Simple Method for Pushover Curves of Asymmetric Structure with Displacement-Dependent Passive Energy Dissipation

Devices Santiago Naranjo Department of Mechanical Engineering University of Florida Gainesville, Florida, USA, 32611 2800 SW Williston Road #1036 Gainesville, FL, USA, 32608 snp1984@ufl.edu Mentor: Dr. Li Hong-Nan, Li Gang State Key Laboratory of Costal and Offshore Engineering Department of Civil Engineering Dalian University of Technology Dalian, Liao Ning, China, 116023

Abstract: This paper presents a simple method to calculate pushover curves for asymmetric structure with displacement-dependent passive energy dissipation devices (DDPEDDs). The method analyzes the deformation of a symmetric structure in translation and in torsion. These results are then combined in order to calculate the pushover curve for an asymmetric structure with DDPEDDs. The numerical results obtained by using the simple analysis method are then compared to the results obtained from the analysis of the models using the software SAP2000. The results show that the simple analysis method can be an effective tool for engineering analysis. Introduction

The technique of passive energy dissipation has been rapidly growing and it is now widely used in many parts of the world. This technique reduces the structure response subjected to wind and earthquake through mounting energy dissipation devices into the buildings (Soong 1997). Contrary to semi-active and active systems, there is no need for an external supply of power. In recent years, variable kinds of dampers have been studied by several researchers and serious efforts have been undertaken to develop the concept of energy dissipation or supplemental damping into a workable technology (Li 2003). A number of these devices, such as passive metallic yielding, visco-elastic, and viscous energy dissipation devices have been installed in structures throughout the world (Soong 1997). The primary objective of adding energy dissipation systems to building frames has been to focus the energy dissipation during an earthquake into disposable elements specifically designed for this purpose, and to substantially reduce energy dissipation in the gravity-load-resisting frame. Moreover, since energy dissipation devices are not part of the gravity-load-resisting frame they can easily be replaced after an earthquake without compromising the structural integrity of the frame. Two different types of passive energy

dissipation systems include the displacement dependent damper and the velocity dependent damper (Code 2001). Due to the material presented here, this paper will focus on displacement dependent dampers, i.e. metallic and friction dampers.

The pushover analysis method, as a foundation for performance-based design has attracted the interest of many researchers. Regarding the pushover analysis procedure of regular structures, there are two widely used methods, i.e. FEMA273 (Federal 1996) and ATC-40 (ATC 1996). These methods are used by most engineers as a standard tool for estimating seismic demands for buildings, because they analyze the progress of the pushover method numerically. The non-linear static pushover analysis is a simple option for estimation the strength capacity in the post-elastic range. This procedure involves applying a predefined lateral load pattern that is distributed along the building height. These methods are considered to be better than other methods such as using linear analysis and ductility-modified response spectra, because they are based on a more accurate estimate of the distributed yielding within a structure, rather than an assumed, uniform ductility. The generation of the pushover curve also provides the engineer with a good feel for the non-linear behavior of the structure under lateral load. However, it is important to remember that pushover methods have no rigorous theoretical basis, and may be inaccurate if the assumed load distribution is incorrect. For example, the use of a load pattern based on the fundamental mode shape may be inaccurate if higher modes are significant, and the use of any fixed load pattern may be unrealistic if yielding is not uniformly distributed, so that the stiffness profile changes as the structure yields.

