Modernizing the System Hierarchy for Tall Buildings: a Data‐Driven Approach to System Characterization

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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

Sally Suzanne Williams

Tracy Kijewski‐Correa, Director

Graduate Program in Civil Engineering and Geological Sciences

Notre Dame, Indiana

April 2014

Sally Suzanne Williams

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

Abstract

Sally Suzanne Williams

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.

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

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

iii

B. 2 Central Plaza Case Study ...... 145 B. 3 CS4 Case Study ...... 151 B. 4 John Hancock Tower Case Study ...... 154

Bibliography ...... 157

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

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 3.9: CS3’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.55

Figure 3.10: CS3’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison...... 56

Figure 3.11: CS4’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.59

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 3.14: CS5’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.63

Figure 3.15: CS5’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison...... 64

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 3.19: CS6’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.69

Figure 3.20: CS6’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison...... 70

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

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 3.24: CS7’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.76

Figure 3.25: CS7’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison...... 77

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 3.29: CS8’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.82

Figure 3.30: CS8’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison...... 83

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

Figure 3.32: CS9’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.86

Figure 3.33: CS9’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison...... 87

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 3.36: CS10’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison...... 91

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 vii

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

viii

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

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.

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).

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

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).

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

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.

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).

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.

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

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.

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:

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.

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)

are commonly fit by a power law expression that is a function of the height (z) to the total building height (H) ratio: (z) (z H) (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.

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

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).

, (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 2 (z)  (z H) 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,

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.

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

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.

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.

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)

32 B1 0.1964 0.2404 0.3287 0.4269 0.5151 0.5296 0.5747 0.6390 0.6824 0.6911 0.9985 (1 0000) (1 0470) (1 1420) (1 2506) (1 3831) (1 4055) (1 4754) (1 5886) (1 7143) (1 7473) (2 0000) B2 0.1979 0.2603 0.3538 0.4651 0.5553 0.5705 0.6168 0.6820 0.7179 0.7549 0.9965 (1 0000) (1 0674) (1 1684) (1 2945) (1 4120) (1 4338) (1 5006) (1 6383) (1 7284) (1 8420) (2 0000) B3 0.1751 0.2209 0.3151 0.4308 0.5307 0.5536 0.6132 0.6558 0.7176 0.7378 0.9894 (1 0000) (1 0474) (1 1482) (1 2908) (1 4278) (1 4620) (1 5599) (1 6390) (1 8154) (1 8768) (2 0000) Avg. 0.1898 0.2405 0.3325 0.4409 0.5337 0.5512 0.6016 0.6589 0.7089 0.7250 0.9948 (1 0000) (1 0538) (1 1531) (1 2801) (1 4080) (1 4341) (1 5116) (1 6192) (1 7632) (1 8341) (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).

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 33

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%

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

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 0.6949 0.6386 0.5435 0.4671 0.4229 0.4084 0.3673 0.3243 0.2798 0.1964 37

(2.0000) (1.7823) (1.5878) (1.4270) (1.3109) (1.2460) (1.2300) (1.1846) (1.1371) (1.0892) (1.0000)

0.9965 0.7597 0.6329 0.5416 0.4657 0.4216 0.4069 0.3656 0.3223 0.2790 0.1979 B2 (2.0000) (1.8575) (1.5341) (1.3923) (1.2952) (1.2415) (1.2256) (1.1811) (1.1344) (1.0876) (1.0000)

0.9894 0.7302 0.6101 0.5124 0.4367 0.3963 0.3785 0.3429 0.3024 0.2592 0.1751 B3 (2.0000) (1.8609) (1.5544) (1.4005) (1.2985) (1.2460) (1.2246) (1.1817) (1.1329) (1.0871) (1.0000)

0.9948 0.7283 0.6272 0.5325 0.4565 0.4136 0.3979 0.3586 0.3163 0.2727 0.1898 Avg. (2.0000) (1.8486) (1.5597) (1.4062) (1.3010) (1.2445) (1.2268) (1.1825) (1.1348) (1.0880) (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).

TABLE 2.3 (CONTINUED)

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

38 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%

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.

