Designing 2 Storey Building for Sustainability Designing for Life

Designing 2 storey building

Running Head: Designing 2 storey building for sustainability designing for life

[Name of Writer]

[Name of Institution]

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ContentsCHAPTER I: INTRODUCTION............................................................................................................5

Introduction................................................................................................................................5

Statement of the problem..........................................................................................................7

Purpose of the study...................................................................................................................7

Objectives and scope..................................................................................................................7

CHAPTER II: REVIEW OF THE LITERATURE.......................................................................................9

Introduction................................................................................................................................9

Reinforced Concrete Building Frames.........................................................................................9

Building design............................................................................................................................9

Number of Stories.....................................................................................................................13

Models of Material Behavior....................................................................................................14

Number of Seismic Events Used...............................................................................................15

Number of Seismic Components Used.....................................................................................16

CHAPTER 3....................................................................................................................................19

MULTI-STORY BUILDING MODEL..................................................................................................19

Introduction..............................................................................................................................19

Prototype Building Model.........................................................................................................19

Layout of Frames......................................................................................................................20

Prototype Frame.......................................................................................................................21

Frame Configuration.................................................................................................................21

Plastic Mechanisms Developed by a Frame..............................................................................22

Mathematical Model of Frame.................................................................................................23

F-function for a Frame..............................................................................................................23

Modification of Stiffness or Strength of Frame.........................................................................24

Mathematical Model of the Building........................................................................................25

CHAPTER 4....................................................................................................................................26

DESIGN OF PROTOTYPE BUILDINGS..............................................................................................26

4.1 Introduction........................................................................................................................26

4.2 One Story Building..............................................................................................................28

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Target Backbone Shape........................................................................................................28

4.3 Assembling the PB..............................................................................................................31

4.4 Maximum Elastic Demand and Evaluation of Strength Factor (fu)......................................32

Verification of Developed Plastic Mechanism...........................................................................32

Two Story Building....................................................................................................................33

Target Natural Period...............................................................................................................34

Target Backbone Shape............................................................................................................34

Parameters Fixed in the Frame Definition................................................................................35

Assembling the PB....................................................................................................................36

Maximum Elastic Demand and Evaluation of Strength Factor (fu)............................................37

Buildings with Varying Properties.............................................................................................37

CHAPTER 5....................................................................................................................................39

PARAMETRIC STUDY OF THREE-STORY BUILDINGS.......................................................................39

Introduction..............................................................................................................................39

Parameters Studied..................................................................................................................39

Effect of Mass Eccentricity........................................................................................................39

For the CP and the VP buildings................................................................................................42

Effect of Stiffness Eccentricity...................................................................................................42

Effect of Strength Asymmetry...................................................................................................43

Conclusions...............................................................................................................................44

CHAPTER SUMMARY AND CONCLUSIONS...................................................................................46

Research Summary...................................................................................................................46

Conclusions...............................................................................................................................47

Recommendations for Future Research...................................................................................48

Implications for Concrete Two-Storey Design...........................................................................50

References................................................................................................................................55

Appendix A Figures...................................................................................................................57

List of FigureFigure 1: Story plan view and location of the reference nodes.....................................................20Figure 2: Prototype Frame Configuration.....................................................................................22Figure 3: Static degrees of freedom in the frame..........................................................................24

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Figure 4: Story plan view and location of the reference nodes....................................................28Figure 5: Displacement step.........................................................................................................30Figure 6: Discretized grid of the model domain showing the nominal concrete vault with a half-width of 1,000 cm (10 m).............................................................................................................57

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CHAPTER I: INTRODUCTION

Introduction Two-storey buildings have become an essential part of today’s architects due to

the growing population and limited space available. As innovative building designs are

the prerequisites of any building architect, it is vitally important to design sustainable

two-storey buildings for life. This dissertation offers a critical review of the two-storey

buildings which are innovative, sustainable and sustain earthquake shocks.

Natural calamities are also a matter of great concern for building designs and

materials used in building two-storey buildings. Every time a seismic event occurs

somewhere around the globe, Nature reminds us that there are still many things to be

done by practicing engineers to reach the goal of saving lives and property during these

extreme events. The after quake statistics tell us about large economic loses and

unacceptable number of lives lost. Many reasons can be found or argued about the causes

that lead to the collapse of some structures in these seismic events, ranging them from

plain negligence up to limitations on the knowledge the way structures behave during

strong seismic events. Therefore, there is a constant challenge to every engineer for a

better understanding of the patterns of behavior of building structures, knowledge that

hopefully can be applied in the design and construction of the reliable structures that

people needs. The intention of this dissertation is to focus on the design and materials

needed to sustainable for life two-storey buildings.

When a seismic action induces a horizontal rotation about a vertical axis in a

building floor/roof, that behavior is called torsional response. The effect of this

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phenomenon is to induce additional demands of strength and ductility on some

components of a building that in a hypothetical ideal building would not exist. The

concern arises when these demands were not anticipated by the engineer, and no

measures were taken to accommodate them. Even if the engineer clearly foresee that this

high strength or ductility demands would be induced, he has to deal with the current code

recommendations to handle this issue, and arguably they are insufficient in some design

situations. An example of these national recommendations can be found in the "Minimum

Design Loads for Buildings and Other Structures" code, in the USA, and the Building

Code for the Federal District, in Mexico City. Experiences from past seismic events

around the globe, have taught us that a significant number of collapsed structures had

serious problems in handling the torsion induced demands. Escobar and Ayala (1998)

report that a significant proportion of the observed damage in building structures during

the 1985 Mexican earthquake can be attributed to ill torsional behavior. They explain that

the eccentricity between the building's strength center and its mass center was relatively

large in some of these collapsed buildings, and the worst, surprisingly the eccentricities

were unexpectedly large.

There is a tendency to believe that a nominally symmetric building cannot

respond in a torsion mode of vibration. This is a naive expectation. Actually, buildings

are not perfectly symmetric and as some researchers (DeLaLlera and Chopra, 1994) have

shown before, accidental eccentricity and rotational components of the ground motions

induce torsional response in symmetric buildings.

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Therefore, we must be prepared to confront this truth: all buildings will respond in

torsion during a seismic event, thus we need to know how to evaluate its magnitude and

how to control its effects in a rational and reliable way.

Statement of the problem In the context of the growing rainfalls and other natural calamities like tsunami,

sustainable two-storey buildings have been used and will continue to be used for

providing shelters, safety and conform to its lodgers. However, the building design is an

important consideration that affects the ability of the building to isolate the waste and

protect the environment.

Purpose of the studyThe main purpose of this study is to evaluate the sustainable two-storey building

design, the choice of materials ( recycles, reusable), environmental impact, adaptability /

flexibility of the design and the choice of construction methods ( single connection ,

elements easy to dismantle)

Objectives and scope The overall objective of this research is to gain a better understanding of the

materials used and the design of a sustainable for life two-storey building. For this

purpose, this research focuses on the behavior of buildings designed with the criteria of

the philosophy of Capacity Design, i.e., buildings are capable of developing plastic

mechanisms in the response to the maximum considered earthquake.

In the design practice a variety of structural systems are used. However it is not

realistic to attempt finding patterns of behavior for each one of these structural systems,

in the limited time available for this research. Thus, the research is restricted to study the

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behavior of framed buildings. Simplified framed buildings are adopted to study this type

of structural system.

Within this context, the following specific objectives are pursued herein:

Description of the mathematical model of a one story and a multi story prototype

buildings.

Design of a two-story prototype buildings.

Search for patterns of behavior of buildings responding in a torsional mode.

Perform parametric studies of the one-story and the three-story buildings.

Establish the conclusions of this research.