In order to extend the pushover analysis into more areas of application, various researchers have studied the pushover method for asymmetric structures. Kilar and Fajfar (1997) present a simple method for the non-linear static analysis of complex buildings subjected to monotonically increasing horizontal loading. Their method is designed to be a part of new methodologies for the seismic design and evaluation of structures. It is based on the extension of a pseudo-three-dimensional mathematical model of a building structure into the non-linear range. The structure consists of planar macro elements. For each planar macro element, a simple bilinear or multi linear base shear-top displacement relationship is assumed. Furthermore, Chopra and Goel (2004) developed a modal pushover analysis based on the structural dynamics theory and extended this method to asymmetric–plan buildings. In this method, the seismic demand due to individual terms in the modal expansion of the effective earthquake forces is determined by non-linear static analysis using the inertia force distribution for each mode, which for asymmetric buildings includes two lateral forces and torque at each floor level. At last, by the CQC rule to obtain an estimate of the total seismic demand for structure Barros and Almeida (2005) simplify the process for the static non-linear dynamic response of a structure, utilizing a load pattern proportional to the shape of the fundamental mode of vibration of the structure. In this paper, a simple method to calculate pushover curves for asymmetric structure with DDPEDDs is presented. Basic Theory Asymmetric structures are characterized for having their center of stiffness and center of mass located in different positions. Due to this characteristic, it is rather complex to analyze the effect of lateral forces on such structures because the structure will react both

in torsion and translation to such forces. In this paper, a simple method for calculating the effect of lateral loads on an asymmetric structure with DDPEDDs is presented. Such technique relies on decomposing the deformation of an asymmetric structure into translation and torsion on symmetric structures and then combining the results. Related basic theory is introduced as follows. Structural Deformation

In order to analyze the deformation of an asymmetric structure (3-D) under lateral loading, two reactions, translation and rotation for a symmetric structure (2-D), need to be taken into consideration.

Torsion effects have a considerable impact on the response of a structure reacting to lateral loading as shown in Fig 1. Furthermore, this shows the importance of taking rotational effects on asymmetric structures into consideration for engineering analysis.

Fig 1 shows the difference on the pushover curve after taking torsion effects into

consideration

The deformation imposed on an asymmetric structure due to lateral loading can be calculated by the following method. First, calculate the translational effect of the lateral loading on a planar symmetric structure. Second, calculate rotational effects on the planar symmetric structure by applying torques on the structure. Finally, combine both effects as shown in Fig 2 to obtain the effect that the loading would have on an asymmetric structure.

(a) Combined effect of asymmetric (b) Translation of symmetric (c) Torsion of symmetric

Fig 2 simple method for estimating the effect of an earthquake on an asymmetric structure

Dynamic Analysis of Asymmetric Structure Equations of asymmetric structure with DDPEDDs:

The equation of motion for the multiple degrees of freedom, MDOF, structures with passive energy dissipative devices can be written as

[ ]{ } [ ]{ } [ ]{ } { } [ ][ ]{ }gURMFUKUCUM &&&&& −=+++ (1)

where 、 and represent the mass, damping, and stiffness matrices respectively, while { symbolizes a matrix of force offered by DDPEDDs. Furthermore, the vectors

][M ][C ][K}F

{ }U&& ,{ }U& and { }U represent the acceleration, velocity and displacement respectively. Moreover, these parameters are defined as vector quantities i.e. { } { }θ,,vuU = ，where ， ，u v θ represent the magnitude of the displacement in the different directions. In addition ( )gU&& represents the acceleration contributed by the

earthquake, and represents a matrix of units. Finally, ][R { } { }Tgggg vuU θ&&&&&&&& ,,= represents the acceleration contributed by the earthquake. The matrices for the different parameters are shown below

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

][]][[]][[]][[][]0[]][[]0[][

][

0JmXmYXmmYmm

M

mm

m

m

， ， ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

θθθ

θ

θ

][][][][][][][][][

][KKKKKKKKK

K

yx

yyyx

xxyx

[ ] , { } [ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ] ⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

0000

0

m

m

XIYI

R⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

θTFF

F y

x

where [ and represent matrices of the distance between the center of mass and

the center of stiffness for the different floors. In addition,

] ]mX [ mY

[ ] TI ]111[= represents a unit matrix, and represents the moment of inertia of the structure. Moreover,

represents the moment of inertia of the floor with being the shortest distance between the center of mass and the center of stiffness,

