Ottimizzazione
di Strutture per Edifici Alti
1
Elena Mele
Dottorato di Ricerca in
INGEGNERIA STRUTTURALE E GEOTECNICA
A.A. 2020/21
Corso di Ottimizzazione Strutturale
30 gennaio – 2 febbraio 2023

• efficiency - accomplishment of performance targets with min structural material use
• importance of efficiency - large scale structures, tall buildings
• outcome of optimization process, more or less explicit and rigorous, different levels
2
structural efficiency and optimization of TBs

optimization of building form - geometrical shape of the building mass and volume
3
Prof. Yukio Tamura
Tokyo Polytechnic University
wind tunnel tests and
CFD numerical analyses
for sculpting the optimal
profile of tall buildings
reducing wind
actions/effects
stiffness, strength and
damping demands
structural efficiency and optimization of TBs

• optimization of building structural system
structural system concept → defines global stiffness (and strength)
capacity
structural pattern → translates concept in a specific arrangement
of members, defines local deformation modes and resisting
mechanisms, and strength demand distribution among members
• member size optimization
given the pattern, the geometrical properties of member cross
sections (shape, area, inertia, strength modulus…) define the local
strength and stiffness capacity
4
Mark
Sarkisian
-
SOM
-
San
Francisco
structural efficiency and optimization of TBs

• efficiency - accomplishment of performance targets with min structural material use
• importance of efficiency - large scale structures, tall buildings
• outcome of optimization process, more or less explicit and rigorous, different levels
5
structural efficiency and optimization of TBs
• conscious search for efficiency in tall buildings structural design from the 50’s
• informal process of optimization triggered by the deep insight into the
inherent mechanical aspect of the problem
• invention of modern 3D structural systems, conceived for the very nature of TBs

6
a history of Engineers and Ideas
in search of structural efficiency
for skyscrapers design

engineers and skyscrapers
W. F. Baker (SOM Chicago)
engineering an idea
F. Khan (SOM Chicago)
a pragmatic visionary
Robert Le Ricolais
the structure of the structure
W. Le Baron Jennney
the steel skeleton
Edward C. Shankland
MRF and
Gray columns
Gunvald Aus
wind bracing
Myron Goldsmith (SOM Chicago)
quiet poet of American architecture
L. Robertson (LERA NYC)
skyscraper superstar
W. Le Messurier: search for
ideal is pursuit of elegance
P.L.Nervi: good design
requires patience and LOVE
Mark Sarkisian (SOM San Francisco)
architecture invented from structure

skeleton
core, outrigger & belt
optimization and stress lines
triangulation and patterns
skyscrapers and ideas
bracing
tubes
building
=
cantilever
beam
effect of scale

“…by taking something as large and complex as a huge skyscraper, something formed by
thousands of individual beams, columns, walls and slabs, and imagining it as one simple
entity, the designer can design these giant structures rationally
in a certain sense, the tall building is the simplest of all structural problems
the art and science of tall building design lies in selecting appropriate system to
carry the forces from where they occur in the air down to the structure foundations”
William F. Baker (SOM), 2010
the Idea: the tall building structure as a cantilever beam
engineering this idea :
conceptual definition of structural systems
accounting for the effect of scale

the idea : the tall building structure as a cantilever beam
when a bldg dimensions change, structural engineers must
ask themselves: how will the behaviour of structural system
change?
when the scale changes they must view the system in a
totally different light and look for things that are less
obvious in their normal scale of experience
the first to think the tall building
this way is Fazlur Khan (1929-1982)
- a pragmatic visionary

structural design - effect of scale
gravity G
shear V=wh
moment M=wh2/2
drift Δ=wh4/8EI
V
→ 2G
→ 2 V
→ 4 M
→ 16 Δ
doubling height
16 Δ
G
2G
2V
vertical cantilever beam under uniform transversal load →
simplified model of building behavior under wind actions
4M
M
Δ
• effect of wind load predominant on gravity
• bending behaviour predominat on shear
• stiffness demand predominant on strength
{
w

the height, or better, the slenderness of skyscraper makes it
particularly sensitive to lateral loads, such that a premium is
associated with its lateral load resisting system
effect of scale = premium for height
structural design - effect of scale

premium for height
differential between material for
gravity only and actual load designs
premium
for height
GLRS
GLRS
LLRS
LLRS
Engineers & Ideas
F. Khan

