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POLITECNICO DI MILANO
SCUOLA DI INGEGNERIA CIVILE, AMBIENTALE E TERRITORIALE
Corso di Laurea in Ingegneria Civile
Integration of New Typologies
in Design and Analysis
Relatore: Prof. Franco MOLA
Correlatore: Ing. David SHOOK
Tesi di Laurea di:
Luca LUPI Matr. 782484
Stefano RIVA Matr. 784139
Anno Accademico 2012 – 2013
Summary
PREFACE .................................................................................................................. 13
ABSTRACT ................................................................................................................ 15
ABSTRACT ................................................................................................................ 17
ESTRATTO DELLA TESI ............................................................................................... 19
CHAPTER 1 INTRODUCTION ................................................................................ 27
CHAPTER 2 GENERAL ASPECTS ............................................................................ 31
2.1 GLOBAL TOPOLOGY OPTIMIZATION ............................................................................ 33
2.2 REFINED TOPOLOGY OPTIMIZATION ............................................................................ 34
2.3 INTERPRETATION OF TOPOLOGY OPTIMIZATION ............................................................ 37
2.4 MEMBER SIZE OPTIMIZATION .................................................................................... 37
CHAPTER 3 TOPOLOGY OPTIMIZATION ................................................................ 41
3.1 MATHEMATICAL BACKGROUND.................................................................................. 41
3.2 STRUCTURAL TOPOLOGY OPTIMIZATION ...................................................................... 44
3.2.1 Minimum compliance design ....................................................................... 45
3.2.2 Density method – SIMP ................................................................................ 49
3.3 OPTIMIZATION PARAMETERS ..................................................................................... 52
3.3.1 Design Domain and Non Design Space ........................................................ 52
3.3.2 Loads and Boundary Conditions ................................................................... 53
3.3.3 Optimization Constraints: Fraction Volume and Maximum Admissible Stress
54
CHAPTER 4 ANALYTICAL INTERPRETATIONS......................................................... 57
4.1 MAXWELL’S THEOREM ON LOAD PATHS ...................................................................... 57
4.2 MICHELL TRUSS ....................................................................................................... 64
CHAPTER 5 DISCRETE OPTIMIZATION .................................................................. 69
5.1 RESPONSE SURFACE METHODS .................................................................................. 70
Summary
5.2 GENETIC ALGORITHMS .............................................................................................. 75
5.3 GLOBAL RESPONSE SURFACE METHOD ........................................................................ 78
CHAPTER 6 CASE STUDY: UNITED STATES COURTHOUSE...................................... 81
6.1 OVERVIEW OF THE BUILDING ...................................................................................... 81
6.2 STRUCTURAL SYSTEM ................................................................................................ 86
6.3 OPTIMIZATION PROCESS ............................................................................................ 92
6.3.1 Global Topology Optimization ...................................................................... 92
6.3.2 Refined Topology Optimization .................................................................... 96
6.3.1 Interpretation of the optimization ............................................................. 100
6.3.2 AISC specifications for structural steel ....................................................... 106
6.3.3 Load combinations ..................................................................................... 110
6.3.4 Member size optimization .......................................................................... 113
6.3.5 Consolidating the methodology ................................................................. 123
CHAPTER 7 CASE STUDY: 111 SOUTH MAIN ........................................................ 137
7.1 PROJECT OVERVIEW ............................................................................................... 137
7.1.1 Architectural Description ............................................................................ 137
7.1.2 Structural Description ................................................................................. 139
7.1.3 Project Site Conditions ................................................................................ 143
7.2 REFINED TOPOLOGY OPTIMIZATION .......................................................................... 148
7.2.1 Boundary Conditions .................................................................................. 151
7.2.2 Static Load Cases ........................................................................................ 152
7.2.3 Topology Optimization Parameters Set-Up ................................................ 154
7.2.4 Analysis Results .......................................................................................... 156
7.2.5 Results Interpretation ................................................................................. 160
7.2.6 Architectural and Structural Considerations .............................................. 166
7.3 MEMBER SIZE OPTIMIZATION .................................................................................. 167
7.3.1 Geometry Description ................................................................................. 167
7.3.2 Iterative Process: Strength Design ............................................................. 169
7.3.3 Diagonals Design ........................................................................................ 171
7.3.4 Beams Design ............................................................................................. 172
7.3.5 Columns Design .......................................................................................... 173
7.3.6 Serviceability checks ................................................................................... 174
7.3.7 Weight Comparison .................................................................................... 183
Summary
7.3.8 Analytical study: Alternative Scheme 2 ...................................................... 184
CHAPTER 8 CONCLUSIONS ................................................................................ 189
APPENDIX A CASE STUDY: US COURTHOUSE ...................................................... 195
A.1 STRENGTH CHECKS ................................................................................................ 195
A.2 OPTIMUM POSITION ALGORITHM ............................................................................ 195
APPENDIX B CASE STUDY: 111 SOUTH MAIN ...................................................... 217
B.1 DESIGN OF DIAGONAL MEMBERS ............................................................................. 217
B.2 DESIGN OF STEEL BEAMS ........................................................................................ 217
B.3 DESIGN OF STEEL COLUMNS .................................................................................... 217
BIBLIOGRAPHY ....................................................................................................... 245
Index of Figures
FIGURE 2.1 EXAMPLE OF INSPIRE MODEL [3] ............................................................................................. 34
FIGURE 2.2. EXAMPLE OF HYPERMESH MODEL AND MESH SET-UP [4] ........................................................... 36
FIGURE 2.3 RESULTS PLOT – HYPERVIEW [4] ............................................................................................. 36
FIGURE 2.4 MEMBER SIZE OPTIMIZATION PROCESS .................................................................................... 38
FIGURE 2.5 FLOW CHART OF THE OPTIMIZATION PROCESS ........................................................................... 39
FIGURE 3.1 STRUCTURAL TOPOLOGY OPTIMIZATION ................................................................................... 45
FIGURE 3.2 OPTIMIZATION DESIGN PROBLEM ............................................................................................ 46
FIGURE 3.3 COMPUTATIONAL PROCESS FOR TOPOLOGY OPTIMIZATION [9] ...................................................... 48
FIGURE 3.4 DENSITY MODIFICATION OVER THE DESIGN DOMAIN .................................................................. 49
FIGURE 3.5 RELATION STIFFNESS - DENSITY VARYING PENALIZATION FACTOR ................................................... 51
FIGURE 3.6 POISSON AND YOUNG MODULUS ............................................................................................ 52
FIGURE 3.7 DESIGN DOMAIN AND NON-DESIGN SPACE .............................................................................. 53
FIGURE 3.8 RELATION VOLUME FRACTION - COMPLIANCE ........................................................................... 55
FIGURE 4.1 3:1 CANTILEVER SPACE ......................................................................................................... 60
FIGURE 4.2 TRUSS GEOMETRY SELECTED FOR SHORTEST PATH, WHICH COINCIDES WITH THE MOMENT DIAGRAM .... 61
FIGURE 4.3 GEOMETRY OF A PRATT TRUSS ................................................................................................ 61
FIGURE 4.4 GEOMETRY OF A WARREN TRUSS ............................................................................................ 61
FIGURE 4.5 BOUNDED OPTIMAL TRUSS WITH 12 MEMBERS .......................................................................... 62
FIGURE 4.6 CANTILEVER WITH ONLY COMPRESSION CHORD .......................................................................... 62
FIGURE 4.7 MICHELL OPTIMAL SOLUTION, EXAMPLE 1 ................................................................................ 66
FIGURE 4.8 MICHELL OPTIMAL SOLUTION, EXAMPLE 2 ................................................................................ 67
FIGURE 5.1 RSM APPROXIMATION. FINDING THE MAXIMUM POINT OF THE SOLUTIONS SURFACE ........................ 72
FIGURE 5.2 ARSM APPROXIMATION. FINDING THE MAXIMUM POINT OF THE SOLUTIONS SURFACE ...................... 72
FIGURE 5.3 RSM WORKFLOW ............................................................................................................... 74
FIGURE 5.4 GENETIC ALGORITHM WORKFLOW .......................................................................................... 77
FIGURE 5.5 GRSM METHODOLOGY ......................................................................................................... 80
FIGURE 6.1 COURTHOUSE OVERVIEW (SOM, 2013) ................................................................................. 82
FIGURE 6.2 COURTHOUSE OVERVIEW - FRONT VIEW (SOM, 2013) ............................................................. 83
FIGURE 6.3 COURTHOUSE OVERVIEW – DETAIL (SOM,2013) ..................................................................... 83
FIGURE 6.4 COURTHOUSE OVERVIEW – ATRIUM (SOM, 2013) ................................................................... 84
FIGURE 6.5 LIGHT COURT SOLAR DIAGRAM (SOM, 2013) ........................................................................... 85
Index of Figures
FIGURE 6.6 COURTHOUSE OVERVIEW – AXONOMETRIC VIEW (SOM,2013) ................................................... 86
FIGURE 6.7 STRUCTURAL SYSTEM OVERVIEW ............................................................................................ 87
FIGURE 6.8 TYPICAL PLAN VIEW ............................................................................................................. 88
FIGURE 6.9 LOAD PATH SCHEME (SOM, 2013) ........................................................................................ 89
FIGURE 6.10 MAT FOUNDATION ............................................................................................................. 91
FIGURE 6.11 ISOMETRIC VIEW OF THE MODEL ............................................................................................ 93
FIGURE 6.12 LOADS AND SUPPORTS ........................................................................................................ 94
FIGURE 6.13 OPTIMIZED SHAPE .............................................................................................................. 95
FIGURE 6.14 PLAN VIEW ....................................................................................................................... 96
FIGURE 6.15 ELEVATION ....................................................................................................................... 97
FIGURE 6.16 PLAN BOTTOM VIEW ........................................................................................................... 98
FIGURE 6.17 REFINED OPTIMIZATION RESULTS .........................................................................................100
FIGURE 6.18 TOPOLOGY OPTIMIZATION RESULTS AND TRUSS INTERPRETATION ...............................................101
FIGURE 6.19 DETAIL OF THE TENSION AND COMPRESSION MEMBERS ............................................................102
FIGURE 6.20 FOUR OPTIONS, 3D VIEW ..................................................................................................103
FIGURE 6.21 FOUR OPTIONS, PLAN VIEWS...............................................................................................103
FIGURE 6.22 TYPICAL LOAD PATH SCHEME ..............................................................................................104
FIGURE 6.23 CHOSEN SET UP, 3D VIEW ..................................................................................................105
FIGURE 6.24 CHOSEN SET UP, PLAN VIEWS ..............................................................................................105
FIGURE 6.25 TYPICAL LOAD PATH SCHEME ..............................................................................................106
FIGURE 6.26 ESTIMATED MEMBER SIZES, 3D VIEW ...................................................................................115
FIGURE 6.27 LOAD INTENSITY FOR STRENGHT COMBINATION (KIP) ...............................................................116
FIGURE 6.28 LOAD INTENSITY FOR STIFFNESS CONTROL (KIP) ......................................................................117
FIGURE 6.29 FINAL GEOMETRY, 3D VIEW ...............................................................................................118
FIGURE 6.30 DEFLECTION DIAGRAM ......................................................................................................119
FIGURE 6.31 GRSM CONVERGENCE CURVES ...........................................................................................122
FIGURE 6.32 CONSTRAINT VIOLATION, ELEVATION DETAIL ..........................................................................124
FIGURE 6.33 TOP CHORD - PLAN VIEW ...................................................................................................125
FIGURE 6.34 BOTTOM CHORD - PLAN VIEW .............................................................................................125
FIGURE 6.35 TYPICAL MODULUS OF THE TRUSS ........................................................................................126
FIGURE 6.36 2D MODEL......................................................................................................................127
FIGURE 6.37 2D OPTIMIZATION RESULT .................................................................................................128
FIGURE 6.38 INTERPRETATION OF THE OPTIMIZATION RESULT .....................................................................128
FIGURE 6.39 OPTIMIZED SHAPE AND MICHELL SOLUTION...........................................................................129
FIGURE 6.40 2D OPTIMIZED SHAPE - POINTS A AND B ...............................................................................130
FIGURE 6.41 IDEALIZED PROBLEM: ISOSTATIC FRAME ................................................................................131
FIGURE 6.42 TYPICAL ELEVATION (SOM,2013) ......................................................................................134
Index of Figures
FIGURE 6.43 ELEVATION (SOM, 2013) ................................................................................................ 135
FIGURE 6.44 TRUSS DETAILS (SOM, 2013) ............................................................................................ 135
FIGURE 7.1 SALT LAKE CITY, UTAH ........................................................................................................ 137
FIGURE 7.2 111 SOUTH MAIN (SOM,2013) ......................................................................................... 138
FIGURE 7.3 TYPICAL FLOOR FROM GROUND LEVEL TO L4 (LEFT) AND LEVEL 05-PARAPET (RIGHT) ..................... 139
FIGURE 7.4 CANTILEVER FLOOR AND CONFERENCE HALL DETAIL (SOM, 2013) ............................................ 140
FIGURE 7.5 STRUCTURAL DRAWINGS: NORTH-SOUTH ELEVATION (LEFT) AND SOUTH FAÇADE (RIGHT), (SOM,
2013) .................................................................................................................................... 141
FIGURE 7.6 BASELINE STRUCTURAL MECHANISM FROM ETABS .................................................................. 142
FIGURE 7.7 PEAK GROUND ACCELERATION WITH 2% IN 50 YEARS (USGS) ................................................... 144
FIGURE 7.8 PGA WITH 2% IN 50 YEARS (USGS) – UTAH .......................................................................... 144
FIGURE 7.9 DESIGN RESPONSE SPECTRUM (DE) ...................................................................................... 146
FIGURE 7.10 MAXIMUM CONSIDERED EARTHQUAKE SPECTRUM (MCE) ...................................................... 146
FIGURE 7.11 BASIC WIND SPEED FOR OCCUPANCY CATEGORY II [MPH] AS PER ASCE 7-10 [14] ..................... 147
FIGURE 7.12 OPTIMIZATION MODEL ..................................................................................................... 148
FIGURE 7.13 PROBLEM DIMENSIONS ..................................................................................................... 149
FIGURE 7.14 COMPOSITE METAL DECK AND STEEL BEAMS ........................................................................ 150
FIGURE 7.15 FINITE ELEMENT ANALYSIS MODEL FOR TOPOLOGY OPTIMIZATION ........................................... 152
FIGURE 7.16 DEFINITION OF DESIGN DOMAIN AND NON-DESIGN SPACE ...................................................... 155
FIGURE 7.17 SINGLE DOMAIN RESULTS .................................................................................................. 157
FIGURE 7.18 MULTIPLE DOMAIN MODEL ............................................................................................... 158
FIGURE 7.19 OPTIMIZATION RESULTS FOR 3 DESIGN DOMAIN MODEL ........................................................ 159
FIGURE 7.20 STRESSES DISTRIBUTION ( COMPRESSION IN BLUE COLOR AND TENSION IN RED) ........................... 161
FIGURE 7.21 STRUCTURAL INTERPRETATION OF OPTIMIZATION RESULTS ...................................................... 162
FIGURE 7.22 LOAD PATH .................................................................................................................... 163
FIGURE 7.23 OPTION 1: TYPICAL FLOOR ................................................................................................ 163
FIGURE 7.24 OPTION 1: EAST (LEFT) AND SOUTH (RIGHT) FAÇADES ............................................................. 164
FIGURE 7.25 OPTION 2: EAST FAÇADE (LEFT) AND TYPICAL FLOOR (RIGHT) ................................................... 165
FIGURE 7.26 PRIMARY MODE SHAPES: BASELINE SCHEME (LEFT), OPTION 1 (MIDDLE) AND OPTION 2 (RIGHT) .. 166
FIGURE 7.27 TYPICAL PLAN (RIGHT) AND A-A SIDE ELEVATION................................................................... 168
FIGURE 7.28 TYPICAL PLAN (RIGHT) AND B-B SIDE ELEVATION ................................................................... 168
FIGURE 7.29 TYPICAL PLAN (RIGHT) AND C-C SIDE ELEVATION (LEFT) .......................................................... 169
FIGURE 7.30 INTERACTION BETWEEN DIAGONALS AND BEAMS .................................................................... 170
FIGURE 7.31 COLUMNS LOAD PATH ....................................................................................................... 173
FIGURE 7.32 PRIMARY MODE SHAPES COMPARISON: BASELINE SCHEME (TOP) AND ALTERNATIVE SCHEME(BOTTOM)
............................................................................................................................................. 177
FIGURE 7.33 STORY DRIFT DETERMINATION [14] .................................................................................... 178
Index of Figures
FIGURE 7.34 STORY DRIFT UNDER SEISMIC LOAD COMBINATION ................................................................180
FIGURE 7.35 BASELINE SCHEME - DEAD LOAD AND SUPERIMPOSED DEAD LOAD VERTICAL DISPLACEMENTS (INCHES)
.............................................................................................................................................181
FIGURE 7.36 ALTERNATIVE SCHEME - DEAD LOADS AND SUPERIMPOSED DEAD LOADS VERTICAL DISPLACEMENTS
(INCHES) .................................................................................................................................181
FIGURE 7.37 BASELINE SCHEME - LIVE LOADS VERTICAL DISPLACEMENTS (INCHES).........................................182
FIGURE 7.38 ALTERNATIVE SCHEME - LIVE LOADS VERTICAL DISPLACEMENTS (INCHES) ...................................182
FIGURE 7.39 WEIGHT COMPARISON ......................................................................................................183
FIGURE 7.40 TYPICAL PLAN (RIGHT) AND C-C SIDE ELEVATION ...................................................................184
FIGURE 7.41 CORNER BRACES INCLINATION CHANGE ................................................................................185
FIGURE 7.42 ALTERNATIVE SCHEME 2 - BRACES TOTAL WEIGHT .................................................................186
FIGURE 7.43 ALTERNATIVE SCHEME 2 - WEIGHT COMPARISON ..................................................................187
Index of Tables
TABLE 4.1 LOAD PATH AND DEFLECTION COMPARISON FOR 3:1 CANTILEVER................................................... 63
TABLE 6.1 STEEL SECTIONS .................................................................................................................. 114
TABLE 6.2 POINT COORDINATES ............................................................................................................ 131
TABLE 6.3 DIFFERENCE IN VOLUME BETWEEN OPTIMAL SOLUTIONS .............................................................. 133
TABLE 7.1 CONCRETE PROPERTIES ......................................................................................................... 151
TABLE 7.2 STEEL PROPERTIES ............................................................................................................... 151
TABLE 7.3 SEISMIC STATIC FORCE ......................................................................................................... 154
TABLE 7.4 W14 SHAPES PROPERTIES ..................................................................................................... 171
TABLE 7.5 DETAIL OF DIAGONALS DESIGN ............................................................................................... 171
TABLE 7.6 W21 AND W24 SHAPES PROPERTIES ..................................................................................... 172
TABLE 7.7 DETAIL OF BEAMS DESIGN ..................................................................................................... 173
TABLE 7.8 COLUMN SHAPES PROPERTIES ................................................................................................ 174
TABLE 7.9 DETAIL OF COLUMN DESIGN .................................................................................................. 174
TABLE 7.10 MODAL MASS PARTICIPATION RATIOS .................................................................................... 176
TABLE 7.11 ALLOWABLE STORY DRIFT ΔA [14] ........................................................................................ 178
TABLE 7.12 COMPUTED X AND Y STORY DRIFTS ....................................................................................... 179
Preface
The following work has been possible thanks to prof. Mola and Mark Sarkisian.
Their friendship and professional cooperation has created the possibility of an
experience abroad that allowed us to see different approaches to structural
engineering, to experience different cultures and to learn from real professionals.
A first thanks to prof. Mola and ECSD, in particular to Elena Mola and Laura
Pellegrini. Their support for creating the opportunity of this experience and
throughout all the research work was fundamental. It was especially helpful the
support and the focus that was given during the permanence in San Francisco,
and even more once back in Italy for synthetizing the experience done abroad,
judging it and learning the best from it.
Thanks to Mark Sarkisian and with him to all the Skidmore Owings and Merrill
office of San Francisco. Particularly we want to thanks the people of the studio
we work more closely with, Neville Mathias, David Shook, Eric Long, Peter Lee,
Jeff Keileh, Alessandro Beghini, Andrew Krebs, Alberto Lago, Abel Diaz, Lachezar
Handzhiyski, Ricardo Henoch, Alvin Tsui and all the other members of the
Structural Engineering Team of the San Francisco Office we had the chance to
work with. The meetings, the discussions and their suggestions helped us to learn
how engineers deal with problems. The trust given to us and to the research we
were doing made this work possible.
Abstract
La scelta del sistema strutturale ha un impatto molto elevato nella valutazione
dei costi di una struttura. Questa problematica è ancora più influente quando
l’edificio in esame presenta peculiarità.
Il seguente lavoro pone l’attenzione sulla determinazione di una metodologia
che permetta di integrare all’interno del design di una struttura nuove tipologie.
In particolare si è utilizzata l’ottimizzazione topologica per aver una conoscenza
approfondita del percorso dei carichi. Con tale consapevolezza è possibile
ottenere, mediante la rimozione di materiale, una struttura più rigida.
Si sono definite quattro fasi. Innanzitutto l’ottimizzazione topologica nel
continuo, mediante la quale è possibile avere una conoscenza generale del
percorso dei carichi. Poi, mediante un’ottimizzazione più raffinata, si trova una
soluzione ben definita. Vi è quindi una fase d’interpretazione dei risultati tramite
il passaggio da continuo a discreto, verificando l’attendibilità della soluzione.
Infine, si procede a una fase di ottimizzazione degli elementi strutturali, ai fini di
rispettare le limitazioni delle normative.
La procedura illustrata è stata applicata a due studi: una Corte e Americana e un
edificio in Utah, 111 South Main. Per quanto riguarda la corte e, l’ottimizzazione
è stata usata per trovare una nuova tipologia per la travatura reticolare che
funge da copertura con l’obiettivo di diminuire il più possibile il peso. Su 111
Abstract
South Main, tale metodo è stato applicato per trovare una nuova configurazione
del sistema laterale.
In conclusione, tale approccio è apparso replicabile ed efficiente, fornendo
risultati soddisfacenti. Inoltre, la struttura ottenuta nello studio della corte e è
stata efficace al punto di esser inserita all’interno del detail design.
Si ritiene infine che introdurre questa procedura nel design di un edificio possa
esser molto utile, andando a influire non solo sul sistema strutturale, ma anche
sull’architettura, aiutando a trovare nuove tipologie che siano al contempo belle
ed efficienti. Si consiglia l’utilizzo soprattutto in caso di strutture complesse in cui
un sistema strutturale classico possa essere molto inefficiente.
Abstract
The structural system chosen when designing a building highly influences its cost.
This problem is even more important when the structure has some peculiarities.
The following work focuses on finding a methodology for integrating new
typologies in structural analysis and design. In particular, the attention was
towards topology optimization, in order to understand and replicate the load
path in the structure. In doing so, is possible to obtain a stiffer system by
removing material.
The methodology is divided into four main steps. Starting with a global topology
optimization in the continuum, the load path is generally understood. Moving
into a refined optimization, a more defined solution is obtained. Then the result
is brought from continuum to discrete through its interpretation. During this
step, the system is analyzed and the reliability of the result is verified. Once a
discrete system is refined, the final step consists in optimizing the member sizes
in order to accomplish the requirements of the codes.
The procedure has been applied to two different case studies: a United States
Courthouse and an Office Building, 111 South Main in Utah. The topology
optimization has been used for finding a new typology for the roof truss on the
top of the Building, aiming to diminish the weight of the structure. On the office
Abstract
building instead, it has been used for finding a new typology for the lateral
system, looking for a more efficient solution.
At the end of these processes, the approach has proved to be consistent and
successful. For both the studies, the new system met the requirements and the
objective proposed at the beginning. Furthermore, the optimized structure for
the Courthouse was included into the Detail Design drawings, giving even more
relevance to its efficiency.
As conclusion of this work, including topology optimization into the design of a
building can not only influence the structural system, but also help finding new
architectural solutions that are at the same time beautiful and structurally
efficient. It is suggested to use this approach for special buildings, in which a
more classical system could be inefficient.
Estratto della Tesi
La progettazione strutturale di un edificio è fortemente influenzata dallo sviluppo
architettonico del progetto stesso; negli ultimi decenni quest’affermazione è
ancor più vera riflettendo sull’evoluzione dell’architettura moderna verso forme
sempre più estreme.
Nella maggior parte delle situazioni il sistema strutturale viene definito
all’interno di un set di sistemi tipici in cui il problema fondamentale diviene la
verifica normativa degli elementi. Muovendosi in questo ambito risulta evidente
che questa procedura conduca da un lato ad un notevole risparmio in termini di
tempo necessario per la definizione del sistema strutturale dall’altro alla
possibilità di incorrere in un notevole sovra utilizzo di materiale. Questa doppia
conseguenza è ancor più evidente quando la progettazione riguarda strutture
speciali, ad esempio per effetto della crescente altezza dei palazzi e delle
richieste architettoniche.
Nel corso degli ultimi decenni queste problematiche hanno condotto all’idea di
studiare tipologie strutturali innovative. Il processo può essere sviluppato
integrando il concetto di ottimizzazione topologica all’interno della
progettazione sia ingegneristica che architettonica giungendo in questo modo a
nuove forme.
La possibilità di integrare queste nuove tipologie nella progettazione proviene
dal confronto nato in un periodo di tirocinio presso lo studio di San Francisco
(California, USA) di Skidmore, Owings and Merrill LLP. Nota e diffusa in tutto il
Estratto della Tesi
mondo, SOM rappresenta l’avanguardia mondiale sia in campo architettonico
che in campo ingegneristico lavorando su svariati edifici alti. Le necessità di
minimizzare i costi delle strutture garantendo la massima qualità tecnologica e
sicurezza che caratterizzano SOM, ha spinto la società ad un profondo studio di
queste tematiche. Questa importanza è ancora più sentita nello studio di San
Francisco: lavorando su progetti situati in California, Medio Oriente e Cina, la
richiesta di efficienza strutturale è amplificata dall’elevata sismicità delle aree.
L’integrazione della topologia strutturale nella progettazione può permettere di
cogliere la miglior soluzione in termini di utilizzo di materiale. Questo processo si
può rivelare efficiente a tal punto da poter influenzare sia la progettazione
strutturale che quella architettonica.
Essere all’avanguardia in queste tematiche è di fondamentale importanza per
poter sviluppare nuovi sistemi strutturali e continuare nella crescita tecnologica.
Da questa necessità, il presente elaborato di tesi si prefigge come obiettivo
quello di definire una nuova metodologia che possa implementare
l’ottimizzazione topologica nella progettazione, a partire dalle prime fasi fino allo
sviluppo della fase di progettazione di dettaglio degli elementi strutturali
principali.
Il primo passo della metodologia consiste nella Global Topology Optimization; in
questa fase un modello schematizzato del problema è sviluppato al fine di
studiare il percorso dei carichi all’interno della struttura. A questo livello di
studio la struttura è definita nei suoi elementi generali, i più importanti dei quali
sono il dominio di ottimizzazione e le condizioni al contorno.
A questo punto un analisi ad elementi finiti viene condotta per redistribuire il
materiale imponendo come obbiettivo la massimizzazione della rigidezza della
struttura. Il risultato fornisce un layout strutturale per diversi valori di volume
residuo; il materiale, infatti, è ridistribuito all’interno del volume e concentrato
nelle aree con maggiori sforzi.
Estratto della Tesi
In seguito, viene eseguita la Refined Topology Optimization: questo step consiste
semplicemente in un analisi ad elementi finiti più raffinata della precedente una
volta che il concept architettonico è maggiormente sviluppato: un maggior
numero di parametri devono essere controllati e molti più particolari strutturali
devono essere noti. Il risultato consiste, quindi, in una distribuzione di densità
all’interno del dominio di progettazione che definisce la posizione del materiale
in funzione del percorso dei carichi e della frazione di volume residuo desiderata.
Una volta raggiunta un’adeguata conoscenza della meccanica del problema
strutturale, una profonda interpretazione analitica dei risultati deve essere
eseguita. Quest’analisi ha come obiettivo finale la definizione di un struttura
discreta che individui la posizione, la lunghezza e gli angoli reciproci tra ogni
singolo membro.
In questa fase si introduce la capacità ingegneristica di interpretazione dei
risultati a partire da quelle che erano le ipotesi alla base del problema di
ottimizzazione. Spesso quest’analisi è accompagnata dall’utilizzo di risultati
analitici di letteratura: ad esempio, nel corso dello studio si è fatto ampio
riferimento a studi dell’ingegnere australiano Michell che ha riportato alla luce
studi sulla meccanica di Maxwell.
Il passaggio finale consiste nella Member Size Optimization: vengono definite le
dimensioni sezionali di ogni singolo elemento del modello discretizzato.
Per prima cosa viene costruito un modello utilizzando un software di analisi
strutturale facendo riferimento alle normative vigenti. A questo punto inizia un
processo iterativo che definisce la dimensione di ogni singolo elemento da
sezioni commerciali disponibili e facendo riferimento alle verifiche richieste da
normativa.
Questo passaggio può essere automatizzato attraverso l’implementazione di
algoritmi genetici; in questo modo è possibile ottenere l’ottimizzazione di
strutture con centinaia di elementi riducendo i tempi che sarebbero richiesti da
Estratto della Tesi
un’iterazione manuale. Attraverso algoritmi genetici e l’utilizzo del Global
Response Surface Method è possibile, dunque, ottenere un’ulteriore riduzione
del peso di materiale utilizzato.
L’efficienza di questa metodologia è stata verifica tramite l’applicazione a due
differenti casi di studio: un corte e degli Stati Uniti d’America ed il 111 South
Main, entrambi progetti in corso di sviluppo presso lo studio di SOM di San
Francisco.
Il primo caso di studio fa riferimento ad una corte e il cui aspetto architettonico è
definito da forma cubica con un’altezza totale di 72 metri suddivisi in 10 piani. Il
sistema laterale consiste di 4 core con forma a C e da 4 shear walls interposti;
questo sistema serve anche come sistema per condurre i carichi verticali al
sistema di fondazione interagendo con le colonne perimetrali e il sistema di travi
e metal deck. La particolarità del sistema consiste nel fatto che le colonne,
adempiendo a normative per la sicurezza, non possono proseguire fino alla
fondazione ma sono interrotte all’altezza del primo piano. I carichi gravitazionali
sono condotti tramite le colonne fino all’ultimo livello del palazzo e poi trasferiti
ai core centrali utilizzando una travatura reticolare nel piano di copertura; da
questo meccanismo strutturale consegue un comportamento particolare per le
colonne: lavorano tutte in stato di tensione e non di compressione come per
comuni edifici.
Lo studio di ottimizzazione si è focalizzato sulla travatura reticolare che era stata
inizialmente progettata secondo il modello classico di Warren al fine di ottenere
un minore peso del sistema ed una minore dimensione degli elementi favorendo
la realizzazione delle connessioni. Il successo dell’ottimizzazione è confermato
dal fatto che la struttura reticolare finale è stata inserita nei disegni strutturali
consegnati al contractor nella fase di progettazione di dettaglio.
Estratto della Tesi
Il secondo progetto su cui si è sviluppato lo studio è stata la 111 South Main,
progetto in fase di sviluppo a Salt Lake City (Utah); l’edificio è composto di 24
piani per un’altezza totale pari a 119 metri. La particolarità di questo progetto
consiste nel fatto che all’interno della pianta dell’edificio è in corso di
progettazione una costruzione indipendente, lo Utah Performing Arts Center.
Questo secondo palazzo si estende per un’altezza pari a 5 piani della 111 South
Main rendendo a sbalzo un ampia porzione dei piani dal livello 5 fino alla
copertura. Anche in questo caso di studio le colonne nelle aree tributarie a
sbalzo devono essere sospese non potendo continuare fino al sistema di
fondazione.
Per questo progetto si è pensato di sviluppare un sistema alternativo a quello
precedente che eviti di utilizzare la travatura reticolare per trasmettere i carichi
ai core centrali. L’idea per l’ottimizzazione si è basata sullo sviluppo di un sistema
perimetrale che raccolga i carichi regionalmente alle diverse altezze del palazzo e
li trasmetta direttamente ai core.
In entrambi i casi, poi, lo studio è stato focalizzato all’analisi della struttura dal
punto di vista delle prestazioni di servizio: verifica degli spostamenti verticali e
spostamenti interpiano. Quello che si è trovato è stato un resultato
stupefacente: la rigidezza di un sistema strutturale può essere incrementata
rimuovendo materiale fintanto che questo è posizionato nei punti nevralgici della
struttura. Questo risultato è in qualche modo spiegato dal punto di vista teorico
dagli studi di Michell e Maxwell: integrando questa metodologia nella
progettazione di strutture speciali si può giungere a notevoli risultati in termini di
risparmio di materiale e di capacità strutturali.
Risulta evidente che la metodologia si configura come un sussidio prezioso
specialmente per progetti con caratteristiche particolari e non per comuni edifici.
Questa procedura richiede, infatti, lunghe tempistiche per l’analisi ad elementi
Estratto della Tesi
finiti e professionisti all’interno del team di progettazione che sviluppino
appositamente lo studio.