][ 0J

)( 2220 mjmjjjj YXrmJ ++= thj jr

Basic period and shape of asymmetric structure: The first step in order to find the basic period and shape of the asymmetric structure consists in finding the angular frequency, ω , from Eq. (2a)

[ ] [ ] 02 =− MK ω (2a)

[ ] [ ]{ }{ } { }02 =− φω MK (2b)

Consequently, the value obtained for ω is used to calculate the basic period of oscillation

of the structure by using the relationship ωπ2

=T , where [ ] [ ]Tyx TTTT θ= denotes a

vector quantity. Furthermore, after obtaining the basic period and angular frequency, the interstory drift of the structure can be calculated by using Eq. (2b). [ ] [ ] [ ] [ ][ ]θφφφφ yx= Convert to equivalent SDOF system: The displacement of the floors of the structure is given by

{ } [ ]{ })()( tqtU φ= (3)

where { } is a vector dimension coordinate and [ T

Nqqqq 321 ,,, L= ] [ ]φ is a matrix describing interstory drift displacement. Moreover, the motion equation for the structure is given by

[ ][ ]{ } [ ][ ]{ } [ ][ ]{ } [ ][ ]{ } { }FtURMtqKtqCtqM g −−=++ )()()()( &&&&& φφφ (4)

Following, the motion equation corresponding to the modal is thi

[ ][ ]{ } [ ][ ]{ } [ ][ ]{ } [ ][ ]{ } { }igiiiiii FtURMtqKtqCtqM −−=++ )()()()( &&&&& φφφ (5)

This equation can also be written as

[ ] [ ] [ ][ ][ ] [ ] [ ][ ][ ][ ] [ ][ ] [ ]

{ }{ }{ }

[ ]{ }{ }{ }{ }

[ ] [ ] [ ][ ][ ] [ ] [ ][ ][ ][ ] [ ][ ] [ ]

[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ]

[ ][ ][ ]⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧−

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

i

yi

xi

g

g

g

m

m

mm

m

m

i

i

yi

xi

yx

yyyx

xxyx

iii

i

yi

xi

mm

m

m

FFF

vu

XIYI

JmXmYXmmYmm

qKKKKKKKKK

qCqJmXmYXmmYmm

θ

θθθθ

θ

θ

θ

φ

φφφ

φφφφ

&&&&

&&

&&&

0000

00

0

][][][][][][][][][

00

0

0 (6a)

Disregarding the torsion vector inputted by the earthquake，i.e. and assuming that [ ]

0=gφ&&

{ } { } { } { }[ θ ]φ CCCC vu= , equation (6) can then be written as follows

[ ]{ } [ ][ ]{ }[ ]{ } [ ][ ]{ }

[ ][ ]{ } [ ][ ]{ } [ ]{ }

{ }{ }{ }

[ ] { } [ ] { } [ ] { }[ ] { } [ ] { } [ ] { }[ ] { } [ ] { } [ ] { }

[ ][ ] [ ][ ][ ] [ ]

[ ][ ][ ] [ ][ ][ ] [ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+−−−

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++++++

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−−−

igmgm

yig

xig

i

iyiyxix

iyyiyyxiyx

ixyixyxixx

iv

u

i

iximyim

imyi

imxi

FvImXuImYFvImFuIm

qKKKKKKKKK

qCCC

qJmYmX

XmmYmm

θ

θθθθ

θθ

θθ

θθ

θ

θ

φφφφφφφφφ

φφφφφφφ

&&&&

&&

&&

&&&

0(6b)

Furthermore, assuming that [ ]{ } [ ] [ ] [ ][ ]Tvui mmmm θφ = and that

, Eq. (6b) can then be divided into three equations as shown below [ ]{ } [ ] [ ] [ ][ T

vui KKKK θφ = ]

[ ] [ ] [ ] [ ][ ] [ ]xigiuiuiu FuImqKqCqm −=++ &&&&& (7a) [ ] [ ] [ ] [ ][ ] [ ]yigiviviv FvImqKqCqm −=++ &&&&& (7b)

