basu et al. [4] (=249 m/s), and (iii) Only stationsup to tier 4 are included in the analysis. Since the SDM yieldsone torsional ground motion for each surface station considered,the torsional acceleration history with the highest peak torsionalacceleration is used as the torsional ground motion input in thisstudy. The translational acceleration history is that recorded atthe interior station FA1_1, which is not the station where the peaktorsional acceleration is computed. The translational accelerationhistories along directions x and y, the torsional acceleration andtheir respective 5-percent damped response spectra are shown inFig. 3. As the system is symmetric about the x axis, the transla-tional acceleration in the NS direction is not input to the model.

3.2. Procedure for calculating the accidental eccentricity inconventional approach

The procedure described below represents a systematic ap-proach to quantify the accidental eccentricity for use in dynamicresponse history analysis of structural systems excited with onlytranslational seismic excitation. The accidental eccentricity is

0 10 20 30 40

Time (sec)

-0.2

-0.1

0

0.1

0.2

Acc

eler

atio

n (g

)

(a) NS (x) direction history

0 2 4 6

Period (sec)

0

0.2

0.4

0.6

Spec

tral

acc

eler

atio

n (g

)

(b) NS (x) direction spectrum

0 10 20 30 40

Time (sec)

-0.2

-0.1

0

0.1

0.2

Acc

eler

a tio

n (g

)

(c) EW (y) direction history

0 2 4 6

Period (sec)

0

0.2

0.4

0.6

Spec

tral

acc

eler

atio

n (g

)

(d) EW (y) direction spectrum

0 10 20 30 40

Time (sec)

-0.2

-0.1

0

0.1

0.2

Acc

eler

atio

n (r

ad/s

ec2 )

(e) Torsional history

0 2 4 6

Period (sec)

0

0.1

0.2

0.3

0.4

Spec

tral

acc

eler

atio

n (r

ad/s

ec2 )

(f) Torsional spectrum

Fig. 3. Input ground acceleration data.

4 D. Basu et al. / Engineering Structures 69 (2014) 1–11

established so that the peak corner displacement (along the ydirection) response of the system with only translational excitationis equal to or greater than the peak corner displacement (along they direction) response of the actual system (without accidentaleccentricity) excited by translational and torsional ground excita-tion. The steps of this procedure are:

1. Select values for the normalized parameters xy, X and e/b thatuniquely define the elastic system. Note that these parametersdescribe the uncoupled translational frequency, the ratio ofuncoupled torsional to translational frequency and the actualnormalized eccentricity, respectively.

2. Simultaneously apply the translational and torsional accelera-tion histories and calculate the absolute maximum displace-ment at the farthest element on Side A, Ufl.

3. Repeat Step 2 but apply only the translational acceleration his-tory; let the absolute maximum displacement for the same ele-ment be Ufl; compute the torsional amplification factor asR1 ¼ Ufl=Ufl:

4. Repeat Steps 2 and 3 but reverse the direction of the torsionalacceleration history and compute the torsional amplificationfactor R2; select the target torsional amplification factor asR = max (R1, R2). Note that the target torsional amplification isthe actual torsional amplification.

5. (a) Shift 1: Shift the CM away from the CR by an offset ea (theaccidental eccentricity) and analyze the system by applyingonly the translational acceleration history; let the absolutemaximum displacement at the furthest element on Side A beUfl.(b) Shift 2: Repeat step 5a but shift the CM in the oppositedirection and compare the two values of Ufl; record the greatestvalue.

6. Define the torsional amplification factor associated with offsetea as R� ¼ Ufl=Ufl: This represents the computed torsional amplifi-cation factor. Repeat Step 5 for a range of values of ea and gen-erate the associated torsional amplification factor. The requiredaccidental eccentricity for the system considered is given by theoffset ea for which R* P R.