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

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

iDCA 0.6908 0.7545 0.7379 0.7277 5% 92% 98% (1.7462) (1.8407) (1.8769) (1.8460)

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.

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

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

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

0.1176 0.3803 0.4005 iDCA (<1.0000) (1.2069) (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

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.

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.

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

Heuristic Building Mode Shape Source Primary Material 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 Published Lumped Mass CS3 Model Composite Progressive In‐House FEM Corroborated by CS5 Published Mode Shape Steel Continuous

DIGITIZATION REQURED

Published FEM Calibrated CS8 against Full‐Scale Data Composite Continuous Published FEM Calibrated Intermittent/ CS10 against Full‐Scale Data Composite Progressive Published Lumped Mass CS4 Model Composite Systemic

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).

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.

TABLE 3.2 iDCA AND MS‐DCA FOR FUNDAMENTAL MODES OF CASE STUDY BUILDINGS

Mode Median Shape Mode Mapped Building Identifier Axis MS‐DCA iDCA iDCA EMS‐DCA

1 Y‐Axis 1.7358 1.8085 0.7192 ‐6.3% CS1 2 X‐Axis 1.7394 1.5644 0.6297 ‐4.5% 3 XY‐Axis 1.4118 1.2926 0.4502 ‐11.4% CS2 4 XY‐Axis 1.3835 1.2242 0.3956 ‐16.5% 5 XY‐Axis 1.0796 1.1620 0.3404 0.0% CS3 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% 9 XY‐Axis 1.7674 1.4096 0.5348 ‐23.0% CS5 10 XY‐Axis 1.7649 1.4303 0.5487 ‐23.3% 11 XY‐Axis 1.7867 1.4676 0.5737 ‐18.3% CS5* 12 XY‐Axis 1.7844 1.4876 0.5871 ‐17.6% 13 X‐Axis 1.3829 1.2399 0.4095 ‐7.3% CS6 14 Y‐Axis 1.3742 1.2869 0.4460 ‐17.6% 15 X‐Axis 1.3949 1.2515 0.4196 ‐5.3% CS6* 16 Y‐Axis 1.3978 1.2275 0.3985 ‐11.7% 17 Y‐Axis 1.8185 1.5779 0.6369 35.4% CS7 18 X‐Axis 1.7825 1.5775 0.6367 55.2% 19 Y‐Axis 1.7138 1.5824 0.6393 35.8% CS7* 20 X‐Axis 1.6428 1.6680 0.6771 43.8% 21 XY‐Axis 1.3951 1.3238 0.4735 ‐0.7% CS8 22 XY‐Axis 1.3951 1.3238 0.4735 ‐0.7% CS9 23 X‐Axis 1.3765 1.3694 0.5074 ‐11.4%

24 Y‐Axis 1.3686 1.3659 0.5048 ‐10.6% 25 X‐Axis 1.5380 1.5495 0.6218 ‐0.4% CS10 26 Y‐Axis 1.6426 1.6728 0.6788 23.7% * Modified mode shapes, e.g., cap trusses removed

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.

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

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

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.

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

3.2.2 CS2 Case Study

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.

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

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.

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

3.2.3 CS3 Case Study

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

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.

Figure 3.9: CS3’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

Figure 3.10: CS3’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

3.2.4 CS4 Case Study

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.

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.

Figure 3.11: CS4’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

(a) (b) (c)

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

3.2.5 CS5 Case Study

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.

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.

Figure 3.15: CS5’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

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

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

3.2.6 CS6 Case Study

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).

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.

Figure 3.19: CS6’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

Figure 3.20: CS6’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

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

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

3.2.7 CS7 Case Study

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.

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

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.

Figure 3.24: CS7’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

Figure 3.25: CS7’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

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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.

3.2.8 CS8 Case Study

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‐

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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Figure 3.29: CS8’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

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Figure 3.30: CS8’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

3.2.9 CS9 Case Study

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).

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.

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(a) (b)

(c)

Figure 3.32: CS9’s first mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

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(a) (b) (c)

Figure 3.33: CS9’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

3.2.10 CS10 Case Study

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).