Per le ragioni sopra citate, quando la progettazione richiede di far fronte a
particolari problemi, il risparmio di materiale e l’aumento di rigidezza del sistema
risultano considerevoli e di gran lunga convenienti paragonati ai costi richiesti da
soluzioni tipiche. Per progetti comuni, invece, questa metodologia può essere
sostituita con il giudizio ingegneristico senza ottenere risultati molto differenti da
quelli ottenibili con questa procedura.
Ad esempio, in entrambi i casi di studio la particolarità è stata rappresentata dal
fatto che parte o la totalità delle colonne non potevano giungere alla fondazione
ma dovevano essere supportate da un altro meccanismo.
Allo stesso tempo però, l’utilizzo di questo approccio per la progettazione
consente di allargare il campo dei sistemi strutturali noti, replicabili anche in
maniera più standardizzata ad edifici considerati comuni.
In conclusione, il lavoro di tesi giunge a definire una procedura consistente che
può essere applicata nella pratica progettuale. I passaggi delineano una serie di
step consecutivi che seguono il normale percorso progettuale dalla fase del
concept fino all’ingegneria di dettaglio.
Inoltre, è stato provato come l’ottimizzazione topologica possa essere utilizzata
nella progettazione di differenti componenti strutturali; nella corte e, infatti, è
stata ottimizzata una travatura reticolare mentre nella 111 South Main lo studio
si è focalizzato sullo sviluppo di un sistema strutturale alternativo. La
metodologia può, quindi, essere applicata a differenti problemi senza cambiarne
le ipotesi teoriche di base.
Gli studi sono stati condotti integrando metodi analitici classici e moderni mezzi
computazionali; entrambi sono fondamentali poiché garantiscono di poter
seguire l’evoluzione della progettazione architettonica che molto spesso richiede
Estratto della Tesi
modifiche in corso d’opera: una stretta relazione tra progettazione architettonica
e strutturale è fondamentale poiché in questo modo si può giungere a soluzioni
ancor più efficienti tramite una reciproca influenza. Il metodo, infatti, soddisfa un
duplice compito: segue lo sviluppo architettonico mantenendo un’elevata
efficienza ma ne indirizza anche le modifiche.
Da questo punto di vista l’utilizzo di risultati analitici e della visione ingegneristica
rimane di fondamentale importanza per poter far fronte alle modifiche
architettoniche rapidamente senza dover ripetere completamente lo studio ma
semplicemente modificando i risultati già ottenuti grazie alla comprensione del
comportamento strutturale globale.
Un possibile sviluppo della metodologia consiste nell’introduzione di analisi non
lineari. L’analisi ad elementi finiti per l’ottimizzazione topologica risolve il
problema meccanico di massimizzazione della rigidezza strutturale in campo
elastico lineare con materiale isotropo.
Un ulteriore effetto che dovrebbe essere considerato è quello legato agli effetti
del second’ordine. L’instabilità, infatti, non influenza in alcun modo la
ridistribuzione del materiale e la snellezza degli elementi non è valutata: il
materiale si comporta allo stesso modo, dunque, in compressione e tensione.
Una possibile soluzione potrebbe essere l’introduzione di un fattore di
penalizzazione per gli elementi compressi che descriva l’instabilità secondo la
teoria euleriana. Questo fattore può portare ad una radicale modifica dei
percorsi di carico per minimizzare specificamente il cammino dei carichi in
compressione.
Inoltre, un campo di interesse è rappresentato dalla Multi Objective
Optimization: questo argomento è stato affrontato nel corso dello studio della
111 South Main. In quel caso di studio, il processo di ottimizzazione è stato
condotto sia per carichi gravitazionali che per carichi orizzontali sismici. Il
metodo scelto per la risoluzione del problema è stato l’aggregazione pesata: è
Estratto della Tesi
stato assegnato un peso ad ogni singolo carico e si è proceduto a massimizzare la
rigidezza per il caso di carico ottenuto. Questo ha permesso di trasformare il
problema in un problema di ottimizzazione con Single Objective.
Questo modo di procedere studia un percorso dei carichi unico unendo carichi
gravitazionali e sismici; ulteriori discussioni hanno suggerito che sarebbe
auspicabile sviluppare studi separati in modo da ottenere il comportamento della
struttura in modo chiaro. Una volta che la soluzione è ottenuta per entrambi i
casi la sintesi dovrebbe essere definita nel momento in cui si passa alla
costruzione del modello discreto.
Questa considerazione acquisisce notevole importanza dal momento che il peso
totale della struttura ottimizzata può subire considerevoli incrementi dovuti ai
carichi sismici se questi non vengono considerati negli step iniziali di studio.
In conclusione, implementare l’ottimizzazione topologica può condurre a
modifiche considerevoli nel design strutturale soprattutto per strutture speciali.
L’introduzione della metodologia può portare non solo ad ottenere notevoli
risparmi di materiale, ma anche ad una più profonda conoscenza del
comportamento strutturale globale e dei percorsi dei carichi. Benché la soluzione
non sia applicata direttamente alla progettazione, questa può supportare la
progettazione di altre differenti strutture puntando l’attenzione su quei
componenti che sono stati sviluppati dalla soluzione della procedura di
ottimizzazione.
CHAPTER 1
Introduction
The design of a building is highly influenced by the architectural scheme that
comes from a conceptual design. In this phase, a structural system is thought and
usually it comes from the experience and the understanding that the structural
engineer has of the building’s behavior. After this step, the structure is
developed and the majority of the effort is put in the attempt of fulfill the
concept.
It appears clear that the structural system is usually chosen from a set of
“typical” systems. This choice could sometimes be a double edge sward: if on
one hand preferring a known system can save a lot of time in terms of
understanding and development of the structure, on the other hand it is likely
that a waste of material is necessary in order to accomplish all the checks of the
design.
This double possibility is even more evident when the building to be designed
has some peculiarities that influence the normal load paths. These types of
buildings are nowadays even more frequent due to the increasing height of the
structures and the bizarre architecture.
Chapter 1
28
A way of solving these issues can be found in new typologies. Integration of
Topology optimization software into the structural design can be used in order to
discover these new shapes and systems.
Using topology optimization in structures can turn out to be the best solution in
terms of the economy of materials. Furthermore, from such studies also
innovative shapes and forms can outcome, influencing not only the structural
system, but also the architecture and esthetics if considered in early phases of
the project.
The possibility of integrating those new typologies into the design was given by a
period of internship into a worldwide known firm, Skidmore Owings and Merrill
LLP (USA), which daily faces the challenges of designing tall and innovative
buildings throughout the world. Responding to the growing need of minimizing
the cost of new structures while always guaranteeing the security, the
technologies and the high-performance features of SOM, pushed the firm
towards the investigation of this topic.
Even more importance was given to this subject in the office of San Francisco,
California, USA. This office indeed has to deal not only with the challenges
previously explained, but also the requirements of building in a high seismicity
zone such as California and the states of the West Coast of the United States of
America, together with China and East Asia.
In these conditions, being al'avant-garde on this topic can be critical in order to
have the chance of developing more sophisticated seismic systems.
Starting from this need, the following work aims to define a methodology for
implementing the topology optimization into the design of buildings, since the
early phases until the detailed scheme of the main members of the structure.
The developed methodology has been tested in its consistency thanks to its
application to two different projects, both located in the West Coast of the
Introduction
29
United States. Both those projects, a United States Courthouse and an Office
Tower in Salt Lake City, can be defined as “special buildings” thanks to the
peculiarity that they show. For this reason, the research that has been carried
out is more important to achieve a double goal: satisfying the requirements of
the codes with the least amount of material. Moreover, a new methodology can
develop structures, which are stiffer even though using a lower amount of
material; topology optimization can move material to areas where material is
more influent.
The study could take place thanks to the chances that SOM gave by including the
research directly into the development of the projects, influencing not only the
engineering aspects, but also the architectural reasoning.
The final objective of this work is to give a clear step-by-step procedure that can
be inserted into those projects that more need optimization, both because of
special systems that characterize the buildings and for the expenses control
required by some assignments.
CHAPTER 2
General Aspects
Over the last few decades, an increased attention has been given in recent
research to the problem of optimizing the use of material in structural
engineering design ( [1], [2]). The work is an attempt to unify the methodology of
optimization and design, starting from the concept to the element size definition
in the final discrete structural lay-out. This thesis approach focuses on the overall
development of an efficient process design.
The purpose of the work is to outline how optimization can be implemented in
each step of the design; once a deep understanding of the capabilities of the
method is gained, it is more trivial to take advantage of the method in a variety
of projects, particularly the ones presenting particular design issues, thus
creating a standard optimization procedure.
The efficient use of material is important in many different fields from the
aerospace and automotive industry to the building construction. In particular,
the improvements of the last decades in the optimization of the geometry and
the layout of the structures is related to the development of faster and powerful
computational methods applied in various different software applications able to
discretize the domain and outline the best redistribution of material; these new
Chapter 2
32
tools are able to work with problems having millions degree of freedom but
keeping computational times affordable and results reliable.
The process of optimizing the structure involves all the steps of the design from
the concept to the code checks. In the early steps of design, the topology
optimized shape is defined in terms of material density over the domain and the
geometry is described by what amounts of raster as seen in computer graphics:
this is the concept stage which is greatly affecting the structural efficiency. Later
modifications on the concept of the project will have a lower impact on the
efficiency of the adopted solution design.
Since the aim of this work is to define a complete design methodology, it is
necessary to couple a computational and an analytical approaches to achieve a
deeper understanding of the structural behavior; finally, the size optimization is
focused on the code checks to define members shapes to be used which are not
exactly respecting the material requirements in each structure points because of
the fixed standard dimensions of commercial elements.
A focus is conducted on the analytical studies; the first studies were developed
by Michell (1904) recovering works done by Maxwell in the field of the structural
mechanics. In the early sixties, it was recognized that the approach developed by
Michell could be applied to large-scale problems with millions degrees of
freedom developing specialized algorithms.
The importance of finding the optimal solution for the structural system can be
seen from various point of view. Firstly, a better use of material can lead to
better structural performance: it can be proved that using a lower amount of
material it is possible to obtain a structure which is stiffer. This has been
observed in the analytical studies by the Australian engineer Michell developing
the Maxwell’s Theorem: he discovered a relation between the volume of
material and the tip deflection in a truss system. For different schemed he found
out that the lightest truss is also the one with lowest vertical deflection.
General Aspects
33
On the other hand, a better structural layouts help designers to achieve great
savings in terms of material use; material savings are even more significant in
steel structures where the material cost is predominant compared to assembly
cost because steel elements are not cast in-place. Moreover, in modern
structural design an important variable is the building sustainability and the
effects on the environment: such problems are the object of several studies and
significant results in literature point out that one of the main contribution to the
carbon emission is due to material consumption considering production and
transportation.
For the above stated reasons in modern architectural and engineering design,
topology optimization tools have started to gain a central position in the design.
It is so important to note that these tools are meant to be utilized with a critical
engineering view and compared to analytical studies and theoretical background.
Nevertheless, applying topology optimization in the concept stage of the design
requires a close cooperation between designer and analysis engineer.
In the present chapter it is described the complete methodology, following the
design process step by step.
2.1 Global Topology Optimization
The first step consists in the Global Topology Optimization; the study helps
developing a general understanding of the load path in the structure at the
concept level. One of the most commonly used tools is solidThinking Inspire [3]
by Altair; this kind of software allow a 3D finite elements analysis with a
simplified structural model. Inspire helps to develop a new material lay-out
within a design domain only defining boundary conditions and loads applied. The
material is described as linear elastic with isotropic behavior; it is not possible to
define multiple materials and for this reason no material properties are needed.
In Figure 2.1 an example of Inspire model is shown:
Chapter 2
34
Figure 2.1 Example of Inspire model [3]
The updated layout can be shown for different thresholds of fraction volume left
of the design domain (shadow volume in Figure 2.1). It is possible to define a
Non-Design Space which can be used to apply loads or boundary conditions or to
model the stiffness of elements that cannot be removed due to architectural
constraints (dark grey elements in the above figure). At this level of study it is not
necessary to precisely define the magnitude of loads applied but it is sufficient to
know if they are surface or concentrated loads and their location; the magnitude
can be assumed to have unitary value.
2.2 Refined Topology Optimization
Once the system global behavior is clearly defined, a deeper study is to be
conducted which is called Refined Topology Optimization. The goal of this step is
to achieve a robust understanding of the optimized solution by varying more
parameters. The analysis is performed using a different kind of software. In the
Altair suite, the tools used for the finite elements modeling is Hypermesh [4] and
the one to solve the optimization problem is Optistruct [5].
General Aspects
35
This software allows a FE analysis with a wide set of parameters to be varied in
order to capture the real structural behavior. First, it is necessary to define if the
study is meant to be conducted as a plane or a volume problem; the software, in
fact, allows the use of both 2D and 3D finite elements with quadrangular and
triangular shape. There are problems in which two dimensions are significantly
greater than the third so that they can be described with plane shall elements; if
all the three dimensions are comparable to each other, then 3D element are to
be used. There is also the possibility of customizing the mesh and refining it in
areas of higher stress concentrations.
The geometry is to be defined setting by material properties and boundary
conditions; the material is modeled as linear elastic but different materials can
be defined so that properties are to be input (Young’s Modulus, Poisson Ratio
and Shear Modulus) in order to catch the correct stiffness of each members.
Furthermore, loads are to be defined with more accuracy than in the global
topology optimization because the structural stiffness is better described in the
model. It is possible to define the magnitude of the leading load combination as
per ASCE Code (American Society of Civil Enginnering) or by implementing
multiple load combinations in case of multiple objective optimization.
The optimization set-up consists of objective function (f.e. minimum
compliance), structural constraints (f.e. maximum tip deflection or maximum
allowable stress) and maximum convergence tolerance. An example of
Hypermesh [4] model is shown in Figure 2.2:
Chapter 2
36
Figure 2.2. Example of Hypermesh model and Mesh Set-Up [4]
Results can be shown in terms of material density and stress\pressure for each
iteration of the optimization process. The software allows a fast set-up of the
analysis so that a parametric study can be easily conducted in order to
investigate the influence of the different parameters on the load path.
Figure 2.3 Results plot – Hyperview [4]
General Aspects
37
2.3 Interpretation of Topology Optimization
The next step consists of the interpretation of the results outlined in the
topology optimization in order to define the discretized shape of the structure;
the output is the definitive discrete geometry defining position, length and
mutual angles of each member.
At this level it is very important to use sound engineering judgment to come to a
feasible solution; for instance, since the topology optimization analysis catches
only the linear elastic behavior of material without penalization for buckling, it
could be possible to add or flip over some members to prevent lateral buckling.
Furthermore, the interpretation of the results can be founded on analytical
studies from the theory of the elasticity; important works from Rankine, Maxwell
and Michell can be combined to the computational analysis. Moreover, in many
cases the nature of this studies is theoretical and so this concepts have to match
with constructability considerations.
2.4 Member Size Optimization
The last step is called Member Size Optimization. This step is necessary to choose
the members that are minimizing the total weight of the structure; steel shapes
are picked out from approved and registered shapes in the AISC Code (American
Institute of Steel Construction).
The analysis starts assigning a shape from the code to the discrete members
defined in the interpretation of the results; this first assignment is done choosing
a category of shapes (f.e. W14 or W21) and selecting heavier sections for
element placed where the topology optimization returned higher density
concentrations. The shape category is chosen in consideration of the more
severe requirement for the member: for compression/tension members strength
is leading the design and so square section are preferred, on the contrary for
Chapter 2
38
beams deflection are to be controlled and so deep section are preferred for the
higher moment of inertia. From this first attempt, a structural model is defined in
FE analysis software (f.e. ETABS, SAP2000): the structural behavior is investigated
applying all the load combinations as per ASCE 7-10 and not only those for which
the topology optimization was conducted.
Once the structural model set-up is completed, an iterative process starts in
order to verify the structural efficiency of the model; at every step two kind of
checks are conducted: every member needs to satisfy strength design criteria as
per chapter C and D of AISC360-05 related to compression and tension members;
on the other hand, serviceability constraints are to be satisfied such as maximum
tip deflection or mode shapes respecting comfort requirements. For the code
checks database software is used in order to automate the process.
The process is over once the difference in weight of the current scheme
compared to the previous is smaller than a fixed convergence tolerance and all
the serviceability requirements are satisfied.
Figure 2.4 Member Size Optimization Process
At this point the member size optimization can be refined using algorithms
implemented using suited such as Hyperworks, which is based on the Global
Response Surface Method. This step is useful when designing structures with
thousands of members to be changed at each iteration such as in a truss system.
The final scheme of the member size optimization is summarized in Figure 2.4.
General Aspects
39
At this point the optimization process is concluded and the structural behavior is
defined; the next step would be going over constructions details relates with
joints and studying the interaction with other part of the structure.
In Figure 2.5 a flow chart is reported representing the overview on the entire
optimization process:
Figure 2.5 Flow Chart of the Optimization Process
CHAPTER 3
Topology Optimization
The present chapter is discussing the theoretical hypothesis for structural
topology optimization, in example the material redistribution on a design domain
in order to achieve predetermined structural goals with respect to posed
constraints.
3.1 Mathematical Background
The fundamental concept of optimization is to determine the best feasible
solution for a problem under given constraints. In this way the optimization
problem can be formulated in his general aspect and refers to a wide number of
problems. For example, the optimal solution for a factory manufacturing a
particular product is finding the process involving the shortest time and lowest
production time. In this sense the priority aspect for optimization is the
definition of the fundamental parameters describing the problem: in the
previous example, the parameters could be production time, cost or the number
of employee involved. These parameters should be enough to univoquely
Chapter 3
42
describe the problem but not too many to make the variation of the solution
uneasy to be understood among the parameters.
From a mathematical point of view, optimization consists on the research of
stationary point in the function describing the problem; to reach the maximal or
the minimal solution it is necessary to define a mathematical function. This
function is generally named “cost-function” and depends on design variables. An
optimization problem could be a maximization, as in the case of structural
stiffness, or minimization, as in the case of production cost in the previous
example.
The general formulation for the optimization of the cost-function subject to
constraint functions and is expressed as following [6]:
( ) ( ( ))
( )
( )
where is the vector describing design variables, ( ) represents the cost-
function, ( ) ( ) are the functions imposing constraints to the fields
where the variable can be searched.
In many cases the goal to be reached in the optimization process involves
multiple objectives. For example, the best car to be bought is the most powerful,
cheap and energy efficient; for sure there is no car which maximizes/minimizes
all these parameters at the same time.
In structural engineering, a typical optimization problem consists in the
redistribution of material in the design domain in order to achieve the most stiff
solution. Multiple objective optimization is very important for this kind of
Topology Optimization
43
problem; for instance, considering a building to be constructed in an highly
seismicity area, this will be subjected to two types of loads: gravity and lateral
seismic loads. Ideally the best solution is the one maximizing stiffness both under
seismic and gravity loads. From a general point of view, the solution for the two
load cases is different.
An important concept for multiple objectives optimization is the Pareto
Optimality [7], [8]. A solution for the previous optimization problem is Pareto
Optimal if there is no other admissible solution which decreases one of its
parameters without increasing one of the other objective functions.
The Pareto Optimality leads to the definition of a vector of solution
corresponding to the minimization of one single parameter; the set of solutions
is called Pareto Front and its dimension is equal to the number of objective
functions. The solution of a multiple objective optimization problem is more
difficult to be achieved because there is no unique solution, so the goal of the
process is to find the Pareto Front among which choosing the most suitable one.
The classical approaches for the solution consists on converting MO (multiple-
objective) problem into SO (Single-Objective); this can be made using some
scalar techniques:
a) Weighted Aggregation: this method converts MO problems into SO
problems by applying a function vector to the objective vector, for
example a linear combination of the objective functions. For this
purpose the functions should be homogeneous and a set of weights
need to be defined a priori. The problem can be formulated as
following:
∑ ( )
∑
Eq 3.1
Chapter 3
44
where the weight represents the importance given to each of the
objective functions and should be known a priori. These values need to be
positive and the total weight is unitary.
b) Goal Programming: this method is a variation of the previous and consists
on fixing specific goals for each cost-function and minimizing the
deviation from that value. The problem formulation can be written as:
∑ | ( ) |
Eq 3.2
where is the target value for each objective function.
c) ϵ-constraints: with this method the optimal solution is sought by
optimizing for one function and treating all the others as constraints
bound by some allowable range ϵ.
These solutions techniques require a priori knowledge (such as the relative
importance between objective functions or goal values) and lead to solution in
which trade-off between objectives is not easily evaluated.
3.2 Structural Topology Optimization
Structural Optimization represent an application of the mathematical theory
explained above; the purpose of structural optimization consists in the
redistribution of material over a design domain in order to achieve the highest
possible stiffness for the structure. The problem starts from a uniform material
distribution and the process moves material to region with higher stresses
concentration [9], [10].
Topology Optimization
45
The only known quantities are applied loads, boundary conditions and design
domain; the material layout is the variable for the problem represented by the
density in the volume [11]. An explanation is described in Figure 3.1 for the case
of a simply supported beam:
Figure 3.1 Structural Topology Optimization
3.2.1 Minimum compliance design
The structural optimization problem can be formulated as a compliance
minimization problem. Compliance is the inverse function of stiffness and for this
reason the formulation corresponds to the maximization of stiffness.
The formulation set-up considers a domain Ω in R2 or in R3 defined a priori
following boundary condition (Γu), loads applied (Γf) and architectural constraints
imposing fixed material region (see Figure 3.2).
Chapter 3
46
Figure 3.2 Optimization design problem
Material is considered linear elastic with stiffness defined by the tensor Eijhk(x)
variable in the domain with x. The variable x is the density over the domain and
so the structural stiffness is a function of the density distribution.
From the solid mechanics, it is possible to define the energy bilinear form (Eq
3.3):
( ) ∫ ( ) ( )
( )
Eq 3.3 Internal Work
where is the equilibrium and is the arbitrary virtual displacement and with
linearized strains ( )
(
).
Defining also the load linear form as (Eq 3.4):
( ) ∫
∫
Eq 3.4 Load Linear form
the minimum compliance problem has the hereafter form (Eq 3.5):
Topology Optimization
47
( )
( ) ( )
Eq 3.5 Structural optimization problem
The expression in the second line of Eq 3.5 is the equilibrium equation written in
the energetic form known as the weak variational. Furthermore, the is
expressing the domain of all kinematically admissible displacement field and
the bilinear form depending on the design variable (the stiffness tensor
( )); for this reason, the equilibrium is to be checked by the algorithm at
every iteration. is the space containing all the admissible stiffness tensor
attaining the isotropic material properties.
The typical approach to the solution involves the discretization of the problem
using finite elements and the problem in Eq 3.5 is written in the discretized form.
The FE analysis creates an iterative process (Figure 3.3):
- define the initial design with the homogeneous distribution over the
domain, applying loads and boundary conditions;
- using the linear elastic FE analysis, compute displacements, strains and
stresses;
- compute the compliance of the design; if marginal improvements are
obtained over the previous design, and constraints functions are satisfied,
iterations can be stopped;
- otherwise, compute the update of the density variable and run iterations
until the solution converges.
Chapter 3
48
- finally, the results can be plotted in terms of material density and stress
values and interpreted in a discrete form with CAD drawings.
Figure 3.3 Computational process for topology optimization [9]
Topology Optimization
49
3.2.2 Density method – SIMP
The structural optimization problem consists of the determination of the optimal
placement for material and, as a consequence, in modifying the material density
over the domain. Therefore, it is possible to think of the design domain as a finite
amount of pixels which varies in density during the iterative process following
load patterns (Figure 3.4).
Figure 3.4 Density Modification over the Design Domain
The design variable is the density and it defines the existence or non-existence
of material in every domain point. The problem can be seen as seeking the
optimal material distribution Ωmat ϵ Ω and in this way the set of admissible
tensors consists of those for which :
∫ ( )
Eq 3.6 Stiffness tensor definition
The last inequality expresses the limit on the volume V for the problem. is
the initial stiffness for the isotropic material distribution.
Chapter 3
50
The density method SIMP (Solid Isotropic Material with Penalization) uses
previously described studies modifying the definition on the stiffness tensor
applying intermediate density values(as for [9], [12] and [13]). This modification
consists in transforming the problem variable from an integer to a continuous
and this solves many practical problems: in many cases, in fact, the material
distribution with the integer variable gives spread and undefined results.
With the continuous density definition, it is also possible to have a penalization
on elements with lower density in order to achieve a more feasible material
distribution. In the common optimizations the material density-stiffness relation
is linear but this can be modified introducing a penalization factor (see Figure
3.5).
The stiffness definition is to be changed as following:
( )
∫ ( )
( )
Note that the assumptions of isotropic material is a simplification of the behavior
of the material as for the anisotropic materials the placement of principal
directions of the material should be also considered a design variable. The
consideration can be a valid approximation for most construction material such
as concrete and steel.
Topology Optimization
51
Figure 3.5 Relation stiffness - Density varying penalization factor
In the SIMP method choosing intermediate densities are penalized
because the obtained stiffness is small compared to the cost (in terms of volume)
of the material and the variation is more than linear. For common problems
where volume constraints are active, a penalization factor of 3 returns sufficient
clear results in terms of a feasible interpretation. A discussion on the physical
interpretation of the factor; recent studies prove that the SIMP method can
describe a material model if the power satisfies the following constraints:
In the 2D problem the penalization factor consists only of a penalization in the
thickness of the shell elements; therefore, the problem variable can be
interpreted as thickness instead of density as in the 3D problem.
Chapter 3
52
3.3 Optimization Parameters
Performing a topology optimization there are several different parameters
affecting the final result. To get an increased understanding of the structural
behavior a parametric study is conducted as done in literature [6].
3.3.1 Design Domain and Non Design Space
The optimization problem starts from the definition of a domain Ω ϵ R2\R3; the
design domain consists of all the material points where loads are applied,
boundary conditions are defined and the material is meant to be redistributed to
achieve the optimal structural stiffness. Therefore, it is necessary to assign
material properties as Young’s modulus, Poisson’s ratio and rigidity modulus; this
three material parameters can completely describe Hooke’s law and the stress-
strain relation. In certain cases it is necessary to assign also the yield stress to fix
a constraint to optimization process. The variable describing the domain is the
material density meant to be a continuous variable.
In the Figure 3.6 Poisson and Young modulus are shown for typical construction
material:
MATERIAL ν E [MPa]
STEEL 0.3 206,000 -210,000
CONCRETE 0.2 25,000 -42,000
WOOD 0,29 6,000 - 16,000
Figure 3.6 Poisson and Young Modulus
The design domain definition is assigned from the architectural building shape; it
is important to define the space dimension (R2, R3) which better describes the
optimization problem. For instance, the building façade is better described by a
2D design domain, as opposite to the volume in which a truss scheme is to be
designed, which can be typically defined as a 3D volume.
Topology Optimization
53
Figure 3.7 Design Domain and Non-Design Space
From the architectural shape, Non-Design Spaces are defined to describe areas
where the redistribution is not to be applied. This spaces are openings (no
material is defined, density is null) or fixed domain (the material is to be applied
to model the structure stiffness, density is unitary); to Non-Design Spaces all
properties can be applied as load combinations and boundary conditions.
Both Design Domain and Non-Design Spaces are to be model with a Finite
Element mesh to discretize the problem.
3.3.2 Loads and Boundary Conditions
Other parameters to be defined are loads and boundary conditions. Loads can be
determined in two ways: using an arbitrary load combination reflecting only the
magnitude of real values or applying load combinations from the ASCE Code [14].
For the optimization problem, in most cases, the dominant load is to be applied
and the results represent the optimal load path for that single load case; since all
the other load cases have lower influence on the structural behavior the solution
can be taken as a significant study to understand the general load path. In some
case, it is necessary to take into account multiple load cases: this is the situation
Chapter 3
54
for the study of the general behavior of a tall building in an high seismicity zone.
Both vertical gravity and lateral seismic loads are to be studied.
Boundary conditions are the second design parameters to be defined. The
definition of boundary conditions has great influence when considering a portion
of the structure; they should represent the stiffness of the rest of the structure in
order to capture the correct load path. It is possible to define both linear or point
support and there is the possibility of fixing both translations and rotations.
For example, considering the optimization problem for a truss system, boundary
conditions should represent the rotational interaction between steel volume and
columns and shear walls. An incorrect definition can move the load path because
an increased stiffness can capture higher amount of load.
A parametric study can be conducted on the boundary condition, for example,
investigating the influence on the overall load pattern comparing determined
and undetermined structures. For instance, the optimal solution for structures
subject to thermal loads can be studied comparing the modification in the results
increasing the number of undetermined reactions.
3.3.3 Optimization Constraints: Fraction Volume
and Maximum Admissible Stress
There are several type of constraints which can be applied to the optimization
process: maximum deflection in fixed point, fraction of volume left in the domain
or maximum stress.
In the present case the fraction of volume was used as a constraint; this
parameter controls volume fraction of the initial design domain shall be used in
the final optimized structure. This threshold of fraction volume can be found
from literature in comparison with previous data or from an agreement with the
contractor on the total amount of material to be used. The volume of material, in
fact, is strictly related to the cost of the structure specially for steel construction.
Topology Optimization
55
The volume fraction left can be chosen from previous data available as per the
volume of the gravity frame in tall buildings. For some case studies, it is
interesting to conduct a multi-objective optimization studying volume fraction
and stiffness understanding if an increase in stiffness is worthy in terms of
increase of material used (Figure 3.8)
Figure 3.8 Relation Volume Fraction - Compliance
Stress constraint is an extremely important topic concerning topology
optimization. The goal of reducing the mass in the structure has to deal with the
mechanical constraints of not exceeding the design stress limit for the material.
For the 0-1 formulation of the design problem a stress constraint is always well-
defined, but changing to intermediate density (continuous variable) it is
necessary to introduce a different definition.
For the physical relevance, it is reasonable that the criterion should mimic
microstructural considerations. The stress constraint can be defined for the SIMP
method model (with p exponent) as a constraint on the Von Mises equivalent
stress [9]:
Chapter 3
56
Eq 3.7 Stress Constraints
This constraint describes the strength attenuation of a porous material that
arises when an average strength is distributed in a local space. We can see the
factor as a stress reduction factor.
CHAPTER 4
Analytical Interpretations
4.1 Maxwell’s Theorem on Load Paths
The problem of minimizing the weight of a structure has always been a key point
in the design phase. A number of researchers have approached this theme
throughout the years. [15] [16]
Recent studies in the field of structural analysis have rediscovered the
importance of a theorem developed by Maxwell, [17] [18].
In his paper “On Reciprocal Figures and Diagrams of Forces”, Maxwell
investigated the importance of considering the length of the load path when
thinking of the layout of a structure.
Maxwell’s theorem states that the sum of a structure’s tension load paths minus
the sum of the compression load path is equal to a value related to the applied
external forces, including the reactions. The expression “load path” is meant as
the sum of the axial force acting in a member times its length. Expressed as an
equation, it can be written as
∑ ∑ ∑
eq 4.1
Chapter 4
58
On the left-hand side of the equation there is the sum of load paths, on the right-
hand side there is the dot product of all the external forces with position vectors
from an arbitrary origin. This term represents the work done by the external
forces, it can therefore be assumed as a constant value once the position of all
the loads and reactions is stated.
Taken that the external work is a constant, the power of Maxwell’s theorem can
be better appreciated if expressed as done by Baker [19]: the longer the total
tension path, the longer compression load path must be for a set of external
loads of given magnitude, direction and position. Stated in another way, if the
tension (or compression) load path is longer than the necessary, the inefficiency
of the solution will be paid double, once in compression and once in tension. On
the other hand, if we minimize the load path either in tension or compression, it
will automatically be minimized also for the counterpart.
It comes as an immediate consequence that, if the truss only has tension
elements (or compression), it is a structure of minimal load path.
Furthermore, the terms used for calculating Maxwell’s constant can be directly
related to the deflection of the truss and to its volume.
Taking an external load P, using the principle of Virtual Works, it can be stated
that:
∑
eq 4.2
Where Δ is the deflection of the truss, F is the axial force in the elements and n is
the axial force when a unitary load is applied to the structure.
Analytical Interpretations
59
Therefore
eq 4.3
∑
eq 4.4
If it is assumed that the stress F/A in each element is kept constant, it results
∑
(∑ ∑ )
eq 4.5
At the same time, since
eq 4.6
∑ ∑ ∑
eq 4.7
∑ ∑
eq 4.8
∑ ∑
eq 4.9
Chapter 4
60
If comparing eq 4.9 with eq 4.5, it can be observed that the volume and the
deflection are directly related to each other, because both depend on the sum of
the tension and compression load paths.
In Baker [19], an example of the application of Maxwell’s theorem and all the
related properties has been developed. It is briefly summarized hereinafter.
The starting truss space is a 3:1 cantilever. The applied load is P, which generates
a the reaction at the supports shown in Figure 4.1
Figure 4.1 3:1 Cantilever Space
Five different trusses have been developed in order to solve this problem, for
each one of those it has been calculated the Tensile Load Path, the Compressive
Load Path, the Difference in Load Paths which is constant according to Maxwell’s
theorem, the Sum of Load Paths which is used to calculate the expected
deflection of the truss.