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CHAPTER II: REVIEW OF THE LITERATURE

Introduction The life cycle of concrete buildings is usually 40 to 90 years. However, during

this life cycle, buildings will often meet some circumstances, such as disasters, changing

functions, city reconstruction, or higher demand for the residence etc.; all of these

circumstances will lead to demolition or reconstruction of buildings. Moreover, many of

Reinforced Concrete (RC) buildings in Asia or Europe built after World War II have been

used for about 50 years or more. It is obvious that such buildings, if built with low quality

material, lacking proper construction design or unsuitable geographical location are prone

to deterioration and as a result need demolition.

Reinforced Concrete Building FramesThere are different conduction approaches applied while designing and

constructing a building. In general, reinforced concrete structures are designed to be

durable, serviceable, and attractive. Structural elements composing a reinforced concrete

system may be broadly classified into floor slabs, beams, columns, walls, and

foundations.

Building design Earthquake Engineering is a body of knowledge that has been in continuous

development during a long period of time. This progress has been an evolutionary

process, most of the time, with occasional revolutionary jumps ahead, and sometimes

backwards.

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The occurrence of some catastrophic seismic events around the world were the

source of most of the new knowledge, and a strong motivation to work on the problem.

Structural solutions used by the practicing engineers (designers), are tested by Nature

during their occurrence. Afterward, it appears that some ideas seemingly very good,

turned out to be bad solutions. Thus, academics and researchers have contributed to the

body of knowledge by reviewing the behavior of the solutions used by the engineers,

learning from the good behavior, but much more from the bad behavior and failures of

some of these structures.

So, it has been cycles were designers have tried new structural solutions without

having a complete knowledge of their future behavior, and later researchers helped to

understand why their solutions failed or behaved in unexpected ways. This way, the

original solutions have been repeatedly amended, improved or abandoned as a result of

these cycles. Each cycle that, besides the personal worries and interests of engineers,

academics and researchers, has had a high cost in terms of lives and economic loses

around the globe.

One revolutionary event in the history of the Earthquake Engineering was the

proposal by Park and Paulay (1975) of a systematic method of design, based on the

accumulated observations of the behavior, during strong seismic events, of many

structures around the world. Some of these structures surprised engineers and researchers

due to their capacity to resist unexpected extreme seismic events. These structures

showed extensive damage in non structural elements and localized plasticization, despite

this, the structures survived these events.

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From these accumulated knowledge by the engineering community, emerges the

systematic proposal by Park and Paulay. They called their design approach as the

philosophy of Capacity Design. With the publication of their book in 1975, this

philosophy started to spread out. De Buen (2004) tells that the Mexico City building code

was modified to include in its published 1976 edition, recommendations and procedures

to provide structures with the capacity to dissipate energy by developing non linear

behavior. That was the first time these recommendations appeared.

In this context, it becomes clear why many works published before the 1970's

focused on the elastic behavior of buildings, and after that time, the research community

shifted their interests to study the non linear behavior of structures.

Another milestone is marked by the introduction of computer machines to the

practice and research of Earthquake Engineering. In their early times, the computation

power provided by these machines was relatively small, but during the last 50 years, the

computer power has increased steadily. Wilson (2007) presents a short table describing

how the relative speed of different computer systems has changed with time.

The scope of the research made during this long period has been strongly

influenced by the available computer machines. This way, many early researches focused

on one degree of freedom structures, or one story buildings. Adopting hypothesis of

linear behavior and using single seismic records, applied to the models in only one

direction. The available computer power was a severe limitation.

In more recent times, published works reporting studies of multi story buildings

have emerged, considering non linear behavior of the material and using multiple seismic

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records. This widening in scope occurred hand on hand with the increase in computer

power.

About the first half of the XX century, it was very common to have more

partitions and facade mansory walls in buildings. Also, most of the buildings were

relatively short in height. In those times, the torsional response was not an issue, as it is

nowadays. The "non structural" walls provided a large torsional strength and stiffness to

those buildings.

In the last 50 years, the new buildings have had fewer non structural walls and/or

new materials have substituted the masonry walls by light and fragile materials, like the

gypsum panel boards attached to wood or metallic studs. Also, first story walls (facing

the street side) have disappeared to give way to the wide windows required by modern

commercial buildings.

All these changes have had the effect of changing radically the actual lateral

strength, stiffness, and energy dissipation capacity of the modern buildings. More

important, these changes opened the window to conditions that amplified the problem of

the torsional response of buildings, turning this phenomenon into the big problem that has

been observed in recent strong seismic events. This way, the engineers, pushed by the

necessities and demands of the contemporary world, have contributed to unleash a

structural problem.

For the purposes of this research, the topics investigated by the researchers

interested in the problem of torsional response of buildings were reviewed. From the

perspective of the present time, it is easy to underestimate the importance of the work

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done by the first researchers. However, it is unfair to judge their achievements based on

the current knowledge.

Number of StoriesThe natural interest of researchers and designers is to study the dynamic response

of multistory buildings. The interest on these buildings is due to their use for residential,

commercial, and industrial applications. These buildings must be designed to withstand

the design spectrum accepted by the building code applicable to the site.

When the Earthquake Engineering started its development as a scientific

discipline, arguably in the 1950's, the solution of the mathematical model of multistory

buildings was impractical. Researchers had to use numerical methods to solve the

ordinary differential equations derived for the dynamic system, excited by an earthquake

time-history. Despite that linear behavior of the models was assumed, the amount of

computational work required to solve the numerical problem was huge.

Thus, researchers had to limit their studies to one story buildings. The one story

models were simple versions of realistic buildings. Usually, complete plane frames of the

building had to be modeled with a single shear beam (SB) element, and the layout of

frames reduced to four elements, two on each orthogonal direction.

Another reason to study one story buildings was to start learning from these

simpler cases, to move later to more complex cases (multistory buildings). Since that

time, one story models have been studied by different researchers. At the beginning, the

technological conditions forced researchers to use one story models, but in more recent

times, when the technological conditions have changed radically, disappearing many

restrictions, the continuing use of these simple models is harder to justify.

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As early as 1969, Anderson and Bertero worked with multistory steel frames

(1999) to observe their seismic behavior, when the frames can develop strong column-

weak beam plastic mechanisms. They did not study the torsional response of the building.

To the best knowledge of the author, there are scarce studies of torsional response of

multistory buildings done from the 1960's to the present time. Some studies were done

considering elastic behavior of the building.

Models of Material BehaviorIn the first epoch of the Earthquake Engineering, the most used model of material

behavior was the linear elastic. Gradually, models of nonlinear behavior started to be

used in studies of dynamic response of buildings. The use of non linear models would be

the second epoch. This second epoch overlaps with the first.

Nowadays, linear elastic models are rarely used to evaluate the dynamic response

of buildings covered by the ASCE 7-05 design code.

The first models of non linear behavior were simple models. One story buildings

were idealized with four frames, where each frame is substituted by a SB element. These

SB have an elastic perfect-plastic behavior. Many studies were done with this type of

models.

The main issue with these models is that the philosophy of Capacity Design

postulates strong column-weak beam plastic mechanisms as a design goal. Most recent

studies idealize framed structures with bar elements, to model columns and beams

individually. The non linear behavior is introduced into these elements using the concept

of plastic hinge (PH). The PH is a convenient simplification that transforms a complex

local phenomenon into a mathematical model easier to handle. The behavior of these PH

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is modeled with an elastic perfect-plastic relationship (as used in this Dissertation). Other

models of material behavior that could be used are the bilinear relationships with strain

hardening, trilinear relationships, etc.

In recent studies, more sophisticated models are used. The advances in computer

machines and structural analysis software, has given to researchers the possibility of

including strength and stiffness degradation models. The inconvenient of these models is

that their reliability is limited, specially for reinforced concrete structures. However, it

seems reasonable to expect some changes in the observed patterns of behavior, with

respect to the ones observed for the less sophisticated models, e.g., the elastic perfect-

plastic relationship.

