development of a displacement based design method for steel frame - rc wall, 2007 (garcia)

8/12/2019 Development of a Displacement Based Design Method for Steel Frame - RC Wall, 2007 (Garcia)

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Istituto Universitario

di Studi Superiori

Universit degli

Studi di Pavia

EUROPEAN SCHOOL FOR ADVANCED STUDIES INREDUCTION OF SEISMIC RISK

ROSE SCHOOL

DEVELOPMENT OF A DISPLACEMENT BASED DESIGN

METHOD FOR STEEL FRAME-RC WALL BUILDINGS

A Dissertation Submitted in Partial

Fulfilment of the Requirements for the Master Degree in

EARTHQUAKE ENGINEERING

by

REYES GARCIA LOPEZ

Supervisor 1:Dr TIMOTHY J. SULLIVAN

Supervisor 2:Dr. GAETANO DELLA CORTE

May, 2007


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The dissertation entitled Development of a Displacement Based Design Method for Steel

Frame-RC Wall Buildings, by Reyes Garcia Lopez, has been approved in partial fulfilment

of the requirements for the Master Degree in Earthquake Engineering.

Timothy J. Sullivan

Gaetano Della Corte


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Acknowledgements

ACKNOWLEDGEMENTS

This dissertation is the result of many peoples work who, directly or indirectly, have strongly

influenced my professional and academic life. Firstly, I really would like to thank my advisor, Dr Tim

Sullivan from Buro Happold for his patience and advice. I am pretty sure this work would have not

been possible without his help. His encouraging words were a constant motivation during all the

research process. I also would like to thank my second advisor, Prof. Gaetano Della Corte from the

University of Naples, Italy. His knowledge and corresponding advice in steel design and modelling

were invaluable. I also want to thank Prof. Robert Englekirk from the Englekirk Partners Inc., whose

advice and help at the early stage of this research were really helpful. A special acknowledgment for

the academic and administrative staffs in ROSE School, particularly to Dr Calvi, Dr Pinho, Saverio

and Sandra.

This work was also possible thanks to my wife, Rebeca. Thanks for your sacrifice and for being with

me in the hard moments. Thanks also to my family in Mexico, who gave support during my studies

regardless of the distance. I would like to give a special mention to my mentor, Dr Jara Guerrero from

the University of Michoacan, Mexico. Thanks to him I was able to hear about the DBD methods and

the MEEES Programme. I express my gratitude to the Erasmus Mundus Consortium that provided the

funding for my studies and research in Europe. Finally, thanks to God for giving me this opportunity.

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Index

TABLE OF CONTENTS

Page

ABSTRACT ............................................................................................................................................i

ACKNOWLEDGEMENTS....................................................................................................................ii

TABLE OF CONTENTS ......................................................................................................................iii

LIST OF FIGURES ................................................................................................................................v

LIST OF TABLES................................................................................................................................vii

1. INTRODUCTION.............................................................................................................................1

1.1 Generalities of frame-wall structures.........................................................................................1

1.2 Fundamentals of direct displacement-based design...................................................................2

1.3 Current means of performing DBD of frame wall-structures ....................................................6

1.3.1 Introduction......................................................................................................................6

1.3.2 Assignment of strength proportions to establish the wall inflection point.......................6

1.3.3 Yield deformation of walls and frames............................................................................8

1.3.4 Design displacement profile.............................................................................................8

1.3.5 Design ductility values, effective period and equivalent viscous damping. ....................9

1.4 Application of the existing frame-wall design procedure to structures with steel frames and RC

walls..................................................................................................................................................12

1.4.1 Yield deformation of steel frames..................................................................................12

1.4.2 Equivalent viscous damping in steel frames ..................................................................15

2. DEVELOPMENT OF AN APPROPRIATE EXPRESSION FOR THE YIELD DRIFT OF STEEL

FRAMES ...17

2.1 Introduction..............................................................................................................................17

2.2 Current methods for estimating the yield drift in steel frames.................................................17

2.3 Calibration of the proposed expression to estimate the yield drift...........................................182.4 Factors affecting the accuracy of the proposed expression......................................................20

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Index

3. PROPOSED DBD METHOD FOR STEEL FRAME-RC WALL STRUCTURES .......................24

4. VERIFICATION OF THE DBD METHODOLOGY.....................................................................27

4.1 Selection of case study structures ............................................................................................27

4.2 Design criteria and main assumptions made in the design ......................................................28

4.3 Summary of case study design results .....................................................................................30

4.4 Non-linear time history analysis verification procedure..........................................................33

4.5 Results of analysis....................................................................................................................35

5. SUMMARY AND CONCLUSIONS ..............................................................................................43

REFERENCES .....................................................................................................................................45

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Index

Figure 4.5. Average lateral displacements (left) and average recorded drifts (right) compared with

target displacements and drifts for the 4 storey structure.................................................................38

Figure 4.6. Maximum lateral displacements (left) and maximum recorded drifts (right) for the 8 storey

structure......................................................................................................................................39

Figure 4.7. Average recorded drifts (left) and average lateral displacements (right) compared with

target drifts and displacements for the 8 storey structure.................................................................39

Figure 4.8. Maximum lateral displacements (left) and maximum recorded drifts (right) for the 12

storey structure............................................................................................................................40


target drifts and displacements for the 12 storey structure...............................................................40









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Index

LIST OF TABLES

Page

Table 1.1. Z/I values for some AISC W shapes........................................................................15

Table 2.1. Accuracy of Equation 2.2 vs Gupta and Krawinkler expression.............................20

Table 4.1. Characteristics of frame-wall structures ..................................................................30

Table 4.2. Intermediate design results for frame-wall structures..............................................32

Table 4.3. Final design strengths for frame-wall buildings (in kN-m) .....................................33

Table 4.4. First mode elastic viscous damping values for time-history analysis......................35

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Chapter 1. Introduction

1.INTRODUCTIONDuring the last decade, seismic design of structures has experienced a revaluation due to the

evolution of performance-based design methodologies and the encouraging analytical results

given by time-history inelastic analysis. Most of the research has been conducted, however,

toward the development and verification of design methods for reinforced concrete (RC), steel

moment resisting frames or RC structural walls, whereas less research effort had been

directed to the analysis of more complex systems such as combined frame-wall structures.

Among the several performance-based seismic design methodologies recently developed,

direct displacement-based design (DBD) has demonstrated to be a rational and effective

technique to control structural displacements and thus structural damage. Based on the

concepts of DBD, Sullivan [2005] has developed an innovative seismic design methodology

for RC frame-wall buildings, regular both in plan and elevation.

The methodology has been originally developed for RC frame-wall systems. Nevertheless,due to the versatility of the method its recommendations should also be applicable to steel

frame-RC wall structures. The main scope of this work is to verify the applicability and

effectiveness of the new methodology in terms of control of the storey drifts and maximum

storey displacements of buildings with steel frame-RC wall.

1.1 Generalities of frame-wall structuresFrame-wall systems (also called hybrid or dual systems) are an attractive solution as

earthquake resisting structures which combine the structural advantages of frames and walls.

During a seismic attack, frames usually restrain deformation in the upper storeys of the

building and possess a large capacity of deformation. Since frames are highly redundant, they

can act as a second line of defence in a very strong earthquake in case that walls lose a

significant part of their strength and stiffness. On the other hand, walls provide high stiffness

to the building, being then suitably to control displacements and drifts in the lower levels of

the building. Additionally, due to the intrinsic characteristics of functionality and service,

layouts of buildings are usually required to include walls to form stair wells and lift shafts,

being convenient to use them also as earthquake resistant members.

When acting under seismic attack, an isolated frame system will typically behave in a shear

mode with a concave shape [Pettinga and Priestley, 2006], whereas an isolated wall deforms

as a vertical cantilever with a convex shape (Figure 1.1). When frames and walls are coupled,

the lateral displacements in both structural systems is similar [Paulay and Priestley, 1992].

