Copyright Sally · In the mid‐1960s, Fazlur Khan created a hierarchy of structural systems, ranging from two‐ dimensional moment resisting frames to three‐dimensional tubular

MODERNIZING THE SYSTEM HIERARCHY FOR TALL BUILDINGS: A DATA‐DRIVEN APPROACH TO

SYSTEM CHARACTERIZATION

A Thesis

Submitted to the Graduate School

of the University of Notre Dame

in Partial Fulfillment of the Requirements

for the Degree of

Master of Science in Civil Engineering

by

Sally Suzanne Williams

Tracy Kijewski‐Correa, Director

Graduate Program in Civil Engineering and Geological Sciences

Notre Dame, Indiana

April 2014

© Copyright 2014


MODERNIZING THE SYSTEM HIERARCHY FOR TALL BUILDINGS: A DATA‐DRIVEN APPROACH TO

SYSTEM CHARACTERIZATION

Abstract

by


In the mid‐1960s, Fazlur Khan created a hierarchy of structural systems, ranging from two‐

dimensional moment resisting frames to three‐dimensional tubular systems, to aid designers in

making efficient choices to resist lateral loads. While this hierarchy has historically been a

valuable tool for designers, the ever‐advancing modeling and computational capabilities have

enabled far more exotic structures to become inhabitable possibilities. This implies that few

modern systems obey this classical hierarchy, requiring a new approach to classify structural

systems and their applicability to modern practice as both a design aid and educational tool for

future designers. Therefore, this thesis will respond to this need by modernizing the hierarchy,

not from first principles or theory, but actually from practice by mining the attributes of

constructed systems already in existence. The result of this thesis is a newly proposed system

descriptor, a database structure and procedure to generate modern hierarchies that can be

dynamically updated with time.

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CONTENTS

Figures .............................................................................................................................................. v

Tables ix

Acknowledgments............................................................................................................................ x

Chapter 1: Introduction ................................................................................................................... 1

1.1 Motivation .................................................................................................................... 1

1.2 Overview of Traditional System Classification .............................................................. 2

1.2.1 Overview of Historical System Hierarchy...................................................... 3

1.2.2 Limitations of Historical System Hierarchy ................................................... 5

1.3 Need for Updated Hierarchy ......................................................................................... 7

1.4 Parameterizing System Databases ................................................................................ 8

1.5 Research Objectives .................................................................................................... 10

Chapter 2: Formalizing A New Descriptor of System Behavior ..................................................... 12

2.1 Historical DCA Measures ............................................................................................. 12

2.2 iDCA Development ...................................................................................................... 15

2.2.1 iDCA Calibration: Continuous Mode Shapes ............................................... 16

2.2.2 iDCA Calibration: Discontinuous Mode Shapes .......................................... 19

2.3 Demonstrative Example .............................................................................................. 31

2.4 Summary ..................................................................................................................... 34

Chapter 3: DCA Validation Through Case Studies .......................................................................... 36

3.1 Introduction ................................................................................................................ 36

3.2 Results ......................................................................................................................... 38

3.2.1 CS1 Case Study ............................................................................................ 43

3.2.2 CS2 Case Study ............................................................................................ 48

3.2.3 CS3 Case Study ............................................................................................ 53

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3.2.4 CS4 Case Study ............................................................................................ 57

3.2.5 CS5 Case Study ............................................................................................ 61

3.2.6 CS6 Case Study ............................................................................................ 67

3.2.7 CS7 Case Study ............................................................................................ 73

3.2.8 CS8 Case Study ............................................................................................ 80

3.2.9 CS9 Case Study ............................................................................................ 84

3.2.10 CS10 Case Study ........................................................................................ 88

3.3 Summary ..................................................................................................................... 92

Chapter 4: Database Population and Mining ................................................................................. 96

4.1 Introduction ................................................................................................................ 96

4.2 Data‐Driven Hierarchy for Modern Systems ............................................................. 108

4.2.1 Geometric Descriptors: Height ................................................................. 109

4.2.2 Geometric Descriptors: Aspect Ratio ........................................................ 114

4.2.3 Behavioral Descriptors: MS‐DCA .............................................................. 118

4.2.4 Behavior Descriptors: iDCA ....................................................................... 121

4.3 Modern System Hierarchies ...................................................................................... 127

4.4 Summary ................................................................................................................... 131

Chapter 5: Conclusions and Future Work .................................................................................... 133

5.1 Research Summary ................................................................................................... 133

5.2 DCA Development ..................................................................................................... 133

5.2.1 iDCA Verification ....................................................................................... 134

5.3 Database Population ................................................................................................. 134

5.3.1 Modernized System Hierarchies ............................................................... 135

5.4 Future Work .............................................................................................................. 136

5.4.1 iDCA Refinement ....................................................................................... 136

5.4.2 Database Expansion and Virtualization .................................................... 136

Appendix A: iDCA Mapping .......................................................................................................... 138

Appendix B: Supplementary Case Studies ................................................................................... 142

B. 1 CS3 Case Study ......................................................................................................... 142

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B. 2 Central Plaza Case Study .......................................................................................... 145

B. 3 CS4 Case Study ......................................................................................................... 151

B. 4 John Hancock Tower Case Study ............................................................................. 154

Bibliography ................................................................................................................................. 157

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FIGURES

Figure 1.1: Premium for height for high‐rise structures (Zils and Viise 2003). ................................ 2

Figure 1.2: Khan’s hierarchal comparison of structural systems (CTBUH 1980). ............................ 3

Figure 1.3: Steel, reinforced concrete and composite companions to Khan’s structural system hierarchy (Sarkisian 2011). ................................................................................................. 4

Figure 1.4: Function of the 100 tallest buildings, per decade (CTBUH 2011). ................................. 6

Figure 1.5: Breakdown of existing tall buildings by (a) region, (b) function and (c) material (CTBUH 2013). ..................................................................................................................... 6

Figure 1.6: Story heights of existing buildings as a function of structural system type (CTBUH 2010). .................................................................................................................................. 8

Figure 1.7: Damping as a function of DCA quantified by mode shape power (a) distinguished by material, (b) distinguished by system classification, (c) best‐fit linear trend by material, (d) best‐fit exponential (all points) (Williams et al. 2013). ................................................. 9

Figure 2.1: Degree of reliability of DCA measures (Williams et al. 2013). ..................................... 12

Figure 2.2: Mode shape power system classification (Bentz 2012). .............................................. 14

Figure 2.3: Comparison of two distributions of extracted slopes used in iDCA: (a) ideal cantilever [target distribution] and (b) mode shape in question. ..................................................... 16

Figure 2.4: Normalized mode shapes for Buildings (a) 1, (b) 2, (c) 3 for case of = 2. ................. 17

Figure 2.5: Examples of vertically discontinuous mode shapes for Building 2: C‐60 (left) and S‐60 (right) with cantilever and shear ideals as well as best‐fit power law. ............................ 20

Figure 2.6: Normalized mode shape with outriggers circled for Buildings (a) 1, (b) 2, (c) 3, with shear and cantilever ideals as well as best‐fit power law provided for comparison. ....... 30

Figure 2.7: Finite element models for the three MRFs with aspect ratios of (a) 1, (b) 5, and (c) 10.32

Figure 2.8: Normalized mode shapes for the three MRFs with aspect ratios of (a) 1, (b) 5, and (c) 10, with ideal shear and cantilever mode shapes and best‐fit power law shown for comparison. ...................................................................................................................... 33

Figure 3.1: Comparison of MS‐DCA (squares) and iDCA (stars) for case study buildings. ............. 39

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Figure 3.2: Example of graphical display used in building case studies (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison. ........................................................................... 43

Figure 3.3: CS1 first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison. . 46

Figure 3.4: CS1 second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.47

Figure 3.5: Mode shape displacement with regards to axis assignment of the CS2. .................... 49

Figure 3.6: CS2’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.51

Figure 3.7: CS2’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison. ...................................................................................................................... 52

Figure 3.8: Axis assignment of the CS3 (Bentz 2012). .................................................................... 53




Figure 3.12: CS4’s first mode without the cap truss (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.................................................................................................... 60

Figure 3.13: CS5’s modal directions (Bentz 2012). ........................................................................ 62



Figure 3.16: CS5’s first mode without space truss (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison........................................................................................................... 65

Figure 3.17: CS5’s second mode without space truss (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.................................................................................................... 66

Figure 3.18: CS6’s general floor plans (CTBUH 1995). ................................................................... 67



Figure 3.21: CS6’s first mode without cap truss (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison................................................................................................................ 71

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Figure 3.22: CS6’s second mode without cap truss (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison........................................................................................................... 72

Figure 3.23: General floor plan of CS7 (Courtesy of RWDI). .......................................................... 74



Figure 3.26: CS7’s first mode without pinnacle/spire (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.................................................................................................... 78

Figure 3.27: CS7’s second mode without pinnacle/spire (a) mode shape with power fit, (b) EMS‐

DCA, and (c) DCA comparison. ............................................................................................ 79

Figure 3.28: CS8’s general floor plan (Carden and Brownjohn 2008). ........................................... 80



Figure 3.31: CS9’s general floor plan (Abdelrazaq et al. 2004). ..................................................... 84



Figure 3.34: CS10’s general floor plan (Li and Wu 2004). .............................................................. 88

Figure 3.35: CS10’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison. ...................................................................................................................... 90


Figure 3.37: Comparison of errors and system behavior for case study buildings. ....................... 95

Figure 4.1: Relationship between height and structural system, distinguished by source fidelity.110

Figure 4.2: Relationship between height and structural system, distinguished by material. ..... 111

Figure 4.3: Relationship between aspect ratio and structural system, distinguished by source fidelity. ............................................................................................................................ 115

Figure 4.4: Relationship between aspect ratio and structural system, distinguished by material.116

Figure 4.5: Relationship between MS‐DCA and structural system, distinguished by material. .. 119

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Figure 4.6: Relationship between iDCA and structural system, distinguished by material. ........ 122

Figure 4.7: Relationship between iDCA and structural system, distinguished by material (including Chapter 3 Case Studies). ................................................................................ 124

Figure 4.8: Modernized hierarchy, parameterized by height. ..................................................... 128

Figure 4.9: Modernized hierarchy parameterized by aspect ratio (slenderness). ....................... 129

Figure 4.10: Modernized hierarchy parameterized by degree of cantilever action (iDCA). ........ 131

Figure B.1: CS3’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.143

Figure B.2: CS3’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison. .................................................................................................................... 144

Figure B.3: Axis assignment of Central Plaza (Bentz 2012). ........................................................ 146

Figure B.4: Central Plaza’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison. .................................................................................................................... 147

Figure B.5: Central Plaza’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison. .................................................................................................................... 148

Figure B.6: Central Plaza’s first mode without cap truss (a) mode shape with power fit, (b) EMS‐

DCA, and (c) DCA comparison. .......................................................................................... 149

Figure B.7: Central Plaza’s second mode without cap truss (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison. .................................................................................... 150

Figure B.8: CS4’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.152

Figure B.9: CS4’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison. .................................................................................................................... 153

Figure B.10: John Hancock Tower’s general floor plan (Blanchet 2013). ................................... 154

Figure B.11: John Hancock Tower’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.............................................................................................................. 155

Figure B.12: John Hancock Tower’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison......................................................................................................... 156

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TABLES

Table 2.1 Comparison of iDCA Values for Vertically Continuous Systems .................................... 18

Table 2.2 DCA Sensitivity to Vertical Discontinuity: Progression 1 ................................................ 22

Table 2.3 DCA Sensitivity to Vertical Discontinuity: Progression 2 ................................................ 26

Table 2.4 DCA Sensitivity to Vertical Discontinuity: Outriggers .................................................... 30

Table 2.5 Application of DCAs to MRFs of Varying Aspect Ratio .................................................. 33

Table 3.1 Key Characteristics of Case Study Buildings ................................................................... 37

Table 3.2 iDCA and MS‐DCA for Fundamental Modes of Case Study Buildings ............................ 40

Table 3.3 Comparison of iDCA and MS‐DCA for Case Study Buildings .......................................... 93

Table 4.1 Buildings Used in Proposed Database with Sources for the Systems and Aspect Ratios Data ................................................................................................................................... 98

Table 4.2 Verification of Google Earth Measurements with Published Aspect Ratios ................ 105

Table 4.3 Numerical Identifier for Each System Type .................................................................. 106

Table 4.4 Height as Geometric System Descriptor: Statistics by System Type ............................ 113

Table 4.5 Aspect Ratio as Geometric System Descriptor: Statistics by System Type .................. 117

Table 4.6 MS‐DCA as Behavioral System Descriptor: Statistics by System Type ......................... 119

Table 4.7 iDCA as Behavioral System Descriptor: Statistics by System Type ............................... 122

Table 4.8 System Classification of Chapter 3 Case Studies .......................................................... 124

Table 4.9 MS‐DCA as Behavioral System Descriptor: Statistics by System Type (Including Chapter 3 Case Studies) ................................................................................................................ 126

Table A.1 Look‐Up Table for MS‐DCA to iDCA Mapping .............................................................. 139

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ACKNOWLEDGMENTS

Firstly, thank you to my committee, Dr. Kareem, Dr. Khandelwal, and Dr. Kijewski‐Correa for

their time and input regarding this research. I would like to gratefully acknowledge all the design

firms who contributed to this work, Arup, Magnussen Klemencic Associates, Skidmore Owings

and Merrill, and Thornton Tomasetti, as well as the undergraduate students who helped to

acquire and organize data, specifically Cara Quigley, Tara Rabinek and Dylan Scarpato. Their

assistance and contributions were instrumental in this work, as was the work of Audrey Bentz, a

past DYNAMO member whose research laid the groundwork for this thesis. She has been

wonderful, offering to assist and lend advice wherever needed and being a great mentor in

general.

Furthermore, I would like to thank those in my Notre Dame community who have helped

support me throughout the completion of this project: labmates Andrew Bartolini, Dustin Mix,

and Tara Weigand, Dr. Alex Taflanidis, and fellow graduates Melissa Cheng and Nicholas

Schappler. Their support, encouragement and excitement for my studies were essential at

stressful times and wonderful in moments of success. My time in South Bend would not have

been as great without them.

I cannot go without mentioning how wonderfully compassionate and reassuring my

large family was during my graduate studies. It makes me rather emotional to think how my

grandmother, parents, and siblings all supported me in their unique ways and I could not

imagine going through the process to create this thesis without them.

Most importantly I would like to thank my advisor, Dr. Tracy Kijewski‐Correa. The first

time we met, the passion and creativity she applies to the field of structural engineering was

immediately apparent and intoxicating. All her long hours dedicated to my research and texts of

excitement about its progress were more than appreciated. She is inspirational in so many ways

and I owe her more thanks than I can properly translate to paper.

I am incredibly blessed to have all these diverse and generous people in my life including

the many that brevity keeps me from stating specifically.

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CHAPTER 1: INTRODUCTION

1.1 Motivation

In the case of low to mid‐rise buildings, the selection of structural system is often a trivial

matter, complicated only when designing in zones of high seismicity, where restrictions and

even incentives may drive more careful system choices. However, for high‐rise development,

system choice, regardless of seismic design category, has significant implications that drive

project economy and efficiency (Zils and Viise 2003). Even for elements of the gravity system,

the cost of poor choices that fail to minimize floor‐to‐floor height are quickly compounded over

20, 30 or even 100 stories. More importantly, inefficient lateral systems dramatically increase

requisite member sizing through the so‐called “Premium for Height,” driving up not only project

cost for the members themselves but also overall structural weight and demands on foundation

systems. As evident in Figure 1.1, the amount of material increases linearly for the gravity

system with building height. The same is not true of the lateral system, due the nonlinear

increases in lateral loads with height. Thus there is a significant “Premium for Height” (Zils and

Viise 2003). It is for these reasons that in supertall buildings, structural system

conceptualization is often regarded as an art, with principles whose demands far outshadow

that required for low‐rise buildings (Halvorson 2008). Thus there is considerable interest in

developing tools that can guide system selection. As such, the Council on Tall Buildings and

Urban Habitat (CTBUH) has called for renewed research efforts to database tall buildings in

conjunction with full‐scale monitoring to compare in‐situ behavior with predicted behaviors,

with particular emphasis on “the design and performance of structural systems for complex tall

building forms and geometries,” (Oldfield et al. 2014).

2

Figure 1.1: Premium for height for high‐rise structures (Zils and Viise 2003).

1.2 Overview of Traditional System Classification

Historically, “tall” buildings kept their gravity and lateral systems separate. Lateral systems were

variations on moment resisting frames (MRFs), a so‐called two‐dimensional system that was

well understood and allowed frames at individual column lines to be readily analyzed using

approximate methods by hand. There was little need for system selection guidelines at this

time, but by the 1960s, there was a paradigm shift in tall building systems toward three

dimensional systems that engaged perimeter frames in both directions simultaneously to

behave much like a hollow, thin‐walled cantilever beam. Many of the advancements in this era

can be credited to Fazlur Khan of Skidmore, Owings and Merrill (SOM) in Chicago, who

embraced emerging computational capabilities in the design of these new structural forms:

“[Khan’s] design for the 100‐story, 1,127‐foot CS1 of 1965, put [the] ‘tube system’ to a

spectacular test. Khan used computer analysis to determine exactly how a tube supported by

columns in conjunction with giant cross‐braces would respond three‐dimensionally and

dynamically to the forces of the wind” (Fenske 2013). By the time the World Trade Center, the

CS1, and CS2 were completed (mid‐1970s), the possibilities for system typology had radically

expanded, giving structural engineers newfound choices and analysis capabilities and freeing the

3

vision of architects to conceive forms that deviated significantly from the traditional rectangular

plan supported by planar frames.

1.2.1 Overview of Historical System Hierarchy

Owing to his innovations in both system conceptualization as well as modeling, Khan is often

regarded as the Father of Skyscrapers, who passed on to his descendants a means to navigate

this new landscape in structural systems. His vision produced what is perhaps one of the most

referenced conceptual design aids for tall building systems: the hierarchy in Figure 1.2. The

hierarchy represented a spectrum of steel tall building systems from MRFs to tubular systems.

For each class of system, Khan indicated a number of stories beyond which, based upon his

experience, the system was no longer efficient and a transition to a new system typology was

warranted (CTBUH 1995). Over the years, designers have modified and expanded this chart,

including creating companion charts for concrete and composite structures (McNamara 2005;

Sarkisian 2011; Taranath 1998; Taranath 2012; Zils and Viise 2003), though they have remained

married to number of stories (a rough surrogate for height) for the chart’s parameterization (see

Figure 1.3). Designers have utilized the hierarchy and its successors for decades as a rule of

thumb, but also as an important educational tool to train young designers in the philosophy of

system design.

Figure 1.2: Khan’s hierarchal comparison of structural systems (CTBUH 1980).

4

Figure 1.3: Steel, reinforced concrete and composite companions to Khan’s structural system hierarchy (Sarkisian 2011).

5

1.2.2 Limitations of Historical System Hierarchy

While these charts have been beneficial for designers in the past, the ever‐advancing modeling

and computational capabilities available to designers as well as the free‐form architecture

movement have necessitated many more exotic system typologies than the general classes

encompassed by these hierarchies. These modern structures are in stark contrast to the

homogeneous systems in Khan’s hierarchy, which are continuous vertically not due to a lack of

imagination but to enable analysis by hand or simplified computer programs (McNamara 2005).

Today’s computational freedom has given designers far greater liberty to employ mixed systems

and entirely new classes of structural systems that yield even greater efficiency than Khan’s

tubes, e.g., external diagrids which offer a more efficient use of material than trussed tubes

(Tomasetti et al. 2013), as well as completely new classes like mega systems (superframes) and

stayed/buttressed masts shown in the expanded hierarchies of Figure 1.3.

The diversification of system typologies from Khan’s hierarchy has not only been facilitated by

new computational capabilities but also by functional necessity due to shifts in tall building

occupancy and locale, as structural form is now a “product of characteristics of the developing

countries where these projects are located, cost, change in function, [and] increased

performance [requirements] of structure at great height” (Wood 2013). Historically, office

buildings have dominated the tallest 100 buildings, but as Figure 1.4 demonstrates, the last

decade has witnessed a shift towards residential/mixed‐use/hotel developments, which now

dominate the top 100 projects (Wood 2013) and are likely to continue to grow in demand due to

growing trends in urbanization. Particularly for mixed‐use developments, which updated

projections peg at approximately 30% of the tallest buildings presently (Figure 1.5‐b), different

systems often need be employed, resulting in vertically discontinuous systems for which Khan’s

hierarchy no longer applies. Moreover, tall buildings projects have migrated heavily overseas,

see Figure 1.5‐a, with emphasis in Asia and the Middle East, where construction material

availability and workforce skill set constraints have led to an overwhelming bias toward concrete

and composite construction (Figure 1.5‐c). Thus the homogeneous steel systems within Khan’s

hierarchy have limited capability of encapsulating modern tall buildings.

6

Figure 1.4: Function of the 100 tallest buildings, per decade (CTBUH 2011).

Figure 1.5: Breakdown of existing tall buildings by (a) region, (b) function and (c) material (CTBUH 2013).

7

1.3 Need for Updated Hierarchy

Figure 1.6 provides an excellent example of how modern practice has deviated from Khan’s

hierarchy and more importantly why height or number of stories is not an effective

parameterization for modern system typologies. Consider the outrigger system, previously

deemed suitable by Khan only up to 60 stories, it has been proven effective for super tall towers

due to the development of high‐strength concrete shear wall cores, making it one of the most

popular systems (Tomasetti et al. 2013). Moreover, within a given height range in Figure 1.6,

e.g., the 50‐60 story range previously defined as the regime of the interactive/outrigger system,

there is now a myriad of systems that have been proven in practice to be effective in this height

range. This is thanks to advances in material technology as well as improved understanding of

structural behavior, modeling and analysis. While modern designers may have conceptually

referred to Khan’s hierarchy when conceiving these systems, the final result has evolved as the

result of heuristic assumptions coupled with repeated iterations of design concepts using

computational models until target limit states were satisfied. This process has generated a great

collective wisdom in the tall buildings community regarding how to conceive and design modern

systems. That knowledge, however, has not been captured in such a way that it can be a

reference for current designs and an educational resource for future designers. As such, not only

is a new classification of structural system typologies warranted, one that captures the

community’s collective wisdom in the same way Khan captured his own wisdom back in the

1970s, but one that is capable of accommodating heterogeneous systems. In fact, as previously

stated, CTBUH’s recent research needs report has specifically called for a tall buildings

databasing effort. Therefore, this thesis will respond to this need by modernizing the

hierarchy, not from first principles or theory, but actually from practice by mining the

attributes of constructed systems already in existence.

8

Figure 1.6: Story heights of existing buildings as a function of structural system type (CTBUH 2010).