Number of Seismic Events UsedSome decades ago, when Earthquake Engineering started to develop, the amount and

quality of the available recorded seismic events was limited. During those years, the "El

Centro 1940" earthquake became the prototype of an intense event, and it was used by

different researchers to do their investigations. Since the 70's the amount of recorded

events has steadily increased. Nowadays, researchers have a large set of these recorded

events. In this set, there are many events of much larger magnitude or duration than the

"El Centro 1940" seismic record.

Older published papers and technical reports show results of analysis made using only

one earthquake record. Parametric studies were done using single seismic events, and

these results were, in some form, incorporated into the design codes of different

countries. The problem with this research results is that the accumulated evidence

demonstrates that different seismic events can have quite different characteristics.

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To illustrate this point, in chapter 3 are presented the Fourier Spectra of the seismic

events included in the data set used in this Dissertation. These Spectra show clearly that

the frequency contents vary considerably for each event. There are other important

parameters that differ, i.e., total duration of the event, magnitude, distance to the

epicenter, etc. Nowadays, the research community recognizes the importance of using

more than one seismic event for performing basic research. Thus, it has become the norm

to use a set with many and diverse modified historic seismic events.

Number of Seismic Components Used. The electronic devices used to record ground accelerations are capable of recording three

orthogonal components of the ground acceleration. Two horizontal and one vertical. One

or more of these components have been used to perform dynamic analysis of buildings.

Historically, these seismic records were included in the mathematical model in three

different variations:

1. Unidirectional seismic excitation Only one horizontal component of the seismic record

is used for the time history analysis. 2. Bidirectional seismic excitation The two

horizontal orthogonal components of the seismic record are used for the time history

analysis.

3. Tridirectional seismic excitation The two horizontal and the vertical orthogonal

components of the seismic record are used for the time history analysis. Many studies

about torsional behavior of buildings were performed using unidirectional seismic

excitations. Typically, the studied buildings have two or more parallel frames on one

direction, X, and two or more parallel frames in the orthogonal direction, Y. To handle

the fact that real earthquake ground motions have more than one component of

acceleration, researchers have had to create peculiar concepts such as:

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• Torsionally unrestrained is defined as a building where the unidirectional seismic

action is on Y direction and the orthogonal frames (on the X direction) do not provide

any opposition to the story rotation. The frames parallel to the earthquake action can

behave non linearly.

• Torsionally restrained is defined as a building where the unidirectional seismic action is

on Y direction and the orthogonal frames (on the X direction) provide opposition to the

story rotation. The frames parallel to the earthquake action can behave non linearly and

the orthogonal frames are in their elastic range of deformations.

These concepts were created to make sense of results obtained from unrealistic seismic

excitations. Only in the context of unidirectional seismic actions these concepts make

sense.

More recently, bidirectional seismic actions have been used to perform studies on

torsional behavior of buildings. For this research, this is the type of seismic actions used.

When bidirectional seismic actions are used, instead of unidirectional, the results

obtained can be significantly different. One of the interesting implications of using

bidirectional seismic actions is that the concepts of torsionally unrestrained and restrained

are not necessary anymore. And, as it is demonstrated in chapter 8, new patterns of

behavior can be observed. Patterns that cannot exist in buildings under unidirectional

seismic actions. Of course, bidirectional seismic actions are closer to a realistic modeling

of the seismic action than the unidirectional action.

The author did not find published research on torsional behavior of buildings, using

tridirectional seismic actions. However, in the near future, the inclusion of the vertical

component of the ground acceleration will become a requisite, to allow the evaluation of

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changes of axial load in columns, due to the vertical accelerations, andtheir effects on the

global response of the building.

Angle of Incidence of the Seismic Action. Seismology cannot provide to the designers

with a reliable prediction of the direction where the ground waves will come from in the

next earthquake. This direction is called the angle of attack, or the angle of incidence, of

the seismic event. Therefore, for design or research purposes, a decision needs to be

taken with respect to this parameter. Historically, there are three approaches used by

researchers with respect to the selection of the angle of incidence used in their

investigations:

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CHAPTER 3

MULTI-STORY BUILDING MODELIntroduction

This chapter covers the development of the mathematical model of a two storey

building. The mathematical model is developed considering that it can be applied to

buildings with any number of stories. The developed mathematical model is intended to

represent a particular type of structural system used in buildings. The following sections

describe this model in detail.

Prototype Building ModelIn this section, the differences between the one story prototype building (PB)

model and the multi story PB model are clarified. The multi story PB is defined by a set

of parameters described in this section. For the one story PB there is only one story level.

Thus only one mass center needs to be defined. For the multi story PB case, there are two

or more story levels and each one can have its mass center at a different location, respect

to the other story levels. Figure 1 shows the same conventions used for the one-story PB.

The story mass can have any distribution on the story layout. Thus, the location of the

mass center on the n-th story level is defined by the distances XcMn and YcMm

measured from the mass center to the conventional location of the master node. The

master node coincides with the origin of coordinates X-Y and is defined at the geometric

center of the postulated rectangular plan layout, for all the story levels.

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Figure 1: Story plan view and location of the reference nodes

The kinematic state of the master node at the n-th story level is defined by the set

of story DOF: dxn,dyn,8n. This is an arrangement convenient for the development of

the algorithm.

Layout of FramesThe structural system is idealized as made of two parallel plane frames in X

direction and two parallel plane frames in Y direction. The location of each frame is

defined in Table 6.1. The coordinates (Xj,Yj) correspond to an arbitrary point in the

vertical plane of the j-th frame. The fij angle is the angle between the j-th frame vertical

plane and the X axis. The prototype building is defined with frames parallel to the X or

the Y axis.

Table 6.1. Frame Location Frame Location

Table 1 1

Frame Location Β Angle F1 ((o,y1) 0F2 (o,y2) 0F3 X3,0) 90F4 (X4,0) 90

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The four plane frames interact only through their connection with the floor

system, which is idealized as behaving as a rigid diaphragm in the horizontal plane. As a

consequence of these idealizations, columns and beams bend only in the vertical plane of

the corresponding frame, i.e., biaxial bending is ignored. The location of the static center

of rigidity is established with the same criteria used for the one-story PB.

The same procedure used to set up the static center of rigidity of one story

buildings, is used for multi story buildings.

Prototype Frame. The prototype frame is idealized as an interconnected set of columns and beams

contained in the same vertical plane, interacting to resist the inertia forces induced by the

seismic event. No gravitational load effects are considered in the evaluation of the state

of strain and stress in the material. The purpose of this assumption is to simplify the

mathematical model and to avoid the difficulty in establishing the most representative

initial state of stress and strain at the instant when the seismic event starts.

In general, the geometric and mechanical properties of the two frames in the X

direction can be different to the two frames in the Y direction, but the two frames in the

same direction have identical properties, except for their location.

Frame Configuration. The prototype frame is defined by two continuous lines of columns and one

beam at each story level, as shown in figure 2. The columns at one story level have

identical geometric and mechanical properties, but they can be different to the pair of

columns at the adjacent levels. The columns at the fist level are assumed to be fixed at

their lower end, and have a continuous moment and shear connection with the beam

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located at their top. This simple structural system was chosen to facilitate the parametric

study.

Figure 2: Prototype Frame Configuration

The interstory heights, have different values. The columns at each story level have

identical properties, but the columns or beams at different levels can have different

properties.

Plastic Mechanisms Developed by a FrameAs an extension of the criteria adopted for one story frames, the multi story

prototype frame is dimensioned with the intention of enforcing a strong column-weak

beam (SCWB) plastic mechanism. Plastic hinges (PH) are used to model the plastic

deformations that could be developed at some beams or column ends. The occurrence of

these PH’s providing the global non linear behavior of the frame.

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For one story frames, it is possible to enforce the developing of a SCWB plastic

mechanism, as imposed by the designer, but for multistory frames it is not possible to

attain this mechanism for all the possible lateral displacement patterns, which could be

generated during an arbitrary seismic event. Therefore, a different approach must be

taken to enforce the SCWB mechanism.

The approach used hereafter is a trial and error procedure to dimension the frame

and get the desired type of plastic mechanism.