From Figure 4.2 to Figure 4.6 are shown the different truss solutions. The
solution of Figure 4.6 has been chosen in order to accomplish the goal of having
only a compression chord.
Analytical Interpretations
61
Figure 4.2 Truss geometry selected for shortest path, which coincides with the moment diagram
Figure 4.3 Geometry of a Pratt truss
Figure 4.4 Geometry of a Warren truss
Chapter 4
62
Figure 4.5 Bounded optimal truss with 12 members
Figure 4.6 Cantilever with only compression chord
Table 4.1 shows the summary of the calculated values:
Analytical Interpretations
63
Table 4.1 Load Path and Deflection Comparison for 3:1 cantilever
The third column of the table clearly shows Maxwell’s theorem. The fourth and
fifth columns highlight the efficiency of the different structures in terms of
volume (column four) and deflection. As previously stated, volume and
deflection are strictly related.
Furthermore, it appears that the lighter structure is also the stiffer, and is not the
one with the shortest path in terms of length.
It must be observed though that the minimum load path is not the only factor to
take into account when selecting a final solution. The designer needs to consider
more factors such as the complexity of the structure, the cost, usability,
aesthetics and more.
A final consideration can be done starting from Maxwell’s theorem. If a structure
is uniformly stressed, the relative volume of steel needed by alternate truss
geometries to achieve a target deflection can be proved equal to the square of
the ratio of the load paths.
If the volume of any two structures in
Table 4.1 are compared one another, we have:
Chapter 4
64
eq 4.10
eq 4.11
eq 4.12
( )
eq 4.13
Therefore, load path is very important also in the consideration of deflection-
controlled design.
4.2 Michell Truss
In the design of a structure, the objective is usually minimizing the weight while
maximizing the stiffness. Looking at the results of Maxwell’s theorem, a question
arises: how low can we go? If the goal is to meet a certain deflection target, what
is the lightest structure that can be built? How far is the chosen configuration
from the optimum?
As stated by Mazurek et al. [20] and Baker [19], the starting point for answering
all those questions can be found in a paper published by Michell in 1904 [21], in
Analytical Interpretations
65
which he explores the problem of achieving an optimal configuration, given a set
of loads.
Michell’s treatise is based on the assumption of constant allowable stresses and
does not consider the effect of member weight upon the design. Furthermore,
Michell analysis is carried out in a continuum, whereas a real structure is clearly
discrete. The importance of this study lies in it being a benchmark for
understanding the level optimality of the configuration that will be chosen.
Michell states, “a frame attains the limits of economy of material possible in any
frame-structure under the same applied forces, if the space occupied by it can be
subjected to an appropriate small deformation, such that the strains in all the
bars of the frame are increased by equal fractions of their lengths, not less than
the fractional change of length of any element of the space.”
This condition can be satisfied if all the members have the same stress.
A general class of frames that satisfies this condition consists of those whose
bars, both before and after the appropriate deformation, form curves of
orthogonal systems.
The main two cases studied by Michell, have been reported in this chapter in
order to show the benchmarks that will be used in this work. Following the
complex calculation carried out by Michell, the shape of the optimum solution is
shown in the following figures, where the thick line represents the compression
bars, the tension bars are indicated by fine lines and the portions of principal
strain on which material bars are not required are shown by dotted lines.
The first case is the one of a single force F applied at a point A, and acting at right
angles to the line AB. The constraint in B is the one of a fix point.
Chapter 4
66
Figure 4.7 Michell optimal solution, example 1
The minimum frame is formed of two similar equiangular logarithmic spirals
having their origin at B and intersecting orthogonally at A, together with all the
other spirals orthogonal to these and enclosed between them. Considering that
in B the reactions are distributed over a circle of radius r0 , the necessary volume
is
eq 4.14
Where a = AB.
The second case is the one of a single force F applied at the center of a line AB of
length 2a. In this case the minimum volume is:
eq 4.15
Analytical Interpretations
67
Figure 4.8 Michell optimal solution, example 2
The shape is the one of a semicircle on AB and all its radii. All the radii are
compressed whereas the semicircle is a tension member as shown in Figure 4.8.
Further studies have been developed after Michell truss in order to turn the ideal
and continuous solution into a discrete and designable one. [22] [23] [24]
Albeit this procedure brings to a less optimal solution, it allows a step towards
the real problem. Linear programming has been used in the search of this
solution, mainly using pin-joint approximations in order to be as close as possible
to the ideal one. Pin-joint truss are indeed determined structures subjected only
to axial force.
CHAPTER 5
Discrete Optimization
In the structural optimization field one of the hardest problem to face after the
definition of the configuration, is the section choice. This step requires the
discrete approach. A variety of methods and algorithms have been developed in
the last decade, with the aim of finding a way of exploring as best as possible all
the field of possibility. This problem is often significantly big and its cost can turn
out to be prohibitive.
On one side, Anderson-Cook et al. [25], Jones [26], Roux et al. [27] and Carley et
al. [28] and others have used Response Surface Approximation to develop their
search.
On the other hand, Kripakaran et al. [29], Camp et al. [30], Liu et al. [31], Pezeshk
et al. [32], Murren et al. [33] and others have approached this problem basing
their studies on Genetic Algorithms.
In this chapter, a brief introduction to Genetic Algorithms and Response Surface
Methods will introduce the explanation of the Global Response Surface Method
(GRSM).
Chapter 5
70
GRSM is a method developed by Altair, which tries to explain the benefits of the
two different approaches in order to find a more complete, consistent and less
time-expensive method, which can face problems with a huge number of
variables.
5.1 Response Surface Methods
Response surface methods (RSM) are a collection of statistical and mathematical
techniques useful for developing, improving, and optimizing processes. RSM are
particularly used when several input variables influence the performance of the
process object of the study.
The idea that underlies these methods is to find the surface f that describes all
the possible solutions of the problem. Because the form of the true response is
unknown, it must be approximated. The efficiency of this method is therefore
strictly related to the quality of the approximation of the response is done.
Usually, low order polynomial is a good approximation in a small region of the
independent variable space. For this reason it is often found that either a first-
order or a second-order polynomial are accurate enough.
The first-order polynomial is accurate if the correct response surface is searched
in a small region, where f has a small curvature. For a two independent variable
space, the first-order approximate surface is of the from:
eq 5.1
Where the second order term is to take into account the possible interaction
between the two variables.
The general first-order polynomial model is described by the following equation:
Discrete Optimization
71
eq 5.2
The second-order surface in used when the field of interest is a larger space,
because it turns to be really flexible, taking a wide variety of functional forms.
In the two dimensional space the surface can be described by the eq 5.3.
eq 5.3
The generalization of the second-order is shown in eq 5.4.
∑
∑
∑∑
eq 5.4
An interesting development of the RSM is the Adaptive Response Surface
Method (ARSM). In addition to the normal RSM, this method internally builds a
response surface and adaptively updates this response with new points. ARSM
uses a very efficient algorithm to estimate a response surface to be closer to a
certain design of interest. It also uses moving limits to make the optimization
algorithm robust.
In Figure 5.1 there is the representation of RSM in a 1D problem. RSM finds the
exact value of the surface in two points; it then approximates the surface
according to those points and then maximizes the approximate surface.
In Figure 5.2 ARSM methodology can be seen. It finds the exact value of two
points, and then finds the approximate surface with them. It thereafter adapts
itself by adding a third point starting from the x-coordinate of the maximum
point of the approximate surface. From that it finds the exact value
Chapter 5
72
correspondent to this x coordinate, and generates a new interpolating surface.
This procedure is repeated iteratively until the error between the value found on
the approximate surface and the real one is less than the accepted error ϵ.
Figure 5.1 RSM approximation. Finding the maximum point of the solutions surface
Figure 5.2 ARSM approximation. Finding the maximum point of the solutions surface
Discrete Optimization
73
The most important advantage in this approach is that the dimension of the
error changes accordingly to the level of precision that is required by the user,
i.e. the number of iterations needed to achieve the desired approximation of the
real solution.
Chapter 5
74
Figure 5.3 RSM Workflow
Discrete Optimization
75
5.2 Genetic Algorithms
Another approach to structural optimization is the one followed by the non-
deterministic methods, in particular the Genetic Algorithms (GA).
This approach overcomes some of the critics moved to the RSM, such as the time
expense needed to calculate the gradient of the surface and the incapability of
calculating a surface if the variables are of a large number or of different nature
(discrete and continuous).
A Genetic Algorithm is a machine learning technique modeled after the
evolutionary process theory. It follows Darwin’s principle of survival of the fittest,
where designs with higher fitness values have a higher probability of being
selected for mating purposes to produce the next generation of candidate
solutions.
The process starts with a randomly generated population of design solutions.
The fittest members of this population are evaluated through a fitness function,
which estimates those elements that are closer to the constraints of the
optimization. Those are chosen as the ‘parents’ and a new generation is created
through mutation or crossover of the parents’ value of the parameter. The
fitness of this new generation is calculated. The number of member needed to
generate a new population is chosen by the user, and stored in the algorithm.
Let’s call this percentage k. The fittest k members are then chosen and, if the
termination criteria are not met (i.e. maximum number of iteration reached or
optimization criteria met), the process is repeated. In order to better generate,
every time a new population is created, not only the best members of the last
generation are used as ‘parents’, but also the two best of the older iteration are
kept as part of the k used for the mutation/crossover. By doing so, performing
designs are never lost and the evolution is quicker and more accurate.
Furthermore, in order to guarantee the exploration of the entire search field, a
Chapter 5
76
number of randomly generated solutions are always included into each new
population.
The process previously described can be seen in Figure 5.4.
A problem with Genetic Algorithms is that they do not show the typical
convergence of the other optimization algorithms. Users typically select a
maximum number of iterations (generations) to be evaluated. A number of
solver runs is executed in each generation, with each run representing a member
of the population.
Another issue with this method is that the starting generation is chosen
randomly; therefore the velocity of convergence is highly dependent on the
fitness of the first random population. For this reason, in order to make sure that
the optimum solution is a real absolute optimum; at least two runs are required.
Sophisticated GA have been developed, such as design-driven harmony search
algorithm [33], in order to intelligently move towards the optimum, gaining a lot
in terms of time consumption. Albeit this developments, Genetic Algorithms are
still not considered reliable or consistent by a number of experts.
Discrete Optimization
77
Figure 5.4 Genetic Algorithm Workflow
Chapter 5
78
5.3 Global Response Surface Method
Global Response Surface Method (GRSM) is an algorithm developed by Altair
Engineering, which tries to synthetize the benefit of the RSM and GA and at the
same time overcome the limits of both.
At the beginning of the optimization process there is a phase of calibration of the
algorithm, i.e. one variable at the time is changed and the weight of that change
on the solution is evaluated. In this way the algorithm orders the variable
accordingly to their importance in meeting the goal that the user states. By doing
so, when the permutation is done, the values of the variables are changed
accordingly to their importance.
Once the calibration phase is ended, the real optimization process starts.
A first generation of randomly generated solutions is created. After the analysis,
the fittest members of this population are selected and used as known points in
order to create the response surface. As for RSM, the response surface is created
as a polynomial of second order in order to be as flexible as possible. The
minimum/maximum of the surface is found and is stored as best solution.
The idea that underlies the way this first set of solutions is generated comes
from the studies done with the Genetic Algorithms, whereas the creation of the
surface is the step with which the two methods are integrated.
The fittest members and the optimum of the surface are used as parents of the
new generation, mixed with the fittest members of a randomly generated new
generation in order to always guarantee the complete exploration of the space.
If the minimum of the new surface is better than the minimum previously stored,
the optimum is substituted; otherwise, the best solution remains the same of the
older generation.
Discrete Optimization
79
By doing so, the surface is always improved. Furthermore, once the members of
the last iteration meet the constraints, the convergence of the method is really
fast. The methodology is shown in Figure 5.5.
As for the GAs, theoretically also the GRSM never stops going and keeps
improving the solution. In fact, even though the curve of the iteration shows
overall the typical convergence trend, since every time the fittest members are
permutated, it is always possible to improve in the solution. On the other hand
though this trend is characterized by a step-function because it is never
guaranteed that the best solution could change. The run can be stopped if a
maximum number of runs without finding a new best solution is reached, or if a
number of iterations considered significant is completed.
The GRSM has proved to be useful in problems with a big number of variables,
thanks to its calibration phase that guarantees an intelligent permutation of the
parents.
Chapter 5
80
Figure 5.5 GRSM methodology
CHAPTER 6
Case Study: United States Courthouse
6.1 Overview of the building
The object of this study is a United States Courthouse in a high seismicity zone.
This is a LEED (Leadership in Energy and Environmental Design) platinum
building, which means that, accordingly to the U.S Green Building Council, it
succeeded in meeting the strictest requirements. Among those, there is water
and energy efficiency, reduction of material waste, and other criteria.
Chapter 6
82
Figure 6.1 Courthouse Overview (SOM, 2013)
The building consists of a cube, which represents a simple and elegant shape
with a strong civic presence rooted in classic principles. The cube in fact offers
the most efficient floor plates for the courts and provides the best floor-to-skin
ratio. Furthermore, it allows the courts to be placed around a central light
atrium, which provides for a clear understanding of the building organization
that greatly assists in way finding, and brings daylight deep into the center of the
building. Thanks to an innovative structural engineering concept that allows the
cubic courthouse volume to appear to float over its stone base, the court is one
of the Nation’s safest buildings relating to earthquakes and bomb threats.
This shape gives to the building a unique lightness that can be better appreciated
thanks to the glass façade that covers all the over-grade levels but the first one.
As an overall judgment, the cube is a compelling form that fits well on the site
and provides a strong presence and the desired gravitas in the Civic Center
without being overbearing.
Case Study: United States Courthouse
83
Looking deeper into the building architecture, the serrated glass façade is
designed to achieve north/south orientation, which will maximize views and
minimize the electricity waste thanks to the solar gain that can be obtained due
to the panel system.
Figure 6.2 Courthouse Overview - Front View (SOM, 2013)
Figure 6.3 Courthouse Overview – Detail (SOM,2013)
Chapter 6
84
Figure 6.4 Courthouse Overview – Atrium (SOM, 2013)
The total height of the building is 243’-6” in the imperial unit, which corresponds
to 74.2 meters. The space is divided into ten stories and has one below-grade
level. The center of the building presents a big opening that runs through all the
building height. Thanks to a number of mirrors located onto the roof, the
sunlight is distributed to all the public spaces that face the big opening, reaching
the same goal met by the façade for the rooms on the perimeter of the building.
(Figure 6.5)
Case Study: United States Courthouse
85
Figure 6.5 Light court solar diagram (SOM, 2013)
The nine levels of the court have been dimensioned in order to achieve specific
goals of space and accommodate at the same time courtrooms, judges’ offices
and public spaces. Each floor has a total height of 25’ (7.6 meters). The first,
smaller, floor has a height of 36’ (10.97 meters), has a smaller plan for blast
protection, whereas the last floor has a smaller height, only 18’-6” (5.64 meters),
and accommodates all the mechanical equipment.
In order to give to the building the cube shape previously described, the
dimensions of the typical floor are 222’x222’ (67.67x67.67 meters), whereas the
first smaller and does not have the exterior curtain wall.
June 21st September 21st
December 21st March 21st
Chapter 6
86
Figure 6.6 Courthouse Overview – Axonometric view (SOM,2013)
6.2 Structural system
The structural system of this United States Courthouse is very peculiar.
The lateral system consists of four c-shaped concrete core walls and four
concrete shear walls (in blue in Figure 6.8). This system also acts as the gravity
system, together with the columns placed along the perimeter. At the roof level
the columns and the cores are linked together with a steel roof truss, which acts
not only as a load transfer, but also contributes to the lateral system, adding
stiffness to the cores. The roof truss is initially designed as a typical Warren truss
in both directions.
In Figure 6.7 and Figure 6.9 the truss can be seen. In order to guarantee the
uniformity of the deflection along the perimeter, a belt truss is provided.
Furthermore, the upper level of the truss is 6’-6” (1.98 meters) outer than the
lower level. This is due to the columns on the four corners, which lie on the very
corner of the building.
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Figure 6.7 Structural System Overview
Figure 6.8 shows the typical plan. The 28 columns (in red in the figure) peculiar
feature of the structural system lies in their being suspended. This is to facilitate
the protections towards terroristic or bomb attacks. In fact a requirement for
high-level security buildings, such as courts, is the total security of the building in
case of any kind of attack. This requirement is satisfied if the building is
completely safe even if anyone of the ground columns is taken out. By taking out
from the beginning every one of the columns along the perimeter, the goal is
met.
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Figure 6.8 Typical Plan View
This solution completely changes the usual way of thinking at the columns.
Instead of being in compression, the columns are hanging from the rooftop,
turning into tension elements. This change does not influence the capacity of the
columns, for steel has the same behavior in tension and compression. The big
change is in terms of the load path. Instead of bringing the weight coming from
each floor to the ground, it is brought all the way up to the roof and down again
through the central cores. In order to solve this load path, the cores need to be
big structural elements.
COREANDSHEARWALLS
COLUMNS
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Figure 6.9 Load Path Scheme (SOM, 2013)
If on one side taking away the first floor columns is really smart, on the other
hand this design choice brings along several structural challenges to be solved.
The bigger ones are the deflection of the perimeter points and the high stresses
on the foundation.
The deflection grows from floor to floor. This problem turned to be really hard to
solve, especially because of the differential deflection between the columns,
which could give uneven floor level. This problem has been faced by introducing
a jacking system on the top of each column, with which the deflection due to
dead load and superimposed dead load can be evened out, reducing the final
deflection of each floor only to a small percentage of the initial one.
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In order to build such a structure, it is necessary to define a precise construction
sequence that guarantees the correct construction of the building. At the
beginning temporary columns are built in order to fill the gap between the
columns and the ground. In such a way during the construction sequence the
columns behave as normal compressed column. All the structure is built as a
steel only structure, and in place of the cores, only the embedded columns are
placed. This configuration is kept until the steel structure reaches the roof. At
this point the concrete is poured, always pouring the core wall in between the
floor and then the deck. Once the concrete reaches the roof, the jacking system
on the top of the columns is activated, the load is gradually taken off from the
temporary columns, the deflection is evened out and, once the jacking is
completed, the columns turn into tension only members and the temporary
columns naturally fall from their position. From this time on the structural
system behaves in the way introduced in design.
The foundation system consists of a concrete mat foundation with variable
thickness. Figure 6.10 shows the foundation, with different colors for the
different thicknesses. Under the cores, in red, the thickness is of 6’ (1.82 meters),
whereas on the other side it is smaller and of only 3’ (91 centimeters), shown in
yellow in the figure. This change in thickness is because under the cores the load
intensity is really high, whereas farther from the cores the pressure diminishes
and therefore the thickness of the mat can decrease as well.
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Figure 6.10 Mat Foundation
It is important to note that the optimization process has been carried out while
the project was being developed. When the approach presented hereafter
started, the project was in a Schematic Design phase. This means that the overall
geometry of the building was defined, but the spaces, loads and structural
systems where still subject to changes. The optimization process followed
therefore the different phases in terms of geometry as well.
The structure as previously presented represents the final configuration at the
end of the Detail Design phase, when the engineering was brought to detail
precision, the structural systems were defined and the architectural spaces
where mostly defined.
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6.3 Optimization process
The object of the optimization will be the roof truss. As previously explained, the
loads coming from the floors are forced to go through the columns, reach the
roof and be transferred to the cores.
Since it is a very critical part of the building in terms of security of the structure
and of weight of the roof-truss itself, the optimization process aims at improving
both those characteristics.
The steps of the optimization procedure, recalled in the General Aspects, have
been applied and will be presented in the following.
6.3.1 Global Topology Optimization
Modeling the roof truss sets the first step of optimization. The aim of this phase
is to get a general idea of the preferred path for transferring the columns’ loads.
For this reason it has been chosen to model the roof truss as a cube of elastic
material (defined with the properties of structural steel). None of the elements
has been modeled in order to be the least biased during this optimization.
The cube’s side is 228’ (69,49 meters) long with a height of 25’ (7.62 meters).
This dimensions are bigger than the ones presented before, this is because the
side’s length has been changed throughout the different design phases. The big
atrium opening of the building has been modeled and is sized 120’x52’ (36.58 x
15.85 meters).
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Figure 6.11 Isometric view of the model
The software used for this step is really intuitive that allows the user to build a
3D model and optimize it in a quick way, given the constraints, the loads, the
amount of material required at the end of the process and the maximum
dimension of the mesh elements. The mesh characteristics cannot be defined.
The supports to be modeled are the four central c-shaped cores together with a
column in the middle of the core that acts as a support for the truss as well, and
the four shear walls. In this phase those have been modeled as 3D elements
defined as non-design space. In such a way the software assigns infinite stiffness
to those elements and therefore the loads are attracted towards those
members. On the bottom of the cores, modeled with a height of 3’ (91 cm), x,y
and z pin supports have been defined in order to prevent the cores and shear
walls from moving in any of the three directions.
The columns instead have been modeled as point loads, applied where the
columns meet the roof metal deck. For this stage of the optimization it has been
chosen not to apply the real loads coming from the building, but to use a
reasonable approximation of them. This has been done keeping in mind that this
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step is only to give a general idea of the load path, it was therefore useless to
calculate the real loads. To strengthen this choice it has been considered also
that at this phase of the project, the loads distributed throughout the building
where still subject to a lot of changing due to architectural reasons.
The loads applied to this model are all of the same intensity, which has been
chosen equal to 100 kip (444.82 kN). It is important to notice that, since the
loads have been taken all of the same intensity, it is not important how big they
are.
Figure 6.12 Loads and Supports
The residual amount of material imposed is equal to the 25% of the starting
volume. The maximum size for the mesh elements is 1 ft (30 cm). Given the
dimensions of this model, the parameters have been chosen in order to allow
the software to work flawless. Even if the 25% of material at the end of the
process is a really big amount, it was not possible to run the software with a
smaller percentage. This has not been considered a problem. In fact, given the
aim of this step, the final volume is not a big objection to the value of the study.
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The solution given by Inspire is showed hereafter (Figure 6.13). It is really
important to have a close look to this figure, because the result underlines the
importance of this approach, especially in a schematic design phase in which
changes are still possible.
Figure 6.13 Optimized shape
The solution represents a load path that is completely different from the roof
truss scheme of a series of Warren trusses. As can be seen there is a big amount
of material on the top of the structure, that creates a tension loop, links together
all the diagonals and at the same time spreads the larger stresses towards less
utilized material.
The way in which the loads are transfer to the cores is through tension diagonals
that take the point load, bring it to the top chord. The load is then brought down
through compression diagonals, which create a triangle together with the
tension members on the front. Those triangles are not parallel though but are all
directed towards the corner of the cores. This is to satisfy the principle of the
shortest load path. The solution therefore prefers the cores, while completely
avoids the shear walls. This solution is reasonable, in fact the shear walls have no
out of plane stiffness and therefore avoiding them allows saving a lot amount of
material.
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Finally, the most unpredictable result is that the top chord is not going across the
structure to connect with its equal and opposite. The four sides of the loop are
therefore not coupled. This solution could look awkward at first sight. If a deeper
reasoning is taken it is clear that the principle that brought to create a structure
that crosses the space from side to side is the one of coupling the opposite. This
same idea is kept in the optimized solution. This time though instead of crossing
the entire length of the cube, which represent a long path, the opposites are
coupled on the corners, creating savings in terms of material because the length
of the members, and therefore of the load path, is shorter.
6.3.2 Refined Topology Optimization
Staring from the interesting results of the global topology optimization, the
second step is taken. This time more sophisticated software is used for the
optimization, which takes the geometry from a finite element mesh generator.
The geometry is imported from the previous analysis.
Figure 6.14 Plan view
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At the beginning of this step, some architectural changes have been produced.
The model has been changed accordingly to those mutations in the geometry.
The new design space is a cube with the same dimensions of before, but with a
height of 18’-6” (5.6 meters). Under the structural point of view, embedded
columns are inserted inside the core walls to help the stiffness and the
construction sequence. Consequently the constraints in the new model have
been changed.
Figure 6.15 Elevation
As Figure 6.16 shows, the constraints are not anymore 3D elements defined as
non-design space, but only point x-y-z pin supports. Those pins are located in the
same position where the embedded columns will be. This change is the
consequence of a deeper thought regarding the constructability of the structure.
In fact, once the final optimal geometry will be defined, the elements that will
create the truss will be steel elements. This means that a link will be created
between the cores and the truss, and that this link will most likely be between
the steel embedded columns and the truss in order to transfer the loads from
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the truss to the cores in the more secure way possible. For this reason, model
the supports in this new way appeared to be more realistic.
In Figure 6.16 the pins are represented in red, whereas in black the shape of the
cores is underlined.
Figure 6.16 Plan bottom view
Also the load intensity has been changed in this step, because the new software
allows a more customize optimization process. It is in fact possible to define
constraints that are related to the material while the structure is optimized. The
material has been defined as steel and it has been imposed a yielding limit of 50
ksi (344.7 MPa) for each element.
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In order to let this constraint be useful, the load intensity had to be increased in
order to be closer to the real load acting on the truss. The loads are increased
from 100 kip (444.82 kN) to 1000 kip (4448.2 kN).
The objective of the optimization is to minimize the compliance, which
corresponds to maximize the stiffness of the roof truss. In order to have, at the
end of the process, members that can be brought back to steel elements, a
maximum member size of 1 ft (30 cm) is imposed.
Furthermore, a maximum amount of material at the end of the process equal to
5% of the starting cube is imposed. This 5% is equal to 10000 tons of the imperial
unit (9000 tons in the international system) that is a lot higher than the aimed
weight. Also during this step, the percentage has been chosen because of the
limits of the software.
Since the software used for this phase carries out the optimization through the
SIMP methodology, a penalization factor of 3 has been used. In literature [9] [6],
it has been proved that for steel 3D problems, the factor of 3 is the one that
penalizes the densities without problems of checkerboard.
Results (Figure 6.17) confirm key load patterns observed in the global topology
optimization step are viable. The loads are carried to the supports through
material configures in equilateral triangles, with two members in compression
and one in tension. A large tension member on the top chord is kept in order to
keep the equilibrium of the structure and prevent the triangle to overturn. Again,
the top chord is not going across the opening but is diagonally reaching across
the cores, in order to take the shortest path possible for coupling the loads.
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Figure 6.17 Refined optimization results
6.3.1 Interpretation of the optimization
A crucial step of the optimization is the one that brings from a continuum
structure as the one seen before, to a discrete one.
Looking at Figure 6.17 it can be clearly seen the load path and, thanks to the
maximum dimension of 1 ft imposed during the optimization, the members are
not difficult to be discretized.
A first step for discretizing the optimized shape is to identify and underline,
through lines, the load path. Figure 6.18 shows this process.
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Figure 6.18 Topology optimization results and truss interpretation
In this figure is clearer what stated in the previous paragraph. Looking only at
half of the structure, 14 equilateral triangles can be recognized. All of those
triangles are made of one tension and two compression members (Figure 6.19).
The triangles are linked together with a top chord of tension members. It is
interesting to notice that the top chord is not a typical belt truss running through
the entire perimeter. Members that diagonally link the two center triangles, the
ones at the shear walls location, instead make it. In the corners, a higher
concentration of triangles is due to the attempt of the truss to solve the load
path directly on the cores. Furthermore, the top chord is not continuous along
the perimeter. In fact the optimized shape attempts to divide into two
symmetrical halves the space. The reason for this is that the big atrium develops
itself mostly along one direction, leaving a limited design space for evolving the
diagonals on the other dimension.
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Figure 6.19 Detail of the tension and compression members
The interpretation of the results has the aim of creating a discrete truss that can
be actually built. After this first step of tracing the path with lines, it is important
to take a further step and try to look at those lines under this new point of view.
For this reason, a model is built in software specifically used for the analysis of
structures.
This move brings the study into a specific structural engineering field.
Starting from Figure 6.18, four different possible interpretations of the results
are developed and designed into the structural analysis software. As shown in
Figure 6.21, the different options are the attempt to replicate, as close as
possible, the scheme coming from the discretization previously done. Additional
members were added to the initial truss to ensure stability of the structure.
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Figure 6.20 Four options, 3D view
Figure 6.21 Four options, plan views
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Figure 6.22 Typical load path scheme
Since the analysis has now turned into a structural engineering one, the
members are not anymore undefined amount of steel, but need to have a
specific shape. For now, all the members have been assigned the same shape
properties, equivalent to a W14x342 of the AISC manual [34] for steel
constructions.
A finite element analysis is run with a load applied at the columns equal to 1000
kip, in order to choose the configuration that better performs among the four
identified.
Once the most reasonable layout has been chosen, a model has been built.
(Figure 6.24)
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Figure 6.23 Chosen set up, 3D view
Figure 6.24 Chosen set up, plan views
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Figure 6.25 Typical load path scheme
6.3.2 AISC specifications for structural steel
The design of steel members is conducted as per Specifications for Structural
Steel Buildings in AISC that apply for the erection of steel systems with elements
defined in the AISC Code of Standard Practice for Steel Buildings and Bridges.
General design requirements for structural design are defined in Chapter B of
Specifications for Structural Steel Buildings; in particular, strength design for steel
elements can be determined according with two different methods: Load and
Resistance Factor Design (LRFD) or Allowable Strength Design (ASD).
The first difference between ASD and LRFD, historically, has been that the first
method compares actual and allowable stresses while LRFD required strength
with actual. The difference is not significant since it can be covered multiplying or
dividing both sides of the limit states inequalities by section properties.
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The second major difference is the manner in which the relationship between
applied loads and member capacities are handled. The LRFD specification
accounts separately for the predictability of applied loads and for material and
construction variabilities. The first is introduced through load factors applied to
the required strength, the second through resistance factors on the nominal
strength. The ASD specification combines the two factors into a single one of
safety. By breaking the factor of safety apart into independent load and
resistance factors (as done in the LRFD approach) a more consistent effective
factor of safety is obtained and can result in safer or lighter structures,
depending on the predictability of the load types being used. Therefore, different
modifications factor are applied in the two methods.
In the design of tension and compression members, the current work referenced
to the Load and Resistance Factor Design (LRFD) where design shall be
performed in accordance with equation B3-1 of AISC:
eq 6.1
where
( )
As far as the design is concerned, the references are Chapter D and E of
Specifications for Structural Steel Buildings tension and compression members
respectively.
For members subject to axial tension caused by static force, the design follows
the same equation, independently from the shape of the member itself. The
design tensile strength, , of tension members shall be the lower value obtain
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according to the limit state of tensile yielding in the gross section and tensile
rupture in the net section.
a. For tensile yielding in the gross section:
eq 6.2
( )
eq 6.3
b. For tensile rupture in the net section:
eq 6.4
( )
eq 6.5
where
Net area of a member is the sum of the products of thickness with the net width
of each element computed taking into account bolt holes and connections as
reported in section D3. The effective net area is also modified with the
parameter U, called shear leg factor.
The design for compressive strength, , is determined as explain hereafter.
The nominal compressive strength, , shall be the lowest value obtained
according to the limit states of flexural buckling, torsional buckling and flexural-
torsional buckling.
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a. For doubly symmetrical and singly symmetrical members the limit state
of flexural buckling is applicable;
b. For singly symmetrical and unsymmetrical members the limit states of
torsional and flexural-torsional buckling are also applicable.
( )
eq 6.6
The effective length factor, K, for calculation of column slenderness, KL/r,
shall be determined in accordance with Chapter C,
where
The compressive strength for flexural buckling of members without slender
element depends on section properties. For compact and non-compact sections,
as defined in Section B4, the nominal compressive strength shall be
determined based on the limit state of flexural buckling.
eq 6.7
The flexural buckling stress, Fcr, is determined as follows:
[
] when
√
when
√
eq 6.8
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where Fe is the elastic buckling stress determined according to:
( )
eq 6.9
As mentioned before, sections are defined in the AISC Code of Standard Practice
for Steel Buildings and Bridges.
In particular, for the studies W- shapes are applied; these sections are
characterized by parallel inner and outer flange surfaces. All these shapes are
designated by the mark W, the nominal depth (in.) and nominal weight (lb/ft).
For example, W24x55 is a W-shape that is nominally 24 inches deep and weight
55 lb/ft.
6.3.3 Load combinations
In order to design safe and secure structures, it is responsibility of the structural
engineer to predict the magnitude of the various loads that are likely to be
applied to the structure over its lifetime. It must be also taken into account for
the probability of the simultaneous application of the various load types.
In order to give consistency to this prediction has been adopted standards for
the loads and their probable combination that must be used in the design. The
standard are stated in the ASCE 7-10, Minimum Design Loads for Building and
Other Structures [14].
The principle load types are six and will briefly explained hereafter:
Dead Load, D: it includes the weight of all items that are attached to the
structure and are likely to remain in the as-built location throughout the
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life of the structure. Beams, columns, floor slabs, exterior walls, roofs,
mechanical equipment, and the like are all considered being dead load on
a structure.
Live Load, L: includes anything that can possibly be moved in or out the
structure over the course of its life. This includes people, furniture,
equipment and other similar items. Predicting the live load it’s highly
dependent on the structure’s occupancy. The occupancy of a structure
may vary over its lifetime; therefore reasonable assumptions about its
future must be made. Also, different part of a structure may have
different occupancy. For this reason, different live load is assigned to
each part of the structure and is connected with the usage that is
architecturally predicted for each space. The code requires that live load
is to be placed for maximum effect, generally this means that multiple
load cases need to be solved in order to find the envelope of required
strength values needed in order to design a safe structure.