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

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Figure 3.36: CS10’s second mode (a) mode shape with power fit, (b) EMS‐DCA, and (c) DCA comparison.

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

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

Mode System Shape Behavior Building Number iDCA MS‐DCA DCA

9 29.5 11.6 61% CS5 10 28.5 11.8 59% 11 26.6 10.7 60% Continuous CS5* 12 25.6 10.8 58% 21 33.8 30.2 11% CS8 22 33.8 30.2 11% Progressively CS2 3 35.4 29.4 17%

Discontinuous 4 38.8 30.8 21% 5 41.9 46 ‐10% CS3 6 35.3 43.6 ‐24% 13 38 30.9 19% CS6 14 35.7 31.3 12% 15 37.4 30.3 19% CS6* 16 38.6 30.1 22% 23 31.5 31.2 1% CS9 24 31.7 31.6 0% CS10 26 16.4 17.9 ‐9% CS10 25 22.5 23.1 ‐3% Intermittently Discontinuous 1 9.6 13.2 ‐38% CS1 2 21.8 13 40% CS4 7 24.2 32.7 ‐35% CS4* 8 21.8 30.8 ‐41% Systemically 17 21.1 9.1 57% Discontinuous CS7 18 21.1 10.9 49% 19 20.9 14.3 31% CS7* 20 16.6 17.9 ‐8% * Modified mode shapes, e.g. cap trusses removed

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

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

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

Aspect Ratio Structural System

Quality Quality Building Name Designer Index Source Index Source

111 South Wacker Goettsch Partners 2 GE: Chicago 1 Becker (2006)

23 Marina KEO International Consultants 1 Colaco (2005) 1 EMPORIS (2014) 124

300 North Lasalle Pickard Chilton 2 GE: Chicago 1 Ascribe (2012)

Skidmore Owings & Merrill GE: New York 383 Madison Ave (SOM) 2 City

Al Hamra Firdous Tower SOM 2 GE: Kuwait City 1 CTBUH (2013)

Shahdadpuri Almas Tower The Taisei Corporation 1 Scott (2011) 1 (2007)

Magnussen Klemencic Aqua 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

Metals in GE: New York Construction Bank of America Tower Severud Associates 2 City 1 (2008)

Burj Khalifa* SOM 1 SOM 1 SOM

Lakota and Lakota and Caja Madrid Foster + Partners 1 Alarcon (2008) 1 Alarcon (2008)

Stardubtsev et Capital City Moscow Tower Arup 2 GE: Moscow 1 al. (2011)

TABLE 4.1 (CONTINUED)

Aspect Ratio Structural System

Building Name Designer QI Source QI Source

DesignBuild China World Tower Arup 2 GE: Beijing 1 (2010)

Comcast Center Thornton Tomasetti 1 Milinichik (2006) 1 Milinichik (2006)

Diwang International CityMark Architects and Commerce Center Engineers 2 GE: Nanning 2 LERA (2009)

Doosan Haeundae We've the Thornton Tomasetti; New Thornton Thornton Zenith Tower A* Engineering Consultant, Inc. 1 Tomasetti 1 Tomasetti

Doosan Haeundae We've the Thornton Tomasetti; New Thornton Thornton Zenith Tower B* Engineering Consultant, Inc. 1 Tomasetti 1 Tomasetti

Eight Spruce St / Beekman GE: New York Marcus and 125

Tower Gehry Partners, LLP 2 City 1 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)

Etihad Towers T2 Aurecon 1 Aurecon (2014) 1 Aurecon (2014)

Dean et al. Eureka Tower Connell Mott MacDonald 1 (2001) 1 Grocon (2006)

Excellence Century Plaza Tower China Construction Design 1 International 2 GE: Shenzhen 2 Le Berre (2009)

Guangzhou International Finance Center Arup 2 GE: Guangzhou 1 Wilkinson (2012)

Haeundae I Park Marina Tower Arup; DongYang Structural 2 Engineers 2 GE: Busan

Al Hashemi / Farayand HHHR Tower Architectural Engineering 2 GE: Dubai 1 EMPORIS (2014)