Mathematical Model of FrameAs with one story frames, a F function must be defined. A similar logic deduction

process is followed to obtain this function, which is a matrix function. This function

evaluates the internal reaction forces in the building. The F function is composed by the

addition of reaction forces at each one of the four frames. Therefore, the F function must

be defined for an isolated frame and then added with the other three frame F functions.

The march in time algorithm requires this functional relation to solve the IVP ODE

problem.

F-function for a FrameThe direct stiffness method is used for the development of these mathematical

models. There are too many details that need to be solved to implement all these ideas in

a computer program; It is out of this research scope to present them completely and

exhaustively. Therefore, only some general details are presented in this chapter, to

illustrate some additional considerations required to implement the non linear dynamic

analysis of multi story buildings. One of the most obvious differences is that three DOF

are required per story level to define the kinematic state of the frame model. Figure 3a

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shows these DOF. In the following deductions, the aim is to find the F-function for the

prototype frame, although the mathematical relations are developed for the equivalent

system.

Figure 3: Static degrees of freedom in the frame

The mathematical relations shown in equations 5.1 are applicable. The difference

is that Atii must be substituted by the corresponding Aun, which is calculated with the

equation 6.2. Aun is defined as the interstory drift. Aun = un-un-i (6.2) Another

important difference is that equation 5.3 must be adjusted to include the n stories.

Applying DSM standard procedures, the new relation between changes in the reaction

forces and the change in interstory displacements can be found. After these adjustments,

the remaining procedure is essentially the same used for one story frames.

Modification of Stiffness or Strength of FrameThe two parameters introduced for one story frames are used to modify the lateral

stiffness and the ultimate lateral strength of the prototype frame. They are: K = stiffness

factor. fn = strength factor. The same relations 5.16 are applicable to multistory frames.

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Mathematical Model of the BuildingThe procedure followed to elaborate the mathematical model of the building is the

same used for one story buildings, but considering the additional degrees of freedom at

each additional story level. The original plus the additional degrees of freedom are stated

in equation 1. Taking in consideration the general details mentioned in this chapter, it is

possible to develop the computer algorithm for the non linear dynamic analysis of

multistory buildings.

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CHAPTER 4

DESIGN OF PROTOTYPE BUILDINGS4.1 Introduction

The purpose of this chapter is to show the arguments and procedures used to

define the dimensions of the prototype buildings (PB).

National building codes, like the ASCE 7-05 code, and other national codes

implement the philosophy of Capacity Design to dimension buildings capable of

withstanding the Maximum Considered Earthquake (MCE) in a particular location. Park

(2006) refers that this philosophy of design is the brainchild of John Hollings, a New

Zealander design engineer. Holling's original proposal was extended and refined by

Robert Park and Tom Paulay (2006). Paulay and Priestley (1992) did a further extension

of the explanation of this approach in their book.

In the spirit of the philosophy of Capacity Design, the designer decides the

locations where ductility is allowed to develop, and the magnitude of seismic induced

lateral force that the building must take. Under the designer's control, the building will

respond to the future seismic events in a mode planned by the designer. With this

approach, the maximum lateral load that the building can take is limited by design.

Although it is expected that seismic events larger than the MCE will produce larger

demands of ductility at the designated locations, and that the developed PH will have

enough ductility capacity. The last statement requires careful analysis, because it does not

define how much larger should the expected capacity be.

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A very important requisite for this approach to work is that all the remaining

elements of the building, which are not permitted to develop ductility, stay in the elastic

range. Park and Paulay (2006) present strong arguments to support the idea of not let the

columns in a framed building to develop ductility. They argue that column sidesway

plastic mechanisms (weak stories) must be avoided, and beam sidesway plastic

mechanisms enforced. Nowadays, there are widely recognized the limitations to develop

curvature ductility by the columns, and their reparation after the earthquake. Therefore,

the plastic mechanism commonly known as strong column-weak beam (SCWB) is

enforced for the design of the prototype buildings in this research.

The SCWB plastic mechanism is expected to develop rotational ductility at some,

or all beams. And the columns, to stay in the elastic range of strains. Design codes

(American Society of Civil Engineers, 2006) prescribe some minimum values of the ratio of

∑ Mp columns/ ∑ Mp beams for columns and beams interconnected at a frame node. The

intention of these rules is to avoid the possibility of the columns to reach their plastic

moment and, after that, the developing of ductility demands. Paulay (1992) evaluates

these code rules and gives arguments to support his conclusion about the limited

effectivity of these rules. He emphasizes that there are diverse situations where these

rules fail, and lead to designs where columns will have unexpected ductility demands.

In the section 4.2, the one-story PB is designed, and in section 4.3 the two story

PB. An alternative procedure is proposed in these sections to design the beams and

columns of these buildings. The intention of this procedure is to propose designs, which

can develop the SCWB plastic mechanism, before reaching the ultimate base shear force

of the building.

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4.2 One Story Building

Target Backbone Shape. Figure 4 shows the backbone of the lateral force versus roof displacement for a

single story frame. As it is known, this shape is determined by the type of plastic

mechanism developed by the frame and by how much ductility capacity it has. The

overstrength factor, Ω0, and the deflection amplification factor, Cd, have a definitive

influence on the total ductility demand that potentially can be developed in the beam,

before reaching the plastic moment in the columns. If the beam can supply the demanded

curvature ductility, the frame withstands the MCE by developing a SCWB plastic

mechanism, as it is desired by the designer.

Figure 4: Story plan view and location of the reference nodes

It is not the aim of this research to study buildings that comply strictly with any

design code, but to study prototype buildings whose behavior can be approximately

compared with buildings designed following recommendations of the design code.

28


These four parameters define completely the backbone curve for the one story

prototype frame. The dimensioned frame is warranted to develop a SCWB plastic

mechanism when required by any extreme seismic event. The prototype frame is

calibrated to develop an arbitrary ultimate lateral load, and Fu is tied to the used Mpb (an

arbitrarily chosen value). The Fu value is adjusted in section 4.2, such that the prototype

building can take the actual maximum demand of lateral load, induced by the earthquake

data set. Herein is introduced an alternative method to evaluate the backbone curve,

including the values that define the kinks. This method is named the Hysteretic

Displacement Method (HDM). The HDM evaluates the relation between the applied

lateral displacement and the lateral load response. The intention of this procedure is to

identify the plastic mechanism developed by the frame. The HDM purpose is equivalent

to the purpose of the nonlinear static method of analysis, best known as the Pushover

method (PM) (Krawinkler, 1998; American Society of Civil Engineers, 2000). A

fundamental difference between both methods is that the PM evaluates the nonlinear

response to a fixed pattern of lateral load, applied at the story level, and the HDM

evaluates the nonlinear response to a fixed pattern of lateral displacement. Another

important difference is that the HDM lets to evaluate the response for reversible cyclic

displacements. The HDM has the steps described below:

Step 1. A sequence of discrete displacement steps is created using the equation 1.

Equation 1

Ds = Dmax sin (2π/Steps - s )

Where:

29


Steps = Total number of displacement steps

s = displacement step => s G 0,1, , Steps

Ds = Displacement corresponding to the s-th displacement step

Dmax = Maximum lateral displacement

Figure 5 shows the displacement step history. The frame is under the action of a

forced roof lateral displacement of magnitude Ds. The total number of displacement

steps, required to complete a cycle, is selected to minimize the errors due to local changes

in stiffness of beams and columns, during calculation in a displacement step. Typically,

this is a large integer number. Though easy to handle by the computer. Using a trial and

error procedure, Dmax is increased up to the value where the frame reaches its ultimate

lateral load and starts yielding in a plastic collapse mechanism.

Figure 5: Displacement step

Step 2. The mathematical model is used herein to calculate the response of the

prototype frame. The F function is used to calculate the lateral response.