Empire State
Building
John Hancock
Center
psf
x
5
≈
kg/m
2
≈325 kg/m2
≈125 kg/m2
minimizing the premium for height
premium
for height
1931
1970

15
Home Insurance Building
1890s
Empire State Building
1930s
John Hancok Center
1970s
• same struct. system
• different height
• low efficiency
• same height
• different struct. system
• high efficiency
40 years 40 years
effect of scale

16
steel
interior
r.c.
exterior
F.Khan Ali & Moon

tall building as cantilever beam, bent under lateral loads
structural design: IDEAS for modern skyscrapers
beam: cross section moment of inertia and strength modulus
→ max by centrifugation of areas from centroid
buildings: moment of inertia calculated considering all vertical
elements (core walls and columns) continuous along elevation
and active in the lateral system
increasing of building stiffness requires the placement of
vertical elements as far as possible from the plan centroid, i.e.
at building perimeter

two major milestones
structural design: IDEAS for modern skyscrapers
• the tube: vertical columns and
bracings are moved to perimeter
to maximize the moment of
inertia of the equivalent beam
cross section
• the core plus outrigger system: with
central location for core structure, the
system maximizes the lever arm of the
equivalent beam cross section by inserting
a very stiff interconnection (outrigger)
between core and perimeter columns
only gravity load
Lateral Load Res. System
Lateral Load Res. System

tube

beams
beam cross section: moment of inertia and strength modulus
→ max by centrifugation of areas from centroid
W strength
modulus
I inertia
moment
3,5
> 3,5
6,2
> 6,2
2,3
> 2,3
structural design: maximizing flexural stiffness and strength

the tube is a fully 3D system that utilizes the entire building perimeter to resist lateral loads
the tube concept – bending rigidity
walls connected as a tube → a hollow box section is created, the walls participate together
as a 3D system in resisting loads → the bending effectiveness increases significantly
unconnected perimeter walls
connected perimeter walls → tube
𝐼 = 2 ∙
𝑡𝑏3
12
+ 2 ∙
ℎ−2𝑡 𝑡3
12
For example: h = b = 10 m, t = 0.25 m
2 ∙
0.25∙103
12
+ 2 ∙
10−2∙0.25 0.253
12
= 42 𝑚3
𝐼 =
ℎ𝑏3
12
-
ℎ−2𝑡 ∙ 𝑏−2𝑡 3
12
For example: h = b = 10 m, t = 0.25 m
10∙103
12
-
10−2∙0.25 ∙ 10−2∙0.25 3
12
= 155 𝑚3

22
A B
D
C
B-D face
B D
in its simplest terms, the tube system can be defined as a fully 3D system that utilizes the
entire building perimeter to resist lateral loads
frame tube
the tube concept – bending behavior
flange facades: 2D MRFs under vertical loads - web facades: 2D MRFs under lateral load

23
H/B ≥ 10
δbending >> δshear
beams: the role of shear rigidity

• for max (potential) bending efficiency → continuous vertical elements (columns or
walls) located at the farthest extremity from geometric centre of the bldg, thus at the
plan perimeter
• further, a shear resisting system is essential for carrying the lateral loads to the vertical
elements that, in turn, resist the overturning moments on the cantilever
• crucial point of the tube system: columns should work as elements of an integrated
system (a single giant cantilever beam) rather than an aggregation of individual elements
or subsystems, thus obtaining the ideal bending efficiency → it is necessary to
interconnect columns with an effective (stiff) shear-resisting system
shear rigidity allows for exploitation of potential bending efficiency
bending and shear rigidity

idea of tube → actually a revolution in structural conception
25
Rectangular
beam-columngrid
the tube – shear deformation and shear lag
adoption of traditional rectangular pattern
(Framed Tube) strongly undermined the closeness
of the idea to its concrete realization, due to
racking deformation and shear lag effect
Flange facades: 2D MRFs under vertical loads - Web facades: 2D MRFs under lateral load