[ ] [ ] [ ] [ ][ ][ ] [ ][ ][ ] [ igmgmiii FvImXuImYqKqCqm θθθθ − ]+−=++ &&&&&&& (7c)

Now, multiplying the left and right hand side of Eq. (7a) by { }Txiφ gives

{ } [ ] { } [ ] { } [ ] { } [ ][ ] { } { } [ xi

Txi

Txig

Txiiu

Txiiu

Txiiu

Txi FuImqKqCqm φφφφφφ −=++ &&&&& ] (8)

Moreover, assuming that and solving for xixi qq Γ= xΓ gives

{ } [ ][ ]{ } [ ]

{ } [ ][ ]{ } [ ]{ } { } [ ][ ]{ }im

Txixi

Txi

Txi

uT

xi

Txi

x YmmIm

mIm

θφφφφφ

φφ

−==Γ (9)

Finally, Eq. (8) can be re-written for the X direction as follows

[ ]xigxxixxixxix FuMqKqCqM −=++ &&&&& **** (10) where,

{ } [ ][ ]ImM Txix φ=* (11a)

{ } [ ] xuT

xix CC Γ= φ* (11b)

{ } [ ] xuT

xix KK Γ= φ* (11c)

Likewise, the motion equation for the Y direction and angle θ can be re-written as

[ ]yigyyiyyiyyiy FuMqKqCqM −=++ &&&&& **** (12a)

where， { } [ ][ ]ImM T

yiy φ=* (12a)

{ } [ ] yvT

yiy CC Γ= φ* (12b)

{ } [ ] yvT

yiy KK Γ= φ* (12c) and

[ ]igiiyi TuMqKqCqM θθθθθθθ −=++ &&&&& **** (13)

where， { } [ ][ ]ImM T

iθθ φ=* (13a)

{ } [ ] θθθθ φ Γ= CC Ti

* (13b)

{ } [ ] θθθθ φ Γ= KK Ti

* (13c)

Simple Pushover Analysis Method

The simple method described in this section is used for asymmetric structures with DDPEDDs; where the structural parameters i.e. interstory stiffness and interstory yield displacement, have known values. For structures with DDPEDDs in 2-D

In order to calculate the pushover curve for a structure under lateral loading, symmetric structures with DDPEDDs are simplified into 2-D (planar structure). In this section, a method developed by Oscar Ramirez is introduced in order to find a pushover curve relating the roof displacement with the shear stress at the base of the structure.

The analysis method developed by Ramirez (2000) is based on calculating the coordinates of a couple of significant points that will help create an estimate for the pushover curve. The different points that need to be calculated can be seen in Fig. 3. In this figure, the dashed line is defined by points

DCBA ,,,A and B . Point A is defined to

be the point where all the DDPEDDs have yielded.

Fig 3 Displacement pushover curve

Moreover, point B is defined to be the point where all the structural elements have yielded. After defining points A and B , the multi-line can be obtained. Referring to FEMA273, the tri-linear curve is converted to the bi-linear curve by defining points C and D. The following sections describe how to calculate all of the defining points.

Calculating point A : (1) Define the DDPEDDs yielding displacement to be yjΔ (2) Consider that a frame with displacement DDPEDDs is pushed laterally by loads that

are proportional to the first mode resulting in displacement . The displacement of floor in the

iD

iD X direction can be related to the roof displacement by roofixi DD ⋅= ,φ . Now it is shown that 00, =xφ and that the structure has a modal drift jx,φ in the X

direction at the floor. Accordingly the interstory drift of a structure in the floor in the

thj thjX direction can be written as

( ) roofrjxroofjxjxjjj DDDD ,1,,1 φφφ =−=−=Δ −− , where [ ]xφ is the modal drift matrix of the structure in the X direction .Here, the displacement of the roof when yielding of the devices at level j occurs can be expressed as

rjx

jroofyjD

,φΔ

=

When all the DDPEDDs have yielded, it can be assumed that yjj Δ=Δ At this point the displacement of the roof can be defined as the maximum value of the displacement obtained by the equation above

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ=

rjx

yj

jydD,

maxφ

(14)

(3) To obtain the base shear of the building when all the damping devices yielded, use the

following equation

11

21

24x

x

yd

xydfd W

DgT

DKV ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛Γ

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=+

π (15)

where is the fundamental period of translation of the system in 1xT X direction , 1xΓ is the first modal participation factor of translation of the system in the X direction and 1xW is the first modal weight of translation of the system in the X direction.

