Class of Irregularity

Engineering Structures 30 (2008) 1272–1291www.elsevier.com/locate/engstruct

Analysis for preliminary design of a class of torsionally coupled buildingswith horizontal setbacks

Dhiman Basu∗, N. Gopalakrishnan

Structural Dynamics Lab, Structural Engineering Research Center, Chennai 600 113, India

Received 24 July 2006; received in revised form 1 July 2007; accepted 3 July 2007Available online 4 September 2007

Abstract

Simplified method of analysis of a special class of torsionally coupled buildings with horizontal setbacks is developed that can be executedwith a plane frame analysis by means of a personal computer. Since most buildings may not exactly satisfy all the classification criteria, it is shownthat an averaging technique may be used in such cases up to a certain limit. Perturbation analysis is carried out in determining such a limit, andnumerical examples are presented to validate this. As a whole, the proposed simplified analysis may be used as a convenient and offhand tool atthe preliminary stage of design.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Special class of buildings; Torsional coupling; Seismic analysis; Proportional stiffness; Horizontal setbacks; Dynamic analysis; Preliminary design

1.0. Introduction

Buildings are, hardly ever, truly symmetric. Consequently,lateral vibration of buildings during seismic excitation isalways coupled with torsional vibration. A number of studieshave been reported in the literature addressing the issueof lateral–torsional coupling ([3,6–8,10–13,15,17] and manymore). Further, most seismic codes often suggest an equivalentstatic analysis against a specified lateral load profile for regularand nominally irregular buildings taking into account thetorsional effects. Therefore, behavior of asymmetric buildingsunder monotonic or equivalent static loading has also receivedadequate attention ([1,2,4,5] and many more). On the otherhand, when irregularity exceeds certain nominal limit, forexample, building with horizontal setbacks, complete dynamicanalysis is a must according to most seismic codes.

Instead of complete dynamic analysis, simplified dynamicanalysis is often preferred, especially, at the stage ofpreliminary design. In this connection, a special class oftorsionally coupled buildings has been reported in the literature[7,8,11,12] wherein the lateral stiffness and mass are distributedthroughout the building in a specific way. Kan and Chopra [11,12] reported that a multi-story building belongs to the category

∗ Corresponding author. Tel.: +91 44 2254 9147; fax: +91 44 2254 1508.E-mail address: dhiman [email protected] (D. Basu).

0141-0296/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2007.07.013

of a special class if (i) the center of mass (CM) of all the floorslie on one vertical line and radius of gyration about the verticalaxis passing through the CM is the same for all the floors,(ii) principal planes of all the lateral load-resisting elementsform an orthogonal grid system and (iii) lateral stiffnessmatrices of all the lateral load-resisting elements oriented alongeither of the two orthogonal directions are proportional toa characteristic lateral stiffness matrix along that direction;however, these two characteristic lateral stiffness matrices maynot be identical. Further, it has been shown [11,12] that theresponse behavior of such a shear building may be obtainedthrough appropriate combination of the results calculated fromthe analysis of two smaller systems, namely, (a) correspondingtorsionally uncoupled building and (b) equivalent one-storytorsionally coupled building. Hejal and Chopra [7,8] extendedthe concept of this simplified analysis to buildings comprisingmoment-resisting frames (MRFs) with the imposition ofan additional constraint, which states identical characteristiclateral stiffness matrices along both the orthogonal directions.In other words, lateral stiffness matrices of all the lateral load-resisting elements are proportional.

However, conditions of the existence of the special class ofbuildings are hardly, ever, truly satisfied in practice, e.g., (i)lateral stiffness matrices of the constituting frames may notbe exactly proportional and (ii) CM of all the floors may not

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D. Basu, N. Gopalakrishnan / Engineering Structures 30 (2008) 1272–1291 1273

be located exactly on a single vertical line. None of the twoprevious studies [11,12,7,8] has investigated the applicabilityof the simplified procedure in such cases. In the first study [11,12], shear building is assumed and the existence of which isin itself questionable. In the second study [7,8], numericallyproportional lateral stiffness matrices for the MRFs are directlyassumed. This is because MRFs do not, in general, yield toexactly proportional lateral stiffness matrices. Furthermore,in order to proportion the MRFs so as to yield proportionallateral stiffness matrices, Hejal and Chopra [7,8] made anattempt by introducing a factor, called the ‘joint rotation index’,which is based on the assumption of uniform story height anduniform bay width, but these conditions are hardly, if ever,met in practice. Surprisingly, in both the previous studies,orthogonality of the building was assumed and that seems tobe superfluous.

A building is said to be with horizontal setbacks if thereexists at least two points within its plan area when joinedthrough a straight line, the line runs out of the plan area.In the absence of a stiff shear wall, the lateral stiffnessmatrices of the constituting frames of a building with horizontalsetbacks closely satisfy the desired proportionality criterionof the special class of buildings. Consequently, any buildingwithout stiff shear walls may also be treated as satisfying allthe classification criteria if the CM of all the floors lie on asingle vertical line and the radius of gyration about this verticalline is the same for all the floors. The objective of this paper istwo-fold: first, to explore the concept of this simplified analysisto the special class of buildings with horizontal setbacksand second, to investigate the applicability of the procedurewhere mass proportionality criteria of the special class are notexactly satisfied. In order to meet these objectives, first, theformulation presented in Hejal and Chopra [7,8] is extendedto the special class of buildings with horizontal setbacks,as reported herein followed by a numerical example on aten-storied C-shape building with MRFs. Second, a rigorousperturbation analysis is carried out in order to restrict thelimit of the scattering of the floor CMs (from the vertical linepassing through the average CM of the building) up to whichthe simplified analysis can be applied with an acceptable errorfor all practical purposes. Numerical example on the same C-shape-MRFs building, but with vertically-nonaligned CMs, isthen presented to substantiate the results of the perturbationanalysis. Finally, the same C-shape-MRFs building, but withanother two sets of vertically-nonaligned CMs, are analyzed soas to assess the conservativeness in the limiting criteria derivedfrom the perturbation analysis. In all the numerical examplespresented, results of the simplified method are compared withthat calculated using SAP2000 [14].

The methodology developed in this paper does not imposeany constraint over the shape of the diaphragm and theorientation of the lateral load-resisting elements. Therefore, theproposed approach is equally applicable to buildings of V -, Y -etc. shape provided that the necessary conditions are satisfied.However, without losing the generality, C-shaped buildingis chosen for illustration. Further, 3D FEM analysis of thespecial class of torsionally coupled buildings with horizontal

setbacks is no longer impossible in the present era of computeradvancement. Consequently, one may argue against the utilityof this simplified analysis in the present scenario. Nevertheless,such a simplified analysis may always be preferred to assessthe design force resultants at preliminary stage of design;3D FEM analysis may be carried out at the final stage ofverification. Therefore, the proposed simplified analysis canbe applied as a convenient and offhand tool with sufficientaccuracy at the preliminary stage and requires only plane frameanalysis.

2.0. Development of the methodology

To formulate the methodology presented in this paper, anarbitrarily shaped diaphragm of a typical floor of an N -storybuilding comprising different wings is considered. CM of allthe floors is assumed to be lying on the same vertical line; theradius of gyration about the vertical axis passing through theCM is assumed to be the same for all the floors. Defining thedegrees of freedom at the CM of respective floors, mass matrixof the building can be expressed as the direct product of twosmaller matrices as follows:

3N×3N [M] = 3×3[CM]B ⊗ N×N [m] where, (1)

[CM]B =[1 1 r2]

diag. (2)

Here, [m] is a diagonal matrix with elements as the lumpedmass at the respective floor levels and may be consideredas the mass matrix of the characteristic frame; r the radiusof gyration of a typical floor about the vertical axis passingthrough the CM; and [CM]B may be considered as the massmatrix of an equivalent one-story coupled building. Similarly,considering the proportionality of the lateral stiffness matricesof the constituting frames and assuming (i) i th frame of the qthwing with stiffness proportionality constant as Ciq is locatedat a distance diq from the geometric center of gravity of thewing and oriented at an angle αiq in counter clockwise directionwith respect to the longitudinal axis of the wing, (ii) qth wingof the building, comprising of NEQ number of frames, islocated from the CM of the building at distances dLq and dSqalong the longitudinal and transverse directions of the wings,respectively, and oriented at an angle βq in counter clockwisedirection with respect to the global X -axis and (iii) [K ∗

] is thestiffness matrix of the characteristic frame while N W is thetotal number of wings, stiffness matrix of the building may alsobe expressed as the direct product of two smaller matrices asfollows:

3N×3N [K B] = 3×3[C L]B ⊗ N×N [K ∗] where, (3)

[C L]B =

N W∑q=1

([C L]q B

);

[C L]q B = [T ]Tq [C L]q [T ]q ;

[C L]q =

N E Q∑i=1

([C L]iq

);

(4a)

1274 D. Basu, N. Gopalakrishnan / Engineering Structures 30 (2008) 1272–1291

[T ]q =

cosβq sinβq ±dSq− sinβq cosβq ±dLq

0 0 1

;

[C L]iq = [Tel ]Tiq Ciq [Tel ]iq ;

(4b)

[Tel ]iq =[cosαiq sinαiq ±diq

]. (4c)

Here, [C L]B may be considered as the lateral stiffnessmatrix of an equivalent one-story coupled building. For sucha proportioned building, it may be shown that, similar to CMs,center of rigidity (CR) of the floors are also located on anothervertical line irrespective of the lateral load profile. Eigenvalueequation for the building is now defined as

[K B] {φ} = ω2 [M] {φ} (5)

where ω, {φ} are the frequency and mode shape, respectively,of the building. Also, define two smaller eigen equations, oneeach for characteristic frame and one-story coupled system asfollows:[K ∗]{η} = σ 2 [m] {η} (6)

[C L]B {ξ} = τ 2 [CM]B {ξ} (7)

where, σ , {η} and τ , {ξ} are the frequencies, mode shape vectorsof the characteristic frame and equivalent one-story coupledbuilding, respectively. Next, assuming

{φ} = {ξ} ⊗ {η} (8)

and utilizing Eqs. (1), (3) and (8) in Eq. (5), thereafter,substituting Eqs. (6) and (7) and carrying out direct product,it may be shown that

ω2i j = τ 2

i · σ 2j or ωi j = τi · σ j where i = 1, . . . 3;

and j = 1, 2, . . . , N . (9)

Therefore, the frequency and mode shape vector of the entirebuilding in ‘i j’th mode are given by the direct product ofthose of equivalent one-story coupled building in ‘i’th modeand characteristic frame in ‘ j’th mode. Such decoupling is notonly valid for frequency and mode shapes, but also holds goodfor any other modal response. For example, shear force sharedby the sth frame in qth wing, with stiffness proportionalityconstant Csq may be expressed as

{V }sqi j = SD(Pi Csq 1×3[Tel ]sq 3×3[T ]q 3×1{ξ}i )

⊗(Pj 1×N {` f }T

N×N [K ∗] N×1{η} j ) (10)

where, Pi and Pj are the participation factor (PF) of the one-story coupled system in ‘i’th mode, and the characteristic framein ‘ j’th mode, respectively, {` f } the influence vector of thecharacteristic frame and SD the spectral displacement. In Eq.(10), expressions within the first braces may be identified ascontributions from one-story coupled system in ‘i’th mode andcharacteristic frame in ‘ j’th mode.

Nonproportionality of lateral stiffness matrices of constitut-ing MRFs affects the estimation of mode shapes and hence, theframe shear force, in a relatively adverse way than that of fre-

quency. Thus, only the expression of frame shear force Eq. (10)needs to be modified as follows before applying the simplifiedmethod to buildings with MRFs.

{V }sqi j = SD(Pi 1×3[Tel ]sq 3×3[T ]q 3×1{ξ}i )

⊗(Pj 1×N {` f }TCsq N×N [K ∗

] N×1{η} j )

= SD(Pi 1×3[Tel ]sq 3×3[T ]q 3×1{ξ}i )

⊗(Pj 1×N {` f }T

N×N [Ksq ] N×1{η} j ) (11)

where, [Ksq ] is the lateral stiffness matrix of the frame underconsideration. This is to minimize the error in the story-wisedistribution of the frame shear force, only.