3.3. Results and discussions

The procedure outlined above was applied to a variety of elasticsystems selected by varying the three normalized parametersxy, X and e/b. For torsionally flexible systems (X < 1.0), the proce-dure presented above is appropriately modified so the element re-sponse on Side B is chosen for the calculation of the torsionalamplification. In each system, the aspect ratio of the deck andthe location of the elements with respect to the CM of the deckwere selected to be sx = 1, sy = 1, and sa = 0.5. Damping in this threedegree-of-freedom (DOF) system was described by Rayleigh damp-ing with 5% damping ratio in the first and third modes. Analysis ofeach system was carried out and the calculated ‘‘target’’ (or actual)and ‘‘computed’’ torsional amplification factors were compared tocalculate the required accidental eccentricity ea. Results for twocases (X > 1.0 and X < 1.0) are presented below; other results arepresented in Basu et al. [4].

For the presented case, the uncoupled translational period is1.0 s and the ratio of the uncoupled torsional to translational fre-quency (X) is 1.25. The target (actual) torsional amplification cal-culated in the procedure with Shift 1 is presented in Fig. 4 foractual eccentricities in the range of 0–0.3. It may be seen thatthe target (or actual) torsional amplification is nearly constant atjust less than 1.1. Fig. 4 also presents the ‘‘computed’’ torsionalamplification (for only translational excitation) at various valuesof accidental eccentricity in the range of 0.01 6 ea/b 6 0.05 (panela) and 0.46 6 ea/b 6 0.49 (panel b). It may be seen in Fig. 4 thatthe computed torsional amplification fluctuates randomly and itis apparent that it is not possible to select a single value of acciden-tal eccentricity ea/b to ‘‘match’’ the ‘‘target’’ and the computed tor-sional amplification factors.

It is not possible to select a value for the accidental eccentricityand perform response-history analysis with only translationalexcitation to correctly capture the effects of translational and tor-sional ground excitation. The increase in displacement response isnot monotonic with increasing accidental eccentricity as shown inFig. 4c. (A shift in the CM of greater than 0.5b is meaningless fornearly all framing systems).

0 0.1 0.2 0.3

Normalized eccentricity (e/b)

0.8

0.9

1

1.1

1.2

1.3

Tor

sion

al a

mpl

ific

atio

n Target

ea/b=0.01

ea/b=0.02

ea/b=0.03

ea/b=0.04

ea/b=0.05

(a) 0.01 0.05ae b≤ ≤

0 0.1 0.2 0.3

Normalized eccentricity (e/b)

0.7

0.8

0.9

1

1.1

1.2

1.3

Target

ea/b=0.46

ea/b=0.47

ea/b=0.48

ea/b=0.49

(b) 0.46 0.49ae b≤ ≤

0 0.1 0.2 0.3 0.4 0.5

Accidental eccentricity ( ea /b)

0.6

0.8

1

1.2

1.4

Tor

sion

al a

mpl

ific

atio

n

e/b=0.002

e/b=0.05

e/b=0.10

e/b=0.15

e/b=0.20

e/b=0.25

e/b=0.30

(c) 0 0.30e b≤ ≤

Tor

sion

al a

mpl

ific

atio

n

Fig. 4. Variation of torsional amplification, Shift 1, conventional approach.

0 0.1 0.2 0.3

Normalized eccentricity (e/b)

0.9

1

1.1

1.2

1.3

Tor

sion

al a

mpl

ific

atio

n

Target

ea/b=0.01

ea/b=0.02

ea/b=0.03

ea/b=0.04

ea/b=0.05

(a) 0.01 0.05ae b≤ ≤

0 0.1 0.2 0.3

Normalized eccentricity (e/b)

0.6

0.8

1

1.2

1.4

1.6

Tor

sion

al a

mpl

ific

atio

n

Target

ea/b=0.11

ea/b=0.12

ea/b=0.13

ea/b=0.14

ea/b=0.15

(b) 0.11 0.15ae b≤ ≤

0 0.1 0.2 0.3 0.4 0.5

Accidental eccentricity (ea/b)

0.6

0.8

1

1.2

1.4

1.6

Tor

sion

al a

mpl

ific

atio

n

e/b=0.002

e/b=0.05

e/b=0.10

e/b=0.15

e/b=0.20

e/b=0.25

e/b=0.30

(c) 0 0.30e b≤ ≤

Fig. 5. Variation of torsional amplification, Shift 2, conventional approach.