Roof Live Load, Lf: it is generally associated with the loads that the roof
structure will see during construction and later during maintenance.
These loads are of short duration and generally much smaller than
normal live loads.
Snow Loads, S: it is considered to be everywhere present at a given time.
It is highly depended on the location of the structure.
Wind Load, W: this type of load is a very dynamic event for which static
approximations can be made. The approximate methods for determining
wind load are generally considered to be conservative for a given
predicted wind speed.
Earthquake Load (or Seismic Load), E: this is a very dynamic event. For
certain types of structures a static equivalent method may be used to
estimate the forces applied to the structure. For more complex structures
numerical methods that solve the dynamic problem must be used.
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Specific prescriptions for the seismic determination of loads are explained
in chapters 11 and 12 of the code.
The way in which the loads are combined depends on the approach to follow.
The two contemplated approaches are strength design and allowable stress
design.
Combinations for Strength Design:
The code states that: “structures, components, and foundations shall be
designed so that their design strength equals or exceeds the effects of the
factored loads in the following combinations”:
1. 1.4D
2. 1.4D + 1.6L + 0.5(Lr or S or R)
3. 1.2D + 1.6(Lr or S or R) + (L or 0,5W)
4. 1.2D + 1.0W + L + 0.5(Lr or S or R)
5. 1.2D + 1.0E + L + 0.2S
6. 0.9D + 1.0W
7. 0.9D + 1.0E
Combinations for Allowable Stress Design
The code introducing this type of combinations with the following sentences:
“Loads listed herein shall be considered to act in the following combinations;
whichever produces the most unfavorable effect in the building, foundation, or
structural member being considered. Effects of one or more loads not acting
shall be considered.”
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1. D
2. D + L
3. D + (Lr or S or R)
4. D + 0.75L + 0.75(Lr or S or R)
5. D + (0.6W or 0.7E)
6. D + 0.75L + 0.75(0.6W) + 0.75(Lr or S or R) / D + 0.75L + 0.75(0.7E) + 0.75S
7. 0.6D + 0.6W
8. 0.6D + 0.7E
More comprehensive combinations can be used, if specific proofs of their
reliability are provide.
6.3.4 Member size optimization
Once the configuration is chosen, a specific size is assigned to each element. For
this study the choice of the sections was limited between elements from W14x90
till W14x665. (Table 6.1)
The limitation was chosen looking at the member sizes of the steel elements
throughout the entire project. Limiting the sections in such a way helped also in
the optimization process, giving a smaller space where to search for the optimal
solution.
SectionName Material t3 t2 tf tw t2b tfb Area Weight
Text Text in in in in in in in2 lb/ft2
W14X90 A992Fy50 14 14.5 0.71 0.44 14.5 0.71 26.5 90
W14X99 A992Fy50 14.2 14.6 0.78 0.485 14.6 0.78 29.1 99
W14X109 A992Fy50 14.3 14.6 0.86 0.525 14.6 0.86 32 109
W14X120 A992Fy50 14.5 14.7 0.94 0.59 14.7 0.94 35.3 120
W14X132 A992Fy50 14.7 14.7 1.03 0.645 14.7 1.03 38.8 132
W14X145 A992Fy50 14.8 15.5 1.09 0.68 15.5 1.09 42.7 145
W14X159 A992Fy50 15 15.6 1.19 0.745 15.6 1.19 46.7 159
W14X176 A992Fy50 15.2 15.7 1.31 0.83 15.7 1.31 51.8 176
W14X193 A992Fy50 15.5 15.7 1.44 0.89 15.7 1.44 56.8 193
W14X211 A992Fy50 15.7 15.8 1.56 0.98 15.8 1.56 62 211
W14X233 A992Fy50 16 15.9 1.72 1.07 15.9 1.72 68.5 233
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114
SectionName Material t3 t2 tf tw t2b tfb Area Weight
Text Text in in in in in in in2 lb/ft2
W14X257 A992Fy50 16.4 16 1.89 1.18 16 1.89 75.6 257
W14X283 A992Fy50 16.7 16.1 2.07 1.29 16.1 2.07 83.3 283
W14X311 A992Fy50 17.1 16.2 2.26 1.41 16.2 2.26 91.4 311
W14X342 A992Fy50 17.5 16.4 2.47 1.54 16.4 2.47 101 342
W14X370 A992Fy50 17.9 16.5 2.66 1.66 16.5 2.66 109 370
W14X398 A992Fy50 18.3 16.6 2.85 1.77 16.6 2.85 117 398
W14X426 A992Fy50 18.7 16.7 3.04 1.88 16.7 3.04 125 426
W14X455 A992Fy50 19 16.8 3.21 2.02 16.8 3.21 134 455
W14X500 A992Fy50 19.6 17 3.5 2.19 17 3.5 147 500
W14X550 A992Fy50 20.2 17.2 3.82 2.38 17.2 3.82 162 550
W14X605 A992Fy50 20.9 17.4 4.16 2.6 17.4 4.16 178 605
W14X665 A992Fy50 21.6 17.7 4.52 2.83 17.7 4.52 196 665
W14X730 A992Fy50 22.4 17.9 4.91 3.07 17.9 4.91 215 730
Table 6.1 Steel sections
The first part of this process was developed manually. This phase is important in
order to understand the relation between the members, the different influence
of the elements on the load path, on the stiffness and on the strength of the
truss.
Having a deep understanding of the behavior of the structure is key before
getting into the automatic process.
Starting from the density of material saw in Figure 6.17, a different member size
for each element of Figure 6.23 was chosen. The structure obtained can be seen
in Figure 6.26.
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Figure 6.26 Estimated member sizes, 3D view
In order to correctly choose the sections of the truss’ members, the real loads
need to be applied.
The AISC manual [34] requires two different requirements to be satisfied for
steel members: one is for strength and one for stiffness.
The intensity of the loads was taken from a different structural software where
the entire building was modeled and the load combinations where added
accordingly with the ASCE code [14] chapter 2. (see 6.3.3 )
For the strength check, the load combination taken into account was:
eq 6.10
Where D is the dead load, L the live load and LR is the roof live load.
As Figure 6.27 shows, since the building is not completely symmetrical in the
distribution of the spaces throughout the floors, the loads are not all of the same
intensity.
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Under this load combination, the code requires the ratio between capacity and
required strength to be less than one for each member. The capacity has to be
determined correspondingly with chapter D of part 16 in the AISC code [34] for
members subject to a tension force and chapter E of part 16 in the AISC code
[34] for the ones subject to compression force. (see 6.3.2 )
Figure 6.27 Load intensity for strenght combination (kip)
A different load combination was used for the stiffness check:
eq 6.11
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it can be immediately noticed that the intensity of the load for this combination
is much smaller than the one before.
Figure 6.28 Load intensity for stiffness control (kip)
The code requires, under this combination, the deflection criteria to be satisfied.
In this study, the constraint on the deflection were more strict than the ones
coming from the code in order to control the slope of each floor hanging from
the roof. The goal deflection was of 1,5” (3.8 cm) uniform along the perimeter.
The criteria for changing the sizes were not only the satisfaction of the code
requirements. In addition to the deflection, the weight was aimed not to exceed
the 650 tons (589x103 kg) limit.
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This process was an iterative one. In fact every time the size of an element is
modified, the relatively stiffness of adjacent element changed, bringing to a
different distribution of forces between the elements.
This process brought to further alteration to the structure, and a definitive
geometry is stated, together with element size for each member of the truss.
Figure 6.29 shows the final geometry.
Figure 6.29 Final geometry, 3D view
The checks done during this phase are shown in Appendix A.
At the end of this process the total weight of the truss is of 600 tons (544x103
kg), with a typical deflection of 0.86” (2 cm) and a maximum deflection of 1.1”
(2.7 cm) in the corners. The element size varies from a maximum of W14x664 to
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a minimum of W14x90. The diagram of the deflections for the stiffness check is
shown in Figure 6.30.
Figure 6.30 Deflection diagram
With this understanding of the behavior of the new configuration, a further step
has been taken. The member size optimization process has been automatized.
The algorithm chosen for this phase is the GRSM method (see chapter “
Discrete Optimization”) due to the huge dimensions of this problem. The truss is
in fact made of 300 members, each of which can vary between 28 different
shapes (W14x60 till W14x730). As it can be seen, the difference between the
bigger and the smaller section used is larger than before, this is in order to
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accommodate all the constraints in a quicker way. The dimension of this problem
is 30300.
The optimization objective was set as minimizing the weight of the truss. Three
constraints were given to the optimization field: a maximum deflection less than
¾” (1.9 cm), a differential deflection less than 0.2” (0.5 cm) and the forces in
each elements must submit to the limits of the AISC explained in the previous
paragraph.
In order to cut down the dimensions of the problem and speed up the
optimization process, some symmetry constraints have been imposed. Firstly,
even though the problem is not completely symmetrical, has been chosen to
divide the structure into four quarters, each of them containing one corner.
Furthermore, inside each quarter the elements have been divided into groups,
using the engineering judgment and the knowledge acquired during the manual
process.
This simplification of the problem, even if bias the problem toward symmetrical
solutions, has been considered a judicious compromise taking into account not
only the dimensions of the problem, but also the detail design cost of each
member and the constructability of the structure on the site. In terms of
dimension of the problem, by taking two axes of symmetry, the dimension of the
problem is not only four times smaller, it is indeed 3075 which is much smaller
than 30300.
Once the problem was completely set, the process was started. Even though the
constraints are really strict, the algorithm proved to be consistent and a solution
was found. The convergence curves are shown in Figure 6.31. In particular, it is
interesting to notice that the mass takes some iteration to start dropping. Once
started to drop, it decreases in a really fast way, until it stabilizes itself in a
plateau area going towards the end of the process. This curve is similar to the
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convergence curves found in literature. The maximum displacement curve and
the differential deflection (DeltaDeflection) one present the same trend of the
mass, whereas the minimum deflection increases in order to minimize the
differential and meet the constraints.
The final configuration had a total weight of 580 tons (526x103 kg), with a typical
deflection of 0.7” (1.8 cm) and a maximum differential deflection of 0.19” (0.48
cm).
Considering then that this optimization process was successful and no substantial
improvement could be made with process, the structure was imported into the
complete model of the building. This showed some changes into the responses
of the structure to the loads. Those changes are to be imputed to the general
behavior of the entire structure, which takes into account rotation and deflection
of the walls due to concrete creep, other than additional loads and elements
added to the truss configuration due to the metal deck at the top of the roof.
Chapter 6
Figure 6.31 GRSM convergence curves
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6.3.5 Consolidating the methodology
While this study was carried on, the project moved into the detail design phase.
Meanwhile, some architectural modifications happened to the building, among
those the more significant were a decrease in the plan dimension and a switch in
the occupancy of the roof floor. This was firstly though as a space not accessible
to the people unless for structural checks on the truss itself. Due to the smaller
space throughout the building, it has been chosen to use the roof floor for
arrange the mechanical equipment and therefore benefit of more space for the
court on the 9th floor.
The alteration influenced the truss not only in the design space, but also adding
some architectural constraints to the space occupied by the members of the
truss.
If the decrease of the space did not really influence the configuration of the
truss, the occupancy of the space brought to the requirement of guarantee a
corridor along the perimeter in order to access the equipment. Figure 6.32 shows
the elevation of the roof truss. The triangle typical of the optimized configuration
can be seen in the cantilever part. The violation of the constraint is identified by
the black rectangle, which represent the corridor space that need to be kept
free.
This situation gave the possibility of consolidating the procedure previously
shown, by applying it to the new configuration.
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Figure 6.32 Constraint violation, elevation detail
For this second part of the optimization, it has been chosen to develop a 2D
study, focusing only on the cantilever part of the truss. It has been observed that
the most interesting part of the optimized solution is in fact the cantilever part
with the triangular solution of the load path. As far as the rest of the truss is
concerned, it has been decided to keep it faithful to the initial configuration,
consisting in a typical series of parallel modula crossing orthogonally and creating
a grid along the space (Figure 6.33 and Figure 6.35 show top and bottom chord,
in blue the grid lines of the modulus can be seen). A top and a bottom chord
define the modula linking the opposite sides of the cube, the diagonals between
the chords have been defined following the scheme of a typical Warren truss.
(Figure 6.35)
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Figure 6.33 Top chord - plan view
Figure 6.34 Bottom chord - plan view
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126
Figure 6.35 Typical modulus of the truss
Since the problem was turned into a 2D problem, it has been chosen to analyze a
space that can be defined as typical. For this reason, the corridor non-design
space has been introduced both in the cantilever part and in the inside, because
its position may vary accordingly to the necessity. It can be seen in Figure 6.32.
This typical space does not represent the situation of the corners. It has been
decided though to take out the corner column, giving a bigger architectural space
on each floor and simplifying the load path on the roof truss. The corner in fact
were the most problematic part in terms of deflection, dimension of members
and connection between them.
In Figure 6.36 the 2D model is shown. In particular it is important to point out a
substantial difference between the previous optimizations and this model: the
column load, brought back to 1000 kip, coherently with the generalization of the
typical cantilever, has been applied to the top of the design space and not to the
bottom.
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Figure 6.36 2D model
This comes from a study of the load path. In fact the triangular shape is the
consequence of Maxwell’s theorem of minimum load path (chapter “
Analytical Interpretations”). If the opening is introduced, the triangle is not
possible anymore. Maxwell’s theorem underlines the importance of having equal
length in compression and tension in order to optimize the load path. For doing
that, if the load is kept on the bottom, the bigger amount of force passes through
tension members that need therefore to be shorter for stability problems.
Furthermore, it is difficult to solve the compression member without interfering
with the opening. Whereas, if the load is brought to the top of the design space,
by only extending the column all the way up to the top, the higher force is solved
in compression, through members that can be longer and decreasing the number
of members in tension.
Furthermore, the design space has been increased of 3 ft (0.76 cm) under the
floor level. This is in order to solve the loads and at the same time avoid the
opening.
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The design space is the entire cantilever, but the opening that has been defined
as non-design space. The objective of the process has been kept the same of the
previous step: minimizing the compliance while the constraints are 5% of
material left and yielding limit of steel. The penalization factor for the SIMP has
been assigned equal to 2, since the problem is dimensional this time.
Figure 6.37 2D optimization result
Figure 6.38 Interpretation of the optimization result
The interpretation of the results shows a shape that recalls what shown by
Michell [21]. – shows this parallelism.
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Figure 6.39 Optimized shape and Michell solution
The forces pointed out in Michell’s truss are the same of the scheme object of
study, in fact if the reactions at the supports on the top of the design space are
shown the behavior is the same of Michell’s.
Figure 6.39 is significant because gives a theoretical background to the
optimization solution coming from the automatize process. Since Michell’s truss
is a continuum solution, it is different from the optimized solution, but gives an
important benchmark in order to understand the optimality of the truss
configuration chosen.
Having understood the optimal shape, the problem moved to finding the optimal
position of points A and B of Figure 6.40. Despite point B can be taken as fixed,
due to the geometrical constraints of floor level and position of the embedded
column inside the wall, point A still has two degrees of freedom (x and z
direction’s displacement).
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Figure 6.40 2D optimized shape - points A and B
In order to solve this new problem, it has been implemented a function. The
objective has been kept the same, minimizing the weight of the configuration.
Since this time no software has been used for the optimization of the position of
the point, a quick way of calculating the weight of the structure had to be found.
For solving this issue, Maxwell’s theorem has been used (see chapter
Analytical Interpretations). The formula for this calculation is recalled in –
hereafter.
∑ ∑
eq 6.12
The optimized shape has been modeled as an isostatic frame constrained with a
a hinge in point D and E; the frame is composed by seven elements connected
each other through rotational hinges.
The load F, equal to 1000 kip, is applied in point B.
A B
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Figure 6.41 Idealized problem: isostatic frame
POINT X COORDINATE [ft - in] Z COORDINATE [ft - in]
A x Z
B 45’- 0” 24’- 0”
C 18’- 0” 24’- 0”
D 0’- 0” 6’- 10”
E 18’- 0” 0’- 0”
Table 6.2 Point coordinates
The analysis of the structure has been manually done, keeping the coordinates of
point A in the literal form. Once the force in each element and the length of
them was calculated, a function was used for calculating the optimum position.
This function was implemented in order to calculate the Maxwell’s volume of
each possible position and return the minimum volume of the configuration.
For having a benchmark, the theoretical minimum volume possible was
evaluated through Michell’s equation (chapter
Analytical Interpretations).
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(
)
eq 6.13
Where F is the force, a is the distance BC, P is the greater tensile stress allowable
and Q is the greater compressive stress allowable. For this analysis, P and Q have
been taken equal even though this assumption is not true due to stability
problems for compression members.
A first search of the optimum position has been run with no constraint to the
position of the point A. This is actually the optimal position and it is only 0.3%
less efficient than Michell’s.
This solution does not meet the constructability constraints. In fact, since the
design space has been increased, point E lies under the floor level. If point A lies
where this optimization suggested, element AE needs to pass through the floor
deck, increasing the difficulty of production. Furthermore, having A in not on the
floor level, creates problem of out-of-plane stability, introducing the need of an
additional belt truss at the level of A.
For this reason, a second optimization has been run, this time optimizing only the
x coordinate of point A, while constraining the z coordinate to the same height of
the floor level. This solution is a little more further from the benchmark. The
benefits in terms of constructability are such that the decrease of efficiency does
not represent an issue.
Finally, a third optimization was run. This time the z coordinate has been kept at
the floor level, but a different stress capacity for tension and compression has
been run. The solution is 7% off from Michell’s. This percentage is not reliable
though because Michell’s solution has been calculated with the same capacity
both in tension and compression.
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Case Volume Difference from Michell’s
[%]
Michell 23.562
Unconstraint 23.652 0.3
Floor level constraint 23.978 2
Different capacities 25.275 7
Table 6.3 Difference in volume between optimal solutions
From the three solutions, it has been chosen the last configuration, which takes
into account a more realistic capacity of the members. The position of point A
was therefore takes at (9.18”; 6.75”).
Once the configuration was chosen, it has been modeled and imported into the
entire building model. From the model the section of each member was defined.
It has been chosen not to run an optimization but to assign the section to each
member manually. The criteria for the selection of the member’s section were
the same criteria imposed by the AISC code [34] that have been used in the
previous steps.
Figure 6.42 shows the optimized truss in the typical elevation of the building. At
the end of the process the total weight of the truss is of 650 tons (589x103 kg),
with a net saving of 10% of the total amount of steel compared with the initial
truss. The average deflection is of 1,3” (3,3 cm).
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Figure 6.42 Typical elevation (SOM,2013)
The following figures show the drawings of the 50% Detail Design. Figure 6.43
shows the entire building whereas Figure 6.44 shows the cantilever part of the
truss with the construction details.
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Figure 6.43 Elevation (SOM, 2013)
Figure 6.44 Truss details (SOM, 2013)
CHAPTER 7
Case Study: 111 South Main
7.1 Project Overview
7.1.1 Architectural Description
The 111 South Main is the exclusive office building which is under construction in
Salt Lake City (Utah) (Figure 7.1). The tower is a 24-stories building with a total
gross area of 462,350 gsf; the tower will be 119 m (390’-9’’ ft) high. Below grade
is a single basement level 5.5 m (18 ft) deep.
Figure 7.1 Salt Lake City, Utah
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Figure 7.2 111 South Main (SOM,2013)
The design of the tower has to take into account the performance hall being
currently designed which extends into the footprint of the 111 South Main,
precisely in the Southern portion. Partnering with the Utah Performing Arts
Center, both projects complements each other and will serve as critical elements
in the Salt Lake City’s continued revitalization.
The architectural concept is creating a vibrant office and theatrical experience
both day and night; the two projects will augment and draw on the resources
already in place in the City Creek Development immediately to the north as well
as the Gallivan projects to south. 111 South Main will be a critical resource in
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support of Salt Lake City’s commitment to increased urban density and long-term
sustainability; the building is targeting, in fact, the Gold Level LEED designation,
as per United States Green Building Council specifications.
The performance hall is meant to be extended from ground level up to the 5th
floor of the 111 South Main and it creates a condition where the columns cannot
continue to the foundation system but are to be supported by another
mechanism. The interaction between the two building is described in Figure 7.3
and Figure 7.4:
Figure 7.3 Typical Floor from Ground Level to L4 (left) and Level 05-Parapet (right)
In Figure 7.3 it can be observed the typical floors; in the right picture it is shown
floors from ground level to Level 04 with the 111 South Main area highlighted in
blue. The reduced floor is 39,6m (130 ft) in the East-West direction and 15,2 m
(50 ft) in the other. Left figure represents the typical tower floor (from level 05 to
roof level) with same dimension of the previous in EW direction and 41,6 m (150
ft) in NS direction.
7.1.2 Structural Description
The development of the project is meeting the requirements of the 2012
International Building Code (IBC) [35] and ASCE 7-10 provisions [14].
39.6 m
45.7 m
39.6 m
30.4 m
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140
In the baseline structural scheme the lateral force resisting system consists of
ductile reinforced concrete core shear walls extending from a pile and pile cap
supported foundation to below the penthouse level. RC shear walls and link
beams construction are expected to be 75 cm (30 inches) thick typically. The core
system consists of three shear walls in North-South Direction and two elements
in East-West; the shear walls create a closed space used for elevators and stairs.
The gravity system of the superstructure is using steel framing and composite
metal deck. The gravity system consists of perimeter girders that span between
W14 columns located every 9.15 m (30 ft) and W14-W24 composite beams
spaced typically 3m (10 ft) spanning between the perimeter girders and the
central core. The gravity system is shown in Figure 7.5.
The foundation system will consist of cast-in-place reinforced concrete 25 cm (10
inches) slab-on-grade spanning to grade beams and pile caps. The building
foundation loads will be transferred to the subgrades below the pile cap through
a deep foundation piling system.
Figure 7.4 Cantilever Floor and Conference Hall Detail (SOM, 2013)
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Figure 7.5 Structural Drawings: North-South Elevation (left) and South Façade (right), (SOM, 2013)
In Figure 7.5 it is shown the elevation of the East Façade: steel columns and
beams are creating the gravity system; the steel framing is not moment resisting
and the connection are all considered with moment releases in the structural
analysis. Columns in the South façade are stopped at level 05 and are meant to
avoid interaction with the below building to be constructed. The key issue for the
problem is represented by the collection of gravity loads in the cantilever part of
the building. The baseline structural system is described in Figure 7.6: gravity
loads in the cantilever part of typical tower floors are collected by steel columns
and carried vertically to the steel truss in the roof level. This element is in charge
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of transferring loads to the central core wall and to the columns at the opposite
side of the floor.
Figure 7.6 Baseline Structural Mechanism from ETABS
Therefore, developing this solution other significant issues are to be solved: the
complexity of the truss system supporting gravity loads from the suspended
tributary area related to 21 stories; the high demand on the truss system
creating concerns about the constructability of the system, specifically in the
connections design; significant overturning moment on the foundation system
because of the unbalanced support condition in the North-South direction;
severe serviceability requirements related to the vertical displacement control to
prevent any possible interaction between the tower and the below building.
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Finally, a design issue is represented by the inefficient mass distribution over the
building height with a meaningful jump at Level 05.
It is important to note that the optimization study followed the evolution of the
project starting in the Concept Design and then proceeding in the Schematic
Design; during the process further discussions were developed about the general
mechanism of the building and different solution were investigated: a balanced
structural scheme has been created with all perimeter columns stopped al Level
05 in order to create symmetrical load conditions on the core wall even though
stresses are increased. The two different models were compared; deeper
considerations are presented in the following paragraphs.
7.1.3 Project Site Conditions
The project location is significant for the development of the topology
optimization study; lateral forces, in fact, have high magnitudes because of both
seismic and wind forces. This led to the consideration that the structural system
shall be optimized both for gravity loads and lateral forces.
The project area, in fact, is an high seismic hazard area and this consideration
had significant influence in the study as it will be explained further in the next
paragraphs.
The seismic hazard is taken into account using design values by the U.S.
Geological Survey (USGS) National Seismic Hazard Maps [36]. The archive
displays earthquake ground motions for various probability levels across the
United States and are applied in seismic provisions for building codes, insurance
assessments and other public policy. The resulting maps are derived from seismic
hazard curves calculated on a grid of sites that describes the frequency of
exceeding a set of ground motions. In Figure 7.7 it is shown the peak ground
acceleration with a 2% probability of excess in 50 years; Figure 7.8 is a detailed
map of Utah state where 111 South Main will be located.
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Figure 7.7 Peak Ground Acceleration with 2% in 50 years (USGS)
Figure 7.8 PGA with 2% in 50 years (USGS) – Utah
In particular, seismic loads were defined as per ASCE 7-05 Chapter 11 & 12 [14]
and following geotechnical report. ASCE design parameters based on USGS maps
are reported:
SALT LAKE CITY
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a. Latitude 40.76°
Longitude 111.89°
b. Seismic Site Class D
c. MCE Ground Acceleration at 0.2s Ss = 1.467 g
d. MCE Ground Acceleration at 1s S1 = 0.538 g
e. Site Coefficient Fa = 1
f. Site Coefficient Fv = 1.5
g. Maximum Considered Spectral Response SMS = Fa ∙ Ss = 1.503 g
S1 = Fv ∙ S1 =0.807 g
h. Design Spectral Acceleration Response SDS = 0.978 g
SD1 = 0.538 g
i. Occupancy Category II
j. Seismic Design Category D
k. Superstructure Response Modification Factor
Special Reinforced Concrete Shear Walls R = 5; Ω0 = 2.5; Cd = 5
l. Importance Factor I = 1
In Figure 7.9 and Figure 7.10 response spectra are represented; they are referred
to data previously exposed and are the response spectra implemented in the
finite element model to perform dynamic analysis.
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146
Figure 7.9 Design Response Spectrum (DE)
Figure 7.10 Maximum Considered Earthquake Spectrum (MCE)
The site location is also a medium-high wind speed area; in fact, in accordance
with ASCE 7-10 Chapter 26 the wind coefficients are:
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a. Basic Wind Speed (3 sec gust) 185 km/h (115 mph)
b. Exposure Category B
c. Surface Roughness B
d. Importance Factor I = 1
e. Topographical Factor kzt = 1
Figure 7.11 Basic Wind Speed for Occupancy Category II [mph] as per ASCE 7-10 [14]
The goal of topology optimization study is to define a possible alternative
structural scheme avoiding the use of a roof truss system in order to create a
shorter load path minimizing the use of material in the building. The new
structural system is developed following the architectural progress and facing
with its constraints.
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The starting concept for the system is to define a steel system using perimeter
axial only members directly exposed in the façade. This should create a system
resisting both lateral and gravity loads cooperating with the central core wall
system which will not be removed.
7.2 Refined Topology Optimization
Topology optimization study started with the Refined Topology Optimization;
this choice was taken considering two aspect of the project case: the great
dimension of the problem and the predominance of membrane elements.
Figure 7.12 Optimization Model
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The dimensions of the model representing the problem are reported in Figure
7.13: typical floor is 45.73x39.63m (150x130 ft), the cantilever portion of the
floor is spanning 15.24m (50 ft) out of the building footprint; the total height is
117m. The large scale of the problem does not permit the use of 3D mesh
elements which would requires too large mesh size compromising results
precision. Moreover, the global structural description is more efficient using shell
element because shear-wall, floor and perimeter system are 2D elements since
two dimensions are significantly greater than the elements thickness.
Figure 7.13 Problem Dimensions
The geometry of the building is taken from the structural baseline making
reference to the middle line of each element as reported in the Figure 7.12 and
Figure 7.13; the roof level is not modeled since the truss is meant to be removed
but loads are applied to the floor slab because the level is occupied by
mechanical system.
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150
Furthermore, it has been very important to have a correct description of the
stiffness of each member in order to capture the real interaction between each
structural element which influences the load path. The slab thickness is defined
from the value in the baseline scheme which was using steel metal deck filled
with concrete and steel beams (Figure 7.14); in the model for the optimization
analysis it has been used a 15 cm (6 inches) slab.
Figure 7.14 Composite Metal Deck and Steel Beams
The shear walls were modeled as shell elements with a 60 cm (24 inches)
thickness. Moreover, reinforced concrete properties were assigned both to shear
walls and floor slabs:
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REINFORCED CONCRETE
E 25000 Mpa 3605000 psi
υ 0.2
f'c 27.6 Mpa 4000 psi
Table 7.1 Concrete properties
On the other hand, the façade enclosures, where the perimeter system is to be
defined, are modeled as 30 cm (12 inches) because the system is meant to use
W14 elements. Steel properties are shown in Table 7.2
STEEL
E 345 Mpa 29000000 psi
υ 0.3
fy 27.6 Mpa 50000 psi
Table 7.2 Steel properties
7.2.1 Boundary Conditions
The next step for the topology optimization requires the definition of the
boundary conditions for the design domain.
The supports are applied both to shear walls and perimeter shell elements as line
simple supports releasing bending moment and fixing displacements in all
directions. The supports are represented in the following figure with green
triangular shapes:
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Figure 7.15 Finite Element Analysis Model for Topology Optimization
7.2.2 Static Load Cases
Two different load cases were defined for the optimization, gravity and lateral
seismic loads.
First, the analysis is conducted assigning the minimum Live Loads and
Superimposed Dead Loads expected by the intended use or occupancy, in
accordance with IBC 2012 [35] and ASCE 7-10 [14]. The Dead Load are estimated
with hand calculation in accordance with typical members and expected
concrete to be used.
Typical gravity loads applied are:
- Dead Loads (DD) 2.11 kN/m2 44 psf
- Superimposed Dead Load (SDL) 1.91 kN/m2 40 psf
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- Live Load (LL) 3.83 kN/m2 80 psf
Total 7.85 kN/m2 164 psf
The above magnitude for gravity loads are calculated for the typical tower floor
and applied as a constant to each floor not taking into account the possible
presence of reducible live loads in order to obtain a uniform force distribution
along the building height. This choice is taken considering that at this step of the
study the magnitude of loads is to be considered not precise because no code
check are to be conducted. Moreover, gravity loads are applied as surface loads
in the negative z-direction to the shell elements modeling the tower floors.
Another important load case to be considered is the horizontal force. For this
reason, the second load case considered is the static equivalent seismic force;
the earthquake force is applied to the structure only in its horizontal contribution
in two perpendicular directions, instead the vertical acceleration is not taken into
account. The contribution from wind forces was not considered for two reasons:
it has a lower effect on the structure compared to the seismic forces and the
need not to complicate the model behavior under lateral loads.
The magnitude of the seismic forces is taken from the ETABS model of the
baseline structure and values are reported for each floor in Table 7.3:
STATIC SEISMIC FORCE
E-W STORY FORCE N-S STORY FORCE
kN/m2 psf kN/m2 psf
PARAPET 0.53 11.0 0.48 10.1
ROOF 1.49 31.1 1.36 28.4
L24 0.81 17.0 0.74 15.5
L23 0.75 15.7 0.69 14.4
L22 0.70 14.6 0.64 13.3
L21 0.64 13.4 0.58 12.2
L20 0.59 12.3 0.54 11.2
L19 0.54 11.2 0.49 10.2
L18 0.49 10.2 0.45 9.3
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STATIC SEISMIC FORCE
L17 0.44 9.2 0.40 8.4
L16 0.39 8.2 0.36 7.5
L15 0.35 7.4 0.33 6.8
L14 0.32 6.6 0.28 5.9
L13 0.28 5.8 0.25 5.3
L12 0.24 5.1 0.22 4.6
L11 0.21 4.4 0.19 4.0
L10 0.18 3.8 0.16 3.4
L09 0.15 3.2 0.14 2.9
L08 0.13 2.7 0.12 2.5
L07 0.10 2.1 0.09 1.8
L06 0.08 1.7 0.08 1.6
L05 0.07 1.4 0.06 1.3
L04 0.07 1.4 0.06 1.3
L03 0.05 1.0 0.04 0.9
L02 0.07 1.4 0.06 1.3
L01 0.00 0.1 0.00 0.1
Table 7.3 Seismic Static Force
Static seismic forces are applied in x and y direction as uniform surface loads
because shell elements are rigid in the middle line so that it is possible to avoid
the determination of the center of mass to apply a concentrated load for each
floor.
7.2.3 Topology Optimization Parameters Set-Up
The optimization is conducted fixing parameters as explained in the previous
chapter “Optimization Theory”.
First, the façade enclosures are defined as Design Domain; on the contrary, floor
slab and core wall system is defined as Non-Design Space as it has been
explained in the Optimization Theory chapter. Therefore, floors and shear walls
will not be affected by material redistribution.
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Figure 7.16 Definition of Design Domain and Non-Design Space
The finite element mesh is defined as 2D quadrangular elements with 15 cm (0.5
ft) size.
