Creating Diagrid Structure System Design Tool

Changheon Yi

A project submitted to the faculty of

Brigham Young University

In partial fulfillment of the requirements for the degree of

Master of Science

Richard J. Balling, Chair

Fernando S. Fonseca

Paul W. Richards

Department of Civil & Environmental Engineering

Brigham Young University

April 2015

Copyright © 2015 Changheon Yi

All Rights Reserved

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Table of Contents

1 Introduction .........................................................................................................................3

2 Literature review ..................................................................................................................8

3 Simplified Skyscraper Model ............................................................................................. 10

3.1 Core Super Element .................................................................................................... 11

3.2 Diagonal Areas ........................................................................................................... 13

3.3 Diagonal Stiffness ....................................................................................................... 15

3.4 Stiffness Matrix .......................................................................................................... 17

4 Results ............................................................................................................................... 22

5 Conclusion ......................................................................................................................... 26

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1 Introduction

The objective of this project is to create a design tool for diagrid skyscraper

structural systems. When the height of the buildings increases, the lateral load resisting

technologies require extra consideration. The diagrid structural system is widely used for

skyscrapers because of its structural efficiency and unique geometric shape

(Alaghmandan, Bahrami & Elnimeiri, 2014). According to Panchal and Patel (2014), “the

difference between conventional exterior-braced frame structures and current diagrid

structures is that, for diagrid structures, almost all the conventional vertical columns are

eliminated.” This system is becoming more commonly used as a new aesthetic

architectural/structural concept for tall buildings (Alaghmandan, M., Pehlivan, N. Al, &

Elnimeiri, M., 2014). The journal article of Dr. Richard J. Balling and Jacob Lee showed

great work for designing skyscraper structure system which has mega – columns,

outriggers, and belt trusses. (Balling, R. and Lee, J., 2014). This project is an adaptation

to their work by adding the ability to model diagrid structures. Figure 1-1 shows the

famous examples of diagrid structures in the world. The name of these buildings are the

(a) Swiss Re in London, (b) Hearst Tower in New York, (c) Cyclone Tower in Asan, and

(d) Capital Gate Tower in Abu Dhabi.

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Figure 1-1: Examples of diagrid structures

A new simplified skyscraper analysis model (SSAM) is described herein. The

model has been implemented on a spreadsheet. The building which was modeled in the

spreadsheet is the International Finance Center (IFC) in Guangzhou, China. The IFC

adopted the diagrid system and does not have mega columns, belt trusses or outriggers. It

has a reinforced concrete core and a diagrid external lattice. The diagonals are concrete-

filled steel tubes. The Guangzhou International Finance Center is a 440m high skyscraper

with 103 stories. This building is the tallest building with the diagrid system, and it is also

11th tallest building in the world. The fact that such prominent structures used this type of

structural system is the reason why the SSAM needed to be reconfigured to analyze

diagrids.

For the project, the building needed to be simplified. First, the building has

trochoidal triangular shape, but the simplified model used straight lines for the building

perimeter on each floor. The plan shape of the building was modelled as an equilateral

triangle. Second, the model adopted 20 stories as one interval, and total number of floors

was reduced from 103 to 100. Twenty stories comprising one interval is reasonable to use

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in the spreadsheet because it has 35 X 35 stiffness matrix. Had this model adopted 10

stories in one interval, the matrix would have been 100 X 100 which is very difficult to

work in a spreadsheet. In the real building, it is difficult to define the intervals because

the diagonals are continuous from the 1st floor to the roof. Third, the outer shape of the

model has straight lines in one interval even though the IFC has one curved line from the

bottom to the top. Last, the model transformed all members to reinforced concrete by

employing the ratio of moduli of elasticity between the steel and concrete. These

modifications made it much faster to create the spreadsheet without losing accuracy.

Figure 1-2: International Finance Center Diagrid System

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Figure 1-3: Guangzhou International Finance Center

Figure 1-2 and 1-3 are pictures Guangzhou International Finance Center. These

pictures shows that how the structure of this building consists of. Each diagonal is a part

of one diagonal line. This is more explicit in the inside picture.