D. Basu et al. / Engineering Structures 69 (2014) 1–11 5

Fig. 5 presents the results of Fig. 4 but for Shift 2, which betterrepresents the mandatory language of Section 12.8.4.2 of ASCE 7-10. It is clear from the results of Figs. 4(c) and 5(c) that there is

no trend of increasing torsional amplification with increasing acci-dental eccentricity in response-history analysis using only transla-tional excitation.

6 D. Basu et al. / Engineering Structures 69 (2014) 1–11

Figs. 6 and 7 present a similar study on a set of torsionally flex-ible systems characterized by X = 0.8 and with the uncoupledtranslational period being unity as in the previous studies. Notethe maximum range of e/b is now 0 6 e/b 6 0.03 for the system

0 0.01 0.02 0.03

Normalized eccentricity (e/b)

1

1.1

1.2

1.3

1.4T

orsi

onal

am

plif

icat

ion Target

ea/b=0.01

ea/b=0.02

ea/b=0.03

ea/b=0.04

ea/b=0.05

(a) 0.01 0.05ae b≤ ≤

0 0.1 0.2

Accidental eccentric

0.4

0.6

0.8

1

1.2

1.4

1.6

Tor

sion

al a

mpl

ific

atio

n

(c) 0 0e b≤ ≤

Fig. 6. Variation of torsional amplification in Sid

0 0.01 0.02 0.03

Normalized eccentricity (e/b)

1

1.1

1.2

1.3

1.4

Tor

sion

al a

mpl

ific

atio

n

Target

ea/b=0.01

ea/b=0.02

ea/b=0.03

ea/b=0.04

ea/b=0.05

(a) 0.01 0.05ae b≤ ≤

0 0.1 0.2

Accidental eccentrici

0.8

1

1.2

1.4

1.6

Tor

sion

al a

mpl

ific

atio

n

(c) 0 e b≤

Fig. 7. Variation of torsional amplification in Sid

to have practical significance. The torsional amplification is nowbased on the response of the element located at Side B. Observa-tions are similar to those presented above for the torsionally stiffsystems.

0 0.01 0.02 0.03

Normalized eccentricity (e/b)

0.5

0.6

0.7

0.8

0.9

1

1.1

Tor

sion

al a

mpl

ific

atio

n

Target

ea/b=0.46

ea/b=0.47

ea/b=0.48

ea/b=0.49

(b) 0.46 0.49ae b≤ ≤

0.3 0.4 0.5

ity (ea /b)

e/b=0.002

e/b=0.01

e/b=0.02

e/b=0.03

.03

e-B element, shift-1, conventional approach.

0 0.01 0.02 0.03

Normalized eccentricity (e/b)

0.88

0.92

0.96

1

1.04

1.08

1.12

Tor

sion

al a

mpl

ific

atio

n

Target

ea/b=0.46

ea/b=0.47

ea/b=0.48

ea/b=0.49

(b) 0.46 0.49ae b≤ ≤

0.3 0.4 0.5

ty (ea/b)

e/b=0.002

e/b=0.01

e/b=0.02

e/b=0.03

0.03≤

e-B element, shift-2, conventional approach.

D. Basu et al. / Engineering Structures 69 (2014) 1–11 7

The relationship between torsional amplification and accidentaleccentricity shown in these figures (panel c of Figs. 4–7) is inde-pendent of the torsional acceleration history used for analysisand is rather the result of changes in the dynamic characteristicsof the system when the CM is shifted. Shifting the CM in re-sponse-history analysis does not deliver the expected increase indisplacement response.

4. Alternate definition of accidental eccentricity

Recognizing that the current representation of accidental eccen-tricity does not achieve the desired goal with response-historyanalysis, an alternate definition is proposed here that can be sche-matically described through Fig. 8 in terms of application of theinertial force. A torsionally stiff system is considered in Fig. 8 forthe purpose of illustration. However, the concept applies to tor-sionally flexible systems as well, but with appropriatemodifications.

