application of a simplified skyscraper model to the shanghai tower
TRANSCRIPT
Application of a Simplified Skyscraper Model to the Shanghai Tower
Nicholas Kainoa Simon
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
Paul Richards
Fernando S. Fonseca
Department of Civil Engineering
Brigham Young University
April 2016
Copyright © 2016 Nicholas Kainoa Simon
All Rights Reserved
ABSTRACT
Application of a Simplified Skyscraper Model to the Shanghai Tower
Nicholas Kainoa Simon
Department of Civil and Environmental Engineering, BYU
Master of Science
The Simplified Skyscraper Model (SSM) developed by Balling and Lee (2014) is adapted
to the Shanghai Tower to show that it can be applied to skyscrapers of varying geometries. The
Shanghai Tower is the third of three very tall skyscraper in Shanghai and the second tallest in the
world. It employs a structural system consisting of a core inner wall tube, mega-columns,
outriggers, and belt trusses. The Shanghai Tower is built in subject to typhoon-level wind forces
and the design in controlled by the wind drifts. The SSM is a powerful tool for preliminary
design and project estimation. The SSM uses dominant degrees of freedom at certain levels of
the structure, and designs the super-elements at those levels. A stiffness matrix is used to
calculate lateral forces, lateral displacements, rotations, and drifts.
Keywords: SSM, Shanghai Tower, core inner wall tube, mega-columns, outrigger, belt truss
iv
TABLE OF CONTENTS
LIST OF EQUATIONS ............................................................................................................... vi
1 Introduction ........................................................................................................................... 1
2 Design and Construction of the Shanghai Tower .............................................................. 3
2.1 Architecture .................................................................................................................... 3
2.2 Structural System ............................................................................................................ 5
2.2.1 Core Wall Inner Tube System ..................................................................................... 5
2.2.2 Mega-Column System ................................................................................................ 6
2.2.3 Outrigger System ........................................................................................................ 7
2.2.4 Belt Truss System ....................................................................................................... 8
2.2.5 Seismic Analysis ......................................................................................................... 9
2.2.6 Wind Analysis ........................................................................................................... 10
2.3 Foundation .................................................................................................................... 12
2.4 Construction .................................................................................................................. 13
3 Simplified Model for Analysis: Shanghai Tower ............................................................. 15
3.1 Constants ....................................................................................................................... 15
3.2 Geometry ...................................................................................................................... 16
3.3 Super-elements Sheet .................................................................................................... 18
3.4 Matrices Sheet ............................................................................................................... 21
3.5 Wind and Seismic Sheets .............................................................................................. 24
3.6 Stress Sheet ................................................................................................................... 26
3.7 Optimization ................................................................................................................. 26
3.8 Graphs ........................................................................................................................... 29
4 Conclusion ........................................................................................................................... 33
v
References .................................................................................................................................... 35
vi
LIST OF TABLES
Table 2-1: Shanghai Tower Floor Use (Wood., 2014) ..........................................................4
Table 3-1: Constants Sheet ....................................................................................................16
Table 3-2: Zone Information ..................................................................................................17
Table 3-3: Core Super-element ..............................................................................................19
Table 3-4: Outrigger Super-element ......................................................................................19
Table 3-5: Outrigger Super-element ......................................................................................20
Table 3-6: Core Thickness, Outrigger Volume, and Belt Volume ........................................27
Table 3-7: Column Areas .......................................................................................................28
Table 3-8: Design Constraints ...............................................................................................28
Table 3-9: Design Objectives.................................................................................................28
v
LIST OF FIGURES
Figure 1-1: Shanghai Tower ..................................................................................................2
Figure 2-1: Varying Core Inner Wall Tube Geometry ..........................................................6
Figure 2-2: Mega-column and Belt Trusses...........................................................................7
Figure 2-3: Shanghai Tower Outrigger Truss ........................................................................8
Figure 2-4: Shanghai Tower Single Belt Truss Geometry .....................................................9
Figure 2-5: Analytical 3-D Model in Abaqus ........................................................................10
Figure 2-6: Wind Tunnel Model ............................................................................................11
Figure 2-7: Vortex Shedding .................................................................................................12
Figure 2-8: Concrete Pore of the Six Meter Concrete Mat ....................................................12