3.0. Numerical example-1

A ten-storied C-shaped building (Fig. 1) is consideredfor this example-problem. Column and beam dimensions areshown in Fig. 1. Story height is considered 3.5 m. Modulus ofelasticity of concrete is adopted as 2.55 × 107 kN/m2 while thePoisson’s ratio is taken as 0.2. The building is considered to belocated in zone-V of IS 1893-2002 [9] and ratio of ‘importancefactor’ to ‘response reduction factor’ is taken as unity. Soilcondition is assumed as medium and acceleration spectrumis chosen in compliance with IS 1893-2002 [9]. Intensity ofuniform mass distribution is taken as 1.0 t/m2, 0.9 t/m2,0.8 t/m2 and 0.7 t/m2 for the floors 1–3, 4–6, 7–9 and rooflevel, respectively, which leads to lumped mass of 2400 t,2160 t, 1920 t and 1680 t, respectively. CM is calculated fora typical floor and X -, Y -coordinates of which are 32.656 m,18.594 m, respectively, with respect to point H (Fig. 1) asorigin. Radius of gyration about the vertical axis passingthrough the CM is calculated for a typical floor as 24.639 m.Considering frame A1 as the characteristic frame, stiffnessproportionality constants for the frames A1–A6, A7–A14,B1–B6, B7–B10 and B11–B16 are evaluated, using planeframe analysis, as 1.0, 0.2975492, 0.610739, 0.2032498 and0.4907527, respectively. The building is then analyzed usingthe proposed method. Frequency and modal mass participationratio (MMPR) for the excitation along X - and Y -directions,for the first 15 modes are presented in Table 1 (columns (2),(5) and (6), respectively). Similar responses are also presented(Table 1, columns (3) (7) and (8) respectively) when frame B1 isused as the characteristic frame. Shear force induced in framesA1, A11, A14, B1 and B16 are presented in Table 2 (columns(2), (5), (8), (11) and (17), respectively) when excitationis considered along X -direction and frame A1 is taken asthe characteristic frame. Moreover, for frames B1 and B16,shear forces are also presented (Table 2, columns (14) and(20), respectively) when the excitation is considered along Y -direction and characteristic frame is chosen as B1. In order tocompare these responses, the example-building is then analyzedusing SAP2000 and the responses of interest are tabulated(Tables 1 and 2).

4.0. Range of applicability of the proposed method usingperturbation analysis

It may be intuitively judged that, when CM of all the floorsdo not lie on a single vertical line but the scattering in their


Table 1Comparison of modal responses calculated using the proposed method and SAP2000 for example-1

Modenumber

Frequency (rad/s) Modal mass participation ratio (%)

Proposed SAP2000 Proposed SAP2000Characteristic frameA1

Characteristic frameB1

Characteristic frame A1 Characteristic frame B1 X -direction

Y -direction

X -direction

Y -direction

X -direction

Y -direction

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)1 4.82 4.83 4.84 0.3 79.1 0.3 78.9 0.5 78.32 4.92 4.93 4.95 42.1 0.9 42.0 0.9 45.9 1.43 5.34 5.35 5.37 37.6 0.1 37.5 0.1 33.7 0.14 14.30 14.35 14.38 0.0 10.3 0.0 10.4 0.2 10.25 14.61 14.66 14.65 5.5 0.1 5.5 0.1 6.1 0.46 15.85 15.91 15.94 4.9 0.0 4.9 0.0 4.3 0.07 24.79 24.94 24.98 0.0 4.0 0.0 4.0 0.2 3.58 25.31 25.47 25.36 2.1 0.1 2.1 0.1 2.2 0.59 27.47 27.64 27.65 1.9 0.0 1.9 0.0 1.5 0.0

10 36.81 37.10 37.04 0.00 2.1 0.0 2.1 0.4 1.411 37.59 37.88 37.53 1.1 0.0 1.1 0.0 0.9 0.712 40.80 41.12 41.14 1.0 0.0 1.0 0.0 0.8 0.013 49.52 50.00 49.62 0.0 1.4 0.0 1.4 0.5 0.514 50.56 51.10 50.42 0.8 0.0 0.8 0.0 0.4 0.915 54.88 55.42 55.46 0.7 0.0 0.7 0.0 0.5 0.0

Fig. 1. Typical floor plan of the example-building.

locations is small, the proposed method may still be appliedby using the average location of CM and the average radiusof gyration, provided the stiffness proportionality criterion ofthe constituting frames are closely satisfied. A perturbationanalysis will now be carried out in order to ascertain the limit

of such scattering; up to this limit, the proposed method may beapplied with an acceptable margin of error at the preliminarystage of design. Since the building considered in this paper doesnot comprise shear wall, stiffness proportionality criterion ofthe constituting frames are closely satisfied and hence, while


Table 2Comparison of frame shear force calculated using the proposed method and SAP2000 for example-1

Floor Frame-A1 Frame-A11 Frame-A14X -direction excitation X -direction excitation X -direction excitationProposed SAP %error Proposed SAP %error Proposed SAP %error

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)10 607.9 598.5 1.6 240.3 262.8 −8.6 196.3 215.7 −9.0

9 1154.1 1112.0 3.8 460.2 488.6 −5.8 376.1 401.4 −6.38 1595.6 1535.9 3.9 643.9 679.7 −5.3 526.3 557.8 −5.67 1952.8 1877.2 4.0 792.3 830.6 −4.6 647.7 681.2 −4.96 2288.5 2197.7 4.1 929.1 968.6 −4.1 759.6 794.1 −4.35 2582.5 2476.9 4.3 1051.6 1090.9 −3.6 859.9 894.0 −3.84 2846.0 2725.2 4.4 1165.3 1203.0 −3.1 952.9 985.7 −3.33 3106.8 2968.8 4.7 1275.8 1312.8 −2.8 1043.2 1075.9 −3.02 3312.3 3159.2 4.8 1353.0 1395.4 −3.0 1106.3 1144.6 −3.41 3417.8 3272.5 4.4 1204.4 1284.3 −6.2 984.8 1061.1 −7.2

Frame-B1 Frame-B16X -direction excitation Y -direction excitation X -direction excitation Y -direction excitationProposed SAP2000 %error Proposed SAP2000 %error Proposed SAP2000 %error Proposed SAP2000 %error

(11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)382.0 401.8 −4.9 561.3 577.5 −2.8 403.2 441.6 −8.7 338.2 327.5 3.2747.4 729.7 2.4 1050.8 1093.3 −3.9 784.5 831.0 −5.6 656.5 610.9 7.5

1030.5 1007.1 2.3 1436.4 1483.1 −3.2 1081.4 1130.9 −4.4 898.1 846.2 6.11258.7 1230.4 2.3 1738.5 1783.5 −2.5 1320.6 1367.3 −3.4 1087.5 1033.6 5.21472.8 1438.7 2.4 2016.8 2059.8 −2.1 1546.5 1588.0 −2.6 1263.4 1206.6 4.71660.2 1619.1 2.5 2261.2 2302.1 −1.8 1743.9 1780.4 −2.1 1417.6 1357.4 4.41830.5 1780.1 2.8 2486.1 2525.5 −1.6 1924.2 1954.4 −1.5 1560.0 1494.8 4.42004.4 1940.7 3.3 2718.1 2760.5 −1.5 2113.2 2133.8 −0.9 1710.2 1634.9 4.62139.9 2063.2 3.7 2908.1 2963.4 −1.9 2268.1 2285.6 −0.8 1839.9 1744.4 5.5

carrying out the perturbation analysis, perturbation is givenonly in the mass matrix. Moreover, in perturbation analysis,eigenvalue problem is formulated with respect to the CR of therespective floors as the reference point.

4.1. Perturbation equation from the first principle

Eigenvalue problems of the unperturbed and perturbedsystems are given by

γ0 Kψ = Mψ and (12a)

γ Kφ = (M + εB) φ (12b)

where, K is the stiffness matrix; M , (M + εB) the massmatrices, γ0, γ the eigenvalues, and ψ , φ the eigenvectorsof the unperturbed and perturbed systems, respectively. Here,perturbation in the mass matrix may be defined as E = εB.Now, using the fundamental principle of perturbation [16],expressions for the perturbed eigenvalues and eigenvectors maybe arrived at. Resulting expressions from the second orderperturbation are as follows:

γi = γ0i + γ0i

(ψT

0i Eψ0i

) n∑j=1j 6=i

γ0iγ0 j

γ0i − γ0 j

(ψT

0i Eψ0 j

)2(13)

φi (ε) = ψ0i +

n∑j=1j 6=i

[(γ0 j

γ0i − γ0 j

)(ψT

0 j Eψ0i

)

×

{1 +

(1

γ0i − γ0 j

)[γ0 j

(ψT

0 j Eψ0 j

)

− γ0i

(ψT

0i Eψ0i

)]}]ψ0 j . (14)

4.2. Error estimation by perturbation analysis

Let mi be the lumped mass at the CM of the i th floor and ribe its radius of gyration about the vertical axis passing throughthe CM. Therefore, the average location of the CM of the entirebuilding may be calculated as

XCM =

∑mi xi∑mi

and (15a)

YCM =

∑mi yi∑mi

. (15b)

Here, xi and yi are measured with respect to an arbitrarilychosen origin. Radius of gyration of the i th floor about thevertical axis passing through the average CM is then givenby,

r ′2

i = ri2+ b2

xi+ b2

yi(16)

where bxi and byi are the distances between the actual and av-erage CMs at the i th floor level along the X - and Y -directions,respectively. Therefore, average radius of gyration for the entirebuilding is given by

r̃2=

∑miri

′2∑

mi=

∑mi

(r2

i + b2xi

+ b2yi

)∑

mi. (17)


This is the radius of gyration assumed for all the floors in thesimplified analysis because (i) bxi and byi are small enoughwhen compared to the dimensions of the building and (ii) r2

idoes not differ much from floor-to-floor.

Stiffness matrix of the building with respect to the CR canbe expressed as

[K ] =

Kxx Kxy 0K yx K yy 0

0 0 K rθθ

. (18)

Since, CRs and CMs of all the floors are lying on two differentvertical lines for the model assumed in the simplified analysis(average CM model), denoting the distance between these twovertical lines along the X - and Y -directions as ex and ey ,respectively, mass matrix of the actual building with respect tothe CR may be expressed in Box I

However, in the simplified analysis, the stiffness matrix isassumed to be the same as that of the actual building while themass matrix is assumed to be

[M]B =

[mi ] 0 −[mi ey]

0 [mi ] [mi ex ]

−[mi ey

][mi ex ] [mi

(r̃2

+ e2x + e2

y

)]

. (19)

Let us now denote the actual building as system-A andthat assumed in the simplified analysis as system-B. Forconvenience in the perturbation analysis, in addition, system-Cis assumed wherein the stiffness matrix is considered the sameas before while the mass matrix is assumed to be,

[M]C =

[mi ] 0 00 [mi ] 0

0 0[mi

(r̃2

+ e2x + e2

y

)] . (20)

The system-C , hereafter, may be referred to as the unperturbedsystem. Further, comparing the i th diagonal entry of the Mθθ

sub-matrices of system-A and system-B (or C), the differencemay be noted as

mi d2i = mi

[bxi

(bxi + 2ex

)+ byi

(byi + 2ey

)][as r2

i ≈ r̃2].

(21)

Moreover, denoting L and S to be the maximum dimensionsof the plan area of the building along the X - and Y -directions,

respectively, and letting exL = αx ; ey

L = αy ;bxiL = βxi

;byiL = β yi

and r̃2= εL L2

= εS S2, it may be shown that

d2i

r̃2 =

β2xi

+ 2αx

εL+β

2yi

+ 2αyβ yi

εS

. (22)

Typically, βxiand β yi

are of the order 10−1 (or even 10−2); αx ,αy of the order 10−1 in case of highly eccentric system; and

εL ,εS of the order 10−1. Therefore, it may be shown thatd2

ir̃2

is of the order 10−1 to 10−3; this result will be used in thesubsequent stages of the perturbation analysis.