Calculating point B : (1) The base shear strength of the system, V is equal to the sum of the yield strength of

the frame, , and the strength of the devices in the first floor , , when yielding of both the frame and the devices occurs

yV 1dV

dly VVV += (16)

(2)To find the roof displacement at point B , , use the following equations yfD 1,,, −−=Δ jyfjyfjyf DD

⎟⎟⎠

⎞⎜⎜⎝

⎛

Φ

Δ=⎟

⎟⎠

⎞⎜⎜⎝

⎛

Φ−Φ

Δ=

− rjx

jyf

jjxjx

jyf

jyfD,

,

1,,

, maxmax (17)

where, [ represents the modal drift of the frame of the structure. Furthermore, ]xΦ

jyfD , and represent the yielding displacement and interstory yielding

displacement of the structural elements at the floor. Finally, refers to the

amplitude of the basic modal in the X direction at the floor.

jyf ,Δthj jx,Φ

thj Calculating point C : After obtaining points A and B , a pushover curve can be developed. The effective initial stiffness of the equivalent elasto-plastic representation can be defined as the slope of the straight line that intersects the tri-linear curve a the base shear equal to bV , where

. Knowing that point has to be on line AB, the roof displacement for this point can be easily obtained.

6.0=b C

Calculating point : D

After obtaining point , line can be created. In order to find point , line OC is extended up. Point is then positioned at a base shear stress equal to the one obtained for point

C OC DD

B and laying on line . OC

Torsion energy spectrum of symmetric structure with DDPEDDs In this section, a method similar to the one presented above will be used in order to

calculate the reaction of the symmetric structure with DDPEDDs under the application of a moment force. By using this method, a graph of the torsion at the base versus the angle of the roof θ−M will be obtained. As shown in Fig 4, the pushover curve will be

Fig 4 Torsion curve

defined through making use of points . HGFE ,,, Calculating point E : (1) Define to be the DDPEDDs yielding angle. From this it can be shown that the

angle difference between two different floors is yjΔ

d

yjyj r

Δ=θ (18)

where is defined to be the distance from the damper to the center of stiffness. dr(2) Define [ ]θφ to be the modal torsion of the structure. The angle at any given floor iθ

can be related to the angle of the roof by the relationship roofii θφθ θ ⋅= , , where

00, =θφ and j,θφ is the modal torsion of the structure. Furthermore, at the floor the interstory angle difference can be found by using the equation

thj

( ) roofrjroofjjjjj θφθφφθθ θθθ ,1,,1 =−=−=Ω −− . Here, the angle of the roof when

yielding occurs at the floor can be expressed as thj

rj

jroofyj

,θφθ

Ω=

When all of the damping devices have yielded the angle of the roof can be defined as the maximum value of the angle obtained by the equation above

⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω=

rj

yj

jyd,

maxθφ

θ (19)

(3) In order to find the torque at the base at the moment where all of the damping devices

have yielded, the following equation should be used

11

21

24θ

θ

θπθ WgT

KM ydydtfd ⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛Γ

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=+ (20)

where represents the fundamental period of torsion, 1θT 1θΓ represents the first modal participation factor of the torsion and 1θW represents the first modal weight of the system.