Figure 2-9: The Unsupported Retaining Wall for Excavation ...............................................13
Figure 2-10: The Hydraulic Jack-up Steel Formwork ...........................................................14
Figure 3-1: Shanghai Tower Floor Plans for Zones 1-4 ........................................................17
Figure 3-2: Shanghai Tower Floor Plans for Zones 5-7 ........................................................18
Figure 3-3: The First Half of the Stiffness Matrix of the Shanghai Tower ...........................23
Figure 3-4: The Second Half of the Stiffness Matrix for the Shanghai Tower ......................23
Figure 3-5: Shanghai Tower Floor Plans for Zones 5-7 ........................................................24
Figure 3-4: Forces and Moments ...........................................................................................25
Figure 3-5: PΔ Forces and Moments .....................................................................................25
Figure 3-6: Lateral Force .......................................................................................................29
Figure 3-7: Lateral Displacement ..........................................................................................30
Figure 3-8: Rotation ...............................................................................................................30
Figure 3-9: Drift .....................................................................................................................31
vi
LIST OF EQUATIONS
Equation 3-1 ...........................................................................................................................21
Equation 3-2 ...........................................................................................................................21
Equation 3-3 ...........................................................................................................................21
Equation 3-4 ...........................................................................................................................21
Equation 3-5 ...........................................................................................................................21
Equation 3-6 ...........................................................................................................................21
Equation 3-7 ...........................................................................................................................21
Equation 3-8 ...........................................................................................................................21
Equation 3-9 ...........................................................................................................................21
Equation 3-10 .........................................................................................................................22
Equation 3-11 .........................................................................................................................22
Equation 3-12 .........................................................................................................................22
Equation 3-13 .........................................................................................................................22
Equation 3-14 .........................................................................................................................22
Equation 3-15 .........................................................................................................................23
Equation 3-16 .........................................................................................................................25
Equation 3-17 .........................................................................................................................25
Equation 3-18 .........................................................................................................................26
Equation 3-19 .........................................................................................................................26
Equation 3-20 .........................................................................................................................26
Equation 3-21 .........................................................................................................................26
Equation 3-22 .........................................................................................................................26
Equation 3-23 .........................................................................................................................26
1
1 INTRODUCTION
The modern skyscraper consists of four main elements: a core, mega-columns, outrigger
trusses, and belt trusses. This system allows for very tall buildings that can have over a hundred
stories with very open floor plans. The Simplified Skyscrpaer Model (SSM) was developed by
Balling and Lee (2014) to analyze and optimize a skyscraper design with the four main elements
previously mentioned. The SSM uses dominant degrees of freedom and super-elements to
represent the core, columns, outrigger trusses, and belt trusses. This report details the application
of the SSM to the design of the Shanghai Tower. This project shows that the SSM can be applied
to diverse skyscraper geometries, and will be used to teach structural engineering students about
the structural elements of skyscrapers.
The Shanghai tower is located in Pudong, Shanghai, China and is the tallest of three adjacent
skyscaper including the Jin Mao Tower and Shanghai World Financial Center. Figure 1-1 shows
the three adjacent skyscrapers in the Lujiazui financial district of Shanghai. The Shanghai
Tower, designed by Gensle Architects and Thornton and Tomasetti Structural Engineers, is the
second tallest building in the world, and is only surpassed by the Burj Khalifa. The Shanghai
Tower tops out at a height of 632 meters with 123 stories above ground. The core is 30 meters
square constructed with reinforced concrete. The mega-columns are composite elements made up
of concrete-encased steel sections (Xia, 2010). The outrigger trusses and belt trusses are made of
structural steel. The whole structural frame sits on a six meter thick concrete mat supported by
947 bore piles. The perimeter mega-columns are arranged in a circular plan, radial trusses extend
2
outward from the mega-columns to support an asymmetrical glass façade. Figures 1-1 shows the
asymmetrical, twisting glass façade of the Shanghai Tower. The area between the façade and
perimeter mega-columns is used as open atria. The Shanghai Tower is an engineering feat as the
designers were faced with challenges of a typhoon laden climate, an active earthquake zone, and
clay-based soils.