Here, the objective is to relate the order of error in theestimated frame shear forces when using system-B in lieu

of system-A with the order of the scattering in location ofthe floor CMs. This is done in a step-by-step fashion: First,using the general expressions for the perturbed eigenvaluesand eigenvectors Eqs. (13) and (14), error equations forthe eigenvalues and eigenvectors are formulated. Second,these error equations are greatly simplified using certainpertinent approximations. Third, likelihood order of the errorin eigenvalues and eigenvectors are related with the order of thescattering in location of the floor CMs. Finally, expected orderof the error in the estimation of the frame shear forces is relatedwith the order of the scattering in location of the floor CMs.The complete derivation has been presented in Appendix. Thisis because, (i) readers of this paper have to deal with only theend results of the perturbation analysis and (ii) entire derivationprocess is extremely rigorous and likely to distract the readersfrom the main stream.

In Appendix, it has been shown that the error in frame shearforces, estimated using the simplified procedure, on accountof the vertically-nonaligned floor CMs might be restricted to±10% (which might be considered reasonable at the stageof preliminary design) if the following two conditions aresatisfied:

|βx | ≤ 0.002 and (23a)∣∣βy∣∣ =≤ 0.002. (23b)

Here, βx = {u}T1 [

bxi mi

r̃ ]diag {u}1, βy = {u}T1 [

byi mi

r̃ ]diag {u}1are the indices for assessing the applicability of the simplifiedanalysis and {u}1 the mass normalized fundamental mode shapeof the characteristic frame. If βx = C1 × 10−ρ and βY =

C2 × 10−ρ with |C1|, |C2| close to unity, e.g., 1.0 to 2.0, it hasbeen further shown that the maximum expected difference forthe eigenvalues (γ ) while assuming system-B in lieu of system-A is of the order 10−ρ and that for the eigenvectors is of theorder 10−ρ+1 {u}1.

5.0. Numerical example-2: Validation of the proposedmethod in building with scattered CM

The building considered for this example is the same as thatconsidered in example-1 except the location of CMs. Offsetsin the location of CMs from that considered in example-1 areshown in Table 3 (columns (2), (3)). These locations of theCMs are chosen in such a way that location of the averageCM coincides with that considered in example-1. Also, averageof square of radius of gyration about the vertical axis passingthrough the average CM is calculated as 609.9 m2, which isnearly the same as that considered in example-1. Hence, thebuilding considered in example-1 nearly represents the buildingidealized in the proposed method (average CM model) whenapplied to this example-problem. Therefore, response of the‘average CM model’ associated with the building consideredin this example may be considered the same as that obtainedin example-1. Next, considering frame A1 as the characteristicframe, using Eqs. (23a) and (23b), βx and βy are evaluatedfor this building as 0.00076 and 0.00129, respectively. Clearly,Eqs. (23a) and (23b) are satisfied for this example-building. The


[M]A =

[mi ] 0 −[(

ey + byi

)mi]

0 [mi ][(

ex + bxi

)mi]

−[(

ey + byi)

mi]

[(ex + bxi )mi ][mi

{r2

i +(bxi + ex

)2+(byi + ey

)2}] .

Box I.

Table 3Offsets in CM for example-problems-2, -3 and -4 with respect to example-problem-1

Floor Offset in CM for example-2 Offset in CM for example-3a Offset in CM for example-3bX -direction (m) Y -direction (m) X -direction (m) Y -direction (m) X -direction (m) Y -direction (m)

(1) (2) (3) (4) (5) (6) (7)1 1.0 0.5 2.0 0.5 6.0 −3.52 1.5 −1.0 3.0 −1.0 −1.5 −1.03 −2.5 0.5 −5.0 0.5 −4.5 4.54 0.5 −1.0 1.0 −1.0 3.0 −2.55 −1.5 1.5 −3.0 1.5 1.5 −2.56 1.0 −0.5 2.0 −0.5 −4.5 5.07 −1.5 1.5 −3.0 1.5 5.0 −4.08 −0.5 −0.5 −1.0 −0.5 −3.0 −2.09 2.0 −1.0 4.0 −1.0 −2.0 6.0

10 0.0 0.0 0.0 0.0 0.0 0.0

Table 4Comparison of frame shear force of the perturbed and the average CM model calculated using SAP2000 for example-2

Floor Frame-A1 Frame-A11 Frame-A14X -direction excitation X -direction excitation X -direction excitationPerturbed Average CM %error Perturbed Average CM %error Perturbed Average CM %error

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)10 602.4 598.5 −0.7 257.8 262.8 1.9 211.7 215.7 1.9

9 1140.6 1112.0 −2.5 474.7 488.6 2.9 389.9 401.4 2.98 1576.1 1535.9 −2.6 661.3 679.7 2.8 542.8 557.8 2.87 1900.8 1877.2 −1.2 819.9 830.6 1.3 672.1 681.2 1.46 2229.3 2197.7 −1.4 956.7 968.6 1.3 783.9 794.1 1.35 2494.3 2476.9 −0.7 1085.4 1090.9 0.5 888.7 894.0 0.64 2753.6 2725.2 −1.0 1194.2 1203.0 0.7 977.7 985.7 0.83 2995.8 2968.8 −0.9 1304.3 1312.8 0.7 1068.1 1075.9 0.72 3196.0 3159.2 −1.2 1382.3 1395.4 1.0 1133.1 1144.6 1.01 3306.5 3272.5 −1.0 1272.0 1284.3 1.0 1050.3 1061.1 1.0

Frame-B1 Frame-B16X -direction excitation Y -direction excitation X -direction excitation Y -direction excitationPerturbed Average CM %error Perturbed Average CM %error Perturbed Average CM %error Perturbed Average CM %error

(11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)400.2 401.8 0.4 583.4 577.5 −1.0 440.6 441.6 0.2 323.4 327.5 1.3703.9 729.7 3.7 1057.3 1093.3 3.4 854.2 831.0 −2.7 636.2 610.9 −4.0985.7 1007.1 2.2 1461.7 1483.1 1.5 1145.9 1130.9 −1.3 860.6 846.2 −1.7

1231.7 1230.4 −0.1 1785.3 1783.5 −0.1 1371.1 1367.3 −0.3 1032.7 1033.6 0.11439.1 1438.7 0.0 2042.1 2059.8 0.9 1592.4 1588.0 −0.3 1226.5 1206.6 −1.61638.8 1619.1 −1.2 2295.8 2302.1 0.3 1776.5 1780.4 0.2 1368.5 1357.4 −0.81798.4 1780.1 −1.0 2507.7 2525.5 0.7 1950.2 1954.4 0.2 1516.4 1494.8 −1.41974.9 1940.7 −1.7 2774.2 2760.5 −0.5 2117.7 2133.8 0.8 1631.3 1634.9 0.22088.0 2063.2 −1.2 2965.6 2963.4 −0.1 2276.9 2285.6 0.4 1747.2 1744.4 −0.2

example-building, also referred to as the ‘perturbed model’, isthen analyzed using SAP2000. Resulting shear forces shared bythe frames A1, A11, A14, B1 and B16 due to excitation alongX -direction are presented in Table 4 (columns (2), (5), (8),(11)and (17), respectively). For frames B1 and B16, inducedshear forces due to Y -direction excitation are also presented inTable 4 (columns (14) and (20), respectively). Similar responses

for the ‘average CM model’ calculated using SAP2000 areagain presented in Table 4 for ready reference and comparison.Finally, frame shear forces calculated for the ‘average CMmodel’ using the proposed method are compared with thatcalculated for the ‘perturbed model’ using SAP2000 and theresulting relative errors are presented in Table 5 for the selectedframes.


Table 5Error in the calculation of frame shear force when the proposed method is used for the building with perturbed CM (example-2) (compared with the results ofSAP2000)

Floor Frame-A1 Frame-A11 Frame-A14 Frame-B1 Frame-B16X -direction excitation X -direction excitation Y -direction excitation X -direction excitation Y -direction excitation

(1) (2) (3) (4) (5) (6) (7) (8)10 0.9 −6.8 −7.3 −4.6 −3.8 −8.5 4.6

9 1.2 −3.1 −3.5 6.2 −0.6 −8.2 3.28 1.2 −2.6 −3.0 4.6 −1.7 −5.6 4.47 2.7 −3.4 −3.6 2.2 −2.6 −3.7 5.36 2.7 −2.9 −3.1 2.3 −1.2 −2.9 3.05 3.5 −3.1 −3.2 1.3 −1.5 −1.8 3.64 3.4 −2.4 −2.5 1.8 −0.9 −1.3 2.93 3.7 −2.2 −2.3 1.5 −2.0 −0.2 4.82 3.6 −2.1 −2.4 2.5 −1.9 −0.4 5.31 3.4 −5.3 −6.2 2.3 −4.1 1.3 5.4

6.0. Numerical example-3: Validation of the range ofapplicability

Two example-problems are furnished for this purpose,namely, examples-3a and 3b. Both the example-buildings differfrom that considered in example-1 only in their locations of thefloor CMs. Offsets in location of CMs from that considered inexample-1 are shown in Table 3 (columns 4–7). Moreover, thelocation of the CMs for both the examples are chosen in such away that example-1 nearly represents the ‘average CM model’in either case. Considering frame A1 as the characteristic frame,βx and βy are then calculated (Eqs. (23a) and (23b)) as 1.527×

10−3 and −1.288 × 10−3, respectively, for the example-3a and−2.357×10−2 and 2.423×10−2, respectively, for the example-3b. Clearly, the simplified method should be acceptable for theexample-3a but not for the example-3b. This will be furtherverified by the analysis using SAP2000.

A. Example-3a: Scattering in location of the CMS within theacceptable limit

When the excitation is considered along the X -direction,shear forces shared by the frames A1, A11, A14, B1 andB16 are presented in Table 6 (columns (2), (5), (8), (11) and(17), respectively). Further, when the excitation is consideredalong the Y -direction, induced shear forces for the framesB1 and B16 are also presented in Table 6 (columns (14)and (20), respectively). Moreover, first three mass normalizedmode shapes are presented in Table 7 (columns 5, 9 and 13,respectively).

B. Example-3b: Scattering in location of the CMS beyondthe acceptable limit

Frequency and, MMPR associated with the excitation alongthe X - and Y -directions, are presented in Table 8 (columns(2), (4) and (5), respectively) for the first 15 modes. Whenthe excitation is considered along the X -direction, shear forcesshared by the frames A1, A11, A14, B1 and B16 are presentedin Table 9 (columns (2), (5), (8), (11) and (17), respectively).For the frames B1 and B16, induced shear force due to theexcitation along the Y -direction are also presented in Table 9(columns (14) and (20), respectively). Moreover, first threemass normalized mode shapes are presented in Table 10(columns 5, 9 and 13, respectively).

For the purpose of direct comparison, results of the ‘averageCM model’ (example-1) are again presented in Tables 6–10.

7.0. Results and discussions

In the case of building with vertically-aligned CMs(example-1), close agreement in frequencies and MMPRsbetween the simplified analysis and SAP2000 may be seen(Table 1). It may also be noted that the simplified analysisyields consistent modal response irrespective of the selectionof characteristic frame as A1 or B1 (Table 1). However, thestiffest frame may always be preferred for the selection of thecharacteristic frame. Further, comparing the simplified analysiswith SAP2000 for the frame shear forces, maximum error isnoted as ±9%. More precisely, as the design force resultantoften governs at the ground-story level, the error at that levelmay be a better option for such comparison, which is noted as±7%.

In the case of example-2, scattering in location of CMs issuch that βx = 7.6 × 10−4 and βy = 1.29 × 10−3. Modalresponses calculated for the perturbed and ‘average CM model’using SAP2000 are found well in agreement with each other;these results are, however, not presented for brevity. When theframe shear forces are compared, maximum error of ±4 % isnoted for the upper-story levels while only ±1% is observedat the base level (Table 4). Since, SAP2000 is used for boththe models, these comparisons truly reflect the effect of thescattering of CMs irrespective of the proportionality of thelateral stiffness matrices of the constituting MRFs. Moreover,while applying the simplified analysis to the perturbed modeloverall error in the estimation of frame shear force is notedas ±7% for the upper-story level and ±6% for the base levelwhen compared to the response of the perturbed model usingSAP2000 (Table 5). It may be noted that, this error includes theeffect of nonproportionality of lateral stiffness matrices of theconstituting MRFs also.

