push 29 2013.pdf

8/10/2019 push 29 2013.pdf

1/13

An adaptive pushover procedure based on effective modal mass

combination rule

Reza Abbasnia , Alireza Tajik Davoudi, Mohammad M. Maddah

Department of Civil Engineering, Iran University of Science and Technology, Tehran 1684613114, Iran

a r t i c l e i n f o

Article history:

Received 22 April 2012

Revised 13 March 2013

Accepted 19 March 2013

Available online 24 April 2013

Keywords:

Adaptive pushover

Higher modes

Sign reversal

Modal mass combination rule

a b s t r a c t

In order to overcome the major drawbacks of conventional pushover methods, researchers have recently

been motivated to develop adaptive pushover procedures by which effect of higher modes as well as pro-

gressive damage accumulation are taken into account. In spite of their vigorous theory, these novel meth-

ods suffer from the quadratic modal combination rules, in which the sign reversals of load vectors in

higher modes are neglected and consequently lead to a positive load pattern. In this paper, a displace-

ment-based adaptive modal pushover method, called APAM, based on effective modal mass combination

rule is developed in order to include the sign reversals in the load vectors. In this combination rule a mod-

ification factor associated to each mode of interest is determined and applied to the corresponding load

vector. The modified modal load vectors are algebraically added and subtracted and result in a range of

load pattern and thus, multiple pushover analysis is required. These load patterns are independently

applied to the structure within an adaptive framework and the envelope of demand values is considered.

These modification factors are updated proportional to the instantaneous dynamic characteristic of struc-

ture in each step. Another novel aspect of the proposed method is that the target displacement is esti-

mated during the analysis by implementing the concept of capacity spectrum method recommended

by ATC 40. In order to assess the accuracy of this method in predicting the seismic responses, the pro-

posed methodology is applied to three different moment-frame buildings. The obtained results demon-

strate that APAM procedure provides well estimation of important seismic demand parameters.2013 Elsevier Ltd. All rights reserved.

1. Introduction

Earlier versions of pushover methods presented in different

code provisions such as FEMA-356 [1], EuroCode-8[2]and ATC-

40[3], are limited to fundamental mode of the structures. In these

procedures, the structure is subjected to an invariant load pattern

until a predetermined target displacement is achieved or collapse

occurs. There are two major drawbacks in the conventional push-

over methods: (I) neglect of the higher mode effects [410] and

(II) neglect the changes in the dynamic properties of the structures

that leads to a continuously altered loading pattern[1114]. Due to

these important shortcomings, conventional pushover cannot

accurately predict the response of structures when the higher

mode effects are considerable[410].

Extensive research has been conducted in recent years to over-

come the aforementioned deficiencies. Some researchers are lim-

ited to taking into account the higher mode effects while the

modal load vectors are constant during the analysis (first category).

Some others consider both the higher mode effects as well as

changing in the dynamic properties of the structure which is so

called adaptive pushover analysis (second category). In the former

category, the earliest attempt was conducted by Paret et al.[4]who

introduced the multi-modal pushover procedure. One of the most

famous procedures in this category of pushover procedures is mod-

al pushover analysis (MPA) of building was developed by Chopra

and Goel[8]. In this method, which is a multi-run procedure, the

structure is subjected to different load vectors (proportion to each

mode) and the modal responses are combined with quadratic mod-

al combination rules such as the square root of the sum of the

squares (SRSS) and the complete quadratic combination (CQC).

Since the responses of the structure subjected to higher mode load

vectors are within the elastic range, the modified modal pushover

analysis (MMPA) procedure [15] was proposed in which the re-

sponse of higher modes can be calculated by a response spectrum

analysis.

Although the methods that consider the higher mode contribu-

tions present a better estimation of seismic response of structure in

comparison to the conventional pushover methods, the effect of

the progressive damage accumulation and the subsequent modifi-

cation of the load pattern is not taken into account.

In order to reflect the progressive stiffness degradation of struc-

ture on the load vectors in an inelastic analysis, the adaptive push-

over analysis methods have been introduced in recent years. The

0141-0296/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2013.03.029

Corresponding author. Tel.: +98 9122186903; fax: +98 2177240398.

E-mail address:[email protected](R. Abbasnia).

Engineering Structures 52 (2013) 654666

Contents lists available at SciVerse ScienceDirect

Engineering Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n g s t r u c t
http://dx.doi.org/10.1016/j.engstruct.2013.03.029mailto:[email protected]://dx.doi.org/10.1016/j.engstruct.2013.03.029http://www.sciencedirect.com/science/journal/01410296http://www.elsevier.com/locate/engstructhttp://www.elsevier.com/locate/engstructhttp://www.sciencedirect.com/science/journal/01410296http://dx.doi.org/10.1016/j.engstruct.2013.03.029mailto:[email protected]://dx.doi.org/10.1016/j.engstruct.2013.03.029http://crossmark.dyndns.org/dialog/?doi=10.1016/j.engstruct.2013.03.029&domain=pdf

8/10/2019 push 29 2013.pdf

2/13

earliest version of adaptive pushover was presented by Reinhorn

[11]and Bracci et al.[12]in which the load pattern is updated from

the instantaneous base shear and story resistance of the previous

load step. Gupta and Kunnath[13]presented a force-based adap-

tive pushover procedure (FAP) in which the applied load pattern

is a function of mass-normalized mode shape, modal participation

factor, spectral amplification of considered modes and weight of

story. In this regard, an eigenvalue analysis is performed to calcu-late mode shapes in each step. The load vector related to each

mode is applied to the structure independently and the response

in each step is calculated by the quadratic mode combination rule.