Figure 1-1: Shanghai Tower (Wood, 2014)
3
2 DESIGN AND CONSTRUCTION OF THE SHANGHAI TOWER
2.1 Architecture
Shanghai has undergone very fast growth, and the need for high density housing to support
the growing population has become a very important issue. Gensler used the traditional lane
houses of China to implement “… new planning and design strategies to address the need for
high density development on one hand and ‘breathing room’ on the other” (Xia, 2010). The
architects designed the Shanghai tower to have a vertical floor space in the inner cylindrical
building to mimic the traditional lane houses, and a garden atrium between the inner cylinder and
the façade to act as a park and to facilitate the feeling of community like the communal open
space that the lane houses were centered around.
The Shanghai Tower is split up into seven different vertical zones with varying uses. Table
2-1 is the explanation of floor use. Each zone contains an atrium and a sky lobby that includes
shops, restaurants, and urban amenities to cater to the neighborhood’s daily needs. These vertical
zones and garden atriums create a city within a city without the urban sprawl associated with
many large cities around the world.
4
Table 2-1: Shanghai Tower Floor Use (Wood., 2014)
Floors Use
111-117 Boutique office
84-110 Hotel
101 Hotel Sky lobby
81-83 Mechanical
61-80 Office
60 Sky lobby
58-59 Mechanical
35-57 Office
34 Sky lobby
31-33 Mechanical
8-30 Office
6-7 Mechanical
B2-5 Retail
B5-B3 Parking
Sustainability was a main concern in the design of the Shanghai Tower. The Shanghai
government required that the skyscraper grounds were to be 33 percent green space (Wood,
2014). Gensler designed the ground to meet this requirement and added extensive landscaping to
cool the grounds from the heat of the massive city of Shanghai. The double façade was also
designed to reduce the need for electric lighting by admitting the maximum amount of daylight,
and reducing the amount of heating and cooling by allowing the buffer space in between the two
facades to act like an insulating layer. The design implemented 43 sustainable technologies in the
design of the Shanghai Tower to reduce the energy consumption by 21 percent (Zhao, 2011).
5
2.2 Structural System
The structural engineers at Thornton and Tomasetti implemented the traditional skyscraper
structural system of a core wall inner tube, mega-columns, outriggers, and belt trusses to support
the very tall and slender design and the many design challenges presented by a skyscraper in
typhoon-level winds, an active earthquake zone, and poor soil. The gravity loads are transferred
to and handled by the core and mega-columns. The primary lateral load system consists of the
core, mega-columns, and outriggers, and a secondary lateral load system consists of the mega-
columns and belt trusses. The building is split up into seven different zones with two-story
outriggers and belt trusses at the top of each zone.
2.2.1 Core Wall Inner Tube System
The core of a skyscraper typically has a three dimensional space structure of frames or
shear walls to form a tube like structural system that acts like a vertical cantilever to withstand
lateral loads imposed on the structure (Jiang, 2008). The core also carries the gravity loads along
with the mega-columns. These tube cores allow for fewer interior columns and create more open
floor space. The Shanghai Tower has a core wall inner tube system made up of nine square cells
that combine to make a square reinforced concrete core. The core wall inner tube is not uniform
all the way up the tower and changes geometry as the height of the tower increase as shown in
Figure 2-1. The core is a 30 meter by 30 meter square up until zone four. The core geometry then
changes and the four corner cells become triangles cutting off the corners of the square up until
zone six. At zone six the core becomes a cross as the four corner cells drop off entirely. The core
was designed this way to reduce the core thickness and increase its ductility.
6
Figure 2-1: Varying Core Inner Wall Tube Geometry (Hong Kong Polytechnic University)
2.2.2 Mega-Column System
The mega-columns are the second structural system that helps the core wall inner tube
system carry the gravity loads of the building. There are eight main mega-columns and four
supporting corner columns as shown in Figure 2-2. The main mega-columns extend all the way
up the height of the building and are the main contributors to the primary structural system. The
four corner columns are designed to reduce the length of the belt trusses at lower levels and take
only a fraction of the loads on the building. For this reason, the corner columns only extend up to
zone five.