Validation of the limit predicted by the perturbationformulation is substantiated through examples-3a and 3b.Example-3a is considered to furnish the case wherein scatteringin location of the CMs remains within the prescribed limit; onthe other hand, example-3b represents the case wherein the


Table 6Comparison of frame shear force of the perturbed and the average CM model calculated using SAP2000—-examples-3a and -1


(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)10 618.3 598.5 −3.2 247.1 262.8 6.3 203 215.7 6.3

9 1172 1112 −5.1 457.2 488.6 6.9 375 401.4 78 1618.5 1535.9 −5.1 638 679.7 6.5 523.3 557.8 6.67 1952.4 1877.2 −3.9 794.1 830.6 4.6 650.6 681.2 4.76 2288.3 2197.7 −4 929.8 968.6 4.2 761.4 794.1 4.35 2560.7 2476.9 −3.3 1058.2 1090.8 3.1 865.8 894 3.34 2825.6 2725.2 −3.6 1164.7 1203 3.3 953 985.7 3.43 3076.3 2968.8 −3.5 1271.9 1312.8 3.2 1040.9 1075.9 3.42 3281.8 3159.2 −3.7 1345.9 1395.4 3.7 1102.5 1144.6 3.81 3390.4 3272.5 −3.5 1237.5 1284.3 3.8 1021.6 1061.1 3.9


(11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)393.9 401.8 2 588.8 577.5 −1.9 436.8 441.6 1.1 324.4 327.5 1679 729.7 7.5 1024.3 1093.3 6.7 885.1 831 −6.1 657.9 610.9 −7.1963.8 1007.1 4.5 1446.1 1483.1 2.6 1169.7 1130.9 −3.3 870.1 846.2 −2.7

1226.5 1230.4 0.3 1796.1 1783.5 −0.7 1380.6 1367.2 −1 1023.2 1033.6 11432.5 1438.7 0.4 2037.5 2059.8 1.1 1603.8 1588 −1 1234.9 1206.6 −2.31649.6 1619.1 −1.8 2302 2302.1 0 1776.3 1780.4 0.2 1366 1357.3 −0.61807 1780.1 −1.5 2506.1 2525.5 0.8 1952.6 1954.4 0.1 1519 1494.8 −1.62000.3 1940.7 −3 2805.4 2760.5 −1.6 2110.3 2133.8 1.1 1608.6 1634.9 1.6

prescribed limit is exceeded. Clearly, example-2 would alsosuffice the purpose of example-3a. Nevertheless, example-3a isconsidered, as the βx in example-2 is one order less than theprescribed limit.

In the case of example-3a, βx = 1.527 × 10−3 and βy =

−1.288 × 10−3, and hence, ρ = 3. Therefore, maximumexpected difference in eigenvalues (reciprocal of the squareof frequency in this paper) between the perturbed (example-3a) and average CM (example-1) models should be of theorder of 10−3, especially, in the first three modes. Similarly,maximum expected difference in the mass normalized modeshapes, especially, in the first three modes should be of the orderof 10−2 times the mass normalized first characteristic modeshape. Further, maximum expected error in the estimation offrame shear force (as derived in Appendix) should be of theorder of 100, i.e., 0%–10%.

Comparison of modal responses, e.g., frequency, MMPRetc., shows close agreement between the perturbed and‘average CM model’; these results are not presented forbrevity. However, first three mass normalized mode shapesare compared numerically and graphically. First, comparingcolumns 4, 8, and 12 with columns 5, 9 and 13, respectively,of Table 7, it may be seen that first three mass normalizedmode shapes do match well in perturbed and ‘average CM’models. Moreover, order of their difference (columns 7, 11 and15, Table 7) is also well below the predicted maximum limitexcept at a few places indicated in italic font. On the otherhand, Figs. 2–4 show a graphical comparison for the first threemass normalized mode shapes, respectively. It may be noted,as evident from the modal mass participation ratio in example-

1, first mode shape is predominantly along Y -direction whilesecond and third mode shapes are dominant in X -direction.Therefore, the small jaggedness observed in the Y -componentof the second and third mode shapes (Figs. 3(b) and 4(b)),respectively) does not contribute much to the overall response.Next, columns (4), (7), (10), (13), (16), (19) and (22) of Table 6show that the errors in the estimation of the frame shear forcesfor the selected frames are also well within the maximumpredicted limit of 10%; based on ground-story level, maximumerror in the estimation of frame shear force may be consideredas around 4%.

In order to assess the conservativeness in the limit ofscattering predicted by the perturbation formulation (Eqs.(23a) and (23b)), example-3b is considered. Here, βx =

2.357 × 10−2, βy = 2.423 × 10−2 and hence, ρ =

2. Clearly, the proposed method should not be applied tothis building. Nevertheless, if applied (i) maximum expecteddifference in eigenvalues between the perturbed (example-3b) and average CM (example-1) models should be of theorder 10−2, especially, in the first three modes, (ii) maximumexpected difference in the mass normalized mode shapes,especially, in the first three modes should be of the order 10−1

times the mass normalized first characteristic mode shape and(iii) maximum expected error in the estimation of the frameshear force should be of the order 101, i.e., 10%–100%.

From columns (4), (7), (10), (13), (16), (19) and (22) ofTable 9, maximum error in the estimation of the frame shearforce may be noted to be as much as about 50%. This is incompliance with what is predicted by the perturbation analysis.Clearly, the proposed method cannot be applied to example-3b.


Tabl

e7

Err

orin

mod

esh

apes

onus

ing

aver

age

cent

erof

mas

sm

odel

and

pert

urbe

dm

odel

—ex

ampl

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aan

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Dir

ectio

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leve

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Mod

e-1

Mod

e-2

Mod

e-3

Ave

rage

Pert

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dE

rror

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vera

gePe

rtur

bed

Err

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der

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rage

Pert

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dE

rror

orde

r1

23

45

6=(5

−4)/3

78

910

1112

1314

15

X-d

irec

tion

1−

9.25

E−

04−

7.97

E−

05−

7.79

E−

05−

1.95

E−

03−

3−

7.26

E−

04−

7.15

E−

04−

1.19

E−

02−

2−

6.07

E−

04−

6.21

E−

041.

51E−

02−

22

−2.

41E−

03−

2.00

E−

04−

1.79

E−

04−

8.73

E−

03−

3−

1.85

E−

03−

1.72

E−

03−

5.24

E−

02−

2−

1.57

E−

03−

1.71

E−

035.

82E−

02−

23

−3.

92E−

03−

3.22

E−

04−

3.15

E−

04−

1.79

E−

03−

3−

2.98

E−

03−

2.94

E−

03−

1.12

E−

02−

2−

2.55

E−

03−

2.60

E−

031.

35E−

02−

24

−5.

34E−

03−

4.35

E−

04−

3.88

E−

04−

8.81

E−

03−

3−

4.04

E−

03−

3.76

E−

03−

5.17

E−

02−

2−

3.47

E−

03−

3.77

E−

035.

75E−

02−

25

−6.

61E−

03−

5.38

E−

04−

5.52

E−

042.

12E−

03−

3−

5.00

E−

03−

5.11

E−

031.

62E−

02−

2−

4.29

E−

03−

4.19

E−

03−

1.63

E−

02−

26

−7.

72E−

03−

6.26

E−

04−

5.73

E−

04−

6.87

E−

03−

3−

5.83

E−

03−

5.53

E−

03−

3.84

E−

02−

2−

5.01

E−

03−

5.34

E−

034.

30E−

02−

27

−8.

62E−

03−

6.99

E−

04−

7.12

E−

041.

51E−

03−

3−

6.51

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03−

6.64

E−

031.

55E−

02−

2−

5.59

E−

03−

5.46

E−

03−

1.57

E−

02−

28

−9.

31E−

03−

7.56

E−

04−

6.88

E−

04−

7.30

E−

03−

3−

7.04

E−

03−

6.67

E−

03−

3.95

E−

02−

2−

6.04

E−

03−

6.46

E−

034.

44E−

02−

29

−9.

79E−

03−

7.94

E−

04−

7.01

E−

04−

9.50

E−

03−

3−

7.40

E−

03−

6.88

E−

03−

5.36

E−

02−

2−

6.35

E−

03−

6.94

E−

035.

99E−

02−

210

−1.

00E−

02−

8.16

E−

04−

7.60

E−

04−

5.57

E−

03−

3−

7.61

E−

03−

7.34

E−

03−

2.66

E−

02−

2−

6.52

E−

03−

6.83

E−

033.

05E−

02−

2

Y-d

irec

tion

1−

9.25

E−

04−

9.03

E−

04−

8.95

E−

04−

8.65

E−

03−

31.

12E−

041.

51E−

04−

4.22

E−

02−

2−

1.86

E−

05−

5.90

E−

054.

37E−

02−

22

−2.

41E−

03−

2.36

E−

03−

2.33

E−

03−

1.25

E−

02−

23.

04E−

044.

67E−

04−

6.77

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02−

2−

5.90

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05−

2.34

E−

047.

27E−

02−

23

−3.

92E−

03−

3.85

E−

03−

3.95

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032.

53E−

02−

25.

04E−

04−

8.10

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051.

49E−

01−

1−

1.05

E−

045.

32E−

04−

1.63

E−

01−

14

−5.

34E−

03−

5.26

E−

03−

5.24

E−

03−

2.81

E−

03−

36.

94E−

047.

72E−

04−

1.46

E−

02−

2−

1.50

E−

04−

2.34

E−

041.

57E−

02−

25

−6.

61E−

03−

6.53

E−

03−

6.63

E−

031.

48E−

02−

28.

67E−

042.

56E−

049.

24E−

02−

2−

1.92

E−

044.

73E−

04−

1.01

E−

01−

16

−7.

72E−

03−

7.63

E−

03−

7.57

E−

03−

7.39

E−

03−

31.

02E−

031.

36E−

03−

4.47

E−

02−

2−

2.30

E−

04−

6.10

E−

044.

92E−

02−

27

−8.

62E−

03−

8.54

E−

03−

8.65

E−

031.

37E−

02−

21.

14E−

033.

72E−

048.

93E−

02−

2−

2.61

E−

045.

73E−

04−

9.68

E−

02−

28

−9.

31E−

03−

9.24

E−

03−

9.28

E−

034.

73E−

03−

31.

24E−

039.

31E−

043.

31E−

02−

2−

2.87

E−

043.

94E−

05−

3.51

E−

02−

29

−9.

79E−

03−

9.72

E−

03−

9.57

E−

03−

1.55

E−

02−

21.

31E−

032.

33E−

03−

1.04

E−

01−

1−

3.06

E−

04−

1.44

E−

031.

16E−

01−

110

−1.

00E−

02−

1.00

E−

02−

1.00

E−

022.

99E−

04−

41.

35E−

031.

31E−

033.

48E−

03−

3−

3.19

E−

04−

2.97

E−

04−

2.19

E−

03−

3

θ-d

irec

tion

1−

9.25

E−

044.

52E−

064.

58E−

06−

6.49

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05−

52.

43E−

052.

53E−

05−

1.08

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03−

3−

2.80

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05−

2.68

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05−

1.30

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03−

32

−2.

41E−

031.

12E−

051.

14E−

05−

8.31

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05−

56.

27E−

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53E−

05−

1.08

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3−

7.35

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7.04

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1.29

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03−

33

−3.

92E−

031.

79E−

051.

83E−

05−

1.02

E−

04−

41.

02E−

041.

06E−

04−

1.02

E−

03−

3−

1.20

E−

04−

1.15

E−

04−

1.28

E−

03−

34

−5.

34E−

032.

40E−

052.

42E−

05−

3.75

E−

05−

51.