The spreadsheet follows the basic structure that was created in journal article

“Simplified Model for Analysis and Optimization of Skyscrapers with Outrigger and Belt

Trusses” (Balling, R. and Lee, J., 2014). It consists of constants, superelements, matrices,

wind, seismic, and stress pages. The constants page includes material properties, weight

data and wind and seismic data. The superelements page has all the building design

variables and member properties for the core and diagonals. The stiffness matrix is

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contained in the matrices page, and the page also includes wind and seismic force and

displacement vectors. The wind and the seismic pages contain the data for the lateral

wind and seismic forces calculated for each story. The stress page has the resulting

stresses in structural members.

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2 Literature review

Balling, R. and Lee, J. (2014). "Simplified Model for Analysis and Optimization of

Skyscrapers with Outrigger and Belt Trusses." J. Struct.

Eng. , 10.1061/(ASCE)ST.1943-541X.0001210 , 04014231.

This article guided how to design and analyze tall skyscrapers with outriggers and

belt trusses in simplified model. Even though my project is limited to external diagrid

system, this article helped to learn how to design core of the buildings in the project.

Also the relationship between core thickness and mega columns is helpful to derive

the relationship between core and diagonals.

Wilkinson, C. (2012). Guanzhou Finance Centre: An Elegant Simplicity of Form.

CTBUH 9th World Congress Shanghai.

This article is very helpful to understand how external diagrid system respond to

horizontal lateral force. Unlike outriggers and belt trusses system, the external diagrid

systems transfers horizontal lateral force to its axial force. It is quite interesting

because there is no bending moment on the building except the core. Also this writing

was helpful to construct the stiffness matrix.

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Lee, Jacob Scott, "Accuracy of a Simplified Analysis Model for Modern

Skyscrapers" (2013). All Theses and Dissertations. Paper 4055.

http://scholarsarchive.byu.edu/etd/4055

This thesis would be the main resource for my project. The author researched for

many different structure systems, and found out his excel spread sheet finely works

for the systems but the buildings which have diagonal members. He already derived

many equations such as vertical gravity load, tributary areas, and the relationship

between the core thickness and column areas. I think I can start deriving other

equations that I need from this thesis. Also the spreadsheet he made became the very

base of my project output.

Boake, Terri Meyer. "Diagrids the New Stability System: Combining Architecture

with Engineering." Architectural Engineering Institute Conference (2013).

This article shows how architects view the diagrid systems and the history of the

system. The best thing of this writing is that this thesis referred me to reach many

related articles. It was quite astonishing that the first time the diagrid system was

adopted to real building was 1960s, but it had not been used until 2000. Now this

system is considered as an innovative approach.

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3 Simplified Skyscraper Model

The simplified 100-story skyscraper used in this project is shown in Figure 3-1

and 3 – 2. There are 18 diagonals surrounding a hexagonal concrete core. Each of the

diagonals have different locations on the floor plan at each end. For example, the

diagonal connected to the point A from the 1st floor in the bottom interval connects to the

point B on the 20th floor. Also, for the spread sheet, two different floor areas were used. In

the superelement page, all the floors in same interval have same areas. However, in the

wind and the seismic pages, all the floors have different areas to make the final results

graph have smooth lines. Figure 3-1 shows this idea. In other words, the bottom floor of

the each interval in the superelememt page and the other two have same values, but the

floors between the bottom floors of the each intervals have different areas. Following this

different floor areas between the intervals, the slope of the diagonals are different among

the different intervals.

Figure 3-1: simplified model images

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Figure 3-2: Plan and Side View of the Skyscraper in 2D Dimension Together

3.1 Core Super Element

The first properties calculated in the SSAM are the core section properties. In

order to get the moment of inertia of the core, the method used in the article “Simplified

Model for Analysis and Optimization of Skyscrapers with Outrigger and Belt Trusses”

(Balling, R. and Lee, J., 2014) was used again. Figure 3-3 is example core of the

mentioned article and the core shape for the project model.

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Figure 3-3: Example core model in the article and the core shape for the project model.