7
Figure 2-2: Mega-column and Belt Trusses (Balling & Lee, 2014)
2.2.3 Outrigger System
The outrigger system is the tie between the core inner wall tube system and the mega-
column system to create the primary lateral force resisting system. The outriggers add a substantial
amount of rigidity to the building by allowing the mega-columns to engage in the primary
structural system and reduce the overall deformation. A building that employs the use of outriggers
can reduce the core overturning moment by 40 percent compared to a free cantilever, as well as a
significant reduction in lateral drift (Kian, 2004). This is achieved by applying forces from the
columns on the core that counteract the rotations and overturning.
The Shanghai Tower uses a single brace geometry for the outrigger trusses as shown in
Figure 2-3. Gensler decided to use a two-story outrigger element that is hidden in the mechanical
floors at the top of every zone. Outriggers are placed at the top of each zone to brace the core and
mega-columns laterally and torsionally as the height increases. The outriggers are placed at
8
specific heights of the building to maximize the reduction of rotations and overturning moments
while not applying too much shear force to the core.
Figure 2-3: Shanghai Tower Outrigger Truss
2.2.4 Belt Truss System
The belt truss systems create a frame system between all of the mega-columns that act as
a supplement to the primary lateral force resisting system of the core inner wall tube, mega-
columns, and outriggers. All columns are utilized instead of just the columns attached to the
outriggers when a belt truss system is implemented. The Shanghai Tower has belt trusses between
each column at the top of every zone as shown in Figure 2-2. Figure 2-4 depicts the geometry of a
single belt truss connected to two megacolumns used in the Shanghai Tower.
9
Figure 2-4: Shanghai Tower Single Belt Truss Geometry
2.2.5 Seismic Analysis
The Shanghai Tower was designed to meet the performance based design requirements as
specified in the China Seismic Design Code. Thornton Tomasetti designed for this by creating a
three-dimensional, finite element model in the program called Abaqus to determine the non-
linear response of the members and connections. Moment, axial force, and deformation were
determined from the analysis of the 3-D model and the performance of the structural system was
analyzed with seismic time history graphs for a soil similar to the one at the site. The analysis
showed that the seismic response was less than the maximum drift ratio specified by the China
Seismic Design Code and the core inner wall tube, mega-columns, outriggers, and belts remained
within the elastic range. Plastic hinge rotations for the four elements remained within the limits
for life safety set by the China Seismic Design Code. Figure 2-5 shows the 3-D Abaqus model
of the Shanghai Tower in Arqus.
10
Figure 2-5: Analytical 3-D Model in Abaqus (Poon, 2011)
2.2.6 Wind Analysis
The typhoon-level winds in Shanghai create a very difficult problem for a skyscraper as
tall as the Shanghai Tower. Thornton Tomasetti teamed up with the engineers at RWDI to
conduct wind tunnel testing to simulate the typhoon-level wind loads and design the façade and
structural system accordingly. The wind tunnel model for the Reynolds Correction test is shown
in Figure 2-6. The wind loading became the controlling factor in the design of the structural
system and façade.
11
Figure 2-6: Wind Tunnel Model (Weismantle et al., 2007)
The Shanghai Tower was subjected to wind tunnel testing procedures set forth by Section
6.6 of the ASCE 7-05 standards and the Load Code for the Design of Building Structures GB
50009-2001 for the P.R.C. The engineers also combined the wind tunnel data with a statistical
model of local wind climate to capture the typhoon-level winds in Shanghai. All the tests were
performed on a 1:500 model except for the Reynolds Correction test which was performed on a
1:85 model for more precise testing on loading and the impact of wind vortices on the building.
The wind vortices of the Shanghai Tower is depicted in Figure 2-7. To achieve the best rotation
design of the façade Thornton Tomasetti tested the model with façade rotations of 90, 120, 150,
180 and 210 degrees and a taper scaling factor of 25, 40, 55, 70, and 85 percent. The testing
determined an optimized design of 120 degree rotation and 55 percent taper scaling, which
resulted in $50 million U.S. dollars in the structural system.
12
Figure 2-7: Vortex Shedding (Zhao, 2011)
2.3 Foundation
The foundation of the Shanghai tower is a concrete mat support by bore piles to address
the poor soil conditions of the site which was a type IV soil under the China Building Code and a
Class F equivalent in the IBC Code (Wood, 2014). The concrete mat shown in Figure 2-8 is six
meters thick and supported by 947 bore piles that are one meter in diameter and 52 to 56 meters
long (Su, 2013). The soil and foundation design was incorporated into the 3-D, finite element
Abaqus model previously mentioned.