1.4 Parameterizing System Databases

As the previous section demonstrated, number of stories or height is no longer sufficient to

parameterize modern structural systems. Thus the generation of a modern system hierarchy will

also need to explore a more robust means to classify systems that are often vertically

discontinuous. By determining a more accurate means to describe and classify structural

systems, not only can new system hierarchies be developed, which is the primary objective of

this thesis, but such system descriptors can also be used in the development of empirical models

critical for tall building design. For example, Bentz (2012) demonstrated that a more effective

classifier of modern systems (other than height) could be used in the development of models to

predict their dynamic properties. Specifically, her research proposed that the structure’s Degree

of Cantilever Action (DCA) could be used to predict inherent damping levels, as shown in Figure

1.7, as well as the degree of fidelity required in finite element modeling to achieve an accurate

prediction of in‐situ periods. Both of these are of particular importance to designers of modern

9

tall buildings, known to be dynamically sensitive under the action of wind and for which

accurate prediction of dynamic responses characterized by mass, stiffness, and damping

becomes especially critical to ensuring that governing habitability states can be satisfied.

This realization that height has limited utility in describing system behavior should not be

surprising, as height is not truly what defines the system, but rather is the performance

objective the system enables. In other words, a tube is not a tube due to the fact that its height

is over 60 stories; it is a tube due to the unique lateral load path it engages and the degree of

efficiency it achieves in that load transfer. One can view this efficiency as each system’s ability to

approach the ideal cantilever behavior, hence motivating the DCA parameter first introduced by

Bentz (2012). Based on these findings, it is worthwhile to further explore the concept of system

behavior quantified by DCA for the description and subsequent parameterization of a modern

tall building system database by evaluating various measures of cantilever action, as well as new

measures that overcome identified limitations noted for mixed and discontinuous systems

(Williams et al. 2013). Consequently, the primary objective of this research is to create a

modernized hierarchy of structural systems with a sufficiently robust parameterization for

system behavior derived from the DCA concept.

Figure 1.7: Damping as a function of DCA quantified by mode shape power (a) distinguished by material, (b) distinguished by system classification, (c) best‐fit linear trend by material, (d) best‐fit

exponential (all points) (Williams et al. 2013).

10

1.5 Research Objectives

The creation of a modernized system hierarchy in this thesis has two major phases:

determination of a robust system descriptor to parameterize the database and population and

mining of the database. The former task will require the formalization of a new descriptor, and

its evaluation against common descriptors used to parameterize system databases in the

literature, e.g., height and aspect ratio as well as the DCAs proposed by Bentz (2012). The latter

task will require the assembly of a database of actual building properties, in cooperation with

engineers of record, and its parameterization by each descriptor (established and newly

proposed) to identify clusters and trends, isolating for other variables including material,

function and continent. This will allow the identification of the most appropriate database

parameterization to reveal trends within modern system hierarchies and ultimately offer a

guideline for future system selection that measures system behvaior and mines trends from

structural practice as opposed to subjective opinion of designers (Sarkisian 2011; Taranath 1998;

Taranath 2012).

Accordingly, the primary objectives of this research are:

1. Develop a robust descriptor suitable for heterogeneous systems that is

simple to extract, i.e., requires little effort on the part of cooperating

designers

2. Validate the proposed descriptors against previous DCAs using case studies

of existing tall buildings

3. Populate a comprehensive database of recently built tall buildings with

diverse systems, including significant details that are publically available, as

well as geometric (height, aspect ratio) and DCA descriptors explored in

Objective 2

4. Create a modern hierarchy of systems by mining the database assembled in

Objective 3, revealing underlying trends that can guide future system

selection.

These objectives conveniently map to the subsequent chapters of this thesis:

11

1. Chapter 2 describes the conceptual development and verification of the

new DCA measure (Objective 1)

2. Chapter 3 provides detailed case studies comparing the newly proposed

DCA against the historical DCA measure (Objective 2)

3. Chapter 4 introduces and mines the assembled database using the various

descriptors (Objectives 3 and 4)

4. Chapter 5 concludes the thesis with a summary of major findings and

discussion of future work.

12

CHAPTER 2: FORMALIZING A NEW DESCRIPTOR OF SYSTEM BEHAVIOR

2.1 Historical DCA Measures

As discussed in Chapter 1, to effectively parameterize any database of structural systems, a

robust descriptor of structural system behavior is required. Bentz (2012) first noted the

limitations of historical descriptors, e.g., geometric descriptors, and instead introduced the

concept of the degree of cantilever action (DCA) as an alternative means of classifying tall

building systems for the purposes of predicting in‐situ dynamic properties. Previously, databases

classified structures by primary construction material, then later by the fundamental period or

height (Davenport and Hill‐Carroll 1986; Jeary 1986; Lagomarsino 1993; Satake et al. 2003).

While geometric parameterizations like height or even aspect ratio may be effective for a

collection of buildings with similar typologies, e.g., when comparing a collection of steel MRFs

with similar details, Bentz (2012) found these parameters to be ineffective in facilitating

comparisons across system classes. This was especially evident when moving into the range of

structural systems espoused by modern tall buildings, which are often hybrids of the general

system classes in Figure 1.2 and Figure 1.3. Thus while geometric characterizations of systems

may be the easiest to generate, as they can be simply extracted from publically available data,

they were found to be the lowest fidelity descriptor of structural systems (Williams et al. 2013),

marking the starting point on a reliability progression visualized in Figure 2.1.

Figure 2.1: Degree of reliability of DCA measures (Williams et al. 2013).

As a result, Bentz (2012) went on to propose the DCA as a counter to geometric descriptors,

deriving the DCA initially from the structure’s fundamental sway mode shapes. Mode shapes (ϕ)

13

are commonly fit by a power law expression that is a function of the height (z) to the total

building height (H) ratio:

)()( Hzz (2.1)

The mode shape power () was obtained by a least squares fit. By noting the correlation between the degree of cantilever behavior and the mode shape power, Bentz (2012) used the

mode shape power to classify systems as axial‐ or shear‐dominated. This classification noted

that buildings with fundamental sway mode shapes that obey a linear distribution with height

(=1) can be classified as “shear buildings,” whose frame action is characterized by the local

flexure of beams and columns as their primary (75‐80%) mechanism for lateral force transfer

within the system (Taranath 1998). Similarly, “cantilever buildings” whose lateral load transfer is

increasingly reliant on axial pathways will manifest a quadratic (=2) fundamental sway mode

shape. Based on these bounding limits, Bentz (2012) proposed Figure 2.2 for structural system

classification using mode shape powers as the descriptor. While Bentz (2012) classified

interactive structures as producing a mode shape power between 1.25 and 1.5, this thesis has

broadened that range to include DCAs of 1.25 to 1.75 to achieve symmetry in this subjective

classification, with values above 1.75 being defined as pure cantilever or axial‐dominated

structures. Owing to the fact that this descriptor did consider system behavior rather than

geometry, it is considered to be more reliable, as visualized in the progression in Figure 2.1.

14

Figure 2.2: Mode shape power system classification (Bentz 2012).

Unfortunately, this approach, which we will term MS‐DCA due to its reliance on a global mean‐

square (MS) fit, has some limitations in the case of mixed systems or systems with vertical

discontinuities, e.g., outriggers and cap trusses, in which case the mode shapes are often not

smooth, continuous curves that can be well described by Equation 2.1. Despite these limitations,

the MS‐DCA was attractive since it did not require interrogation of the full finite element model

(FEM) and only used an artifact from that model (fundamental sway mode shape) that is

commonly published in building case studies.

Noting the aforementioned limitations, Bentz (2012) went on to propose a DCA that was derived

from first principles: a cumulative ratio of axial to shear forces within the FEM’s members. While

one may argue this is a potentially more accurate assessment of the behavior (degree of axial

engagement) within a system, its extraction proved to be quite cumbersome for large models,

as it required inventorying forces in every member (Williams et al. 2013). Furthermore, while

the extraction technique developed by Bentz (2012) was fairly straightforward for steel

structures, the same techniques could not be extended easily to concrete structures. Even after

successfully extracting these member forces from concrete elements, there was some question

of how to appropriately normalize this DCA to allow cross‐comparisons between concrete and

steel structures and how to minimize sensitivity to the number of members. The increased

challenge in extracting this DCA from concrete structures was particularly concerning

considering that this is the prevailing material for modern tall buildings. Moreover, this DCA

required interrogation of the original FEM. On one hand, while the research team would be

willing to conduct this interrogation, to relieve the burden on busy engineering firms, it is

unlikely that firms would be willing to share their FEMs. That then implies that the extraction

would need to be executed by the design firms themselves, which was not desirable given the

cumbersome nature of the extraction process, especially for concrete buildings. Still, it was

15

considered to be potentially the most reliable descriptor of cantilever action, as visualized in

Figure 2.1.

2.2 iDCA Development

Clearly the MS‐DCA’s fidelity is questionable for systems with vertical discontinuities, since it is a

global fit; however, using the mode shape to define the DCA measure is ideal since it would be

more likely to be shared by design firms as it is commonly published and presented when a new

building project is introduced. This low barrier to access makes a mode‐shape‐based DCA

preferable. Therefore, in this thesis the fundamental sway mode shapes will be retained as the

basis, though a more reliable means to quantify the DCA from this design artifact is required.

Since the MS‐DCA used a best‐fit power law, it was unable to detect subtle modulations in the

mode shapes associated with phenomena like shear lag1 and could be readily biased by sharp

discontinuities in the system, rendering it incapable of fully capturing the system’s behavior. To

correct this flaw, a new DCA measure, dubbed the integral DCA (iDCA), is now proposed. As

opposed to a single parameter, global best‐fit, the iDCA compares the mode shape to an ideal

cantilever mode shape using their slopes calculated at each floor. A number of measures were

initially explored to quantify the degree of agreement between the extracted slopes of the

mode shape in question and an ideal cantilever. As the process results in a probability

distribution of the slope values, these evaluations included statistical measures like mean,

median and mode as a simple basis for comparison between the distributions, as well as the

probability distribution’s values at various percentiles. Unfortunately, these did not prove robust

enough to capture the behavior, suffering from some of the same potentials for bias of other

“global” measures such as the MS‐DCA. Instead, ways to compare the distribution of the mode

shape’s slopes to the distribution of the cantilever’s slopes were explored. Ultimately,

established methods to quantify the similarity between two probability distributions proved to

be the most fruitful DCA basis, specifically relative entropy and the Hellinger distance.

Relative entropy, or Kullback‐Leibler divergence, relies on the logarithmic difference of the two

distributions, which becomes problematic when only one distribution is zero (Yamano 2009).

The relative entropy, KL, is expressed in terms of ideal cantilever slope density, π(m), and mode

shape slope density, ρ(m), where m is the vector of slopes:

, ln (2.2)

This requires at least one distribution to be broader than the other, which could not be

guaranteed. Thus this measure was deemed too unreliable.

On the other hand, the Hellinger distance, DH, is again expressed in terms of ideal cantilever’s

slope density, π(m), and mode shape in question’s slope density, ρ(m):

1 Shear lag (def): Axial forces in the perimeter columns are transferred by flexure of the beams. In this process, a “lag” in the force distribution is often witnessed due to the inefficiency of the beams in this transfer, thereby causing non‐uniform axial loading of the columns (Iyengar 2000).

16

, (2.3)

This measure is bounded, 0 , 1, resulting in zero if and only if ρ(m) and

π(m) are identical and one if π(m) takes on zero values at every value that ρ(m) is greater than

zero (Chen et al. 2005). This means, if the mode shape in question is an ideal cantilever, the

Hellinger distance will result in a value of zero, and if the mode shape has ideal shear behavior,

it will result in a value near one. However, from an intuitive perspective, a system that is

cantilever‐dominated should have a large DCA value, thus the iDCA will be defined as:

1 , (2.4)

This equation results in theoretical iDCA values of one for ideal cantilever behavior and close to

zero for ideal shear behaviors. Figure 2.3‐a shows an example of the distribution of the slopes

extracted at each floor for the target mode shape: an ideal cantilever mode shape where

and an arbitrary mode shape in question (Figure 2.3‐b), which for this example is

a vertically continuous mode shape with = 1.875 in Equation 2.1. From these visualizations, it

is clear that the density of the slopes changes significantly even for a minor deviation in that would still be classified as cantilever‐dominated. Therefore it is expected that this measure will

be sensitive enough to discern even minor levels of shear lag. A quantitative assessment of this

capability now follows.

Figure 2.3: Comparison of two distributions of extracted slopes used in iDCA: (a) ideal cantilever [target distribution] and (b) mode shape in

question.

2.2.1 iDCA Calibration: Continuous Mode Shapes

It is important to first verify the insensitivity of the iDCA to system geometry, e.g., height or

floor‐to‐floor height, using vertically continuous systems. To do so, three representative

“buildings” are proposed: 60‐story and 100‐story buildings with uniform floor‐to‐floor height of

12 feet called Building 1 (B1) and Building 2 (B2), respectively, and a 100‐story building with

varying floor‐to‐floor heights ranging from 12 to 16 feet called Building 3 (B3). The three

building’s normalized mode shapes are shown in Figure 2.4 for the case of = 2, where the consequence of normalization, a standard practice, is evident: B1 has a coarser discretization,

while B3 has a finer resolution near the base and between 0.65H and 0.75H. For each building,

2)()( Hzz

17

eight different mode shapes were generated by varying the powers in Equation 2.1 between 1

and 2 in increments of 0.125. The resulting iDCA values are reported in Table 2.1, along with the

average and coefficient of variation (CoV) of the iDCAs for each mode shape power and the

ratios between the iDCAs for Building 1 to Building 2 and Building 3 to Building 2. For

completeness, each simulated mode shape was best‐fit using the mean‐square approach from

Bentz (2012); the resulting from that process was identical to the used to simulate the mode

shape, as one may expect, showing that for ideal mode shapes, the process used to extract and thus estimate MS‐DCA, introduces no additional bias.

Figure 2.4: Normalized mode shapes for Buildings (a) 1, (b) 2, (c) 3 for

case of = 2.

18

TABLE 2.1

COMPARISON OF iDCA VALUES FOR VERTICALLY CONTINUOUS SYSTEMS

Mode Shape Powers ()

1 1.125 1.25 1.375 1.5 1.625 1.75 1.875 2

B1 0.1964 0.3133 0.4265 0.5099 0.5906 0.6589 0.6918 0.7038 0.9985

B2 0.1979 0.3136 0.4295 0.5296 0.6165 0.6767 0.7265 0.7651 0.9969

B3 0.1751 0.2958 0.3996 0.4953 0.5791 0.6503 0.6995 0.7341 0.9894

Avg. 0.1898 0.3076 0.4185 0.5116 0.5954 0.6620 0.7059 0.7343 0.9949

CoV 7% 3% 4% 3% 3% 2% 3% 4% 0%

B1/B2 99% 100% 99% 96% 96% 97% 95% 92% 100%

B3/B2 88% 94% 93% 94% 94% 96% 96% 96% 99%

Note: Classification used in this thesis: ≤1.25, shear building; 1.25<<��, interactive building (in bold); ≥1.75, cantilever building.

From Table 2.1, note first that for each case, the iDCA values for the ideal cantilever (=2) and ideal shear (=1) are as what are expected, approaching one and zero, respectively. The CoVs indicate that the iDCA is fairly insensitive to variations in building and floor‐to‐floor height in

general, though some variation is expected. As such, the iDCA values for each Building in this

table will be used to create a mapping with their “equivalent” mode shape power, ��so MS‐

DCA and iDCA results can be effectively compared in this chapter for each Building. To

generalize this mapping, the average of the iDCAs for each mode shape power from Table 2.1

were linearly interpolated to create a look‐up table that can be used to convert iDCA measures

into their MS‐DCA equivalents. This is provided in Appendix A and will be especially important to

compare the two DCAs for real buildings in Chapter 3.

Isolating for the effect of differences in overall height and thus in the discretization of a

normalized mode shape, the ratio of iDCAs for Buildings 1 and 2 shows that differences are less

than 10% and are generally smaller for more shear‐type buildings, as one would expect since the

19

more linear a curve is, the coarser the discretization it can accommodate and still be accurately

described. Next isolating for the effect of differences in floor‐to‐floor height and thus variable

discretization, the ratio of the iDCAs for Buildings 3 and 2 shows greater sensitivity, with one

instance of a difference exceeding 10% in the case of the pure shear building. While this larger

difference may be surprising due to a shear building’s linear form being very insensitive to the

coarseness of the discretization, it is important to keep in mind that each slope is being

compared to the ideal cantilever discretized in the same manner. That being said, since Building

3 has a finer discretization (0.008H vs. 0.01H) near the base, where the cantilever slopes are

steep, B3 benefits from the higher discretization in this critical regime. Therefore, the slope

distribution for the ideal cantilever, π(m) from Equation 2.3, is greater at these large slope

values for B3 than the other two buildings. This effect is especially pronounced in the ideal

shear case, where all the slope distributions tend to cluster near one. However, this effect

diminishes with even the slightest introduction of nonlinearity to the mode shape and since

perfect shear mode shapes are not expected in tall buildings, this effect is of little concern. Still,

the B3 example helps to underscore the influence of the mode shape slopes at the base of the

structure on the iDCA – an issue that will resurface especially in Chapter 3.

2.2.2 iDCA Calibration: Discontinuous Mode Shapes

The next verification will examine the iDCA’s robustness to vertical discontinuities. For this

examination, the same three case study buildings are used, with different discontinuities

introduced. The first will vary the mode shape between ideal cantilever (=2) and ideal shear (=1) along the height according to two progressions. In Progression 1 (P1), the mode shape

evolves from cantilever to shear with height. In each P1 case, the percentage of the mode shape

that is cantilever increases from 0 to 100% in increments of 10%. The notation C‐X indicates P1

with only the bottom X% of the mode shape being an ideal cantilever, e.g., C‐40 would have the

bottom 40% of the mode shape be an ideal cantilever and the remaining 60% at the upper

elevations has an ideal shear behavior. The limits of this progression (C‐0 and C‐100) would be a

pure shear and pure cantilever mode shape, respectively. Progression 2 (P2) uses the same

increments but with the opposite trend, moving from shear at the base toward cantilever with

height. A similar notation is introduced, S‐X, indicating P2 with the bottom X% of the mode

shape being ideal shear, e.g., S‐30 would have the bottom 30% of the mode shape be an ideal

shear and the remaining 70% at the upper elevations has an ideal cantilever behavior. The limits

of this progression (S‐0 and S‐100) would be a pure cantilever and pure shear mode shape,

respectively. Examples of these two progressions are shown as the black line in Figure 2.5, along

with the shear and cantilever ideal and the best‐fit of Equation 2.1 (red dashed line) to

demonstrate the potential shortcomings of the MS‐DCA that is based upon it. Subsequent

quantitative assessments will now consider both the iDCA and MS‐DCA extracted from these

simulated mode shapes. To facilitate comparison, the iDCA values will also be mapped to their

MS‐DCA equivalent using the unique mapping for each building listed in Table 2.1.

20

Figure 2.5: Examples of vertically discontinuous mode shapes for Building 2: C‐60 (left) and S‐60 (right) with cantilever and shear ideals as

well as best‐fit power law.

21

The iDCA and MS‐DCA values for Progression 1 are shown in Table 2.2, along with their

respective averages and coefficients of variation (CoV), as well as the ratios of the DCAs for

Building 1 to Building 2 and Building 3 to Building 2. Similar to the vertically continuous cases,

the CoVs for the iDCAs are all less than 10% for each case and generally show less variation as

the cantilever degree increases, showing the robustness necessary to capture behaviors of even

abruptly discontinuous systems. The MS‐DCA measures show even smaller CoVs (less than 1%),

which may initially be perceived as a strength but instead reiterates its insensitivity to subtle

variations, which actually will prove to be a determent later. To again isolate the effect of

varying building height and thus mode shape discretization, the B1/B2 iDCA ratios are all within

10%. The influence of non‐uniform story height (discretization) evidenced by the B3/B2 ratios

shows slightly greater deviation for the vertically discontinuous systems than in the vertically

continuous systems of Table 2.1, with differences as great as 15%; again showing greater

sensitivity to story height irregularities in shear‐type buildings, for the same reasons articulated

previously. On the other hand, the MS‐DCA shows no sensitivity to discretization, with all the

ratios falling within 1%, again due its general lack of sensitivity to minor variations in the mode

shapes as a global best‐fit measure. From Table 2.1, interactive systems, shown in bold, would

encompass the cases where the cantilever portion is approximately the bottom quarter to three

quarters of the mode shape.

TABLE 2.2

DCA SENSITIVITY TO VERTICAL DISCONTINUITY: PROGRESSION 1

C‐0 C‐10 C‐20 C‐30 C‐40 C‐50 C‐60 C‐70 C‐80 C‐90 C‐100

iDCA

(MS‐DCA equivalent)B1 0.1964

(1 0000)

0.2404

(1 0470)

0.3287

(1 1420)

0.4269

(1 2506)

0.5151

(1 3831)

0.5296

(1 4055)

0.5747

(1 4754)

0.6390

(1 5886)

0.6824

(1 7143)

0.6911

(1 7473)

0.9985

(2 0000)B2 0.1979

(1 0000)

0.2603

(1 0674)

0.3538

(1 1684)

0.4651

(1 2945)

0.5553

(1 4120)

0.5705

(1 4338)

0.6168

(1 5006)

0.6820

(1 6383)

0.7179

(1 7284)

0.7549

(1 8420)

0.9965

(2 0000)B3 0.1751

(1 0000)

0.2209

(1 0474)

0.3151

(1 1482)

0.4308

(1 2908)

0.5307

(1 4278)

0.5536

(1 4620)

0.6132

(1 5599)

0.6558

(1 6390)

0.7176

(1 8154)

0.7378

(1 8768)

0.9894

(2 0000)Avg. 0.1898

(1 0000)

0.2405

(1 0538)

0.3325

(1 1531)

0.4409

(1 2801)

0.5337

(1 4080)

0.5512

(1 4341)

0.6016

(1 5116)

0.6589

(1 6192)

0.7089

(1 7632)

0.7250

(1 8341)

0.9948

(2 0000)CoV 7% 8% 6% 5% 4% 4% 4% 3% 2% 5% 0%

B1/B2 99% 92% 93% 92% 93% 93% 93% 94% 96% 90% 100%

B3/B2 88% 85% 89% 93% 96% 97% 99% 96% 100% 98% 99%

Note: Interactive systems (in bold) from Table 2.1 are: B1 = (0.5099, 0.6589), B2 = (0.5296, 0.6767), B3 = (0.4953, 0.6503), and Avg. = (0.5116, 0.6620).