TABLE 4.1 (CONTINUED)

Aspect Ratio Structural System

Building Name Designer 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 DongYang Structural Skyscraper Page Tower Engineers 2 GE: Hanoi 2 (2011)

Khalid Al Attar Tower 2 Eng. Adnan Saffarini 2 GE: Dubai 2 Solt (2010)

KK100 Development 1 Farrell (2011) 1 RBS Architectural Engineering Hernandez

Design Associates (2012)

126 Leatop Plaza* MKA 2 GE: Guangzhou 1 MKA

Skyscraper City Longxi International Hotel A&E Design 2 GE: Jiangyin 2 (2010)

Wikimedia Marina Pinnacle National Engineering Bureau 2 GE: Dubai 2 Commons (2007)

Millennium Tower e.Construct 2 GE: Dubai 1 EMPORIS (2014)

Skyscraper City Minsheng Bank Building Wuhan Architectural Institute 2 GE: Wuhan 2 (2006)

Metals in Metals in Construction Construction New York Times Tower Thornton Tomasetti 1 (2006) 2 (2006)

Nina Tower Arup 2 GE: Hong Kong 2 Johnson (2010)

Arup; DongYang Structural Chung et al. Northeast Asia Trade Tower Engineers 2 GE: Incheon 1 (2008)

Ocean Heights Meinhardt 2 GE: Dubai 1 Blackman (2010)

TABLE 4.1 (CONTINUED)

Aspect Ratio Structural System

Building Name Designer QI Source QI Source

One Island East Centre Arup 2 GE: Hong Kong 1 ISE (2009)

One World Trade Center SOM 2 1 Gonchar (2011) GE: New York

City

Frechette III and Pearl River Tower SOM 2 GE: Guangzhou 1 Gilchrist (2009)

Princess Tower Eng. Adnan Saffarini 2 GE: Dubai 1 Ephgrave (2012)

Suzhou RunHua Global Building A ECADI 2 GE: Suzhou

McMorrow

127 The Address Atkins 2 GE: Dubai 2 (2012)

The Domain Foster + Partners 2 GE: Abu Dhabi 2 Jimaa (2011)

Halvorson and Partners; The Index Bruechle; Gilchrist & Evans 2 GE: Dubai 1 Halvorson 2008)

Guangzhou Hanhua Architects Skyscraper City

The Pinnacle & Engineers 2 GE: Guangzhou 2 (2010)

Renzo Piano Building

The Shard 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 GE: New York

Tower SOM 2 City 1 SOM

Time Warner Center South GE: New York

Tower SOM 2 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)

TABLE 4.1 (CONTINUED)

Aspect Ratio Structural System

Building Name Designer QI Source QI Source

Pinzon Lozano & Asociados Torre Vitri Arquitectos 2 GE: Panama City

Pinzón Lozano & Asociados Tower Financial Center Arquitectos 2 GE: Panama City

Tower Palace III* SOM 2 GE: Seoul 1 SOM

Trump International Hotel and 128 Tower Halcrow Yolles 2 GE: Toronto 2 Conway (2010)

Baker et al. Trump International Tower SOM 2 GE: Chicago 1 (2009)

Costas Kondylis & Partners

Trump World Tower LLP Architects 1 Seinuk (2013) 1 Seinuk (2013)

Shanghai Institute of Architectural Design & Skyscraper City Wenzhou Trade Center Research 2 GE: Wenzhou 2 (2010)

Yingli International Finance Chongqing Yingli Real Estate Skyscraper City Centre Development 2 GE: Chongqing 2 (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

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.

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

Published Values Google Earth Measurements

Building Name 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

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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

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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

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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.

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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.

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

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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

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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).