Figure 2-8: Concrete pore of the six meter concrete mat (Wood, 2014)
13
2.4 Construction
A narrow construction site and a complex surrounding environment presented many
challenges to the construction and management of the Shanghai Tower. To address these
challenges the engineer implemented an unsupported circular retaining wall, a hydraulic jack-up
steel platform, a template scaffold system for the core construction, Building Information
Modeling software, and a whip type tower crane.
A combination of island excavation and basin excavation was used to create the 1.2 meter
thick, unsupported circular retaining wall. At a depth of 50 meters, the deep bored piles were
driven into the soft soil foundation. Then the six meter thick concrete mat was poured with C50
concrete in a continuous pour of concrete without cooling pipes, breaking the world record for
continuous volume pour. Then the core was constructed using the hydraulic jack-up steel
platform and the structural steel frame around the core as the hydraulic jack-up steel platform
rose. Design management teams resolved conflicts with the detailed design while a three-level
schedule management and monitoring system was established to achieve every target on
schedule. Figure 2-9 shows the unsupported retaining wall and Figure 2-10 show the hydraulic
jack up steel platform and steel framing.
Figure 2-9: The unsupported retaining wall for excavation (Wood, 2014)
14
Figure 2-10: The hydraulic jack-up steel formwork (Wood, 2014)
15
3 SIMPLIFIED MODEL FOR ANALYSIS: SHANGHAI TOWER
The SSM is a way to analyze skyscrapers for gravity, wind and seismic loading without the
complexity of 3-D models. The SSM produces a preliminary design and is beneficial in project
estimation and for design constraint consideration. The Shanghai Tower is perfect for analysis by
the SSM as it consists of a structural system with a concrete core, mega-columns, outriggers, and
belt trusses. The SSM is implemented in a spreadsheet with seven pages: constants,
superelements, matrices, wind, seismic, stress and graphics sheet.
3.1 Constants
The constants used in the constants page are shown in Table 3-1. These constants match
the wind and seismic testing done in the design process of the Shanghai Tower.
16
Table 3-1: Constants Sheet
Concrete
allowable stress (KPa) 48000
modulus (KPa) 43400000
density (KN/m^3) 21.7
cost ($/m^3) 157
Steel
allowable stress (KPa) 207000
modulus (KPa) 200000000
density (KN/m^3) 77
cost ($/m^3) 5390
Weight Data
floor dead load (KPa) 4.34
floor live load (KPa) 2.4
cladding weight (KPa) 1.3
Wind Data
speed (m/s) 55
air density (Kg/m^3) 1.226
reference height (m) 900
exponent 9.5
drift allowable 0.002
Seismic Data
spectral acceleration (g) 0.22
ductility factor 3
exponent 2
drift allowable 0.01
3.2 Geometry
The Shanghai Tower was split into 7 vertical zones for the analysis in the SSM. Outriggers
and belt trusses are located at the top of these zones. Table 3-2 shows the story ranges for each
zone, number of stories, and the total height of each zone.
17
Table 3-2: Zone Information
Zone Dimensions
stories # stories Height (m)
101 to 117 17 76.6
84 to 100 17 72.1
68 to 83 16 79.2
52 to 67 16 74.7
37 to 51 15 65.7
22 to 36 15 74.7
1 to 21 21 99
The varying floorplans for zones one through four are shown in Figure 3-1. Note that
these zones have twelve mega-columns. The varying floor plans for zones five through seven are
shown in Figure 3-2. These zones have eight mega-columns. Outrigger and belt trusses
dimensions were also determined from these geometries.
Figure 3-1: Shanghai Tower floor plans for zones 1-4
18
Figure 3-2: Shanghai Tower floor plans for zones 5-7
3.3 Super-elements Sheet
The super-elements page is used to calculate the size and stiffness for the individual
elements of the structural system. This page starts by listing the core thicknesses, and volumes of
steel for the outriggers, and volumes of steel for belt trusses. The core section properties are
shown in Table 3-3. The moment of inertia is calculated with the section properties of the
varying core configurations (Equation 3-1). The outrigger super-element section properties are
shown in Table 3-4 are listed on the super-element sheet. The outriggers are two story outriggers
with a height of 9.9 meters but the length varies from story to story because of the building taper.