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Convex

shape

Deformed shapeof isolated walls

Deformed shapeof isolated frames

Concave

shape

Figure 1.1. Behaviour of frame-wall structures

As such, the structural system composed of frames and walls must resist and share the seismic

lateral loads and, as a consequence, seismic overturning moments. Paulay and Priestley

[1992] found that as the stiffness of the wall increases, the contribution of the walls to resist

the overturning moment increases. Furthermore, they found that beyond the midheight of the

building the contribution of the walls to resist moment is negligible, and that this contribution

depends on the flexibility of the walls. In 1998, Priestley and Kowalsky demonstrated that

stiffness and flexural strength are not independent, and instead stiffness is directly

proportional to strength. Therefore, as the wall is stiffer, its capacity to resist moment

increases.

One of the main advantages provided by dual systems is, therefore, that walls give enough

lateral stiffness to control displacements and give designer some freedom in the assignment of

the frame shear, and can be used to resist most of the lateral load induced by the ground

motion to the building. The two systems interact to give an approximate linear displacement

shape. As a consequence, sections and amount of steel in beams and columns of dual systems

can be smaller compared with those of moment frame buildings, which can represent

important savings in economic terms.

1.2 Fundamentals of direct displacement-based designThe objective of this section is to establish the fundamentals of the Direct Displacement-based

seismic design (DBD) of frame buildings. Some additional particularities concerning its

application to frame-wall buildings will be discussed in detail in subsequent sections.

The Direct DBD procedure has been developed over the last decade [Priestley, 2003] in

recognition of the deficiencies of current force-based seismic approaches. This alternative

method utilises the substitute structure approach developed by Shibata and Sozen [1976] and

characterises the performance of the structure by a single-degree-of-freedom (SDOF) system

at its maximum response. The fundamentals of the method can be seen graphically in Figure

1.2 and the design process is briefly described in the next paragraphs.

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The building to be designed is represented by an equivalent SDOF system with effective mass

me, and effective height he (Figure 1.2a), and by a secant stiffness Ke, at the maximum

displacement d (Figure 1.2b). The maximum or design displacement d can be set by the

designer and is commonly defined by setting displaced shape to a design drift d, chosen to

ensure acceptable levels of displacement for a given risk event. As it will be seen in

subsections 1.3.3 and 1.3.4, the design displacement shape for frame-wall buildings can be

calculated by using a series of equations in a straightforward manner.

Once the displaced profile of the structure at the maximum displacement is known, then the

design displacement dat the effective height heof the equivalent SDOF can be defined using

the next expression:

( ) ( ) = =

=i i

iiiid mm1 1

2

N N

)

(1.1)

WhereNis the total number of storeys of the building, miis the mass of each storey, and iis

the design displacement for the storey i.

The effective height he, is also a function of the displaced shape of the masses at maximum

response, in addition to the storey height hi, and is calculated according to the Equation 1.2.

( ) ( = =

=i i

iiiiie mhmh1 1

N N

(1.2)

Where all the terms involved in the equation have been already described in previous

paragraphs. To calculate the effective mass of the system me, the participation of the

fundamental mode of vibration at maximum response is considered. As such, the effective

mass can be estimated with the Equation 1.3.

( )=

=i

diie mm1

N

(1.3)

Since in the DBD methodology the actual response of the structure is predominately non-

linear, the effect of ductility in the system is through an equivalent viscous dampingcoefficient SDOF, which includes both elastic and hysteretic damping components, i.e. the

total amount of energy absorbed during the seismic response. Recent research [Grant et al,

2005; Blandon and Priestley, 2006] has found, however, that the amount of equivalent viscous

damping is also dependant on the effective period of the substitute structure, Te. These two

factors affecting the level of equivalent viscous damping will be discussed in detail in the

subsection 1.3.5. Presently, it is sufficient to recognise that for the same level of ductility

demand, the level of equivalent damping assigned to a steel frame building possessing

compact sections is higher than the level of equivalent damping assigned to a RC frame

building (Figure 1.2c), as a consequence of the larger capacity to dissipate energy of the steel

members during the cyclic response.

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As such, the ductility demand of the substitute structure can be calculated with the Equation

1.4.

y

d

=

(1.4)

Where is the displacement ductility demand of the equivalent SDOF system, and dand y

are the maximum displacement and yield displacement of the equivalent SDOF system,

respectively (see Figure 1.2b). The value yis a function of the yield curvature y, which is

dependent on the section geometry and material properties of the component members of the

original building. Based on the type of structural member to be analysed, Priestley [2003] has

proposed the use of the appropriate version of the Equation 1.5 to calculate the approximate

yield curvature y.

Circular concrete columns Dyy 2.25= (1.5a)

Rectangular concrete columnscyy h2.10= (1.5b)

Rectangular concrete wallswyy l2.00= (1.5c)

Symmetrical steel sectionssyy h2.10= (1.5d)

Flanged concrete beamsbyy h 1.70= (1.5e)

Where yis the yield strain of the flexural reinforcement,D, hc, lw, hs, and hbare the depths of

circular column, rectangular column, rectangular wall, steel shape and flanged concrete beam

section, respectively. Paulay [2002] has also found similar equations to approximate the yield

curvature of members. Once the yield curvature is known, the yield displacement (or drift) of

RC and steel frames can be calculated with relatively simple expressions.

Having established the design displacement and mass of the equivalent SDOF system and the

corresponding damping expected, for the expected displacement ductility demand, the effectiveperiod Te can be read from a displacement spectrum appropriate for the level of equivalent

viscous damping (Figure 1.2d). The period Tof a SDOF system can be defined in terms of its

massM, and stiffnessK, for the next relation:

K

MT 2= (1.6)

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(c) Equivalent damping vs. ductility

Period (sec)

Displacement(m)

Te

d

5%

10%

15%

20%

30%

Ductility

Damping(%)

(d) Design displacement spectra

0 1 32 54 6

0

10

20

30

40

50

60 Elasto-plastic

Steel frame

RC frame

Unbonded prestressing

(a) Equivalent SDOF system

Fd

d

rK

(b) Effective stiffness, K e

i

iKKe

Fn

y

F

em

he

Displacement

Force

RC walls Frames

4321

d

Vb

Figure 1.2. Fundamentals of Direct Displacement Based Design [from Priestley, 2003]

Finally, the design lateral force Fd, equivalent to the design base shear Vb, is given by

Equation 1.8.

debd KVF == (1.8)

The shear force Vbcan be distributed over the height of the building as a function of the mass

mi, and the design displacement iof each storey. Thus, the corresponding force for the storey

ican be defined by

( ) ( )=

=i

iiiibi mmVF1

N

(1.9)

Where all the terms have been defined in preceding paragraphs. This expression is similar to

that proposed in a force-based approach. However, the difference is that Equation 1.9 utilises

an inelastic displacement profile, rather than a displacement profile proportional to the

building height as is done by current force-based approaches.

Due to its versatility, the procedure can be applied to any type of structural system. However,

to be used in frame-wall structures some additional considerations must be done. Theseconsiderations are discussed in the following sections.

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1.3 Current means of performing DBD of frame wall-structures1.3.1 IntroductionAn innovative DBD procedure of dual systems was recently developed by Sullivan [2005],

and has proved its effectiveness in terms of controlling the maximum displacements and inter-

storey drifts in regular RC frame-wall structures. The method starts by assigning strength

proportions to frames and walls, and by calculating the inflection height of the wall at which

the moment and curvature are zero. Secondly, yield deformation of walls and frames are

established to calculate the design displacement profile of the building. After this, the

characteristics of the substitute structure are calculated and the design procedure can be

carried out. These steps are described thoroughly in the following subsections.