The long sides of the hexagon is 41m and the short dimension is 7. The project model

was forced by the horizontal wind and seismic force from the left side. The axis of

bending is perpendicular to axis of bending of the example core. The following table

shows the values of the each core wall properties.

stories d y sin n d y sin n

81 to 100 41 16 0 1 7 13 0.87 2

61 to 80 41 16 0 1 7 13 0.87 2

41 to 60 41 16 0 1 7 13 0.87 2

21 to 40 41 16 0 1 7 13 0.87 2

1 to 20 41 16 0 1 7 13 0.87 2

stories d y sin n d y sin n

81 to 100 41 8 0.87 2 7 25.5 0 1

61 to 80 41 8 0.87 2 7 25.5 0 1

41 to 60 41 8 0.87 2 7 25.5 0 1

21 to 40 41 8 0.87 2 7 25.5 0 1

1 to 20 41 8 0.87 2 7 25.5 0 1

Table 3-1: Core Section Properties

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The moment of inertia has been calculated by using the following equation. Then,

elements for the stiffness matrix from the core were calculated.

𝐼𝑖𝑐𝑜𝑟𝑒 = 𝑡𝑖

𝑐𝑜𝑟𝑒 ∑[𝑑𝑖𝑗(𝑦𝑖

𝑗)2 +

(𝑑𝑖 𝑗)3𝑠𝑖𝑛2𝜑𝑖

𝑗

12]

𝑗

3.2 Diagonal Areas

For the diagonal members, one critical assumption was made. The tributary areas

and perimeters are equal for all diagonals on one interval. In other words, all diagonals on

interval i have same cross-sectional areas. This assumption makes the design and analysis

process much simpler. Then, the equation to calculate the area of the diagonals in one

interval is derived from the idea that the vertical strains for concrete core and diagonals

must be same. The Figure 3-3 and following equations show this concepts.

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Figure 3-4: Diagonal Assumption

𝜀 = 𝐹𝑖

𝑐𝑜𝑟𝑒+ϒ𝑐𝑜𝑟𝑒 𝐴𝑖𝑐𝑜𝑟𝑒 ℎ𝑖

𝐸𝑐𝑜𝑟𝑒 𝐴𝑖𝑐𝑜𝑟𝑒 =

𝐹𝑖𝑑𝑖𝑎𝑔

2+ϒ𝑑𝑖𝑎𝑔 𝐴𝑖

𝑑𝑖𝑎𝑔 ℎ𝑖𝑆𝑖

𝐸𝑑𝑖𝑎𝑔 𝐴𝑖𝑑𝑖𝑎𝑔

(𝑆𝑖 )3 Equation 1

𝐴𝑖𝑑𝑖𝑎𝑔

= 𝐹𝑖

𝑑𝑖𝑎𝑔𝐸𝑐𝑜𝑟𝑒𝐴𝑖

𝑐𝑜𝑟𝑒

2𝐸𝑑𝑖𝑎𝑔(𝑆𝑖 )3(𝐹𝑖𝑐𝑜𝑟𝑒+ϒ𝑐𝑜𝑟𝑒 𝐴𝑖

𝑐𝑜𝑟𝑒 ℎ𝑖(1−ϒ𝑑𝑖𝑎𝑔𝐸𝑐𝑜𝑟𝑒

ϒ𝑐𝑜𝑟𝑒𝐸𝑑𝑖𝑎𝑔(𝑆𝑖 )4))

Equation 2

Symbols ε, F, A, h, S, E, ϒ, i indicate strain, force, area, height of an interval,

sineϕ, elasticity of modulus, density and interval i each. The diagonal’s area can be

derived from the top equation that left side is the strain of the core in interval i and the

right side is the strain of a diagonal in interval i. The first equation indicates that the

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strain in the core and the strain in the diagonals must be equal. Solving for the area of

diagonal from equation 1 yields equation 2.

3.3 Diagonal Stiffness

In order to create the stiffness matrix, the contribution of the each diagonal is

assessed. Unlike the vertical members such as columns, the diagonal members have two

kinds of stiffness; lateral and vertical stiffness. Also the diagonals in different intervals

have different angles from horizontal lines, and the general equation should be derived

for all the intervals. The following figures and equations are the general concepts and

equation for the diagonal’s stiffness.