The belt super-element section properties shown in Table 3-5 are listed on the super-elements
sheet. The belt truss heights are also 9.9 meters because they are also two stories high. Axial
forces for the core and two columns are calculated on the super-elements sheets based on a
quarter of the building because of symmetry. Equations 3-2, 3-3, and 3-4 are used to solve for
column areas (Equation 3-5). These equations assume that the axial strains in the columns are the
same as the axial strain in the core under gravitational loads (Equation 3-6).All of these
properties are used to calculate the stiffness of the core, mega-columns, outriggers and belt
trusses (Equations 3-7, 3-8, 3-9).
19
Table 3-3: Core Super-element
Core Section Properties Zones 6-7
Stories Area Inertia
101 to 117 53.75534 3573.594514
84 to 100 53.75534 3573.594514 Core Section Properties Zones 4-5
Stories Area Inertia
68 to 83 72.76072 5625.372852
52 to 67 72.76072 5625.372852 Core Section Properties Zones 1-3
Stories Area Inertia
37 to 51 80.63301 8048.158818
22 to 36 80.63301 8048.158818
1 to 21 154.9442 15465.3243
Table 3-4: Outrigger Super-element
Outrigger Superelement
story w h mem area stiffness mem sine mem length
117 8.63543632 9.9 0.0012557 25129.8069 0.75359673 13.1369997
100 10.5260717 9.9 0.61996689 9895342.34 0.68511178 14.4501967
83 12.7773858 9.9 0.4234257 5072518.94 0.61247604 16.1638977
67 15.1986394 9.9 0.33169431 2915322.82 0.5457975 18.1385953
51 17.8007228 9.9 0.60154202 3834291.81 0.48604466 20.3684985
36 20.4767148 9.9 0.10393298 484934.802 0.43527277 22.7443586
21 23.4312096 9.9 0 0 0.38919966 25.4368155
20
Table 3-5: Outrigger Super-element
Belt Superelement
Story w h mem area stiffness mem sine mem length
117 Belt AC 7.425291312 9.9 0.269901149 1395793 0.799989 12.37517479
Belt C 7.425291312 9.9 0.269901149 1395793 0.799989 12.37517479
100 Belt AC 8.019251936 9.9 0.288457935 1242484 0.777054 12.74042392
Belt C 8.019251936 9.9 0.072114484 1242484 0.777054 12.74042392
83 Belt AC 8.726523129 9.9 0.269495919 1079241 0.750167 13.19705293
Belt C 8.726523129 9.9 0.06737398 1079241 0.750167 13.19705293
67 Belt AB 9.487182381 9.9 0.117070374 926048.9 0.722 13.7119156
Belt BC 9.487182381 9.9 0.117070374 926048.9 0.722 13.7119156
Belt C 9.487182381 9.9 0.058535187 926048.9 0.722 13.7119156
51 Belt AB 10.30465097 9.9 0.108076778 785114 0.692806 14.28971069
Belt BC 10.30465097 9.9 0.108076778 785114 0.692806 14.28971069
Belt C 10.30465097 9.9 0.054038389 785114 0.692806 14.28971069
36 Belt AB 11.14533865 9.9 0.105779107 662867.4 0.664103 14.90733288
Belt BC 11.14533865 9.9 0.105779107 662867.4 0.664103 14.90733288
Belt C 11.14533865 9.9 0.052889554 662867.4 0.664103 14.90733288
21 Belt AB 12.07352058 9.9 0.101748138 550846.6 0.634069 15.61345251
Belt BC 12.07352058 9.9 0.101748138 550846.6 0.634069 15.61345251
Belt C 12.07352058 9.9 0.050874069 550846.6 0.634069 15.61345251
21
3-1
3-2
3-3
3-4
3-5
3-6
3-7
3-8
3-9
3.4 Matrices Sheet
The matrices sheet contains a stiffness matrix that is used to analyze the performance of
the skyscraper by using the dominant degrees of freedom (DOF’s). The dominant DOF’s at the
top of each zone which are the horizontal displacement, the rotation, and the vertical
displacements of each column.