Fig. 8a shows the inertial force and moment acting through theCM of the considered system when subjected to the translationalacceleration €ugy along the y direction. The inertial force is com-prised of components m€uy and m€ugy in the Y direction and momentmr2€uh, where r is the radius of gyration of the slab. To account forthe effect of accidental torsion in the elements located on Side Aonly, the force m€ugy is shifted away from the CR by a distance ea

(see Fig. 8b). This is equivalent to applying a torsional momentequal to mea€ugy (see Fig. 8c). The inertial force and moment shownin Fig. 8c can be considered as resulting from a set of equivalentground motions acting on the original system as shown inFig. 8d. The equivalent ground motions consist of the originaltranslational motion and a torsional motion calculated by multi-plying the translational motion by an ‘‘arm’’ ea/r2, where ea is theaccidental eccentricity to be determined.

(a) Without accidental torsion (b) fo

(c) Equivalence of (b)

Fig. 8. Schematic representation of the altern

4.1. Procedure for calculating the accidental eccentricity in proposedapproach

The steps for calculating the accidental eccentricity in the pro-posed approach are identical to those described in the conven-tional approach except that step 5 of shifting of the CM isreplaced by the application of a torsional acceleration history asdefined above. Note that the procedure uses two types of torsionalacceleration histories: (i) an actual record (described above underthe heading of ‘‘Ground Motion Considered’’) to compute the target(or actual) torsional amplification, and (ii) an artificial record de-rived by multiplying the translational acceleration history by a fac-tor that is a function of the newly defined accidental eccentricity(ea/r2). The intensity of the artificial torsional acceleration historyis increased by incrementing the accidental eccentricity until thetarget torsional amplification is obtained.

Note that for torsionally flexible systems, the direction of theartificial torsional acceleration is opposite to that for the one fortorsionally stiff systems.

4.2. Scaling of accidental eccentricity

The artificial torsional acceleration history described in Fig. 8dis €ugh ¼ �ðea=r2Þ€ugy, where ea is the accidental eccentricity and €ugy

the translational acceleration history. Alternatively, the torsionalacceleration history may be expressed as

€ugh ¼ �ð�ea=r2Þðb�=rÞ2€ugy ð3Þ

where b⁄ is the greater of the two plan dimensions, a and b, and

�ea ¼ ea=ðb�=rÞ2 ð4Þ

Note that quantity (b*/r)2 is a multiplier larger than unity (for asquare plan, it is equal to 6).

Shifting one part of inertial force to account r accidental torsion in elements located on

Side A

(d) Equivalent ground motion inputs

ative definition of accidental eccentricity.

0 0.01 0.02 0.03

Normalized eccentricity (e/b)

1

2

3

4

Tor

sion

al a

mpl

ific

atio

n Target

ea/b=0.01

ea/b=0.02

ea/b=0.03

ea/b=0.04

ea/b=0.05

ea/b=0.06

ea/b=0.07

ea/b=0.08

ea/b=0.09

ea/b=0.10

(a) 0.01 0.10f≤ ≤

0 0.02 0.04 0.06 0.08 0.1

Accidental eccentricity (f)

1

2

3

4

Tor

sion

al a

mpl

ific

atio

n

e/b=0.002

e/b=0.01

e/b=0.02

e/b=0.03

8 D. Basu et al. / Engineering Structures 69 (2014) 1–11

4.3. Results and discussions

The set of systems analyzed previously by shifting the CM isreanalyzed using the proposed definition of accidental eccentricity(�ea in remainder of this paper). Fig. 9 presents results for the set oftorsionally stiff systems considered earlier. Fig. 9 presents the tor-sional amplification calculated by analysis with translational andthe artificial torsional ground motions. This is compared to the tar-get torsional amplification, which is calculated by analysis withtranslational and the actual torsional ground motions. Fig. 10 pre-sents similar results to those of Fig. 9 but for the set of torsionallyflexible systems considered earlier. Note that Fig. 9(b) andFig. 10(b) show increasing torsional amplification with increasingaccidental eccentricity. In other words, if the structure is ableachieve the required torsional amplification with 2% accidentaleccentricity, it will meet the same with any higher value of acci-dental eccentricity. Such a property was not evident with conven-tional procedure.