32

TABLE 2.2 (CONTINUED)

C‐0 C‐10 C‐20 C‐30 C‐40 C‐50 C‐60 C‐70 C‐80 C‐90 C‐100

MS‐DCA

B1 1.0000 1.8986 1.9878 1.9692 1.9446 1.9359 1.9431 1.9597 1.9786 1.9936 2.0000

B2 1.0000 1.9119 1.9919 1.9722 1.9482 1.9398 1.9466 1.9624 1.9802 1.9942 2.0000

B3 1.0000 1.8980 1.9985 1.9790 1.9517 1.9433 1.9526 1.9671 1.9822 1.9948 2.0000

Avg. 1.0000 1.9028 1.9927 1.9735 1.9482 1.9397 1.9474 1.9631 1.9803 1.9942 2.0000

CoV 0.0% 0.4% 0.3% 0.3% 0.2% 0.2% 0.2% 0.2% 0.1% 0.0% 0.0%

B1/B2 100.0% 99.3% 99.8% 99.8% 99.8% 99.8% 99.8% 99.9% 99.9% 100.0% 100.0%

B3/B2 100.0% 99.3% 100.3% 100.3% 100.2% 100.2% 100.3% 100.2% 100.1% 100.0% 100.0%

33

24

Now comparing the two DCAs, using the iDCA equivalents on the MS‐DCA scale shown in

parentheses in Table 2.2, one will note the iDCA values range from shear‐dominated to

interactive to cantilever‐dominated structures as expected when the cantilever proportion

increases in the mode shapes. On the other hand, while the MS‐DCA perfectly identifies the

ideal shear and ideal cantilever cases, every other assessment is consistently biased toward

cantilever‐dominated in P1. Even in the C‐10 case, where the building is predominantly shear

and only 10% cantilever, the MS‐DCA categorizes the system as highly cantilevered, revealing

that the MS‐DCA is extremely biased by the base behavior in mixed systems. From this

inability to capture the systemic discontinuity of a mode shape varying from cantilever to shear

along its height, it is predicted that the MS‐DCA will similarly fail in capturing the mode shape

behaviors in P2.

The presentation in Table 2.2 is now repeated for P2 in Table 2.3. Again the iDCA CoVs are all

less than 10% amongst the buildings for each case and, consistent with Pr1, show less variation

as the cantilever degree increases (degree of shear decreases), reaffirming the robustness

necessary to capture behaviors of even abruptly discontinuous systems, regardless of the

progression. As observed in the previous progression, the MS‐DCA’s lack of sensitivity in general

leads it to have exceptionally low CoVs, as well as comparable ratios for all buildings and all

cases. This was not expected to vary with progression type. To again isolate the effect of varying

discretization, the B1/B2 iDCA ratios are all within 10%, while the influence of non‐uniform

discretization evidenced by the B3/B2 ratios is less dramatic than for Progression 1 and again

shows greater sensitivity to story height irregularities for shear‐type buildings. Interactive

systems, shown in bold, would approximately encompass the cases when the shear portion is

the bottom fifth to half of the mode shapes (Table 2.3). Thus, when the mode initiates with

shear behavior, it requires a slightly greater proportion of its overall mode shape to be

cantilever in order to achieve an interactive classification. Conversely, as predicted from P1,

the MS‐DCAs are again strongly biased by the base behavior. The MS‐DCA correctly classifies the

ideal shear (S‐100) and cantilever (S‐0) cases, as expected, but again the slightest introduction of

linearity (S‐10) at the base, results in a near perfect shear building classification. A 10% and 90%

shear building are essentially indistinguishable by the MS‐DCA. This bias toward the behavior of

the structure at its base will prove to be a major liability for the MS‐DCA in some of the case

studies in Chapter 3.

It is important to understand that the implications of the iDCA’s sensitivity to progression, e.g.,

the mode shape that has its bottom 10% in shear (S‐10) does not have the identical iDCA as a

system with the top 10% of its mode shape in shear (C‐90). This nuance is more marked for

some cases. For instance, systems with 30% cantilever behavior are classified as interactive

(although marginally) according to the mapping in Table 2.1 when the cantilever is at the base

(C‐30) and classified as shear‐dominated when the portion of the mode shape that is cantilever

is at the top (S‐70). This reiterates the observation that the base system behavior tends to more

strongly influence the classification of systems by iDCA. The rationale for this tendency stems

from how the iDCA has been defined in this chapter. Recall that cantilevers have a defining

characteristic of smaller deflections at the base than at the top. These small deflections result in

comparatively higher‐valued slopes that are not found in the upper sections of the cantilever or

in an idealized shear mode shape. Therefore, the iDCA will detect these missing quintessential

25

slopes of an ideal cantilever in the mode shape distribution and thus classify the mode shape as

more shear‐dominated, even if it has strong cantilever tendencies at its upper floors. This point

will become important to remember in the case studies in Chapter 3. These investigations

reveal an important point: both DCAs are influenced by the behavior of the structure at the

base; however, only the iDCA avoids complete biasing and retains the sensitivity to distinguish

minor variations in the proportion of shear and cantilever action.

TABLE 2.3

DCA SENSITIVITY TO VERTICAL DISCONTINUITY: PROGRESSION 2

S‐0 S‐10 S‐20 S‐30 S‐40 S‐50 S‐60 S‐70 S‐80 S‐90 S‐100

iDCA

(MS‐DCA)

B1 0.9985

(2.0000)

0.6949

(1.7823)

0.6386

(1.5878)

0.5435

(1.4270)

0.4671

(1.3109)

0.4229

(1.2460)

0.4084

(1.2300)

0.3673

(1.1846)

0.3243

(1.1371)

0.2798

(1.0892)

0.1964

(1.0000)

B2

0.9965

(2.0000)

0.7597

(1.8575)

0.6329

(1.5341)

0.5416

(1.3923)

0.4657

(1.2952)

0.4216

(1.2415)

0.4069

(1.2256)

0.3656

(1.1811)

0.3223

(1.1344)

0.2790

(1.0876)

0.1979

(1.0000)

B3

0.9894

(2.0000)

0.7302

(1.8609)

0.6101

(1.5544)

0.5124

(1.4005)

0.4367

(1.2985)

0.3963

(1.2460)

0.3785

(1.2246)

0.3429

(1.1817)

0.3024

(1.1329)

0.2592

(1.0871)

0.1751

(1.0000)

Avg. 0.9948

(2.0000)

0.7283

(1.8486)

0.6272

(1.5597)

0.5325

(1.4062)

0.4565

(1.3010)

0.4136

(1.2445)

0.3979

(1.2268)

0.3586

(1.1825)

0.3163

(1.1348)

0.2727

(1.0880)

0.1898

(1.0000)

CoV 0% 4% 2% 3% 4% 4% 4% 4% 4% 4% 7%

B1/B2 100% 91% 101% 100% 100% 100% 100% 100% 101% 100% 99%

B3/B2 99% 96% 96% 95% 94% 94% 93% 94% 94% 93% 88%

Note: Interactive systems (in bold) from Table 2.1 are: B1 = (0.5099, 0.6589), B2 = (0.5296, 0.6767), B3 = (0.4953, 0.6503), and Avg. = (0.5116, 0.6620).

37


S‐0 S‐10 S‐20 S‐30 S‐40 S‐50 S‐60 S‐70 S‐80 S‐90 S‐100

MS‐DCA

B1 2.0000 1.0140 1.0043 1.0257 1.0416 1.0468 1.0424 1.0316 1.0180 1.0059 1.0000

B2 2.0000 1.0082 1.0021 1.0234 1.0389 1.0439 1.0397 1.0294 1.0165 1.0053 1.0000

B3 2.0000 1.0068 0.9937 1.0185 1.0362 1.0413 1.0355 1.0259 1.0149 1.0047 1.0000

Avg. 2.0000 1.0097 1.0000 1.0225 1.0389 1.0440 1.0392 1.0290 1.0165 1.0053 1.0000

CoV 0.0% 0.4% 0.6% 0.4% 0.3% 0.3% 0.3% 0.3% 0.2% 0.1% 0.0%

B1/B2 100.0% 100.6% 100.2% 100.2% 100.3% 100.3% 100.3% 100.2% 100.1% 100.1% 100.0%

B3/B2 100.0% 99.9% 99.2% 99.5% 99.7% 99.8% 99.6% 99.7% 99.8% 99.9% 100.0%

38

29

While the previous scenario represents one kind of vertical discontinuity, one that we will

characterize as a systemic discontinuity, which may be observed in mixed systems with changes

in occupancy or floor plan with height, another common discontinuity in mode shapes is caused

by the outrigger, which was introduced in Chapter 1 as one of the most popular modern

structural systems. Outriggers cause a vertical jump in the mode shape. To examine this effect,

each of the three buildings defined previously had its mode shape simulated by Equation 2.1 for

a power of =2 (perfect cantilever) with the added feature of two vertical jumps in the mode

shape beginning at 0.2H and 0.6H. While these placements are not considered optimal by

outrigger theory, they were selected to maximize the effect on the slopes at both the lower and

upper halves of the mode shape. Because of the differences in building and floor‐to‐floor height,

these outriggers constitute varying percentages of the total height of Buildings 1, 2 and 3: 6.7%,

6%, 6.5%, respectively. The three simulated mode shapes are shown in Figure 2.6 with the

vertical segment representing the outrigger circled. The cantilever and shear ideals as well as

the best‐fit power law used as the basis of the MS‐DCA are provided for additional illustration.

The results in Table 2.4 show again show relative consistency for the three building cases, with a

CoV of only 5% for iDCA and under 0.5% for MS‐DCA, with the cause of this lower variability

again explained previously. The slight differences that are noted do follow a discernable trend:

the larger the percentage of the height occupied by outriggers, the less cantilever the iDCA

value. This confirms the degree of sensitivity of the iDCA, which can detect subtle differences

among similar systems. Comparatively, the MS‐DCA does vary for each of the three cases,

indicating it is affected by the relative proportion of outrigger height in the mode shape;

however, it shows no clear trend. According to the mapping in Table 2.1, used to report the

mapped iDCAs in parentheses, both DCAs would essentially classify the structure as cantilever‐

dominated2. Recall the mode shape was simulated to be a perfect cantilever and thus the

outriggers will be expected to reduce the iDCA and MS‐DCA from 1 and 2, respectively;

however, when viewing the mapped iDCA values, it is clear that the effect of the outriggers is

more strongly detected by this measure, though showing some sensitivity to the discretization

of the mode shape embodied by the three different buildings. The rationale for the iDCA

“penalizing” for the effect of the outriggers more than MS‐DCA can be observed from Figure 2.6.

The red dashed line shows the MS‐DCA best fit and its near perfect alignment with the

cantilever ideal (solid blue line). The actual mode shape (solid black line) is visibly displaced from

these curves; therefore, suggesting an inconsistent behavior that the iDCA captures through a

measure that is more markedly reduced from the cantilever ideal. We will consider this type of

discontinuity a progressive discontinuity because the behavior of the system remains the same

both before and after the outriggers and the discontinuity repeats over the height. While this

specific discontinuity (outrigger) affects local behavior, the global behavior is only mildly altered,

and this can be clearly discerned in Table 2.4. This demonstrates an important observation:

when discontinuities only mildly distort the mode shape, while preserving a consistent overall

behavior, the MS‐DCA maintains the ability to accurately classify the system, as will be

confirmed in certain case studies in Chapter 3.

2 Note B1 is marginally interactive according to iDCA.

30

Figure 2.6: Normalized mode shape with outriggers circled for Buildings (a) 1, (b) 2, (c) 3, with shear and cantilever ideals as well as best‐fit

power law provided for comparison.

TABLE 2.4

DCA SENSITIVITY TO VERTICAL DISCONTINUITY: OUTRIGGERS

31

B1 B2 B3 Avg. CoV B1/B2 B3/B2

iDCA

0.6908

(1.7462)

0.7545

(1.8407)

0.7379

(1.8769)

0.7277

(1.8460)5% 92% 98%

MS‐DCA 1.9419 1.9515 1.9590 1.9508 0.4% 99.5% 100.4%

Note: iDCA mapped to MS‐DCA equivalent shown in parenthesis.

2.3 Demonstrative Example

Having established the performance of the two DCAs on mode shapes simulated using the

idealized expression in Equation 2.1, it is now pertinent to demonstrate their application to

mode shapes extracted from FEMs. To do so, three two‐dimensional steel MRFs with fully rigid

boundary conditions were created in SAP 2000. The MRFs had 3 bays, spaced 14 feet on center

and a story‐to‐story height of 14 feet. Floors were then simply replicated to achieve target

aspect ratios (H/B) of 1, 5, and 10, consistent with those used in Bentz (2012) for her FEM DCA

study. The three MRFs are shown in Figure 2.7. It is of course acknowledged that MRFs are often

optimized to increase member sizing in the lower floors, where inter‐story shears are high;

however, to facilitate an objective comparison, member sizes were kept uniform, regardless of

the elevation. As such, the absolute behaviors will lack practicality; however, the focus herein is

on relative behavior and more practical applications will follow in Chapter 3. As MRFs manifest

high degrees of frame racking due to reliance on bending of beams and columns as their primary

means of transferring lateral loads, these frames are expected to be classified as shear‐

dominated with a slight increase in cantilever action simply due to their increasing aspect ratio.

The mode shapes from these FEMs are portrayed in Figure 2.8, alongside the shear and

cantilever ideals as well as the best‐fit power law used in MS‐DCA. Note the influence of

discretization apparent as the aspect ratio increases, as well as the exaggerations in the lower

aspect ratio mode shape that are simply a consequence of the standard normalization

procedure required in Equation 2.1, as well as the impracticalities of the simple MRF case study

used. The iDCAs and MS‐DCAs found for each mode shape are reported in Table 2.5. iDCAs

mapped to their MS‐DCA equivalents according to Appendix A are also provided, though H/D=1

could not be mapped since it falls just outside the range of this look‐up tool. For both DCAs, the

degree of cantilever action increases, as expected, with aspect ratio. Looking at the mapped

iDCA values, it is clear that the higher aspect ratio buildings are classified as having increasingly

cantilevered behavior, as one may expect. The MS‐DCA, due to the lack of continuity in the

mode shape, has difficulty in classifying these mode shapes. In all cases, it is “sub‐shear” in its

classification ( < 1). This underscores how actual mode shapes from a FEM can fail to behave in

a smooth and continuous fashion and as a result pose challenges for the MS‐DCA in subsequent

classification, challenges that will now become more apparent in Chapter 3.

32

Figure 2.7: Finite element models for the three MRFs with aspect ratios of (a) 1, (b) 5, and (c) 10.

33

Figure 2.8: Normalized mode shapes for the three MRFs with aspect ratios of (a) 1, (b) 5, and (c) 10, with ideal shear and cantilever mode

shapes and best‐fit power law shown for comparison.

TABLE 2.5

APPLICATION OF DCAs TO MRFs OF VARYING ASPECT RATIO

34

H/B = 1 H/B = 5 H/B = 10

iDCA

0.1176

(<1.0000)

0.3803

(1.2069)

0.4005

(1.2297)

MS‐DCA 0.3843 0.6447 0.8602

Notes: iDCA mapped to MS‐DCA equivalent using Appendix A shown in parenthesis. H/B=1 could not be fully mapped as it falls outside the range of the mapping.

2.4 Summary

This chapter overviewed various DCA measures, discussing their pros and cons. Specifically,

while a DCA measure quantifying the actual distribution of axial and shear forces in members

would have a higher fidelity, the extraction process can be cumbersome and requires access to

the full‐finite element model. Basing DCAs on mode shape offers a reasonable compromise

since mode shapes are sometimes publically available and if not, have a lower barrier to access

than the full FEMs themselves. To overcome the limitations of the previously proposed mean‐

square DCA (MS‐DCA), the author presented a new DCA measure called the integral DCA (iDCA),

which seeks the similitude between the distributions of the floor‐by floor slope of the mode

shape in question and that of an ideal cantilever using the Hellinger distance. By seeking a more

localized measure, this DCA can capture both abrupt and subtle discontinuities in mode shape

that previously could not be resolved by the MS‐DCA.

The robustness of the iDCA measure was tested against smooth, continuous mode shapes for

three case study buildings to explore sensitivity to mode shape resolution as well as the

simulated degree of cantilever action. This also allowed for the creation of a mapping between

the MS‐DCA and iDCA in Appendix A, which will benefit the further comparison of these DCAs in

Chapter 3. In this, as well as explorations of vertical discontinuities with various progressions,

the iDCA proved slightly more sensitive than the MS‐DCA to variations in mode shape

discretization. Interestingly, when exploring the influence of progression, it was found that the

base behavior will generally influence the overall classification by iDCA; when the mode shape

initiates with shear behavior, it requires a slightly greater proportion of the overall mode shape

to be cantilever in order to achieve an interactive classification. Conversely, the MS‐DCA’s

insensitivity to discretization actually was evidence of its overall insensitivity to subtle changes

in mode shapes, leading to a significant bias that misclassified interactive systems. Regardless of

progression or the proportion of a given behavior at the base, the MS‐DCA immediately biased

itself toward the behavior at the base. Thus, while both DCAs were influenced by the behavior

of the structure at the base, only the iDCA avoids complete biasing and retains the sensitivity to

distinguish minor variations in the proportion of shear and cantilever action within similar

systems. Only when the mode shape was completely continuous or when discontinuities were

mild (discrete outriggers) and preserved a consistent overall behavior (so called “progressive

discontinuities”), could the MS‐DCA maintain the ability to accurately classify the system.

Unfortunately, these limitations became even more pronounced when the DCAs were used with

35

mode shapes extracted from FEMs of MRFs with varying aspect ratios. These findings verify that

only the iDCA was robust enough to consistently classify systems with vertical discontinuities

and even sensitive enough to quantify the relative degree of cantilever action created by those

discontinuities.

Thus the outcome of this chapter satisfies the requirement of Objective 1: develop a robust

descriptor suitable for heterogeneous systems that is simple to extract, i.e., requires little effort

on the part of cooperating designers. In the subsequent chapter, the iDCA will be applied to

mode shapes from existing tall buildings to allow a comparison with the MS‐DCA to further

explore its ability to classify modern tall buildings, in support of Objective 2.

36

CHAPTER 3: DCA VALIDATION THROUGH CASE STUDIES

3.1 Introduction

In order to evaluate the performance of two potential system descriptors, the proposed integral

Degree of Cantilever Action (iDCA) introduced in Chapter 2 will be further validated against the

Mean‐Square Degree of Cantilever Action (MS‐DCA) in this chapter using a series of Case Study

Tall Buildings. For each of the buildings, mode shapes in this chapter came from one of two

sources: (1) directly from the designer’s full finite element model or lumped mass model used

for wind tunnel testing or (2) digitized from figures published in the literature (outputs from

finite element or lumped mass models, in some cases calibrated against full‐scale

observations)3. The software package UN‐SCAN‐IT was used to digitize the published mode

shapes, cautioning that this process is limited by the size and resolution of the image and does

not output points at the actual story elevations but rather at a finer resolution automatically

selected by the software. The digitized curves were manually corrected for this over‐digitization

to yield one mode shape value per floor. For a number of the buildings in this chapter, as well as

others not included, the mode shapes derived from in‐house finite element models were also

considered. These were constructed using only publically available details and traditional

assumptions as described in Bentz (2012). Because of questionable mode shape curvatures,

these ultimately were not considered reliable, though their analysis is included in this thesis’s

Appendix B for completeness. All buildings included in this chapter are discussed using the

language of their designers, specifically when discussing system features. This vernacular will

later be unified in Chapter 4 based on the results of the databasing efforts. Table 3.1

summarizes the buildings utilized and the sources of the mode shapes for each. This table also

presents the primary material and heuristic classification of the system. There are four primary

heuristic classifications for discontinuities used in this chapter:

1. Intermittent: where structural features that cause pronounced modulations in

the mode shape recur over the height

2. Progressive: where structural features “reset” the mode shape, yet the

behavior before and after this feature is relatively unchanged

3. Continuous: where there are no distinct structural features that cause

discontinuities, resulting in a smooth, continuous mode shape

3 For CS5, published mode shapes used a rendering of the full FEM that proved too difficult to accurately digitize. However, the output of the in‐house finite element model was compared to these published mode shapes and was found to agree well and thus was used as the surrogate for the digitized mode shape from the literature.

37

4. Systemic: where the building employs at least two distinct structural systems

with a sharp transition somewhere along the height.

These heuristic classifications will be helpful in discerning the situations under which the two

measures perform comparably, presented in the final section of this chapter. The remainder of

this chapter will present the findings of this analysis.

TABLE 3.1

KEY CHARACTERISTICS OF CASE STUDY BUILDINGS

Building Mode Shape Source Primary Material Heuristic

Classification

NO DIGITIZATION REQUIRED

CS1 Designer Steel Intermittent

CS2 Designer Steel Progressive

CS6 Designer Reinforced Concrete Progressive

CS7 Designer Reinforced Concrete Systemic

CS9 Designer Composite Progressive

CS3 Published Lumped Mass

Model Composite Progressive

CS5

In‐House FEM Corroborated by

Published Mode Shape Steel Continuous

DIGITIZATION REQURED

CS8 Published FEM Calibrated

against Full‐Scale Data Composite Continuous

CS10 Published FEM Calibrated

against Full‐Scale Data Composite Intermittent/ Progressive

CS4 Published Lumped Mass

Model Composite Systemic

38

3.2 Results

The two DCA measures explored in Chapter 2 were applied to each available fundamental sway

mode of the buildings. To ensure a fair comparison, the MS‐DCA was not doctored to find the

most accurate fit, i.e., all floors were included in this fit except the first floor in cases where its

displacement was zero. By keeping every floor with even the slightest displacement, an

equivalent comparison of the iDCA and MS‐DCA was achieved, with at most, one floor (the first

floor) excluded from the MS‐DCA (14 of 26 cases).

Each fundamental sway mode shape is assigned a generic number to assist in graphical display

of results in Figure 3.1. This figure incorporates a double y‐axis, displaying the iDCA as stars and

the MS‐DCA as squares. The figure also depicts the classification, introduced previously in

Chapter 2, where MS‐DCAs greater than 1.75 (region above the horizontal red line) are defined

as cantilever or axial‐dominated structures, while those less than 1.25 are defined as shear‐

dominated structures (region below the horizontal blue line). Using the mapping in Appendix A,

these demarcations respectively map to iDCA values of 0.7059 and 0.4185. Through this

graphical display, the differences in how individual modes of each of the case study buildings

would be classified by the two measures can be gauged.

From Figure 3.1, there is a general trend of the MS‐DCA classifying the structures as more

cantilever than their iDCA counterparts (occurring for 16/26 mode shapes considered, circled in

green in Figure 3.1). Only five mode shapes were found to be less cantilever by the MS‐DCA

(circled in red in Figure 3.1). In eleven cases, these discrepancies resulted in a change in the

classification of the building. In five instances, the two DCAs essentially converged (circled in

blue in Figure 3.1). Each case will now be investigated to determine the reasons behind these

differences, with particular focus on instances of convergence (Mode Shapes 19, 23‐26) and

significant deviation (Mode Shapes 9‐12).