The rotation demand at ends of columns and beams is shown in figure 5. The

reported values are in radians. The histories of bending moments at these elements is

30


shown in figure 6. The reported values are normalized dividing them by their

corresponding plastic moment. It is evident that the beam starts to yield plastically under

smaller lateral displacement at the roof level. After the lateral displacement has reached a

larger magnitude, the lower end of the column starts yielding. In between these two

events, the frame is developing a SCWB plastic mechanism, as it is planned. If the

seismic event continues demanding ductility, eventually the columns yield and the frame

reach its ultimate lateral load capacity.

4.3 Assembling the PB. A one story PB is created by assembling four frames. The assembling procedure

is described in the next section. The following plan dimensions are adopted:

b = lin ... length of the side parallel to X axis

d = in ... length of the side parallel to Y axis

For the purpose of this chapter, a doubly symmetric building is defined. The four

frames are identical, and based on a modified prototype frame. The K and factors are fu

factors are used to modify the prototype frame.

The prototype frame is modified to obtain the target natural period, T = 0.22 sec.

The 1/2 factor is introduced to consider that the lateral stiffness is supplied by two

parallel frames in each orthogonal direction. The fu = 10000 is used to have a linear

elastic response for the set of seismic records used in this research. By trial and error, this

value was found to be appropriate. No ductility demand is induced at any element.

31


4.4 Maximum Elastic Demand and Evaluation of Strength Factor (fu). The next step is to find the maximum elastic demand, induced by the action of the

set of ten seismic events used for this research. The possibility that any of the seismic

events attack the building from random directions is considered. The evaluated incidence

angles (directions of attack) go from 0° to 180°, at each 15°, measured respect to the

positive X axis.

The time-history analysis lead to a maximum base shear force demand, Ve =

720.749kip, for each one of the four frames. With this result and the procedure prescribed

by the ASCE 7-05 code, the reduced inelastic base shear force is evaluated as:

The strength factor for one frame is:

Using these results, the fu factors are changed to the value: fu = 0.23777.

Verification of Developed Plastic MechanismThe last stage of the procedure is to verify that the defined prototype building can

develop the SCWB plastic mechanism, for all the seismic events in the set and all the

evaluated incidence angles.

On the other hand, the ductility demand for columns is about 1.2. The expectation

is that this value should be smaller than one, indicating no ductility demand in the

32


columns. These results indicate that the Q factor needs to be a little bit larger than Q0 = 3,

value specified by the code. Anyway, the ductility demand is just above the desired upper

limit (one). Another implication of these results is that for any future seismic event, that

could be larger than the ones used here, the building would respond with a weak column-

weak beam plastic mechanism for the peak demands of the unexpected event. And this is

against the original spirit of the Capacity Design philosophy. As stated by Park and

Paulay in their work. It is important to visualize that this observation emerges from the

analysis of the effect of a swept of seismic events and incidence angles. When only one

seismic event is considered, the observation could be different. The author is convinced

that conclusions base only on one seismic event are fundamentally wrong.

In conclusion, the designed prototype building satisfies the criteria

(approximately) of withstanding the set of seismic actions developing SCWB plastic

mechanisms.

Two Story Building Two different prototype buildings are defined in this section. One has the same

column and beam properties at each story level. The other has columns and beams that

reduce their stiffness and strength with the story level. The building with variation of

properties is used to show what would happen to multistory buildings that have more than

three stories. These buildings usually reduce the size of columns

As with the one story PB, most of the parameters that define a two story PB have

fixed values, to reduce the complexity of the parametric study. Buildings with two or

more story levels require the introduction of new parameters to define the way that the

column and beam properties change from one story level to the next level. It needs to be

33


defined also how the mass is distributed at each story level, and the properties that

characterizes it, like the location of the center of mass, rotational inertia, etc.

Target Natural Period. One critical parameter to be fixed is the first natural period of translation of the

building. The upper modes natural periods are not designed, but implied in the other

parameter values adopted. The equation of the ASCE 7-05 code is used again to evaluate

a representative value. The story height used in the evaluation is 13.123/ft (4m), at the

two stories.

Ta = 0.028 (39.37ft)08 = 0.529sec ... Steel moment-resisting frames (7.3a)

Ta = 0.016 (39.37/i)09 = 0.436sec ... Concrete moment-resisting frames (7.3b)

Ta = 0.1 N = 0.3sec ... a more coarse approximation (7.3c)

A natural period of T = 0.50sec is selected to define the three story PB.

Target Backbone ShapeThe same code-specified values, used for the one story PB, are used here. The

response modification coefficient, R, overstrength factor, Ω0, and the deflection

amplification factor, Cd

R = 8

Ω0 = 3

Cd = 5.5

34


Parameters Fixed in the Frame DefinitionThe values of h and rbs where selected by trial and error, trying different values

until the targeted first natural period was achieved. The ratio rbs controls the relation

between the beam and columns bending inertias, after fixing the ratio of beam to column

lengths.

The reason for this lack of exact analytical expressions is that there is neither a

unique pattern of lateral story forces nor displacements. These patterns, induced by the

seismic action, mainly depend on the earthquake frequency contents, the natural vibration

periods of the building, and the actual values of overstrength and deflection amplification

factors. This situation makes impossible to find the desired analytical expressions. The

approach taken to solve this problem is to use the HDM. The PM is another alternative.

During this procedure, the targeted natural period, the overstrength factor, and

the deflection amplification factor are compared with the actual values found with the

HDM. The procedure is repeated until values close enough to the targeted values are

found. The two steps procedure required to apply the HDM to the one story frame are

required here.

Step 1. Equation 7.4 is used to create the sequence of discrete

displacement steps.

35


The Dmax, Steps, and s variables are the same scalar parameters used in equation

7.2, is the modal shape of the first natural vibration mode of the frame, (pi is a three

elements column-vector. The history of displacement steps is shown in figure 7.5.

Step 2. The F function described in chapter 6 is used to calculate the lateral

response. The hysteretic behavior of the frame is shown in figure 7.6 for the frame with

constant column and beam properties at all story levels, and in figure 7.7 for the frame

with varying properties (different properties at each story level). s 8 Lateral

Displacement First Story (in) Figure 7.6. Hysteresis loop of Constant Properties Frame

Assembling the PB. The two three story PB are created by assembling four frames. Using the results

and observations from Appendix A, the following plan dimensions are adopted:

b = in ... length of the side parallel to X axis

d = in ... length of the side parallel to Y axis

For the purpose of this chapter, doubly symmetric buildings are defined. The four

frames are identical, and based on a modified prototype frame. The K and fu factors are

used to modify the prototype frame. The frame locations and some factors applicable to

each frame are shown in Table 7.7 Considering that the prototype frame has the target

value of the first natural period, and the prototype building is made of two modified

parallel frames, these two frames must have one half of the prototype frame stiffness.

Thus, Kframe = 1/2.

36


Maximum Elastic Demand and Evaluation of Strength Factor (fu)The possibility of seismic events attacking the building from random directions is

considered. The evaluated incidence angles (directions of attack) go from 0° to 180°, at

each 15°, measured respect to the positive X axis.

The maximum elastic demand, induced by the action of the set of seismic events

on each one of the four frames, is evaluated by time-history analysis. The values are

shown in Table 7.8. This maximum demand is taken from chapter 10, for the case of zero

eccentricity.

Buildings with Varying PropertiesThe maximum demand of rotation ductility on beams is about 22 to 43. Rotation

ductility demand on columns is 1.0. The maximum demand on beams is two to four

times the maximum demand for the case of a one story PB. On the other hand, the

maximum demand on columns for the first story is at the limit that separates the SCWB

and the weak column- weak beam plastic mechanisms. The remaining rotation capacity

in columns, before reaching their yield rotation, is significantly smaller for the varying

properties building than for the constant properties building. For any future seismic

event, that is larger than the ones used here, the building would respond with a weak

column-weak beam plastic mechanism for the peak demands of the unexpected event.