Then, three constraints were fixed for the analysis:
- Fraction of Volume Left: it has been defined the amount of material to be
left in the final layout of the process; this threshold is expressed as a
percentage of the initial design domain volume which is initially set at
30% of the façade. This values was fixed from architectural constraints
with the use of literature results [1];
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156
- Stress Constraint: the maximum allowable stress in the final layout is
fixed at fy;
- Maximum Member Size: the maximum size of each continuous member is
set at 30 cm (1 ft);
The penalization method for the Solid Isotropic Method with Penalization (SIMP,
see Chapter Optimization Theory) is fixed at 3, as reported in literature [9].
The convergence tolerance for the analysis is fixed to 10-5: the optimization
process stops when difference between results in two consecutive steps is
smaller than this threshold. This parameter is very important to be fixed because
the software has a maximum allowable number of step and so the results report
shows if convergence is reached or not. This parameter is important to describe
the accuracy and the reliability of the results.
Finally, the optimization objective was defined: the analysis minimize the
weighted compliance. As it has been explained in the paragraph Mathematical
Background in chapter “Optimization Theory”, the problem has been converted
from a Multiple Objective Optimization to a Single Objective Optimization
implementing the method of the Weighted Aggregation. Therefore, the two load
cases previously defined were aggregated assigning a factor equal to 1 and
minimizing the structural compliance for the single load case. The value of the
two weights is equal after considering the relative importance of each single load
case.
7.2.4 Analysis Results
The analysis is initially conducted with a single Design Domain and results are
shown in Figure 7.17 in terms of material density:
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Figure 7.17 Single Domain Results
The results are displayed with a color scale where red represents the unitary
density (no material has been removed) and blue is the null (material completely
removed). All other colors are intermediate densities.
As a consequence of the high overturning moment in the structure it can be
observed that a significant material concentration occurs around the corner in
the cantilever part and in the lower levels. This material has very high values of
stresses and as a consequence of density; the volume fraction fixed a priori is
concentrated in a single location instead of being more distributed over the
design domain. Moreover, the result is missing one constraint imposed which is
the one relate to the maximum size of a single continuous element.
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158
To encourage material distribution the design domain is divided in three parts to
constraint the volume in each domain of the structure separately (Figure 7.18).
Figure 7.18 Multiple Domain Model
This will ensure the material used regionally and avoid the large concentration
observed in Figure 7.17. The volume fraction is fixed as following:
- Domain 1: 30%
- Domain 2: 30%
- Domain 3: 25%
The quantity of volume is smaller in the higher domain in consideration of the
fact that the demand on structural elements decrease along the building height.
In Figure 7.19 results for the 3 Design Domain model are reported:
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Figure 7.19 Optimization Results for 3 Design Domain Model
As observed in Figure 7.19, results show loads path making X braces in the
southern and northern façades of the building with two vertical elements; in the
lateral façade one clear vertical element is recognizable and other inclined
braces are linking that member to front bracing system.
It is possible to observe that elements in the lower portion of the north façade
are more vertical compared to the higher levels member; this distribution is
common in the design for lateral loads and underlines a shear behavior at the
top of the structure, whereas at the base both bending moment and shear forces
are present.
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160
It is possible to observe that results are not completely symmetric; this is due to
the unsymmetrical position of shear walls in the plan section: this is creating a
tolerable difference in the lateral façade results.
7.2.5 Results Interpretation
For a deeper understanding, it is possible to plot results in terms of stress
distribution; in Figure 7.20 the distribution of stresses is represented with
compression in blue and tension in red. Blue and red lines are representing areas
where optimization results returned higher density which are representing
possible discrete members.
From stresses contour, it is recognizable a possible load path where load in the
cantilever part of the system is transferred from central region to lateral thanks
to steel braces; these elements are carrying loads to the two main elements in
the lateral façade: floors interacting with braces in tension are hung from the
system in the lateral façade; on the contrary floors linked to members in
compression are leaning on this system.
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Figure 7.20 Stresses Distribution ( Compression in blue color and Tension in red)
Finally, a focus is done on the East and West façades. Lateral façade stresses plot
reveals that most concentrations of material are in compression; this results are
a consequence of the finite element analysis which is assuming a Linear Elastic
Material and no buckling effects are considered in the analysis. In this way the
shortest load path to bring loads to the supports is the one shown in Figure 7.20.
Therefore, a structural considerations lead to the idea of changing members
inclinations by flipping them as reported in Figure 7.21: since steel is meant to be
the construction material, elements to be used are going to have high
slenderness.
Chapter 7
162
Figure 7.21 Structural Interpretation of Optimization Results
The main consideration shown by the results is a regional collection of gravity
loads which are led to the central compression column by steel inclined braces
hanging the floors (Figure 7.22); the combination of the steel braces and floor
slabs are creating a strut and tie mechanism. As a consequence, the load path for
gravity loads is significantly shorter because loads are not coming all the way up
to the steel truss, back to the shear wall and finally down to the foundation
system.
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Figure 7.22 Load Path
Once a deep understanding of results is achieved, the next step requires the
definition of a discrete system in order to fix the geometry of a structural model
to develop the optimization of the shape section of each member. Two different
discrete models are defined in order to capture the global behavior of each one:
1. The first alternative scheme developed is a perimeter steel system which
is defined as represented in Figure 7.23 and Figure 7.24: red dotted lines in the
typical floor represent truss lines which are distributed only in the South and
lateral façade.
Figure 7.23 Option 1: Typical Floor
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164
Figure 7.24 Option 1: East (left) and South (right) façades
2. The second option is focused on the strut and tie model using the
lay-out of the East and West façades; this system is repeated in order to define
five truss lines in NS direction corresponding to the perimeter columns. The
structural steel framing, in fact, is not moment resisting thus braces are required
to support each column. To meet architectural constraints, which requires great
space to develop office spaces, braces are staggered for each line in order to
achieve better space distribution. The analysis model is shown in Figure 7.25
with the dotted lines representing the truss lines.
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Figure 7.25 Option 2: East Façade (left) and Typical Floor (right)
The first initial study of the two systems is focused only on the understanding if
the structures are meeting general comfort requirements for structures. At this
level of study two parameters are investigated:
- Natural Building Mode Shapes: the first two primary modes should be
translational modes in x and y direction and primary rotational mode
should be third and faraway from other;
- Interstory Drift: this parameter is better explained in the serviceability
checks in the following paragraphs but this is required to be lower than
2%;
Chapter 7
166
Figure 7.26 Primary Mode Shapes: Baseline Scheme (left), Option 1 (Middle) and Option 2 (right)
For this analysis only serviceability load combination are applied to the model; in
this condition both of the model defined are meeting comfort requirements and
are presenting a global behavior similar to the baseline scheme.
7.2.6 Architectural and Structural Considerations
From results explained above, two considerations are developed:
- Balanced structure: from the analysis on the baseline structure it has
been discovered that the unsymmetrical condition on the shear wall
creates high stress concentration at lower levels in the southern portion
of the core system. Nevertheless, vertical displacements are not meeting
requirements imposed from the presence of the conference hall which
need to avoid every possible interaction. For this reasons a balanced
scheme is developed where all perimeter columns are stopped at level 05
in order to create a cantilever system all around core system.
- Façade architectural design: from development in the architectural
design of the building it has been discussed with the Salt Lake City
Municipality about the possibility of makeing use of structural member
directly exposed in the façade in one of the major city building; since Salt
Lake City is a very conservative city from an architectural viewpoint and
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167
many constraints should apply, the solution with internal brace is
preferred.
From this two considerations a final discrete model has been developed to
optimize the members size and to evaluate the variation in steel weight from the
baseline solution to the alternative scheme proposed.
7.3 Member Size Optimization
7.3.1 Geometry Description
The final step consists of optimizing the dimension of steel members in the
discrete model.
Therefore, the preliminary passage is to define the definitive geometry of the
structure and to create the Finite Element analysis model in ETABS. This model is
applying the two consideration explained in the final part of previous paragraph
so that a balanced scheme is implemented using only internal braces. Moreover,
the geometry adopts the strut and tie results observed in Figure 7.21 maintaining
the same inclination but with staggered position for elements along the three
North-South truss lines.
Now a deep description of the geometry is presented; in Figure 7.27 the first two
truss lines in South-North Direction are represented with a side elevation along
A-A section. It is important to note that these two truss lines are supporting the
greatest tributary areas in the typical floor.
Also, it can be observed the balanced scheme: columns are stopped at level 05
and first floors are behaving independently with respect to the tower floors.
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168
Figure 7.27 Typical Plan (right) and A-A Side Elevation
In Figure 7.28 the side elevation B-B is shown: columns in East and West façade
are to be supported as well as others because of the balanced scheme used.
Moreover, steel members in North-South direction are staggered with respect to
the previous shown.
Figure 7.28 Typical Plan (right) and B-B Side Elevation
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The last braces are shown in Figure 7.29: these members support the two
corners columns. It has been chosen to place a single level of braces since
tributary areas are significantly smaller.
Figure 7.29 Typical Plan (right) and C-C Side Elevation (left)
7.3.2 Iterative Process: Strength Design
The Finite Element model is defined and an iterative process starts to define the
optimized size for steel members.
The strength design is conducted using load combination as per Chapter 2.3 in
ASCE Code [14] as explained in previous chapter. All load combinations are
considered: seismic analysis is conducted both with equivalent static forces and
with response spectrum described in the introduction of present chapter; wind
load are applied only as static forces.
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170
The ETABS model is built starting from the baseline scheme which is reflecting
scheme explained in the structural description; in the iterative process starting
member sizes are the same of baseline structural model. Connections between
beams and columns are not moment resisting; in addition, also diagonals
members have moment releases at both ends connecting with shear wall and
beam. Finally, interaction between inclined members and beams at intermediate
levels is not avoided so that axial compression in diagonals is not constant but
presents jumps corresponding with floors intersections.
All design checks are conducted as axial loaded only members: tension members
are checked as per Chapter D in Specification for Structural Steel Buildings,
instead tension as per Chapter E, as explained in the previous chapter.
The design of members is conducted on the global system and not only on
diagonals members since the modification of the global load path is influencing
many different structural elements. First, all columns have been designed since
the load path is significantly changed from the baseline scheme. Then, both
diagonals and beams interacting with them have been design as shown in Figure
7.30. The design starts an iterative process continuously changing members
dimension to achieve a weight reduction; at each step the global weight is
evaluated and iteration stops when difference between current step and
previous is smaller than an assigned threshold.
Figure 7.30 Interaction between diagonals and beams
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7.3.3 Diagonals Design
All diagonals are tension only members since diagonals and beams interact with
vertical and horizontal forces and moment distribution is null. For this reason
sections for this members are chosen from W14 shapes because they have the
most squared section which perform better for axial elements.
In particular, all shapes used are summarized in
Table 7.4:
Section Name
Depth [in]
WidthTop [in]
ThickTop [in]
WebThick [in]
WidthBot [in]
ThickBot [in]
Area [in
2]
TotalWt [lb/ft]
W14X159 15 15.6 1.2 0.7 15.6 1.2 46.7 159.0
W14X193 15.5 15.7 1.4 0.9 15.7 1.4 56.8 193.0
W14X211 15.7 15.8 1.6 1.0 15.8 1.6 62.0 211.0
W14X233 16 15.9 1.7 1.1 15.9 1.7 68.5 233.0
W14X342 17.5 16.4 2.5 1.5 16.4 2.5 101.0 342.0
W14X370 17.9 16.5 2.7 1.7 16.5 2.7 109.0 370.0
W14X605 20.9 17.4 4.2 2.6 17.4 4.2 178.0 605.0
Table 7.4 W14 Shapes properties
In Appendix B, design is reported for each diagonal member, instead in Table 7.5
design of the member with highest DC ratio and with the greatest dimensions is
reported.
DIAGONALS
Element ID Pu,max Pu,min Model Section Length φtPnt φcPnc T/C
Governed DCR
LEVEL 22
D170 6942.9 737.8 W14X605 276 8010.0 6121.3 Tension 0.87
LEVEL 07
D221 2875.5 -491.9 W14X233 323 3082.5 1954.2 Tension 0.93
Table 7.5 Detail of Diagonals design
Chapter 7
172
7.3.4 Beams Design
In addition, all steel beams interacting with diagonals members have been
designed; these elements have a behavior comparable to the bottom chords in a
truss system and are subject to great values of compression axial forces as shown
in Figure 7.30. For every diagonal member, three beams are designed since
interaction is complete at each floor.
At this point of development of the project the design of every single beam is not
complete yet but from early studies it has been observed that for these elements
axial forces are significantly greater than bending moment and is leading the
design. This result can be understood thinking that connections have moment
releases so that bending moment are consequence of surface loads applied to
the single floor.
For this reason beams are designed as axial members only, keeping the Design
Criteria Ratio not greater than 0.7 in order to have a sufficient mark up to cover
bending additional demand. Moreover, beams are designed not considering
buckling effects; lateral stability, in fact, is guaranteed in both directions: studs
for shear resistance in the composite metal deck prevents horizontal and vertical
displacements.
Moreover, section shapes for beam design are taken from W21 and W24 shapes
since the key issue for beam and floor design is vertical displacement and these
shapes have greater web and consequently moment of inertia.
In particular, shapes used are:
Section Name
Depth [in]
WidthTop [in]
ThickTop [in]
WebThick [in]
WidthBot [in]
ThickBot [in]
Area [in
2]
TotalWt [lb/ft]
W21X44 20.7 6.5 0.45 0.35 6.5 0.45 13 44
W21X55 20.8 8.22 0.522 0.375 8.22 0.522 16.2 55
W21X68 21.1 8.27 0.685 0.43 8.27 0.685 20 68
W21X111 21.5 12.3 0.875 0.55 12.3 0.875 32.7 111
W24X146 24.7 12.9 1.09 0.65 12.9 1.09 43 146
Table 7.6 W21 and W24 Shapes Properties
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In Appendix B design for each beam is reported while a detail of the beams with greatest section shape is shown in
Table 7.7.
BEAMS
Element ID Pu,max Pu,min Model Section Ag φtPnt φcPnc T/C Governed
DCR
L07
B112 -27.6 -1301.1 W24X146 43.00 1935.0 302.9 Compression 0.67
B124 -81.2 -1198.3 W24X146 43.00 1935.0 302.9 Compression 0.62
Table 7.7 Detail of Beams design
7.3.5 Columns Design
Finally, columns are designed using W14 shapes as well as for diagonals; columns
are both tension and compression members, in fact, the modulus changes in
correspondence of the connection with diagonals as shown in Figure 7.31.
Figure 7.31 Columns load path
Members applied are reported in Table 7.8while detail of member with highest
DCR is reported inTable 7.9. Complete description of columns design is reported
in Appendix B.
Section Name
Depth [in]
Width Top [in]
Thick Top [in]
Web Thick [in]
Width Bot [in]
Thick Bot [in]
Area [in
2]
Total Wt [lb/ft]
W14X34 14.0 6.8 0.5 0.3 6.8 0.5 10.0 34.0
W14X38 14.1 6.8 0.5 0.3 6.8 0.5 11.2 38.0
Chapter 7
174
Section Name
Depth [in]
Width Top [in]
Thick Top [in]
Web Thick [in]
Width Bot [in]
Thick Bot [in]
Area [in
2]
Total Wt [lb/ft]
W14X48 13.8 8.0 0.6 0.3 8.0 0.6 14.1 48.0
W14X61 13.9 10.0 0.6 0.4 10.0 0.6 17.9 61.0
W12X65 12.1 12.0 0.6 0.4 12.0 0.6 19.1 65.0
W14X68 14.0 10.0 0.7 0.4 10.0 0.7 20.0 68.0
W14X74 14.2 10.1 0.8 0.4 10.1 0.8 21.8 72.0
W14X82 14.3 10.1 0.9 0.5 10.1 0.9 24.0 82.0
W14X90 14.0 14.5 0.7 0.4 14.5 0.7 26.5 90.0
W14X99 14.2 14.6 0.8 0.5 14.6 0.8 29.1 99.0
W14X109 14.3 14.6 0.9 0.5 14.6 0.9 32.0 109.0
W14X120 14.5 14.7 0.9 0.6 14.7 0.9 35.3 120.0
W14X132 14.7 14.7 1.0 0.6 14.7 1.0 38.8 132.0
W14X145 14.8 15.5 1.1 0.7 15.5 1.1 42.7 145.0
W14X159 15.0 15.6 1.2 0.7 15.6 1.2 46.7 159.0
W14X193 15.5 15.7 1.4 0.9 15.7 1.4 56.8 193.0
W14X257 16.4 16.0 1.9 1.2 16.0 1.9 75.6 257.0
Table 7.8 Column shapes properties
COLUMNS
Element ID Pu,max Pu,min Model Section Ag φtPnt φcPnc T/C
Governed DCR
LEVEL 14 300.9 -1020.2 W14X90 26.50 1192.5 1030.8 Compression 0.99
Table 7.9 Detail of Column design
7.3.6 Serviceability checks
Studies on global structural response are conducted at every iteration of the
process to investigate if the building meets serviceability requirements fixed
from code and from project architectural constraints.
Serviceability requirements are more strictly influencing the project in relation
with two features: the project is adopting structural steel framing for gravity
system; second, the presence of Utah Performing Arts Center in the footprint of
111 South Main requires a severe control of vertical deflection at level 05 to
prevent any possible interaction between the two different structures.
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In this paragraph a deep description of the checks is presented and results are
related to the final structural layout once member size are fixed.
The first check is related to the global building behavior; a dynamic analysis is
performed applying to the ETABS model response spectrum as per ASCE 7-05
and USGS maps.
In the dynamic analysis it can be observed that first three mode shapes are the
primary, involving more than the 65% of the mass participating ratio; the first is
in y-direction which correspond to the North-South; the second is in East-West
and the third is rotational mode around z axis.
Moreover, primary translational modes have a substantial separation from the
primary translational mode, which meets comfort requirements for occupants:
Primary Torsional Mode/Primary NS Translational Mode: 2.132s/4.126s=0.52
Primary Torsional Mode/Primary EW Translational Mode: 2.132s/3.445s=0.62
Chapter 7
176
Modes Period
(s)
UX (%) UY (%) RZ (%) Sum UX
(%)
Sum UY
(%)
Sum RZ
(%)
1 4.126 2.6171 62.4291 0.2312 2.6171 62.4291 0.2312
2 3.445 63.0724 2.6301 0.0464 65.6895 65.0592 0.2776
3 2.132 0.0983 0.0623 80.8818 65.7878 65.1215 81.1595
4 0.998 0.0011 0.0009 0.0001 65.7888 65.1224 81.1596
5 0.982 0.0023 0.0015 0.0000 65.7911 65.1239 81.1596
6 0.979 0.1862 0.0090 0.0031 65.9773 65.1329 81.1627
7 0.977 0.0259 0.0668 0.0000 66.0032 65.1997 81.1627
8 0.977 0.0015 0.0020 0.0000 66.0046 65.2017 81.1627
9 0.950 0.0015 0.0010 0.0001 66.0061 65.2027 81.1628
10 0.934 0.0000 0.0000 0.0000 66.0061 65.2027 81.1628
11 0.932 0.0000 0.0000 0.0000 66.0061 65.2027 81.1628
12 0.924 0.3201 0.0239 0.0009 66.3262 65.2266 81.1637
13 0.922 0.0329 0.1742 0.0018 66.3591 65.4008 81.1655
14 0.899 0.0000 0.0000 0.0000 66.3591 65.4008 81.1655
15 0.895 0.0000 0.0000 0.0000 66.3591 65.4008 81.1655
16 0.811 0.0001 0.0000 0.0000 66.3592 65.4008 81.1655
17 0.811 0.0000 0.0000 0.0000 66.3593 65.4008 81.1655
18 0.793 15.2204 0.0796 0.0170 81.5796 65.4804 81.1826
19 0.775 0.0139 15.4195 0.0755 81.5936 80.8999 81.2581
20 0.773 0.0000 0.0005 0.0000 81.5936 80.9004 81.2581
21 0.773 0.0000 0.0002 0.0000 81.5936 80.9006 81.2581
22 0.720 0.0006 0.0002 0.0006 81.5942 80.9008 81.2587
23 0.715 0.0002 0.0010 0.0001 81.5944 80.9018 81.2588
24 0.705 0.0002 0.0008 0.0001 81.5945 80.9026 81.2589
25 0.702 0.0005 0.0001 0.0017 81.5951 80.9027 81.2606
26 0.698 0.0069 0.0045 0.0199 81.6020 80.9071 81.2805
27 0.695 0.0083 0.0139 0.0032 81.6103 80.9211 81.2836
28 0.692 0.0054 0.0126 0.0058 81.6157 80.9336 81.2895
29 0.689 0.0058 0.0000 0.1343 81.6215 80.9336 81.4237
30 0.675 0.1932 0.0812 5.6491 81.8147 81.0148 87.0728
31 0.666 0.0003 0.0600 0.1179 81.8150 81.0748 87.1907
32 0.648 0.0075 0.0076 0.4995 81.8225 81.0824 87.6902
33 0.648 0.0103 0.0093 0.5381 81.8328 81.0917 88.2284
34 0.647 0.0024 0.0027 0.0241 81.8351 81.0944 88.2525
35 0.645 0.0028 0.0002 0.0101 81.8380 81.0947 88.2626
36 0.642 0.0001 0.0000 0.0010 81.8381 81.0947 88.2635
37 0.640 0.0001 0.0006 0.0014 81.8381 81.0953 88.2650
38 0.638 0.0345 0.0717 0.0031 81.8726 81.1669 88.2680
39 0.637 0.0228 0.0136 0.0569 81.8954 81.1805 88.3249
40 0.636 0.0029 0.0020 0.0049 81.8984 81.1825 88.3298
Participation Ratios
Table 7.10 Modal mass participation ratios
Finally, primary translational modes are decoupled because the mass
participating ratio in the opposite direction is less than 3% for both the first two
modes. In Figure 7.32 a comparison between primary mode shapes of baseline
structural system (adopting truss system at roof level) and alternative scheme is
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shown. Global behavior is similar for both structural systems with difference in
natural period lower than 0.01 s.
Figure 7.32 Primary mode shapes comparison: Baseline scheme (top) and Alternative scheme(bottom)
The second check is conducted on story drifts under seismic service load
combinations; story drift checks are required to provide comfort to occupants
Chapter 7
178
and integrity of interior partitions and exterior claddings. As per paragraph
12.8.6 in ASCE 7-10 [14], the design story drift is to computed as the difference
of the deflections at the center of mass at the top and bottom of the story under
consideration. In Figure 7.33 ASCE 7-10 [14] story drift calculation procedure is
explained.
Figure 7.33 Story Drift Determination [14]
Story drift limits are defined as per paragraph 12.12.1 [14]; story drift shall not
exceed the allowable Δa as obtained from Table 7.11 for any story.
Table 7.11 Allowable Story Drift Δa [14]
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In Table 7.12 seismic computed story drifts are reported in both orthogonal
directions; the building is satisfying code prescriptions at each floor.
Max of DriftX Max of DriftY
Story Total Story Total ROOF 0.96% ROOF 1.34%
L24 0.98% L24 1.34%
L23 1.01% L23 1.33%
L22 1.03% L22 1.37%
L21 1.06% L21 1.37%
L20 1.13% L20 1.50%
L19 1.03% L19 1.35%
L18 1.03% L18 1.35%
L17 1.01% L17 1.34%
L16 0.99% L16 1.38%
L15 0.98% L15 1.36%
L14 0.97% L14 1.36%
L13 1.00% L13 1.33%
L12 0.99% L12 1.31%
L11 0.97% L11 1.28%
L10 0.97% L10 1.28%
L09 0.93% L09 1.23%
L08 0.90% L08 1.17%
L07 0.81% L07 1.19%
L06 0.76% L06 1.15%
L05 0.77% L05 1.00%
L04 0.66% L04 0.93%
L03 0.62% L03 0.88%
L02 0.71% L02 0.83%
L01 0.24% L01 0.17%
Table 7.12 Computed x and y Story Drifts
Chapter 7
180
Figure 7.34 Story Drift under Seismic Load Combination
Finally, vertical displacements are compared between baseline structural system
and the alternative scheme at level 05 where there is the separation between
111 South Main and Utah Performing Arts Center.
0
50
100
150
200
250
300
350
400
0.00% 0.50% 1.00% 1.50% 2.00% 2.50%
Sto
ry E
leva
tio
n (
ft)
Story Drift Ratio Under Seismic Action
Drift XDrift YASCE 7-10 2% Limit
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Figure 7.35 Baseline Scheme - Dead Load and Superimposed Dead Load Vertical Displacements (inches)
Figure 7.36 Alternative scheme - Dead Loads and Superimposed Dead Loads Vertical Displacements (inches)
Chapter 7
182
Figure 7.37 Baseline Scheme - Live Loads Vertical Displacements (inches)
Figure 7.38 Alternative Scheme - Live Loads Vertical Displacements (inches)
As shown from Figure 7.35 to Figure 7.38 the alternative scheme is also achieving
a reduction in vertical displacements. The reductions are significant specifically
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for braces in the NS and EW direction with values up to 60%; corner columns
experiences lower reduction with a maximum reduction of 20%.
7.3.7 Weight Comparison
A weight comparison is conducted between baseline and alternative scheme
taking into account all structural members designed and replaced truss system.
In particular, truss system weighted 1125 tons (1240 US tons) against the 290
tons (320 US tons) of diagonal members in the alternative scheme.
From the design of beams, it has been observed an increase of weight in the
alternative scheme equal to 9 tons (10 US tons).
Also, the design of columns led to a final weight of 313 tons (344 US tons) lighter
than the 556 tons (613 US tons) of the baseline scheme.
Finally, in the alternative scheme it should be accounted also the weight of the
floor framing at roof level. Adopting this system, the design of this floor is to be
conducted because floor framing is required to accommodate mechanical
equipment. From the design it was discovered an additional weight of 84 tons
(92 US tons).
A summary is reported in Figure 7.39 with a grand total summation of weight,
leading to a final steel saving equal to 984 tons.
Figure 7.39 Weight Comparison
Chapter 7
184
7.3.8 Analytical study: Alternative Scheme 2
A new analytical study is developed starting from bases of the alternative
scheme design. This study has been developed without implementing any finite
element analysis model but making hand calculations from results obtained in
the previous model.
The analysis starts from the idea of moving corner braces to roof mechanical
level where minor constraints on the geometry are fixed and to free spaces at
intermediate levels.
Figure 7.40 Typical Plan (right) and C-C Side Elevation
The new position of corner braces has lower axial efficiency because the
inclination is smaller than the previous and more distant from the 45°, as shown
in Figure 7.41.
This change requires modification not only in the diagonals members but also to
columns placed in the corners. The load path for loads applied to their tributary
area is similar to the one of the baseline scheme with loads carried up to the roof
level and then transferred to the central core system.
Case Study: 111 South Main
185
Figure 7.41 Corner Braces Inclination Change
The increase in member sizes for corner braces is computed with and hand-
calculation imposing vertical displacements equal to the Alternative Scheme 1.
Eq 7.1 Vertical Displacements
From the equality in vertical displacements the ratio between areas in the two
schemes is obtained:
(
( )
( ))
( ( )
( ))
Eq 7.2 Members Area Ratio
Since areas are linearly proportional to members weight, the same ratio can be
extended to members weight, so that weight in the alternative scheme 2 can be
evaluated from Alternative scheme 1 (90,72 tons) as:
Eq 7.3 Alternative Scheme 2 Corner Braces Weight
Chapter 7
186
In this way, the total braces weight can be evaluated as:
( )
Figure 7.42 Alternative Scheme 2 - Braces Total Weight
Moreover, columns total weight is modified to take into account that moving
braces to roof level is changing load path of gravity loads for corner columns;
with the new layout, load path is similar to the load path of the baseline scheme,
in fact, loads are coming all the way up to the roof level and then back to the
shear walls. The design is assumed to be similar to the baseline scheme for
corner columns so that total columns weight can be computed as:
( )
Eq 7.4 Alternative Scheme 2 - Columns Total Weight
Again a summary of weight changes is reported with a total steel weight saving
equal to 769 tons.
Case Study: 111 South Main
187
Figure 7.43 Alternative Scheme 2 - Weight Comparison
The new system is in this way less efficient compared to the first alternative
scheme but anyway achieve a significant saving compared to the baseline model.
Moreover the new position of corner braces in the mechanical level allow an
increased freedom to change members and so deeper studies can be
implemented to define a new more optimized geometry.
CHAPTER 8
Conclusions
As a conclusion of the studies, the innovative methodology described along the
thesis has been found to be solid and applicable to the design in the engineering
practice. It defines a sequence of consecutive steps that shadow the
development of the project from the initial architectural concept to the final
documents of the design. All the methodology was developed and refined by the
application on two different case studies: a roof truss system in a U.S.
Courthouse and a strut and tie system for the 111 South Main in Salt Lake City.
This method can be fully implemented in the design of building, especially if the
typology is special or has peculiar requirements. The case studies shown that
following this methodology, a new typology for the structural system was found
that resulted more efficient. In particular, the optimized truss that was presented
in this work was included in the U.S. Courthouse calculation book and structural
drawings presented to the contractor for the Detailed Design.
Furthermore, it has been proved that the process can be applied to different
structural components. In fact, the two projects had different structural concept
to replicate, but in both cases the optimization process was successful. In U.S.
Courthouse the study has been focused on the truss system in order to achieve
Chapter 8
190
an optimized truss minimizing the material waste while maximizing the stiffness;
on the other hand, in 111 South Main the study started from a different
consideration so that the idea of removing the hat truss lead to the 2D
optimization of the façade enclosures to develop an alternative structural
system. These two examples underline the consistency of the methodology,
which can be applied to any kind of problem without changing the theoretical
approach.
From the studies, it has been found that this methodology integrates the
classical approach of the structural mechanics and innovative powerful
computational tools following the steps of the design. Related to the previous
consideration, a strict relation between architectural and structural design is
necessary for finding fast solution to design issues. Since the architectural design
can last months, many changes can be applied to the optimization parameters
starting from the Design Domain for the problem.
For example in the U.S. Courthouse, this is represented by additional Non-Design
Spaces due to corridor to be guaranteed because of changes in the roof
occupancy; in these spaces elements were placed in the first optimization
analysis, which had to be modified in the following optimized layout . In 111
South Main the starting concept was focused on the definition of a perimeter
steel system; architectural constraints were imposed by Salt Lake City
municipality which required not to use exterior steel members in the façade of
the major building in the city renovation.
The method accomplishes a double task: it follows the architectural
developments while maintaining a high structural efficiency and suggests
possible architectural modifications to achieve better solutions.
The optimization design process is suggested for projects presenting peculiar and
specific design issues. Its innovative development requires long time to run
analysis, powerful and expensive tools to perform topology optimization analysis
Conclusions
191
and designers inside the project team in charge of developing the optimization
study. For all of these reasons, the presented methodology can represent a
worthy opportunity to develop an efficient design when dealing with big and
complex problems. In design for typical projects, engineering judgment by itself
can lead to results similar to the one achievable from the optimization process
with great savings in terms of time.
For example, in both case studies presented the peculiarity of the project is
represented by the fact that all columns in the structure cannot continue to the
foundation system as in typical buildings but are to be supported by other
mechanism.
At the same time, using this approach for “special” building can broaden the field
of structural system known, giving a better chance when facing typical buildings
as well.
Finally, something that can look like a paradox was found: the stiffness of a
structure can be maximized by removing material, as long as the material is
placed in the correct position. The more the solution is far from this
configuration, the bigger is the waste of material.
Thanks to the methodology, it is possible to define in each step a new layout that
can improve the stiffness of the structure reducing displacements and to reduce
the amount of material used.
To reach this goal all steps of the process are fundamental: in the topology
optimization process the material is redistributed identifying areas with higher
stress concentrations. Then, in member size optimization elements are designed
through code checks and members are iteratively modified: this process is not
assigning the smallest section admissible for each section but the one required to
minimize the total structural weight; in this way, it could be necessary to assign
bigger sections to particular members because they have a greater influence in
the structural stiffness so that they allow a global weight reduction.
Chapter 8
192
Further developments that can get the methodology better are in introducing
non-linear analysis. The finite element analysis used during the process up to
now, work in the linear elastic field, with isotropic material.
Also second order analysis must be implemented into the methodology. In fact,
buckling effects are not influencing the redistribution of material and member’s
slenderness is not evaluated so that members in compression behave in the
exactly same way of members in tension.
A possible way of introducing this problem could be define a penalization factor
that can be applied in order to penalize the compression to represent the
Eulerian instability. This modification can modify load path in order to minimize
the compression load path and not only the global load path.
Secondly, a field of interest is the Multi Objective optimization problem: this
topic was faced during the study of the 111 South Main. In that case study, the
optimization process had to be applied both for gravity and seismic lateral loads.
The method chosen for solving the Multi Objective optimization was the weight
aggregation: this assigns a weight to each of the load cases. By doing so, the
problem turns into a Single Objective Optimization problem.
This way of proceeding investigates a single load path for gravity and lateral
loads; further discussions suggested to develop separate studies for the different
load case. Once the optimized solution is found for each case, a synthesis of the
concept shown by the solution can be applied in the definition of the final
discrete model. This approach can lead to a better understanding of the global
behavior of the structure.
This consideration is very important because the global weight of the optimized
solution can experience significant increases due to seismic lateral load if this
was not considered in the early step of the study.