39

Figure 3.1: Comparison of MS‐DCA (squares) and iDCA (stars) for case study buildings.

Table 3.2 presents the iDCA and MS‐DCA results for the two fundamental sway modes for each

of the case study buildings. Since the two DCA values operate on different scales, a mapped

iDCA is also presented that translates the iDCA to its equivalent on the MS‐DCA scale, using the

mapping in Appendix A. Additionally, the error in the MS‐DCA, EMS‐DCA, along the height, z, of the

building is expressed in terms of the actual mode shape, , and the best‐fit obtained using the power law expression, fit:

φ

(3.1)

Its median value is reported in Table 3.2 as the preferred measure of goodness of fit, since in

many cases there are discontinuities in the mode shapes that cause large localized error values

along the height.

40

TABLE 3.2

iDCA AND MS‐DCA FOR FUNDAMENTAL MODES OF CASE STUDY BUILDINGS

Building

Mode Shape

IdentifierMode Axis MS‐DCA

Mapped iDCA iDCA

Median

EMS‐DCA

CS1

1 Y‐Axis 1.7358 1.8085 0.7192 ‐6.3%

2 X‐Axis 1.7394 1.5644 0.6297 ‐4.5%

CS2

3 XY‐Axis 1.4118 1.2926 0.4502 ‐11.4%

4 XY‐Axis 1.3835 1.2242 0.3956 ‐16.5%

CS3

5 XY‐Axis 1.0796 1.1620 0.3404 0.0%

6 XY‐Axis 1.1275 1.2940 0.4513 5.2%

CS4 7 Y‐Axis 1.3461 1.5169 0.6044 ‐5.8%

CS4* 8 Y‐Axis 1.3845 1.5631 0.6290 ‐6.2%

CS5

9 XY‐Axis 1.7674 1.4096 0.5348 ‐23.0%

10 XY‐Axis 1.7649 1.4303 0.5487 ‐23.3%

CS5*

11 XY‐Axis 1.7867 1.4676 0.5737 ‐18.3%

12 XY‐Axis 1.7844 1.4876 0.5871 ‐17.6%

CS6

13 X‐Axis 1.3829 1.2399 0.4095 ‐7.3%

14 Y‐Axis 1.3742 1.2869 0.4460 ‐17.6%

CS6*

15 X‐Axis 1.3949 1.2515 0.4196 ‐5.3%

16 Y‐Axis 1.3978 1.2275 0.3985 ‐11.7%

CS7

17 Y‐Axis 1.8185 1.5779 0.6369 35.4%

18 X‐Axis 1.7825 1.5775 0.6367 55.2%

CS7*

19 Y‐Axis 1.7138 1.5824 0.6393 35.8%

20 X‐Axis 1.6428 1.6680 0.6771 43.8%

CS8

21 XY‐Axis 1.3951 1.3238 0.4735 ‐0.7%

22 XY‐Axis 1.3951 1.3238 0.4735 ‐0.7%

CS9 23 X‐Axis 1.3765 1.3694 0.5074 ‐11.4%

41

24 Y‐Axis 1.3686 1.3659 0.5048 ‐10.6%

CS10

25 X‐Axis 1.5380 1.5495 0.6218 ‐0.4%

26 Y‐Axis 1.6426 1.6728 0.6788 23.7%

* Modified mode shapes, e.g., cap trusses removed

42

Before introducing these case studies, a primer is provided to explain how the data will be

displayed. As depicted in Figure 3.2, overlaid lines corresponding to key transition points in the

structural system often correlate to features in the fundamental mode shapes, which are

displayed in (a). In this figure, the actual mode shape predicted from finite element modeling is

shown in black, accompanied by the ideal cantilever behavior in blue (=2), the ideal shear behavior in cyan (=1), and the best‐fit of the mode shape as the dashed red line (whose power,

, is the MS‐DCA). The next plot in (b) shows the mode shape power error (defined in Equation

3.1), i.e., goodness of power law mode shape fit, as a function of height. Negative values

(plotted in the red regime) indicate that the structure was actually less cantilever than the MS‐

DCA predicted. The final plot in (c) assesses the DCAs’ classification of the structure in a manner

that allows a relative comparison between the various measures by mapping all measures to the

MS‐DCA scale and denoting the percent deviation from the cantilever ideal. Color‐coding

classifies the regions on this plot as blue for cantilever systems, purple for interactive systems,

and red for shear systems, again based on the conventions adopted in Chapter 2. The square on

this chart indicates the MS‐DCA (in the case of Figure 3.2 this was 13% less than the cantilever

ideal). Next the predicted iDCA is displayed as the solid star. This prediction is made by taking

the MS‐DCA’s percent deviation from the cantilever ideal and then correcting it by the median

of EMS‐DCA from Table 3.2. In this case, the MS‐DCA was 13% and the median error suggests that it

is even less cantilever (by 6.4%). Thus one may predict that the iDCA would be approximately

19.4% less than the cantilever ideal. Note that what is more important here is the general trend

– the expectation that the iDCA will come out even less cantilever than the MS‐DCA, even

though it is unlikely to be by this exact amount. Finally, the actual iDCA is presented as the

hollow star, using the mapped iDCA value from Table 3.2, along with its percent deviation from

the cantilever ideal. This format will be used for all building case studies in this chapter.

43

Figure 3.2: Example of graphical display used in building case studies (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

Additionally, for each case study, the percent difference between the two DCAs will be

quantified using the following expression, where MS‐DCA is the MS‐DCA percent difference from

the cantilever ideal and iDCA is the iDCA percent difference from the

cantilever ideal:

∆ ∆ ∆

∆ (3.2)

According to this definition, a negative value will indicate a more cantilever structure by the

iDCA measure. These various difference measures used in Equation 3.2 as well as percent

difference between the two DCAs determined by Equation 3.2 will be summarized and discussed

later in Table 3.3.

3.2.1 CS1 Case Study

The CS1 is referred to as a braced tube system. This structural system relies on a perimeter of

columns connected by beams to form a tube system that is tied with large diagonal cross‐braces

on both axes. The cross‐braces help to minimize the effects of shear lag common to tube

systems, achieving a near uniform distribution of axial forces in the columns at the windward

44

and leeward faces of the structure. In the CS1, the introduction of these large cross‐braces on

each face of the building effectively resets the distribution of forces at each level where the

braces connect with the corners of the building. At these intersection points, axial forces in the

diagonal braces are transferred into the corner columns and down the building. This helps the

structure to achieve a highly efficient tube behavior that mimics the idealized “vertical

cantilever.”

The mode shape evaluations for the CS1 are presented in Figure 3.3 and Figure 3.4. Here the

semi‐transparent black lines overlaid on the mode shapes facilitate in the visualization of how

the bracing affects the mode shape. From these lines, it is clear that the modulations of the

mode shape correspond to the heights at which the bracing intersects itself or the corner

columns of the building. Since the building utilizes the same structural system for both axes, the

first two mode shapes should be similar in appearance. The shear lag phenomenon being

arrested by the cross braces is more marked for the “long axis” of the building due to the

greater distance over which beams must transfer forces to distribute them among the columns.

As such, one would expect to see more pronounced “modulations” in the second mode (x‐axis)

as the braces are required to “reign in” more of these shear lag effects. This indeed can be

observed in Figure 3.4‐b. It is not surprising to note that EMS‐DCA correlates with these

modulations in the mode shapes themselves, due to the “discontinuities” created by these key

transition points in the bracing scheme.

Beginning with Mode 1, it was found that the MS‐DCA was 13.2% less cantilever than the ideal,

as shown in Figure 3.3‐d. Based on the median EMS‐DCA of ‐6.3%, the MS‐DCA is predicted to be

even less cantilever. The actual iDCA, mapped to the MS‐DSA scale, is only 9.6% less than an

ideal cantilever. In this case, the iDCA does not follow the trend predicted by the median MS‐

DCA error, yet the EMS‐DCA reveals that the modulations do induce errors in the MS‐DCA, which

tends to deviate from the general trend of the mode shape.

Similarly, for Mode 2 shown in Figure 3.4, despite the fact that the modulations in the mode

shape are more pronounced, the MS‐DCA yielded a fit nearly identical to Mode 1, 13.0% less

than the cantilever ideal, revealing its inability to detect obvious variations in similar mode

shapes. Based on the median EMS‐DCA of ‐4.5%, the iDCA is predicted to again be even less

cantilever than the MS‐DCA suggests. The mapped iDCA does indeed confirm that the structure

is less cantilever on this axis than the MS‐DCA predicted, deviating 21.8% from the cantilever

ideal ‐‐ approximately 9% less cantilever than the MS‐DCA suggests.

This presents an interesting scenario where the same structural system is used on both axes of a

building yet the agreement between the two DCAs is greater for one of the axes (Mode 1, x‐axis

sway). The difference in performance can be explained by noting again the greater levels of

modulation in the Mode 2. The shear lag “resets” achieved by the bracing are more pronounced

for this mode shape. While this explains the greater difference in the two DCA measures on this

axis, it does not express which of the measures is more accurate. Since both Mode 1 and Mode

2 had nearly identical MS‐DCA values (=1.7358, 1.7394), it is clear that the MS‐DCA is not

capable of discerning which of the modes has greater shear lag effects and thus which mode is

comparatively less cantilever. The iDCA possesses this sensitivity, correctly detecting the more

pronounced shear lag being reset by the braces in Mode 2, and thus can be considered a

45

superior measure of the degree of cantilever action. For this case study, DCA is near 40% for

both modes, but the first mode iDCA is more cantilever and for Mode 2 the opposite is true.

Thus it can be hypothesized from this case study that the MS‐DCA has limitations when mode

shapes manifest intermittent discontinuities. As the severity of these discontinuities

increases, the MS‐DCA becomes increasingly ineffective in quantifying its degree of cantilever

action.

Figure 3.3: CS1 first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

62

Figure 3.4: CS1 second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

63

48


The CS2 structural system is described as a bundled tube, employing a system of belts and

outriggers at each level where at least one of its tubes is truncated, until only two bundled tubes

remain (top 20 stories). These belts and outriggers help to more uniformly distribute the forces

into the columns at each of these transition zones. The interior grid of column lines created by

the bundling effect aids in the mitigation of shear lag by providing additional links for force

transfer instead of purely through the perimeter column lines. Incidentally, this decreases the

amount of shearing in the perimeter beams and thereby the shear lag problem, allowing for

near uniform loading along column lines on the windward and leeward faces of the building.

Despite this, as the primary mechanism for force transfer around the perimeter is through beam

flexure, bundled tubes still have a large degree of frame action.

A plan view of the roof level deflection in the first two modes is displayed in Figure 3.5, revealing

a strong degree of coupling between the x‐ and y‐axes. There are slightly greater deflections

along the y‐axis, due to the asymmetric nature of the truncations leading to a comparatively

softer structure in that axis, especially at the uppermost elevations. Vertical discontinuities

within the mode shapes correlate to locations where a belt truss occurs, effectively “reigning in”

the deflections of the building, as one would expect. The horizontal lines overlaid in Figure 3.6

and Figure 3.7 designate these locations. The only exception is the second (from the bottom)

overlaid line, which correlates to a tube truncation that does not have a belt truss and was not

discernable in this particular elevation view.

49

Figure 3.5: Mode shape displacement with regards to axis assignment of the CS2.

It is clear for the first mode (Figure 3.6‐b) that the section below the first belt truss correlates

well with the best‐fit power law, but the two progressively deviate at the upper elevations; this

is both due to the vertical discontinuities caused by the belt trusses as well as the increasingly

“shear‐like” behavior manifested by the upper quarter of the building. The second mode (Figure

3.7‐b) agrees with the best‐fit power law only over the lower tenth of the building, after which it

progressively approaches the idealized shear behavior over almost the entire top half of the

building. In both cases, EMS‐DCA affirms that the structure is actually less cantilever than the fit

suggests, even more so in the case of Mode 2. Based on the median errors of these fits, the iDCA

is predicted to be more shear‐dominated than the MS‐DCA originally predicted. Indeed the

actual iDCAs are consistent with this trend, and in the case of Mode 2, actually move the

building out of the interactive system classification into the shear building classification – a fact

that may be surprising to some considering the height of this building, but not unexpected

considering the strong reliance on deep beams as the primary mechanism for load transfer

within each tube. While this structural system is discontinuous, the discontinuity can be

classified as “progressive” (progressive reductions in width with height) as opposed to the

50

intermittent nature of the previous case study. In this case, the MS‐DCA tends to perform

better, though slightly overestimating the degree of cantilever action, with DCA near +20% for

both modes (see Table 3.3). Thus it can be deduced that the MS‐DCA is less affected by

discontinuities that are progressive in nature and more vulnerable to discontinuities that are

intermittent in nature.

Figure 3.6: CS2’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

67

Figure 3.7: CS2’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

68

53


The CS3 is similar structurally to the CS2 in that it is a bundled system with progressive reduction

in cross‐section with height; however instead of framed tubes, this structure utilizes space

trusses (Rastorfer 1987). These trusses force loads to travel through its members axially;

therefore this structure may be expected to have more cantilever action than the CS2. As such,

this case study provides an interesting point of contrast with the previous case study, though

there is a critical difference: in the current case study, truncations are not simply the

discontinuation of a previous structural framing pattern (CS2) but a complete change in the load

path between the mega columns. This includes a major structural discontinuity when the

structure transitions from a square cross‐section at its base to a triangular cross‐section at its

apex, requiring the introduction of a central mega column that continues up the building.

From a dynamic lumped mass model (Spence et al. 2008), it was confirmed that, like the CS2,

the CS3 behaves as a coupled building, as shown by Figure 3.8. The first two mode shapes

predicted by Spence et al.’s (2008) model are shown in Figure 3.9‐b and Figure 3.10‐b. The

overlaid lines denote the mid‐height of each truncation, which mark distinct transitions in the

mode shapes.

Figure 3.8: Axis assignment of the CS3 (Bentz 2012).

The limitations of the MS‐DCA can be observed in this case study, due to the dramatic variations

in the mode shape. As Mode 1 demonstrates, the mode shape translates from one that is

cantilever‐dominated in the lower part of the building to one that is progressively more shear‐

dominated at the upper elevations. This is due to the aspect ratio and cross‐section of the

building sections changing as towers are truncated. The cantilever nature of the building is less

obvious as the sections become shorter, particularly between the second and third overlaid lines

where barely any curvature is discernable (Figure 3.9‐b). The EMS‐DCA (Figure 3.9‐c) is minor,

resulting in a median error of 0.0%, which leads to the expectation that the MS‐DCA is accurate.

This accuracy is not reflected in the actual iDCA, whose deviation from the ideal cantilever is less

than the MS‐DCA predicted. However, as shown by the EMS‐DCA (Figure 3.9‐c), before the

54

introduction of the central column, the error suggests a more cantilevered behavior, which is

indeed reflected in the actual iDCA.

In the case of Mode 2, one can immediately observe a mode shape that is comparatively more

gradual in its variations: it does not possess the sharp transition in curvature near the mid‐

height noted in Mode 1 (see Figure 3.9‐b). This is caused by the truncation of Tower 3, which

contributes significantly to the stiffness of the structure when it deforms along this primary axis

of the tower (the deformation pattern reflected by Mode 1). As such, EMS‐DCA has a median value

of 5.2%, suggesting a slightly more cantilever structure than the MS‐DCA predicts. This is

consistent with the actual iDCA in Figure 3.10‐d. For this case study, DCA values are again less

than 25% for both modes (see Table 3.3) Thus this case study affirms that the MS‐DCA is

marginally affected by mode shapes with progressive discontinuities.


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CS4 employs a highly discontinuous structural system, with the structure terminating its core

around 0.65H to produce a dramatic thirty‐six‐story atrium for its hotel. To accommodate this

architectural requirement, the engineers switched the structural system from the mega‐column

system with a “mega‐reinforced concrete core” to a similar system but with a reconfigured

reinforced concrete core that allowed for the large open‐space in the center of the building

(Sarkisian et al. 2006). It is thus expected that this irregularity in the structural system will cause

a distinct distortion in the mode shapes and reduction in stiffness in the upper elevations. Two

two‐story outrigger trusses will further cause discontinuities in the mode shapes. The structure

terminates with a major steel cap truss supporting the spire, which will induce a near vertical

segment at the top of the mode shape. Thus this system represents a complex combination of

major discontinuities in the primary system, as well as a transition to a new system and

construction material at its top.

The first sway mode shape from a lumped mass model (Peng et al. 2003) is presented in Figure

3.114. The overlaid lines designate critical locations (from bottom: outrigger 1, outrigger 2, start

of atrium, cap truss). As in the previous case study of CS3’s first mode, this mode shape has

highly cantilever behavior at its base transitioning to linear behaviors for the hotel and cap truss

levels. Past case studies have supported the hypothesis that mode shapes with intermittent

irregularities over the height are not well captured by the MS‐DCA. In fact, the cap truss in this

building extends over about 12% of the total height of the building, and thus may have a

pronounced effect on the quality of any best‐fit power law expression. Thus the analysis herein

will consider both the full mode shape and the mode shape of only the primary structure with

the cap truss removed to isolate for this potential effect.

As discussed previously in Chapter 2, the MS‐DCA shows the tendency to be biased by the

behavior of the structure near its base, regardless of whether this behavior continues over a

substantial portion of the height. It is clear from Figure 3.12‐b that the MS‐DCA is indeed

influenced by the office section of the mode shape, despite a fairly modest median EMS‐DCA that

predicted the iDCA to be less cantilever than the MS‐DCA suggests. Instead, the actual iDCA is

more cantilever, with the two DCAs differing by 35%, establishing a stark difference in

classification for this building with a systemic discontinuity. Given knowledge of the structural

transitions, the structure would be assumed to be cantilever dominated over 60% of its height

and thus expected to correlate well with the C‐60 structure in the verification study in Section

2.2.2. For these, the average iDCA from the values presented in Table 2.2 is 0.6016, which is

indeed enveloped by the iDCAs for the CS4 with and without its cap truss (0.6044 and 0.6290).

This provides some assurance of the authenticity of the iDCA. When excluding the cap truss

region from the mode shape, the DCAs all increase (see Figure 3.12‐d), as expected; however

their agreement does not improve and actually slightly worsens, indicating that this feature

does not significantly impact either DCA measure. Thus this case study represents an example

of major structural discontinuities that occur at multiple locations over the height and a

4 The second sway mode shape was not available from the literature.

58

reaffirmation of MS‐DCA’s tendency to be biased by the curvature of the base and thus unable

to capture the behavior at upper elevations.


77

Figure 3.12: CS4’s first mode without the cap truss (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

(a) (b) (c)

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The CS5 was designed as an efficient megasystem of megacolumns, a concrete shear wall core,

outrigger belt trusses and diagonal bracing, which are seamlessly integrated to generate smooth

and nearly continuous mode shapes (Katz et al. 2008). The outrigger belt trusses break the

structure into “modules” over which gravity forces are locally transferred by frame elements

and then redistributed to the megacolumns by the belt trusses to manage the gravity loads

effectively. Between modules, diagonal braces are used to reduce shearing to maintain system

behavior without the need for dense frame elements. The overall concept leads to a light and

elegant structural form. Note that the structure, which begins as a square floor plan at the base,

tapers on the two opposing sides ending in a rectangular shape at the apex (Katz et al. 2008). As

the megasystem holds these outriggers and braces as integral to achieving its overall 3D

behavior, they are less likely to cause the level of mode shape discontinuity noted in past case

studies and are intended to achieve a highly cantilevered behavior with the only major

discontinuity being at the top of the structure, where the core terminates and the system

transitions into a space frame.

A FEM was developed internally based on published descriptions of the structural system and

fundamental periods (Bentz 2012). The mode shapes from this model corroborated well with

the published modes shapes from another finite element model (Shi et al. 2012), and thus the

in‐house finite element model was retained for this analysis. Figure 3.13 shows the top of the

tower in a cross‐sectional view from this model and visualizes the first two modes, oriented at

45° angles from the x and y axes defined for the purposes of model generation. The

fundamental mode shapes generated from this model are shown in Figure 3. 14‐b and Figure

3.15‐b. As anticipated, both fundamental mode shapes are smooth and continuous, strongly

linearizing near the top of the structure when the core terminates and the frame elements

provide the lateral resistance. The second mode shows a slightly greater sensitivity to this

structural discontinuity. It should be noted that the transition from the more cantilever‐

dominated system below the opening to the shear‐dominated top of the structure is very

smooth, speaking to the cohesiveness of the structural system as a whole.

62

Figure 3.13: CS5’s modal directions (Bentz 2012).

While the megasystem is intended to be a highly efficient, cantilever structure, the mode shapes

manifest DCAs that would classify them generally as an interactive system, indicating that the

diagonal bracing between belt outrigger levels may not be capable of fully arresting the effects

of shearing within each “module” of the building and enabling the columns to be completely

engaged axially. With that being said, the mode shapes show strong similarities, as one may

expect for megasystems intended to behave as a three‐dimensional structural system much like

tubes. As such, the MS‐DCA quantifies the two modes with almost identical degrees of

cantilever action. Based on the median EMS‐DCA, iDCAs in both modes are expected to be less

cantilever than the best‐fit power law suggests. This indeed is the case, with DCA of

approximately 60% for both modes. The significant difference between the two measures may

be attributed to the MS‐DCA’s tendency to be biased by base behaviors, as noted in Chapter 2,

misclassifying the building as cantilever‐dominated based solely on the base behavior. Visual

evidence of this overcompensation is apparent in Figure 3. 14 and Figure 3.15.

When the system was analyzed without the opening and space truss at the top of the structure,

all the DCAs correctly detected the increased cantilever action (Figure 3.16 and Figure 3.17).

Still, the agreement between the two measures does not improve significantly, with DCA

remaining near 60% in both modes, reaffirming the observations of the CS4 case study that

these cap truss regions have little effect on the two DCAs. As such, this case study reaffirms

that while the MS‐DCA may be assumed to be reliable for continuous systems, if these

systems have a strong tendency toward a specific mechanism exclusively near the base, the

MS‐DCA will tend to bias its classification toward this mechanism.

Figure 3. 14: CS5’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

52


83

Figure 3.16: CS5’s first mode without space truss (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

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Figure 3.17: CS5’s second mode without space truss (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

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CS6 is a reinforced concrete office building that utilizes a shear core with outrigger beams to link

it to a perimeter frame to handle lateral and gravity loads. It is a rectangular building at its base

that tapers to a square floor plan as it increases in height, topped with a cap truss (CTBUH

1995). The floor plans, depicted in Figure 3.18, show that the core’s strong axis orients with the

x‐direction, with a reliance on the weak axis of the walls interconnected by link beams for the y‐

axis resistance. The x‐axis also benefits from outriggers (CTBUH 1995), which are expected to

pull in the mode shape along this axis.