1.3.2 Assignment of strength proportions to establish the wall inflection pointIn order to develop an accurate SDOF representation of the frame-wall structure, strength

proportions are assigned at the start of the procedure. This involves setting the proportion of

base shear -or overturning resistance- offered by frames and walls, in addition to the relative

strength distribution of yielding elements within the frames. For this purpose, a weak beam-

strong column approach is adopted in such a way that yielding is concentrated at the ends of

beams and at the base of ground storey columns of frames. Hence, any inelastic activity in the

upper storey columns in inhibited. Having established the strength proportions, the shear and

moment profile in the walls enable the calculation of the inflection height, from which the

design displacement profile will be obtained.

The walls are expected to remain elastic in the upper floors through application of suitable of

suitable capacity design guidelines, and a plastic hinge action is expected to take place only at

the base of the walls. In this way, the shear force acting on the walls is dependent on the

strength of the walls, whereas the frame shear is dependent on the strength of the beams.

Since the frame storey shear is dependent only on the strengths of the beams up the building

height, the wall shears can obtained as the difference between the total shear and the frame

shear:

b

framei,

b

totali,

b

walli,

VV

V

V

V=

V (1.10)

where Vb is the total base shear, Vi,wall is the wall shear at level i, Vi,total is the total shear at

level i, and Vi,frameis the frame shear at level i.

The next step is to calculate the inflection height at the wall. For this purpose, a triangular

distribution of the fundamental mode of inertia forces up the height of the building is

assumed. With this approximation, the total storey shear can be obtained as a function of the

total base shear:

)(nnVb

totali,

1

1

+

=)-(iiV 1

(1.11)

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of overturning moment will be used as shown in subsection 1.3.5 in the process for

calculation of the system damping.

1.3.3 Yield deformation of walls and framesThe wall yield curvature and displacement at yield are important parameters for the

development of the design displacement profile because the walls tend to control the response

of the structure. The frame yield displacement (or yield storey drift) is also important to the

design process as it is used to provide an indication of the energy absorbed during the

hysteretic response of the frame.

The yield curvature of the walls can be obtained with the Equation 1.5c, repeated here for

didactical purposes.

wyyWall l 2.00= (1.5c)

Where yis the yield strain of the flexural reinforcement and lwis the wall length.

The design displacement profile of the structure at yield of the wall y, can then be

established using the wall yield curvature, inflection height hinf, and height at the storey of

interest hi, in accordance with the appropriate version of Equation 1.13.

62

infyWalliinfyWall

iy

hhh =

2

, for hi hinf (1.13a)

inf

iyWalliyWall

iyh

hh

62

=

32

, for hi < hinf (1.13b)

The yield drift of a RC frame, yFrame, is obtained in accordance with Equation 1.14 [Priestley,

2003].

b

yb

yFrameh

l =

0.5 (1.14)

Where lb is the average beam length, y is the yielding strain of beam longitudinal

reinforcement and hb is the depth of the beams at the level of interest. This value is used to

calculate the ductility demand and equivalent viscous damping of the frames. It is noteworthy

that the latter equation is only applicable to RC frame structures.

1.3.4 Design displacement profileThe design displacement profile can be calculated with the Equation 1.15.

i

yWall

diyi h

hinf

+= 2

(1.15)

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Where iis the design displacement for level i, iyis the displacement of level iat yielding of

the walls, d is the design storey drift, yWall is the yield curvature of the walls, hinf is the

inflection height, and hiis the height at level i.

Despite of the fact that the first mode shape tends to control the lateral displacement in abuilding, higher modes can have an important effect in increasing the displacement in tall

structures. The design storey drift can be initially taken as the code limit for non-structural

damage. Nevertheless, by considering the results in Sullivan [2005], the code limit drift is

reduced to allow for higher mode effects in accordance with Equation 1.16.

itlim,d

Total

Frameitlim,dd

M

MN

+

= 0.25

100

5)(1 (1.16)

Where N is the total number of stories, MFrame is the overturning resistance offered by theframe and MTotal is the total overturning resistance of the structure. The ratio of frame

overturning to total overturning moment can be calculated in terms of the base shear as was

discussed in previous sections. The design drift d, may be reduced further if it is found that

inelastic demands on walls and/or frames are likely to be excessive. Notice that Equation 1.16

effectively acts as a reduction factor for structures having more than five storeys.

Once that i is found, the design displacement, effective height and effective mass which

characterise the substitute structure can be calculated with Equations 1.1, 1.2 and 1.3,

respectively.

1.3.5 Design ductility values, effective period and equivalent viscous damping.An important issue in the DBD procedure is the calculation of the equivalent viscous

damping. This is mainly a function of the ductility value defined with Equation 2.4, and,

according to recent findings by Grant et al [2005] and Blandon and Priestley [2006], the

effective period of the substitute structure.

In order to use the equivalent viscous damping approach, the ductility demand on the walls

should be calculated using the displacement at the effective height. Therefore, the wall

ductility displacement demand Wall, can be defined as the design displacement divided by the

yield displacement of the walls at the effective height, according to Equation 1.17.

yhe,

dWall

=

(1.17)

Where d is the design displacement calculated with the Equation 1.1, and he,y is the yield

displacement of the wall at the effective height, obtained substituting the effective height into

the appropriate version of Equation 1.13. Note that this expression is in fact Equation 1.4,

now adapted to calculate the wall displacement ductility demand.

The displacement ductility demand on the frame at each level up the height of the buildingcan be estimated using the storey drifts:

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yFrameii

-iii,Frame

hh

1

1

1

=

(1.18)

Where i, i-1, hiand hi-1are the displacements and heights at levels iand i-1, respectively,Frame,iis the frame ductility at level i, and yFrameis the yield drift of the frame calculated with

the Equation 1.14. If beams of equal strength are used up the height of the structure, the

ductility defined by Equation 1.18 for each storey can be averaged to give the frame

displacement ductility demand.

The system ductility demand is found by taking the average of the frame and wall ductilities

weighted by their overturning resistance, as shown in Equation 1.19.

FrameWall

FrameFrameWallWallsys

MM

+

=MM +

(1.19)

WhereMwallandMFrameare the wall and frame overturning resistance, and Walland Frameare

the ductility displacement demands for the wall and frame, respectively.

Although the wall ductility demand given by Equation 1.17 is appropriate for estimation of

the equivalent viscous damping, it is not an accurate measure of the inelastic deformation that

the walls will undergo. A more appropriate parameter is the wall curvature ductility Wall,

which can be obtained in accordance with Equation 1.20.

+=2

11

infyWall

d

yWallp

Wall

h

L

(1.20)

Where Lp is the wall plastic hinge length, d is the design storey drift, yWall is the yield

curvature of the walls, hinfis the inflection height.

The wall plastic hinge length to be used within Equation 1.21 is taken as the minimum of:

infbyp hdfL 0.0540.022 +=

hLL 0.030.2

(1.21a)

infwp += (1.21b)

Where fy is the yield stress expressed in MPa, db the diameter of the longitudinal

reinforcement in the wall (in millimetres), Lw is the wall length and hinf is the inflection

height. These two equations have been adapted from [Priestley, 2003] with the inflection

height substituting the total height.

The curvature ductility capacity of a RC wall will depend on the strain limits selected for the

concrete in compression c, and the longitudinal reinforcement in tension s. For reasonably

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conservative values of c=0.018 and s=0.06, Priestley and Kowalsky [1998] found that the

ultimate curvature of a reinforced concrete wall is well represented by:

wu L

=0.072

(1.22)

Where u is the ultimate curvature and Lw is the wall length. Combining Equation 1.5c and

Equation 1.22, it is found that the curvature ductility capacity is approximately equal to

0.036/y.

If the checks on ductility indicate that the inelastic deformation associated with the design

drift will be excessive, then the design drift must be reduced and the design displacement

profile recomputed as discussed in the previous subsections. If the ductility demands are

sustainable, then the next step in the design procedure is to compute equivalent viscous

damping values.