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TABLE 4.4 HEIGHT AS GEOMETRIC SYSTEM DESCRIPTOR: STATISTICS BY SYSTEM TYPE

Mean [m] (Coefficient of Variation, %)

Sorted By Material

System Only QI = Concrete ID N All 1 Concrete Composite Steel and Steel

308.5 337.0 280.0 337.0 0.0 2 (10.67%) (0.00%)* (0.00%)* (0.00%)* ‐‐‐‐ ‐‐‐‐ 294.0 294.0 291.0 300.0 1.0 3 (3.16%) (3.16%) (3.57%) (0.00%)* ‐‐‐‐ ‐‐‐‐ 322.3 322.3 321.2 323.4 1.1 8 (32.24%) (32.24%) (18.63%) (43.24%) ‐‐‐‐ ‐‐‐‐ 334.9 339.6 367.7 311.8 325.0 228.3 1.2 23 (36.85%) (44.48%) (41.23%) (24.69%) (2.13%) (0.00%)* 328.3 329.0 348.9 314.7 325.0 228.3 All Cores 34 (34.45%) (38.90%) (36.97%) (30.73%) (2.13%) (0.00%)* 330.8 413.0 315.0 280.0 413.0 2.0 4 (16.73%) (0.00%)* (7.70%) (0.00%)* ‐‐‐‐ (0.00%)* 225.0 225.0 225.0 (0.00%) 2.1 1 (0.00%)* (0.00%)* ‐‐‐‐ ‐‐‐‐ * ‐‐‐‐ 307.5 307.5 296.7 329.0 2.2 6 (8.34%) (0.00%)* (8.50%) (0.35%) ‐‐‐‐ ‐‐‐‐

303.0 330.0 303.0 2.3 1 (0.00%)* (0.00%)* ‐‐‐‐ (0.00%)* ‐‐‐‐ ‐‐‐‐ 225.0 308.0 317.8 302.8 310.25 413.0 All (0.00%) Tubes 12 (14.62%) (22.60%) (8.39%) (7.05%) * (0.00%)* 3.0 20 331.2 339.4 309.1 382.8 ‐‐‐‐ ‐‐‐‐

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(20.01%) (22.51%) (9.52%) (25.21%) 439.0 439.0 439.0 4.0 1 (0.00%)* (0.00%)* ‐‐‐‐ (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.

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

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

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TABLE 4.5 ASPECT RATIO AS GEOMETRIC SYSTEM DESCRIPTOR: STATISTICS BY SYSTEM TYPE

Mean (Coefficient of Variation, %)

Sorted By Material

System Only QI Concrete ID N All = 1 Concrete Composite Steel and Steel

6.19 6.76 5.62 0.0 4 (23.14%) ‐‐‐‐ (0.00%)* (39.16%) ‐‐‐‐ ‐‐‐‐ 6.00 7.00 7.00 4.00 1.0 6 (25.82%) (0.00%) (0.00%) (0.00%)* ‐‐‐‐ ‐‐‐‐ 3.40 7.69 8.69 6.71 (54.33% 1.1 16 (36.13%) ) (23.64%) (47.41%) ‐‐‐‐ ‐‐‐‐ 7.80 7.13 7.41 6.39 6.55 8.65 (30.74% 1.2 46 (28.91%) ) (27.62%) (33.79%) (13.51%) (23.09%) 7.61 7.16 7.65 6.30 6.55 8.65 All (33.63% Cores 68 (31.07%) ) (25.92%) (39.65%) (13.51%) (23.09%) 8.07 8.17 6.89 9.05 2.0 8 (26.89%) ‐‐‐‐ (36.37%) (19.23%) ‐‐‐‐ (2.17%) 4.42 4.42 2.1 2 (38.57%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ (38.57%) ‐‐‐‐ 7.27 7.72 5.00 2.2 12 (26.03%) ‐‐‐‐ (18.26%) (11.75%) ‐‐‐‐ ‐‐‐‐ 6.81 7.27 2.3 2 (8.91%) ‐‐‐‐ ‐‐‐‐ (8.91%) ‐‐‐‐ ‐‐‐‐ All Tubes 24 7.07 ‐‐‐‐ 7.87 6.04 4.42 9.05

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(28.55%) (24.51%) (21.66%) (38.57%) (2.17%) 9.40 7.58 7.84 7.00 (37.99% 3.0 40 (39.74%) ) (42.08%) (31.91%) ‐‐‐‐ ‐‐‐‐

8.61 8.61 4.0 2 (0.00%)* ‐‐‐‐ ‐‐‐‐ (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.