In conclusion, considering structural topology optimization can produce
significant changes in the design of the building, especially when special
Conclusions
193
structures are dealt. The introduction of this methodology can bring not only a
gain in terms of material saving, but also a deepening of the understanding of
possible load paths and therefore of the global behavior of the structure. Even if
the solution found through this process is not directly applied, it can help the
design of a different structure focusing the attention towards those members
that were pointed out by the optimization process’ solution.
APPENDIX A
Case Study: US Courthouse
A.1 Strength Checks
A.2 Optimum Position Algorithm
APPENDIX A
STRENGTH CHECKS
Fy 50 ksi
Total weight 603.483 tons
E 29000 ksi
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
kip
in2 in in
kip
1 16D12L05LR -1517.583 W14X283 W14X283 83.30 323.99 4.16 77.92 47.14 32.07 compression 2404.63 ok
2 16D12L05LR -60.432 W14X455 W14X455 134.00 562.85 4.37 128.77 17.26 15.14 compression 1825.57 ok
4 16D12L05LR -1616.203 W14X283 W14X283 83.30 324.00 4.16 77.93 47.13 32.07 compression 2404.53 ok
6 16D12L05LR 1803.573 W14X665 W14X665 196.00 400.84 4.61 86.90 37.90 28.78 tension 8820.00 ok
7 16D12L05LR -3728.846 W14X426 W14X426 125.00 319.02 4.35 73.42 53.10 33.71 compression 3792.72 ok
8 16D12L05LR 1506.513 W14X120 W14X120 35.30 662.31 3.74 176.87 9.15 8.02 tension 1588.50 ok
9 16D12L05LR 1533.342 W14X120 W14X120 35.30 662.31 3.74 176.87 9.15 8.02 tension 1588.50 ok
10 16D12L05LR 1395.591 W14X132 W14X132 38.80 274.82 3.76 73.13 53.52 33.82 tension 1746.00 ok
11 16D12L05LR -2375.282 W14X426 W14X426 125.00 274.82 4.35 63.25 71.55 37.32 compression 4198.46 ok
13 16D12L05LR 1469.203 W14X132 W14X132 38.80 274.82 3.76 73.12 53.53 33.82 tension 1746.00 ok
14 16D12L05LR -2736.032 W14X426 W14X426 125.00 274.82 4.35 63.25 71.55 37.32 compression 4198.46 ok
15 16D12L05LR 1213.158 W14X132 W14X132 38.80 274.82 3.76 73.12 53.53 33.82 tension 1746.00 ok
A1 Strength Checks
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
16 16D12L05LR -2383.01 W14X311 W14X311 91.40 274.82 4.20 65.48 66.75 36.54 compression 3006.10 ok
17 16D12L05LR -699.849 W14X145 W14X145 42.70 324.00 3.98 81.37 43.23 30.81 compression 1184.11 ok
18 16D12L05LR 1190.447 W14X132 W14X132 38.80 274.82 3.76 73.13 53.52 33.82 tension 1746.00 ok
19 16D12L05LR -1780.207 W14X311 W14X311 91.40 274.82 4.20 65.48 66.75 36.54 compression 3006.10 ok
20 16D12L05LR 2612.932 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
21 16D12L05LR 3142.518 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
22 16D12L05LR 2976.701 W14X311 W14X311 91.40 270.00 4.20 64.33 69.16 36.94 tension 4113.00 ok
23 16D12L05LR 494.363 W14X159 W14X159 46.70 336.00 4.00 83.96 40.61 29.86 tension 2101.50 ok
24 16D12L05LR -877.15 W14X283 W14X283 83.30 404.81 4.16 97.36 30.19 25.00 compression 1874.35 ok
26 16D12L05LR -877.226 W14X283 W14X283 83.30 404.83 4.16 97.37 30.19 25.00 compression 1874.24 ok
27 16D12L05LR 1299.834 W14X311 W14X311 91.40 330.44 4.20 78.73 46.17 31.78 tension 4113.00 ok
28 16D12L05LR 208.004 W14X109 W14X109 32.00 434.08 3.74 116.14 21.22 18.61 tension 1440.00 ok
29 16D12L05LR 220.499 W14X120 W14X120 35.30 398.09 3.74 106.31 25.33 21.88 tension 1588.50 ok
30 16D12L05LR 2505.869 W14X398 W14X398 117.00 229.10 4.31 53.20 101.14 40.65 tension 5265.00 ok
31 16D12L05LR -1414.62 W14X211 W14X211 62.00 360.00 4.08 88.32 36.69 28.27 compression 1577.19 ok
32 16D12L05LR -18.174 W14X90 W14X90 26.50 222.00 3.70 60.06 79.33 38.41 compression 916.00 ok
33 16D12L05LR -1123.264 W14X145 W14X145 42.70 324.00 3.98 81.37 43.23 30.81 compression 1184.11 ok
34 16D12L05LR 1062.406 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
APPENDIX A
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
35 16D12L05LR 332.004 W14X145 W14X145 42.70 576.00 3.98 144.66 13.68 12.00 tension 1921.50 ok
36 16D12L05LR 332.868 W14X132 W14X132 38.80 216.00 3.76 57.47 86.65 39.27 tension 1746.00 ok
37 16D12L05LR 1315.784 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
38 16D12L05LR 53.611 W14X132 W14X132 38.80 288.00 3.76 76.63 48.74 32.55 tension 1746.00 ok
39 16D12L05LR -1414.696 W14X176 W14X176 51.80 323.99 4.02 80.55 44.11 31.11 compression 1450.45 ok
40 16D12L05LR 1249.173 W14X398 W14X398 117.00 319.02 4.31 74.08 52.16 33.48 tension 5265.00 ok
41 16D12L05LR 999.788 W14X398 W14X398 117.00 319.01 4.31 74.07 52.16 33.48 tension 5265.00 ok
42 16D12L05LR 686.186 W14X109 W14X109 32.00 434.08 3.74 116.14 21.22 18.61 tension 1440.00 ok
43 16D12L05LR 1173.796 W14X109 W14X109 32.00 398.09 3.74 106.51 25.23 21.81 tension 1440.00 ok
44 16D12L05LR 2475.248 W14X398 W14X398 117.00 638.20 4.31 148.19 13.03 11.43 tension 5265.00 ok
45 16D12L05LR 1509.235 W14X120 W14X120 35.30 349.54 3.74 93.34 32.85 26.44 tension 1588.50 ok
46 16D12L05LR 83.591 W14X120 W14X120 35.30 349.55 3.74 93.34 32.85 26.44 tension 1588.50 ok
47 16D12L05LR 168.915 W14X109 W14X109 32.00 638.20 3.74 170.76 9.82 8.61 tension 1440.00 ok
48 16D12L05LR 766.115 W14X145 W14X145 42.70 422.95 3.98 106.22 25.37 21.91 tension 1921.50 ok
49 16D12L05LR 0 W14X120 W14X120 35.30 336.00 3.74 89.73 35.55 27.75 tension 881.73 ok
50 16D12L05LR 0 W14X120 W14X120 35.30 288.00 3.74 76.91 48.39 32.44 tension 1030.77 ok
51 16D12L05LR 192.024 W14X120 W14X120 35.30 336.00 3.74 89.73 35.55 27.75 tension 1588.50 ok
52 16D12L05LR -229.897 W14X120 W14X120 35.30 288.00 3.74 76.91 48.39 32.44 compression 1030.77 ok
A1 Strength Checks
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
53 16D12L05LR 973.623 W14X233 W14X233 68.50 624.00 4.10 152.29 12.34 10.82 tension 3082.50 ok
54 16D12L05LR -346.407 W14X176 W14X176 51.80 216.00 4.02 53.70 99.24 40.49 compression 1887.82 ok
55 16D12L05LR 585.541 W14X211 W14X211 62.00 576.00 4.08 141.32 14.33 12.57 tension 2790.00 ok
56 16D12L05LR 0 W14X120 W14X120 35.30 216.00 3.74 57.68 86.02 39.20 tension 1245.48 ok
57 16D12L05LR 0 W14X120 W14X120 35.30 576.00 3.74 153.82 12.10 10.61 tension 337.06 ok
58 16D12L05LR 1183.334 W14X145 W14X145 42.70 730.94 3.98 183.57 8.49 7.45 tension 1921.50 ok
59 16D12L05LR 651.977 W14X145 W14X145 42.70 505.75 3.98 127.01 17.74 15.56 tension 1921.50 ok
60 16D12L05LR -18.378 W14X145 W14X145 42.70 389.39 3.98 97.79 29.93 24.85 compression 954.91 ok
61 16D12L05LR 236.265 W14X120 W14X120 35.30 624.00 3.74 166.64 10.31 9.04 tension 1588.50 ok
62 16D12L05LR -642.04 W14X145 W14X145 42.70 433.50 3.98 108.87 24.15 21.02 compression 807.73 ok
63 16D12L05LR -820.045 W14X176 W14X176 51.80 466.77 4.02 116.05 21.25 18.64 compression 868.91 ok
64 16D12L05LR -261.331 W14X90 W14X90 26.50 222.00 3.70 60.06 79.33 38.41 compression 916.00 ok
65 16D12L05LR 1293.974 W14X211 W14X211 62.00 378.00 4.08 92.74 33.28 26.66 tension 2790.00 ok
67 16D12L05LR 836.333 W14X211 W14X211 62.00 288.00 4.08 70.66 57.33 34.71 tension 2790.00 ok
68 16D12L05LR 920.886 W14X211 W14X211 62.00 162.00 4.08 39.75 181.18 44.55 tension 2790.00 ok
71 16D12L05LR -1443.494 W14X500 W14X500 147.00 660.87 4.43 149.30 12.84 11.26 compression 1489.75 ok
78 16D12L05LR 1561.405 W14X211 W14X211 62.00 360.00 4.08 88.32 36.69 28.27 tension 2790.00 ok
80 16D12L05LR -983.621 W14X211 W14X211 62.00 323.99 4.08 79.49 45.30 31.50 compression 1757.79 ok
APPENDIX A
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
81 16D12L05LR 0 W14X120 W14X120 35.30 360.00 3.74 96.14 30.97 25.44 tension 808.18 ok
82 16D12L05LR 1809.11 W14X283 W14X283 83.30 274.82 4.16 66.10 65.52 36.33 tension 3748.50 ok
83 16D12L05LR -1833.326 W14X211 W14X211 62.00 274.82 4.08 67.43 62.96 35.86 compression 2000.96 ok
84 16D12L05LR 1268.682 W14X311 W14X311 91.40 360.00 4.20 85.77 38.90 29.20 tension 4113.00 ok
85 16D12L05LR -985.65 W14X211 W14X211 62.00 323.99 4.08 79.49 45.30 31.50 compression 1757.79 ok
86 16D12L05LR 1819.353 W14X283 W14X283 83.30 274.82 4.16 66.10 65.52 36.33 tension 3748.50 ok
87 16D12L05LR -1843.662 W14X211 W14X211 62.00 274.82 4.08 67.43 62.96 35.86 compression 2000.96 ok
88 16D12L05LR 1563.78 W14X211 W14X211 62.00 360.00 4.08 88.32 36.69 28.27 tension 2790.00 ok
89 16D12L05LR 0 W14X120 W14X120 35.30 360.00 3.74 96.14 30.97 25.44 tension 808.18 ok
90 16D12L05LR 1448.387 W14X120 W14X120 35.30 162.00 3.74 43.26 152.93 43.61 tension 1588.50 ok
91 16D12L05LR 1420.181 W14X109 W14X109 32.00 162.00 3.74 43.34 152.35 43.58 tension 1440.00 ok
92 16D12L05LR -3.973 W14X145 W14X145 42.70 484.32 3.98 121.63 19.35 16.97 compression 652.02 ok
93 16D12L05LR -7.894 W14X145 W14X145 42.70 484.32 3.98 121.63 19.35 16.97 compression 652.02 ok
94 16D12L05LR -1526.371 W14X283 W14X283 83.30 323.99 4.16 77.92 47.14 32.07 compression 2404.63 ok
95 16D12L05LR -52.887 W14X455 W14X455 134.00 562.85 4.37 128.77 17.26 15.14 compression 1825.57 ok
96 16D12L05LR -1626.613 W14X283 W14X283 83.30 324.00 4.16 77.93 47.13 32.07 compression 2404.53 ok
97 16D12L05LR 1801.821 W14X665 W14X665 196.00 400.84 4.61 86.90 37.90 28.78 tension 8820.00 ok
98 16D12L05LR -3740.866 W14X426 W14X426 125.00 319.02 4.35 73.42 53.10 33.71 compression 3792.72 ok
A1 Strength Checks
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
99 16D12L05LR 1407.537 W14X132 W14X132 38.80 274.82 3.76 73.13 53.52 33.82 tension 1746.00 ok
100 16D12L05LR -2375.711 W14X426 W14X426 125.00 274.82 4.35 63.25 71.55 37.32 compression 4198.46 ok
101 16D12L05LR 1475.152 W14X132 W14X132 38.80 274.82 3.76 73.12 53.53 33.82 tension 1746.00 ok
102 16D12L05LR -2747.616 W14X426 W14X426 125.00 274.82 4.35 63.25 71.55 37.32 compression 4198.46 ok
103 16D12L05LR 1226.12 W14X132 W14X132 38.80 274.82 3.76 73.12 53.53 33.82 tension 1746.00 ok
104 16D12L05LR -2395.692 W14X311 W14X311 91.40 274.82 4.20 65.48 66.75 36.54 compression 3006.10 ok
105 16D12L05LR -689.482 W14X145 W14X145 42.70 324.00 3.98 81.37 43.23 30.81 compression 1184.11 ok
106 16D12L05LR 1181.199 W14X132 W14X132 38.80 274.82 3.76 73.13 53.52 33.82 tension 1746.00 ok
107 16D12L05LR -1768.157 W14X311 W14X311 91.40 274.82 4.20 65.48 66.75 36.54 compression 3006.10 ok
108 16D12L05LR 2613.442 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
109 16D12L05LR 3157.498 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
110 16D12L05LR 2983.43 W14X311 W14X311 91.40 270.00 4.20 64.33 69.16 36.94 tension 4113.00 ok
111 16D12L05LR 526.028 W14X159 W14X159 46.70 336.00 4.00 83.96 40.61 29.86 tension 2101.50 ok
112 16D12L05LR -880.842 W14X283 W14X283 83.30 404.81 4.16 97.36 30.19 25.00 compression 1874.35 ok
113 16D12L05LR -880.917 W14X283 W14X283 83.30 404.83 4.16 97.37 30.19 25.00 compression 1874.24 ok
114 16D12L05LR 1313.619 W14X311 W14X311 91.40 330.44 4.20 78.73 46.17 31.78 tension 4113.00 ok
115 16D12L05LR 202.869 W14X109 W14X109 32.00 434.08 3.74 116.14 21.22 18.61 tension 1440.00 ok
116 16D12L05LR 220.493 W14X120 W14X120 35.30 398.09 3.74 106.31 25.33 21.88 tension 1588.50 ok
APPENDIX A
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
117 16D12L05LR 2509.184 W14X398 W14X398 117.00 229.10 4.31 53.20 101.14 40.65 tension 5265.00 ok
118 16D12L05LR -1410.924 W14X211 W14X211 62.00 360.00 4.08 88.32 36.69 28.27 compression 1577.19 ok
119 16D12L05LR -18.174 W14X90 W14X90 26.50 222.00 3.70 60.06 79.33 38.41 compression 916.00 ok
120 16D12L05LR 1066.561 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
121 16D12L05LR 334.85 W14X145 W14X145 42.70 576.00 3.98 144.66 13.68 12.00 tension 1921.50 ok
122 16D12L05LR 329.743 W14X132 W14X132 38.80 216.00 3.76 57.47 86.65 39.27 tension 1746.00 ok
123 16D12L05LR 1317.087 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
124 16D12L05LR 69.834 W14X132 W14X132 38.80 288.00 3.76 76.63 48.74 32.55 tension 1746.00 ok
125 16D12L05LR -1426.32 W14X176 W14X176 51.80 323.99 4.02 80.55 44.11 31.11 compression 1450.45 ok
126 16D12L05LR 1254.476 W14X398 W14X398 117.00 319.02 4.31 74.08 52.16 33.48 tension 5265.00 ok
127 16D12L05LR 1007.898 W14X398 W14X398 117.00 319.01 4.31 74.07 52.16 33.48 tension 5265.00 ok
128 16D12L05LR 686.894 W14X109 W14X109 32.00 434.08 3.74 116.14 21.22 18.61 tension 1440.00 ok
129 16D12L05LR 1157.119 W14X109 W14X109 32.00 398.09 3.74 106.51 25.23 21.81 tension 1440.00 ok
130 16D12L05LR 2555.518 W14X193 W14X193 56.80 274.82 4.05 67.88 62.12 35.70 tension 2556.00 ok
131 16D12L05LR -2573.475 W14X283 W14X283 83.30 274.82 4.16 66.10 65.51 36.33 compression 2723.47 ok
132 16D12L05LR 2474.201 W14X398 W14X398 117.00 638.20 4.31 148.19 13.03 11.43 tension 5265.00 ok
133 16D12L05LR 1517.829 W14X120 W14X120 35.30 349.54 3.74 93.34 32.85 26.44 tension 1588.50 ok
134 16D12L05LR 82.163 W14X120 W14X120 35.30 349.55 3.74 93.34 32.85 26.44 tension 1588.50 ok
A1 Strength Checks
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
135 16D12L05LR 169.31 W14X109 W14X109 32.00 638.20 3.74 170.76 9.82 8.61 tension 1440.00 ok
136 16D12L05LR 766.732 W14X145 W14X145 42.70 422.95 3.98 106.22 25.37 21.91 tension 1921.50 ok
137 16D12L05LR 0 W14X120 W14X120 35.30 336.00 3.74 89.73 35.55 27.75 tension 881.73 ok
138 16D12L05LR 0 W14X120 W14X120 35.30 288.00 3.74 76.91 48.39 32.44 tension 1030.77 ok
139 16D12L05LR 180.686 W14X120 W14X120 35.30 336.00 3.74 89.73 35.55 27.75 tension 1588.50 ok
140 16D12L05LR -230.833 W14X120 W14X120 35.30 288.00 3.74 76.91 48.39 32.44 compression 1030.77 ok
141 16D12L05LR -346.243 W14X176 W14X176 51.80 216.00 4.02 53.70 99.24 40.49 compression 1887.82 ok
142 16D12L05LR 591.016 W14X211 W14X211 62.00 576.00 4.08 141.32 14.33 12.57 tension 2790.00 ok
143 16D12L05LR 0 W14X120 W14X120 35.30 216.00 3.74 57.68 86.02 39.20 tension 1245.48 ok
144 16D12L05LR 0 W14X120 W14X120 35.30 576.00 3.74 153.82 12.10 10.61 tension 337.06 ok
145 16D12L05LR 1185.029 W14X145 W14X145 42.70 730.94 3.98 183.57 8.49 7.45 tension 1921.50 ok
146 16D12L05LR 673.72 W14X145 W14X145 42.70 505.75 3.98 127.01 17.74 15.56 tension 1921.50 ok
147 16D12L05LR -17.791 W14X145 W14X145 42.70 389.39 3.98 97.79 29.93 24.85 compression 954.91 ok
148 16D12L05LR -639.537 W14X145 W14X145 42.70 433.50 3.98 108.87 24.15 21.02 compression 807.73 ok
149 16D12L05LR -829.694 W14X176 W14X176 51.80 466.77 4.02 116.05 21.25 18.64 compression 868.91 ok
150 16D12L05LR -261.46 W14X90 W14X90 26.50 222.00 3.70 60.06 79.33 38.41 compression 916.00 ok
151 16D12L05LR 1276.842 W14X211 W14X211 62.00 378.00 4.08 92.74 33.28 26.66 tension 2790.00 ok
152 16D12L05LR 846.778 W14X211 W14X211 62.00 288.00 4.08 70.66 57.33 34.71 tension 2790.00 ok
APPENDIX A
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
153 16D12L05LR 931.599 W14X211 W14X211 62.00 162.00 4.08 39.75 181.18 44.55 tension 2790.00 ok
154 16D12L05LR -1445.559 W14X500 W14X500 147.00 660.87 4.43 149.30 12.84 11.26 compression 1489.75 ok
155 16D12L05LR 1508.296 W14X120 W14X120 35.30 662.31 3.74 176.87 9.15 8.02 tension 1588.50 ok
156 16D12L05LR 1534.182 W14X120 W14X120 35.30 662.31 3.74 176.87 9.15 8.02 tension 1588.50 ok
194 16D12L05LR 2519.525 W14X193 W14X193 56.80 162.00 4.05 40.01 178.76 44.48 tension 2556.00 ok
195 16D12L05LR 2524.974 W14X193 W14X193 56.80 162.00 4.05 40.01 178.76 44.48 tension 2556.00 ok
200 16D12L05LR 1447.665 W14X120 W14X120 35.30 162.00 3.74 43.26 152.93 43.61 tension 1588.50 ok
201 16D12L05LR 1418.431 W14X109 W14X109 32.00 162.00 3.74 43.34 152.35 43.58 tension 1440.00 ok
211 16D12L05LR -121.711 W14X145 W14X145 42.70 703.10 3.98 176.58 9.18 8.05 compression 309.38 ok
213 16D12L05LR -131.322 W14X145 W14X145 42.70 703.10 3.98 176.58 9.18 8.05 compression 309.38 ok
224 16D12L05LR -20.697 W14X109 W14X109 32.00 389.49 3.74 104.21 26.36 22.60 compression 650.89 ok
225 16D12L05LR -4.451 W14X145 W14X145 42.70 484.32 3.98 121.63 19.35 16.97 compression 652.02 ok
227 16D12L05LR -7.803 W14X145 W14X145 42.70 484.32 3.98 121.63 19.35 16.97 compression 652.02 ok
294 16D12L05LR -20.621 W14X109 W14X109 32.00 389.49 3.74 104.21 26.36 22.60 compression 650.89 ok
301 16D12L05LR 0 W14X120 W14X120 35.30 624.00 3.74 166.64 10.31 9.04 tension 287.20 ok
438 16D12L05LR -1000.31 W14X145 W14X145 42.70 324.00 3.98 81.37 43.23 30.81 compression 1184.11 ok
439 16D12L05LR 2153.385 W14X176 W14X176 51.80 274.82 4.02 68.33 61.30 35.54 tension 2331.00 ok
440 16D12L05LR -2170.61 W14X257 W14X257 75.60 274.82 4.13 66.53 64.66 36.18 compression 2461.39 ok
A1 Strength Checks
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
441 16D12L05LR 973.473 W14X233 W14X233 68.50 624.00 4.10 152.29 12.34 10.82 tension 3082.50 ok
442 16D12L05LR -20.933 W14X109 W14X109 32.00 389.49 3.74 104.21 26.36 22.60 compression 650.89 ok
443 16D12L05LR -24.501 W14X109 W14X109 32.00 389.49 3.74 104.21 26.36 22.60 compression 650.89 ok
444 16D12L05LR 0 W14X120 W14X120 35.30 624.00 3.74 166.64 10.31 9.04 tension 287.20 ok
445 16D12L05LR -980.8 W14X145 W14X145 42.70 324.00 3.98 81.37 43.23 30.81 compression 1184.11 ok
446 16D12L05LR 2067.595 W14X159 W14X159 46.70 274.82 4.00 68.67 60.70 35.42 tension 2101.50 ok
447 16D12L05LR -2085.147 W14X257 W14X257 75.60 274.82 4.13 66.53 64.66 36.18 compression 2461.39 ok
448 16D12L05LR 7.379 W14X109 W14X109 32.00 443.44 3.74 118.65 20.33 17.83 tension 1440.00 ok
449 16D12L05LR 18.992 W14X109 W14X109 32.00 443.44 3.74 118.65 20.33 17.83 tension 1440.00 ok
450 16D12L05LR 255.192 W14X120 W14X120 35.30 624.00 3.74 166.64 10.31 9.04 tension 1588.50 ok
451 16D12L05LR 1983.175 W14X159 W14X159 46.70 162.00 4.00 40.48 174.68 44.35 tension 2101.50 ok
452 16D12L05LR 2060.051 W14X159 W14X159 46.70 162.00 4.00 40.48 174.68 44.35 tension 2101.50 ok
453 16D12L05LR -86.664 W14X145 W14X145 42.70 703.10 3.98 176.58 9.18 8.05 compression 309.38 ok
454 16D12L05LR -90.888 W14X145 W14X145 42.70 703.10 3.98 176.58 9.18 8.05 compression 309.38 ok
455 16D12L05LR -1510.111 W14X283 W14X283 83.30 323.99 4.16 77.92 47.14 32.07 compression 2404.63 ok
456 16D12L05LR -60.878 W14X455 W14X455 134.00 562.85 4.37 128.77 17.26 15.14 compression 1825.57 ok
457 16D12L05LR -1621.576 W14X283 W14X283 83.30 324.00 4.16 77.93 47.13 32.07 compression 2404.53 ok
458 16D12L05LR 1803.266 W14X665 W14X665 196.00 400.84 4.61 86.90 37.90 28.78 tension 8820.00 ok
APPENDIX A
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
459 16D12L05LR -3722.083 W14X426 W14X426 125.00 319.02 4.35 73.42 53.10 33.71 compression 3792.72 ok
460 16D12L05LR 1404.176 W14X132 W14X132 38.80 274.82 3.76 73.13 53.52 33.82 tension 1746.00 ok
461 16D12L05LR -2365.009 W14X426 W14X426 125.00 274.82 4.35 63.25 71.55 37.32 compression 4198.46 ok
462 16D12L05LR 1457.408 W14X132 W14X132 38.80 274.82 3.76 73.12 53.53 33.82 tension 1746.00 ok
463 16D12L05LR -2739.556 W14X426 W14X426 125.00 274.82 4.35 63.25 71.55 37.32 compression 4198.46 ok
464 16D12L05LR 1227.45 W14X132 W14X132 38.80 274.82 3.76 73.12 53.53 33.82 tension 1746.00 ok
465 16D12L05LR -2388.169 W14X311 W14X311 91.40 274.82 4.20 65.48 66.75 36.54 compression 3006.10 ok
466 16D12L05LR -695.746 W14X145 W14X145 42.70 324.00 3.98 81.37 43.23 30.81 compression 1184.11 ok
467 16D12L05LR 1201.856 W14X132 W14X132 38.80 274.82 3.76 73.13 53.52 33.82 tension 1746.00 ok
468 16D12L05LR -1740.993 W14X311 W14X311 91.40 274.82 4.20 65.48 66.75 36.54 compression 3006.10 ok
469 16D12L05LR 2594.424 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
470 16D12L05LR 3151.151 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
471 16D12L05LR 3000.356 W14X311 W14X311 91.40 270.00 4.20 64.33 69.16 36.94 tension 4113.00 ok
472 16D12L05LR 503.762 W14X159 W14X159 46.70 336.00 4.00 83.96 40.61 29.86 tension 2101.50 ok
473 16D12L05LR -876.725 W14X283 W14X283 83.30 404.81 4.16 97.36 30.19 25.00 compression 1874.35 ok
474 16D12L05LR -876.8 W14X283 W14X283 83.30 404.83 4.16 97.37 30.19 25.00 compression 1874.24 ok
475 16D12L05LR 1227.69 W14X311 W14X311 91.40 330.44 4.20 78.73 46.17 31.78 tension 4113.00 ok
476 16D12L05LR 214.869 W14X109 W14X109 32.00 434.08 3.74 116.14 21.22 18.61 tension 1440.00 ok
A1 Strength Checks
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
477 16D12L05LR 228.358 W14X120 W14X120 35.30 398.09 3.74 106.31 25.33 21.88 tension 1588.50 ok
478 16D12L05LR 2508.232 W14X398 W14X398 117.00 229.10 4.31 53.20 101.14 40.65 tension 5265.00 ok
479 16D12L05LR -983.182 W14X211 W14X211 62.00 323.99 4.08 79.49 45.30 31.50 compression 1757.79 ok
480 16D12L05LR -1398.803 W14X211 W14X211 62.00 360.00 4.08 88.32 36.69 28.27 compression 1577.19 ok
481 16D12L05LR -18.174 W14X90 W14X90 26.50 222.00 3.70 60.06 79.33 38.41 compression 916.00 ok
482 16D12L05LR 1053.03 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
483 16D12L05LR -1128.375 W14X145 W14X145 42.70 324.00 3.98 81.37 43.23 30.81 compression 1184.11 ok
484 16D12L05LR 1806.741 W14X283 W14X283 83.30 274.82 4.16 66.10 65.52 36.33 tension 3748.50 ok
485 16D12L05LR -1830.956 W14X211 W14X211 62.00 274.82 4.08 67.43 62.96 35.86 compression 2000.96 ok
486 16D12L05LR 356.835 W14X145 W14X145 42.70 576.00 3.98 144.66 13.68 12.00 tension 1921.50 ok
487 16D12L05LR 319.632 W14X132 W14X132 38.80 216.00 3.76 57.47 86.65 39.27 tension 1746.00 ok
488 16D12L05LR 1314.59 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
489 16D12L05LR 2537.181 W14X193 W14X193 56.80 274.82 4.05 67.88 62.12 35.70 tension 2556.00 ok
490 16D12L05LR -2555.89 W14X283 W14X283 83.30 274.82 4.16 66.10 65.51 36.33 compression 2723.47 ok
491 16D12L05LR 63.997 W14X132 W14X132 38.80 288.00 3.76 76.63 48.74 32.55 tension 1746.00 ok
492 16D12L05LR -1434.39 W14X176 W14X176 51.80 323.99 4.02 80.55 44.11 31.11 compression 1450.45 ok
493 16D12L05LR 1249.942 W14X398 W14X398 117.00 319.02 4.31 74.08 52.16 33.48 tension 5265.00 ok
494 16D12L05LR 992.501 W14X398 W14X398 117.00 319.01 4.31 74.07 52.16 33.48 tension 5265.00 ok
APPENDIX A
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
495 16D12L05LR 599.362 W14X109 W14X109 32.00 434.08 3.74 116.14 21.22 18.61 tension 1440.00 ok
496 16D12L05LR 1138.62 W14X109 W14X109 32.00 398.09 3.74 106.51 25.23 21.81 tension 1440.00 ok
497 16D12L05LR 2454.096 W14X398 W14X398 117.00 638.20 4.31 148.19 13.03 11.43 tension 5265.00 ok
498 16D12L05LR 1533.552 W14X120 W14X120 35.30 349.54 3.74 93.34 32.85 26.44 tension 1588.50 ok
499 16D12L05LR 78.76 W14X120 W14X120 35.30 349.55 3.74 93.34 32.85 26.44 tension 1588.50 ok
500 16D12L05LR 168.857 W14X109 W14X109 32.00 638.20 3.74 170.76 9.82 8.61 tension 1440.00 ok
501 16D12L05LR 764.821 W14X145 W14X145 42.70 422.95 3.98 106.22 25.37 21.91 tension 1921.50 ok
502 16D12L05LR 0 W14X120 W14X120 35.30 336.00 3.74 89.73 35.55 27.75 tension 881.73 ok
503 16D12L05LR 0 W14X120 W14X120 35.30 288.00 3.74 76.91 48.39 32.44 tension 1030.77 ok
504 16D12L05LR 170.484 W14X120 W14X120 35.30 336.00 3.74 89.73 35.55 27.75 tension 1588.50 ok
505 16D12L05LR -229.088 W14X120 W14X120 35.30 288.00 3.74 76.91 48.39 32.44 compression 1030.77 ok
506 16D12L05LR -351.863 W14X176 W14X176 51.80 216.00 4.02 53.70 99.24 40.49 compression 1887.82 ok
507 16D12L05LR 595.113 W14X211 W14X211 62.00 576.00 4.08 141.32 14.33 12.57 tension 2790.00 ok
508 16D12L05LR 0 W14X120 W14X120 35.30 216.00 3.74 57.68 86.02 39.20 tension 1245.48 ok
509 16D12L05LR 0 W14X120 W14X120 35.30 576.00 3.74 153.82 12.10 10.61 tension 337.06 ok
510 16D12L05LR 1151.775 W14X145 W14X145 42.70 730.95 3.98 183.57 8.49 7.45 tension 1921.50 ok
511 16D12L05LR 627.397 W14X145 W14X145 42.70 505.75 3.98 127.01 17.74 15.56 tension 1921.50 ok
512 16D12L05LR -14.46 W14X145 W14X145 42.70 389.39 3.98 97.79 29.93 24.85 compression 954.91 ok
A1 Strength Checks
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
513 16D12L05LR 6.696 W14X109 W14X109 32.00 443.44 3.74 118.65 20.33 17.83 tension 1440.00 ok
514 16D12L05LR -637.375 W14X145 W14X145 42.70 433.50 3.98 108.87 24.15 21.02 compression 807.73 ok
515 16D12L05LR -615.992 W14X145 W14X145 42.70 466.77 3.98 117.23 20.83 18.27 compression 701.98 ok
516 16D12L05LR -269.341 W14X90 W14X90 26.50 222.00 3.70 60.06 79.33 38.41 compression 916.00 ok
517 16D12L05LR 1277.989 W14X211 W14X211 62.00 378.00 4.08 92.74 33.28 26.66 tension 2790.00 ok
518 16D12L05LR 826.782 W14X211 W14X211 62.00 288.00 4.08 70.66 57.33 34.71 tension 2790.00 ok
519 16D12L05LR 912.804 W14X211 W14X211 62.00 162.00 4.08 39.75 181.18 44.55 tension 2790.00 ok
520 16D12L05LR -1435.4 W14X500 W14X500 147.00 660.87 4.43 149.30 12.84 11.26 compression 1489.75 ok
521 16D12L05LR -1518.37 W14X283 W14X283 83.30 323.99 4.16 77.92 47.14 32.07 compression 2404.63 ok
522 16D12L05LR -53.181 W14X455 W14X455 134.00 562.85 4.37 128.77 17.26 15.14 compression 1825.57 ok
523 16D12L05LR -1622.572 W14X283 W14X283 83.30 324.00 4.16 77.93 47.13 32.07 compression 2404.53 ok
524 16D12L05LR 14.752 W14X109 W14X109 32.00 443.44 3.74 118.65 20.33 17.83 tension 1440.00 ok
525 16D12L05LR 1800.142 W14X665 W14X665 196.00 400.84 4.61 86.90 37.90 28.78 tension 8820.00 ok
526 16D12L05LR -3725.787 W14X426 W14X426 125.00 319.02 4.35 73.42 53.10 33.71 compression 3792.72 ok
527 16D12L05LR 1406.101 W14X132 W14X132 38.80 274.82 3.76 73.13 53.52 33.82 tension 1746.00 ok
528 16D12L05LR -2362.22 W14X426 W14X426 125.00 274.82 4.35 63.25 71.55 37.32 compression 4198.46 ok
529 16D12L05LR 1264.5 W14X311 W14X311 91.40 360.00 4.20 85.77 38.90 29.20 tension 4113.00 ok
530 16D12L05LR 1463.174 W14X132 W14X132 38.80 274.82 3.76 73.12 53.53 33.82 tension 1746.00 ok
APPENDIX A
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
531 16D12L05LR -2748.977 W14X426 W14X426 125.00 274.82 4.35 63.25 71.55 37.32 compression 4198.46 ok
532 16D12L05LR 1244.469 W14X132 W14X132 38.80 274.82 3.76 73.12 53.53 33.82 tension 1746.00 ok
533 16D12L05LR -2404.066 W14X311 W14X311 91.40 274.82 4.20 65.48 66.75 36.54 compression 3006.10 ok
534 16D12L05LR -688.872 W14X145 W14X145 42.70 324.00 3.98 81.37 43.23 30.81 compression 1184.11 ok
535 16D12L05LR 1195.936 W14X132 W14X132 38.80 274.82 3.76 73.13 53.52 33.82 tension 1746.00 ok
536 16D12L05LR -1729.39 W14X311 W14X311 91.40 274.82 4.20 65.48 66.75 36.54 compression 3006.10 ok
537 16D12L05LR 2590.539 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
538 16D12L05LR 3165.336 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
539 16D12L05LR 3003.509 W14X311 W14X311 91.40 270.00 4.20 64.33 69.16 36.94 tension 4113.00 ok
540 16D12L05LR 534.726 W14X159 W14X159 46.70 336.00 4.00 83.96 40.61 29.86 tension 2101.50 ok
541 16D12L05LR -879.815 W14X283 W14X283 83.30 404.81 4.16 97.36 30.19 25.00 compression 1874.35 ok
542 16D12L05LR -879.891 W14X283 W14X283 83.30 404.83 4.16 97.37 30.19 25.00 compression 1874.24 ok
543 16D12L05LR 1243.1 W14X311 W14X311 91.40 330.44 4.20 78.73 46.17 31.78 tension 4113.00 ok
544 16D12L05LR 212.333 W14X109 W14X109 32.00 434.08 3.74 116.14 21.22 18.61 tension 1440.00 ok
545 16D12L05LR 229.093 W14X120 W14X120 35.30 398.09 3.74 106.31 25.33 21.88 tension 1588.50 ok
546 16D12L05LR 2506.741 W14X398 W14X398 117.00 229.10 4.31 53.20 101.14 40.65 tension 5265.00 ok
547 16D12L05LR -1398.293 W14X211 W14X211 62.00 360.00 4.08 88.32 36.69 28.27 compression 1577.19 ok
548 16D12L05LR -18.174 W14X90 W14X90 26.50 222.00 3.70 60.06 79.33 38.41 compression 916.00 ok
A1 Strength Checks
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
549 16D12L05LR 1058.677 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
550 16D12L05LR 355.384 W14X145 W14X145 42.70 576.00 3.98 144.66 13.68 12.00 tension 1921.50 ok
551 16D12L05LR 313.304 W14X132 W14X132 38.80 216.00 3.76 57.47 86.65 39.27 tension 1746.00 ok
552 16D12L05LR 1311.134 W14X311 W14X311 91.40 162.00 4.20 38.60 192.11 44.84 tension 4113.00 ok
553 16D12L05LR 74.583 W14X132 W14X132 38.80 288.00 3.76 76.63 48.74 32.55 tension 1746.00 ok
554 16D12L05LR -1448.115 W14X176 W14X176 51.80 323.99 4.02 80.55 44.11 31.11 compression 1450.45 ok
555 16D12L05LR 1248.555 W14X398 W14X398 117.00 319.02 4.31 74.08 52.16 33.48 tension 5265.00 ok
556 16D12L05LR 1000.077 W14X398 W14X398 117.00 319.01 4.31 74.07 52.16 33.48 tension 5265.00 ok
557 16D12L05LR 592.924 W14X109 W14X109 32.00 434.08 3.74 116.14 21.22 18.61 tension 1440.00 ok
558 16D12L05LR 1131.058 W14X109 W14X109 32.00 398.09 3.74 106.51 25.23 21.81 tension 1440.00 ok
559 16D12L05LR 2452.14 W14X398 W14X398 117.00 638.20 4.31 148.19 13.03 11.43 tension 5265.00 ok
560 16D12L05LR 1541.522 W14X120 W14X120 35.30 349.54 3.74 93.34 32.85 26.44 tension 1588.50 ok
561 16D12L05LR 75.439 W14X120 W14X120 35.30 349.55 3.74 93.34 32.85 26.44 tension 1588.50 ok
562 16D12L05LR 168.203 W14X109 W14X109 32.00 638.20 3.74 170.76 9.82 8.61 tension 1440.00 ok
563 16D12L05LR 764.725 W14X145 W14X145 42.70 422.95 3.98 106.22 25.37 21.91 tension 1921.50 ok
564 16D12L05LR 0 W14X120 W14X120 35.30 336.00 3.74 89.73 35.55 27.75 tension 881.73 ok
565 16D12L05LR 0 W14X120 W14X120 35.30 288.00 3.74 76.91 48.39 32.44 tension 1030.77 ok
566 16D12L05LR 160.998 W14X120 W14X120 35.30 336.00 3.74 89.73 35.55 27.75 tension 1588.50 ok
APPENDIX A
Frame OutputCase P Model Section Design Section Area Length r KL/r Fe Fcr
Max adm force ok?