Equivalent viscous damping has usually been computed as the sum of elastic and hysteretic

components [Priestley, 2003]. For RC structures it is commonly assumed that an elastic

damping component of 5% critical damping is reasonable. Priestley and Grant [2006]

however, have shown that care must be taken to ensure that 5% damping is actually provided

taking account of the changing stiffness characteristics of the structure.

Recent work by Grant et al [2005], recommends that the hysteretic component of the

equivalent viscous damping be computed as a function of the effective period. As this

effective period is unknown at the start of the design process, a trial value can be used and aniterative design process adopted. A reasonable estimate for the trial value of the effective

period Te,trial, can be obtained from Equation 1.23.

systrial,eT 6

=N

(1.23)

WhereNis the total number of storeys and sysis the system ductility. Equation 1.23 is similar

in form to a code-based equation that uses the height -or number of storeys- to estimate the

initial period. The ductility term accounts for the difference between the initial and effective

period, neglecting the effect of the strain hardening of steel. Given the nature of Equation

1.23, trial effective period values may be different that the final effective period; however, by

using such a trial value, it will be found that convergence is attained within two iterations as

maximum.

With the trial effective period and expected ductility values established, the frame and wall

components of equivalent viscous damping can now be calculated. In their research, Grant et

al [2005] have also proposed a series of calibrated equations to calculate the equivalent

viscous damping for different hysteretic behaviours. For design purposes, the frame and wall

equivalent damping can be obtained by using Equations 1.25 and 1.26, respectively

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++

+=

3.2500.527 0.761)(

11

1124.95

trial,eFrame

FrameT

(1.25)

++

+=

3.6070.588 0.848)(

11

1118.35

trial,eWall

WallT

(1.26)

All the above parameters have been defined in previous paragraphs. Once the wall and frame

damping components have been obtained, the damping of the equivalent SDOF system can be

evaluated with:

Frame,T/OWall

FrameFrame,T/OWallWall

SDOFMM +

MM +=

(1.27)

At this point of the design process, all the characteristics of the substitute structure have been

defined. The next step is proceed with the development of the displacement spectrum at the

design level damping of the equivalent SDOF given by Equation 2.27, and use it to read (or

interpolate) the required effective period as shown in Figure 2.3. The effective period read

from the displacement spectrum is then compared with the trial effective period value

(Equation 1.23). If the periods do not match, then the period obtained from the displacement

spectrum replaces the trial period and the design step is repeated. When trial period finally

matches the period read from the displacement spectrum, the effective stiffness and design

base shear can be calculated by using Equations 1.7 and 1.8. Finally, having calculated thedesign base shear, the seismic forces acting at each storey can be calculated and distributed on

the height of the building with the help of Equation 1.9.

1.4 Application of the existing frame-wall design procedure to structures with steelframes and RC walls

Up to this point, the proposed methodology has been focused only on RC frame-wall systems.

Nevertheless, its principles can perfectly be applied to other type of structural component

materials, as steel frame-RC wall buildings. Some modifications should be done, however,

with regard to the calculation of the yield drift of the steel frames, as well as the estimation of

the equivalent viscous damping of the frames. These issues are discussed in the nextparagraphs.

1.4.1 Yield deformation of steel framesIn order to calculate the yield curvature of a steel I-shape, Paulay [2002] has proposed a

similar expression to that proposed by Priestley [2003] (see Equation 1.5d), as a function of

the depth of the frame beam:

byy d2.30= (1.28)

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Where yis the nominal yield curvature of the steel I-beam,yis the yield strain of the steel

and dbis the beam depth. For seismic design purposes, the nominal curvature is defined using

a bi-linear relation of moment-curvature response as shown in Figure 1.3. Nonetheless,

designers must be aware since Sullivan [2005] has found that, even for beams with exactly the

same depth but different weight, calculation of yield drift with Equation 1.28 could only

provide approximate results. Additionally, he also notes that the strength of typical steel

members is changed by changing the section depth. This differs considerably from RC

structures, where the member strength can be controlled by the designer just by modifying the

amount of longitudinal reinforcing steel, without changing the cross section geometry.

Figure 1.3. Definition of nominal yield curvature using a bi-linear representation of response

As a consequence, in order to use depth-dependent yield curvature (or yield displacement)expressions in displacement-based design, an iterative design approach should be adopted.

Since iterative processes are time consuming, Sullivan [2005] has developed an alternative

expression that allows the calculation of the yield curvature in a straightforward way. Based

in the approach showed in Figure 1.3, it is suggested that the yield curvature be calculated by:

EI

M

EIM

M ny

y

n

y ==M

(1.29)

Where y is the nominal yield curvature, E is the modulus of elasticity of the steel,Mnis thenominal strength,Myis the first-yield strength, andIis the moment of inertia of the shape.

For seismic design, the nominal strength of a steel frame can be calculated as Mn=Zfy, where

Zis the plastic modulus of the section andfyis the yield strength of the steel. Substituting this

into Equation 1.29, the yield curvature can be defined as:

yn

yI

Z

EI ==

Zf (1.30)

Where all the parameters have been defined in precedent paragraphs.

13


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An analysis of Figure 1.4 explains the reason why the yield curvature is expressed in terms of

the plastic modulus and moment inertia of the shape. Figure 1.4 is a plot of the plastic section

modulus versus the moment of inertia for different groups of standard AISC [2001] W-shape

sections. It is evident that, for a group of sections with specific depth, the relationship

between the plastic modulus and the moment of inertia can be represented by linear

trendlines.

0

200

400

600

800

1000

1200

1400

0 5000 10000 15000 20000

Moment of Inertia, I (in4)

PlasticM

odulus,Z(in

3)

W14

W18

W16

W21

W24

W30

W27

W33

Figure 1.4. Relation between plastic modulus and moment of inertia for some groups of AISC [2001]

standard W-beams [from Sullivan, 2005]

As can be seen, the relationship between Z and I for a given set of beam shapes is linear;

therefore, from Equation 1.30 it can be inferred that the nominal yield curvature for each

group of sections is rather constant. Notice how these values of nominal yield curvature are

constant despite the changes in section depth. Table 1.1 presents the trendline values for Z/I

obtained for the different groups of AISC [2001] W-beam sections.

In order to optimise the design in terms of amount of steel required to build the structure,

some experience in design of steel frames could be required to propose, at the start of the

process, the beam and column groups to be used in the design. This experience, however, can

be easily achieved once the designer is familiar with the method. Even more, the beam shape

group can be initially calculated based on a control of maximum gravitational deflections,

checking subsequently whether or not the selected shape group satisfies the earthquake

demands. For design purposes, therefore, the engineer should select a beam group at the start

of the design procedure. The appropriate value ofZ/Ican then be multiplied by the yield strain

of the steel to give the nominal yield curvature of the frame element being considered

(Equation 1.30).

For beam and column groups not listed in Table 1.1, the designer can obtain their own values

forZ/I using the property tables provided by AISC [2001], or the local steel supplier.

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Table 1.1. Z/I values for some AISC W shapes.

W-shape

group

Z/I (in-1

) Z/I (m-1

)

W12 0.157 6.19

W14 0.140 5.52

W16 0.136 5.36

W18 0.119 4.67

W21 0.103 4.04

W24 0.087 3.44

W27 0.077 3.04

W30 0.066 2.58

W33 0.072 2.82

W36 0.060 2.35

The yield drift of a steel frame, ySteelFrame, can then be estimated using Equation 1.31

[Sullivan, 2005], where Lb is the beam length and hcol is the column height. y,beamand y,col

are the beam and column nominal yield curvature respectively, obtained using Equation 1.30

and theZ/I values appropriate for the beam and column group being considered.