Elnashai[14]and Antoniou and Pinho[16]elaborated the previous

forced-based adaptive methodologies and developed a single-run

full adaptive pushover method.

In spite of the conceptual superiority of forced-based adaptive

pushover methods, the prediction of seismic responses has not

been significantly improved [1618]. The main reason for this inac-

curacy can be related to the use of the quadratic modal combina-

tion rules[1618]. In these rules, the sign reversals in the modal

load vectors are neglected which results in a constantly positive

load pattern.

Antonio and Pinho [19]have developed a displacement-based

adaptive pushover method (DAP) to improve the prediction of

adaptive pushover methods. The general concept of this method

is the same as FAP procedure. The only important difference is

the implementation of the displacement vector instead of the force

vector. It is confirmed by several researches that DAP procedure

has improved the prediction of seismic demands of the structures

in comparison to FAP method[20,21]. Although the accuracy of the

results has became more satisfactory, the aforementioned prob-

lems arising from SRSS or CQC, still exist. Kalkan and Kunnath

[22]proposed a multi-run adaptive pushover, which is called adap-

tive pushover combination AMC. The authors combined the MPA

method, capacity spectrum method (CSM) and adaptive modal

procedure in which the target displacement is computed during

the analysis. The seismic demand is predicted more accurately in

comparison to the conventional methods[22,23]. However, inter-action between modes in the inelastic range is not considered

and the responses are computed based on SRSS or CQC rules. Re-

cently, Shakeri et al.[24]have proposed a story shear-based adap-

tive pushover method where the applied load pattern is derived

from the instantaneous combined modal story shear profile. The

lateral load pattern is calculated by subtracting the combined

modal shear of consecutive stories. This method underestimates

the crucial seismic demand such as drift.

As mentioned above, quadratic modal combination is a serious

problem in the adaptive modal pushovers which leads to decreas-

ing the precision of seismic demand prediction. In the following,

the research related to the modal combination rules rather than

quadratic modal combinations in a pushover analysis are

presented.

2. Alternative modal combination rules

The crucial shortcoming of the quadratic modal combination

rule is that the signs of modal load vectors are suppressed during

the combination. In other words, the possible negative sign of

modal load vectors is eliminated and inevitably leads to monoton-

ically increasing load vectors[1618].

Matsumori et al. [25] used the alternative modal combination

rule to estimate the response of structure. In this methodology

two independent story shear patterns including the sum and the

difference of two modal story shears, are utilized. Afterward, Kun-

nath[26]presented a pushover method in which multiple invari-ant load patterns are calculated by adding and subtracting modal

story loads. The following expression (Eq.(1)) is used to compute

the story forces in this methodology:

FiXkj1

Rj CjM /ijSajfj; Tj 1

whereFiis the lateral force to be applied at story level i and j stand

for the mode number;Rj, a modification factor to scale the contribu-

tion of each mode; Saj, the spectral acceleration at period Tj formode j and corresponding to the damping ratio fj. / ij, the ith com-

ponent of thejth eigenvector (mode shape) and Cjis the modal par-

ticipation factor for the jth mode.

The obtained story forces (Fi) are applied to the structure inde-

pendently and the envelope of responses is considered as the seis-

mic demand. The challenging matter in this method is how to

determine the modification factor (Rj). However, the author does

not present a distinct rule to compute this modification factor

[27]. Park et al. [27]used the concept of the Kunnath procedure

and developed a novel method whereby modal combination factor

of each mode is presented. The modal combination factors (Rj) are

calculated from a comprehensive set of elastic time history analy-

ses. Although this procedure can accurately predict the seismic de-

mand of the structure in the elastic range, it cannot significantlyimprove the accuracy of the predictions in the inelastic range.

In the present study, in order to overcome the above mentioned

shortcomings arising from the quadratic modal combination rules,

an adaptive pushover analysis based on Modal Mass Participation

(APAM) is developed. The proposed method, which is a displace-

ment-based adaptive procedure, employs an effective modal mass

combination rule (EMMC) to construct the applied load pattern. In

EMMC rule, the sign of each modal load vector is maintained and

unlike the quadratic modal combination rules, the sign reversals

in the load vectors are included. In addition, the proposed tech-

nique employs the concept of CSM [3] and AMC methodology

[22] to estimate the target displacement during the analysis and

therefore, the challenge related to estimating a predefined target

displacement is eliminated. It should be noted that this methodis intended to estimate the seismic demands in the building

frames.

3. Adaptive pushover analysis based on Modal Mass

Participation

The main advantages of APAM procedure, as mentioned previ-

ously, rely upon the methodology which is implemented in defin-

ing the load pattern as well as determination of the target

displacement during the analysis.