Calculating point : F(1) The torsion at the base of the system, M is equal to the sum of the torsion of the

yield strength of the frame, , and the torsion strength of the devices in the first story , when yielding of both the frame and the devices occur; that is

yM

dlM dly MMM += (21)

(2) Based on the structure without passive devices, the yield angle at point and the modal shape can be calculated using the following formulas

F

1,,, −−=Ω jyfjyfjyf θθ

⎟⎟⎠

⎞⎜⎜⎝

⎛

Φ

Ω=⎟

⎟⎠

⎞⎜⎜⎝

⎛

Φ−Φ

Ω=

− rjx

jyf

jjxjx

jyf

jyf,

,

1,,

, maxmaxθ (22)

where, jyf ,θ and represent the yielding angle and interstory angle of the

structure without passive devices at the floor. Moreover,jyf ,Ω

thj j,θΦ refers to the basic

torsion modal amplitude at the floor. thj

Calculating point G : After obtaining points a pushover curve can be developed for the torque at the

base floor against the angle of the roof, FE,

θ−M . The effective initial stiffness of the equivalent elasto-plastic representation can be defined as the slope of the straight line that intersects the tri-linear curve a the base shear equal to bV , where . After obtaining the value for the base torsion, a value for the angle can be obtained by using the fact that point is located on line

6.0=b

G EF .

Calculating point H : After obtaining pointG , line OG can be created. In order to find point H , line OG

is extended up. Point H is then positioned at a base shear stress equal to the one obtained for point and laying on line . F OG Convert θ−M curve to curve : Δ−P In order to convert the torsion pushover curve into a translation curve, the following equations should be used. These equations will convert torques ( M ) into forces ( ) and angles (

Fθ ) into displacements (Δ ).

mXFM ×= (23a) mYFM ×= (23b)

mX×=Δ θ (23c)

mY×=Δ θ (23d) Where and represent the displacement of the center of stiffness and the

center of mass in the mX mY

X direction and Y direction respectively.

Combining the curves Δ−P

After obtaining the torsion and translation pushover curves shown in Fig 5a and Fig 5b, the following equations can be used in order to find point Q which will then help create a pushover curve that takes both the translation and torsion effects into consideration.

1111 XKF ≤ΔΔ= 2222 XKF ≤ΔΔ=

where and stand for the slope of the torsion and translation pushover curves respectively as shown in Fig 5.

1K 2K

FKKKK

21

2121

+=Δ+Δ=Δ

Δ+

=21

21

KKKKF ( )21,min XX≤Δ

Q =( )

⎪⎩

⎪⎨⎧

+=

=

XKK

KKP

XXX

21

21

21,min (24)

After point Q has been found, it is plotted on a graph. Next, lines are drawn from

point Q to the origin and from point Q into the positive X direction as shown in Fig 5c to create the simple method pushover curve.

(a) Torsion curve (b) 2-D translation curve (c) Translation and torsion curve

Fig 5 shows the different pushover curves

Procedure

The simple method for calculating asymmetric structure with displacement dissipation devices can be outlined as follows: (1) Calculate the structure parameters, i.e. interstory stiffness and torsional stiffness,

yield displacement and angle. (2) Calculate the necessary points to find the curve of a planar structure utilizing the

method introduced in the preceding sections. (3) Calculate the necessary points to find the curve for a structure under torsion by the

method explained in the preceding sections. (4) Convert the θ−M curve to a Δ−P curve. (5) Combine the two curves obtained on step (3) and (4) into a single curve. The curve

just obtained takes into consideration both translational and rotational effects.

Numerical example Given information This section provides all of the information necessary in order to calculate the different points needed to build a pushover curve. First, in order to simplify the model, the weight of each of the floors is considered to be concentrated in the center of mass of the structure. Furthermore, the weight of each of the floors is given to be

KNWWWW 30004321 ==== , KNWW 160065 ==

Moreover, Table 1 shows the different parameters used for the dampers in all of the six different floors.

Table 1 Provides the parameters for the damper Metallic damper parameter

Floor Stiffness (KN/mm)

10.50 15.2

Yieldi ment

1－2F

ng displace(mm)

Yielding force (KN) 160

3－4F 7.85 15.2 120 6.1 15.2 97 5－6F

Furthermore, an appropriate model of the structure to be analyzed is provided in Fig . A

6 s can be seen in Fig 6b, the center of stiffness and the center of mass of this structure are located in different positions, making it an asymmetric structure.