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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

Mean

N (Coefficient of Variation, %) System

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ID Sorted By Material

Only QI = Concrete All 1 Concrete Composite Steel and Steel

0.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.09 1.09 1.09 1.0 4 (3.75%) (3.75%) (3.75%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.1 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.68 1.68 1.68 1.2 2 (2.99%) (2.99%) (2.99%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.29 1.29 1.29 All (23.66% Cores 6 ) (23.66%) (23.66%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 2.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 2.1 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 2.2 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.18 1.18 1.18 2.3 2 (2.38%) (2.38%) ‐‐‐‐ (2.38%) ‐‐‐‐ ‐‐‐‐ 1.18 1.18 1.18 All Tubes 2 (2.38%) (2.38%) ‐‐‐‐ (2.38%) ‐‐‐‐ ‐‐‐‐ 1.57 1.57 1.57 (14.48% 3.0 4 ) (14.48%) ‐‐‐‐ (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

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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.

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

TABLE 4.7 iDCA AS BEHAVIORAL SYSTEM DESCRIPTOR: STATISTICS BY SYSTEM TYPE

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Mean (Coefficient of Variation, %)

Sorted By Material

System Only QI = Concrete ID N All 1 Concrete Composite Steel and Steel

0.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.30 1.30 1.30 1.0 4 (9.78%) (9.78%) (9.78%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.1 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.63 1.63 1.63 1.2 2 (3.72%) (3.72%) (3.72%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.41 1.41 1.41 All (13.93% Cores 6 ) (13.93%) (13.93%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 2.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 2.1 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 2.2 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.33 1.33 1.33 2.3 2 (0.97%) (0.97%) ‐‐‐‐ (0.97%) ‐‐‐‐ ‐‐‐‐ 1.33 1.33 1.33 All Tubes 2 (0.97%) (0.97%) ‐‐‐‐ (0.97%) ‐‐‐‐ ‐‐‐‐ 1.39 1.39 1.39 3.0 4 (2.25%) (2.25%) ‐‐‐‐ (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

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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).

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TABLE 4.9 MS‐DCA AS BEHAVIORAL SYSTEM DESCRIPTOR: STATISTICS BY SYSTEM TYPE (INCLUDING CHAPTER 3 CASE STUDIES)

Mean (Coefficient of Variation, %)

Sorted By Material

System Only QI = Concrete and ID N All 1 Concrete Composite Steel Steel

0.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.30 1.30 1.30 1.0 4 (9.78%) (9.78%) (9.78%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.1 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.47 1.47 1.47 1.2 8 (11.71%) (11.71%) (11.71%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.41 1.41 1.41 All Cores 10 (12.25%) (12.25%) (12.25%) ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 2.0 0 ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ ‐‐‐‐ 1.32 1.32 1.32 2.1 2 (0.00%) (0.00%) ‐‐‐‐ (0.00%) ‐‐‐‐ ‐‐‐‐ 1.26 1.26 1.26 2.2 2 (3.84%) (3.84%) ‐‐‐‐ ‐‐‐‐ (3.84%) ‐‐‐‐ 1.51 1.51 1.32 1.69 2.3 4 (15.26%) (15.26%) ‐‐‐‐ (9.69%) (10.23%) ‐‐‐‐ 1.40 1.40 1.33 1.47 All Tubes 8 (13.74%) (13.74%) ‐‐‐‐ (0.58%) (18.20%) ‐‐‐‐ 1.36 1.36 1.36 3.0 5 (10.09%) (10.09%) ‐‐‐‐ (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

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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 dramacally 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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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 paern 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

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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

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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.

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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).

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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

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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

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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

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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).

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179

(a) (b) (c)

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

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(a) (b) (c)

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

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 aDCA 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

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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).

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184

(a) (b) (c)

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

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(a) (b) (c)

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

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(a) (c) (b)

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

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(a) (b) (c)

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

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.

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189

(a) (b) (c)

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

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

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.

xx 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.

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(a) (b) (c)

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

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(a) (b) (c)

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

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