In the following the basic elements of the proposed procedure

are discussed in detail.

3.1. Load pattern

In the APAM procedure the effects of higher modes, the changes

in the dynamic characteristics of structure in the inelastic range

and the effect of the frequency content of a specific response spec-

trum on the load pattern are taken into account. In addition, in or-

der to overcome the known drawbacks in the quadratic

combination rule, herein the EMMC rule for multi-story buildings

is presented. In this combination rule a modification factor associ-

ated to each mode of interest is determined and applied to the cor-

responding load vector. The modified modal load vectors are

algebraically added and subtracted which result in a range of load

pattern and thus, multiple pushover analyses are required. These

load patterns are independently applied to the structure within

an adaptive framework and the envelope of demand values areconsidered. These modification factors are updated proportional

R. Abbasnia et al. / Engineering Structures 52 (2013) 654666 655

8/10/2019 push 29 2013.pdf

3/13

to the instantaneous dynamic characteristic of structure in each

step. The process for determining the applied load pattern at one

step of the proposed method (APAM) is schematically depicted in

Fig. 1.

The story displacement associated to each mode (Fig. 1a) is cal-

culated at each step by the following equation:

Dij Cj /ijSdjfj; Tj 2

where Sdj is the spectral displacement corresponding to the jth

mode and Dij is modal displacement at the story i related to jth

mode.

Now, EMMC rule is implemented to obtain the applied displace-

mentDiat the storyi through:

Di Xnj1

Rj Cj /ijSdjfj; Tj 3

In Eq.(3),Rjis a relative mode contribution factor which is defined

for each mode using the following equations:

Rj ajamax

4

amax Maxa1;a2;. . . ; aj 5

where aj is the modal mass coefficient corresponding to the jthmode and amax is the greatest value between modal mass coeffi-cients for all modes. aj is obtained using Eq. (6):

aj f/ijg

Tmflg2

f/ijgTmf/ijg

Pjmj

6

where {l} and [m] are the influence and mass matrixes, respectively.

If the first three modes are considered, regarding to Eq. (3), the

following four load patterns would be used (Fig. 1c):

1 Di R1 C1 /i1Sd1f1; T1 R2 C2/i2Sd2f2; T2 R3

C3 /i3Sd3f3; T3 7

Fig. 1. The process for determining the applied load pattern at one step of the proposed method (APAM) in comparison to DAP method. (a) Mode shapes and modal loadvectors. (b) Load pattern in DAP method. (c) Load pattern in APAM method.

656 R. Abbasnia et al. / Engineering Structures 52 (2013) 654666

8/10/2019 push 29 2013.pdf

4/13

2 Di R1 C1 /i1Sd1f1; T1 R2 C2 /i2Sd2f2; T2 R3

C3 /i3Sd3f3; T3 8

3 Di R1 C1 /i1Sd1f1; T1 R2 C2 /i2Sd2f2; T2 R3

C3 /i3Sd3f3; T3 9

4 Di R1 C1 /i1Sd1f1; T1 R2 C2 /i2Sd2f2; T2

R3 C3/i3Sd3f3; T3 10

Finally, these load patterns are applied to the structure indepen-

dently, and the envelope of demand values are obtained.

Since using the modal interstory drift for each mode (/ij/(i1)j),

instead of modal displacement (/ij), leads to much more improved

results [19], herein an interstory drift-based scaling technique is

used as shown in Eq.(11)instead of Eq.(3)

Di Xik1

Di Di Xkj1

Rj Cj /ij/i1j Sdjfj; Tj 11

It should be noted that for updating the load vectors at each

step, the algorithm proposed by Antonio and Pinho [19] is used.In this algorithm, considering the stiffness of the structure at the

end of previous step, an Eigenvalue analysis is carried out and peri-

ods and mode shapes are calculated. Based on these mode-shapes,

the load vector for each mode of interest is computed according to

Eq.(2) in each step.

3.2. Determination of target displacement

Determination of the target displacement is a basic element in

each pushover procedure. The ATC-40 document [3] presents an

equivalent linear method (CSM) in which the capacity of the struc-

ture (in the form of a pushover curve) is compared with the de-

mands on the structure (in the form of a response spectrum).

This iterative procedure is limited to the fundamental mode ofvibration[22]. In order to account for the higher modes in the esti-

mation of target displacement, Chopra and Goel[8] employed the

concept CSM procedure and developed modal pushover analysis

(MPA) procedure in which the target displacement of MDOF sys-

tem is estimated through a series of bilinear equivalent single de-

gree of freedom (ESDOF) systems corresponding to the modes of

interest and consequently, using the roof displacement as a con-

version parameter the maximum inelastic displacement of each

ESDOF system transformed back to the roof displacement. The sig-

nificant restriction is that the increase of the roof displacement is

not proportional to the other stories in the higher modes and it

is only meaningful for the first mode. In order to overcome this

limitation, an energy-based method is developed by Hernandez-

Montes et al.[28]in which the incremental displacement of ESDOFsystem is defined by dividing the incremental work done due to

lateral force at step k by the base shear.