6

coly,colby,beam

eySteelFram

hL

+= (1.31)

By using the full yield curvature of the columns in Equation 1.31 in addition to the beam and

column lengths between centrelines, some allowance is made for deformations in the panel

zone and shear deformations along the member lengths. Calibration of Equation 1.31 has not

been done against full-size experimental tests or analytical results, and this will be undertaken

in detail in Chapter 2.

Finally, the beam and column sections that will provide the necessary strength can then be

selected from the steel section groups chosen at the start of the design procedure. One

significant advantage of the method proposed by Sullivan [2005], is that exact section

dimensions do not need to be known at the start of the design process, and therefore an

iterative design is avoided.

1.4.2 Equivalent viscous damping in steel framesIn order to obtain the equivalent viscous damping of the steel frames, the ductility

displacement demand on the frames can be estimated using the design displacement profile

and the yield displacement calculated with Equation 1.31.

For a structural member with bi-linear hysteretic behaviour, the equivalent viscous damping

can be estimated with enough accuracy by Equation 1.32 [Grant et al, 2005], which is adopted

in the method for use with steel frames.

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Chapter 2. Development of an Appropriate Expression for the Yield Drift of Steel Frames

2.DEVELOPMENT OF AN APPROPRIATE EXPRESSIONFOR THE YIELD DRIFT OF STEEL FRAMES

2.1 IntroductionAs was discussed in subsection1.4.1, the proposed expression to calculate the yield drift of

steel frames (Equation 1.31) has not been calibrated against other tests or analytical findings.

In order to be used in the proposed DBD design methodology, a sufficiently accurate

calculation of the yield drift and displacement is essential, since the ductility demand, the

equivalent viscous damping of the steel frame and consequently the design base shear are

dependant on this value. The objective of this chapter, therefore, is the development of a

calibrated expression appropriate for design purposes.

2.2 Current methods for estimating the yield drift in steel framesIn recent years, the calculation of the drift demand of a moment resisting steel frame hasreceived particular attention by the scientific community. Particularly, after the widespread

destruction generated in steel structures by the Northdridge earthquake, many research has

been focused to the estimation of seismic drift demands [FEMA, 1997; Gupta and

Krawinkler, 2000a].

A more recent research has studied the relationship between story drift demand and element

plastic deformation demands [Gupta and Krawinkler, 2002]. By using a series of code

compliant regular moment resisting frames, the authors propose a straightforward

methodology to estimate the total storey drift demand based on the storey yield drift

calculated with the weakest element in a connection (beams and panel zones). To estimate thedemand, the method utilises the storey geometry and member properties that consider the

beam and panel zone plastic deformation demands. By using a series of expressions that

consider explicitly the drift components of beams b, columns c, and panel zones pz, to the

total yield drift of the building, the total yield drift of a regular steel frame can be estimated

with enough accuracy y using Equation 2.1.

h

pzcb

K&y,G

+ += (2.1)

One of the main assumptions made in the calculation of the yield drift is that the columnbehaves elastically. The storey yield drift is only associated with yielding in beams and panel

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zones, so that the columns are protected against plastic hinging. Additionally, the effect of

gravity load in yielding of beams and panel zones is neglected, which is considered as

acceptable in high seismic areas. Another important simplification is that a midpoint

inflection points is considered in the model used to develop their equation, which is a

reasonable assumption provided that structure is regular in storey stiffness. This assumption,

however, is not always valid particularly for the lowermost storey of a building.

In the model used to develop the procedure, beam plastification is considered to occur at the

column face. This is not always the case because new connections after the Northdridge

earthquake have been developed in order to locate and force the plastic hinge to form far from

the column [FEMA 350, 2000]. Fortunately, connections such as those formed with reduced

beam sections (RBS) or connections with cover plates can be considered in the expression

proposed by Gupta and Krawinkler by amplifying the beam plastic moment. Furthermore, it is

also possible to consider the case when different beam shapes are framing in the connection

which can be useful when height limitations exist.

Since the expression considers as weak member only beams and panel zones, a column

hinging case is seen as improbable. However, it is evident that the slab can contribute

significantly to the beam strength. If the bending strength of the beam is underestimated in the

design, undesirable plastic hinging can occur in columns. In this case, the expression ignores

the influence of the concrete slab. Other aspects as partitions walls or secondary girders are

also disregarded.

2.3 Calibration of the proposed expression to estimate the yield driftIn order to obtain an adequate expression to calculate the yield drift of steel frame buildings,

the calibration of Equation 1.31 should be ideally made using experimental results of full-size

steel frame tests. Nonetheless, most of current research on steel frames is done with scaled

models which use small steel shapes which are not included in steel tables. Even more, it was

found that the scientific community tends to direct its attention toward other structural

alternatives, such as braced systems. Therefore, due to the lack of reliable results, in this work

the calibration of Equation 1.31 was done using analytical results of a series of push-over

curves, obtained for the set of model buildings of the SAC Joint Venture programme [FEMA

355C, 2000]. The structures possess 3, 9 and 20 storeys and are located in three different

seismic regions of the United States, so that 9 push-over curves are available. The

characteristics of the buildings as well as the assumptions made for modelling can be

consulted elsewhere [FEMA 355C, 2000].

The push-over curves were idealised by a bi-linear relation representative of the initial elastic

stiffness and the nominal strength of the structure. The yield displacement was then estimated

graphically as the point where the elastic stiffness intersected the nominal strength. An

example of this calculation is presented in Figure 2.1, where the push-over curve and the

corresponding bi-linear idealisation are plotted. In this case, the push-over curve corresponds

to a 3-storey steel frame located in Seattle, USA [FEMA 355C, 2000].

Since a beam yield mechanism is selected in the proposed methodology, the portion of yielddrift of the beam that contributes to the total yield drift of the frame is known. In contrast,

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other contributions which are not actually known can be included in the contribution of the

column to the total yield drift. Due to this fact, any modification should be done to the column

component of the Equation 1.31 since the contribution of the column is not known at this

stage.

Based on the column and beam sections reported in the study, the yield drift of each steel

frame was estimated by using Equation 1.31. It was found that results given by Equation 1.31

were slightly larger compared with those from the push-over curves. Hence, to match the

results between the yield drift from the push-over curves and the yield drift from Equation

1.31, the contribution of the column to the total yield drift was modified by a factor of 0.9.

After this modification, the form of the expression to calculate the total yield drift of a steel

frame is that given by Equation 2.2.

6

coly,colby,beam

eySteelFram

0.9 hL

+=

(2.2)

It is reminded that the yield curvature of beam and column should be calculated with the

Equation 1.30 and the appropriateZ/I values according to the Table 1.1.

Due to the format of the Equation 2.2, it is important to note that the calibration could depend

on the code used for design. Since the yield drift of the column is dependent on the nominal

strength acting on the column, this strength is then affected by the capacity design approach

adopted for design. Furthermore, the factors used for capacity-based design are different from

one code to another and, as a consequence, different proportions of yield curvature could be

appropriate depending on the code used for design. Therefore, the reducing factor affecting

the contribution of the column to the yield drift could be different to the 0.9 proposed in this

work.

Figure 2.1. Example of calculation of the yield drift of steel frames to calibrate Equation 2.2 [adapted

from FEMA 355C, 2000]

Building yield drift, y

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Roof drift angle

Normalisedbase

shearV/N

0.

0.

0.

0.

0

4

3

2

1

Push-over curve

Bi-linear idealisation

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2.4 Factors affecting the accuracy of the proposed expressionIn Section 2.2 the proposed expression to calculate the yield drift of a steel frame has been

calibrated considering only the push-over curves of nine steel frames, and ignoring therefore

possible changes both in the building dimensions and in the beam and column members. The

accuracy of Equation 2.2 in relation with other expressions which are considered to give

results closer to the actual values is therefore studied in this section. Additional factors that

could influence the results of the expression, such as frame geometry, depth of beams and

columns, and moment of inertia of steel members, are also studied.