Therefore, the proposed procedure can be used to develop de-sign recommendations for accidental eccentricity for use with re-sponse-history analysis to account for the effects of torsionalground motion. The recommendation should be based on a com-parison of calculated torsional amplification and the target tor-sional amplification obtained using actual torsional groundmotion input. Table 1 presents values of accidental eccentricity(�ea=b) as a percentage of the plan dimension normal to the direc-tion of excitation, for use in response-history analysis. Note thesevalues are for elastic systems and based on analysis using onlyone record of torsional ground motion.

0 0.1 0.2 0.3

Normalized eccentricity (e/b)

0

1

2

3

4

5

Target

f=0.01

f=0.02

f=0.03

f=0.04

f=0.05

f=0.06

f=0.07

f=0.08

f=0.09

f=0.10

(a) 0.01 0.10f≤ ≤

0 0.02 0.04 0.06 0.08 0.1

Accidental eccentricity (f)

0

1

2

3

4

5

Tor

sion

al a

mpl

ific

atio

nT

orsi

onal

am

plif

icat

ion

e/b=0.002

e/b=0.05

e/b=0.10

e/b=0.15

e/b=0.20

e/b=0.25

e/b=0.30

(b) 0 0.30e b< ≤

Fig. 9. Variation of torsional amplification in torsionally stiff system, proposedapproach, f ¼ �ea=b.

(b) 0 0.30e b< ≤

Fig. 10. Variation of torsional amplification in torsionally flexible system, proposedapproach, f ¼ �ea=b.

Table 1Accidental eccentricity �ea=b for an elastic system.

X Tn (s)

0.5 1.0 1.5 3.0 4.0

0.8 0.01 0.01 0.01 0.03 0.031 0.01 0.01 0.02 0.03 0.031.25 0.01 0.03 0.03 0.03 0.021.5 0.01 0.03 0.02 0.01 0.02

5. Accidental eccentricity in nonlinear isolation systems

The system shown in Fig. 1 is assumed now to represent a rigidstructure that is seismically isolated. The isolation system consistsof six isolators and the system is symmetric about the x axis buthas an eccentricity about the y axis. The mass of the rigid super-structure is lumped at the CM (and geometric center) of the deck.The isolators have the bilinear hysteretic force–displacement rela-tionship shown in Fig. 11, where Q is the strength (force at zerodisplacement), Qy is the yield strength, Y is the yield displacementand Kd is the post-elastic stiffness. Analysis of this nonlinear isola-tion system can be performed using well-established procedures asoutlined in Basu et al. [4].

A broad range of behavior of the isolation system is considered.These systems are characterized as described above in case of elas-tic systems with the exception that the post-elastic stiffness, Kd, isused instead of the elastic stiffness, Ky. The uncoupled translationalperiod based on the post-elastic stiffness is denoted as

Td ¼ 2p=xd ð5Þ

Three isolation systems are considered with periods of Td = 3 s,Td = 4 sand Td = 5 s. The ratios of the uncoupled torsional frequency

Fig. 11. Force–displacement relationship for a typical isolator.

0 0.1 0.2 0.3

Normalized eccentricity (e/b)

1

1.5

2

2.5

3

3.5

Tor

sion

al a

mpl

ific

atio

n Target

f=0.005

f=0.010

f=0.015

f=0.020

f=0.025

f=0.030

f=0.035

f=0.040

f=0.045

f=0.050

Fig. 12. Variation of torsional amplification in Side A element of nonlinear isolationsystem, Td = 4 s, X = 1.25, Q/W = 0.05, Y = 5 mm, f ¼ �ea=b.