Kalkan and Kunnath[22]developed an innovative procedure to

determine the target displacement. In their procedure, a capacity

curve for each mode of interest is computed using the energy ap-

proach. Afterwards, a series of predetermined ductility-level re-

sponse spectra, which are in acceleration displacement response

spectrum (ADRS) format (i.e., spectral acceleration versus spectral

displacement), are determined. The intersection of the capacity

curve of each mode at the current step of analysis and the

predetermined inelastic response spectra is considered as dynamic

target displacement if this point represents the same ductility level

of intersected response spectrum curve (Fig. 2). This methodology

is repeated for all modes considered and the target displacementrelated to each mode is calculated.

In this paper, the proposed methodology is conceptually analo-

gous in approach to Kalkans method[22]for determining the tar-

get displacement, which was described above, but in a single-run

framework. In this regard, a set of inelastic response spectra in

ADRS format with various ductility level are determined. An inter-

val of 0.5 (Dl= 0.5) is generally adequate to generate these re-sponse spectra [22]. Then spectral acceleration is determined

versus spectral displacement curve of ESDOF system according tothe energy approach[28]. In this energy approach, the incremental

displacement of ESDOF system is defined by dividing the incre-

mental work done due to lateral force at step k by the base shear.

In this regard, Eq.(12)through Eq.(14)are utilized for determining

the peak displacement of ESDOF system.

DDk DE

k

Vk

b

12

DEk Xni1

Fk

i Ddk

i 13

Sk

d Sk1

d DDk

14

where Fki is the existing force in the story i at step k; DE(k) the

increment of work done by lateral forces; Vkb

, the base shear at step

k; Ddki , the incremental displacement in the storyiat stepkandS

k

d

is the displacement of the ESDOF system at step k.

The spectral acceleration of ESDOF system is computed at each

stepk by Eq.(15).

Ska

Vk

b

anW 15

whereSka is the spectra acceleration at stepk;W, the total weight;

an is the modal mass coefficient which is obtain by Eq. (6).Since the load pattern in the proposed method, which is a single

run adaptive pushover, is representative of the contribution of all

modes of interest, this load pattern cannot be attributed to any

particular dynamic mode shape. In other words, none of the natu-

ral mode shapes can be visualized in Eqs. (6) and (15)since these

fundamental modes shapes are not compatible with the current

load pattern. In order to solve this problem Casarotti and Pinho

[29]and Shakeri et al.[24]introduced an assumed equivalent fun-

damental mode shape which is derived from the load pattern at

each step using Eq.(16):

f/gk

m1ffg

k16

where ffgk is the vector of the applied force to the structure and

f/gk is the assumed mode shape at step k.

Therefore using this equivalent mode shape, the spectral accel-eration of ESDOF system is computed by Eq.(15).

The obtained capacity curve of ESDOF system in ADRS format is

intersected to the predefined ductility level response spectra and

similar to the procedure proposed by Kalkan and Kunnath [22]

the target displacement is determined. After determining the tar-

get displacement of ESDOF, it shall be back transformed to the

MDOF using Eq.(17):

ur Ck /k Skd 17

where ur is the roof displacement and /k is the assumed mode

shape obtained by Eq. (16).

In the following, a step by step procedure is presented to esti-

mate the seismic demands of a multi-story building as well asthe target displacement, using APAM method:


8/10/2019 push 29 2013.pdf

5/13

3.3. APAM procedure

(1) The modal shapesf/kij g and the natural frequencies fxkj g

of the structure at the current state of analysis are computed

for all the considered modes.

(2) The modal story load corresponding to the jth mode of the

structure is calculated, {Dij}, using Eq.(2).

(3) The relative mode contribution factors (Rk

j ) are calculated

based on EMMC rule, using Eqs. (4)(6). It should be men-

tioned that these coefficients are updated at each step of

analysis.

(4) The load patterns fDk

i g corresponding to all considered

modes are constructed using Eqs.(7)(10).

(5) Nonlinear static analyses (NSA) are independently per-

formed employing the load patterns which are computed

in the previous step. If the first three modes are considered,

four NSA analyses are required.

(6) The spectral displacement of ESDOF system (Sk

d ) corre-

sponding to thekth step of the analysis are computed, using

Eqs. (12)(14).

(7) The assumed mode shapef/gk and the spectral acceleration

of ESDOF system Ska are computed by Eqs. (16) and (15),

respectively.

(8) If the response is inelastic at kth step of the analysis, the

approximate global system ductilityl(k) is calculated, usingEq.(18):

lk Skd

Syield

d

18

(9) The response spectra of a given ground motion are extracted

in ADRS format (spectral acceleration,Sal; f; k, versus spec-tral displacement,Sdl; f; k) for a series of predefined ductil-ity levels.

(10) Ska versus Sk

d (steps 6 and 7) is plotted together with the

inelastic response spectra at different ductility levels (step

9). The target displacement, Sip

d , is considered as the inter-

section of the capacity curve of ESDOF system and the

response spectrum corresponding to the global system duc-

tilityl(k) obtained from step 8.