The core super-element stiffness was added into rows and columns of the stiffness matrix
corresponding to the horizontal and rotational DOF’s at the top of zone i and i+1 as shown in
Equation 3-10. The column super-element stiffness was added into rows and columns of the
22
stiffness matrix corresponding to the vertical DOF’s at the top of zone i and i+1 as shown in the
Equation 3-11. The outrigger super-element stiffness was added into the rows and columns of the
stiffness matrix corresponding to the rotational and vertical DOF’s at the top of zone i as shown
in the Equation 3-12. The 25 meters in Equation 3-12 is replaced by the appropriate
perpendicular distance from the axis of bending to the appropriate mega-column. The belt super-
element stiffness was added into the rows and columns of the stiffness matrix corresponding to
the rotational and vertical DOF’s at the top of the zone i as shown in the Equations 3-13, 3-14,
and 3-15 as appropriate for the specific columns. The 12.5 meters in the Equations 3-13 and 3-15
is replaced by the appropriate component of the belt length parallel to the direction of the lateral
loading. The final stiffness matrix is shown in Figures 3-3 and 3-4.
3-10
3-11
3-12
3-13
3-14
23
3-15
Figure 3-3: The first half of the stiffness matrix of the Shanghai Tower
Figure 3-4: The second half of the stiffness matrix for the Shanghai Tower
The inverse of the stiffness matrix is then multiplied by the wind and seismic force vector
to produce the wind and seismic displacement vector shown in Figure 3-5. The analysis can non-
linear by using a Microsoft Excel macro that automates iterations.
24
Figure 3-5: Shanghai Tower floor plans for zones 5-7
3.5 Wind and Seismic Sheets
These two sheets contain the calculations that produce the wind and seismic forces and
moments on the structure for each of the 123 stories. The stories are listed with the parameters
such as story height, perimeter, floor area, concrete volume, and steel volume corresponding the
zone and story. Lateral forces due to wind and seismic forces are calculated (Equations 3-10, 3-
11). These lateral forces are used calculate forces and moments at the tops and bottoms of each
zone according to the fixed-end forces in Figures 3-1. Finally, displacements and rotations are
calculated for each story using results computed on the Matrices Sheet for each story. PΔ forces
and moments are also calculated for the non-linear analysis option on the Matrices sheet (Figure
3-2).
25
3-16
3-17
Figure 3-6: Forces and Moments
Figure 3-7: PΔ Forces and Moments
26
3.6 Stress Sheet
The stress sheet is essential to the optimization aspect of the SSM. Maximum stress is
calculated for each of the super-element: core inner wall tube, mega-columns, outriggers, and
belt trusses. Stresses for the core and mega-columns are calculated at the bottom of the zones
where gravity load stress is added to lateral load stress (Equations 3-12, 3-13, 3-14). In Equation
3-13 and 3-14, the 12.5 is replaced by the appropriate distance from the core neutral axis to the
core outermost fiber. The outrigger and belt trusses stresses are calculated according to the
lateral forces apply to them (Equation 3-15, 3-16, 3-17). In Equation 3-15, the 25 is replaced by
the distance from the core neutral axis to the mega-columns.
3-18
3-19
3-20
3-21
3-22
3-23
3.7 Optimization
The solver add-in is utilized in the SSM and offers two optimization methods. The first is
called the evolutionary method, which uses a genetic algorithms to change the design inputs of
core thickness, mega-column areas, and steel volumes of outriggers and belt trusses and find an
27
optimum design. The second is called GRG nonlinear which uses a gradient based algorithm to
change design inputs and find an optimum design. Both methods will optimize the inputs to find
the lower lowest cost of materials while keeping the stress and design parameters within acceptable
limits. Table 3-6 shows final and optimized core thickness and steel volumes of the outriggers and
belt trusses. The SSM assumes that the composite steel-concrete mega-columns are converted to
all concrete mega-columns by multiplying the steel area by the ratio of concrete elastic modulus
to the steel elastic modulus. Table 3-7 shows final and optimized areas of the all concrete mega-
columns. The core thickness of the actual Shanghai Tower is 0.5 meters in the top zone and 1.2
meters in the bottom zone (Xia, 2010). The optimized core thickness is 33 percent smaller in the
top zone and 45 percent smaller in the bottom zone. The column areas of the actual Shanghai
Tower is 4.56 square meters in the top zone and 22.79 square meters in the bottom zone (Xia,
2010). When the all concrete mega-columns areas are corrected for a composite steel-concrete
with a third of the area consisting of steel the mega-column areas are 4.98 square meters at the top
zone and 30.9 square meters at the bottom zone. The optimized design has larger mega-column
areas and smaller core thickness to reduce overall cost of materials.