(a)3-D model (b)top view (c)side views

Fig 6 provides different views of the model

Calculations

mics parameters: scribed above, an example will be calculated. From

Structural dynaFollowing the procedures de

equation (2a), the direction X and angle θ of the structure can be calculated. Then, from the specifications of the building, the fundamental period of oscillation is obtained to be

184.11 =xT s. Next, from the fundamental period, the modal translation drift is obtained uation (2b).

[ ] [

by using eq

]Tx 22.041.061.078.091.01=φ

The funda the structure in torsion is found to be 1 =θT

[ ] [ ]37.055.075.086.095.01=θφ

Moreover, n be used in order to find the different para

mental period of oscillation of

934.1 s. By using equation (2b), the modal torsion drift can be obtained

T

equations (9), (11a), and (13a) ca

meters of the structure along the θ,X directions. First, considering the reactions a the long X direction, it is found that the first modal

translation weight is KNWx 269401 = ，and that the fundamental period of translation is 37.11 =Γx

Secondly, reactions along the angle considering the θ , it is found that the first modal torsion weight is mKNW .391001 =θ ，and that the fundamental period of rotation is

27.11 =Γθ 2-D curve:

to create a pushover curve for a planar symmetric structure, the procedures In order described before are followed. (1) Point A

s he devices in the dBecau te ifferent floors have the same dimensions and are made of the same materials, the yield displacement is the same in all the floors. The yield displacement as shown in Table 1 is mmyj 3.15=Δ . As stated above the fundamental period of oscillation in the X direction is

T

[ ] [ ]x 22.041.061.078.091.01=φ

Moreover, the inte

184.11 =x S. Using this period the modal drift vector is found to be

T

rstory modal drift vector is

[ ] [ ]Trx 22.019.020.017.013.009.0, =φ

Using equation（ etallic damper yields

14）and choosing the value at which the last m

it is obtained that

mmDrjx

yj

jyd 7.69max,

=⎟⎟⎠

⎜⎜⎝

=φ

⎞⎛ Δ

Then, by using equation（15）, the base shear stress when all of the damping devices have yielded is found

KNWgT

DKV xx

yd

xydfd 4.3931

121

=⋅⎟⎟⎠

⎜⎜⎝ Γ⋅⎟⎟⎠

⎜⎜⎝

=+

D4 2 ⎞⎛⎞⎛ π

(2) Point shear stress when the damping devices and the frame have yielded is found

KNVVV 750320430

BThe ba seby using equation（16）

dly + = + ==

Then, the displacement of the roof at this point is found through the use of

equation（17）

mmDrjx

jyf

jjxjx

jyf

jyf 22007.0

maxmax,

,

1,,

, ==⎟⎟⎠

⎜⎜⎝ Φ

=⎟⎟⎠

⎜⎜⎝ Φ−Φ

=−

4.15⎞⎛ Δ⎞⎛ Δ

(3) Point C shear stress at this point is found to be KNbVV 4507506.0 ＝＝The ba se = ×c . Then,

point C is positioned on line AB . Subsequently, at this point is found to be m2.950＝D

the displacement of the roof.

(4) Point D is extended up until it reaches the same base shear stress as that for point

m

Line OCB . Po D is then located at a base shear stress of KNVVV dyD 7501int =+= . Then the

f displ ement can be obtained by analyzing found to be mmDy 7.158= .

roo ac the graph and it is

(5) Curve g all the points in a graph, the translation pushover curve is developed as Plottin

shown in Fig 7.