Steps 110 are repeated until the target displacement iscalculated with a reasonable approximation. When the target

displacement is determined, the responses of the MDOF structure

are calculated using the assumed equivalent fundamental mode

shape concept (Eq.(17)). As mentioned previously, if the first three

modes are taken into account, four pushover analyses are indepen-

dently performed employing the load patterns of Eqs.(7)(10)and

the envelopes of obtained results are considered as seismic de-

mands. It is noteworthy to mention that for estimation of the

capacity curve and the target displacement, only the first load

pattern (Eq.( 7)) is adequate. The reason is discussed later in the

Section4.3.1.

4. Validation of the proposed method

The proposed procedure is verified for three typical concrete

moment-resisting frames with different heights and twenty strong

ground motion records. The seismic response of the mentioned

structures is predicted by the APAM procedure, the displace-

ment-based adaptive pushover (DAP) method, and also the con-

ventional pushover analysis with triangular and uniform load

pattern. These seismic responses are then compared to benchmark

results obtained from incremental dynamic analysis (IDA). The to-

tal drift, inter-story drift and capacity curve are considered to eval-

uate the accuracy of each procedure in comparison to IDA method.

4.1. Numerical models

In this paper a 3-story building as a low rise, a 9-story buildingas a mid rise and a 20-story building as a high rise are selected to

evaluate the accuracy of the proposed procedure. The plans and

elevation views of these buildings are shown in Fig. 3. The total

heights of the structures are 9.6 m, 28.8 m and 64 m for 3, 9, and

20 story buildings, respectively. These buildings are designed for

site class C according to ACI 318-08[30]and ASCE 7-05[31]. The

specified compressive strength of concrete and the yield stress

for all reinforcement are assumed as 30 MPa and 400 MPa, respec-

tively. The uniform dead and live loads equal to 6 kN/m2 and 2 kN/

m2 are applied to all floors, respectively. For the seismic design of

these buildings, the response modification coefficient for special

reinforced concrete moment frames is 8 according to ASCE 7-05

[28]. Also, the importance factor is considered to be 1. Effective

seismic weight includes total dead load and no contribution ofthe live load. Accidental torsion equal to five percent of dimension

Fig. 2. Determination of the target displacement.


8/10/2019 push 29 2013.pdf

6/13

of the structure perpendicular to the direction of the applied forces

is considered for the seismic design of buildings.

Since the structural systems are regular in plan and elevation

[31], it is permitted to analyze one typical frame (two planer mod-

els) in each main direction. In this regard, a typical frame in X

direction is selected. In the numerical analyses, the column bases

are assumed fixed. The distributed vertical load on beams is

30 kN/m by considering a tributary width of 5 m. Nonlinear push-

over and incremental dynamic analyses are performed using the

Opensees software[32]. The model takes into account geometrical

nonlinearity and material inelasticity. Material inelasticity is

explicitly considered by employing a fiber modeling approach.

Beams and columns are modeled as finite elements with distrib-

uted inelasticity. The proposed model by Mander et al. [33] isemployed for confined concrete zone to take into account the

confining effect. A bilinear model with kinematic strain hardening

is employed to represent the reinforcing steel bars. Five Gauss inte-

gration points are used for each element. The first three elastic

periods of the structural frames are T= (0.79, 0.2, 0.09) s, (1.6,

0.54, 0.3) s and (2.5, 0.93, 0.54) s for 3-story, 9-story and 20-story

buildings, respectively.

4.2. Ground motion data base

In the present work, a total of twenty strong earthquake ground

motions are compiled in order to develop a reliable set of bench-

mark responses. The main purpose of selecting this ensemble is

to take into account the variation in term of magnitude, source

to site distance and PGA. These motions were recorded during seis-mic events with moment magnitude 5.8

8/10/2019 push 29 2013.pdf

7/13

distance 1

8/10/2019 push 29 2013.pdf

8/13

8/10/2019 push 29 2013.pdf

9/13

4.3.1. Capacity curve

Fig. 5illustrates the capacity curves of three case studies, which

are obtained by incremental dynamic analysis (IDA) procedure and

different pushover methods. In this study, capacity curve consists

of the maximum total drift versus maximum base shear.

The capacity curve discrepancy factor (CCDF) is used to measure

the accuracy of each pushover method[17]. This parameter defines

the difference between the ordinates of points on the pushover

curve in comparison to the corresponding IDA points. The CCDF

is computed by the following equation.

CCDF

1

nXnk1

abs VkPUSH VkIDA

VkIDA 20

whereVkIDA and VkPUSHare maximum base shears at step k obtained

from IDA method and different pushover procedures, respectively.

It should be mentioned that the CCDF is computed for control

node displacements ranging between zero and 150% of the target

displacement[1].

As observed inFig. 5, APAM and DAP procedures provide a clo-

ser fit to the dynamic analyses envelops than those of the conven-

tional methods in all studied buildings. Also, the error of the APAM

method is less than the error of the DAP procedure in all structural

frames (Fig. 6). As anticipated, the triangular load pattern provides

an admirable estimate of capacity curve only for three story build-

ing where the effect of higher modes is negligible. The accuracy of

this method is significantly decreased as the higher mode effectsare increased in 9 and 20 story buildings. The uniform load pattern

Fig. 8. The target displacement determination in the APAM method.