Figure 3-5: Stiffness of the Diagonals under Lateral and Vertical Forces

KLV

KLL

KLL

KVV

KLV

KLV

KLV

KVV

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For the lateral stiffness, horizontal stiffness is defined as KLL, and vertical stiffness is

defined as KVV. Also, there are two elements for vertical stiffness in the lateral

displacement case, and lateral stiffness in the vertical displacement case KLV. The

equations are defined as the following.

For lateral stiffness,

KLL = 𝐸𝐴

ℎ𝑆(1 − 𝑆2)

KLV = 𝐸𝐴

ℎ𝑆2√1 − 𝑆2

For vertical stiffness,

KVV = 𝐸𝐴

ℎ𝑆3

KLV = 𝐸𝐴

ℎ𝑆2√1 − 𝑆2

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The above equations are for the tensioned diagonals. The compressed members have

opposite signs for KLV. The S in the equations indicate the ratio of interval height to

diagonal length.

3.4 Stiffness Matrix

Lateral load analysis in the example model is performed by creating a stiffness

matrix in the spreadsheet for the skyscraper. The degree of freedom consist of the

horizontal displacement of the core at the top of the each interval, the rotation of the core

at the top of the each interval, and the vertical displacement of the each points that two

diagonals meet at the top of the each interval. The Figure from the thesis “Accuracy of a

Simplified Analysis Model for Modern Skyscrapers (Lee, 2013).” shows basic concepts of

the displacements.

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Figure 3-6: Displacement Figure of the Model Building

Since the floor plan shape of the skyscraper is an equilateral triangle, the diagonals which

are connected to points A, B, C and D should be multiplied by proper cosine values. In

this example model, the value is cosine 30 degree. Especially, the KLL values need to be

multiplied by (cosine 30)2 because the force and diagonal’s physical direction.

In order to construct the stiffness matrix, the work need to be divided into two

parts. The first part is the core contribution and the other one is the diagonal contribution.

Since the core is the only element that contributes to the rotation, it is much easier to

finish core contribution. Following table shows equations for core contribution elements

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in the matrix for top 2 intervals. The 8 greyed cells are overlapping parts between two

intervals, so these cells need be added together because each interval have core

contribution at top and bottom.

hor 100 rot 100 hor 80 rot 80

hor 100

12𝐸𝐼

𝐿3

−6𝐸𝐼

𝐿2

−12𝐸𝐼

𝐿3

−6𝐸𝐼

𝐿2

rot 100

−6𝐸𝐼

𝐿2

4𝐸𝐼

𝐿

6𝐸𝐼

𝐿2

2𝐸𝐼

𝐿

hor 80

−12𝐸𝐼

𝐿3

6𝐸𝐼

𝐿2

12𝐸𝐼

𝐿3

6𝐸𝐼

𝐿2

rot 80

−6𝐸𝐼

𝐿2

2𝐸𝐼

𝐿

6𝐸𝐼

𝐿2

4𝐸𝐼

𝐿

hor 80 rot 80 hor 60 rot 60

hor 80

12𝐸𝐼

𝐿3

−6𝐸𝐼

𝐿2

−12𝐸𝐼

𝐿3

−6𝐸𝐼

𝐿2

rot 80

−6𝐸𝐼

𝐿2

4𝐸𝐼

𝐿

6𝐸𝐼

𝐿2

2𝐸𝐼

𝐿

hor 60

−12𝐸𝐼

𝐿3

6𝐸𝐼

𝐿2

12𝐸𝐼

𝐿3

6𝐸𝐼

𝐿2

rot 60

−6𝐸𝐼

𝐿2

2𝐸𝐼

𝐿

6𝐸𝐼

𝐿2

4𝐸𝐼

𝐿

Table 3-2: Core Contributions in the Matrix

Second part of constructing the matrix is the contribution of the diagonals. Since

not only each diagonal has different action to one direction force because of its location,

but also there are two different internal force in the members; tension and compression,

each contribution of each diagonal should be analyzed. For example, the diagonals in the

side that faces to lateral force directly provide only vertical stiffness, Kvv. The following

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figure and the each matrix for the diagonals demonstrate how the stiffness matrix was

constructed.