38E−

041.

44E−

04−

1.12

E−

03−

3−

1.64

E−

04−

1.57

E−

04−

1.31

E−

03−

35

−6.

61E−

032.

95E−

052.

93E−

053.

02E−

05−

51.

72E−

041.

79E−

04−

1.06

E−

03−

3−

2.04

E−

04−

1.95

E−

04−

1.36

E−

03−

36

−7.

72E−

033.

42E−

053.

32E−

051.

30E−

04−

42.

00E−

042.

09E−

04−

1.17

E−

03−

3−

2.38

E−

04−

2.28

E−

04−

1.30

E−

03−

37

−8.

62E−

033.

80E−

053.

64E−

051.

86E−

04−

42.

24E−

042.

33E−

04−

1.04

E−

03−

3−

2.66

E−

04−

2.55

E−

04−

1.28

E−

03−

38

−9.

31E−

034.

09E−

053.

81E−

053.

01E−

04−

42.

42E−

042.

52E−

04−

1.07

E−

03−

3−

2.88

E−

04−

2.76

E−

04−

1.29

E−

03−

39

−9.

79E−

034.

29E−

053.

89E−

054.

09E−

04−

42.

55E−

042.

65E−

04−

1.02

E−

03−

3−

3.02

E−

04−

2.90

E−

04−

1.23

E−

03−

310

−1.

00E−

024.

39E−

053.

96E−

054.

28E−

04−

42.

62E−

042.

73E−

04−

1.09

E−

03−

3−

3.10

E−

04−

2.98

E−

04−

1.19

E−

03−

3


Table 8Comparison of modal responses of perturbed and average CM model calculated using SAP2000—examples-3b and -1

Mode number Frequency (rad/s) Modal mass participation ratio (%)Perturbed model Average CM model Perturbed model Average CM model

X -direction Y -direction X -direction Y -direction

(1) (2) (3) (4) (5) (6) (7)1 4.782 4.838 14.6 41.7 0.5 78.32 4.894 4.946 24.5 37.3 45.9 1.43 5.38 5.365 41 0.7 33.7 0.14 14.137 14.376 2.5 4.6 0.2 10.25 14.539 14.653 3 5.8 6.1 0.46 16.071 15.937 5 0.1 4.3 0.07 24.313 24.977 0.9 1.2 0.2 3.58 25.29 25.362 1 2.7 2.2 0.59 27.836 27.654 2.1 0.2 1.5 0.0

10 37.077 37.043 0 2.2 0.4 1.311 38.731 37.533 1.8 0 0.9 0.712 39.387 41.137 0.3 0 0.8 0.013 49.184 49.618 0.4 0.3 0.5 0.514 50.599 50.419 0.4 1 0.4 0.915 55.188 55.458 0.6 0.1 0.5 0.0

Table 9Comparison of frame shear force of the perturbed and the average CM model calculated using SAP2000—examples-3b and -1


(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)10 662.9 598.5 9.7 187 262.8 40.5 156 215.7 38.3

9 1102 1112 −0.9 358.8 488.6 36.2 297 401.4 35.18 1607.8 1535.9 4.5 467.9 679.7 45.3 389.3 557.8 43.37 2060.4 1877.2 8.9 555.8 830.6 49.4 464.4 681.2 46.76 2357.8 2197.7 6.8 656 968.6 47.7 547.8 794.1 455 2698.5 2476.9 8.2 725.8 1090.8 50.3 607.8 894 47.14 3005.2 2725.2 9.3 793.8 1203. 51.5 666.6 985.7 47.93 3227.6 2968.8 8 875.3 1312.8 50 734.4 1075.9 46.52 3437.9 3159.2 8.1 926.7 1395.4 50.6 779.2 1144.6 46.91 3557.1 3272.5 8 852.4 1284.3 50.7 725.3 1061.1 46.3


(11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)403.7 401.8 −0.5 477.9 577.5 20.8 418.1 441.6 5.6 413.7 327.5 −20.8772.3 729.7 −5.5 993 1093.3 10.1 778.7 831 6.7 794.7 610.9 −23.1

1082.4 1007.1 −7 1375.9 1483.1 7.8 1050.6 1130.9 7.6 1005.9 846.2 −15.91266.5 1230.4 −2.9 1579.9 1783.5 12.9 1355.6 1367.2 0.9 1237.1 1033.6 −16.51486.2 1438.7 −3.2 1912.1 2059.8 7.7 1545.4 1588. 2.8 1417.8 1206.6 −14.91650.5 1619.1 −1.9 2113.6 2302.1 8.9 1757.7 1780.4 1.3 1576.8 1357.3 −13.91789.4 1780.1 −0.5 2278.2 2525.5 10.9 1961.1 1954.4 −0.3 1740 1494.8 −14.11977.6 1940.7 −1.9 2538.8 2760.5 8.7 2113.1 2133.8 1 1892.9 1634.9 −13.6

However, frequencies of the perturbed and ‘average CM’models are still in close agreement with each other (columns(2) and (3) of Table 8). On the other hand, MMPRs in both themodels are significantly different (columns (4)–(7) of Table 8).Moreover, maximum difference in mode shapes for the firstthree modes is noted to be of the order of 10−1 times the massnormalized first characteristic mode shape (columns 7, 11 and15 of Table 10), which is also predicted by the perturbationanalysis. Such a large variation in first three mass normalized

mode shapes is also evident from the graphical comparisonpresented in Figs. 5–7.

Therefore, these results show that upper bound of the orderof expected error in the estimation of the eigenvalues, modeshapes and frame shear forces predicted by the perturbationanalysis is accurate enough. Thus, the proposed conditions(Eqs. (23a) and (23b)) may be used to assess the applicability ofthe simplified analysis wherein the conditions of special classare not truly met. However, it may be noted that the perturbed


Tabl

e10

Err

orin

mod

esh

apes

onus

ing

aver

age

cent

erof

mas

sm

odel

and

pert

urbe

dm

odel

—ex

ampl

es-3

ban

d-1

Dir

ectio

nFl

oor

leve

lC

hara

cter

istic

Mod

e-1

Mod

e-2

Mod

e-3

Ave

rage

Pert

urbe

dE

rror

orde

rA

vera

gePe

rtur

bed

Err

oror

der

Ave

rage

Pert

urbe

dE

rror

orde

r1

23

45

6=(5

−4)/3

78

910

1112

1314

15

X-d

irec

tion

1−

9.25

E−

04−

7.97

E−

05−

3.30

E−

042.

71E−

01−

1−

7.26

E−

04−

4.59

E−

04−

2.89

E−

01−

1−

6.07

E−

04−

7.74

E−

041.

81E−

01−

12

−2.

41E−

03−

2.00

E−

04−

9.65

E−

043.

18E−

01−

1−

1.85

E−

03−

1.28

E−

03−

2.36

E−

01−

1−

1.57

E−

03−

1.84

E−

031.

11E−

01−

13

−3.

92E−

03−

3.22

E−

04−

2.02

E−

034.

32E−

01−

1−

2.98

E−

03−

2.47

E−

03−

1.31

E−

01−

1−

2.55

E−

03−

2.38

E−

03−

4.29

E−

02−

24

−5.

34E−

03−

4.35

E−

04−

1.93

E−

032.

81E−

01−

1−

4.04

E−

03−

2.65

E−

03−

2.61

E−

01−

1−

3.47

E−

03−

4.28

E−

031.

52E−

01−

15

−6.

61E−

03−

5.38

E−

04−

2.39

E−

032.

80E−

01−

1−

5.00

E−

03−

3.28

E−

03−

2.61

E−

01−

1−

4.29

E−

03−

5.29

E−

031.

51E−

01−

16

−7.

72E−

03−

6.26

E−

04−

4.03

E−

034.

41E−

01−

1−

5.83

E−

03−

4.92

E−

03−

1.18

E−

01−

1−

5.01

E−

03−

4.55

E−

03−

5.93

E−

02−

27

−8.

62E−

03−

6.99

E−

04−

2.85

E−

032.

49E−

01−

1−

6.51

E−

03−

4.03

E−

03−

2.88

E−

01−

1−

5.59

E−

03−

7.24

E−

031.

91E−

01−

18

−9.

31E−

03−

7.56

E−

04−

3.49

E−

032.

93E−

01−

1−

7.04

E−

03−

4.72

E−

03−

2.49

E−

01−

1−

6.04

E−

03−

7.28

E−

031.

33E−

01−

19

−9.

79E−

03−

7.94

E−

04−

5.37

E−

034.

67E−

01−

1−

7.40

E−

03−

6.46

E−

03−

9.63

E−

02−

2−

6.35

E−

03−

5.47

E−

03−

8.99

E−

02−

210

−1.

00E−

02−

8.16

E−

04−

4.22

E−

033.

39E−

01−

1−

7.61

E−

03−

5.50

E−

03−

2.10

E−

01−

1−

6.52

E−

03−

7.28

E−

037.

55E−

02−

2

Y-d

irec

tion

1−

9.25

E−

04−

9.03

E−

04−

5.33

E−

04−

4.00

E−

01−

11.

12E−

047.

30E−

04−

6.68

E−

01−

1−

1.86

E−

05−

2.46

E−

042.

46E−

01−

12

−2.

41E−

03−

2.36

E−

03−

1.78

E−

03−

2.41

E−

01−

13.

04E−

041.

58E−

03−

5.29

E−

01−

1−

5.90

E−

05−

1.55

E−

043.

99E−

02−

23

−3.

92E−

03−

3.85

E−

03−

3.15

E−

03−

1.79

E−

01−

15.

04E−

042.

36E−

03−

4.74

E−

01−

1−

1.05

E−

046.

33E−

05−

4.30

E−

02−

24

−5.

34E−

03−

5.26

E−

03−

3.44

E−

03−

3.41

E−

01−

16.

94E−

043.

98E−

03−

6.16

E−

01−

1−

1.50

E−

04−

1.04

E−

031.

67E−

01−

15

−6.

61E−

03−

6.53

E−

03−

4.48

E−

03−

3.10

E−

01−

18.

67E−

044.

76E−

03−

5.88

E−

01−

1−

1.92

E−

04−

1.01

E−

031.

24E−

01−

16

−7.

72E−

03−

7.63

E−

03−

6.22

E−

03−

1.82

E−

01−

11.

02E−

034.

69E−

03−

4.76

E−

01−

1−

2.30

E−

041.

08E−

04−

4.38

E−

02−

27

−8.

62E−

03−

8.54

E−

03−

5.21

E−

03−

3.86

E−

01−

11.

14E−

036.

79E−

03−

6.55

E−

01−

1−

2.61

E−

04−

2.16

E−

032.

20E−

01−

18

−9.

31E−

03−

9.24

E−

03−

7.24

E−

03−

2.14

E−

01−

11.

24E−

035.

94E−

03−

5.04

E−

01−

1−

2.87

E−

04−

2.55

E−

04−

3.44

E−

03−

39

−9.

79E−

03−

9.72

E−

03−

7.41

E−

03−

2.36

E−

01−

11.

31E−

036.

43E−

03−

5.23

E−

01−

1−

3.06

E−

04−

5.39

E−

042.

38E−

02−

210

−1.

00E−

02−

1.00

E−

02−

7.19

E−

03−

2.79

E−

01−

11.

35E−

037.

00E−

03−

5.62

E−

01−

1−

3.19

E−

04−

1.12

E−

037.

92E−

02−

2

θ-d

irec

tion

1−

9.25

E−

044.

52E−

062.

01E−

05−

1.68

E−

02−

22.

43E−

051.

73E−

057.

57E−

03−

3−

2.80

E−

05−

2.52

E−

05−

3.03

E−

03−

32

−2.

41E−

031.

12E−

055.

17E−

05−

1.68

E−

02−

26.

27E−

054.

50E−

057.

36E−

03−

3−

7.35

E−

05−

6.63

E−

05−

2.99

E−

03−

33

−3.

92E−

031.

79E−

058.

38E−

05−

1.68

E−

02−

21.

02E−

047.

34E−

057.