Table 3-6: Core Thickness, Outrigger Volume, and Belt Volume
Design Variables
stories core t outrig V belt V
101 to 117 0.33597088 0.30546644 133.602956
84 to 100 0.33597088 176.082204 41.5075504
68 to 83 0.33597088 141.318055 41.3003832
52 to 67 0.33597088 128.792586 38.2659506
37 to 51 0.33597088 269.346184 37.7177284
22 to 36 0.33597088 52.9624088 39.347723
1 to 21 0.64560091 0 40.455562
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Table 3-7: Column Areas
Column Area
column A/C column B
6.73177556 0
10.41233 0
16.6613005 0
18.8865842 0.70619022
22.1073772 4.45300572
22.3093628 7.20539542
41.7348878 20.3183975
The design constraints are shown in Table 3-8. All of these constraints must be under one
to within the acceptable design parameters. There are six constraints that are used in the SSM:
wind drift, seismic drift, core stress, column stress, outrigger stress, and belt stress are shown in
Table 3-8. The optimized design objectives were concrete cost, steel cost, and total cost and are
shown in Table 3-9.
Table 3-8: Design Constraints
Design Constraints
wind drift 1.00000003
seismic drift 0.61465534
core stress 0.8413984
column stress 0.66779505
outrigger stress 1.00000103
belt stress 1.00000008
Table 3-9: Design Objectives
Design Objective
concrete cost $ 15,944,974.58
steel cost $ 6,150,015.64
total cost $ 22,094,990.22
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3.8 Graphs
This tab contains graphs that describe the analysis of the skyscraper in four graphics. The
four graphs are of the lateral force (Figure 3-6), lateral displacement (Figure 3-7), rotation
(Figure 3-8), and lateral drift (Figure 3-9) from the base of the skyscraper to the top. The lateral
drift is a finite approximation of the rotation. Both the seismic and wind parameters are graphed
on each graph to allow for easy comparison between the two lateral forces.
Figure 3-8: Lateral Force
0
100
200
300
400
500
0 2000 4000 6000 8000 10000
Heig
ht
(m)
Lateral Force (KN)
Wind
Seismic
30
Figure 3-9: Lateral Displacement
Figure 3-10: Rotation
0
100
200
300
400
500
0 0.5 1 1.5 2 2.5 3
Heig
ht
(m)
Lateral Displacement (m)
Wind
Seismic
0
100
200
300
400
500
0 0.002 0.004 0.006 0.008
Heig
ht
(m)
Rotation (rad)
Wind
Seismic
31
Figure 3-11: Drift
0
100
200
300
400
500
0 0.002 0.004 0.006 0.008
Heig
ht
(m)
Drift
Wind
Seismic
32
33
4 CONCLUSION
The SSM is a powerful but yet simple tool to analyze and optimize wind, seismic, and
gravity loading on the Shanghai Tower. The circular geometry and tapering structural system
was easily captured within the scope of the SSM. The SSM reasonably optimized the cost of the
material while keeping design parameter within acceptable limits. The SSM is a great tool to use
for project estimation because of its simplicity and power. Further analysis with detailed output
should accompany the SSM later in the design process but a feasible design can be tested before
extensive time and resources are used in 3-D modeling software. The SSM was successfully
applied to the Shanghai Tower, and it can be applied to other skyscrapers with varying plan
geometries consisting of a core, mega-columns, outriggers, and belt trusses. The SSM is a great
tool to teach students about the principles of skyscraper design, and this project will expand their
vision to how the SSM can be applied to many other skyscraper designs while providing
reasonable results.
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35
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