The same observation is made for one story buildings. In conclusion, the designed

prototype buildings satisfy the criteria of withstanding the set of seismic actions

37


developing SCWB plastic mechanisms. And for one story or three story buildings, the

proposed procedure to dimension the buildings is working acceptably.

38


CHAPTER 5

PARAMETRIC STUDY OF THREE-STORY BUILDINGSIntroduction

The study of multi story buildings is limited to two story buildings. These

buildings are characterized by the parameters presented in chapters 3 and 4. The purpose

of this chapter is to do a parametric study to evaluate the effect on the torsional response

of the building of different values for some critical parameters. The same approach used

in chapter 9 for one story buildings is used herein. In general, the same criteria used in

that chapter are applied for the three story buildings. When a different consideration is

done, it is explained in the corresponding section.

In the previous chapter, two cases are considered: all angles of incidence are equal

to zero (IA = 0°), and the evaluations are done using the specified set of twelve incidence

angles (I A = ALL). In this chapter, only the case with IA = ALL is evaluated.

Parameters Studiedhe sources of eccentricity described in section 8 are evaluated in this parametric

study, i.e., the location of the CM, the location of the static Center of Rigidity, and the

strength asymmetry induced by an accidental strength reduction of one frame with

respect to its parallel frame. The results of the parametric studies are presented only for

the building with non linear behavior.

Effect of Mass Eccentricity. Only static eccentricities on the X axis are considered for this study. The static

eccentricity in the Y direction is taken as zero. The static eccentricities on the X axis are

39


induced by distributing the mass such that its CM has different locations. The static

eccentricities used in this study are expressed as ratios of emx/b and the adopted values,

V, are in the set: V* € 0,0.05,0.10,0.15.

The buildings have two story levels and each one have the CM at a location that

can be different with respect to the other levels. The author considered convenient to

include the following three potential patterns of CM locations in the parametric study:

Pattern 1: All CM are located on the same side of the master node and at the

same distance.

Pattern 2: Three different cases are generated. The first case has the location of

the first story CM at the point defined by the Vi value, and the other story levels have

zero eccentricity. The second case has the eccentricity only in the second story level. And

the third case has the eccentricity only in the third story level.

Pattern 3: The location of the CM at each story level is alternated on the X axis.

The three patterns are postulated with the intention of studying more possible locations of

the CM and their impact on results. The maximum demands are calculated using the

algorithm presented in section 4.That algorithm is modified to include the evaluations for

the three patterns defined above. Two different prototype buildings are studied herein.

These prototypes are designed in section 7.3. One has the same column and beam

properties at each story level, and is called the case with Constant properties. Its results

are shown in figures 10.1 to 10.12. The other prototype building has columns and beams

that reduce their stiffness and strength with the story level, and is called the case with

Varying properties. Its results are shown in figures 10.13 to 10.24.

40


All the figures use the same convention of symbols to identify the results

corresponding to each frame. The continuous lines with squares show results for frame 1,

dashed lines with squares for frame 2, continuous lines with circles for frame 3, and

dashed lines with circles for frame 4. This convention is used consistently through all the

graphs.

The type of plastic mechanism developed by the two different prototype buildings

is identified by inspection of the maximum demands of rotation ductility in beams and

columns. For the constant properties building, CP, and the varying properties building,

VP, only the first level columns reach their yield rotation. See figures 10.10 and 10.22.

This data is not enough to know if columns are yielding at both ends or only at one end.

Though, the second and third level columns do not yield, as seen in figures 10.11 and

10.12 for the CP building, and in figures 10.23 and 10.24 for the VP building. All the

beams yield at a certain point in time, as seen in figures 10.7 to 10.9 for the CP building,

and in figures 10.19 to 10.21 for the VP building. For multi story buildings, additional

information is necessary to identify the type of plastic mechanism developed.

Figures 10.4 and 10.16 demonstrate that frame 4 can develop larger demands of

normalized story shear force, for emx/b — 0.15. This can happen only if the frame has

not developed a column sway mechanism. Therefore, the CP and the VP building

develop SCWB plastic mechanisms. In conclusion, the procedure proposed in this

research, to design the three story buildings, created designs that comply with the

postulated design goals of the philosophy of Capacity Design. From these results can be

inferred that the buildings have additional ductility capacity, though the margin cannot

be estimated. However, it can be stated that the used seismic performance factors are

41


satisfactory. The analysis of the results of each individual frame reveals that frame 4 has

the highest response demands.

For the CP and the VP buildingsThis frame is the one that is closer to the CM. Comparing results of the CP and

the VP buildings, it can be observed that the CP building has smaller demands of rotation

ductility for the beams in the three levels. About 29.0 for the critical beam in the CP

(second level beam) versus about 45.0 for the critical beam in the VP (third level beam).

And the contrast is more noticeable when comparing maximum demands of rotation

ductility in columns. Thus, the CP building has a better behavior, in terms of the ductility

demands, than the VP building. However, a commonly seen solution, in the

professional practice, it is to reduce sizes of columns in upper levels. From this point of

view, the VP building behavior is representative of more buildings actually built.

Effect of Stiffness EccentricityOnly static eccentricities on the X axis are considered for this study. The static

eccentricity on the Y direction is taken as zero, and the CM is at the master node. The

static eccentricities on the X axis are induced by changing the location of the frame

number 3, to force a change of the location of the static Center of Rigidity. The static

eccentricities are expressed as ratios of ex/b and the adopted values, V, are in the set:

V e 0, 0.05,0.10, 0.15,0.20, 0.25. In this section, the parametric study is done only

for the VP building, and its results are shown in figures 10.25 to 10.36. The maximum

demands are calculated using the algorithm in section 9.2.1. The figures are organized in

a similar way as in section 10.2.1, and use the same symbol conventions. The type of

plastic mechanism developed by the building is identified using the same arguments used

for the mass eccentricity case. The conclusion is that the building develops a SCWB type

42


of plastic mechanism. Therefore, the design procedure used, works satisfactorily. For

one story buildings, frame 3 (closer to the CM) has the tendency to have the largest

maximum demands, but for the three story building, frame 1 and 2 tend to have the

maximum demands. This pattern of behavior could not be extrapolated from the one

story buildings behavior. An interpretation of this pattern of behavior is that frames 3

and 4 reduce their contribution to the total instant Me//, and frames 1 and 2 increase their

contribution to balance the demand. The reduction of frame 3 and 4 would be due to a

reduction in their separation for larger static eccentricities.

Effect of Strength AsymmetryThis section studies the effect of the strength asymmetries on the torsional

response. The evaluated buildings are doubly symmetric in stiffness and mass, i.e., their

static center of rigidity and their mass center are located at the master node. The strength

asymmetry is introduced through strength reductions, Rsi, from 0% up to 20% of the

original strength of beams and columns in the frame 3 only. The strength reductions are

handled through a "fraction of original strength" factor. This factor is evaluated as Vi = 1

— Rsi. The set of used factors, V, are: Vi € 1.0,0.95,0.90,0.85,0.80. This Vi values

are applied to frame 3 in three different ways, creating three different cases of buildings

to be evaluated: 1. Case 1. Original plastic moment, MP, of columns and beam in story

1 is multiplied by Vi. 2. Case 2. Original MP of columns and beam in story 2 is

multiplied by Vi. 3. Case 3. Original MP of columns and beam in story 3 is multiplied

by Vi. The maximum demands are calculated using the algorithm in section 9.2.1, and

the results are shown in figures 10.37 to 10.72, for the three cases described above.

Reviewing the results of the three postulated cases, it is evident that an increase of both

beam and column rotation ductility is concentrated at the respective story level, e.g., at

43


level 1, for case 1, the increment of maximum demand on the beam is 110% and on the

column is 260%. For a Vi = 1.0 (without strength reduction), the building is identical to

the building studied in the mass eccentricity case for emx/b = 0. Thus, a SCWB plastic

mechanism is developed by the building. The results shown in the figures, do not include

enough information to prove that a SCWB is developed for the other V values.