For comparative purposes and with the aim of evaluating the accuracy of Equation 2.2, the

yield drift of three 10-storey steel frames studied by Gupta and Krawinkler [2002] is

calculated with Equation 2.2. The frames possess constant storey height of 3.6 m and bay

length of 9.0 m. Steel Grade 50 is used for the columns, and Grade 36 is used for the beams.

The results of such comparison are shown in Table 2.1. Based on the results of the Table 2.1,

it can be concluded that results of yield drift given by Equation 2.2 are not far from those

given by Equation 2.1 proposed by Gupta and Krawinkler.

Table 2.1. Accuracy of Equation 2.2 vs Gupta and Krawinkler expression.

BuildingBeams

(inlb/ft)

Columns

(inlb/ft)y,G&K y,Equation 2.1

1 W36150 W14500 0.0093 0.0095

2 W3099 W14283 0.0101 0.0099

3 W2784 W14257 0.0106 0.0108

It is recognised that the expressions suggested by Gupta and Krawinkler provide a result

closer to the actual value of the yield drift and, as such, it is also recognised the Equation 2.2

gives only an approximate estimation of the yield drift. To investigate the factors that mainly

influence the accuracy of Equation 2.2, some geometric parameters as span length to storey

height ratio, beam to column depths ratio, and beam to column inertia ratio are modified in

the set of steel frames studied in Table 2.1. The results are discussed in the subsequent

paragraphs.

Figure 2.2 presents results about the accuracy of Equation 2.2 as a function of the beam length

and the storey height (Lb/hcol) ratio. All the other geometrical parameters are kept constant in

the calculation process. The vertical axis of Figure 2.2 presents the ratio between the yield

drift obtained from Equation 2.2 and the yield drift calculated with Equation 2.1. The value of

the Lb/hcol ratio is varied from 1 to 4. In real structures considered as typical cases, this

relationship commonly varies from 2 to 3 which implies that the bay length is 2 to 3 times

greater than the storey height.

The average results of the three studied frames indicate that Equation 2.2 marginally

overestimates the yield drift as the value ofLb/hcoltends to be smaller, whereas the yield drift

is underestimated as the value Lb/hcoltends to larger values. Note also that for typical values

ofLb/hcolgoing from 2 to 3, the yield drift calculated by both equations is quite similar. As a

consequence of the small influence of the Lb/hcol relationship in the calculation of the yield

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drift, it is considered that Equation 2.2 can be used with confidence in the range of beam

length to column height ratios commonly used in actual structures.

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

0 1 2 3 4 5

L b /h col

y,Equation(2.2

)/y,G

&K

Bldg 1

Bldg 2

Bldg 3

Avg

Figure 2.2. Accuracy of the Equation 2.2 as a function of the beam length to column height ratio

The accuracy of Equation 2.2 in calculation of the yield drift of steel frames was also

evaluated as a function of the beam depth to column depth ratio db/dcol, and as a function of

the relationship between beam Ib, to column inertia Icol. For this evaluation, a series of real

steel shapes taken from the AISC tables are used, combining the section properties of possible

solutions for beams and columns shapes. In this case it is not possible to set constant all thevalues because in symmetrical I-shaped sections the value of the plastic modulus is always

directly proportional to the inertia. For steel sections which are symmetrical about their major

axis, the plastic and elastic neutral axis coincide, and therefore the inertia and plastic modulus

of an I-shape section are related by Equation 2.3.

Z

Is =2

h (2.3)

Where hsis the depth of the steel shape, and IandZare the corresponding moment of inertia

and section modulus. As a result, in this evaluation both the plastic modulus and moment of

inertia are modified. When designing a steel frame subjected to seismic loads, it is unlikely

that only the beam section be required to change without changing also the column section,

generating thus that the inertia of the columns increases or decreases proportionally to the

inertia of the beams. Based on this, it is considered that the actual relationship between beam

and column inertia is likely to be between 1 and 2.

The results of the calculations are presented in Figures 2.3 and 2.4. It was decided to vary the

value db/dcol from 1 to 3 because in actual steel buildings this ratio is commonly between 1

and 2. Analysing Figure 2.3, it can be noticed that the best match is obtained for a db/dcolratio

close to 1.3, where the average results yielded by Equations 2.1 and 2.2 are practically the

same. For ratios db/dcol larger than 1.5, the actual yield drift is underestimated, implying

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therefore that results between Equations 2.1 and 2.2 diverge significantly as the ratio db/dcolis

considerably far from a value of 1.5.

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

0.5 1 1.5 2 2.5 3

db /dcol

y,

Eq.

(2.2

)/y,G

&K

Figure 2.3. Accuracy of the Equation 2.2 as a function of the ratio between beam and column depths

Finally, in Figure 2.4, Equation 2.2 is evaluated as a function of the relationship between

beamIb, to column inertiaIcol. The average results from Figure 2.4 follow a similar tendency

to those of Figure 2.3, and indicate that Equation 2.2 provides accurate values of yield drift

when the Ib/Icol relationship is between 0.5 and 1.0, where most of the results are located.

Values ofIb/Icolfrom 0.5 to 1.0 are not atypical in actual steel frames and therefore Equation2.2 can be used with relatively good confidence between these ranges.

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

0 0.5 1 1.5 2 2.5 3 3.5

Ib /Icol

y,E

q.

(2.2

)/y,G

&K

Figure 2.4. Accuracy of the Equation 2.2 as a function of the ratio between beam and column inertia

Figures 2.3 and 2.4 show a large scatter in the calculation of the yield drift of a steel frame byusing Equation 2.2 in comparison with the expression suggested by Gupta and Krawinkler.

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Differences of around 50% are important and can have a significant impact in the design

process. The results yielded by the plots allow an insight on the main factors affecting the

accuracy of the proposed expression. Whereas the ratio of the beam length and the storey

height (Lb/hcol) seems to have little influence on the process, the beam to column depth ratio

(db/dcol) and beam to column inertia (Ib/Icol) have an important influence on the results.

Considering the limits of this investigation and the small number of structures studied,

Equation 2.2 used in conjunction with the values of Z/I included in Table 1.1 can lead to

reasonable estimates of the yield drift of moment steel frames. Factors affecting the accuracy

of Equation 2.2 as such studied in this section impact in a significant manner the results only

for cases seen as uncommon. An uncommon case, for instance, could be the use of large depth

steel beams combined with small depth columns (or vice versa) that will surely produce

unreal estimates of the yield drift. These extreme cases should be then avoided for the

designer if he or she wishes to use Equation 2.2.

It is important to recognise that the expression proposed by Gupta and Krawinkler (Equation

2.1) represents an attractive approach to estimate the yield drift of a steel moment frame. With

the proper calibration, it is thought that Equation 2.2 can also provide a good estimate of the

yield drift value of that type of structures. The small number of study cases used in the

calibration of Equation 2.2 and the use of push-over curves to assist the calibration can be

considered as a possible limitation of this research. Nevertheless, by carrying an appropriate

calibration process based on results coming from full-scale tests of steel frames, it is expected

that both equations can provide satisfactory results. As such, the designer can select the

approach that he or she prefers to estimate the yield drift of a frame, provided that the study

case is congruent with the assumptions made in the development of the selected approach.

Consequently and considering its simplicity, in this work will be used the Equation 2.2 for the

evaluation of the yield drift of steel frames.

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Chapter 3. Proposed DBD Method for Steel Frame-RC Wall Structures

3.PROPOSED DBD METHOD FOR STEEL FRAME-RC WALLSTRUCTURES

The proposed DBD method to design RC dual systems has been thoroughly discussed in

Chapter 1. With the aim of applying the current method to structures formed with steel frames

and RC walls, some additional modifications were introduced in Section 1.4. In this chapter, a

synthesis of the method is briefly presented.

The flowchart of the proposed methodology by Sullivan [2005] is presented in Figure 3.1.