567 16D12L05LR -232.83 W14X120 W14X120 35.30 288.00 3.74 76.91 48.39 32.44 compression 1030.77 ok
568 16D12L05LR -351.353 W14X176 W14X176 51.80 216.00 4.02 53.70 99.24 40.49 compression 1887.82 ok
569 16D12L05LR 600.547 W14X211 W14X211 62.00 576.00 4.08 141.32 14.33 12.57 tension 2790.00 ok
570 16D12L05LR 0 W14X120 W14X120 35.30 216.00 3.74 57.68 86.02 39.20 tension 1245.48 ok
571 16D12L05LR 0 W14X120 W14X120 35.30 576.00 3.74 153.82 12.10 10.61 tension 337.06 ok
572 16D12L05LR 1158.923 W14X145 W14X145 42.70 730.95 3.98 183.57 8.49 7.45 tension 1921.50 ok
573 16D12L05LR 660.401 W14X145 W14X145 42.70 505.75 3.98 127.01 17.74 15.56 tension 1921.50 ok
574 16D12L05LR -13.921 W14X145 W14X145 42.70 389.39 3.98 97.79 29.93 24.85 compression 954.91 ok
575 16D12L05LR -637.785 W14X145 W14X145 42.70 433.50 3.98 108.87 24.15 21.02 compression 807.73 ok
576 16D12L05LR -635.614 W14X145 W14X145 42.70 466.77 3.98 117.23 20.83 18.27 compression 701.98 ok
577 16D12L05LR -267.877 W14X90 W14X90 26.50 222.00 3.70 60.06 79.33 38.41 compression 916.00 ok
578 16D12L05LR 1258.648 W14X211 W14X211 62.00 378.00 4.08 92.74 33.28 26.66 tension 2790.00 ok
579 16D12L05LR 843.9 W14X211 W14X211 62.00 288.00 4.08 70.66 57.33 34.71 tension 2790.00 ok
580 16D12L05LR 929.984 W14X211 W14X211 62.00 162.00 4.08 39.75 181.18 44.55 tension 2790.00 ok
581 16D12L05LR -1436.963 W14X500 W14X500 147.00 660.87 4.43 149.30 12.84 11.26 compression 1489.75 ok
597 16D12L05LR -984.412 W14X211 W14X211 62.00 323.99 4.08 79.49 45.30 31.50 compression 1757.79 ok
602 16D12L05LR 1815.008 W14X283 W14X283 83.30 274.82 4.16 66.10 65.52 36.33 tension 3748.50 ok
603 16D12L05LR -1839.317 W14X211 W14X211 62.00 274.82 4.08 67.43 62.96 35.86 compression 2000.96 ok
A2 Optimum Position Algorithm
Optimum Position
%Calculate the position
clc
clear all
%% Calculate Michell's Volume
a=27;
P=24.5*12^2;
Q=24.5*12^2;
F=1000;
Michell_volume=F*a*pi/2*(1/P+1/Q)
%% Define the geometry
H=24;
B=27;
H2=6.75;
B2=18;
P=1000;
npoints=100;
sigma=2*24.5*12^2;
xstep=B/npoints;
ystep=H/npoints;
%% Calculate optimum position - no constraints
Volume= zeros(1,npoints);
cx=zeros(1,npoints);
cy=zeros(1,npoints);
for i=1:npoints
cx(i)=i*xstep;
for k=1:npoints
cy(k)=k*ystep;
if cy(k) < H/B*cx(i)
Volume(i,k)=maxwell_3(cx(i),cy(k),H,B,P,H2,B2);
else
Volume(i,k)=10^6;
APPENDIX A
end
end
end
minvalue=min(Volume(:));
[r,c]= find(Volume==minvalue);
x_min=r*xstep
y_min=c*ystep
minvalue=min(Volume(:))/sigma
maxwell_excedance=(minvalue-
Michell_volume)/Michell_volume
%% Calculate optimum position - y constraint at floor
level
Volume2= zeros(1,npoints);
cx2=zeros(1,npoints);
cy2 = 6.75;
for i=1:npoints
cx2(i)=i*xstep;
if cy2 < H/B*cx2(i)
if cx2(i)> cy2*B/H
Volume2(i)=maxwell_3(cx2(i),cy2,H,B,P,H2,B2);
else
Volume2(i)=10^6;
end
else
Volume2(i)=10^6;
end
end
minvalue2=min(Volume2);
r2= find(Volume2==minvalue2);
A2 Optimum Position Algorithm
x_min2=r2*xstep
y_min2=cy2
minvalue2=min(Volume2)/sigma
maxwell_excedance=(minvalue2-
Michell_volume)/Michell_volume
Maxwell Function
function maxwell = maxwell_3(x,y,H,B,P,H2,B2)
F= zeros(1,7);
L= zeros(1,7);
maxwell=0;
X=[x,y];
a = atan( X(2)/X(1));
b = atan((H-X(2))/(B-X(1)));
c = atan((H-X(2))/X(1));
d = atan(H2/B2);
e = atan((H-H2)/B2);
V2=P*B/B2;
V1=P+V2;
% F3 F4 F6 F1 positive compression, F5 F2 F7 positive
tension
F(6) = V2/(sin(d)+cos(d)*tan(e));
F(7) = F(6)*cos(d)/cos(e);
F(4) = F(6)*cos(d)/cos(a);
F(1) = V1 - F(4)*sin(a) - F(6)*sin(d);
F(3) = P/sin(b);
F(2) = F(3)*cos(b);
F(5)= (F(3)*sin(b)-F(4)*sin(a))/sin(c);
eq = F(3)*cos(b)+F(5)*cos(c)-F(4)*cos(a);
n=length(F);
L(1) = H;
L(2) = B;
L(3) = (H-X(2))/ sin(atan(b));
L(4) = X(2)/ sin(atan(a));
L(5) = (H-X(2))/ sin(atan(c));
APPENDIX A
L(6) = B2/cos(atan(d));
L(7) = B2/cos(atan(e));
for i=1:n
if F(i)< 0
F(i)=-F(i);
end
% F(i)=abs(F(i));
maxwell = maxwell + F(i)* L(i);
end
end
APPENDIX B Case Study: 111 South Main
B.1 Design of Diagonal Members
B.2 Design of Steel Beams
B.3 Design of Steel Columns
APPENDIX B
Appendix B1 – Design of DIAGONALS MEMBERS
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
ROOF 1440.1 -208.0
D128 1343.8 -141.6 W14X159 W14X159 46.70 248 62.03 50 2101.5 1586.1 Tension 0.64
D132 1403.9 -208.0 W14X159 W14X159 46.70 248 62.03 50 2101.5 1586.1 Tension 0.67
D146 1440.1 126.0 W14X159 W14X159 46.70 234 58.39 50 2101.5 1637.8 Tension 0.69
D150 1346.6 175.6 W14X159 W14X159 46.70 234 58.39 50 2101.5 1637.8 Tension 0.64
L24 1356.0 81.3
D127 1250.9 81.3 W14X159 W14X159 46.70 248 62.03 50 2101.5 1586.1 Tension 0.60
D131 1312.4 146.1 W14X159 W14X159 46.70 248 62.03 50 2101.5 1586.1 Tension 0.62
D145 1356.0 96.4 W14X159 W14X159 46.70 234 58.39 50 2101.5 1637.8 Tension 0.65
D149 1262.2 145.7 W14X159 W14X159 46.70 234 58.39 50 2101.5 1637.8 Tension 0.60
L23 1266.8 50.4
D126 1168.8 50.4 W14X159 W14X159 46.70 248 62.03 50 2101.5 1586.1 Tension 0.56
D130 1232.3 116.6 W14X159 W14X159 46.70 248 62.03 50 2101.5 1586.1 Tension 0.59
D144 1266.8 64.8 W14X159 W14X159 46.70 234 58.39 50 2101.5 1637.8 Tension 0.60
D148 1173.3 113.7 W14X159 W14X159 46.70 234 58.39 50 2101.5 1637.8 Tension 0.56
L22 6982.0 -126.2
D154 6456.8 952.2 W14X605 W14X605 178.00 263 57.78 50 8010.0 6275.2 Tension 0.81
D158 1751.8 309.2 W14X193 W14X193 56.80 259 63.90 50 2556.0 1896.2 Tension 0.69
D162 1809.3 90.0 W14X193 W14X193 56.80 234 57.72 50 2556.0 2003.3 Tension 0.71
D166 1843.9 266.2 W14X193 W14X193 56.80 266 65.70 50 2556.0 1864.2 Tension 0.72
D170 6942.9 737.8 W14X605 W14X605 178.00 276 60.64 50 8010.0 6121.3 Tension 0.87
D174 1714.3 -126.2 W14X193 W14X193 56.80 248 61.32 50 2556.0 1941.6 Tension 0.67
D178 6982.0 787.9 W14X605 W14X605 178.00 276 60.64 50 8010.0 6121.3 Tension 0.87
D182 1807.3 311.3 W14X193 W14X193 56.80 266 65.70 50 2556.0 1864.2 Tension 0.71
D186 1762.9 308.5 W14X193 W14X193 56.80 259 63.90 50 2556.0 1896.2 Tension 0.69
D190 6432.4 984.7 W14X605 W14X605 178.00 263 57.78 50 8010.0 6275.2 Tension 0.80
B.1 Design of Diagonal Members
Appendix B1 – Design of DIAGONALS MEMBERS
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L21 6955.3 42.2
D153 6430.8 940.6 W14X605 W14X605 178.00 263 57.78 50 8010.0 6275.2 Tension 0.80
D157 1423.6 227.1 W14X193 W14X193 56.80 259 63.90 50 2556.0 1896.2 Tension 0.56
D161 1721.5 59.2 W14X193 W14X193 56.80 234 57.72 50 2556.0 2003.3 Tension 0.67
D165 1469.8 174.2 W14X193 W14X193 56.80 266 65.70 50 2556.0 1864.2 Tension 0.58
D169 6914.7 728.7 W14X605 W14X605 178.00 276 60.64 50 8010.0 6121.3 Tension 0.86
D173 1625.4 42.2 W14X193 W14X193 56.80 248 61.32 50 2556.0 1941.6 Tension 0.64
D177 6955.3 779.7 W14X605 W14X605 178.00 276 60.64 50 8010.0 6121.3 Tension 0.87
D181 1435.8 216.1 W14X193 W14X193 56.80 266 65.70 50 2556.0 1864.2 Tension 0.56
D185 1432.6 225.8 W14X193 W14X193 56.80 259 63.90 50 2556.0 1896.2 Tension 0.56
D189 6406.3 973.8 W14X605 W14X605 178.00 263 57.78 50 8010.0 6275.2 Tension 0.80
L20 6781.3 11.9
D152 6289.5 912.5 W14X605 W14X605 178.00 263 57.78 50 8010.0 6275.2 Tension 0.79
D156 1341.2 205.7 W14X193 W14X193 56.80 259 63.90 50 2556.0 1896.2 Tension 0.52
D160 1631.7 28.2 W14X193 W14X193 56.80 234 57.72 50 2556.0 2003.3 Tension 0.64
D164 1415.3 158.7 W14X193 W14X193 56.80 266 65.70 50 2556.0 1864.2 Tension 0.55
D168 6741.6 696.8 W14X605 W14X605 178.00 276 60.64 50 8010.0 6121.3 Tension 0.84
D172 1543.4 11.9 W14X193 W14X193 56.80 248 61.32 50 2556.0 1941.6 Tension 0.60
D176 6781.3 748.8 W14X605 W14X605 178.00 276 60.64 50 8010.0 6121.3 Tension 0.85
D180 1381.8 199.7 W14X193 W14X193 56.80 266 65.70 50 2556.0 1864.2 Tension 0.54
D184 1349.9 204.3 W14X193 W14X193 56.80 259 63.90 50 2556.0 1896.2 Tension 0.53
D188 6265.4 944.8 W14X605 W14X605 178.00 263 57.78 50 8010.0 6275.2 Tension 0.78
L19 1444.8 -405.9
D128 1342.5 -335.5 W14X159 W14X159 46.70 248 62.03 50 2101.5 1586.1 Tension 0.64
D132 1369.0 -405.9 W14X159 W14X159 46.70 248 62.03 50 2101.5 1586.1 Tension 0.65
D146 1444.8 29.0 W14X159 W14X159 46.70 234 58.39 50 2101.5 1637.8 Tension 0.69
APPENDIX B
Appendix B1 – Design of DIAGONALS MEMBERS
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
D150 1386.1 82.1 W14X159 W14X159 46.70 234 58.39 50 2101.5 1637.8 Tension 0.66
L18 1365.2 2.4
D127 1256.8 34.9 W14X159 W14X159 46.70 248 62.03 50 2101.5 1586.1 Tension 0.60
D131 1283.2 77.2 W14X159 W14X159 46.70 248 62.03 50 2101.5 1586.1 Tension 0.61
D145 1365.2 2.4 W14X159 W14X159 46.70 234 58.39 50 2101.5 1637.8 Tension 0.65
D149 1306.4 55.3 W14X159 W14X159 46.70 234 58.39 50 2101.5 1637.8 Tension 0.62
L17 1273.0 -27.7
D126 1171.5 6.1 W14X159 W14X159 46.70 248 62.03 50 2101.5 1586.1 Tension 0.56
D130 1197.9 48.2 W14X159 W14X159 46.70 248 62.03 50 2101.5 1586.1 Tension 0.57
D144 1273.0 -27.7 W14X159 W14X159 46.70 234 58.39 50 2101.5 1637.8 Tension 0.61
D148 1214.6 24.6 W14X159 W14X159 46.70 234 58.39 50 2101.5 1637.8 Tension 0.58
L16 4211.9 -507.3
D162 1553.3 289.6 W14X193 W14X193 56.80 234 57.72 50 2556.0 2003.3 Tension 0.61
D174 1661.7 -507.3 W14X193 W14X193 56.80 248 61.32 50 2556.0 1941.6 Tension 0.65
D194 4125.8 776.5 W14X370 W14X370 109.00 245 57.25 50 4905.0 3859.8 Tension 0.84
D198 4211.9 888.7 W14X370 W14X370 109.00 247 57.72 50 4905.0 3844.7 Tension 0.86
D202 4173.9 879.9 W14X370 W14X370 109.00 247 57.72 50 4905.0 3844.7 Tension 0.85
D206 4046.5 806.6 W14X370 W14X370 109.00 245 57.25 50 4905.0 3859.8 Tension 0.82
L15 3761.4 261.6
D161 1464.2 261.6 W14X193 W14X193 56.80 234 57.72 50 2556.0 2003.3 Tension 0.57
D173 1567.7 281.1 W14X193 W14X193 56.80 248 61.32 50 2556.0 1941.6 Tension 0.61
D193 3718.3 658.2 W14X370 W14X370 109.00 245 57.25 50 4905.0 3859.8 Tension 0.76
D197 3761.4 761.2 W14X370 W14X370 109.00 247 57.72 50 4905.0 3844.7 Tension 0.77
D201 3724.9 751.1 W14X370 W14X370 109.00 247 57.72 50 4905.0 3844.7 Tension 0.76
D205 3642.2 688.1 W14X370 W14X370 109.00 245 57.25 50 4905.0 3859.8 Tension 0.74
L14 3294.6 232.6
B.1 Design of Diagonal Members
Appendix B1 – Design of DIAGONALS MEMBERS
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
D160 1372.2 232.6 W14X193 W14X193 56.80 234 57.72 50 2556.0 2003.3 Tension 0.54
D172 1481.6 254.0 W14X193 W14X193 56.80 248 61.32 50 2556.0 1941.6 Tension 0.58
D192 3277.9 530.8 W14X370 W14X370 109.00 245 57.25 50 4905.0 3859.8 Tension 0.67
D196 3294.6 623.9 W14X370 W14X370 109.00 247 57.72 50 4905.0 3844.7 Tension 0.67
D200 3264.2 613.1 W14X370 W14X370 109.00 247 57.72 50 4905.0 3844.7 Tension 0.67
D204 3206.4 560.7 W14X370 W14X370 109.00 245 57.25 50 4905.0 3859.8 Tension 0.65
L13 1846.1 -771.9
D128 1846.1 -771.9 W14X211 W14X211 62.00 248 60.91 50 2790.0 2127.1 Tension 0.66
D132 1814.4 -725.6 W14X211 W14X211 62.00 248 60.91 50 2790.0 2127.1 Tension 0.65
D146 1622.5 144.1 W14X211 W14X211 62.00 234 57.34 50 2790.0 2193.9 Tension 0.58
D150 1631.4 179.0 W14X211 W14X211 62.00 234 57.34 50 2790.0 2193.9 Tension 0.58
L12 1757.9 116.7
D127 1757.9 166.1 W14X211 W14X211 62.00 248 60.91 50 2790.0 2127.1 Tension 0.63
D131 1724.6 177.7 W14X211 W14X211 62.00 248 60.91 50 2790.0 2127.1 Tension 0.62
D145 1537.1 116.7 W14X211 W14X211 62.00 234 57.34 50 2790.0 2193.9 Tension 0.55
D149 1544.3 151.7 W14X211 W14X211 62.00 234 57.34 50 2790.0 2193.9 Tension 0.55
L11 1668.7 86.9
D126 1668.7 137.9 W14X211 W14X211 62.00 248 60.91 50 2790.0 2127.1 Tension 0.60
D130 1634.8 150.1 W14X211 W14X211 62.00 248 60.91 50 2790.0 2127.1 Tension 0.59
D144 1442.0 86.9 W14X211 W14X211 62.00 234 57.34 50 2790.0 2193.9 Tension 0.52
D148 1448.5 122.1 W14X211 W14X211 62.00 234 57.34 50 2790.0 2193.9 Tension 0.52
L10 4170.0 -875.8
D162 2372.7 -58.7 W14X233 W14X233 68.50 234 57.04 50 3082.5 2430.0 Tension 0.77
D174 2847.3 -875.8 W14X233 W14X233 68.50 248 60.59 50 3082.5 2356.8 Tension 0.92
D194 4121.4 557.4 W14X342 W14X342 101.00 245 57.78 50 4545.0 3560.5 Tension 0.91
D198 4085.4 585.0 W14X342 W14X342 101.00 247 58.25 50 4545.0 3546.3 Tension 0.90
APPENDIX B
Appendix B1 – Design of DIAGONALS MEMBERS
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
D202 4126.3 587.9 W14X342 W14X342 101.00 247 58.25 50 4545.0 3546.3 Tension 0.91
D206 4170.0 566.6 W14X342 W14X342 101.00 245 57.78 50 4545.0 3560.5 Tension 0.92
L09 3762.6 -87.6
D161 2281.0 -87.6 W14X233 W14X233 68.50 234 57.04 50 3082.5 2430.0 Tension 0.74
D173 2763.2 18.2 W14X233 W14X233 68.50 248 60.59 50 3082.5 2356.8 Tension 0.90
D193 3715.8 445.9 W14X342 W14X342 101.00 245 57.78 50 4545.0 3560.5 Tension 0.82
D197 3652.8 465.7 W14X342 W14X342 101.00 247 58.25 50 4545.0 3546.3 Tension 0.80
D201 3697.5 468.2 W14X342 W14X342 101.00 247 58.25 50 4545.0 3546.3 Tension 0.81
D205 3762.6 457.3 W14X342 W14X342 101.00 245 57.78 50 4545.0 3560.5 Tension 0.83
L08 3320.8 -116.7
D160 2184.4 -116.7 W14X233 W14X233 68.50 234 57.04 50 3082.5 2430.0 Tension 0.71
D172 2675.6 -10.8 W14X233 W14X233 68.50 248 60.59 50 3082.5 2356.8 Tension 0.87
D192 3275.9 327.5 W14X342 W14X342 101.00 245 57.78 50 4545.0 3560.5 Tension 0.72
D196 3185.6 339.8 W14X342 W14X342 101.00 247 58.25 50 4545.0 3546.3 Tension 0.70
D200 3232.7 342.3 W14X342 W14X342 101.00 247 58.25 50 4545.0 3546.3 Tension 0.71
D204 3320.8 341.6 W14X342 W14X342 101.00 245 57.78 50 4545.0 3560.5 Tension 0.73
L07 2875.5 -554.3
D221 2875.5 -491.9 W14X233 W14X233 68.50 323 78.95 50 3082.5 1954.2 Tension 0.93
D224 2819.5 -554.3 W14X233 W14X233 68.50 323 78.95 50 3082.5 1954.2 Tension 0.91
D227 2481.4 -465.8 W14X233 W14X233 68.50 298 72.75 50 3082.5 2093.3 Tension 0.80
D230 2392.6 -384.9 W14X233 W14X233 68.50 298 72.75 50 3082.5 2093.3 Tension 0.78
L06 2771.4 -499.6
D220 2771.4 -289.5 W14X233 W14X233 68.50 323 78.95 50 3082.5 1954.2 Tension 0.90
D223 2714.2 -324.9 W14X233 W14X233 68.50 323 78.95 50 3082.5 1954.2 Tension 0.88
D226 2369.8 -499.6 W14X233 W14X233 68.50 298 72.75 50 3082.5 2093.3 Tension 0.77
D229 2279.4 -419.0 W14X233 W14X233 68.50 298 72.75 50 3082.5 2093.3 Tension 0.74
B.2 Design of Steel Beams
Appendix B2 Design of Steel Beams
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L24 65.8 -56.4
B71 65.8 -35.7 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.11
B73 58.8 -56.4 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.10
B93 55.0 -40.1 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.09
B96 49.6 -48.8 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.08
L23 18.6 -59.3
B71 15.0 -56.7 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.10
B73 14.6 -59.3 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.10
B93 16.8 -58.1 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.10
B96 18.6 -56.3 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.10
L22 104.9 -315.0
B71 74.9 -315.0 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.54
B73 104.9 -299.7 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.51
B93 -29.1 -292.8 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.50
B96 -16.2 -311.8 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.53
L21 60.6 -50.8
B69 33.0 -50.8 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.09
B94 27.7 -49.8 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.09
B113 45.8 -18.5 W21X44 W21X44 13.00 600 475.49 50 585.0 13.0 Compression 0.08
B158 46.8 -14.5 W21X44 W21X44 13.00 600 475.49 50 585.0 13.0 Compression 0.08
B185 60.6 -15.1 W21X44 W21X44 13.00 600 475.49 50 585.0 13.0 Compression 0.10
B188 59.2 -16.5 W21X44 W21X44 13.00 600 475.49 50 585.0 13.0 Compression 0.10
L20 32.7 -100.0
B69 25.3 -45.9 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.08
B94 25.7 -47.5 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.08
B115 27.3 -73.0 W21X44 W21X44 13.00 600 475.49 50 585.0 13.0 Compression 0.12
APPENDIX B
Appendix B2 Design of Steel Beams
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
B120 10.2 -22.8 W21X44 W21X44 13.00 600 475.49 50 585.0 13.0 Compression 0.04
B154 10.1 -22.7 W21X44 W21X44 13.00 600 475.49 50 585.0 13.0 Compression 0.04
B156 27.2 -72.8 W21X44 W21X44 13.00 600 475.49 50 585.0 13.0 Compression 0.12
B181 31.7 -99.8 W21X44 W21X44 13.00 600 475.49 50 585.0 13.0 Compression 0.17
B182 5.8 -26.9 W21X44 W21X44 13.00 600 475.49 50 585.0 13.0 Compression 0.05
B186 6.1 -27.1 W21X44 W21X44 13.00 600 475.49 50 585.0 13.0 Compression 0.05
B187 32.7 -100.0 W21X44 W21X44 13.00 600 475.49 50 585.0 13.0 Compression 0.17
L19 19.0 -310.4
B69 19.0 -297.4 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.51
B94 -31.0 -310.4 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.53
L18 46.2 -67.3
B71 46.2 -57.4 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.10
B73 32.2 -67.3 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.12
B93 34.9 -48.1 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.08
B96 32.2 -54.3 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.09
L17 46.9 -58.9
B71 46.9 -58.2 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.10
B73 33.7 -58.5 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.10
B93 16.7 -58.7 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.10
B96 16.6 -58.9 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.10
L16 41.1 -306.1
B71 41.1 -300.2 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.51
B73 27.1 -294.5 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.50
B93 -2.9 -292.6 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.50
B96 6.2 -306.1 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.52
L15 140.1 -235.0
B.2 Design of Steel Beams
Appendix B2 Design of Steel Beams
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
B53 128.8 -224.5 W24X146 W24X146 43.00 542 179.66 50 1935.0 301.0 Compression 0.12
B69 30.8 -72.6 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.12
B94 29.9 -59.4 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.10
B99 131.2 -234.3 W24X146 W24X146 43.00 550 182.34 50 1935.0 292.2 Compression 0.12
B197 112.3 -235.0 W24X146 W24X146 43.00 542 179.66 50 1935.0 301.0 Compression 0.12
B206 140.1 -217.5 W24X146 W24X146 43.00 550 182.34 50 1935.0 292.2 Compression 0.11
L14 36.1 -236.8
B53 -4.0 -217.8 W24X146 W24X146 43.00 542 179.66 50 1935.0 301.0 Compression 0.11
B69 36.1 -52.3 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.09
B94 25.4 -49.8 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.09
B99 -5.1 -236.8 W24X146 W24X146 43.00 550 182.34 50 1935.0 292.2 Compression 0.12
B197 -2.9 -223.2 W24X146 W24X146 43.00 542 179.66 50 1935.0 301.0 Compression 0.12
B206 -1.8 -227.0 W24X146 W24X146 43.00 550 182.34 50 1935.0 292.2 Compression 0.12
L13 38.7 -1296.5
B69 38.7 -326.0 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.56
B94 -26.5 -299.4 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.51
B112 -39.6 -1286.2 W24X146 W24X146 43.00 540 179.08 50 1935.0 302.9 Compression 0.66
B124 -175.5 -1117.2 W24X146 W24X146 43.00 540 179.08 50 1935.0 302.9 Compression 0.58
B134 -166.6 -1140.1 W24X146 W24X146 43.00 540 179.08 50 1935.0 302.9 Compression 0.59
B151 -50.3 -1296.5 W24X146 W24X146 43.00 540 179.08 50 1935.0 302.9 Compression 0.67
L12 74.8 -66.6
B71 74.8 -60.7 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.13
B73 51.5 -66.6 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.11
B93 34.0 -64.1 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.11
B96 35.2 -62.2 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.11
L11 77.9 -62.6
APPENDIX B
Appendix B2 Design of Steel Beams
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
B71 77.9 -62.2 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.13
B73 53.9 -62.6 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.11
B93 17.8 -62.2 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.11
B96 15.8 -62.3 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.11
L10 87.9 -476.9
B71 87.9 -468.7 W21X55 W21X55 16.20 557 321.96 50 729.0 35.3 Compression 0.64
B73 51.1 -476.9 W21X55 W21X55 16.20 557 321.96 50 729.0 35.3 Compression 0.65
B93 -0.3 -352.2 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.60
B96 -4.9 -350.2 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.60
L09 131.9 -247.1
B53 117.7 -233.2 W24X146 W24X146 43.00 542 179.66 50 1935.0 301.0 Compression 0.12
B69 52.9 -72.3 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.12
B94 19.1 -71.8 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.12
B99 129.6 -247.1 W24X146 W24X146 43.00 550 182.34 50 1935.0 292.2 Compression 0.13
B197 106.0 -240.5 W24X146 W24X146 43.00 542 179.66 50 1935.0 301.0 Compression 0.12
B206 131.9 -231.9 W24X146 W24X146 43.00 550 182.34 50 1935.0 292.2 Compression 0.12
L08 62.3 -219.5
B53 13.9 -212.4 W24X146 W24X146 43.00 542 179.66 50 1935.0 301.0 Compression 0.11
B69 62.3 -52.8 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.11
B94 26.0 -54.7 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.09
B99 8.8 -219.5 W24X146 W24X146 43.00 550 182.34 50 1935.0 292.2 Compression 0.11
B197 6.8 -218.6 W24X146 W24X146 43.00 542 179.66 50 1935.0 301.0 Compression 0.11
B206 7.9 -202.0 W24X146 W24X146 43.00 550 182.34 50 1935.0 292.2 Compression 0.10
L07 82.7 -1301.1
B69 82.7 -801.9 W21X68 W21X68 20.00 557 309.41 50 900.0 47.2 Compression 0.89
B94 39.0 -501.2 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.86
B.2 Design of Steel Beams
Appendix B2 Design of Steel Beams
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
B112 -27.6 -1301.1 W24X146 W24X146 43.00 540 179.08 50 1935.0 302.9 Compression 0.67
B124 -81.2 -1198.3 W24X146 W24X146 43.00 540 179.08 50 1935.0 302.9 Compression 0.62
B134 -77.5 -1180.1 W24X146 W24X146 43.00 540 179.08 50 1935.0 302.9 Compression 0.61
B151 -33.3 -1285.9 W24X146 W24X146 43.00 540 179.08 50 1935.0 302.9 Compression 0.66
L06 117.0 -79.4
B71 117.0 -68.2 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.20
B73 74.5 -79.4 W21X44 W21X44 13.00 557 441.01 50 585.0 15.1 Compression 0.14
B93 68.2 -66.5 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.12
B96 69.1 -72.2 W21X44 W21X44 13.00 497 393.46 50 585.0 19.0 Compression 0.12
L05 219.5 -1206.0
B71 210.1 -1178.1 W21X111 W21X111 32.70 557 192.25 50 1471.5 199.9 Compression 0.80
B73 130.5 -1206.0 W21X111 W21X111 32.70 557 192.25 50 1471.5 199.9 Compression 0.82
B93 188.3 -973.2 W21X111 W21X111 32.70 497 171.52 50 1471.5 251.1 Compression 0.66
B96 219.5 -1010.2 W21X111 W21X111 32.70 497 171.52 50 1471.5 251.1 Compression 0.69
APPENDIX B
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
ROOFC2 -59.0 -281.3 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Compression 0.43
ROOFC3 -109.3 -377.2 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.20
ROOFC7 37.6 -154.8 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.22
ROOFC12 20.3 -198.1 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.28
ROOFC13 -36.5 -259.7 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.25
ROOFC18 -48.0 -270.8 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.26
ROOFC24 -37.0 -172.2 W14X61 W14X61 17.90 165 67.49 50 805.5 577.4 Compression 0.30
ROOFC19 -41.8 -201.8 W14X61 W14X61 17.90 165 67.49 50 805.5 577.4 Compression 0.35
ROOFC25 -57.2 -310.3 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.30
ROOFC30 -50.6 -263.4 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.26
ROOFC31 24.7 -193.7 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.27
ROOFC36 28.1 -205.6 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.29
ROOFC38 -66.5 -279.3 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Compression 0.43
ROOFC39 -100.9 -377.0 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.20
ROOFC40 -97.1 -374.1 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.20
ROOFC41 -65.6 -271.4 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Compression 0.42
ROOFC4 -99.4 -373.7 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.20
ROOFC5 -65.0 -267.1 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Compression 0.41
L24C2 -63.3 -310.9 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Compression 0.48
L24C3 -153.1 -597.5 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.32
L24C7 0.7 -341.3 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.48
L24C12 -19.9 -385.1 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.54
L24C13 -67.1 -412.7 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.40
L24C18 -78.2 -409.7 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.40
L24C24 -70.9 -325.8 W14X61 W14X61 17.90 165 67.49 50 805.5 577.4 Compression 0.56
L24C19 -74.3 -360.1 W14X61 W14X61 17.90 165 67.49 50 805.5 577.4 Compression 0.62
B.3 Design of Steel Columns
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L24C25 -86.7 -463.8 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.45
L24C30 -80.1 -403.4 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.39
L24C31 -8.6 -384.6 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.54
L24C36 -13.0 -391.3 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.55
L24C38 -71.6 -308.1 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Compression 0.48
L24C39 -143.4 -598.5 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.32
L24C40 -143.4 -591.8 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.32
L24C41 -70.3 -300.7 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Compression 0.46
L24C4 -147.9 -589.3 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.32
L24C5 -69.9 -296.2 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Compression 0.46
L23C2 -67.5 -340.5 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Compression 0.53
L23C3 -195.2 -819.8 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.44
L23C7 -22.3 -545.6 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.77
L23C12 -54.8 -580.7 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.82
L23C13 -91.2 -543.7 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.53
L23C18 -100.9 -525.1 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.51
L23C24 -103.9 -480.1 W14X61 W14X61 17.90 165 67.49 50 805.5 577.4 Compression 0.83
L23C19 -105.6 -519.3 W14X61 W14X61 17.90 165 67.49 50 805.5 577.4 Compression 0.90
L23C25 -109.9 -596.3 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.58
L23C30 -102.3 -520.1 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.50
L23C31 -32.4 -588.5 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.83
L23C36 -48.2 -585.8 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.83
L23C38 -76.6 -336.9 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Compression 0.52
L23C39 -184.7 -821.5 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.44
L23C40 -187.4 -812.2 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.44
L23C41 -75.0 -329.8 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Compression 0.51
APPENDIX B
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L23C4 -192.2 -809.2 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.44
L23C5 -74.8 -325.0 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Compression 0.50
L22C2 -71.6 -370.1 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.52
L22C3 -237.0 -1042.2 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.56
L22C7 -43.5 -752.3 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.60
L22C12 -85.5 -780.8 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.63
L22C13 238.6 -255.3 W14X82 W14X82 24.00 165 66.44 50 1080.0 782.0 Compression 0.33
L22C18 359.7 -239.0 W14X82 W14X82 24.00 165 66.44 50 1080.0 782.0 Tension 0.33
L22C24 -137.4 -637.1 W14X99 W14X99 29.10 165 44.39 50 1309.5 1133.8 Compression 0.56
L22C19 -137.1 -681.4 W14X99 W14X99 29.10 165 44.39 50 1309.5 1133.8 Compression 0.60
L22C25 232.0 -234.9 W14X82 W14X82 24.00 165 66.44 50 1080.0 782.0 Compression 0.30
L22C30 299.1 -207.4 W14X82 W14X82 24.00 165 66.44 50 1080.0 782.0 Tension 0.28
L22C31 -55.4 -792.9 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.63
L22C36 -81.7 -782.5 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.63
L22C38 -81.4 -365.7 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.52
L22C39 -226.2 -1044.8 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.56
L22C40 -230.8 -1033.1 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.56
L22C41 -79.6 -358.8 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.51
L22C4 -235.1 -1030.6 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.56
L22C5 -79.7 -353.8 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.50
L21C2 -75.9 -400.1 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.56
L21C3 -268.9 -1206.9 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.65
L21C7 -63.5 -919.1 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.74
L21C12 -110.5 -939.7 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.75
L21C13 185.3 -371.7 W14X82 W14X82 24.00 165 66.44 50 1080.0 782.0 Compression 0.48