Table 2

Target displacement obtained by APAM method in comparison with the NTH method.

ESDOF target displacements (cm)

NTH Buildings APAM Error (%)

Buildings Rec.1 Rec.2 Rec.3 Rec.4 Rec.5 Rec.6 Rec.7 Rec.8 Rec.9 Rec.10

3 Story 20.27 21.22 17.76 19.76 18.76 16.76 17.28 20.72 17.06 19.77

9 Story 37.91 31.07 44.48 39.67 42.24 40.69 33.76 46.94 49.62 44.95 3 Story 18.6 3.3020 Story 64.21 66.78 62.29 66.51 51.07 47.56 55.42 52.24 58.01 62.67

Buildings Rec.11 Rec.12 Rec.13 Rec.14 Rec.15 Rec.16 Rec.17 Rec.18 Rec.19 Rec.20 9 Story 38.9 3.40

3 Story 24.37 22.86 23.85 21.39 20.89 17.32 22.01 21.78 19.61 21.71

9 Story 44.25 32.41 32.06 40.34 39.07 46.44 49.35 33.68 44.82 38.16 20 Story 55.1 3.46

20 Story 65.09 63.37 66.14 64.96 47.2 57.01 64.41 66.93 69.35 60.16

Fig. 9. Mean peak inter-story drift profiles resulting from the different NSPs and the IDA analysis for the buildings studied.


8/10/2019 push 29 2013.pdf

10/13

could not provide suitable results in all studied buildings except in

20 story building where results are somewhat similar to IDA

outputs.

Another important observation is that load pattern 1 (Eq.(7))

dominate the responses in APAM method for estimation of the

capacity curve in all case studies. As it can be seen from Fig. 7,

the capacity curve due to load pattern 1 lay beyond the other

capacity curves due to load patterns 24, throughout the entire

deformation range. Therefore, in order to determine the capacity

curve of a structure in APAM method, it is only required that the

first load pattern (Eq.(7)) apply to the structure. It should be noted

that the drop observed in the capacity curve of 20 story building is

due to the significant contribution of the higher modes in load

combinations 3 and 4 which will be explained in Section4.3.5.

4.3.2. Target displacement

As mentioned previously, one of the main advantages of the

proposed methodology is determination of the target displacement

during the analysis. Fig. 8 illustrates the target displacements of

ESDOF systems and the pertinent ductility ratios. In order to eval-

uate the APAM procedure efficiency in estimating the target dis-

placement, these target displacements are compared with the

maximum inelastic displacement of ESDOF system obtained from

the nonlinear time history (NTH) analysis. In this regard, the de-

rived capacity curve of ESDOF system is idealized as a bilinear

curve. A NTH analysis was performed for each record using this

bilinear curve and the maximum inelastic displacement is com-

puted as target displacement of ESDOF system. The target displace-

ments, which are obtained by NTH analyses as well as APAM

Fig. 10. Observed mean errors of peak inter-story drift in different NSPs for the buildings studied.

Fig. 11. Mean peak total drift profiles resulting from the different NSPs and the IDA analysis for the buildings studied.

Fig. 12. Observed mean errors of total drift in different NSPs for the buildings studied.


8/10/2019 push 29 2013.pdf

11/13

Fig. 13. The variation of relative mode contribution (Rj) of each mode within the analysis.

Fig. 14. (a) The relative mode contribution of each mode corresponding to the peak interstory drift of each story for 20-story building. (b) The interstory drift profile related toeach load combinations of APAM method.


8/10/2019 push 29 2013.pdf

12/13

method for each building and each record, are presented inTable 2.

The mean errors of target displacement are calculated by the fol-

lowing equation.

Error% 100 1

n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXni1

TDNTHTDAPAMTDNTH

2s 21

whereTDNTHand TDAPAM are target displacements which are com-puted by NTH analysis and by APAM procedure, respectively.

As it can be seen fromTable 2, APAM estimates the target dis-

placement with an admirable accuracy. Therefore, the inter-story

drift and total-drift of structure are evaluated at the target dis-

placement which is computed by APAM procedure.

4.3.3. Inter-story drift

The peak inter-story drift profile for the buildings studied is

presented in Fig. 9. Where it can be seen that APAM produces

structural response which is similar to the IDA results in all build-

ings. In order to measure the accuracy of different nonlinear static

pushover (NSPs) methods, Eq.(22)is used[24].

Error% 100 1n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXni1

Di-PUSH Di-IDADi-IDA

2s 22

In this expression Di-PUSHand Di-IDA are inter-story drifts at level i

obtained by pushover analysis and IDA, respectively. The mean ob-

served errors in the different NSPs are revealed inFig. 10. In the case

of 3-story building, APAM, triangular pattern and DAP procedure

provide well estimation of inter-story drift. This observation is jus-

tifiable since the effects of higher modes are negligible in this low-

rise building and the governing mode is the first mode. Therefore

the triangular load pattern can predict this parameter accurately.