Figure 3-7: Diagonal Members are Labeled with Numbers

DIAGONAL 1, 3, 5 DIAGONAL 2, 4, 6

H100 H80 A100 B80 H100 H80 B100 A80

H100 KLL -KLL KLV -KLV H100 KLL -KLL -KLV KLV

H80 -KLL KLL -KLV KLV H80 -KLL KLL KLV -KLV

A100 KLV -KLV KVV -KVV B100 -KLV KLV KVV -KVV

B80 -KLV KLV -KVV KVV A80 KLV -KLV -KVV KVV

DIAGONAL 7, 8, 9

H100 H80 D100 E80

H100 0 0 0 0

H80 0 0 0 0

D100 0 0 KVV -KVV

E80 0 0 -KVV KVV

Table 3-3: Piece Matrix of the Diagonals

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In addition to the values in the table above, cosine 30 degree value should be multiplied

to each diagonal not perfectly parallel to the load. KLL values need to be multiplied by

cosine 30 degree twice because of its physical location and force, and KLV should be

multiplied by cosine 30 degree because the force be applied vertically. After the core

contributions and diagonal contribution are made, these two elements need to be added

together. The rest of the spreadsheet follows the original method implemented by Jacob

Lee.

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4 Results

All the properties of the modeled building were inserted to the wind and seismic

pages in the spreadsheet such as floor areas, material volume, building perimeter etc.

Then by using the wind and the seismic force from the ASCE 7-05 ((ASCE), 2006), the

displacement and the rotation of the core and vertical displacements for each of the 5

points are calculated.

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wind disp seismic disp

hor 100 1.32311809 2.20656724

rot 100 0.0039962 0.00705392

ver A 100 -0.1694609 -0.2941344

ver B 100 -0.0816701 -0.1405128

ver C 100 0.00271791 0.00545283

ver D 100 0.08473044 0.1470672

ver E 100 0.07895223 0.13506001

hor 80 0.96897679 1.58189782

rot 80 0.00401586 0.00704673

ver A 80 -0.1662409 -0.286312

ver B 80 -0.0776846 -0.1322504

ver C 80 0.0029006 0.00528636

ver D 80 0.08312044 0.14315601

ver E 80 0.07478401 0.126964

hor 60 0.62578744 0.98823148

rot 60 0.00369805 0.00626367

ver A 60 -0.1508963 -0.2534492

ver B 60 -0.0677115 -0.1133143

ver C 60 0.00355863 0.00602375

ver D 60 0.07544815 0.12672458

ver E 60 0.06415283 0.10729056

hor 40 0.32784648 0.4967111

rot 40 0.00297733 0.00474282

ver A 40 -0.1173296 -0.1897664

ver B 40 -0.0502496 -0.0819543

ver C 40 0.00345708 0.00512883

ver D 40 0.05866479 0.0948832

ver E 40 0.0467925 0.07682545

hor 20 0.10292231 0.14896866

rot 20 0.00205087 0.00304907

ver A 20 -0.0707383 -0.1103136

ver B 20 -0.0290153 -0.0470466

ver C 20 0.00214424 0.00329436

ver D 20 0.03536915 0.05515678

ver E 20 0.02687105 0.04375223

Table 4-1: Wind and Seismic Displacement

The displacements results are reasonable because all the vertical displacements of the

points A and B have negative signs and the other 3 points have positive signs. Those 2 and 3

points are located in opposite side of the axis of bending. Since the load is put on the left

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side of the building in Figure 3-5, the points A and B should be in compression with

negative displacement and the points C, D and E are in tension with positive displacement.

This result also verify the Figure 3-3. Also the lateral displacement, lateral force and rotation

are plotted.