Figure 3.18: CS6’s general floor plans (CTBUH 1995).

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As the setbacks are mainly in the y‐axis, they contribute to only small modulations in the first

mode, Figure 3.19‐b and Figure 3.21‐b, which is dominantly in the x‐sway direction. The

outrigger levels correspond to the top and center overlaid lines. The outrigger near the mid

height of the building has the more pronounced “resetting” effect on the mode shape, while the

top outrigger and cap truss result in shear‐like behavior for the upper 15% of the building. As in

past cases with cap trusses, the mode shapes are analyzed for the full structure and the

structure without its cap truss to isolate the influences of this major structural discontinuity.

Despite both the action of the stiff core and the outriggers, Mode 1 is characterized by the MS‐

DCA as an interactive system, even when the cap truss is excluded (Figure 3.19 and Figure 3.21).

In fact, compensation for the cap truss makes no significant difference in either DCA measure,

consistent with past case studies (see Table 3.3). The EMS‐DCA indicates deviations particularly in

the middle third of the building suggesting a structure that is even less cantilever. The respective

7.3% and 5.3% median values of EMS‐DCA lead to a predicted iDCA that is even more shear‐

dominated. This trend is confirmed by the actual iDCA. As this mode shape does not appear to

be strongly influenced by the outriggers, the MS‐DCA and iDCA are in fairly good agreement,

within 20% of one another, even when the cap truss is included.

In Mode 2, the dominantly y‐axis sway mode relies on the linked core walls along their weak axis

and only the frame action of the slab to engage perimeter columns. Modulations within this

mode tend to correlate with the setback scheme along this axis and the effect of the cap truss

can be clearly seen in creating a near vertical profile within the mode shape (Figure 3.20‐b), thus

having a more dramatic influence upon this mode in comparison with Mode 1. This explains why

the DCA in Mode 1 is largely unaffected by the cap truss, while in the effect is more pronounced

in Mode 2, with agreement between the two measures actually diminishing (DCA goes from 12%

to 22%). Surprisingly, this mode does not have a lesser degree of cantilever action than Mode 1,

according to the MS‐DCA; however, both Mode 2 cases have median EMS‐DCA values that predict

the iDCA be more shear‐dominated (Figure 3.20‐d and Figure 3.22‐d). This trend is not observed

for the full structure, as the iDCA actually maintains the system classification as an interactive

system, but with the cap truss removed, the iDCA does indeed classify the structure as

marginally shear‐dominated. Still, for this mode, the two measures are again quite consistent

(see Table 3.3). Thus, as is the case of CS2 and CS3, this case study reaffirms that structural

systems with progressive discontinuities are well represented by the MS‐DCA.


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90

Figure 3.21: CS6’s first mode without cap truss (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

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Figure 3.22: CS6’s second mode without cap truss (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

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CS7 employs a structural system of a hexagonal core restrained with hammer head walls along

the three axes of its Y‐shaped floor plate shown in Figure 3.23. Through careful management of

gravity loads in the hammerhead system, the building efficiently counteracts overturning under

lateral loads. The structure introduces setbacks in a spiraling fashion to “confuse” the wind by

disorganizing vortex shedding over significant sections of the structure. Such coherence of the

different structural features working together leads to a very efficient system (BurjKhalifa.ae

2013). While the description of the primary lateral system would suggest that the building has a

high DCA, since the tower employs a steel space frame over roughly the top 10% of the tower,

there will be a significant shift in the mode shape curvature toward a shear‐dominated behavior

whose impacts will be explored in this section. Moreover, the considerable aspect ratio

variations (which cause commensurate reduction of the inertia of the structural system) will

similarly affect mode shape curvature along the height. The tower’s base, with the full length of

hammer head walls to restrain the core, is relatively rigid, creating a near vertical mode shape at

the base. As the walls truncate to create a subtle progressive discontinuity, the system becomes

increasingly flexible due to its effective reduction to just a core. By removing the ends of the

wings gradually with height, CS7 has a much more subtle truncation strategy than bundled tubes

employ. Eventually the structure becomes a core with barely any restraint indicating that the

stiffness in the lower 40% of the structure is dramatically greater than the upper 60%.

Moreover, these terminations create subtle progressive modulations that are not easily

detectable by the eye when viewing the mode shape, but will be noticed by the DCA measures.

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Figure 3.23: General floor plan of CS7 (Courtesy of RWDI).

From the designer’s original FEM, the fundamental mode shapes were obtained, revealing

coupled modes that vibrate dominantly in the y‐axis in its first mode and then the x‐axis in its

second mode, with similar characteristics as shown in Figure 3.24‐b and Figure 3.25‐b, as well as

in Figure 3.26‐b and Figure 3.27‐b where the steel spire/pinnacle region of the structure is

removed to explore the influence of this major structural discontinuity. Note that the y‐axis

(Mode 1) essentially aligns with Wing A denoted in Figure 3.23. When including the spire, it

estimates a cantilever structure in both modes, while the MS‐DCA estimates a marginally

interactive system (though borderline cantilever) for both modes when the spire/pinnacle is

discounted. This reflects the significant influence of the spire/pinnacle, in contrast with past

case studies, whose “whiplash” effect exaggerates the tip deflections relative to the

displacements of the rest of the tower. However, the EMS‐DCA for both pairs of modes suggests a

strong overestimation of the degree of cantilever action near the base and a consistent

underestimation in the upper elevations. The errors are particularly exaggerated below the

bottom semi‐transparent line, due to the high rigidity of the restrained regions and the

essentially negligible deflections in this region. Based on these median errors, the iDCA was

predicted to be even more cantilever than the MS‐DCA. Interestingly, the opposite is observed,

as the iDCA actually suggests the tower is more of an interactive system, to varying degrees in

each mode. This degree of shear behavior may be somewhat counterintuitive given the

understanding of the primary lateral system, but may be explained by the loss of inertia as the

shear walls truncate.

The abrupt transition to a steel space frame at the uppermost elevations (see top overlaid

horizontal line for Figure 3.24‐b and Figure 3.25‐b) results in a strongly linear regime that its

designers have suggested is “equivalent to having a 30‐story steel building sitting atop a 160‐

story concrete skyscraper.” But this appears to have a more marked effect on the classification

of Mode 2. The iDCA is identical for both Modes 1 and 2 when the pinnacle/spire is included.

When it is removed, the Mode 1 iDCA shows no significant change, while Mode 2 increases in its

degree of cantilever action by 20%. When comparing the two DCAs, the DCA is considerable,

approximately 60% in Mode 1 and 50% in Mode 2, when the pinnacle/spire is included (see

Table 3.3). Once this segment is removed, the DCA diminishes in Mode 1 to approximately 30%,

but experiences a more dramatic improvement in Mode 2, where the two measures are now

within 10%. Two observations are apparent: iDCA has greater immunity to the effects of the

pinnacle/spire region, and Mode 2 is more sensitive to the effect of the pinnacle/spire region.

Reasons for this consistent pattern of “immunity” will be discussed in the summary section of

this chapter.

As observed in previous case studies, the MS‐DCA is adversely affected by major structural

discontinuities like the pinnacle/spire region and again becomes biased by the base

characteristics, in this case a highly restrained (rigid) core at the lower elevations resulting in

almost “’hyper cantilever” behavior. While removal of the discontinuous region at the top did

help to improve performance in Mode 1 (reduced DCA from 57% to 31%), the disparate rigidity

75

of the shear walls on levels below the bottom semi‐transparent line results in a biasing that MS‐

DCA cannot fully overcome. It is presumed that the lack of improvement in Mode 1’s MS‐DCA

once the spire/pinnacle is removed is likely due to the fact that Mode 1’s dominant axis aligns

with Wing A of the tower and thus the effect of restrained core and potential for base biasing

may be very pronounced for this mode and less so for Mode 2, which does not align with any of

the wings explicitly. Therefore, this case study reaffirms that the MS‐DCA cannot accurately

measure a structure with systemic discontinuous behaviors that affect major portions of the

structure, as well as the iDCAs relative immunity to such discontinuities.


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Figure 3.26: CS7’s first mode without pinnacle/spire (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

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Figure 3.27: CS7’s second mode without pinnacle/spire (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

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CS8 employs a framed tube system that connects its 16 exterior columns to a reinforced

concrete core by beams at every floor to achieve a so‐called “tube in tube” system. The

structure tapers with increasing height, but keeps the same general, tapered square floor plan

(see Figure 3.28) and incorporates outriggers on two floors (Brownjohn et al. 2006), which are

assumed to be a secondary measure to enhance the engagement of the core with the perimeter

columns. As these discrete elements that are not integral to the behavior of the primary framed

tube, these outriggers are expected to cause minor discontinuities in the mode shapes at these

two locations. The mode shapes obtained from finite element models calibrated against full‐

scale data were digitized from Brownjohn et al. (1998), noting, “the orientation of lateral modes

did not coincide with the natural geometric axes of the building” (Carden and Brownjohn 2008);

however, the exact directionality of the vibrations were not specified.

Figure 3.28: CS8’s general floor plan (Carden and Brownjohn 2008).

As one may expect for a framed perimeter tube with fairly wide column spacing and no

additional measures to reduce shear lag, both mode shapes were identical and characterized

with the same MS‐DCA as interactive systems. From the EMS‐DCA in Figure 3.29‐c and Figure 3.30‐

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c, these fits are quite accurate with only a median error of ‐0.7%. As such, the iDCA would be

expected to show strong agreement with the MS‐DCA and if anything a slightly less cantilever

structure. This is exactly the case with this example; the iDCAs are found to be 11% less

cantilever than the MS‐DCA, though still classified as interactive systems. This case further

reaffirms that the MS‐DCA and iDCA show strong agreement for vertically continuous systems

without strong tendencies toward a particular deformation mode exclusively at their base.


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106

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CS9 is considered the inspiration to the restrained core system used in CS7. CS9 similarly utilizes

a hexagonal core linked to exterior column lines as its lateral load resisting system with

outrigger belt walls connecting the core and the columns at two mechanical levels (Abdelrazaq

et al. 2004). Another similarity is the truncating Y‐shaped floor plan (Figure 3.31) although this

structure differs from CS7 by truncating each tower fully, one at a time, more consistent with

bundled tubes.

Figure 3.31: CS9’s general floor plan (Abdelrazaq et al. 2004).

85

The fundamental mode shapes from the designer’s FEM displayed coupled modes that vibrate

dominantly in the direction of the shortest tower (x‐axis) and then towards the tallest in its

second mode (y‐axis). The overlaid semi‐transparent lines in Figure 3.32 and Figure 3.33

represent the outrigger levels (first and third lines from the bottom), as well as where the

building truncates for the first two modes. Both modes, as expected with a coupled, continuous

system, show similar behavior with vertical hitches at the outrigger levels and other trademarks

of progressively discontinuous systems like the CS2. Therefore, it is expected that the MS‐DCA

will similarly be capable of capturing the degree of cantilever action for the modes.

While the median EMS‐DCA for both cases indicated a significant decline in degree of cantilever

action, predicting the iDCA will classify the system as shear‐dominated instead of interactive

(Figure 3.32‐c and Figure 3.33‐c), the actual iDCA values show exceptional agreement with the

MS‐DCA (see Table 3.3). Thus, this case study further reaffirms the accuracy of the MS‐DCA

when classifying systems with progressive discontinuities.


(a) (b)

(c)

109


(a) (b)

(c)

110

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The structural system of CS10 includes a composite frame system with a concrete core and

outrigger‐belt systems at four levels to restrict the overall deflections of the building and

increase the stiffness (Li and Wu 2004). This structure has a unique shape: a rectangle with two

semi‐circular sections attached to the two short faces (see Figure 3.34). From this image, one

can deduce that the structure will derive its y‐axis resistance from the strong axis bending of the

core shear walls and engagement of perimeter columns by the outriggers, whereas the x‐axis

resistance is derived from the composite action of weak axis core walls connected by link beams

and the perimeter MRF.

Figure 3.34: CS10’s general floor plan (Li and Wu 2004).

Published mode shapes from finite element models calibrated against full‐scale data from the

CS10 (Li and Wu 2004) were digitized and analyzed for this case study, with Mode 1

representing y‐axis sway and Mode 2 corresponding to x‐axis sway. Overlaid horizontal lines

mark the levels of the outriggers and additional vertical bracing. Starting with the first mode,

Figure 3.35‐b, the mode shape shows significant modulations; some coinciding with the

outrigger and vertical bracing levels, embodying a mild form of intermittent discontinuity

observed in previous case studies (see CS1) that are known to adversely affect the MS‐DCA. The

median EMS‐DCA suggests the iDCA may be more cantilever than the MS‐DCA; true to form, the

iDCA reveals that the building is approximately 9.0% more cantilever than the MS‐DCA along this

axis. Since there are only two clear instances of intermittent discontinuity, the effects are less

severe than the CS1 where the discontinuities continued repeatedly all along the height, thus

explaining why the two measures are within 10% of one another (see Table 3.3).

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Conversely, for the second mode in Figure 3.36‐b, the modulations in the mode shapes due to

the outriggers are subtler than expected. Moreover, the mode shape exhibits a high degree of

cantilever action with no discernable “resets,” which suggests that the deflections are strongly

core dominated with the outriggers playing a very modest role; this is expected since the

literature denotes the outriggers are running perpendicular to the longitudinal direction and

thereby would have little impact on this direction’s behavior. Therefore, the second mode

exhibits progressively discontinuous behavior similar to the last case study, CS9, leading to the

belief that the MS‐DCA and iDCA will show reasonable agreement. This is confirmed with the

best‐fit power underestimating the mode shape slightly from 0.25H to 0.7H, easily visualized on

Figure 3.36‐c. From the median of EMS‐DCA, the iDCA is expected to be more cantilever than the

MS‐DCA, with the actual iDCA being slightly more cantilever, but essentially showing identical

classifications by the two DCAs (see Table 3.3). Therefore, as the discontinuities in this case

study are comparatively mild, the mode shapes behave essentially like a continuous system,

particularly in Mode 2, for which the two DCA measures are consistent.


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115

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3.3 Summary

This chapter presented a validation of the proposed iDCA through comparisons with the MS‐

DCA for several real‐world case study buildings. Table 3.3 summarizes the differences in the two

DCA measures, as calculated by Equation 3.2, as well as the variables used in that calculation,

with the case studies categorized by the heuristic system behaviors observed throughout this

chapter. The errors are also graphically displayed in Figure 3.37 according to the mode shape

number assigned earlier in the chapter (see Table 3.2) and repeated in Table 3.3. The light green

shaded region represents a region of strong agreement between the two DCA measures. These

case studies and the visual representation enabled by this figure help to reveal a number of

recurring themes:

1. Consistent with the findings of Chapter 2, MS‐DCA shows a greater likelihood of

being biased when the base behaviors of a system have a strong tendency

toward a specific mechanism exclusively in that region. This was commonly

manifested by base behaviors that were strongly cantilever in comparison to the

rest of the structure. Such is the case in systems with systemic discontinuities,

as well as some instances of continuous systems, e.g., CS5.

2. Systems that are continuous (without the base behaviors described above) or

have progressive discontinuities were consistently characterized by both DCAs

(differences of 25% or less), whereas the MS‐DCA proved to be incapable of

detecting the effects of strong, intermittent discontinuities such as those in the

CS1.

3. iDCA showed superior ability to capture subtle variations in the degree of

cantilever action in mode shapes with similar behaviors and an overall

consistency with heuristic understanding of system behavior, reaffirming the

findings of Chapter 2.

4. Traditional cap trusses were not found to significantly impact the DCAs, except

when they support major architectural features, such as the large spire in CS7.

In this case, the MS‐DCA showed greater sensitivity to these features, while

iDCA remained comparatively immune.

One way to explain the consistent “immunity” of the iDCA to cap truss and architectural

features is due to the fact that it is derived based on the “distribution of slopes” and not

affected by the spatial location of those slopes. On the other hand, a best‐fit curve, and thus the

MS‐DCA, is highly sensitive to not only the curvature of the line but also the spatial variation of

this curvature, as evidenced in the case of Mode 2 in CS7. It is worthwhile, at this point, to

consider when attempting to quantify the fundamental deformation mechanisms of the primary

lateral system, particularly for the purposes of classifying systems or predicting dynamic

properties such as energy dissipation potential, whether architectural features be included.

Even though it was shown that in most cases, the cap truss’s omission had minor effects on both

DCAs; the iDCA is sensitive enough to detect minor variations in the degree of cantilever action

and will thus increase slightly when cap trusses supporting architectural features are omitted. As

such, does the inclusion of the structure supporting architectural features unnecessarily bias the

93

DCA measure and potentially distort our understanding of the primary system behavior? Some

may argue that it indeed may be warranted to extract DCA measures from truncated mode

shapes that remove major cap trusses supporting spires and pinnacles. However, since three out

of the four case studies involving cap trusses did not drastically affect the DCA measures, the full

mode shapes will be employed for the population of the database presented in Chapter 4.

Thus, it is concluded from the case studies in this chapter that the MS‐DCA measure is not

robust enough to capture the mode shape behaviors of a wide array of structural systems with

varying classes of discontinuity. As the iDCA measure proposed in this thesis has proven to

effectively capture behaviors consistent with heuristic understanding and ample sensitivity

while demonstrating sufficient robustness for diverse classes of discontinuities in real‐world

structures, it will be recommended as the preferred measure of the degree of cantilever action.

Having fully vetted this DCA, it will now be applied in the next chapter, along with other

geometric descriptors, to describe the system characteristics within a database of modern tall

buildings.

TABLE 3.3

COMPARISON OF iDCA AND MS‐DCA FOR CASE STUDY BUILDINGS

System Behavior Building

Mode Shape Number iDCA MS‐DCA DCA

Continuous

CS5 9 29.5 11.6 61%

10 28.5 11.8 59%

CS5* 11 26.6 10.7 60%

12 25.6 10.8 58%

CS8 21 33.8 30.2 11%

22 33.8 30.2 11%

Progressively CS2 3 35.4 29.4 17%

94

Discontinuous 4 38.8 30.8 21%

CS3 5 41.9 46 ‐10%

6 35.3 43.6 ‐24%

CS6 13 38 30.9 19%

14 35.7 31.3 12%

CS6* 15 37.4 30.3 19%

16 38.6 30.1 22%

CS9 23 31.5 31.2 1%

24 31.7 31.6 0%

CS10 26 16.4 17.9 ‐9%

Intermittently Discontinuous

CS10 25 22.5 23.1 ‐3%

CS1 1 9.6 13.2 ‐38%

2 21.8 13 40%

Systemically Discontinuous

CS4 7 24.2 32.7 ‐35%

CS4* 8 21.8 30.8 ‐41%

CS7 17 21.1 9.1 57%

18 21.1 10.9 49%

CS7* 19 20.9 14.3 31%

20 16.6 17.9 ‐8%

* Modified mode shapes, e.g. cap trusses removed

95

Figure 3.37: Comparison of errors and system behavior for case study buildings.

96

CHAPTER 4: DATABASE POPULATION AND MINING

4.1 Introduction

Having validated the proposed descriptor of system behavior, the integral degree of cantilever

action, against mode shapes from actual buildings in the previous chapter, as well as against

simulated mode shapes in Chapter 2, it will now be used in this chapter as the preferred means

to characterize the behaviors of tall buildings in the current databasing effort. This database was

created to include tall buildings (defined as buildings with heights of at least 200 m) completed

since 2002, so as to maximize the availability of FEMs or mode shapes required for the iDCA

calculation. The CTBUH Skyscraper Center and Emporis websites (CTBUH 2014; EMPORIS 2014)

were utilized to identify the tall buildings meeting these criteria, recording their height and

construction material in the proposed database. A total of 75 buildings were identified, the

shortest of which being 207.1 m. The buildings included in the database, sorted alphabetically,

are listed in Table 4.1. The location (city) of each building was also recorded in the database, and

each building was assigned a randomized numerical identifier to preserve their anonymity in

subsequent analyses. The aspect ratios (for both primary building axes) and primary lateral

system (also distinguished for each axis) were also included in the database for each building.

Unfortunately, such details are not readily available from the CTBUH and Emporis websites. As a

result, a variety of secondary sources were consulted.

Table 4.1 lists the sources used for each building. Wherever possible, published aspect ratios

and lateral system descriptions were used. Buildings for which such sources were available are

marked with a Quality Index (QI) of 1 in Table 4.1. In instances where this information was not

reported in the literature by the designer, a secondary approach was employed. It is first noted

that the aspect ratio (height/width) can be difficult to quantify given the complex geometries of

modern tall buildings. In fact, 95% of tall buildings have varying widths along the height even in

just one axis; some taper as they climb while others have “bellies”, where the largest widths are

not at the base and instead somewhere in the midsection of the building (Ho 2014). As such, for

height we will adopt the CTBUH Skyscraper Center convention, used to crown the world’s tallest

building: “measured from the level of the lowest, significant, open‐air, pedestrian entrance to

the architectural top of the building, including spires, but not including antennae, signage, flag

poles or other functional‐technical equipment” (CTBUH 2014). The mode of the width will be

taken as the “width” in the aspect ratio calculation, thus eliminating concerns over buildings

with sculpted tops and/or podium levels. This mode was secondarily determined from Google

Earth by the following procedure. From the downloaded Google Earth software, each building

was located and the software’s ruler tool was used to measure the widths of the extruded

buildings in each direction. To verify accuracy of this process, four buildings with published

aspect ratios were measured via Google Earth and found to match nearly identically (the largest

97

discrepancy being only 0.12 m), as shown in Table 4.2. This process was required for 57 of the 75

buildings, thereby assigned a Quality Index of 2 in Table 4.1. In the cases where only one aspect

ratio was published, the published floor areas were used to estimate the length of the building

in the opposite direction. This is also suboptimal since it assumes a rectangular geometry;

therefore the Google Earth techniques were used to verify these calculated lengths. Such

instances are also awarded a Quality Index of 2 in Table 4.1.