D. Basu et al. / Engineering Structures 69 (2014) 1–11 9

to translational frequency, based on the post-elastic stiffness, X, areconsidered as 1.0, 1.25 and 1.5 to cover a wide range of torsionallystiff isolation systems. The normalized natural eccentricity (e/b) isincreased from zero, in increments of 0.05, until the maximum va-lue possible for which the system has physical meaning. Further,X = 0.8 is also considered to study the torsional flexible isolationsystems and in such a case e/b is increased from zero, in incrementsof 0.01, until the maximum value possible for which the system hasphysical meaning. The yield displacement of each isolator was setequal to 1, 5 and 10 mm. The ratio of characteristic strength to sup-ported weight (Q/W) for each isolator was set equal to 0.04, 0.05,0.06 and 0.07. The combination of parameters used to describethe isolation system represents a range of Friction Pendulum andlead-rubber bearing isolation systems, but using yield displace-ments appropriate for each type of system: = 1 mm for FrictionPendulum bearings and Y = 10 mm for lead-rubber bearings.

Table 2Accidental eccentricity �ea=b in nonlinear isolation system, Y = 1 mm.

X Td = 3 s Td = 4 sj = Q/W (%) j = Q/W (%)

4 5 6 7 4 5

0.8 0.005 0.005 0.005 0.005 0.005 0.0051 0.005 0.005 0.005 0.005 0.005 0.0051.25 0.005 0.005 0.005 0.005 0.005 0.0051.5 0.005 0.005 0.005 0.005 0.005 0.005

Table 3Accidental eccentricity �ea=b in nonlinear isolation system, Y = 5 mm.

X Td = 3 s Td = 4 sj = Q/W (%) j = Q/W (%)

4 5 6 7 4 5

0.8 0.005 0.005 0.01 0.01 0.005 0.0051 0.01 0.005 0.01 0.01 0.01 0.0051.25 0.01 0.005 0.01 0.01 0.01 0.0051.5 0.01 0.005 0.01 0.01 0.01 0.005

Table 4Accidental eccentricity �ea=b in nonlinear isolation system, Y = 10 mm.

X Td = 3 s Td = 4 sj = Q/W (%) j = Q/W (%)

4 5 6 7 4 5

0.8 0.01 0.01 0.005 0.005 0.01 0.011 0.01 0.01 0.01 0.005 0.01 0.011.25 0.01 0.01 0.01 0.005 0.01 0.011.5 0.01 0.01 0.01 0.005 0.01 0.01

5.1. Results and discussions

The accidental eccentricity computed for all systems consideredin this study are reported in Tables 2–4. Results from one of thesecases, Td = 4 s, Y = 5 mm, X = 1.25 and Q/W = 0.05 are presented inFig. 12. This is an example of torsionally stiff isolation systems,where it is demonstrated that the accidental eccentricity is nearlyindependent of the actual eccentricity. A similar observation wasmade for a torsionally flexible isolation system with Td = 4 s,Y = 5 mm, X = 0.8 and Q/W = 0.05, for which results are presentedin Fig. 13. The results presented here and additional results in Basuet al. [4] suggest that the required normalized eccentricity per Eq.(3) is in the range of 0.005–0.01 with the yield displacement

Td = 5 sj = Q/W (%)

6 7 4 5 6 7

0.005 0.005 0.005 0.005 0.005 0.0050.005 0.005 0.005 0.005 0.005 0.0050.005 0.005 0.005 0.005 0.005 0.0050.005 0.005 0.005 0.005 0.005 0.005

Td = 5 sj = Q/W (%)

6 7 4 5 6 7

0.01 0.01 0.005 0.005 0.01 0.010.01 0.01 0.01 0.005 0.01 0.010.01 0.01 0.01 0.005 0.01 0.010.01 0.01 0.01 0.005 0.01 0.01

Td = 5 sj = Q/W (%)

6 7 4 5 6 7

0.005 0.005 0.01 0.01 0.005 0.0050.01 0.005 0.01 0.01 0.01 0.0050.01 0.005 0.01 0.01 0.01 0.0050.01 0.005 0.01 0.01 0.01 0.005

0 0.01 0.02 0.03

Normalized eccentricity (e/b)

1

1.5

2

2.5

3

3.5

Tor

sion

al a

mpl

ific

atio

n Target

f=0.005

f=0.010

f=0.015

f=0.020

f=0.025

f=0.030

f=0.035

f=0.040

f=0.045

f=0.050

Fig. 13. Variation of torsional amplification in side B element of nonlinear isolationsystem, Td = 4 s, X = 0.8, Q/W = 0.05, Y = 5 mm, f ¼ �ea=b.