In the case of 9-story and 20-story buildings the errors of APAM

procedure are significantly less than the errors of the other proce-

dures. Since the effect of higher modes in the 9-story and particu-

larly, in the 20-story building is significant, the accuracy of

triangular method is significantly decreased. On the other hand,

the APAM method takes into account the contribution of higher

modes as well as the sign reversals in the modal load vectors and

results in well estimations of responses. Although DAP method pro-

vides well estimation of inter-story drift in comparison to the con-

ventional pushover, this procedure almost underestimates the

inter-story drift in all stories for 9 and 20 story buildings.

4.3.4. Total drift

InFig. 11the total drift obtained by NSPs and IDA methods for

studied buildings are depicted. In order to compute the errors of

the different NSPs, the Eq. (22) is used. As presented inFigs. 11

and 12, the accuracy of APAM procedure is much more than the

accuracy of other methods. The triangular pattern provides reason-able responses in all buildings particularly in 3 and 9 story build-

ings. DAP method underestimates the total drift in all studied

buildings except in 3-story building. It is surprising that the errors

of triangular pattern is less than that of DAP method.

4.3.5. Discussion

As stated in the previous sections, the APAM method provides

admirable estimation of seismic demands for all case studies. The

much more improved predictions by the APAM method in compar-

ison with those of the DAP method, is due to employing the EMMC

rule. In the EMMC rule, not only the relative mode contribution (Rj)

is computed based on the modal mass contribution, but also it is

updated throughout the analysis proportional to the instantaneous

dynamic properties of structures. This implies that Rj for eachmode is continuously altered within the inelastic range rather than

a constant value.Fig. 13 illustrates the variation of relative mode

contribution (Rj) for the first three modes.

As seen from this figure, the relative mode contribution of the

first mode in the load combinations 1 and 2 is increased, while

in the load combinations 3 and 4, the contribution of the first mode

is significantly decreased. Inversely, the contributions of the sec-

ond and the third modes are increased in load combinations 3

and 4. Also, in Fig. 14 the peak interstory drift profiles predictedby different load combinations of the APAM method for 20-story

building (Fig. 14a) as well as the corresponding relative mode con-

tribution of each mode (Fig. 14b) are depicted. As shown, the load

combinations 1 and 2 provide well estimation in the lower stories.

In these load combinations, the contribution of the first mode is

about 90% and the higher mode contributions are not significant

(less than 10%). On the other hand, the load combinations 3 and

4 provide reasonable estimations in the upper stories. In this case,

the contribution of the second mode is increased to about 35%

while the contribution of the first mode is decreased to about 60%.

These observations are compatible with the relative contribu-

tion of each mode during a NTH analysis. The study of the relative

contribution variations of each mode in the NTH analyses revealed

the following phenomena[23]:

I. Approximately, the peak interstory drifts in the lower stories

occur in the first inelastic excursion in the system. In this sit-

uation the first mode dominates the responses and the con-

tribution of higher modes are not significant.

II. The peak interstory drifts in the upper stories occur later and

the effects of higher modes significantly increase.

Therefore, the EMMC rule can estimate the relative contribution

of each mode in a manner consistent with the NTH analysis.

5. Conclusions

In the present study, an alternative displacement-based adap-

tive pushover is developed based on the effective modal mass com-

bination rule. In addition to the higher mode effects consideration

and the progressive changes in the dynamic characteristics of the

structures, this procedure utilize an effective modal mass combina-

tion rule in order to take into account the sign reversals of the ap-

plied load vector in the higher modes. In this regard, a relative

mode contribution factor, which is updated proportional to the

instantaneous dynamic characteristic of structure, is applied to

each modal load vector. The modified modal load vectors are alge-

braically added and subtracted and consequent load patterns are

independently applied to the structure within an adaptive frame-

work and the envelope of demand values are considered. Also,

the proposed methodology can estimate the target displacement

during the analysis using the energy approach and CSM concept

proposed in ATC40. The accuracy of APAM method is evaluatedthrough three concrete moment-resisting frames under four

near-fault different ground motions.

The obtained results illustrate that APAM method can capture

the results of IDA analysis with a reasonable accuracy in all case

studies. Comparisons indicate that APAM method is able to repro-

duce the capacity curve obtained by IDA method with enough

accuracy. Moreover, well estimation of inter-story drift profiles, a

critical parameter in seismic evaluation, as well as total drift pro-

files feature the high ability of APAM method to reproduce IDA

envelops.

References

[1] FEMA. Prestandard and commentary for the seismic rehabilitation of buildings,FEMA 356. Washington (DC): Federal Emergency Management Agency; 2000.


8/10/2019 push 29 2013.pdf

13/13

[2] CEN. Eurocode 8: design of structures for earthquake resistance, Part 1: Pr-EN1998-1, Final Draf. Brussels (BEL): European Committee forStandardization; 2003.

[3] ATC. Seismic evaluation and retrofit of concrete buildings. Report ATC-40.Redwood City (CA): Applied Technology Council; 1996.

[4] Paret TF, Sasaki KK, Elibeck DH, Freeman SA. Approximate inelastic proceduresto identify failure mechanism from higher mode effects. In: Proceeding of theeleventh world conference on earthquake engineering; 1996.

[5] Krawinkler H, Seneviratna GDPK. Pros and cons of a pushover analysis ofseismic performance evaluation. Eng Struct 1998;20(46):45264.