TABLE 4.1

BUILDINGS USED IN PROPOSED DATABASE WITH SOURCES FOR THE SYSTEMS AND ASPECT RATIOS DATA

Building Name Designer

Aspect Ratio Structural System

Quality Index Source

Quality Index Source

111 South Wacker Goettsch Partners 2 GE: Chicago 1 Becker (2006)

23 Marina KEO International Consultants 1 Colaco (2005) 1 EMPORIS (2014)

300 North Lasalle Pickard Chilton 2 GE: Chicago 1 Ascribe (2012)

383 Madison Ave Skidmore Owings & Merrill

(SOM) 2 GE: New York

City

Al Hamra Firdous Tower SOM 2 GE: Kuwait City 1 CTBUH (2013)

Almas Tower The Taisei Corporation 1 Scott (2011) 1 Shahdadpuri

(2007)

Aqua Magnussen Klemencic

Associates (MKA) 2 GE: Chicago 1 MKA (2012)

Arraya Tower Pan Arab Consulting Engineers 2 GE: Kuwait City 1 WAN (2010)

Aspire Tower Arup 2 GE: Doha 1 Arup

Bank of America Tower Severud Associates 2 GE: New York

City 1

Metals in Construction

(2008)

124

Burj Khalifa* SOM 1 SOM 1 SOM

Caja Madrid Foster + Partners 1 Lakota and

Alarcon (2008) 1 Lakota and

Alarcon (2008)

Capital City Moscow Tower Arup 2 GE: Moscow 1 Stardubtsev et

al. (2011)




QI Source QI Source

China World Tower Arup 2 GE: Beijing 1 DesignBuild

(2010)

Comcast Center Thornton Tomasetti 1 Milinichik (2006) 1 Milinichik (2006)

Diwang International Commerce Center

CityMark Architects and Engineers 2 GE: Nanning 2 LERA (2009)

Doosan Haeundae We've the Zenith Tower A*

Thornton Tomasetti; New Engineering Consultant, Inc. 1

Thornton Tomasetti 1

Thornton Tomasetti

Doosan Haeundae We've the Zenith Tower B*

Thornton Tomasetti; New Engineering Consultant, Inc. 1

Thornton Tomasetti 1

Thornton Tomasetti

Eight Spruce St / Beekman Tower Gehry Partners, LLP 2

GE: New York City 1

Marcus and Hamos (2013)

Elite Residence Eng. Adnan Saffarini 2 GE: Dubai 2 Solt (2010)

Emirates Crown Design & Architecture Bureau 2 GE: Dubai 2 Solt (2010)

Etihad Towers T1 Aurecon 1 Aurecon (2014) 1 Aurecon (2014)

125

Etihad Towers T2 Aurecon 1 Aurecon (2014) 1 Aurecon (2014)

Eureka Tower Connell Mott MacDonald 1 Dean et al.

(2001) 1 Grocon (2006)

Excellence Century Plaza Tower 1

China Construction Design International 2 GE: Shenzhen 2 Le Berre (2009)

Guangzhou International Finance Center Arup 2 GE: Guangzhou 1 Wilkinson (2012)

Haeundae I Park Marina Tower 2

Arup; DongYang Structural Engineers 2 GE: Busan

HHHR Tower Al Hashemi / Farayand

Architectural Engineering 2 GE: Dubai 1 EMPORIS (2014)




QI Source QI Source

Highcliff DLN Architects & Engineers 1 Kostura (2008) 1 Kostura (2008)

Hyatt Center Pei Cobb Freed & Partners 1 Hopple (2005) 1 Hopple (2005)

International Commerce Centre Arup 2 GE: Hong Kong 1 KPF (2012)

Keangnam Hanoi Landmark Tower

DongYang Structural Engineers 2 GE: Hanoi 2

Skyscraper Page (2011)

Khalid Al Attar Tower 2 Eng. Adnan Saffarini 2 GE: Dubai 2 Solt (2010)

KK100 Development RBS Architectural Engineering

1 Farrell (2011) 1 Hernandez

Design Associates (2012)

Leatop Plaza* MKA 2 GE: Guangzhou 1 MKA

Longxi International Hotel A&E Design 2 GE: Jiangyin 2 Skyscraper City

(2010)

Marina Pinnacle National Engineering Bureau 2 GE: Dubai 2 Wikimedia

Commons (2007)

Millennium Tower e.Construct 2 GE: Dubai 1 EMPORIS (2014)

Minsheng Bank Building Wuhan Architectural Institute 2 GE: Wuhan 2 Skyscraper City

(2006)

New York Times Tower Thornton Tomasetti 1


(2006) 2


(2006)

Nina Tower Arup 2 GE: Hong Kong 2 Johnson (2010)

Northeast Asia Trade Tower Arup; DongYang Structural

Engineers 2 GE: Incheon 1 Chung et al.

(2008)

Ocean Heights Meinhardt 2 GE: Dubai 1 Blackman (2010)




QI Source QI Source

One Island East Centre Arup 2 GE: Hong Kong 1 ISE (2009)

One World Trade Center SOM 2 GE: New York

1 Gonchar (2011)

126

City

Pearl River Tower SOM 2 GE: Guangzhou 1 Frechette III and Gilchrist (2009)

Princess Tower Eng. Adnan Saffarini 2 GE: Dubai 1 Ephgrave (2012)

Suzhou RunHua Global Building A ECADI 2 GE: Suzhou

The Address Atkins 2 GE: Dubai 2 McMorrow

(2012)

The Domain Foster + Partners 2 GE: Abu Dhabi 2 Jimaa (2011)

The Index Halvorson and Partners;

Bruechle; Gilchrist & Evans 2 GE: Dubai 1 Halvorson 2008)

The Pinnacle Guangzhou Hanhua Architects

& Engineers 2 GE: Guangzhou 2 Skyscraper City

(2010)

The Shard Renzo Piano Building

Workshop 2 GE: London 1 Pearson (2012)

The Torch Khatib & Alami 1 Nair (2011) 1 Nair (2011)

Tianjin Global Financial Center SOM 2 GE: Tianjin 1 Griffith (2012)

Time Warner Center North Tower SOM 2

GE: New York City 1 SOM

Time Warner Center South Tower SOM 2

GE: New York City 1 SOM

Tomorrow Square John Portman & Associates 2 GE: Shanghai 2 Cichy (2012)

Torre Mayor Zeidler Partnership Architects 2 GE: Mexico City 1 Taylor (2004)

127




QI Source QI Source

Torre Vitri Pinzon Lozano & Asociados

Arquitectos 2 GE: Panama City

Tower Financial Center Pinzón Lozano & Asociados

Arquitectos 2 GE: Panama City

Tower Palace III* SOM 2 GE: Seoul 1 SOM

Trump International Hotel and Tower Halcrow Yolles 2 GE: Toronto 2 Conway (2010)

Trump International Tower SOM 2 GE: Chicago 1 Baker et al.

(2009)

Trump World Tower Costas Kondylis & Partners

LLP Architects 1 Seinuk (2013) 1 Seinuk (2013)

Wenzhou Trade Center

Shanghai Institute of Architectural Design &

Research 2 GE: Wenzhou 2 Skyscraper City

(2010)

Yingli International Finance Centre

Chongqing Yingli Real Estate Development 2 GE: Chongqing 2

Skyscraper City (2011)

Zifeng Tower SOM 2 GE: Nanning 1 CTBUH (2013)

Notes: * indicates participating design firm supplied actual mode shape data. QI = Quality Index: 1 if obtained from

128

designer or published literature, 2 if estimated from Google Earth (Aspect Ratio) or from construction photos (Structural System). Entries shaded in grey omitted from final database due to lack of Structural System information.

105

Even when published descriptions of a lateral system are available, the lack of a unified

vernacular for structural systems leads to differing terminology essentially describing the same

fundamental system concept. As such, a vernacular is proposed here to group and order the

systems, building off of the general classes used in the historical hierarchies in Chapter 1. Each

class of systems will be accompanied by common augmentations based on heuristic

understanding of iterative system design. Each general class will be numbered with a system

identifier (System ID) to facilitate their presentation in subsequent tables. These identifiers are

summarized in Table 4.3. Because of the explicit focus on tall buildings, there will be little

representation of foundational or Basic systems like exterior shear walls or moment resisting

frames or combinations thereof, as such these are assigned a System ID of 0.0 in Table 4.3.

TABLE 4.2

VERIFICATION OF GOOGLE EARTH MEASUREMENTS WITH PUBLISHED ASPECT RATIOS

Building Name

Published Values Google Earth Measurements

Length (m) Width (m) Length (m) Width (m)

23 Marina 41.60 41.60 41.66 41.66

Almas Tower 64.00 42.00 63.98 42.09

The Torch 35.00 35.00 34.88 34.88

Pearl River Tower 68.89 33.70* 68.92 33.63

Notes: Sources of published values reported previously in Table 4.1

* Calculated from height divided by published aspect ratio

The first major family of tall building systems will be Core systems (System ID = 1.0). These are

systems that rely primarily on a stiff interior core for lateral resistance. A progressive sequence

of augmentations is commonly observed, the first creating a dual system through exterior lateral

load resisting elements. These are termed Core + Ext. System (System ID = 1.1). These exterior

systems may be distributed or focused at the perimeter only and include moment resisting

frames (with and without braces), shear walls, or combinations of these elements. The next

augmentation commonly observed takes this dual system and adds explicit linkages between

the core and perimeter elements. These Core + Ext. System & Link (System ID = 1.2)

traditionally employ discrete outriggers, though continuous buttressing of cores has recently

surfaced, best exemplified by the world’s tallest building: Burj Khalifa.

The next system family is the Tube, whose basic form is a purely perimeter moment resisting

frame with closely spaced columns and deep beams (System ID = 2.0). By definition, it places all

106

lateral resistance at the exterior of the structure creating great flexibility in interior planning,

directly opposing the Core family’s philosophy. A series of augmentations have been observed

for the tubes, noting that these are not progressive (cumulative) as was the case in the Core

family. The first possible augmentation is the creation of the so‐called “tube‐in‐tube” through

the addition of a stiff interior core. This Tube + Core is given System ID of 2.1. The next possible

augmentation is the addition of explicit linkages within the tube. These can take on various

forms, from the most simplified case of discrete outriggers to distributed linkages providing

interior pathways by bundling a series of smaller tubes, best exemplified by the iconic Sears

Tower. The Tube + Link (System ID = 2.2) therefore draws attractive parallels to the Core + Ext.

System + Link (System ID = 1.2) by creating distributed linkages that enable super tall forms.

The third augmentation in tubes is the addition of exterior bracing, perhaps most classically

embodied by the John Hancock Center in Chicago (the so‐called braced tube). This Tube + Braces

system is assigned a System ID of 2.3.

The final system family is the Megasystem (System ID = 3.0). The classical definition of a

megasystem actually comes from Fazlur Khan’s concept of a modular system that manages

gravity loads over each module; pushing them out by transfer elements to megacolumns on the

perimeter. This modular format implies that this same structural load path is repeated over the

height of the structure many times. Megasystems have three critical ingredients: a stiff interior

core, stiff perimeter elements (generally megacolumns) and a mechanism to link them together,

often accomplished through outriggers and belt trusses. Additional perimeter bracing is often

introduced to further arrest the effects of shear lag over the modules to maintain highly

cantilever system behavior. Shanghai World Financial Center is the exemplar for this concept.

However, there are a number of modern systems that use this formula (core + megacolumn +

link) without highly modularizing the systems to repeatedly manage the gravity loads as Khan

envisioned. Thus Khan’s system may be considered the purist form of what has become an

increasingly common strategy for tall buildings. As such, the Megasystem family will include any

structures that employ megacolumns with linkages as simple as the slabs at each floor and as

robust as discrete outriggers, which may appear at many levels. These may also include

additional bracing or distributed shear walls to further stiffen the system. Because this essential

formula sees many variations within modern systems, it is not further distinguished by

subclasses. Finally, a fourth evolving class of systems is the exterior Diagrid (System ID = 4.0), a

highly efficient exterior frame system with distributed resistance that was popularized by

architect Sir Norman Foster, perhaps best visualized by 30 St. Mary Axe in London.

TABLE 4.3

NUMERICAL IDENTIFIER FOR EACH SYSTEM TYPE

Structural System System ID

Basic 0.0

107

Core 1.0

+ Exterior System (+ Ext.) 1.1

+ Exterior System & Link (+ Ext. & Link) 1.2

Tube 2.0

+ Core 2.1

+ Link 2.2

+ Braces 2.3

Megasystem 3.0

Diagrid 4.0

Note: Abbreviations used in figures shown in parenthesis.

The assignment of each building into the families of systems outlined in Table 4.3 utilized

published descriptions of the system whenever possible (QI = 1). In instances where lateral

system descriptions were not publically available, available photos of the structure during

construction before adding any cladding or interior finishes were inspected. Note that while this

reveals the primary structure, some features may be obscured. It should also be noted that

while cores are discernable in some construction photos, it is not always obvious whether they

are part of the lateral load resisting system or part of a separate gravity system. To err on the

side of caution, all cores detected in construction photos were considered to be part of the

lateral system, as modern systems typically no longer make such strong divisions between

gravity and lateral systems. 23 of the 75 buildings required construction photos to classify their

structural system, each denoted with Quality Indexes of 2 in Table 4.1; unfortunately, another 5

buildings did not have published structural systems or construction photos available and were

ultimately excluded from the final database. These are shaded in grey in Table 4.1.

Next, the database was populated with each building’s DCA value. Securing mode shape data

proved to be the most challenging aspect of the database population, since it largely relied on

designers to supply the information, as mode shapes were rarely published. Unfortunately, such

voluntary participation of firms can be difficult to secure, and in some cases, although willing,

the firms may not deliver the required information in a timely fashion. In order to maximize the

potential for success, the firms with multiple buildings in the database were identified (Arup,

Foster + Partners, Magnusson Klemencic Associates, Halvorson and Partners, Skidmore Owings

and Merrill, and Thornton Tomasetti), so that mode shape information could be secured for up

to 27 buildings through a concerted campaign of engagement. Of these, two declined to

participate, two provided the requested information, and two others had yet to provide agreed

upon information at the time this chapter was written. This effort yielded mode shapes for three

more buildings, with roughly twenty more to be provided within the foreseeable future.

Additionally, three of the buildings’ mode shape information was already available from Chapter

108

35. For each of the buildings with designer‐supplied mode shape information, denoted by

asterisks in Table 4.1, the MS‐DCAs and iDCAs were calculated and included in the database.

4.2 Data‐Driven Hierarchy for Modern Systems

Having populated the database to the greatest extent possible, four different system descriptors

will now be explored for the remainder of this chapter to enable the construction of a data‐

driven hierarchy for modern systems. Two geometric descriptors: height and aspect ratio will be

considered; along with two behavior descriptors: the degree of cantilever measures, MS‐DCA

and iDCA. Note that while the limited number of mode shapes secured will limit the number of

buildings for which DCAs can be calculated and therefore the inferences that can be drawn from

these figures, the primary intent of this thesis is to establish an appropriate parameterization,

database structure, and process for extracting the hierarchy, which can be dynamically updated

with time, as discussed later in Chapter 5.

An initial statistical analysis will be conducted on each of the four descriptors to extract

information relevant to the hierarchy’s construction. To support this process two figures will be

created for each of the four system descriptors. In these figures, the systems are ordered along

the x‐axis according to the progression assumed in historical charts like Figure 1.3. By adopting

this convention, trendlines will readily underscore whether the historical conceptualization of

system evolution holds true. The parent system is displayed in large bold font followed by any

augmentations (see Table 4.3) in smaller, regular fonts, retaining the color‐coding in that table

throughout so families of systems can be clearly identified. The first figure in the two‐figure

sequence for each descriptor will consider the Quality Index of the data. Purely black squares in

this figure have a Quality Index of 1 for both system and aspect ratio. If either data source has a

Quality Index of 2, the data point will be outlined (in orange for aspect ratios; in red for

systems). For each system, the mean of the descriptor is calculated and displayed on the figure

as a hollow circle, with error bars designating ± one standard deviation from the mean. A

dashed linear trend line through the mean values is also displayed.

The second figure in the two‐figure sequence will sub‐classify the data by the primary material

used in the lateral system, maintaining the classifications in Emporis: reinforced concrete (blue

square), steel and concrete (red square), steel (green square) and composite (purple square).

The distinctions between steel and concrete and composite are not clearly articulated by

Emporis, so for the purposes of interpretation and discussion, they will be viewed as one in the

same in this chapter. The Quality index will not be distinguished in this second set of figures,

since it is depicted in the first figure in the sequence. For each material, the mean value is again

presented as a circle. In instances where more than one observation for that material within

that system class is available, error bars signifying ± one standard deviation from the mean are

also presented. When at least three systems have mean values for a given material, a dashed

linear trendline will be presented.

5 Due to the limited number of mode shapes available for modern systems, additional iDCAs will be imported from older buildings showcased in Chapter 3 for a demonstrative analysis in Section 0.

109

Additionally, a series of tables will present both mean values as well as coefficients of variation

(CoVs) to reveal scatter within each system. To discern the influence of lower fidelity data

sources, variation within each system is first documented across all data points, then retaining

only data points with QI = 1. The entirety of the data is also independently sub‐classified by

material (retaining both QI = 1 and 2 data). Statistics are presented first individually for each

system defined in Table 4.3 and then across each of the families of systems (bolded in the

tables). The geometric descriptors will be discussed first, followed by the behavioral descriptors.

4.2.1 Geometric Descriptors: Height

Height is explored as the first geometric descriptor of structural systems. Recall from Chapter 1

that this parameter has been historically utilized, serving as the basis for Fazlur Khan’s system

hierarchy, and was shown previously not to correlate with system type (see Figure 1.6). This is

reinforced by the current database (through Figure 4.1 and Figure 4.2). To further underscore

the differences between constructed tall buildings and conceptual hierarchies, the

recommended limits of general system classes from Figure 1.3 were converted to heights

(assuming a 4 m floor‐to‐floor height) and superimposed on Figure 4.1 and Figure 4.2 as the grey

shaded region, with the upper bound associated with steel‐based systems and the lower bound

with reinforced concrete‐based systems. It is interesting to note that while material choice is

reported as negligible by these historical hierarchies for half the systems, there is a clear

bifurcation beginning at the Tubes. Interestingly, while the historical hierarchy correlates well

with linked tubes, Megasystems are clearly employed at much lower heights than the hierarchy

suggests. Conversely, core systems dramatically exceed the heights assumed in the historical

hierarchy, further motivating the need to modernize our understanding of the heights over

which systems can be effectively applied; they are clearly “over‐performing” when considering

their hypothesized limits. Figure 4.1 suggests a weak trend of increasing height (NS = 2.48%)6

along the historical system progression displayed on the x‐axis, though again when considering

the grey shaded region showing the conceptual correlation between height and system

classification, modern tubes and megasystems are considerably shorter than the range over

which the historical hierarchy suggested they would be applied. To explore the influence of

material, Figure 4.2 shows trendlines for both composite and reinforced concrete systems.

Interestingly, reinforced concrete systems effectively “flatline,” showing no strong increase in

their mean heights along the historical system evolution (NS = 0.50%). On the other hand, the

composite systems show a stronger correlation between height and system progression (NS =

3.21%), though megasystems still are being used at heights much lower than historically

expected.

6 NS: Normalized slope is defined as the slope of the linear regression divided by its y‐intercept; permits cross comparison between descriptors.

110

Figure 4.1: Relationship between height and structural system, distinguished by source fidelity.

111

Figure 4.2: Relationship between height and structural system, distinguished by material.

Table 4.4 helps to further underscore the variability in height within each system classification,

as well as important general trends. Core‐based systems prove to be most popular for lateral

resistance (N=34, where N is the number of data points for a given system type), showing

effective use of a feature all tall buildings inherently require to support elevators and other

services. Within the Core family, those employing linkages are clearly the dominant typology

(N=23/34), most commonly using outriggers, which is consistent with the trends observed by

CTBUH in Figure 1.6. This trend towards cores is further substantiated by also considering that

the second most prevalent system family, Megasystems (N=20), also relies heavily on a

structural core. When considering the effect of reduced fidelity observations on each system

family, all CoVs if anything increased when only highest quality data was retained, indicating

that the questionable quality data, if anything, tended to cluster around the general trend for

that system class and thus did not cause outliers that may bias interpretation significantly.

Breaking down CoVs by material or even across classes provides limited opportunity for

meaningful analysis, due to the low number of observations in some sub‐categories. Cores and

megasystems have the highest CoVs, likely due to the relatively larger number of data points in

these classes. Because of this fact, these will be examined in depth for each descriptor, tracking

the CoVs of these two classes of systems. Between the two, the doubly augmented cores

(System ID = 1.2) show appreciably higher scatter (36.85% vs. 20.01%). The core family is the

112

only progressive augmentation scheme. As the augmentations progress, system behavior is

expected to be enhanced and thus be increasingly warranted as height increases. Examining the

mean heights across these augmentations, the basic core has a mean height of 294 m and its

augmentations progressively increase the mean height to 322.3 m and 334.9 m respectively,

thus being consistent with our heuristic understanding of core progression. Due to the limited

number of observations, the same cannot be reliably tracked for tubes, though the observations

in Table 4.4 at least suggest that the augmentations, like System ID = 2.2, do not correlate with

an appreciable increase in height over the basic tube (307.5 m vs. 330.8 m, respectively).

113

TABLE 4.4

HEIGHT AS GEOMETRIC SYSTEM DESCRIPTOR: STATISTICS BY SYSTEM TYPE

System ID

N

Mean [m]

(Coefficient of Variation, %)

All Only QI =

1

Sorted By Material

Concrete Composite Steel Concrete and Steel

0.0 2

308.5

(10.67%)

337.0

(0.00%)*

280.0

(0.00%)*

337.0

(0.00%)* ‐‐‐‐ ‐‐‐‐

1.0 3

294.0

(3.16%)

294.0

(3.16%)

291.0

(3.57%)

300.0

(0.00%)* ‐‐‐‐ ‐‐‐‐

1.1 8

322.3

(32.24%)

322.3

(32.24%)

321.2

(18.63%)

323.4

(43.24%) ‐‐‐‐ ‐‐‐‐

1.2 23

334.9

(36.85%)

339.6

(44.48%)

367.7

(41.23%)

311.8

(24.69%)

325.0

(2.13%)

228.3

(0.00%)*

All Cores 34

328.3

(34.45%)

329.0

(38.90%)

348.9

(36.97%)

314.7

(30.73%)

325.0

(2.13%)

228.3

(0.00%)*

2.0 4

330.8

(16.73%)

413.0

(0.00%)*

315.0

(7.70%)

280.0

(0.00%)* ‐‐‐‐

413.0

(0.00%)*

2.1 1

225.0

(0.00%)*

225.0

(0.00%)* ‐‐‐‐ ‐‐‐‐

225.0

(0.00%)* ‐‐‐‐

2.2 6

307.5

(8.34%)

307.5

(0.00%)*

296.7

(8.50%)

329.0

(0.35%) ‐‐‐‐ ‐‐‐‐

2.3 1

303.0

(0.00%)*

330.0

(0.00%)* ‐‐‐‐

303.0

(0.00%)* ‐‐‐‐ ‐‐‐‐

All Tubes 12

308.0

(14.62%)

317.8

(22.60%)

302.8

(8.39%)

310.25

(7.05%)

225.0

(0.00%)*

413.0

(0.00%)*

3.0 20 331.2 339.4 309.1 382.8 ‐‐‐‐ ‐‐‐‐

114

(20.01%) (22.51%) (9.52%) (25.21%)

4.0 1

439.0

(0.00%)*

439.0

(0.00%)* ‐‐‐‐

439.0

(0.00%)* ‐‐‐‐ ‐‐‐‐

Notes: N is number of data points for that system type; ‐‐‐‐ indicates no data points in that category; CoV of 0.00%* indicates a single data point in that category

4.2.2 Geometric Descriptors: Aspect Ratio

The second and final geometric descriptor included in the database is the aspect ratio. Unlike

height, aspect ratios can be unique to each axis since most buildings are not geometrically (or

structurally) symmetric. This gave the database twice the number of measures for every system.