The first steps in the method deals with the substitution of the original building by the

equivalent SDOF system required to implement DBD seismic design. This is achieved by

assigning strength proportions to walls and frames, and subsequently using the moment

profile in the walls to establish a displacement shape. Additionally, a steel beam group must

be selected from charts, based on the designers experience and/or maximum allowed

deflection under gravity loads. The second steps are aimed to determine the required effective

period of the equivalent SDOF system and its corresponding effective stiffness. The design

base shear is then obtained by multiplying the necessary effective stiffness by the design

displacement. Finally, the strength of individual structural elements is set taking care to

ensure that initial strength assignments are maintained.

It is noteworthy that in this work the procedure described in Figure 3.1 is carried out up to the

steps before the capacity-based design of members that will be protected against premature

failure, since the interest is mainly focused on the effectiveness of the proposed DBD

methodology in control the maximum displacements and inter-storey drifts.

The design procedure can be optimised to achieve the lowest possible design base shear byaltering the strength proportions for given storey drift and curvature ductility limits. The

lowest design base shear will occur when the design drift and the curvature ductility design

drift are both at their maximum values.

In the proposed procedure, the designer is free to assign any value of inflection height by

changing the relative strengths of the frames and walls. For a given design drift, if the

designer uses a large value of inflection height it may be that the curvature ductility capacity

of the walls is not fully utilised. Hence, the designer could opt by increasing the proportion of

overturning moment resisted by the frames to reduce the inflection height and increase the

value of the curvature ductility demand in the walls. Alternatively, if the designer uses a lowvalue of inflection height, the curvature ductility in the walls can go beyond reasonable limits.

24


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Chapter 3. Proposed DBD Method for Steel Frame-RC Wall Structures

In this case, the designer should reduce the value of design storey drift to maintain a curvature

ductility limit.

The inflection height is dependent on the frame and wall strength proportions. Based on the

results of his research, Sullivan [2005] found that the required frame strength is relativelyindependent of the number of storeys of the building. This means that, for a given value of

inflection height, the strength proportions carried by walls and frames is relatively constant,

being only slightly dependent on the total height of the structure. In order to assist in the

design procedure, Sullivan [2005] also provides a series of graphics that can be easily used to

assist in the selection of the appropriate inflection height for a given ductility curvature limit.

In his work it is also provided a plot that can be useful to select the proportion of base shear to

be carried by the frame as a function of the total base shear. The designer can then select the

inflection height of the wall and estimate the frame strength proportion associated to that

inflection height, or set the frame strength proportion and calculate the inflection height.

It should be recognised that, in general, it is difficult to maximise design drift and curvature

ductility limit both at the same time. Some restrictions as maximum reinforcement contents or

dimensional limitations can influence the design choices, forcing the designer to propose

solutions that are not necessarily optimal. The designer should then use his or her criteria and

assign strength proportions and curvature ductility demands to achieve the most desirable

design solutions. Furthermore, the design solution can be influenced by more aspects than the

purely structural. As it will be discussed in Section 4.3, the economic feature is also important

and influences the assignment of strength proportions, having therefore an important impact

on the final choice.

25


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Chapter 4. Verification of the DBD Methodology

4.VERIFICATION OF THE DBD METHODOLOGY4.1 Selection of case study structuresFive buildings with 4, 8, 12, 16 and 20 storey are selected for verification of the proposed

methodology. The structures have a regular layout both in plan and elevation, and their

general characteristics are presented in Figure 4.1. The earthquake lateral resistant system is

formed by two RC walls and two lateral steel frames in each orthogonal direction. The storey

height is constant over the building and equal to 4 m (157 in), whereas the bay length is 8 m

(315 in). The length of each wall is equal to 4 m (157 in) for the 4 storey building, 6 m (236

in) for the 8 and 12 storey structures, and 8 m (315 in) for the 16 and 20 storey buildings; the

wall thickness is equal to 0.35 m (14 in). Notice that the proposed layout tries to reflect an

authentic case by including areas for shafts, stairs, etc.

Figure 4.4. Plan and elevation of frame-wall structures selected for the evaluation.

The lateral seismic force is considered to act parallel to the short dimension of the building.

Additional beams are considered as vertical load carrying members, connected to the walls

and steel frames in a way that only vertical shear load can be transferred. It is assumed that a

series of composite steel decks act as a rigid diaphragm in the horizontal direction at each

level. Additionally, the foundations were considered as fixed.

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4.2 Design criteria and main assumptions made in the designThe design spectrum selected for the DBD corresponds to the spectrum type 1 and soil type C

according to the EC8 [CEN, 2003]. The level of ground acceleration used for design is 0.5g.

In Figure 4.2 it can be seen the acceleration and displacement design spectra for a 5% of

elastic damping, in accordance with the parameters given by the Eurocode EC8. The

displacement design spectrum was developed using the relation between the acceleration and

displacement, sD=(T/2)2 sA. Although it is accepted that at high periods the spectral

displacements can be considered as independent on the period value, in this work it is decided

to simply extrapolate the initial linear spectrum without applying a cut-off period.

The concrete and reinforcement properties considered for the structures are, for concrete

fc=30 MPa (4.35 ksi) and Ec=25740 MPa (3730 ksi), while for reinforcement fy=400 MPa

(58 ksi) and Es=200000 MPa (29000 ksi). Note that these are expected values of strength and

stiffness, and therefore are not factored.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5 2 2.5 3 3.5 4

Period (s)

Spectralacceleration(g)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6 7 8

Period (s)

Spectraldisplacement(m

Figure 4.5. Design spectrum (left) and displacement design spectrum (right) for 5% of elastic damping

The seismic weight of the concrete deck was calculated considering a concrete density of 24.5

kN/m3(156 lb/ft3) and a slab thickness of 200 mm (8 in). A super-imposed dead load of 1 kPa

(20.8 lb/ft2), a reduced live-load of 1 kPa and a loaded floor area of 982 m2(10565 ft2) at each

level are also considered.

Since the proposed procedure requires the spectral values for levels of viscous damping larger

than 5%, these values are obtained by using the damping correction factor, , suggested by theEC8, in accordance with Equation 4.1.

( )0.55

5

10

+=

(4.1)

The initial storey drift selected for design purposes was 2.5%. This limit intends to control

damage of non-structural elements in the building. Damage to structural items was controlled

by imposing strain limits on the concrete and reinforcing steel. Ultimate compressive strains

of 0.018 for the concrete and 0.06 for the reinforcing steel were deemed appropriate for these

case studies. Priestley and Kowalsky [1998] have argued that these strain limits are

28


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welded to beams and columns in the joint zone, and that the panel zone has only a limited

contribution to the lateral deformation of the frame.

The selection of a beam section group was initially done by considering deflection control

criteria due to gravity loads. Typically, a beam length to beam depth ratio of 19 (in inches) isconsidered as adequate to satisfy deflections constraints imposed by some codes.

Nevertheless, this value is greatly dependent on the code selected for design and its

corresponding constraints. Furthermore, since the beam to depth ratio is only a rough

approximation for the beam size and does not have a direct influence on the design

methodology, the designer is free to choose the ratio that he or she considers more

appropriate.

For the case study structures considered in this work, a beam length to depth ratio of 15 is

adopted. Hence, for a 8 m (315 in) bay length, a first trial with a W21 (21 inches) depth beam

would be adequate. Initial column sections are selected based on the fact that columns inmodern medium rise steel buildings are commonly built with W14 shapes. Furthermore, the

wide availability of W14 shapes and plastic section modulus, Z, included in this shape group

make them appropriate to be used as column sections.

4.3 Summary of case study design resultsThe general characteristics of the structures are shown in Table 4.1. The dimensions of bays

and the properties of the group shapes selected for design are within the ranges explored in

Chapter 3 where the calibration of Equation 2.2 was carried out. Hence, Equation 2.2 can be

used with relative confidence to estimate the yield drift of the steel frames.