Figure 4-1: Lateral Force on the building

0

50

100

150

200

250

300

350

400

0 1000 2000 3000 4000

Heig

ht

(m)

Lateral Force (KN)

Wind

Seismic

25

Figure 4-2: Lateral Displacement of the Building

Figure 4-3: Rotation of the Building

0

50

100

150

200

250

300

350

400

0 0.5 1 1.5 2 2.5

Heig

ht

(m)

Lateral Displacement (m)

Wind

Seismic

0

50

100

150

200

250

300

350

400

0 0.002 0.004 0.006 0.008

Heig

ht

(m)

Rotation (rad)

Wind

Seismic

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5 Conclusion

In this project, analysis and design of 100 story diagrid structure system is

presented in detail. The stiffness matrix has been constructed by considering the

geometry of the building and general structural analysis method. The wind and seismic

forces were determined by using equation of ASCE 7-05. The resulting displacements for

the core and 5 peripheral points show very reasonable values. Also, the maximum

displacements of top of the building caused by the wind and the seismic are 0.3% and

0.5% of the building height. The total cross-sectional area of the diagrid is about 2.5 m2 at

the first interval, and 1 m2 at the top interval. These analysis data and the diagonal design

area almost match with the area of the diagrid in Guangzhou International Finance

Center. This fact explains the project is successful.

The diagrid system has a recognizable appearance and it is structurally efficient.

When the writer visited Guangzhou, China, the International Finance Center gave an

unforgettable impression. Also, according to Leonard, “Perimeter “diagrid” system saves

approximately 20 percent of the structural steel weight when compared to a conventional

moment-frame structure” (Leonard, 2007). This system is worth further research and

there is much possibility to satisfy the trend of the architectural design.

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Figures

Figure 1-1: Examples of diagrid structures ..................................................................................4

Figure 1-2: International Finance Center Diagrid System ............................................................5

Figure 1-3: Guangzhou International Finance Center ...................................................................6

Figure 3-1: Simplified model images ......................................................................................... 10

Figure 3-2: Plan and Side View of the Skyscraper in 2D Dimension Together ........................... 11

Figure 3-3: Example core model in the article and the core shape for the project model............. 12

Figure 3-4: Diagonal Assumption .............................................................................................. 14

Figure 3-5: Stiffness of the Diagonals under Lateral and Vertical Forces ................................... 15

Figure 3-6: Displacement Figure of the Model Building ............................................................ 18

Figure 3-7: Diagonal Members are Labeled with Numbers ........................................................ 20

Figure 4-1: Lateral Force on the building .................................................................................. 24

Figure 4-2: Lateral Displacement of the Building ...................................................................... 25

Figure 4-3: Rotation of the Building .......................................................................................... 25

Tables

Table 3-1: Core Section Properties ............................................................................................ 12

Table 3-2: Core Contributions in the Matrix .............................................................................. 19

Table 3-3: Piece Matrix of the Diagonals .................................................................................. 20

Table 4-1: Wind and Seismic Displacement .............................................................................. 23

28

References

American Society of Civil Engineers (ASCE), A. S. (2006). Minimum design loads for buildings

and other structures., (pp. ASCE 7-05). New York.

Alaghmandan, Bahrami & Elnimeiri. (2014). The Future Trend of Architectural Form and

Structural System in High-Rise Buildings. http://journal.sapub.org/arch.

Alaghmandan, M., Pehlivan, N. Al, & Elnimeiri, M. (2014). Architectural and Structural

Development of Tall Buildings. 3rd Annual International Conference on Architecture.

Athen, Greece.

Balling, R. and Lee, J. (2014). Simplified Model for Analysis and Optimization of Skyscrapers

with Outrigger and Belt Trusses. J. Struct. Eng.

Boake, T. (2013). Diagrids the New Stability System: Combining Architecture with Engineering.

Architectural Engineering Institute Conference.

Lee, J. (2013). Accuracy of a Simplified Analysis Model for Modern Skyscrapers. Retrieved from

All Thesis and Dissertations: http://scholarsarchive.byu.edu/etd/4055

Leonard, J. (2007). Investigation of Shear Lag Effect in High-Rise Buildings with Diagrid

System. M.S. thesis,.

Panchal N & Patel W. (2014). Diagrid Structural system:strategies to reduce lateral forces on

high-rise buildings. international journal of research in Engineering and technology.

Wilkinson, C. (2012). Guangzhou Finance Centre: An Elegant Simplicity of Form. CTBUH 9th

World Congress. Shanghai: CTBUH.