However, due to the concentration of ductility demand on the frame elements located at

one level, the conditions are set to develop a column sway mechanism at that level. It is

necessary to study this possibility with more detail in future investigations. Another

pattern of behavior is that frame 3 is the most demanded frame and its parallel frame

tends to increment its response. Though, at a much smaller ratio. And the frames 1 and 2

are sligthly affected.

ConclusionsThe main purpose of this chapter is to evaluate the effect of some parameters on

the torsional response of three story buildings. These parameters are the static Center of

Rigidity, the location of the Center of Mass, and the magnitude of the strength-

asymmetry in parallel frames. The effect is evaluated through a parametric study, which

evaluates the maximum response demands.

The following list summarizes the main findings from the analysis of the results

of these parametric studies:

1. Two types of building were studied herein. A building with constant properties

(CP) and a building with varying properties (VP). These buildings were designed

using the code-prescribed seismic performance factors. The R, the Q,0, and the d

factors. For the cases of mass and stiffness eccentricity, these buildings were able

44


to develop a strong column-weak beam type of plastic mechanism. A reserve of

capacity to resist unexpected seismic events, larger than the MCE, is expected.

Though, it cannot be quantified with the available information from this research.

2. The CP building kept the maximum demands of ductility at smaller levels than

the VP building. This results suggest that CP configurations should be considered

when designing new buildings. Despite that more VP buildings are built than CP

buildings.

3. When the effect of accidental variations of strength on the torsional response are

evaluated, a significant increase of maximum demands is found. These variations

in strength are localized in an small part of a building. The studied cases

considered these variations in one story of a frame at a time. For the largest

strength reduction studied, the increases in maximum demands of rotation

ductility for beams are 110% and for columns is 260%.

45


CHAPTER SUMMARY AND CONCLUSIONS

Research SummaryDuring large seismic events, a significant number of buildings have suffered from severe

damage to complete collapse. Some researchers have found that a significant percentage

of these buildings had excessive torsional response. Many of these damaged buildings

were built before the development of the modern practice of earthquake engineering.

Thus, the design limitations of some decades ago were the source of most of these

problematic structures.

Nowadays, the knowledge of torsional behavior has evolved notably, but it still needs

improvement. There are several aspects of this behavior that are not well known.

Although, many researchers have worked on this topic for long time, more research needs

to be done for multi story buildings, especially when they develop a non linear behavior.

The main intention of this research is to study the torsional response of simplified framed

buildings with one and three stories.

1. Their response is studied to find the patterns of behavior characteristic of this

phenomenon. In the pursue of this goal, this research was done. The present study

consists of the following:

2. Description of the mathematical model of a one-story and two-story prototype

buildings

3. Design of an one-story and a three-story prototype building

4. Search for patterns of behavior of buildings responding in a torsional mode

5. Perform parametric studies of the one-story and the three-story buildings

46


ConclusionsThe main conclusions of the research presented herein are:

1. A building with static eccentricity only on the X axis, shows instant eccentricity

in both orthogonal directions X and Y during its dynamic response to a

bidirectional seismic action. The common procedures used to evaluate the

torsional moment do not consider the existence of an instant eccentricity on the Y

direction

2. The instant eccentricity can have a magnitude quite different with respect to the

static eccentricities

3. In a torsionally unbalanced building, the instant resultant of reaction forces can

pass through the location of the mass center during some instants, but most of the

time it moves to other locations. However, there is a tendency to align itself with

the mass center

4. The nominal torsional moment is a poor predictor of the maximum effective

torsional moment, and it is unconservative in some of the studied cases. Thus, the

concept of nominal torsional moment requires a critical revision

5. The existence of accidental strength eccentricities, in a torsionally balanced

building, can induce relatively large torsional responses. The nominal torsional

moment cannot account for this effect

6. The strength eccentricity has the potential to render a carefully designed building

to develop strong column-weak beam plastic mechanisms, into an unintended

condition were a column sway mechanism can form at a story level

7. For the cases in the parametric study, it was found that ignoring the possibility of

diverse incidence angles leads to unconservative predictions of the maximum

47


demands. Therefore, the use of a finite set of incidence angles is recommended

for analysis

8. Seismic performance factors play a central role in the control of the plastic

mechanisms that can be developed in a building.

Recommendations for Future Research Based on this study, the following additional studies are recommended:

1. The possibility of having different incidence angles, other than on the orthogonal

principal axes, needs to be considered in design of buildings. For research

purposes, the author believes that it is a must. For practical applications to the

design of new buildings it is necessary, but a strong opposition from practicing

engineers can be anticipated. As an alternative approach, that could be used in an

engineering office, a procedure that includes analysis with IA=0 and IA=90, plus

a carefully evaluated correction factor, could be developed

2. The author found that the seismic performance factors are not well understood by

a significant number of practicing engineers and researchers. It is true that there

are engineers and researchers that understand very clearly these factors and their

implications for the dynamic behavior of a building. But, these factors and the

philosophy of Capacity Design need to be taught in a more comprehensive way.

The teaching of these topics in a prescriptive style limit the future engineers of a

sound understanding of these crucial concepts for the successful design of

buildings in seismic zones. Therefore, work need in this area on how to teach

these concepts in a more clear and conceptually useful way

48


3. The eccentricity induced by planned or accidental differences in strength of the

frames of a building has important effects on the maximum demands. Arguably,

the accidental eccentricity prescribed by the codes already includes these effects.

Intriguingly, the USA and the Mexican design codes use different values of

accidental eccentricity. May be that in the USA the accidental asymmetries in

strength are minimized, with respect to other countries, or may be this issue is not

properly or completely addressed. A rational way to include these effects needs to

be developed

4. The mathematical model prepared for this research included the possibility of

local plasticization of the beams and columns. To simplify the model, the plastic

hinge concept was adopted. Future work need to implement different methods to

consider the non linear behavior of the building. The use of the plastic hinge,

without any strength or stiffness degradations, creates doubts in the conclusions

presented here. Thus, the work done in this research should be expanded to

confirm the generalization of the conclusions obtained here

5. In the few last years, the computer hardware capabilities has improved at a much

faster pace than structural engineers have improved their computer algorithms

and approaches to the solution of the complex problems in seismic design.

Engineers have to move faster to be able to use all the computer power available

today (year 2009), developing more ambitious algorithms to solve the most

demanding problems they face. In this way, approaches of solution that seemed

impossible some years ago, today they have become possible. The current trend to

emphasize approximate analysis methods in earthquake engineering practice

49


misuses all the available computer power. The capacities of the structural

analysis software available in the market, has been dictated more by the

particular needs of the companies that develop this software, than by the

particular needs of the practicing engineers, academics, and researchers. Thus,

engineers have become mostly users of black boxes. Hopefully, everything done

by the software is conceptually understood by the engineer, but experience has

taught the author that there is a big difference between knowing concepts and

mastering them. Thus, engineers need to take the control of the development of

the future software tools. Initiatives like the one implemented by the University of

California at Berkeley seem to be in the right direction. Their Open System for

Earthquake Engineering Simulation (OpenSees), an open-source development,

needs to be supported by more researchers. It could be a contribution to their

work or a parallel development by other research institutions.

Implications for Concrete Two-Storey Design The results displayed in Figures 4 through 15 have implications in the planning and

design of concrete buildings. These implications are oriented toward designing the two-

storey building with the best hydraulic performance as measured by exhibiting the lowest

seepage rate of water through the concrete vault floor. Decreased seepage leads to lower

releases from the disposal and consequently a lower risk of exposure to humans. The

results indicate roof slope is not an important design consideration; however scale effects

(i.e., varying vault half-widths) do affect hydraulic performance.

An alternative concrete vault design suggested by these simulation results is “double

containment.” In this configuration, the outer vault would be similar to a large storage

area that would be filled with containers holding the waste. A larger-scale concrete vault

50


having a clay outer cover layer (minimally) could be utilized as the outer containment.