[6] Kim S, DAmore E. Pushover analysis procedures in earthquake engineering.Earthq Spectra 1999;15(3):41734.

[7] FEMA. Improvement of nonlinear static seismic analysis procedures, FEMA440. Washington (DC): Federal Emergency Management Agency; 2005.

[8] Chopra AK, Goel RK. A modal pushover analysis procedure for estimatingseismic demands for buildings. Earthq Eng Struct Dyn 2002;31:56182.

[9] Kim S, Kurama Y. An alternative pushover analysis procedure to estimateseismic displacement demands. Eng Struct 2008;30:3793807.

[10] Poursha M, Khoshnoudian F, Moghadam AS. A consecutive modal pushoverprocedure for estimating the seismic demands of tall buildings. Eng Struct2009;31:5919.

[11] Reinhorn AM, Inelastic analysis techniques in seismic evaluation. In:Proceedings of the international workshop on seismic design methodologiesfor the next generation of codes; 1997.

[12] Bracci JM, Kunnath SK, Reinhorn AM. Seismic performance and retrofitevaluation of reinforced concrete structures. J Struct Eng, ASCE1997;123(1):310.

[13] Gupta B, Kunnath SK. Adaptive spectra-based pushover procedure for seismicevaluation of structures. Earthq Spectra 2000;16(2):36791.

[14] Elnashai AS. Advanced inelastic static (pushover) analysis for earthquakeapplications. Struct Eng Mech 2001;12(1):5169.

[15] Chopra AK, Goel RK, Chinatanapakdee C. Evaluation of a modified MPAprocedure assuming higher modes as elastic to estimate seismic demands.Earthq Spectra 2004;20(3):75778.

[16] Antoniou S, Pinho R. Advantages and limitations of adaptive and non-adaptiveforce-based pushover procedures. J Earthq Eng 2004;8(4):497522.

[17] Papanikolaou VK, Elnashai AS, Pareja JF. Evaluation of conventional andadaptive pushover analysis II: comparative results. J Earthq Eng2006;10(1):12751.

[18] Lopez-Menjivar MA, Pinho R. A review of existing pushover methods for 2-Dreinforced concrete buildings. Pavia (Italy): Rose School; 2004.

[19] Antonio S, Pinho R. Development and verification of a displacement-basedadaptive pushover procedure. J Earthq Eng 2004;8(5):64361.

[20] Pinho R, Antoniou S, Peierta D. A displacement-based adaptive pushoverfor seismic assessment of steel and reinforced concrete buildings. In:Proceedings of the 8th US national conference on, earthquake engineering;2006.

[21] Ferracuti B, Pinho R, Savoia M, Francia R. Verification of displacement-basedadaptive pushover through multi-ground motion incremental dynamicanalyses. Eng Struct 2009;31:178999.

[22] Kalkan E, Kunnath SK. Adaptive modal combination procedure for nonlinear

static analysis of b uild ing st ructur es. J Struct Eng, ASCE2006;132(11):172131.

[23] Kalkan E, Kunnath SK. Assessment of current nonlinear static procedures forseismic evaluation of buildings. Eng Struct 2007;29(3):30516.

[24] Shakeri K, Shayanfar MA, Kabeyasawa T. A story shear-based adaptivepushover procedure for estimating seismic demands of buildings. Eng Struct2010;32:17483.

[25] Matsumori T, Otani S, Shiohara H, Kabeyasawa T. Earthquake memberdeformation demands in reinforced concrete frame structures. In:Proceeding of the USJapan workshop on performance-based earthquakeengineering methodology for RC building structure; 1999.

[26] Kunnath SK. Identification of modal combination for nonlinear static analysisof building structures. Comput-Aided Civ Eng 2004;19:24659.

[27] Park HG, Eom T, Lee H. Factored modal combination for evaluation ofearthquake load profiles. J Struct Eng, ASCE 2007;133(7):95668.

[28] Hernandez-Montes E, Kwon O-S, Aschheim MA. An energy based formulationfor first and multiple-mode nonlinear static Pushover analyses. J Earthq Eng2004;8:6988.

[29] Casarotti C, Pinho R. An adaptive capacity spectrum method for assessment ofbridges subjected to earthquake action. Bull Earthq Eng 2007;5(3):37790.

[30] ACI. Building code Requirements for structural concrete with commentary. ACI318. American Concrete Institute; 2008.

[31] American Society of Civil Engineers (ASCE). Minimum design loads forbuildings and other structures. ASCE Standard No. ASCE/SEI 7-05; 2006.

[32] OpenSees Development Team (Open Source Project). OpenSees: Open systemfor earthquake engineering simulation. Berkeley (CA): Pacific EarthquakeEngineering Research Center, University of California; 2008. .

[33] Mander JB, Priestley MJN, Park R. Theoretical stressstrain model for confinedconcrete. J Struct Eng 1988;114(8):180426.

http://opensees.berkeley.edu/http://opensees.berkeley.edu/http://opensees.berkeley.edu/http://opensees.berkeley.edu/

push 29 2013.pdf

Documents

Transcript of push 29 2013.pdf