30E−

03−

3−

1.20

E−

04−

1.09

E−

04−

2.81

E−

03−

34

−5.

34E−

032.

40E−

051.

14E−

04−

1.69

E−

02−

21.

38E−

041.

00E−

047.

12E−

03−

3−

1.64

E−

04−

1.48

E−

04−

3.00

E−

03−

35

−6.

61E−

032.

95E−

051.

41E−

04−

1.69

E−

02−

21.

72E−

041.

24E−

047.

26E−

03−

3−

2.04

E−

04−

1.84

E−

04−

3.02

E−

03−

36

−7.

72E−

033.

42E−

051.

65E−

04−

1.69

E−

02−

22.

00E−

041.

45E−

047.

13E−

03−

3−

2.38

E−

04−

2.15

E−

04−

2.98

E−

03−

37

−8.

62E−

033.

80E−

051.

84E−

04−

1.69

E−

02−

22.

24E−

041.

63E−

047.

08E−

03−

3−

2.66

E−

04−

2.40

E−

04−

3.02

E−

03−

38

−9.

31E−

034.

09E−

052.

00E−

04−

1.71

E−

02−

22.

42E−

041.

76E−

047.

09E−

03−

3−

2.88

E−

04−

2.58

E−

04−

3.22

E−

03−

39

−9.

79E−

034.

29E−

052.

10E−

04−

1.71

E−

02−

22.

55E−

041.

86E−

047.

05E−

03−

3−

3.02

E−

04−

2.70

E−

04−

3.27

E−

03−

310

−1.

00E−

024.

39E−

052.

16E−

04−

1.71

E−

02−

22.

62E−

041.

91E−

047.

07E−

03−

3−

3.10

E−

04−

2.77

E−

04−

3.28

E−

03−

3


Fig. 2. Comparison of first mode shape on using the average center of massmodel and the perturbed model—examples-3a and -1; (a) X -direction, (b) Y -direction and (c) theta-direction.

models are used as the basis of estimation of error in frameshear forces in the example-problems (Tables 6 and 9) whilethe ‘average CM model’ is used as the basis for the same in theformulation of perturbation analysis.

It may be noted that the shear wall is not included inthe example-buildings. This is because the simplified analysisassumes the stiffness proportionality of the constituting MRFs.This implies that if one frame is predominantly deflectingin a flexural mode, other frames will also follow the similarpattern. This does not happen when only a few selected framescontain squat shear walls, which force such frames to deflectin a shear type pattern and frame–shear wall interaction comesinto the picture. In such cases, simplified procedure does notwork. However, if the constituting shear walls are of slendertype such that, even in their presence, constituting framescontinue to deflect in a flexural pattern, similar to the frames notcontaining shear walls, simplified procedure and the subsequentperturbation analysis are still applicable. This has been verifiedbut not included in this paper. However, inclusion of suchshear walls requires some more studies and thus, may beexcluded from the scope of this paper, and will be reportedseparately.

Moreover, one may argue about the randomly scattered CMsselected at different floors in a proportionally framed structuralarrangement without shear walls, as evident from the example-problems. However, it may be noted that the scattering assumedin the example-problems is to illustrate the limit of applicability

Fig. 3. Comparison of second mode shape on using the average center of massmodel and the perturbed model—examples-3a and -1; (a) X -direction, (b) Y -direction and (c) theta-direction.

of the proposed procedure and does not necessarily reflect thetrue scattering available in the actual buildings. Further, whenthe degree of scattering is considered, absence of shear wallsmay not play a significant role as the presence of infill masonrywalls nearly compensates it. Also, it is a common practice,unlike the mass, not to rely upon the stiffness provided by theinfill masonry walls, which may not be available after one ortwo cycles of seismic excitation.

Further, this paper deals with the elastic response spectrumanalysis of the MRF-buildings with horizontal setbacks. Incontrast, the MRF-buildings should be designed and detailedfor their ductile behavior because of the possible inelasticexcursion under strong shaking. Consequently, one may argueabout the use of elastic analysis. However, it may be noted thatelastic analysis has its own importance. First, MRF-buildingshould be designed such that it can sustain minor shakingwithout any damage of even the nonstructural components.Hence, an elastic analysis is required to ensure the necessaryelastic limit of the building. On the other hand, ductile design ofthe MRF-buildings following the most seismic codes requiresan elastic (static or dynamic, depending upon the order of massand stiffness irregularity) analysis based on the gross cross-sectional properties of the elements. Member force resultantthus calculated is then reduced by suitable ‘response reductionfactor’ to ascertain the design member force resultant. Capacitydesign principle is then adopted to suppress the undesirableshear failure modes and to detail the constituting members


Fig. 4. Comparison of third mode shape on using the average center of massmodel and the perturbed model—examples-3a and -1; (a) X -direction, (b) Y -direction and (c) theta-direction.

so as to meet the seismic demand. Therefore, following mostseismic codes, it is seen that only elastic analysis is requiredfor the preliminary design of MRF-buildings. Further, as longas the dynamic analysis is limited within the elastic range (asductility is usually incorporated by virtue of code specified‘response reduction factor’), elastic time history analysismay be performed with modal decomposition. Therefore, theproposed simplified method is equally applicable to modal timehistory analysis, which however, has not been illustrated. Since,formulation for perturbation analysis is based on mode-by-mode decomposition, final results are equally applicable to timehistory analysis also.

It may also be noted that the effect of higher modesare neglected in perturbation analysis for its simplicity butnot in the simplified analysis. Further, simplified methoddoes not work in the presence of vertical setbacks. This, insome sense, may also limit the applicability of the proposedsimplified method. However, in practice, buildings with onlyhorizontal setbacks are also seen in quite large numbers.Finally, detailed nonlinear 3D FEM analysis of special classof torsionally coupled buildings with horizontal offsets is nolonger impossible in the present era of computer advancement.Consequently, one may argue against the use of the proposedsimplified method. However, a simplified method is alwayspreferred to assess the design force resultants at the preliminarystage of design; detailed nonlinear analysis may be carried out

Fig. 5. Comparison of first mode shape on using the average center of massmodel and the perturbed model—examples-3b and -1; (a) X -direction, (b) Y -direction and (c) theta-direction.

at the final stage of verification. Therefore, the proposed methodcan be applied as a convenient and offhand tool with sufficientaccuracy at the preliminary stage of design and requires onlythe execution of plane frame analysis with a personal computer.

8.0. Summary and conclusions

Simplified method of analysis is proposed for the specialclass of torsionally coupled buildings with horizontal setbacks.The method can also be applied to the cases wherein all theclassification criteria for the special class of buildings arenot strictly satisfied. Range of applicability of the simplifiedanalysis with an acceptable margin of error is also proposed.

Acknowledgement

This paper is published with the approval of the Director,Structural Engineering Research Center, Chennai, India.

Appendix. Limit of scattering in location of the floor CMSusing perturbation analysis

A.1. Formulation of error equations for eigenvalues andeigenvectors

Here, the objective is to ascertain the error in computationof eigenvalues and eigenvectors if system-B is used in lieu


Fig. 6. Comparison of second mode shape on using the average center of massmodel and the perturbed model—examples-3b and -1; (a) X -direction, (b) Y -direction and (c) theta-direction.

of system-A. For this purpose, system-C is considered as theunperturbed system and the error in using system-C in lieu ofsystem-A is evaluated. Next, the error in using system-C in lieuof system-B is evaluated. Difference between these two errorsmay be considered as the error in using system-B instead ofusing system-A.

A.1.1. Case-1:System-A and system-CEigen equations of system-C and system-A may be

expressed as follows:

γ0 [K ] {ψ} = [M]C {ψ} (A.1)

γ A [K ] {φA} = [M]A {φA

} (A.2)

where, [K ] is the stiffness matrix, [M]C , [M]A the massmatrices, γ0, γ A the eigenvalues, and ψ , φA the eigenvectors ofsystem-C and system-A, respectively. Perturbation in the massmatrix is given by

[E]A= 0 0 −

[(byi + ey

)mi]

0 0[(

bxi + ex)

mi]

−[(

byi + ey)

mi] [(

bxi + ex)

mi]

[mi d2i ]

.(A.3)

It may be noted that the unperturbed system has two distincttypes of eigenvectors, e.g., pure translational (uncoupled from

Fig. 7. Comparison of third mode shape on using the average center of massmodel and the perturbed model—examples-3b and -1; (a) X -direction, (b) Y -direction and (c) theta-direction.

torsional rotation but two translations along any two orthogonallateral directions may be coupled) and pure torsional; 2Nnumber of pure translational and N number of pure torsionaleigenvectors constitute all together 3N number of eigenvectors.

Let ψ0i be the i th eigenvector of the unperturbed system. Ifψ0i is pure translational

{ψ0i } = {ψT L0i }

T= { ψ̄ xT

0i ψ̄yT

0i 0T }. (A.4)

On the other hand, if ψ0i is a pure torsional eigenvector

{ψ0i } = {ψT R0i }

T={0T 0T ψ̄θT

0i

}. (A.5)

Let us now evaluate the expression ψT0i E Aψ0i . In the case of a

pure translational eigenvector, utilizing Eqs. (A.3) and (A.4), itmay be shown that

ψT LT

0i E AψT L0i = 0. (A.6)

Similarly, in the case of pure torsional eigenvector, utilizingEqs. (A.3) and (A.5), in conjunction with mass normalizationcriteria, and noting that (di/r̃)

2 is of the order 10−1 to 10−3

(Eq. (22) of the main text), it may be shown that ψT RT

0i E AψT R0i

is of the order 10−1 to 10−2 and hence, may be approximated aszero. Therefore, it is seen that, irrespective of pure translational


and pure torsional eigenvectors,

ψT0i E Aψ0i = 0. (A.7)

Now using Eq. (A.7) in the expressions of eigenvalue andeigenvector (Eqs. (13) and (14) in the main text), the followingsimplified formulae can be derived:

Eigenvalues:

γi = γ0i +

n∑j=1j 6=i

(γ0iγ0 j

γ0i − γ0 j

)(ψT

0i E Aψ0 j

)2. (A.8)

Eigenvectors:

φi = ψ0i +

n∑j=1j 6=i

(γ0 j

γ0i − γ0 j

)(ψT

0 j E Aψ0i

)ψ0 j . (A.9)

Let us now evaluate the expression ψT0i E Aψ0 j as follows:

(i) When ψ0i and ψ0 j are two different pure translationaleigenvectors: It may be shown that

ψT LT

0i E AψT L0 j = 0. (A.10)

(ii) When ψ0i is a pure translational eigenvector and ψ0 j isa pure torsional eigenvector:

ψT LT

0i E AψT R0 j = ψ̄ xT

0i E A13ψ̄

θ0 j + ψ̄

yT

0i E A23ψ̄

θ0 j . (A.11)

(iii) When ψ0i is a pure torsional eigenvector and ψ0 j is a puretranslational eigenvector:

ψT RT

0i E AψT L0 j = ψ̄θ

T

0i E A13ψ̄

x0 j + ψ̄θ

T

0i E A23ψ̄

y0 j . (A.12)

(iv) When ψ0i and ψ0 j are two different pure torsionaleigenvectors:

ψT RT

0i E AψT R0 j = ψ̄θ

T

0i E A33ψ̄

θ0 j . (A.13)

Further, using the orthogonality criteria and noting that (di/r̃)2

is of the order 10−1 to 10−3 (Eq. (22) of the main text), it maybe shown that

ψT RT

0i E AψT R0 j ≈ 0. (A.14)

Estimation of eigenvalue of the perturbed system (γi ) byexpanding Eq. (A.8)

(i) Expanding Eq. (A.8) and thereafter utilizing Eqs.(A.10) and (A.11), γ A

i associated with any pure translationaleigenvector of the unperturbed system is given by

γ Ai = γ T L

0i +

N∑j=1

(γ T L

0i γT R0 j

γ T L0i − γ T R

0 j

)

×

[ψ̄ xT

0i E A13ψ̄

θ0 j + ψ̄

yT

0i E A23ψ̄

θ0 j

]2. (A.15)