10 D. Basu et al. / Engineering Structures 69 (2014) 1–11

having the largest influence on its value. Additional studies withother torsional motions are needed to better define the values ofnormalized eccentricity.

6. Discussion on the proposed definition of accidentaleccentricity

This paper presents an alternative definition of accidentaleccentricity to account for the effects of torsional ground motion.This new definition of accidental eccentricity applies a torsionalground motion, which is the product of a translational ground mo-tion and a factor that is a function of the proposed accidentaleccentricity. The required accidental eccentricity is that whichachieves the actual torsional amplification as determined by useof the actual torsional ground motion. The displacement of a cornerof the building is used to define the torsional amplification, as hasbeen used in many prior studies (e.g., [8–11]. For multistory sys-tems, the number of response quantities that could be used to de-fine the torsional amplification increases rapidly. Each quantitywould produce a different accidental eccentricity in a multistorybuilding.

While the calculation of the required accidental eccentricity in amultistory system should consider all important response quanti-ties and utilize that which produces the greatest demand, it hasbeen common to define the accidental eccentricity using singlestory models and then verify the validity of the results by analysisof selected multistory structures. For example, De-La-Llera andChopra [9] showed the torsional amplification computed for aone-story system was identical to that in any story of a specialclass of multistory buildings [16,2] if the contributions from thehigher modes are negligible. The design recommendations for acci-dental eccentricity for this special class of structures are then iden-tical to those for one-story systems.

The definition of the accidental eccentricity proposed hereinalso applies to sources of accidental torsion other than the tor-sional component of ground motion. However, the procedure toquantify the value of the accidental eccentricity will differ bysource. For example, the source could be uncertainty in the in-plane stiffness of lateral load resisting elements. Then the proce-dure of De-La-Llera and Chopra [9] could be used together withthe definition of accidental eccentricity proposed here.

Further studies are needed if the definition of accidental eccen-tricity presented above is accepted. Such studies would require theuse of a large number of torsional ground motions and structuralsystems to provide the body of knowledge necessary for the devel-opment of design recommendations and code language.

7. Conclusions

When only considering the effects of torsional ground motionand without consideration of uncertainty in mass distribution,the conventional approach of accounting for the effects of acciden-tal torsion by shifting the CM does not produce the desired effect inresponse-history analysis because the shift changes the dynamiccharacteristics of the structure so that it is possible to have reduceddisplacement demands with increasing eccentricity. An alternativedefinition of accidental eccentricity is proposed wherein accidentaltorsion is accounted for by simultaneously applying torsional andtranslational acceleration histories. The torsional acceleration his-tory is computed as the product of the translational history and ascale factor, which is a function of the proposed accidentaleccentricity.

The proposed procedure has been studied for a broad range ofsingle story elastic systems and nonlinear isolation systems. Tor-sional amplification is predicted correctly, namely, increasing tor-sional response with increasing eccentricity. The studydemonstrates that for nonlinear isolation systems, the required acci-dental eccentricity increases as the yield displacement increases.

Values of the required accidental eccentricity to account for tor-sional ground motion effects are presented (tabulated in Tables 1–4) but are specific to the ground motions considered in this studyand should not be considered as the design recommendations. Thispaper proposes only a methodology of accounting for the acciden-tal eccentricity due to the torsional component of the ground mo-tion. Quantification of such accidental eccentricity requires arigorous application of the proposed methodology based on a largenumber of ground motions recorded at various geologic/siteconditions.

Acknowledgements

The financial support for the studies described herein was pro-vided by MCEER (www.mceer.buffalo.edu) under Thrust Area 3,Innovative Technologies, through a grant from the State of NewYork. The Institute of Earth Science, Academia, Sinica, Taiwan pro-vided the strong motion data. The financial support, technical re-view and provision of data are gratefully acknowledged. Anyopinions, findings, conclusions or recommendations expressed inthis paper are the authors and do not necessarily reflect those ofeither MCEER or the State of New York.

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