Examining the trend line in Figure 4.3, there is a slightly stronger correlation between increasing

aspect ratio and the historical system progression than was observed for height (NS = 2.59% vs.

2.48%). Examining Table 4.5, the influence of lower quality aspect ratio data again appears not

to be significant, as CoVs largely increased when only higher quality data was retained,

indicating that the lower quality observations at least tended to cluster around the mean values

in each system class. The only exception was for megasystems and this was only a minor

decrease in CoV for QI = 1 data. For the two most common systems, doubly augmented cores

(Sys ID = 1.2) and megasystems, the latter, when classified by aspect ratio, was now found to

have the greatest scatter of the two. This is in opposition to what was observed when height

was the descriptor. Moreover, when aspect ratio is used as the descriptor, scatter within the

doubly augmented cores and cores as a whole drops when compared to the use of height (from

36.85% to 28.91% and from 34.35 to 31.07%, respectively). On the other hand, megasystems

experienced an increase in scatter with a change in geometric descriptor (from 20.01% to

39.74%). Moreover, megasystems, as a family, were found on average to support the highest

aspect ratios (7.58) among the tall buildings considered (vs. 7.16 for cores and 7.07 for tubes),

but plain tubes (System ID = 2) were the individual system with the greatest average aspect ratio

(8.07). Looking at the correlation of core augmentations with aspect ratio, one would

hypothesize that larger aspect ratios indicate greater slenderness and likely the need for greater

augmentation; however, the mean aspect ratios do not fully support this hypothesis, though

considering the error bars in Figure 4.4, this slight inconsistency may be due to scatter alone.

While the basic core system has a mean aspect ratio of 6.00 and its two augmentations have

respective aspect ratios of 7.69 and 7.13, revealing that they do support greater slenderness,

but not following the progression one would expect. Examining the tube progression, again

noting the limited observations, the opposite is observed: the base system has larger average

aspect ratio (8.00) than its most common augmentation (System ID = 2.2: 7.27), so the heuristic

understanding of these augmentations being required with increasing slenderness does not

hold. As was the case with height, Figure 4.4 confirms that aspect ratio shows a stronger positive

trend with historical system progression for composite systems (NS = 5.75%) than for concrete

systems (NS=1.32%), which again essentially “flatlines”. The degree of material sensitivity for

aspect ratio is stronger than it was for height, e.g., normalized slopes of composites is 2.2 times

greater than the overall trend for aspect ratio, while it is only 1.3 times greater for height.

115

Figure 4.3: Relationship between aspect ratio and structural system, distinguished by source fidelity.

116

Figure 4.4: Relationship between aspect ratio and structural system, distinguished by material.

117

TABLE 4.5

ASPECT RATIO AS GEOMETRIC SYSTEM DESCRIPTOR: STATISTICS BY SYSTEM TYPE

System ID

N

Mean


All Only QI = 1

Sorted By Material


0.0 4

6.19

(23.14%) ‐‐‐‐

6.76

(0.00%)*

5.62

(39.16%) ‐‐‐‐ ‐‐‐‐

1.0 6

6.00

(25.82%)

7.00

(0.00%)

7.00

(0.00%)

4.00

(0.00%)* ‐‐‐‐ ‐‐‐‐

1.1 16

7.69

(36.13%)

3.40

(54.33%)

8.69

(23.64%)

6.71

(47.41%) ‐‐‐‐ ‐‐‐‐

1.2 46

7.13

(28.91%)

7.80

(30.74%)

7.41

(27.62%)

6.39

(33.79%)

6.55

(13.51%)

8.65

(23.09%)

All Cores 68

7.16

(31.07%)

7.61

(33.63%)

7.65

(25.92%)

6.30

(39.65%)

6.55

(13.51%)

8.65

(23.09%)

2.0 8

8.07

(26.89%) ‐‐‐‐

8.17

(36.37%)

6.89

(19.23%) ‐‐‐‐

9.05

(2.17%)

2.1 2

4.42

(38.57%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

4.42

(38.57%) ‐‐‐‐

2.2 12

7.27

(26.03%) ‐‐‐‐

7.72

(18.26%)

5.00

(11.75%) ‐‐‐‐ ‐‐‐‐

2.3 2

6.81

(8.91%) ‐‐‐‐ ‐‐‐‐

7.27

(8.91%)

‐‐‐‐ ‐‐‐‐

All Tubes 24 7.07 ‐‐‐‐ 7.87 6.04 4.42 9.05

118

(28.55%) (24.51%) (21.66%) (38.57%) (2.17%)

3.0 40

7.58

(39.74%)

9.40

(37.99%)

7.84

(42.08%)

7.00

(31.91%) ‐‐‐‐ ‐‐‐‐

4.0 2

8.61

(0.00%)* ‐‐‐‐ ‐‐‐‐

8.61

(0.00%)* ‐‐‐‐ ‐‐‐‐

Notes: N is number of data points for that system type; ‐‐‐‐ indicates no data points in that category; CoV of 0.00%* indicates a single building with the same axis length

4.2.3 Behavioral Descriptors: MS‐DCA

Whereas the geometric descriptors could be defined for every building in the database, as

stated previously in this chapter, mode shapes are currently available for only six of the

database’s buildings. This translates into only twelve DCA measures total split among 5

structural systems (System IDs = 1.0, 1.2, 2.2, 3.0). As all the DCAs in this chapter are based on

actual designer reported or published mode shapes, all have QI = 1 and thus are presented only

distinguishing by material. However, the depth of analysis or reliability of conclusions will pale in

comparison to those of the geometric descriptors.

Although the accuracy of the MS‐DCA was questioned in Chapters 2 and 3, it is presented here

as the first behavioral descriptor for completeness. Circled pairs of MS‐DCA values in Figure 4.5

indicate that the observations are from the same building, representing its behavior on its two

fundamental sway axes. In general, the degree of cantilever action is quite similar on both axes

for all systems, with slight deviations in the case of the doubly augmented core and braced tube,

though the CoVs in Table 4.6 reiterate that even these differences are quite small. This table

also demonstrates that the scatter within the family of cores is greater than megasystems

(23.66% vs. 14.48%), keeping in mind the limited observations available for both, though each

individual core subclass has a low CoV. Recall from Chapter 2 that low CoVs in the MS‐DCA were

not necessarily a favorable feature, as it indicated a lack of sensitivity in the descriptor. The core

family also allows the examination of progression on MS‐DCA, expecting augmentations to

increase the degree of cantilever action, which is observed between the basic core and the

doubly augmented core (from 1.09 to 1.68). Unfortunately the limited number of observations

does not allow any mean trends to be observed in Figure 4.5.

119

Figure 4.5: Relationship between MS‐DCA and structural system, distinguished by material.

TABLE 4.6

MS‐DCA AS BEHAVIORAL SYSTEM DESCRIPTOR: STATISTICS BY SYSTEM TYPE

System

N

Mean


120

ID

All Only QI =

1

Sorted By Material


0.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

1.0 4

1.09

(3.75%)

1.09

(3.75%)

1.09

(3.75%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

1.1 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

1.2 2

1.68

(2.99%)

1.68

(2.99%)

1.68

(2.99%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

All Cores 6

1.29

(23.66%)

1.29

(23.66%)

1.29

(23.66%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

2.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

2.1 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

2.2 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

2.3 2

1.18

(2.38%)

1.18

(2.38%) ‐‐‐‐

1.18

(2.38%) ‐‐‐‐ ‐‐‐‐

All Tubes 2

1.18

(2.38%)

1.18

(2.38%) ‐‐‐‐

1.18

(2.38%) ‐‐‐‐ ‐‐‐‐

3.0 4

1.57

(14.48%)

1.57

(14.48%) ‐‐‐‐

1.57

(14.48%) ‐‐‐‐ ‐‐‐‐

4.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

Notes: All systems not listed in this table could not have MS‐DCAs calculated due to lack of mode shape information; ‐‐‐‐ indicates no data points in that category

121

4.2.4 Behavior Descriptors: iDCA

The second behavior descriptor, the iDCA, is presented in Figure 4.6 (which plots the mapped

iDCA values using Appendix A). When comparing it against Figure 4.5, it becomes clear that the

iDCA does show greater sensitivity within a given building, underscored most significantly in the

core and megasystem classes. Despite this, Table 4.7 reveals that the CoVs for both the families

of cores and megasystems are lower when iDCA is employed (13.93% vs. 23.66% and 2.25% vs.

14.48%, respectively). In particular, the megasystems essentially collapse when iDCA is used.

Recall from Chapter 3 that the MS‐DCA was challenged in classifying the CS5’s megasystem.

Direct comparisons of the average DCA for each class reveal that basic cores and braced tubes

have greater cantilever action by iDCA, while megasystems have a lower degree of cantilever

action with iDCA, and doubly augmented cores are captured nearly identically by both

measures.

Because the limited number of observations does not allow reliable inferences to be drawn, the

iDCAs from all the case studies of Chapter 3, regardless of building age, are now included in the

following iDCA analysis to demonstrate the insights that can be gained using iDCA on a

descriptor for a more thoroughly populated database. Table 4.8 is included to show how the

systems from the Chapter 3 Case Studies were classified under the unified vernacular presented

in this chapter. As shown in Figure 4.7, the average values for each system class, unlike the

geometric measures, show no discernable trend for the historical progression of systems. What

is important to note is that the added observations maintain a fairly consistent classification of

most of the systems: all cores (mean iDCA = 1.41 vs. 1.41) and megasystems (mean iDCA = 1.36

vs. 1.39), helping affirm the ability of the iDCA to consistently classify different realizations of

the same fundamental system. Moreover, even while increasing the number of data points, the

CoVs in Table 4.9 remain under 15% with megasystems showing the least scatter, suggesting

that iDCA does capture the mean behavior of these systems better than height and aspect ratio,

which had higher CoVs; the only exception being the pair of tubes with braces, which show the

highest CoV (15.26%) and the greatest change in iDCA (1.33 vs. 1.51). Interestingly, the iDCA of

each of the major families show considerable similarities and an interesting progression: Cores

(1.41), Tubes (1.40), and Megasystems (1.36). When considering how the cores were shown

previously in Figure 1.7 to be achieving heights far greater than the historical hierarchies

suggested, while the megasystems were used at lower heights, the relative degrees of cantilever

action achieved in actual systems, as quantified in the iDCA, show consistency with this trend.

The various trends observed in the preceding analyses will now be captured in the following

section through modernized system hierarchies.

122

Figure 4.6: Relationship between iDCA and structural system, distinguished by material.

TABLE 4.7

iDCA AS BEHAVIORAL SYSTEM DESCRIPTOR: STATISTICS BY SYSTEM TYPE

123

System ID

N

Mean


All Only QI =

1

Sorted By Material


0.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

1.0 4

1.30

(9.78%)

1.30

(9.78%)

1.30

(9.78%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

1.1 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

1.2 2

1.63

(3.72%)

1.63

(3.72%)

1.63

(3.72%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

All Cores 6

1.41

(13.93%)

1.41

(13.93%)

1.41

(13.93%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

2.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

2.1 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

2.2 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

2.3 2

1.33

(0.97%)

1.33

(0.97%) ‐‐‐‐

1.33

(0.97%) ‐‐‐‐ ‐‐‐‐

All Tubes 2

1.33

(0.97%)

1.33

(0.97%) ‐‐‐‐

1.33

(0.97%) ‐‐‐‐ ‐‐‐‐

3.0 4

1.39

(2.25%)

1.39

(2.25%) ‐‐‐‐

1.39

(2.25%) ‐‐‐‐ ‐‐‐‐

4.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

Notes: All systems not listed in this table could not have iDCAs calculated due to lack of mode shape information; ‐‐‐‐ indicates no data points in that category

124

TABLE 4.8

SYSTEM CLASSIFICATION OF CHAPTER 3 CASE STUDIES

Building System ID

CS3 3.0

CS7 1.2

CS10 1.2

CS4 3.0

CS1 2.3

CS8 2.1

CS2 2.2

CS5 3.0

CS9 1.2

CS6 1.2

Figure 4.7: Relationship between iDCA and structural system, distinguished by material (including Chapter 3 Case Studies).

125

126

TABLE 4.9

MS‐DCA AS BEHAVIORAL SYSTEM DESCRIPTOR: STATISTICS BY SYSTEM TYPE (INCLUDING

CHAPTER 3 CASE STUDIES)

System ID

N

Mean


All Only QI =

1

Sorted By Material

Concrete Composite Steel Concrete and

Steel

0.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

1.0 4

1.30

(9.78%)

1.30

(9.78%)

1.30

(9.78%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

1.1 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

1.2 8

1.47

(11.71%)

1.47

(11.71%)

1.47

(11.71%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

All Cores 10

1.41

(12.25%)

1.41

(12.25%)

1.41

(12.25%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

2.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

2.1 2

1.32

(0.00%)

1.32

(0.00%) ‐‐‐‐

1.32

(0.00%) ‐‐‐‐ ‐‐‐‐

2.2 2

1.26

(3.84%)

1.26

(3.84%) ‐‐‐‐ ‐‐‐‐

1.26

(3.84%) ‐‐‐‐

2.3 4

1.51

(15.26%)

1.51

(15.26%) ‐‐‐‐

1.32

(9.69%)

1.69

(10.23%) ‐‐‐‐

All Tubes 8

1.40

(13.74%)

1.40

(13.74%) ‐‐‐‐

1.33

(0.58%)

1.47

(18.20%) ‐‐‐‐

3.0 5

1.36

(10.09%)

1.36

(10.09%) ‐‐‐‐

1.36

(10.09%) ‐‐‐‐ ‐‐‐‐

4.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐

Notes: All systems not listed in this table could not have iDCAs calculated due to lack of mode shape information; ‐‐‐‐ indicates no data points in that category

127

4.3 Modern System Hierarchies

The classical system hierarchies proposed a progression of systems based on the number of

stories (see Figure 1.2 and Figure 1.3). The database assembled in this chapter now provides the

ability to examine whether this progression is indeed consistent with modern practice, and what

descriptors best characterize modern systems. For this exercise, both geometric descriptors and

one behavioral descriptor (iDCA, including all Chapter 3 Case Study buildings) will be examined.

The historical hierarchy, essentially parameterizing systems by height, resulted in the

progression of systems captured in Table 4.3 and repeated on the x‐axis of the figures in Section

4.2. The analyses in Section 4.2.1 already underscored major deviations between modern

systems and this height‐based hierarchy. To quantify this further, a modern data‐driven

hierarchy with height as a descriptor is presented in Figure 4.8. Note that in this figure and the

others that follow, each system icon is sized vertically to plot its average value of the descriptor

in question, along with error bars indicating one standard deviation from this mean value. These

charts will not consider the basic system (System ID = 0.0), as it is uncommon for tall buildings. It

should also be cautioned that there is a single diagrid in the database, which is included in the

charts but should be interpreted with caution.

What is most striking from Figure 4.8, compared to the traditional progression of cores → tubes

→ megasystems, is how drama cally cores have advanced in the progression, at the expense of

tubes. Similarly, note how megasystems and doubly augmented cores are essentially side by

side in the modern hierarchy, as companion systems for the tallest of modern buildings, which

makes heuristic sense considering both follow the basic formula of stiff central core linked to

stiff perimeter elements.

128

Figure 4.8: Modernized hierarchy, parameterized by height.

Using aspect ratio, a measure of slenderness, as the descriptor yields a different

hierarchy shown in Figure 4.9. In this progression, the three linked systems cluster together in

the mid‐range of slenderness, while cores and tubes are the preferred systems among the

slenderest systems. The tube‐in‐tube remains the starting point for the progression by aspect

ratio, as was the case with height, while the single diagrid holds the top position in both

progressions.

129

Figure 4.9: Modernized hierarchy parameterized by aspect ratio (slenderness).

Interestingly, moving away from geometric descriptors toward a behavioral descriptor

like iDCA produces a distinctly different hierarchy shown in Figure 4.10. Note that not all

systems could have iDCAs calculated for them. However, these systems are still included in the

hierarchy but in muted tones at hypothetical positions. The most striking observation is the

extent to which megasystems are now viewed as almost a foundational system with respect to

cantilever behavior, even though they were conceived to be one of the most efficient structural

typologies. This could be, in great measure, due to the fact that these growingly popular systems

are not being implemented in the purest sense envisioned by Khan or when doing so, there is a

130

failure to effectively arrest shear lag. Linked tubes similarly lie early in the progression, which is

understandable considering their heavy reliance on deep beams as the transfer mechanism

within the system, generating a great proportion of frame action. The highest cantilevered

systems in practice are actually the braced tube and doubly augmented core systems, each able

to achieve a highly cantilevered behavior dominated by axial shortening. Such elevation of the

core is in striking contrast to the historical hierarchy in Table 4.3, but not unexpected

considering the heights these systems now achieve, most notably through the buttressed core.

This reveals how greatly the engineering of this system and the advances in high strength

concrete have enabled it, in recent decades, to achieve the behavior necessary to rationalize its

use at once unheard of heights.

131

Figure 4.10: Modernized hierarchy parameterized by degree of cantilever action (iDCA).

4.4 Summary

This chapter populated a database with modern tall buildings to explore the ability of both

geometric (height and aspect ratio) and behavioral (DCA) descriptors to classify their lateral

systems and enable the conception of modernized, data‐driven hierarchies for tall buildings.

These were compared to the historical hypothesized progression of cores → tubes →

megasystems, with various intermediate augmentations. The major findings of this chapter can

be summarized as follows:

1. The family of cores was found to consistently “over perform,” realizing heights far

greater than the historical progression of systems ever imagined. All descriptors were

able to capture the heuristic understating of core augmentations facilitating greater

heights, slendernesses and efficiencies, respectively.

132

2. Conversely, the heuristic progression of augmentation within tubes did not correlate

with increasing height or aspect ratio in modern tall buildings, though plain tubes did

realize the greatest average aspect ratio of all systems.

3. The family of megasystems has grown in popularity, but they are being employed at

heights much shorter than historical progression assumed, likely due to their

implementation in forms that deviate from Khan’s purist conception of this system.

4. With respect to material, reinforced concrete showed no correlation between historical

progression and increasing height or aspect ratio; though composite systems did show

positive correlation, more so for aspect ratio.

5. The iDCA proved to have the lowest degree of scatter of the descriptors, maintaining its

consistency even as more buildings were added to each system class. This suggests that

iDCA may indeed characterize systems more effectively than height or aspect ratio, as

hypothesized by this thesis. Moreover, the iDCA was shown to increase for system

families in the following order: megasystems → tubes → cores, which is consistent with

the pattern in average height among modern systems. This suggests that iDCA may

indeed be a direct surrogate for the behavior of these systems, as it corroborates the

companion trend in the heights they are able to realize on average.

6. When exploring the proposed modernized, data‐driven hierarchies, the traditional

progression of cores → tubes → megasystems is certainly not observed. Instead, each

basic system and its augmentations tend to scatter throughout the progressions

suggested by each descriptor. Regardless of the order or the descriptor, what is

important to note is how dramatically cores have advanced in each progression. Height‐

based hierarchies tend toward megasystems and their sister system, the doubly

augmented core, while aspect‐ratio or slenderness‐based hierarchies trend toward

cores and tubes. Meanwhile, behavioral‐based hierarchies, which measure system

cantilever efficiency, perhaps surprisingly to some, begin with linked tubes, while the

highest cantilevered system is the doubly augmented core and the braced tube system.

This last observation is the ultimate testimony to how what was once only a foundational

system in the historical height‐based hierarchy has now become the single most popular system

for tall buildings constructed over the last decade, achieving one of the greatest average

efficiencies to sit atop the modern hierarchy. This has lead to the conclusion that these

hierarchies not only reflect the trends in design based on engineering principles but also

developer trends. Developers want to maximize space for their concrete, mixed‐use buildings

resulting in a return to the core systems to allow for vertically discontinuous systems with space

for large exterior views on residential levels. Chapter 5 will now explore how this database and

its modern hierarchies can be expanded and refined with time to continue to capture the

evolution of modern systems in the future.

133

CHAPTER 5: CONCLUSIONS AND FUTURE WORK

5.1 Research Summary

The main goal of this thesis was to create a modernized hierarchy of structural systems, not

from first principles or theory, but actually from practice by mining the attributes of constructed

systems already in existence. Because of the diversity of modern systems, an amply robust

descriptor of system behavior was required. This research elected to utilize the degree of

cantilever action (DCA) as this descriptor, focusing on alternate methods for its quantification,

validated against previous DCA descriptors. As such, this research had the following objectives:

1. Develop a robust descriptor suitable for heterogeneous systems that is simple

to extract, i.e., requires little effort on the part of cooperating designers

2. Validate the proposed descriptors against previous DCAs using case studies of

existing tall buildings

3. Populate a comprehensive database of recently built tall buildings with diverse

systems, including significant details that are publically available, as well as

geometric (height, aspect ratio) and DCA descriptors explored in Objective 2

4. Create a modern hierarchy of systems by mining the database assembled in

Objective 3, revealing underlying trends that can guide future system selection.

This chapter will summarize the progress made toward each of these objectives.

5.2 DCA Development

In the past, hierarchies used geometric descriptors to quantify a system’s behavior. As expected,

these descriptors (height and aspect ratio) were not sufficiently robust and lacked the sensitivity

necessary to distinguish diverse systems, e.g., a myriad of systems could be used for any given

height. Bentz (2012) proposed the classification of systems based on their degree of cantilever

action, quantified using both fundamental mode shapes and finite element models. Chapter 2 of

this thesis noted that mode shapes, while potentially lower in fidelity than an artifact obtained

directly from the force distribution in the finite element model, required less complicated

extraction procedures and were more readily available. Therefore the mode shape was

recommended in this thesis as the basis of a new DCA, which compared the floor‐to‐floor slope

density of the mode shape in question to that of an ideal cantilever by way of the Hellinger

distance. This measure, the integral DCA (iDCA), was vetted against the historical mean‐square

DCA (MS‐DCA) using both simulated and actual case study mode shapes for tall buildings.