The axial load ratios were computed using the floor weights factored by the tributary area of

floor supported by the individual element, wall or column. Wall axial load ratio is calculated

at the ground level by using the formula N/Agfc for the RC walls, and N/Agfy for the steel

columns.

Table 4.1. Characteristics of frame-wall structures

4 storey 8 storey 12 storey 16 storey 20 storey

Wall length (m) 4.0 6.0 6.0 8.0 8.0

Wall thickness (m) 0.35 0.35 0.35 0.35 0.35

Inter-storey height (m) 4.0 4.0 4.0 4.0 4.0

W-Beam depth (inlb/ft) 2144 2168 2162 2162 2183

Interior W-columns depth (inlb/ft) 14109 14176 14159 14176 14211

Exterior W-columns depth (inlb/ft) 1461 1490 1482 1490 14109

Wall axial load ratio 0.033 0.053 0.085 0.098 0.142

Interior columns axial load ratio 0.035 0.044 0.073 0.088 0.091

Exterior columns axial load ratio 0.033 0.044 0.073 0.088 0.091

Floor seismic weight (kN) 7250 7250 7250 7250 7250

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Intermediate design results can be seen in the Table 4.2. Proportions of shear carried by walls

and frames are assigned in accordance with the optimisation process suggested by Sullivan

[2005], described previously in Chapter 3.Additionally, because structural steel can increase

significantly the cost of an engineering project, an effort in reducing the weight of the steel

sections required in the DBD design of the structure was also done. This fact highlights one

advantage of the proposed DBD method: by assigning the strength proportions at the start of

the design process it is possible to control not only the amount of seismic moment carried by

walls and frames, but also (and directly) the cost of the structure. This economical issue is

particularly significant in those countries where the price of structural steel is an important

factor. By considering the design results for a given structure, a designer could, for instance,

decide to decrease the depth of beams and columns to make the walls more effective at

resisting lateral loads; yet, this would demand an increase in the amount of concrete and/or

steel in the wall. In contrast, if the longitudinal reinforcing ratio in the wall is excessive, the

designer could opt to increase the beam and column sections.

Notice how the frame yield drift is equal for all the buildings due to the fact that W21 beam

and W14 column groups were selected as initial trial sections. Additionally, note that the

design storey drift was not reduced in the 4 storey building since this case study has less than

six storeys; hence, Equation 1.16 does not modify the design storey drift originally selected

for design. Nevertheless, the design drift was effectively reduced in the other case studies to

consider the higher mode effects. For all the study cases, the wall curvature ductility ( )is

limited to values of less than 18.

Wall

It is important to note that, for a design drift of 2.5% which is around the largest acceptable

storey drift limit for life safety events in some seismic codes, the average frame ductility,Frame, is only slightly larger than 1.0 for all the case studies. Since the structures are designed

to remain almost elastic in the DBD procedure, little inelastic activity in the steel frames can

be expected to occur. If low ductility demands are likely to occur in the frames, then the

amount of detailing in connections will not be stringent. Nevertheless, the designer should

keep in mind that the frame ductility demand should match a demand level associated with an

appropriate capacity design, which is commonly related to a capacity design earthquake or

maximum credible earthquake. Hence, the level of ductility capacity must be significantly

larger than that provided by the limit state initially considered in the DBD procedure.

Additionally, notice that the values of wall ductility are significantly larger compared with

those from frames. Because frames and walls must maintain displacement compatibility, thisimplies that frames will undergo low ductility demand even for large values of wall ductility

demands and inelastic deformations. From these points, it seems that any effort to provide

steel frames with a large inelastic capacity should be avoided. Hence, if the ductility

requirements are not very stringent, detailing in steel connections can be reduced, for

instance, from a high structural ductility class (DCH) to a level of intermediate ductility class

(DCM). Furthermore, in terms of force based design, this would suggest that the ductility

dependent reduction factors suggested by the codes could be relatively smaller than those

currently included. This, in turn, is in line with the findings made by Paulay [2002] in frame-

wall structures.

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It is interesting that the average frame ductility tends to be smaller as the height of the

structure increases. This seems to be a direct result of the influence of the walls on the

displacement response. This fact suggests that structural walls could have a larger effect in

controlling drifts and displacements in taller buildings than in lower structures. Taller

structures with a large number of storeys usually provide a good amount of axial load to the

walls. If the walls have a reasonable amount of axial load acting on them and such a load

tends to re-centre them during the seismic event, the frame-wall system will undergo low

damage and residual displacements, particularly under more frequent events. It can be

expected, therefore, that low-rise structures with low load axial load ratios will undergo more

damage and residual displacements in comparison with tall structures with high load axial

load ratios.

A possible disadvantage of such low values of ductility demand is that the steel frames are not

really dissipating much energy even for a large design drift associated to a life-safety level.

Hence, their potential can not be fully used under such levels of shaking. Even more, undermore frequent low intensity earthquakes, it could be expected that the frames may not yield. If

the designer wants the frames to undergo more inelastic activity, it would be necessary to

increase the proportion of strength carried by the frame at the start of the design.

Table 4.2. Intermediate design results for frame-wall structures


Proportion of Vbassigned to walls (%) 60 50 50 50 45

Frame yield drift, ySteelFrame(%) 1.74 1.74 1.74 1.74 1.74

Inflection height, hinf(m) 16.0 24.0 30.83 40.0 46.9Design storey drift, d(%) 2.5 2.44 2.36 2.29 2.19

Design displacement, d(m) 0.26 0.50 0.71 0.90 1.06

Wall curvature ductility,

Wall 14.28 13.82 10.44 10.21 7.80

Wall displacement ductility, Wall 4.78 4.16 2.87 2.84 2.22

Average frame ductility, Frame 1.28 1.29 1.23 1.20 1.13

Frame overturning moment,MFrame(kN-m) 5.33 12.0 17.33 22.67 30.80

Wall overturning moment,MWall(kN-m) 6.77 10.67 16.0 21.33 23.87

System ductility, sys 3.22 2.64 2.02 1.99 1.60

System damping, SDOF 13.0 11.8 10.6 10.4 9.0Effective mass, me(kNs

2/m) 2377 4515 6557 8658 10615

Effective period, Te(s) 1.66 2.76 4.12 5.24 5.85

Final design strengths and longitudinal reinforcement of walls are included in Table 4.3 It is

noteworthy that the reported values of beam and column flexural strength correspond to the

values yielded by the design procedure and not to those corresponding to the actual values

provided by the selected shapes. The actual values of flexural strength are marginally larger,

so that this issue has a negligible influence on the results of next section. In all the cases, wall

longitudinal reinforcement ratios are between the maximum and minimum values suggestedby Paulay and Priestley [1992]; hence, they are considered as realistic.

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Table 4.3. Final design strengths for frame-wall buildings (in kN-m)


Base shear (kN) 9035 11760 10770 11255 12977

Wall strength (kN-m) 29526 61492 84474 117699 151826

Wall longitudinal reinforcement (%) 1.40 1.21 1.62 1.16 1.44

Beam strength (kN-m) 765 1244 1140 1191 1511

Interior column strength (kN-m) 1530 2489 2279 2382 3021

Exterior column strength (kN-m) 765 1244 1140 1191 1511

4.4 Non-linear time history analysis verification procedureIn order to verify the effectiveness of the proposed method in terms of displacement and drift

control, the structures are analysed by performing non-linear time-history analyses in

Ruaumoko [Carr, 2004]. The set of buildings were designed in accordance with the

displacement spectrum derived from the design spectrum of the EC8. Additionally, the

records used for the evaluation must be compatible with the design spectrum used for design.

The selected records correspond to seven code-compatible artificial accelerograms generated

using the program SIMQKE [Carr, 2000]. Values of equivalent viscous damping calculated in

the design process range between 9 and 13%. Fig

development of a displacement based design method for steel frame - rc wall, 2007 (garcia)

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