Fig 7 Translation pushover curve

θ−M Curve:

(1) Point E As shown in Table 1, the yield displacement of the structure is . Furthermore, the angle difference between the floors is calculated from the yield displacement and the distance from the damper to the center of stiffness

mmyj 2.15=Δ

007.0=Δ

=d

yjyj r

θ

With the fundamental period of oscillation for the angle,θ , as S，the modal angle drift vector of the building is

934.11 =θT

[ ] [ 37.055.075.086.095.01=θ ]φ

Moreover, the modal interstory angle drift is

[ ] [ 36.018.020.011.009.005.0=rθ ]φ Using equation（19）and choosing the value at which the last metallic damper yields, the angle of the roof is

02.0max,

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

rj

yj

jydθφθ

θ

Then, by using equation（20）, the torque at the base when all of the damping devices have yielded is found

mmKNWgT

KM ydydtfd ⋅=⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛Γ

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=+ 6.6635634

11

21

2

θθ

θπθ

(2) Point FThe torque at the base when the damping devices and the frame of the structure have yielded is found by using equation（21）

mmKNMMM dly ⋅=+= 2172000

Then, the angle of the roof at this point, yfθ is found by using equation（22）

09.0maxmax,

,

1,,

, =⎟⎟⎠

⎞⎜⎜⎝

⎛

Φ

Ω=⎟

⎟⎠

⎞⎜⎜⎝

⎛

Φ−Φ

Ω=

− rjx

jyf

jjxjx

jyf

jyfθ

(3) Point G

The torque at the base at this point is found to be . Then, point is placed on line OG on the KNbMM G 130320021720006.0 ＝×== G

θ−M curve. Subsequently, the angle of the roof at this point is found to be 05.00 =θ .

(4) Point H Line is then extended up until reaching the same torque as that for point . Point

OG FH is then located at a torque of mmKNM H .2172000= . The roof angle is then

found by analyzing the graph, and it is found to be 05.0=yθ . (5) θ−M curve

Plotting all of the points obtained, a torsion pushover curve is obtained as shown in Fig 8.

Fig 8 Torsion pushover curve

(6) Convert the θ−M curve to a Δ−P curve

Using equations (23a)-(23d), the torsion pushover curve can be converted into a translation pushover curve as shown in Fig 9.

Fig 9 Δ−P curve

Combine the curves:

Using equation（24）the values for point Q are found to be , and . After finding point Q, a combined pushover curve can be created.

mmX 150=KNP 5.473=

Compare to SAP2000 results After obtaining the pushover curve for an asymmetric structure with passive energy

dissipative devices, analysis of the same structure by software SAP2000 can be used in order to compare the results. SAP2000 was used in order to develop two different pushover curves for a symmetric 2-D and an asymmetric 3-D structure as shown in Fig 10. Analyzing the figure, shows that 3-D Pushover analysis are crucial in order to accurately predict the reaction of a structure under earthquake forces. Furthermore, the pushover curve developed through SAP2000 is then compared to the pushover curve obtained from the simple analysis method discussed in the previous sections as shown in Fig 11. Comparing both curves shows that the simple method used to calculate the pushover curve is very successful because it gives values very similar to those given by SAP2000. So, this graph is the validation that the simple method can be used in order to obtain a pushover curve for asymmetric buildings with passive energy dissipative devices.

Fig 10 compares 2-D and 3-D pushover curves Fig 11 compares simple method with sap2000 Conclusion

The results indicate that the simple method explained in this paper in order to

calculate the pushover curve for asymmetric structures with DDPEDDs is capable of accurately predicting the actual structural response of such structures. Furthermore, the similarity between the results obtained from using the simple method and the results obtained using the software SAP2000 proves the accuracy of the simple method at predicting the effect that torsion and translation have on the response of the structure. In conclusion, the simple method presented in this paper can be used in engineering analysis in order to obtain very good estimates for pushover curves of asymmetric structures with DDPEDDs.

Acknowledgments

This material is based on work supported by the Natinal Science Foundation under Grant No. OISE-0229657. Any opinions, findings, and concludions expressed in this material are those of the author and do not necessarrily reflect the views of the National Science Foundation.

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