134

5.2.1 iDCA Verification

Verification studies in Chapters 2 and 3 underscored the limitations posed by the MS‐DCA’s

global best‐fit approach. In contrast, the iDCA offered greater sensitivity to localized variations

in mode shapes, by virtue of its use of the slopes of the mode shape at each floor. By analyzing

these local behaviors, the iDCA was able to detect subtle modulations in the mode shapes that

the MS‐DCA could not. From these verifications, which included simulated systemic

discontinuities progressing from an ideal cantilever to an ideal shear building (and vice versa) in

varying proportions, simulated outriggers, and actual tall building mode shapes, this thesis

formulated four major conclusions:

1. While both DCA measures were influenced by the base behavior of the mode

shape, the MS‐DCA was excessively biased toward the base behavior in its best‐

fit, often leading to a misclassification of the system.

2. Continuous mode shapes or those with progressive discontinuities (modulations

in the mode shape where the behavior before and after the discontinuity is

unchanged) were well classified by both measures.

3. The MS‐DCA’s ability to accurately classify systems was significantly

compromised when the mode shape exhibited strong intermittent

discontinuities (pronounced modulations recurring over height due to certain

repeating structural features) or behavioral change due to large architectural

spires.

4. Overall, the iDCA showed greater sensitivity, detecting slight variations in the

degree of cantilever action within subclasses of systems.

Therefore, from these conclusions, it was determined that the MS‐DCA was not robust enough

to handle a wide range of mode shape behaviors typified by modern structural systems.

Moreover, as proven by a host of both real‐world and simulated examples, the iDCA proved to

not only provide the necessary robustness while still maintaining the sensitivity necessary to

identify subtle differences between similar buildings, but also proved consistent with the

heuristic understanding of structural behavior. Thus, the iDCA was recommended as the

preferred measure of degree of cantilever action.

5.3 Database Population

In Chapter 4, a database of 75 recently constructed tall buildings was populated using various

structural (system classification, material) and geometric descriptors (height, aspect ratio) and

DCA measures derived from mode shapes, in order to identify trends within subsets of modern

systems. A common vernacular was established to group tall building systems into general

families of cores, tubes, megasystems and diagrids, within which various augmentations were

presented, e.g., Core + Exterior System and Tube + Bracing. The database was initially populated

from CTBUH and Emporis skyscraper databases, and supplemented by other publically available

data sources. While height and material were consistently available for each building in the

database, the aspect ratio, system and mode shapes often had to be gathered from unpublished

sources.

135

In such instances, the aspect ratio, which was described in this thesis as the height divided by

the mode of the cross‐sectional width, was found secondarily by extracting scaled building

geometries from Google Earth’s satellite imagery, which was verified in Chapter 4 to be highly

accurate. The secondary method for discerning structural system was to classify the system

based upon images of the structure during construction at stages when primary elements were

visible. Mode shapes were taken from the literature or received directly from participating

designers, though in some cases, their contributions were not received in a timely manner, thus

limiting the number of buildings for which the DCAs could be extracted.

5.3.1 Modernized System Hierarchies

The systems within the database were statistically and graphically analyzed when classified by

their geometric (height and aspect ratio) and behavioral (iDCA) descriptors to observe any

trends relative to Khan’s historical progression (cores → tubes → megasystems). From these

comparisons, this thesis framed four major conclusions:

1. As a family, cores were found to exceed historical expectations: achieving

greater heights, slendernesses, and efficiencies following the heuristic

understanding of the impacts of its progressive augmentations. Conversely, the

tube family did not show increases in the descriptors when augmentations were

used. Lastly, while the family of megasystems has grown in popularity, they are

being employed at heights far shorter than the historical progression assumed.

2. With respect to material, reinforced concrete showed no correlation between

the historical system progression and increasing height or aspect ratio, though

composite systems did show positive correlation, more so for aspect ratio. (Due

to limited number of mode shapes, similar conclusions could not be drawn for

the iDCA).

3. The iDCA proved to have the lowest degree of scatter of the descriptors,

maintaining its consistency even as more buildings (from the Chapter 3 Case

Studies) were added to each system class, suggesting that iDCA may indeed

characterize systems more effectively than height or aspect ratio. Moreover, the

iDCA was shown to increase for system families in the following order:

megasystems → tubes → cores, which is consistent with the pa ern in average

height among modern systems. This suggests that iDCA may indeed be a direct

surrogate for the behavior realized by these systems, as it corroborates the

companion trend in the heights they are able to realize on average.

4. The proposed modern hierarchies did not follow the traditional progression of

systems. Instead, each basic system and its augmentations tended to scatter

throughout the progressions suggested by each descriptor. Regardless of the

order or the descriptor, the most important realization was how dramatically

cores have advanced in each progression. Height‐based hierarchies trended

toward megasystems and their sister system: the doubly augmented core (Core

+ Ext. System + Links), while aspect‐ratio or slenderness‐based hierarchies

trended toward cores and tubes. The behavioral‐based hierarchies trended from

136

linked tubes to megasystems to the doubly augmented core and the braced

tube.

Therefore, while there were not enough mode shapes to establish a fully‐populated modern

hierarchy, these conclusions prove that the parameterization developed in this thesis is capable

of capturing the mean behavior of systems better than the historically popularized geometric

descriptors. Furthermore this thesis has successfully established a database structure and

process for extracting a modern hierarchy that can be dynamically updated in the future.

5.4 Future Work

While many important conclusions were made through the work of this thesis, there are still

many areas to explore to further the community’s knowledge base for modern structural

systems in tall buildings. These areas will now be presented.

5.4.1 iDCA Refinement

While the DCA measure proposed in this thesis was more robust than previous measures, it

required extra steps in extraction that could be refined. In particular, the iDCA produced values

that ranged from roughly 0.2 to 1.0, by definition. While this isn’t necessarily a flaw, it resulted

in the requirement of a mapping, discussed in Chapter 2 and reported in Appendix A, which

could be avoided if its limits were redefined.

Furthermore, while the comparison of the distributions of floor‐to‐floor slopes of the mode

shape in question to the ideal cantilever proved to have higher fidelity than the previously

defined MS‐DCA, exploration into other DCA measures is recommended. The iDCA was very

sensitive to behaviors at the base of the mode shape as this is where the cantilever slopes are

very high, which had the ability to bias the measure slightly. Finding a DCA that achieves a

balance between capturing global behaviors and the sensitivity to capture local features could

result in a DCA less susceptible to that local biasing. Additionally, as the distribution of slopes is

not sensitive to the spatial variation of the slope, changes in concavity or irregular distribution of

curvatures could be erroneously classified by the iDCA. Therefore, alternate definitions of DCA

that are less susceptible to these less common anomalies could be proposed. In particular,

exploration into the use of the Sobolov Norm to find differences between the derivative of the

mode shape in question and that of an ideal cantilever or using influence functions in lieu of the

mode shape to quantify the degree of cantilever action may be worthwhile.

5.4.2 Database Expansion and Virtualization

While the current database encompasses a wide range of heights and systems, more mode

shapes need to be obtained to enhance its utility and the value of the trends it identified or

expose new trends altogether. This would result in a much more qualified data‐driven hierarchy

and further the understanding of system behavior. Additionally, the database should be further

expanded to investigate the influence of building function on trends as well as those due to

region. It is important to note that the true value of this thesis is in the framework it offers and

not necessarily the hierarchy or trends that surfaced from its initial database presented in

137

Chapter 4. Instead, this database should be continually updated as new buildings are

constructed. As such, moving the database into a web environment where it can constantly

evolve and be dynamically explored by users and even directly populated by designers would

create the greatest value and possibility for impact, particularly as paradigm shifts in system

typologies occur.

Furthermore, while this thesis has provided a uniform vernacular for structural systems, criteria

for the vernacular needs to be formalized. Designers populating the database would then be

able to accurately classify the system being uploaded from this criteria resulting in more reliable

input and thus more reliable trends to extract.

138

APPENDIX A:

IDCA MAPPING

To facilitate comparisons between the two DCA measures introduced in Chapter 2, iDCAs are

often mapped to their equivalent MS‐DCA values. This mapping can be facilitated by taking the

average iDCA values from the three test buildings reported for each mode shape power (MS‐

DCA value) in Table 2.1. A look up tool (Table A.1) was then created to map MS‐DCA values at

increments of 0.05 to their iDCA equivalents, linearly interpolating between the values taken

from Table 2.1 (shaded in grey in Table A.1).

139

TABLE A.1

LOOK‐UP TABLE FOR MS‐DCA TO iDCA MAPPING

MS‐DCA iDCA MS‐DCA iDCA MS‐DCA iDCA

1.0000 0.1898 1.1650 0.3431 1.3300 0.4781

1.0050 0.1945 1.1700 0.3475 1.3350 0.4818

1.0100 0.1992 1.1750 0.3520 1.3400 0.4855

1.0150 0.2039 1.1800 0.3564 1.3450 0.4893

1.0200 0.2086 1.1850 0.3608 1.3500 0.4930

1.0250 0.2134 1.1900 0.3653 1.3550 0.4967

1.0300 0.2181 1.1950 0.3697 1.3600 0.5004

1.0350 0.2228 1.2000 0.3741 1.3650 0.5042

1.0400 0.2275 1.2050 0.3786 1.3700 0.5079

1.0450 0.2322 1.2100 0.3830 1.3750 0.5116

1.0500 0.2369 1.2150 0.3874 1.3800 0.5150

1.0550 0.2416 1.2200 0.3919 1.3850 0.5183

1.0600 0.2463 1.2250 0.3963 1.3900 0.5217

1.0650 0.2511 1.2300 0.4008 1.3950 0.5250

1.0700 0.2558 1.2350 0.4052 1.4000 0.5284

1.0750 0.2605 1.2400 0.4096 1.4050 0.5317

1.0800 0.2652 1.2450 0.4141 1.4100 0.5351

1.0850 0.2699 1.2500 0.4185 1.4150 0.5384

1.0900 0.2746 1.2550 0.4222 1.4200 0.5418

1.0950 0.2793 1.2600 0.4259 1.4250 0.5451

1.1000 0.2840 1.2650 0.4297 1.4300 0.5485

1.1050 0.2888 1.2700 0.4334 1.4350 0.5518

1.1100 0.2935 1.2750 0.4371 1.4400 0.5552

1.1150 0.2982 1.2800 0.4408 1.4450 0.5585

140

1.1200 0.3029 1.2850 0.4446 1.4500 0.5619

1.1250 0.3076 1.2900 0.4483 1.4550 0.5652

1.1300 0.3120 1.2950 0.4520 1.4600 0.5686

1.1350 0.3165 1.3000 0.4557 1.4650 0.5719

1.1400 0.3209 1.3050 0.4595 1.4700 0.5753

1.1450 0.3253 1.3100 0.4632 1.4750 0.5786

1.1500 0.3298 1.3150 0.4669 1.4800 0.5820

1.1550 0.3342 1.3200 0.4706 1.4850 0.5853

1.1600 0.3387 1.3250 0.4744 1.4900 0.5887

TABLE A.1 (CON’T)

LOOK‐UP TABLE FOR MS‐DCA TO IDCA MAPPING

MS‐DCA iDCA MS‐DCA iDCA MS‐DCA iDCA

1.4950 0.5920 1.6650 0.6760 1.8350 0.7252

1.5000 0.5954 1.6700 0.6778 1.8400 0.7263

1.5050 0.5981 1.6750 0.6796 1.8450 0.7275

1.5100 0.6007 1.6800 0.6813 1.8500 0.7286

1.5150 0.6034 1.6850 0.6831 1.8550 0.7298

1.5200 0.6061 1.6900 0.6848 1.8600 0.7309

1.5250 0.6087 1.6950 0.6866 1.8650 0.7320

1.5300 0.6114 1.7000 0.6883 1.8700 0.7332

1.5350 0.6140 1.7050 0.6901 1.8750 0.7343

1.5400 0.6167 1.7100 0.6919 1.8800 0.7447

1.5450 0.6194 1.7150 0.6936 1.8850 0.7551

1.5500 0.6220 1.7200 0.6954 1.8900 0.7656

1.5550 0.6247 1.7250 0.6971 1.8950 0.7760

1.5600 0.6274 1.7300 0.6989 1.9000 0.7864

1.5650 0.6300 1.7350 0.7006 1.9050 0.7968

1.5700 0.6327 1.7400 0.7024 1.9100 0.8073

1.5750 0.6354 1.7450 0.7041 1.9150 0.8177

141

1.5800 0.6380 1.7500 0.7059 1.9200 0.8281

1.5850 0.6407 1.7550 0.7070 1.9250 0.8385

1.5900 0.6434 1.7600 0.7082 1.9300 0.8490

1.5950 0.6460 1.7650 0.7093 1.9350 0.8594

1.6000 0.6487 1.7700 0.7104 1.9400 0.8698

1.6050 0.6513 1.7750 0.7116 1.9450 0.8802

1.6100 0.6540 1.7800 0.7127 1.9500 0.8907

1.6150 0.6567 1.7850 0.7139 1.9550 0.9011

1.6200 0.6593 1.7900 0.7150 1.9600 0.9115

1.6250 0.6620 1.7950 0.7161 1.9650 0.9219

1.6300 0.6638 1.8000 0.7173 1.9700 0.9324

1.6350 0.6655 1.8050 0.7184 1.9750 0.9428

1.6400 0.6673 1.8100 0.7195 1.9800 0.9532

1.6450 0.6690 1.8150 0.7207 1.9850 0.9636

1.6500 0.6708 1.8200 0.7218 1.9900 0.9741

1.6550 0.6725 1.8250 0.7229 1.9950 0.9845

1.6600 0.6743 1.8300 0.7241 2.0000 0.9949

142

APPENDIX B:

SUPPLEMENTARY CASE STUDIES

In the process of developing the Chapter 3 case studies, mode shapes from in‐house finite

element models were initially evaluated. Their basis on limited publically available information

raised questions surrounding their accuracy, causing the author to seek other sources of mode

shapes in the literature. If successful, those buildings were retained in the Chapter 3 case

studies using published mode shape data. If unsuccessful, the case study was completely

eliminated from Chapter 3, e.g., Central Plaza and John Hancock Tower (Boston). For

completeness, this appendix contains these initial analyses of in‐house finite element model

mode shapes using the same format introduced previously at the beginning of Section 3.2.

B. 1 CS3 Case Study

The CS3 is included in Chapter 3 (see Section 3.2.3); as such discussion of the structural system is

not repeated. The in‐house finite element model yielded a first mode that is highly cantilever in

the lower half of the building transitioning to one that is progressively more shear dominated at

the upper elevations. This is in significant contrast to the mode shape from the lumped mass

model (see Figure B.1). While the EMS‐DCA (Figure B.1‐c) suggests an overall less cantilever

structure, immediately following the introduction of the central column, it is considerably more

cantilever, until the next truncation. This results in a median error of ‐7.63%, which leads to the

expectation that the iDCA would be less cantilever than the MS‐DCA, a prediction that is indeed

reflected in the actual iDCA, whose deviation from the ideal cantilever is twice as much as the

MS‐DCA’s. In the case of Mode 2, the in‐house model agrees much better with the lumped mass

model mode shape (Figure B.2). This mode shape does not possess the sharp transition in

curvature near the mid‐height noted in the in‐house model’s first mode (see Figure 3.9‐b). As

such, the EMS‐DCA, despite being significantly errant near the base of the structure (Figure B.2‐c),

has a median error of only 4.86%, suggesting a slightly more cantilever structure than the MS‐

DCA predicts. This is consistent with the actual iDCA in Figure B.2‐d. Note that for both modes,

the in‐house model tends to predict a greater degree of cantilever action than was observed in

the lumped mass model’s mode shapes (Figure B.1 and Figure B.2).

Figure B.1: CS3’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

(a) (b)

(c)

179

Figure B.2: CS3’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

(a) (c)

(b)

180

145

B. 2 Central Plaza Case Study

Central Plaza’s structural system consists of a triangular tube with a reinforced concrete core for

a so‐called “tube in tube” system. While Central Plaza does not have any enhancements to

mitigate shear lag in its perimeter tube, any cantilever action is assumed to be strongly derived

from its slender concrete core. The floor plan is consistent for the main section of the office

building, with a larger floor plan at the podium level. The structural system then transitions so

that the upper 15% of the building is a MRF and cap truss system over the mechanical levels,

supporting the mast (Setya et al.), which is expected to cause a discernable shift in the mode

shape curvature. From an internally produced FEM, the first two sway modes were found to be

coupled, responding at 45 inclines with respect to the primary axes assumed when generating

the model, as shown by the plan views in from this FEM in Figure B.3. The mode shapes

generated from this FEM can be seen in Figure B. 4‐b and Figure B.5‐b, but as they could not be

corroborated by other sources, this case study could not be included in Chapter 3. In the case of

both modes, the best‐fit power law biases toward the idealized cantilever form, increasingly

deviating with height from the actual mode shape, which becomes increasingly linear in the

MRF/cap truss region. The overlaid horizontal lines indicate the points of transition in the

structure (podium level to office tower and beginning of MRF). To examine the implications of

the abrupt discontinuity introduced by a fundamental change in structural system and

construction material, the mode shapes are reassessed neglecting the cap truss (Figure B.6 and

Figure B.7), as done to other case studies in Chapter 3. The reanalysis shows that even without

the linear section at the top of the mode shapes, similar to the case studies in Chapter 3, the

removal of the cap truss does not significantly affect the classification of the system by either

DCA.

Note that the EMS‐DCA is quite significant at the lower levels (0.15H to 0.30H), exceeding 800% in

some cases. To avoid biasing the entire plot, these data points are not shown in the current view

in Figure B. 4‐c to Figure B.7‐c. In all modes, the iDCA is predicted to be less cantilever than the

MS‐DCA suggests, moving appropriately out of the “hyper‐cantilever” range and into the range

common to cantilever‐dominated structures. The actual iDCAs are consistent with this trend,

with the iDCAs of the structures without the cap trusses displaying more cantilever behavior, as

observed in Chapter 3. This is to be expected as the cap truss behavior increases the overall

percentage of shear behavior, thus affirming iDCA’s ability to capture these subtleties. Although

the structural system does not have features unique to one axis, thus being a “symmetric”

structural system, the MS‐DCA shows more marked disagreement with the iDCA in Mode 2 with

aDCA of ‐532% in the first mode compared to ‐634% in Mode 2. Interestingly, these error trends

reverse and increase when the cap truss is removed

(‐943% in Mode 1, ‐881% in Mode 2). Thus, even when compensating for the potential effects of

the cap truss, the discrepancies between the two DCAs are still significant and were the worst of

any of the case studies in Chapter 3; they may be a more extreme case of what was observed in

Shanghai World Financial Center and CS7, where there was considerable biasing by base

behaviors, which appears to be evident from subplot (b) in Figure B. 4 to Figure B.7. While the

finite element model is again not reliable enough to officially include in Chapter 3, this case

study had the most significant difference between the DCAs (DCA), reaffirming the conclusion

in Chapter 3 regarding the potential for bias of the MS‐DCA based on behaviors that are

146

unique to the base region, commonly seen in systemically discontinuous systems. From this

case study and that of Shanghai World Financial Center, this appears to be the discontinuity

that has the most severe impact on MS‐DCA performance.

Figure B.3: Axis assignment of Central Plaza (Bentz 2012).

Figure B. 4: Central Plaza’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

(a) (b) (c)

184

Figure B.5: Central Plaza’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

(a) (b) (c)

185

Figure B.6: Central Plaza’s first mode without cap truss (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

(a)

(b)

(c)

186

Figure B.7: Central Plaza’s second mode without cap truss (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

(a) (b) (c)

187

151

B. 3 CS4 Case Study

As CS4 is presented fully in Section 3.2.4, its details will not be repeated here. The mode shapes

from an internally developed FEM without the full cap truss region are presented in Figure B.8

and Figure B.9, where Mode 1 corresponds to lateral sway along the y‐axis and Mode 2 along

the x‐axis. The overlaid lines designate critical locations (from bottom to top: outrigger 1,

outrigger 2, start of atrium, cap truss). Note when compared to Figure 3.15, the mode shapes

from the in‐house finite element model show considerable distortions in the atrium region,

indicating the assumptions made in modeling this particularly complex feature of the building

were not appropriate. Nevertheless, these mode shapes simulate a form of intermittent

discontinuity that previously proved troublesome for the MS‐DCA. This kind of intermittent

discontinuity can be observed in Mode 2 to cause the MS‐DCA to overestimate the degree of

cantilever action at the base and top and underestimate the degree of cantilever action

elsewhere. In contrast, for Mode 1 the MS‐DCA consistently overestimates the degree of

cantilever action. Based on the median EMS‐DCA, Mode 1 was predicted to be less cantilever than

the MS‐DCA suggests; however, the actual iDCA is nearly identical to the MS‐DCA (within 4%).

On the other hand, the median errors in Mode 2 suggest that the MS‐DCA slightly overestimates

the degree of cantilever action, though the resulting iDCA is within 17% of the MS‐DCA. Thus

this case study represents an interesting situation where the MS‐DCA was not expected to

capture the degree of cantilever action accurately, yet agrees quite well with the iDCA, but

reasons for this are difficult to discern since the physical significance of the mode shapes were

questionable.

Figure B.8: CS4’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

(a) (b) (c)

189

Figure B.9: CS4’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

(a) (b)

(c)

190

154

B. 4 John Hancock Tower Case Study

The John Hancock Tower in Boston employs a MRF as its primary lateral system, with a steel

core supporting its gravity loads (Blanchet 2013). The tower has an elongated floor plan with a

unique notched trapezoidal shape, shown in Figure B.10, with the structural system and floor

plan being continuous with height. The more slender aspect ratio for the y‐axis would suggest

the structure would manifest a greater degree of cantilever action in this axis. An internally

generated FEM was created for this building, with modes shown in Figure B.11‐b and Figure

B.12‐b, but unfortunately could not be included in Chapter 3 due to a lack of published evidence

to corroborate them. The first mode interestingly has a best‐fit power less than 1, which was

unexpected and suggested that modeling assumptions were not valid. The second mode is

shows a more reasonable behavior, with shear dominated deformation mechanisms consistent

with an MRF system. Given the concerns surrounding the accuracy of the first mode shape in

this in‐house model, only the second mode will be discussed herein.

Figure B.10: John Hancock Tower’s general floor plan (Blanchet 2013).

Note the consistency of EMS‐DCA over the height (Figure B.12‐c). Based on the median error, the

iDCA is predicted to be more shear dominated than the MS‐DCA, consistent with the actual

iDCA, which was actually in a good agreement with the MS‐DCA (within 13% of one another).

This case study represents a continuous structural system with no irregularities in plan in its

primary office tower (the structure does have a podium level that was not modeled) or heavy

bias toward a particular behavior at its base and thus is an instance where the two DCA

measure are expected to show good agreement, consistent with observations in Chapter 3.

y

x x

y

Figure B.11: John Hancock Tower’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

(a) (c) (b)

193

Figure B.12: John Hancock Tower’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

(a)

(c)

(b)

194

157

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Copyright Sally · In the mid‐1960s, Fazlur Khan created a hierarchy of structural systems, ranging from two‐ dimensional moment resisting frames to three‐dimensional tubular