L21C18 246.0 -291.7 W14X82 W14X82 24.00 165 66.44 50 1080.0 782.0 Compression 0.37
B.3 Design of Steel Columns
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L21C24 -165.6 -771.6 W14X99 W14X99 29.10 165 44.39 50 1309.5 1133.8 Compression 0.68
L21C19 -165.8 -833.0 W14X99 W14X99 29.10 165 44.39 50 1309.5 1133.8 Compression 0.73
L21C25 167.8 -341.4 W14X82 W14X82 24.00 165 66.44 50 1080.0 782.0 Compression 0.44
L21C30 189.5 -264.9 W14X82 W14X82 24.00 165 66.44 50 1080.0 782.0 Compression 0.34
L21C31 -76.7 -957.7 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.77
L21C36 -107.6 -940.1 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.75
L21C38 -86.5 -395.1 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.56
L21C39 -257.7 -1210.0 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.65
L21C40 -265.9 -1206.3 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.65
L21C41 -84.5 -388.4 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.55
L21C4 -271.5 -1202.9 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.65
L21C5 -84.7 -383.2 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.54
L20C2 -79.9 -428.9 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.60
L20C3 -308.8 -1399.8 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.46
L20C7 -77.9 -1064.7 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.85
L20C12 -129.7 -1076.1 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.86
L20C13 156.8 -512.7 W14X82 W14X82 24.00 165 66.44 50 1080.0 782.0 Compression 0.66
L20C18 204.5 -418.0 W14X82 W14X82 24.00 165 66.44 50 1080.0 782.0 Compression 0.53
L20C24 -188.7 -884.8 W14X99 W14X99 29.10 165 44.39 50 1309.5 1133.8 Compression 0.78
L20C19 -190.5 -966.1 W14X99 W14X99 29.10 165 44.39 50 1309.5 1133.8 Compression 0.85
L20C25 140.4 -483.8 W14X82 W14X82 24.00 165 66.44 50 1080.0 782.0 Compression 0.62
L20C30 160.7 -403.6 W14X82 W14X82 24.00 165 66.44 50 1080.0 782.0 Compression 0.52
L20C31 -92.6 -1101.4 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.88
L20C36 -126.1 -1076.5 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.86
L20C38 -91.1 -423.1 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.60
L20C39 -296.3 -1404.0 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.46
APPENDIX B
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L20C40 -309.6 -1392.1 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.46
L20C41 -89.1 -416.3 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.59
L20C4 -317.7 -1386.3 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.46
L20C5 -89.4 -411.2 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Compression 0.58
L19C2 422.5 14.1 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Tension 0.43
L19C3 -362.0 -1624.1 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.54
L19C7 2769.2 291.4 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.81
L19C12 2670.5 394.3 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.78
L19C13 123.2 -674.1 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.54
L19C18 168.9 -577.4 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.46
L19C24 301.9 -397.1 W14X99 W14X99 29.10 165 44.39 50 1309.5 1133.8 Compression 0.35
L19C19 241.5 -561.6 W14X99 W14X99 29.10 165 44.39 50 1309.5 1133.8 Compression 0.50
L19C25 107.6 -646.9 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.52
L19C30 126.3 -564.9 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.45
L19C31 2758.7 305.6 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.81
L19C36 2663.6 409.8 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.78
L19C38 408.6 28.0 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Tension 0.42
L19C39 -346.0 -1632.4 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.54
L19C40 -366.1 -1611.6 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.53
L19C41 417.6 38.3 W14X74 W14X74 21.80 165 66.55 50 981.0 709.6 Tension 0.43
L19C4 -375.7 -1604.0 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.53
L19C5 417.6 38.0 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.66
L18C2 393.3 10.7 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.62
L18C3 -407.2 -1846.8 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.61
L18C7 2570.0 269.8 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.76
L18C12 2477.0 363.9 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.73
B.3 Design of Steel Columns
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L18C13 91.1 -836.0 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.67
L18C18 136.8 -727.6 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.58
L18C24 264.1 -548.6 W14X99 W14X99 29.10 165 44.39 50 1309.5 1133.8 Compression 0.48
L18C19 205.2 -718.2 W14X99 W14X99 29.10 165 44.39 50 1309.5 1133.8 Compression 0.63
L18C25 76.9 -811.2 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.65
L18C30 95.7 -717.1 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.57
L18C31 2562.6 281.9 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.75
L18C36 2471.6 378.5 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.73
L18C38 380.1 24.0 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.60
L18C39 -390.9 -1855.5 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.61
L18C40 -412.3 -1833.2 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.61
L18C41 389.5 34.1 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.61
L18C4 -422.0 -1825.5 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.60
L18C5 390.2 33.4 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.61
L17C2 363.3 7.3 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.57
L17C3 -452.6 -2068.4 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.68
L17C7 2366.0 247.4 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.70
L17C12 2277.6 332.7 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.67
L17C13 65.7 -972.1 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.78
L17C18 113.1 -847.1 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.68
L17C24 226.0 -699.0 W14X99 W14X99 29.10 165 44.39 50 1309.5 1133.8 Compression 0.62
L17C19 168.5 -873.3 W14X99 W14X99 29.10 165 44.39 50 1309.5 1133.8 Compression 0.77
L17C25 53.2 -949.3 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.76
L17C30 73.4 -838.3 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.67
L17C31 2361.1 257.4 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.69
L17C36 2273.9 346.1 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.67
APPENDIX B
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L17C38 350.7 19.9 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.55
L17C39 -436.0 -2077.4 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.69
L17C40 -458.4 -2054.2 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.68
L17C41 360.4 29.7 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.57
L17C4 -468.1 -2046.4 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.68
L17C5 361.4 28.8 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.57
L16C2 333.5 4.5 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.74
L16C3 -498.0 -2290.2 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.76
L16C7 2162.8 225.5 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.64
L16C12 2078.7 301.7 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.61
L16C13 371.7 -695.7 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.67
L16C18 504.1 -558.0 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.54
L16C24 186.7 -852.0 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.50
L16C19 130.5 -1031.4 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.61
L16C25 363.7 -631.3 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.61
L16C30 428.2 -517.9 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.50
L16C31 2159.5 233.7 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.63
L16C36 2076.3 313.9 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.61
L16C38 321.7 16.4 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.71
L16C39 -481.0 -2299.5 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.76
L16C40 -504.3 -2275.4 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.75
L16C41 331.6 25.9 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.74
L16C4 -514.2 -2267.6 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.75
L16C5 332.8 24.7 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.74
L15C2 303.3 1.9 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.67
L15C3 -528.9 -2418.5 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.80
B.3 Design of Steel Columns
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L15C7 1975.3 200.5 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.58
L15C12 1903.8 274.1 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.56
L15C13 336.3 -857.4 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.83
L15C18 461.9 -705.3 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.68
L15C24 152.4 -982.5 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.58
L15C19 94.6 -1177.6 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.69
L15C25 330.8 -795.0 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.77
L15C30 394.7 -672.9 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.65
L15C31 1973.4 207.2 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.58
L15C36 1902.9 284.6 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.56
L15C38 292.2 13.0 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.65
L15C39 -511.9 -2427.2 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.80
L15C40 -535.7 -2404.3 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.79
L15C41 302.2 22.4 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.67
L15C4 -546.1 -2396.9 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.79
L15C5 303.6 21.1 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.67
L14C2 273.0 -0.1 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.61
L14C3 -533.8 -2443.1 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.81
L14C7 1789.6 175.1 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.53
L14C12 1730.5 245.9 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.51
L14C13 300.9 -1020.2 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.99
L14C18 426.3 -859.1 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.83
L14C24 123.8 -1091.5 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.64
L14C19 63.4 -1304.8 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.77
L14C25 297.8 -959.8 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.93
L14C30 361.0 -827.9 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.80
APPENDIX B
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L14C31 1788.7 180.3 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.53
L14C36 1730.5 254.8 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Tension 0.51
L14C38 262.8 10.2 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.58
L14C39 -516.5 -2450.4 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.81
L14C40 -538.5 -2423.9 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.80
L14C41 273.0 19.5 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.61
L14C4 -549.4 -2418.1 W14X257 W14X257 75.60 165 39.94 50 3402.0 3027.4 Compression 0.80
L14C5 274.4 18.0 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.61
L13C2 249.1 0.6 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.55
L13C3 125.4 -833.1 W14X120 W14X120 35.30 165 44.06 50 1588.5 1378.3 Compression 0.60
L13C7 1578.5 154.9 W14X193 W14X193 56.80 165 40.76 50 2556.0 2263.7 Tension 0.62
L13C12 1525.1 215.0 W14X193 W14X193 56.80 165 40.76 50 2556.0 2263.7 Tension 0.60
L13C13 264.3 -1188.5 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.70
L13C18 387.8 -1021.1 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.60
L13C24 428.1 -452.4 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.44
L13C19 339.0 -618.9 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.60
L13C25 264.1 -1130.4 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.67
L13C30 324.7 -991.4 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.58
L13C31 1577.9 159.0 W14X193 W14X193 56.80 165 40.76 50 2556.0 2263.7 Tension 0.62
L13C36 1524.5 223.0 W14X193 W14X193 56.80 165 40.76 50 2556.0 2263.7 Tension 0.60
L13C38 240.1 10.0 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.53
L13C39 146.3 -815.5 W14X120 W14X120 35.30 165 44.06 50 1588.5 1378.3 Compression 0.59
L13C40 121.9 -728.4 W14X120 W14X120 35.30 165 44.06 50 1588.5 1378.3 Compression 0.53
L13C41 243.6 16.5 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.54
L13C4 125.1 -764.1 W14X120 W14X120 35.30 165 44.06 50 1588.5 1378.3 Compression 0.55
L13C5 245.1 15.0 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.54
B.3 Design of Steel Columns
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L12C2 218.0 -0.6 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.48
L12C3 72.5 -1043.3 W14X120 W14X120 35.30 165 44.06 50 1588.5 1378.3 Compression 0.76
L12C7 1381.8 133.8 W14X193 W14X193 56.80 165 40.76 50 2556.0 2263.7 Tension 0.54
L12C12 1332.0 185.4 W14X193 W14X193 56.80 165 40.76 50 2556.0 2263.7 Tension 0.52
L12C13 229.4 -1349.1 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.80
L12C18 353.5 -1168.0 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.69
L12C24 369.4 -582.8 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.57
L12C19 298.2 -769.7 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.75
L12C25 231.8 -1293.1 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.76
L12C30 292.8 -1138.7 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.67
L12C31 1381.7 136.9 W14X193 W14X193 56.80 165 40.76 50 2556.0 2263.7 Tension 0.54
L12C36 1331.1 192.0 W14X193 W14X193 56.80 165 40.76 50 2556.0 2263.7 Tension 0.52
L12C38 210.1 7.8 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.47
L12C39 91.7 -1023.8 W14X120 W14X120 35.30 165 44.06 50 1588.5 1378.3 Compression 0.74
L12C40 69.2 -938.4 W14X120 W14X120 35.30 165 44.06 50 1588.5 1378.3 Compression 0.68
L12C41 213.9 13.7 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.48
L12C4 72.0 -973.5 W14X120 W14X120 35.30 165 44.06 50 1588.5 1378.3 Compression 0.71
L12C5 215.3 12.3 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.48
L11C2 186.3 -1.6 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.41
L11C3 19.8 -1253.6 W14X120 W14X120 35.30 165 44.06 50 1588.5 1378.3 Compression 0.91
L11C7 1187.6 112.7 W14X193 W14X193 56.80 165 40.76 50 2556.0 2263.7 Tension 0.46
L11C12 1141.7 155.6 W14X193 W14X193 56.80 165 40.76 50 2556.0 2263.7 Tension 0.45
L11C13 200.1 -1491.9 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.88
L11C18 325.7 -1293.7 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.76
L11C24 327.3 -729.1 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.71
L11C19 256.8 -919.7 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.89
APPENDIX B
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L11C25 205.1 -1437.6 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.85
L11C30 267.3 -1264.4 W14X145 W14X145 42.70 165 41.44 50 1921.5 1694.8 Compression 0.75
L11C31 1187.3 115.1 W14X193 W14X193 56.80 165 40.76 50 2556.0 2263.7 Tension 0.46
L11C36 1140.0 161.0 W14X193 W14X193 56.80 165 40.76 50 2556.0 2263.7 Tension 0.45
L11C38 179.4 5.7 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.40
L11C39 37.3 -1232.4 W14X120 W14X120 35.30 165 44.06 50 1588.5 1378.3 Compression 0.89
L11C40 17.0 -1149.2 W14X120 W14X120 35.30 165 44.06 50 1588.5 1378.3 Compression 0.83
L11C41 183.8 11.2 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.41
L11C4 19.0 -1183.6 W14X120 W14X120 35.30 165 44.06 50 1588.5 1378.3 Compression 0.86
L11C5 185.1 9.9 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.41
L10C2 154.5 -2.2 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.34
L10C3 -32.7 -1464.7 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.79
L10C7 990.3 92.6 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Tension 0.83
L10C12 948.8 126.3 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Tension 0.80
L10C13 341.3 -637.0 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.62
L10C18 470.4 -564.4 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.55
L10C24 284.0 -878.4 W14X132 W14X132 38.80 165 43.90 50 1746.0 1516.5 Compression 0.58
L10C19 213.8 -1073.6 W14X132 W14X132 38.80 165 43.90 50 1746.0 1516.5 Compression 0.71
L10C25 350.1 -600.4 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.58
L10C30 420.9 -512.9 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.50
L10C31 989.7 94.6 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Tension 0.83
L10C36 946.6 131.0 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Tension 0.79
L10C38 148.7 3.9 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.33
L10C39 -17.0 -1441.7 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.78
L10C40 -34.9 -1360.9 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.73
L10C41 153.6 8.9 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.34
B.3 Design of Steel Columns
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L10C4 -33.7 -1394.7 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.75
L10C5 154.8 7.7 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.34
L09C2 122.2 -2.7 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.27
L09C3 -66.2 -1588.4 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.86
L09C7 805.7 70.7 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Tension 0.68
L09C12 776.1 102.0 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Tension 0.65
L09C13 302.2 -795.8 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.77
L09C18 431.7 -715.2 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.69
L09C24 246.1 -1004.8 W14X132 W14X132 38.80 165 43.90 50 1746.0 1516.5 Compression 0.66
L09C19 173.7 -1215.1 W14X132 W14X132 38.80 165 43.90 50 1746.0 1516.5 Compression 0.80
L09C25 313.0 -760.5 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.74
L09C30 383.5 -664.3 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.64
L09C31 805.2 72.3 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Tension 0.68
L09C36 774.8 105.5 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Tension 0.65
L09C38 117.7 2.2 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.26
L09C39 -51.4 -1564.5 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.84
L09C40 -69.1 -1485.6 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.80
L09C41 122.8 7.0 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.27
L09C4 -68.1 -1519.8 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.82
L09C5 123.8 6.0 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.28
L08C2 90.1 -2.9 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.20
L08C3 -74.4 -1609.3 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.87
L08C7 622.0 48.6 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Tension 0.69
L08C12 604.5 76.9 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Tension 0.67
L08C13 263.0 -954.4 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.93
L08C18 393.2 -865.5 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.84
APPENDIX B
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L08C24 213.8 -1109.1 W14X132 W14X132 38.80 165 43.90 50 1746.0 1516.5 Compression 0.73
L08C19 138.5 -1337.7 W14X132 W14X132 38.80 165 43.90 50 1746.0 1516.5 Compression 0.88
L08C25 275.6 -920.4 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.89
L08C30 345.9 -815.2 W14X90 W14X90 26.50 165 44.64 50 1192.5 1030.8 Compression 0.79
L08C31 621.5 49.7 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Tension 0.69
L08C36 604.0 79.4 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Tension 0.67
L08C38 86.6 0.8 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.19
L08C39 -59.7 -1584.8 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.85
L08C40 -76.0 -1502.6 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.81
L08C41 92.1 5.4 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.20
L08C4 -75.1 -1537.5 W14X159 W14X159 46.70 165 41.23 50 2101.5 1855.9 Compression 0.83
L08C5 92.8 4.6 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.21
L07C2 65.1 -0.2 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.14
L07C3 440.8 87.9 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.69
L07C7 408.6 32.5 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Tension 0.45
L07C12 399.4 49.7 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Tension 0.44
L07C13 222.5 -1117.1 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.89
L07C18 351.7 -1024.5 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.82
L07C24 318.3 76.4 W14X38 W14X38 11.20 165 106.87 50 504.0 218.7 Tension 0.63
L07C19 325.2 79.8 W14X38 W14X38 11.20 165 106.87 50 504.0 218.7 Tension 0.65
L07C25 237.1 -1085.1 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.87
L07C30 305.5 -974.9 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.78
L07C31 408.2 33.5 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Tension 0.45
L07C36 398.8 51.8 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Tension 0.44
L07C38 62.8 2.1 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.14
L07C39 440.5 86.3 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.69
B.3 Design of Steel Columns
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L07C40 440.1 89.7 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.69
L07C41 61.4 3.5 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.14
L07C4 439.9 86.5 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.69
L07C5 62.0 2.9 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.14
L06C2 32.5 -0.2 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.07
L06C3 218.9 43.0 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.35
L06C7 213.0 14.6 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Tension 0.24
L06C12 207.5 22.8 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Tension 0.23
L06C13 187.5 -1266.5 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 1.01
L06C18 318.1 -1160.4 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.93
L06C24 160.5 38.0 W14X38 W14X38 11.20 165 106.87 50 504.0 218.7 Tension 0.32
L06C19 165.1 40.0 W14X38 W14X38 11.20 165 106.87 50 504.0 218.7 Tension 0.33
L06C25 204.2 -1236.1 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.99
L06C30 272.8 -1110.3 W14X109 W14X109 32.00 165 44.15 50 1440.0 1248.7 Compression 0.89
L06C31 212.5 15.4 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Tension 0.24
L06C36 206.6 24.3 W14X68 W14X68 20.00 165 67.08 50 900.0 647.7 Tension 0.23
L06C38 31.3 1.0 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.07
L06C39 218.7 42.0 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.34
L06C40 218.6 43.9 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.34
L06C41 30.6 1.6 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.07
L06C4 218.5 42.3 W14X48 W14X48 14.10 165 86.42 50 634.5 367.5 Tension 0.34
L06C5 30.9 1.3 W14X34 W14X34 10.00 165 108.10 50 450.0 191.5 Tension 0.07
L04C3 -23.7 -89.4 W12X65 W12X65 19.10 165 54.67 50 859.5 690.8 Compression 0.13
L04C12 -56.3 -204.2 W12X65 W12X65 19.10 165 54.67 50 859.5 690.8 Compression 0.30
L04C18 -39.6 -149.8 W12X65 W12X65 19.10 165 54.67 50 859.5 690.8 Compression 0.22
L04C24 -43.4 -158.5 W12X65 W12X65 19.10 165 54.67 50 859.5 690.8 Compression 0.23
APPENDIX B
Appendix B3 – Design of Steel Columns
Element ID Pu,max Pu,min Model Section Design Section Ag L kL/rmin Fy φtPnt φcPnc T/C
Governed DCR
L04C30 -39.5 -149.7 W12X65 W12X65 19.10 165 54.67 50 859.5 690.8 Compression 0.22
L04C36 -56.3 -204.3 W12X65 W12X65 19.10 165 54.67 50 859.5 690.8 Compression 0.30
L04C39 -23.5 -89.4 W12X65 W12X65 19.10 165 54.67 50 859.5 690.8 Compression 0.13
L04C40 -60.8 -221.6 W12X65 W12X65 19.10 165 54.67 50 859.5 690.8 Compression 0.32
L04C41 -8.7 -24.6 W12X65 W12X65 19.10 165 54.67 50 859.5 690.8 Compression 0.04
L04C4 -60.4 -221.5 W12X65 W12X65 19.10 165 54.67 50 859.5 690.8 Compression 0.32
L04C5 -8.7 -24.6 W12X65 W12X65 19.10 165 54.67 50 859.5 690.8 Compression 0.04
Bibliography
1 Beghini, A., Stromberg, L., Baker, W., Pulino, G. and Mazurek, A. A New
Frontier in Modern Architecture: Optimal Structural Topologies. International
Association for Shell and Spatial Structures (2011).
2 Mazurek, A., Baker, W. and Tort, C. Geometrical Aspects of Optimum Truss
like Structures. Structural and Multidisciplinary Optimization (2011), 231-242.
3 ALTAIR ENGINEERING. Inspire 9.0. 2012.
4 ALTAIR ENGINEERING. Hypermesh. 2013.
5 ALTAIR ENGINEERING. Optistruct. 2013.
6 Olason, T. and Tidman, D. Methodology for Topology and Shape Optimization
in the Design Process. Chalmers University of Technology, Goteborg, 2010.
7 Caramia, M. and Dell'Olmo. In Multiobjective management in freight logistics:
increasing capacity, service level and safety with optimization algorithms.
Springer, London, 2008.
8 Ngatchou, P., Zarei, A. and El Sharkawi, M.A. Pareto Multi Objective
Optimization. ISAP (2005).
9 Bendsoe, M.P. and Sigmund, S.P. In Topology Optimization: Theory, Methods
and Applications. Springer, Berlin, 2002.
10 Christensen, P. and Klarbring, A. An Introduction to Structural Optimization.
Springer, Linkoping (Sweden), 2009.
11 Olhoff, H. and Schnell, W. Applied Structural Mechanics. Springer, 1997.
Bibliography
12 Zhou, X., Chen, L. and Huang, Z. The SIMP-SRV Method for Stiffness Topology
Optimization. International Journal of CAD/CAM (2007).
13 Bourdin, B. Filters in Topology Optimization. International Journal for
Numerical Methods (2001).
14 AMERICAN SOCIETY OF CIVIL ENGINEERING. Minimum Design Loads for
Buildings and Other Structures. 2010.
15 Cox, H.L. The Design of Structures of Least Weight. Pergamon Press (1965).
16 Bow, R.H. Economics of Construction in Relation to Frame Structures. ICE
Publishing, London, 1873.
17 Maxwell, J.C. On Reciprocal Figures and Diagrams of Forces. Philosophical
Magazine (1864), 250-261.
18 Maxwell, J.C. On Reciprocal Figures, Frames and Diagrams of Forces.
Edinbrook Royal Society Proceedings, 7 (1870), 160-208.
19 Baker, W.F. Structural Innovation: Combining Classical Theories with New
Technologies. The Higgins Lecture (2013).
20 Mazurek, A., Baker, W.F., Tort, C. Geometrical Aspects of Optimum Truss-Like
Structures. Structural and Multidisciplinary Optimization , 43, 2 (2011), 231-
242.
21 Michell, A. G. M. The Limts of Economy of Material in Frame-Structures.
Philosophical Magazine, 8, 47 (November 1904).
22 Chan, A.S.L. The Design of Michell Optimum Structures. Ministry of Aviation
Aeronautical Research Council Report, 3303 (1960), 1-40.
23 Sokol, T. A 99 Line Code for Discretized Michell Truss Optimization Written in
Mathematica. Structural and Multidisciplinary Optimization, 43 (2011), 181-
190.
24 Almegaard, H. On Geometry of Discrete Michell Trusses. IABSE-IASS
Symposium - Proceedings (2011).
25 Anderson-Cook, C.M., Montgomery, D.c:, Myers, R.H. Response Surface
Bibliography
Methodology: Process and Product in Optimization Using Designed
Experiments. John Wiley and Sons (2009).
26 Jones, D.R. A Taxonomy of Global Optimization Methods Based on Response
Surface. Journal of Global Optimization (2001).
27 Roux, W.J., Haftka, R.T. Response Surface Approximations for Structural
Optimization. International Journal for Numerical Methods in Engineering, 42
(June 1998), 517-534.
28 Carley, K.M., Kamneva, N.Y., Reminga, J. Response Surface Methodology
(2004).
29 Kripakaran, P., Hall, B., Gupta, A. A Genetic Algorithm fro Design of Moment-
Resisting Steel Frames. Structural and Multidisciplinary Optimization (2011).
30 Camp, C.V., Bichon, B.J., Stovall, S.P. Design of Steel Frames Using Ant Colony
Optimization. Journal of Structural Engineering (2005).
31 Liu, M., Burns, S.A:, Wen, Y.K. Genetic Algorithm Based on Construction-
Conscious Minimim Weight Design of Seismic Steel Moment-Resisting Frames.
Journal of Structural Engineering (2006).
32 Pezeshk, S., Camp, C.V., Chen, D. Design of Nonlinear Frames Using Genetic
Optimization. Journal of Structural Engineering (2000).
33 Murren, P., Khandelwal, K. Optimization of Large Steel Frames Using a Design-
Driven Harmony Search. Structures Congress (2011), 2508-2519.
34 AMERICAN INSTITUTION OF STEEL CONSTRUCTION INC.. Steel Construction
Manual.
35 Council, International Code. Internatonal Building Code. 2012.
36 (USGS), United States Geological Survey. National Seismic Hazard Maps. 2008.