(ii) Expanding Eq. (A.8) and thereafter utilizing Eqs. (A.12)and (A.14), γ A

i associated with any pure torsional eigenvectorof the unperturbed system is given by

γ Ai = γ T R

0i +

2N∑j=1

(γ T R

0i γ T L0 j

γ T R0i − γ T L

0 j

)

×

[ψ̄θ

T

0i E A13ψ̄

x0 j + ψ̄θ

T

0i E A23ψ̄

y0 j

]2. (A.16)

Estimation of the eigenvector of the perturbed system (φi )byexpanding Eq. (A.9)

(i) Expanding Eq. (A.9) and thereafter utilizing Eqs.(A.10) and (A.12), φA

i associated with any pure translationaleigenvector of the unperturbed system is given by

φAi = ψT L

0i +

N∑j=1

(γ T R

0 j

γ T L0i − γ T R

0 j

)

×

[ψ̄θ

T

0 j E A13ψ̄

x0i + ψ̄θ

T

0 j E A23ψ̄

y0i

]ψT R

0 j . (A.17)

(ii) Expanding Eq. (A.8) and thereafter utilizing Eqs. (A.11)and (A.14), φA

i associated with any pure torsional eigenvectorof the unperturbed system is given by

φAi = γ T R

0i +

2N∑j=1

(γ T L

0 j

γ T R0i − γ T L

0 j

)

×

[ψ̄ xT

0 j E A13ψ̄

θ0i + ψ̄

yT

0 j E A23ψ̄

θ0i

]ψT L

0 j . (A.18)

A.1.2. Case-2: System-B and system-CLet us now consider the unperturbed system as system-C

as before but the perturbed system is system-B. Therefore,perturbation in the mass matrix can be expressed as

[E B] =

[0] [0] −[eymi ]

[0] [0] [ex mi ]−[eymi ] [ex mi ] [0]

. (A.19)

Noting the similarity between [E B] and [E A

], results forsystem-B can be directly expressed by the formulae alreadyderived for system-A by replacing [E A

] by [E B]. These results

are, however, not presented for brevity.

A.1.3. Case-3: System-A and System-BNow, from the results of the perturbation analysis of

systems-A and -B with respect to the unperturbed system-C ,one may write

For eigenvalues (i)(γ A

i − γ Bi

)associated with any pure

translational eigenvector of the unperturbed system is

γ Ai − γ B

i =

N∑j=1

(γ T L

0i γT R0 j

γ T L0i − γ T R

0 j

)

×

[ψ̄ xT

0i

(E A

13 + E B13

)ψ̄θ0 j + ψ̄

yT

0i

(E A

23 + E B23

)ψ̄θ0 j

]×

[ψ̄ xT

0i

(E A

13 − E B13

)ψ̄θ0 j + ψ̄

yT

0i

(E A

23 − E B23

)ψ̄θ0 j

]. (A.20)

(ii)(γ A

i − γ Bi

)associated with any pure torsional eigenvector

of the unperturbed system is

γ Ai − γ B

i = γ T R0i +

2N∑j=1

(γ T R

0i γ T L0 j

γ T R0i − γ T L

0 j

)

×

[ψ̄θ

T

0i

(E A

13 + E B13

)ψ̄ x

0 j + ψ̄θT

0i

(E A

23 + E B23

)ψ̄

y0 j

]


×

[ψ̄θ

T

0i

(E A

13 − E B13

)ψ̄ x

0 j + ψ̄θT

0i

(E A

23 − E B23

)ψ̄

y0 j

].

(A.21)

For eigenvectors(i)

(φA

i − φBi

)associated with any pure translational

eigenvector of the unperturbed system is

φAi − φB

i = γ T L0i +

N∑j=1

(γ T R

0 j

γ T L0i − γ T R

0 j

)

×

[ψ̄θ

T

0 j

(E A

13 − E B13

)ψ̄ x

0i + ψ̄θT

0 j

(E A

23 − E B23

)ψ̄

y0i

]ψT R

0 j .

(A.22)

(ii)(φA

i − φBi

)associated with any pure torsional eigenvec-

tor of the unperturbed system is

φAi − φB

i = γ T R0i +

2N∑j=1

(γ T L

0 j

γ T R0i − γ T L

0 j

)

×

[ψ̄ xT

0 j

(E A

13 − E B13

)ψ̄θ0i + ψ̄

yT

0 j

(E A

23 − E B23

)ψ̄θ0i

]ψT L

0 j .

(A.23)

A.2. Simplification of error equations for the eigenvalues andeigenvectors

The contribution from the higher modes may be neglectedin the perturbation analysis and therefore, it may be sufficientto account for the effects of the first three modes. Thesethree modes show predominantly translation along X -direction,predominantly translation along Y -direction and predominantlytorsional behavior. However, in each of these three modes, anytypical constituting frame vibrates in its first mode pattern,i.e., without any point of contraflexure. Therefore, it may besufficient to take i = 1 and 2 in Eq. (A.20) and i = 1 inEq. (A.21) for the eigenvalue perturbation. Similarly for theeigenvectors, one may consider i = 1 and 2 in Eq. (A.22) andi = 1 in Eq. (A.23). Further, it may be noted that the factorin terms of γ within the first brace in Eqs. (A.20)–(A.23) diesout rapidly if j differs from i . Therefore, it may be sufficient toconsider j = 1 in Eqs. (A.20)–(A.22) and j = 1 and 2 in Eqs.(A.22) and (A.23) for all practical purposes.

Utilizing these simplifications to the eigenvalue perturba-tion, one may write from Eq. (A.20)

γ A1 − γ B

1 =

(γ T L

01 γT R01

γ T L01 − γ T R

01

)×

[ψ̄ xT

01

(E A

13 + E B13

)ψ̄θ01 + ψ̄

yT

01

(E A

23 + E B23

)ψ̄θ01

]×

[ψ̄ xT

01

(E A

13 − E B13

)ψ̄θ01 + ψ̄

yT

01

(E A

23 − E B23

)ψ̄θ01

](A.24)

γ A2 − γ B

2 =

(γ T L

02 γT R01

γ T L02 − γ T R

01

)×

[ψ̄ xT

02

(E A

13 + E B13

)ψ̄θ01 + ψ̄

yT

02

(E A

23 + E B23

)ψ̄θ01

]∗

[ψ̄ xT

02

(E A

13 − E B13

)ψ̄θ01 + ψ̄

yT

02

(E A

23 − E B23

)ψ̄θ01

](A.25)

and from Eq. (A.21),

γ A1 − γ B

1 =

(γ T R

01 γ T L01

γ T R01 − γ T L

01

)×

[ψ̄θ

T

01

(E A

13 + E B13

)ψ̄ x

01 + ψ̄θT

01

(E A

23 + E B23

)ψ̄

y01

]∗

[ψ̄θ

T

01

(E A

13 − E B13

)ψ̄ x

01 + ψ̄θT

01

(E A

23 − E B23

)ψ̄

y01

]+

(γ T R

01 γ T L02

γ T R01 − γ T L

02

)×

[ψ̄θ

T

01

(E A

13 + E B13

)ψ̄ x

02 + ψ̄θT

01

(E A

23 + E B23

)ψ̄

y02

]∗

[ψ̄θ

T

01

(E A

13 − E B13

)ψ̄ x

02 + ψ̄θT

01

(E A

23 − E B23

)ψ̄

y02

]. (A.26)

Similarly, for the eigenvectors, one may write from Eq. (A.22)

φA1 − φB

1 =

(γ T R

01

γ T L01 − γ T R

01

)×

[ψ̄θ

T

01

(E A

13 − E B13

)ψ̄ x

01 + ψ̄θT

01

(E A

23 − E B23

)ψ̄

y01

]ψT R

01

(A.27)

φA2 − φB

2 =

(γ T R

01

γ T L02 − γ T R

01

)×

[ψ̄θ

T

01

(E A

13 − E B13

)ψ̄ x

02 + ψ̄θT

01

(E A

23 − E B23

)ψ̄

y02

]ψT R

01

(A.28)

and from Eq. (A.23),

φA1 − φB

1 =

(γ T L

01

γ T R01 − γ T L

01

)×

[ψ̄ xT

01

(E A

13 − E B13

)ψ̄θ01 + ψ̄

yT

01

(E A

23 − E B23

)ψ̄θ01

]ψT L

01

+

(γ T L

02

γ T R01 − γ T L

02

)[ψ̄ xT

02

(E A

13 − E B13

)ψ̄θ01

+ ψ̄yT

02

(E A

23 − E B23

)ψ̄θ01

]ψT L

02 . (A.29)

At this stage, it may be noted that

E A13 + E B

13 = −[(

byi + 2ey)

mi]

diag ; (A.30a)

E A13 − E B

13 = −[byi mi

]diag ; (A.30b)

E A23 + E B

23 = −[(

bxi + 2ex)

mi]

diag ; (A.31a)

E A23 − E B

23 = −[bxi mi

]diag . (A.31b)

It is now required to investigate the first three eigenvectorsof the unperturbed system-C . In doing so, eigenvalue equationof the unperturbed system may be expressed as

γ0

Kxx Kxy 0Kxy K yy 0

0 0 Kθθ

ψ̄ x

0ψ̄

y0ψ̄θ0


=

[mi ] 0 00 [mi ] 00 0 [mi (r̃

2+ e2

x + e2y)]

ψ̄ x

0ψ̄

y0ψ̄θ0

. (A.32)

Stiffness proportionality of the constituting frames anduncoupled nature of the mass matrix enable us to decouplethe eigenvalue problem into two smaller eigenvalue problems,namely, characteristic frame problem and equivalent one-storytorsionally uncoupled problem. Letting σ , {u} be the eigenvalueand eigenvector of the characteristic frame; τ , α =

{αxαyαθ

}T

the eigenvalue and eigenvector of the equivalent one-storysystem; and γ0, ψ0 =

{ψ̄ x

0 ψ̄y0 ψ̄

θ0

}Tthe eigenvalue and

eigenvector of the entire building, it may be shown that

γ0 = τ ⊗ σ (A.33)

ψ0 ={ψ̄ x

0 ψ̄y0 ψ̄

θ0

}T={αxαyαθ

}T⊗ {u} . (A.34)

Further, mass normalization of the ‘i j’th eigenvector leads to[α2

xi + α2yi + α2

θ i

(r̃2

+ e2x + e2

y

)]⊗ {u}

Tj [mi ] {u} j = 1.

(A.35)

Let us now normalize the eigenvector of the characteristic frameand that of one-story system as

α2xi + α2

yi + α2θ i

(r̃2

+ e2x + e2

y

)= 1 (A.36)

and

{u}Tj [mi ] {u} j = 1. (A.37)

Therefore, if i is a pure translational mode, αθ i = 0 andconsequently,

α2xi + α2

yi = 1. (A.38)

Otherwise, if i is a pure torsional mode, αxi = αyi = 0, andconsequently,

α2θ i = (r̃2

+ e2x + e2

y)−1. (A.39)

It may be noted that, for the first three modes of the building,eigenvector of the characteristic frame is essentially its firsteigenvector {u}1. In other words, floor-wise variation of thetranslations along X - and Y -directions and torsional rotation isthe same in the first three modes of the unperturbed system.

Utilizing this simplification along with Eqs. (A.30a),(A.30b), (A.31a) and (A.31b), the following simplificationsmay be carried out for Eqs. (A.24)–(A.29).

For eigenvalues(i)(γ A

1 − γ B1

)associated with the first pure translational

eigenvector of the unperturbed system is given by (from Eq.(A.24))

γ A1 − γ B

1 =

(λT R

01 − λT L01

)−1 [αx1αθ1Y + 2eyαx1αθ1

+ αy1αθ1 X + 2exαy1αθ1]∗[αx1αθ1Y + αy1αθ1 X

]. (A.40)

(ii)(γ A

2 − γ B2

)associated with the second pure translational

eigenvector of the unperturbed system is given by (from Eq.