The outer containment would initially reduce the flow of water through the vault.

However, as discussed previously, the outer layer would degrade. Water passing through

this degraded outer concrete vault would be conditioned (e.g., the pH would increase),

thus protecting the inner containment that holds the waste from concrete degradation.

Furthermore, as indicated by the results presented in this paper, the smaller scale inner

containment would better divert flow towards the side of that inner containment.

Another design consideration is suggested by these simulation results. Engineered covers

are designed for placement near the ground surface and away from the below ground

concrete vault (DOE 2000). Because of the location, these ground surface covers are

susceptible to failure from activities related to plant growth, animals, and humans. Failure

at the ground surface could redirect infiltration toward the below ground concrete vault. It

is recommended that the cover by moved away from the surface and instead be placed

adjacent to the concrete vault. The simulations results show the positive effects on

hydraulic performance of using clay as a cover layer for the concrete vault and the

negative effects on hydraulic performance of using loam and sand (i.e., backfill) as a

cover layer for the concrete vault.

There is a significant scale effect for both degraded and intact concrete at lower

infiltration rates. At higher infiltration rates water is perched on the vault roof and vault

half-width does not affect the seepage rate. The magnitude of the scale effect increases

with decreasing infiltration rate. The scale effect is greater for intact concrete than for

degraded concrete, because there is greater variability in seepage rates over the range of

vault half-widths with intact concrete than with degraded concrete. These results suggest

51


that although present for both degraded and intact concrete, the scale effect is greater

initially and decreases as the concrete degrades.

For both intact and degraded concrete, perched water exists on the vault roof at higher

infiltration rates and is evident when the seepage rate reaches a maximum value as

infiltration increases. Therefore, once there is perched water, water is diverted around the

vault and the vault hydraulic performance is unchanged even though infiltration

increases. Water perches for intact concrete at lower infiltration rates than for degraded

concrete. Additionally, intact concrete vault has a higher saturation than the degraded

concrete vault. Paradoxically, the low permeability of intact concrete and good hydraulic

performance promotes a greater degradation rate.

A summary of above conclusions from these modeling simulations is provided below.

Clay layers placed adjacent to the concrete were found to lower water flow

through the vault and enhance hydraulic performance.

Smaller vault sizes result in lower flow rates and indicate a scale effect.

Roof slope has a relatively small influence on hydraulic performance.

Although not evaluated in this work, the simulation results suggest that a “double

containment” system comprised of outer and inner containments would yield an

enhanced hydraulic performance over a single concrete vault. The outer vault would

initially provide a low permeability barrier to flow. Once the outer vault degrades, the

water passing inside of the outer vault would be conditioned and then contact a smaller-

scale concrete vault. As indicated in the results from this paper for scale effects, the

52


smaller vault would have more of a capability to divert flow towards the sides of the

vault. Future work will evaluate this design.

There is a significant scale effect for both degraded and intact concrete at lower

infiltration rates. At higher infiltration rates water is perched on the vault roof and vault

half-width does not affect the seepage rate. The magnitude of the scale effect increases

with decreasing infiltration rate. The scale effect is greater for intact concrete than for

degraded concrete, because there is greater variability in seepage rates over the range of

vault half-widths with intact concrete than with degraded concrete. These results suggest

that although present for both degraded and intact concrete, the scale effect is greater

initially and decreases as the concrete degrades.

For both intact and degraded concrete, perched water exists on the vault roof at higher

infiltration rates and is evident when the seepage rate reaches a maximum value as

infiltration increases. Therefore, once there is perched water, water is diverted around the

vault and the vault hydraulic performance is unchanged even though infiltration

increases. Water perches for intact concrete at lower infiltration rates than for degraded

concrete. Additionally, intact concrete vault has a higher saturation than the degraded

concrete vault. Paradoxically, the low permeability of intact concrete and good hydraulic

performance promotes a greater degradation rate.

A summary of above conclusions from these modeling simulations is provided below.

Clay layers placed adjacent to the concrete were found to lower water flow

through the vault and enhance hydraulic performance.

53


Smaller vault sizes result in lower flow rates and indicate a scale effect.

Roof slope has a relatively small influence on hydraulic performance.

Although not evaluated in this work, the simulation results suggest that a “double

containment” system comprised of outer and inner containments would yield an

enhanced hydraulic performance over a single concrete vault. The outer vault would

initially provide a low permeability barrier to flow. Once the outer vault degrades, the

water passing inside of the outer vault would be conditioned and then contact a smaller-

scale concrete vault. As indicated in the results from this paper for scale effects, the

smaller vault would have more of a capability to divert flow towards the sides of the

vault. Future work will evaluate this design.

54


References

Ahmed, S. ‘‘Analysis of Fluid Flow Through Intact and Degraded Below Ground Vaults,’’ Master of Science Project, University of Texas at El Paso, 1995.

Booth, C. J. and B. C. Price, ‘‘Infiltration, Soil Moisture, and Related Measurements at a Landfill with a Fractured Cover,’’ Illinois, Journal of Hydrology. 108. (1-4). p. 175-188, 1989.

Brun, P., M. Audiguier, J. Billiotte, M. Deveughele, ‘‘Experimental and Numerical Study of the Infiltration Phenomena in a Compacted Clay Designed for the Deposit of Short Period Radioactive Wastes,’’ Engineering-Geology. 37. (2). p. 123-136, 1994.

Clifton, J. R. and L. I. Knab, ‘‘Service Life of Concrete,’’ NUREG/CR-5466, 1989.

Jury, W.A., Gardner, W.R., and Gardner, W.H., “Soil Physics,” John Wiley and Sons, New York, 1991.

Kacimov, A. R. and Y. V. Obnosov, “Steady Water Flow Around Parabolic Cavities and Through Parabolic Inclusions in Unsaturated and Saturated Soils”, Journal of Hydrology, Vol. 238 p. 65-77, 2000.

Nichols, W. F. and P. D. Meyer, ‘‘Multidimensional Water Flow in a Low-Level Waste Isolation Barrier’’, Groundwater, Vol. 34(4) p. 659-665, July-August 1996.

Shahjahan, A.S. ‘‘Flow Through Degraded Below Ground Concrete Vault,’’ Master of Science Project, University of Texas at El Paso, 1995.

van der Sloot, H. A., ‘‘Characterization of the Leaching Behaviour of Concrete Mortars and of Cement-Stabilized Wastes with Different Waste Loading for Long Term Environmental Assessment’’, Waste Management, Vol. 22 p. 181-186, 2002.

Walton, J. C. and R. R. Seitz, ‘‘Fluid Flow Through Fractures in Below Ground Concrete Vaults, Waste Management,’’ Vol. 12 p. 179-187, 1992.

Walton, J. C., L. E. Plansky, and R. W. Smith, ‘‘Models for Estimation of Service Life of Concrete Barriers in Low-Level Radioactive Waste Disposal,’’ NUREG/CR-5542, 1990.

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Warrick, A. W., Wierenga, P. J., and L. Pan, “Downward Water Flow Through Sloping Layers in the Vadose Zone: Analytical Solutions for Diversions”, Journal of Hydrology, Vol. 192 p. 321-337, 1997.

Weeks, O. L., R. S. Mansell, and S. W. McCallister, ‘‘Evaluation of Soil Top-cover Systems to Minimize Infiltration Into a Sanitary Landfill; A Case Study.’’ Environmental-Geology-and-Water-Sciences. 20. (2). p. 139-151, 1992.

Zhang, X., Bengough, A. G., Crawford, J. W., and I. M. Young, ‘‘Efficient Methods for Solving Water Flow in Variably Saturated Soils Under Prescribed Flux Infiltration’, Journal of Hydrology, Vol. 260 p. 75-87, 2002.

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Appendix A Figures

Figure 6: Discretized grid of the model domain showing the nominal concrete vault with a half-width of 1,000 cm (10 m).

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