(A.25))

γ A2 − γ B

2 =

(λT R

01 − λT L02

)−1 [αx2αθ1Y + 2eyαx2αθ1

+ αy2αθ1 X + 2exαy2αθ1]∗[αx2αθ1Y + αy2αθ1 X

]. (A.41)

(iii)(γ A

1 − γ B1

)associated with the first pure torsional

eigenvector of the unperturbed system is given by (from Eq.(A.26))

γ A1 − γ B

1 =

(λT L

01 − λT R01

)−1

×[αθ1αx1Y + 2eyαθ1αx1 + αθαy1 X + 2exαθ1αy1

]∗[αθ1αx1Y + αθ1αy1 X

]+

(λT L

02 − λT R01

)−1

×[αθ1αx2Y + 2eyαθ1αx2 + αθ1αy2 X + 2exαθ1αy2

]∗[αθ1αx2Y + αθ1αy2 X

]. (A.42)

For eigenvectors(i)

(φA

1 − φB1

)associated with the first pure translational

eigenvector of the unperturbed system is given by (from Eq.(A.27))(φA

1 − φB1

)= −

(λT R

01

λT L01

− 1

)−1

×[αθ1αx1Y + αθ1αy1 X

]ψT R

01 . (A.43)

(ii)(φA

2 − φB2

)associated with the second pure translational


2 − φB2

)= −

(λT R

01

λT L02

− 1

)−1

×[αθ1αx2Y + αθ1αy2 X

]ψT R

01 . (A.44)

(iii)(φA

1 − φB1

)associated with the first pure torsional


1 − φB1

)= −

(λT L

01

λT R01

− 1

)−1 [αx1αθ1Y + αy1αθ1 X

]ψT L

01

+

(λT L

02

λT R01

− 1

)−1 [αx2αθ1Y + αy2αθ1 X

]ψT L

02 . (A.45)

Here, in Eqs. (A.40)–(A.45), it is assumed that

λ0 =1γ0

; (A.46a)

Y = {u}T1

[byi mi

]diag {u}1 ; (A.46b)

X = {u}T1

[bxi mi

]diag {u}1 . (A.46c)

A.3. Estimation of error in eigenvalues and eigenvectors

From Eqs. (A.40)–(A.45), it may be seen that the differencein eigenvalues and in eigenvectors between system-A and


system-B is zero if X = Y = 0. Moreover, for a given orderof X and Y , it may also be possible to ascertain the order ofthe differences in eigenvalues and in eigenvectors between thesystems -A and -B. For this purpose, let us define,

Y = βy r̃ and (A.47a)

X = βx r̃ (A.47b)

where,

βy = {u}T1

[byi

r̃mi

]diag

{u}1 and (A.48a)

βx = {u}T1

[bxi

r̃mi

]diag

{u}1 . (A.48b)

Further, letting βx = C1×10−ρ and βY = C2×10−ρ with |C1|,|C2| close to unity, e.g., 1.0–2.0, the order of the differencein eigenvalues and eigenvectors may be evaluated from Eqs.(A.40)–(A.45) as follows:

Utilizing Eqs. (A.47) and (A.48) in Eq. (A.40), it may beshown that

γ A1 − γ B

1 =

(λT R

01 − λT L01

)−1{

1 +

(ex

r̃

)2+

(ey

r̃

)2}−1

[αx1βy + 2

ey

r̃αx1 + αy1βx + 2

ex

r̃αy1

]∗[αx1βy + αy1βx

]. (A.49)

It may be noted that α2x1, α

2y1 are of the order 10−1 as evident

from Eq. (A.38). Thus, αx1, αy1 are also of the order 10−1 butmore close to unity. Therefore, last term in Eq. (A.49) can beof the order maximum 10−ρ . Further, ex

r̃ , eyr̃ are of the order

10−1 and can be even close to 0.5 for a highly torsionallycoupled building. Therefore, if ρ ≥ 1, third term in Eq. (A.49)then may be of the order maximum 10−1 but close to unity oreven slightly more than unity. Similarly, second term in Eq.(A.49) is also of the order 10−1 but may not be very close tounity. Finally, in a torsionally coupled building, the first threefrequencies are very close to each other and thus, first termin Eq. (A.49) is of the order 10−1. Therefore, it is seen thatγ A

1 − γ B1 may be of the order maximum 10−(ρ+1). Similarly,

Eq. (A.41) may be expressed as

γ A2 − γ B

2 =

(1

λT R01 − λT L

02

)[1 +

(ex

r̃

)2+

(ey

r̃

)2]−2

×

[αx2βy + 2

ey

r̃αx2 + αy2βx + 2

ex

r̃αy2

]∗[αx2βy + αy2βx

]. (A.50)

Following the same reasoning as before, it may be shown thatγ A

2 − γ B2 may be of the order maximum 10−(ρ+1). Similarly

from Eq. (A.42) one may write,

γ A1 − γ B

1 =

(λT L

01 − λT R01

)−1[

1 +

(ex

r̃

)2+

(ey

r̃

)2]−1

×

[αx1βy +

2ey

r̃αx1 + αy1βx +

2ex

r̃αy1

] [αx1βy + αy1βx

]

+

(λT L

02 − λT R01

)−1[

1 +

(ex

r̃

)2+

(ey

r̃

)2]−1

×

[αx2βy +

2ey

r̃αx2 + αy2βx +

2ex

r̃αy2

]×[αx2βy + αy2βx

]. (A.51)

Following the same reasoning as before, it may be shown thatγ A

1 − γ B1 may be of the order maximum 10−ρ .

Similarly, from Eq. (A.43) it may be shown for theeigenvector that

(φA

1 − φB1

)= −

(λT R

01

λT L01

− 1

)−1 [αx1βy + αy1βx

]×

[1 +

(ex

r̃

)2+

(ey

r̃

)2]−

12

ψT R01 . (A.52)

Note that, ψT R01 =

{0 0 αθ1

}⊗ {u}1 and αθ1 is of the order

10−2or at least 10−1, as evident from Eq. (A.39), the maximumorder of

(φA

1 − φB1

)may be calculated as 10−ρ {u}1. Similarly,

Eq. (A.44) may be written as

(φA

2 − φB2

)= −

(λT R

01

λT L02

− 1

)−1 [αx2βy + αy2βx

]×

[1 +

(ex

r̃

)2+

(ey

r̃

)2]−

12 {

0 0 αθ1}T

⊗ {u}1 (A.53)

and the maximum order of(φA

2 − φB2

)may be shown as

10−ρ {u}1. Similarly, from Eq. (A.45)

(φA

1 − φB1

)= −

(λT L

01

λT R01

− 1

)−1 [1 +

(ex

r̃

)2+

(ey

r̃

)2]−

12

×[αx2βy + αy1βx

] {αx1 αy1 0

}T⊗ {u}1

−

(λT L

02

λT R01

− 1

)−1 [1 +

(ex

r̃

)2+

(ey

r̃

)2]−

12

×[αx2βy + αy2βx

] {αx2 αy2 0

}T⊗ {u}1 (A.54)

and the maximum order of(φA

1 − φB1

)can be calculated as

10−ρ+1 {u}1.Therefore, it is seen that, if system-B is used in lieu of

system-A, difference in the eigenvalues (γ ) is of the ordermaximum 10−ρ and that for the eigenvectors is of the ordermaximum10−ρ+1 {u}1 where {u}1 is the mass normalized firstmode shape of the characteristic frame.

A.4. Estimation of error in frame shear forces

Let the value of ρ be chosen such that error in eigenvalue(or frequency) estimation is small enough to make a significantchange in the estimation of the spectral ordinate. Further,resulting change in participation factor is also assumed smallenough for all practical purposes. In view of these, error inthe estimation of the frame shear force in any of the first threemodes of the building is attributed only from the error in the


estimation of the eigenvectors. Let φ ={φi x φiy φiθ

}T

be the eigenvector in any (say, i th) of the first three modesof the average CM model, i.e., system-B. Similarly, let φ/ bethe associated eigenvector of the actual (perturbed) system-A.Therefore, error in the estimation of this mode shape is givenby(φ/ − φ

)={±Si x ±Siy ±Siθ

}T(

10−ρ+1)

⊗ {u}1 (A.55)

where Si x , Siy and Siθ are of the order maximum 100. Let ηbe the inclination, measured in the counter clockwise direction,of any frame with respect to the X -axis. Therefore, error inthe estimation of the modal shear force for the frame may beexpressed as(

F/ − F)

= Sdi Pi(±Si x cos η ± Siy sin η ± DSiθ

)×

(10−ρ+1

)[K ] j {u}1 (A.56)

where D is the distance of the frame from the CR, Sdi thespectral displacement, Pi the participation factor, and [K ] j thestiffness matrix of the frame under consideration.

In the case of average CM model, i.e., system-B, since CMsof all the floors are located on one vertical line and all the CRslie on another vertical line, it may be shown that

φT= { hx {u}

T1 hy {u}

T1 hθ {u}

T1 } (A.57)

where, hx , hy and hθ are scalar quantities. Moreover, massnormalization criteria lead to

h2x + h2

y + h2θ (r̃

2+ e2

x + e2y)

− 2eyhx hθ + 2ex hyhθ = 1. (A.58)

Therefore, if the mode shows translation predominantly alongX -direction, hx → 1, hy → 0, and hθ → 0. Similarly, if themode is predominantly translational along Y -direction, hx →

0, hy → 1, and hθ → 0; and, for a predominantly torsionalmode, hx → 0, hy → 0, and hθ → (r̃2

+ e2x + e2

y)−1.

Finally, considering system-B as the basis, percentage errorin the estimation of the modal frame shear force may be shownas[(

F/ − F)/F]× 100 = C0 × 10−ρ+3 (A.59)

where

C0 =(±Si x cos η ± Siy sin η ± DSiθ

)/(

±hx cos η ± hy sin η ± Dhθ). (A.60)

Since, Si x , Siy are of the order 100, DSiθ usually of the order10−1 or even less, (hx , hy) of the order 10−1, Dhθ of the order10−2 or even less, and most importantly, signs of the respectiveterms in the numerator and the denominator are consistent, itmay be easily seen that C0 is of the order of 100. For example,if ρ is taken as 3, maximum error in the estimation of the frame

shear force is of the order 100, i.e., 0%–10%; for ρ equals to 2,the maximum error is increased to 101order, i.e., 10%–100%.The formulation is also valid for the remaining two of the firstthree modes of the building and the resultant upper bound ofthe error is valid even after applying modal combination too.Therefore, the applicability conditions of the simplified methodto a building with vertically-nonaligned CMs may be stated as

|βx | ≤ 0.002 and (A.61a)∣∣βy∣∣ =≤ 0.002. (A.61b)

Here, βx = {u}T1

[bxi mi

r̃

]diag

{u}1, βy = {u}T1

[byi mi

r̃

]diag

{u}1

are the indices for assessing the applicability of the simpli-fied procedure and {u}1 the mass normalized fundamental modeshape of the characteristic frame.

References

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[7] Hejal R, Chopra AK. Earthquake response of torsionally-coupled build-ings. Report no. UCB/EERC-87/20. Berkeley: Earthquake EngineeringResearch Center, Univ. of Calif. December 1987.

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[9] IS: 1893 (Part 1)-2002, Criteria for earthquake resistant design ofstructures: General provisions and buildings. New Delhi: Bureau of IndianStandards. 2002.

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[11] Kan CL, Chopra AK. Coupled lateral torsional response of buildingsto ground shaking. Report no. UCB/EERC-76/13. Berkeley: EarthquakeEngineering Research Center, Univ. of Calif. May 1976.

[12] Kan CL, Chopra AK. Torsional coupling and earthquake response ofsimple elastic and inelastic systems. Journal of Structural Engineering-ASCE 1981;107(ST8):1569–88.

[13] Nelson TK, Wilson JL, Hutchinson GL. Review of the torsional couplingof asymmetrical wall-frame buildings. Engineering Structures 1997;19(3):233–46.

[14] SAP2000. Structural analysis program. Berkeley (CA): Computer andStructures, Inc; 1995.

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[16] Wilkinson JH. The algebraic eigenvalue problem. Clarendon Press; 1965.[17] Yoon YS, Smith BS. Estimating period ratio for predicting torsional

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