F. Khan (1967): in a frame structure the total lateral drift is caused by 3 primary factors
MRF deformation modes under lateral loads (web façade)
column shortening
and elongation
• bending deformations in girders (65%)
• bending deformations in columns (15%)
• axial deformations in columns (20%)
column and beam
bending deformations
shear sway component → 80%
cantilever component → 20%
the first two factors represent the frame action, which can be manipulated
the third factor is based on statics (cantilever action) and cannot be altered
connecting columns by very rigid beams → structure exhibit only 20% of its original drift

“a high-rise structure wants to react to applied load as a cantilever element:
not only it is inaccurate to overlook cantilever action of the building form in the analysis,
but cantilever action is the mode of response that a designer ought to strive to achieve”
Khan is guided by a strong intellectual footing, and by an
empathy with the structure under load
when he was a student at University of Illinois, he heard
the admonition of H. Cross:
“you must learn to think as the structure thinks”
Fazlur Khan: the tall building structure as a cantilever beam

28
optimization of framed tube – F. Khan, 1973
it is obviously advisable to optimize the subsystem made of 2 columns and 1 spandrel, to
arrive at optimum spacing of columns and proportion of spandrels and columns

MRF (2D): shear rigidity depends on beam length/depth ratio
→ frame with deep beams connecting closely spaced columns
on the four faces of a square bldg
FT (3D): efficient bending configuration with high shear rigidity
shear stiffness
shear stiffness
ext. columns spaced at 1.20m – 3.00 m, max 4.50 m
spandrel beams depth 600 - 1200 mm
“… necessary to keep the proper balance of stiffness between
the spandrel elements and the column elements so that
neither of the elements presents inefficiency in terms of
stiffness of the structure against lateral sway, or in terms of
the overall strength of the tube system against lateral forces”
Fazlur R. Khan, 1973

FT: first applications
“This system was probably first applied on the design of
the 43-st. DeWitt Chestnut Apartment Bldg in Chicago
in 1963
Since then, the system has received wide acceptance
among the designers all over the world, and many
variations are being used in a number of buildings
under construction
In the DeWitt Chestnut Apartment Building, the
columns were spaced at 1.65 m centres and the
spandrels were 600 mm deep
The spacing of the columns in this case was also related
to the module for interior planning of the apartment
floors”

design phase: 1962
excavation: 1966
construction: 1968
inauguration: 1973
Architect: Minoru Yamasaki,
Engineers: Worthington, Skilling, Helle & Jackson
(John Skilling e Leslie E. Robertson)
WTC towers of innovation
WTC site
the towers are a major milestone in structural steel design
Rapp, 1964
from the structural point of view, the WTC towers
featured several innovations at the time of their
design/construction - Robertson, 2002
110 st.
Framed Tube

“there has been much discussion of the identity of the engineer who
first conceived the structural system that we call the ‘tube’ – with a
handful of engineers being so named
it seems to me that it was an idea in the need of realization –
making it not unlikely that several engineers more-or-less
simultaneously came up with the same concept
in any event, I do not lay claim to authorship nor do I disclaim it
believing instead that such ideas are creatures of their time
– not of an individual”
Leslie E. Robertson
FT: first applications
concept of multiple discovery, simultaneous invention)

• structural system: 1st (?) external frame tube + central core large column-free floor area
• use of column-tree prefabricated panels for the perimeter structure
• use of 12 different steel grades, optimised for minimizing structural weight
• floor structural system: grid of orthogonal beam truss
• the first bldg “in wind tunnel”
• the first bldg with dampers for mitigating wind vibrations
• foundation structures, elevators, ….
WTC: towers of innovation
the concentration of invention and integration across disciplines as well as the minimalist
sculptural presence of the towers had not been rivalled in NYC

59 perimeter columns along each face of the bldg, spaced at 1016 mm (in alternate stories,
column at center of each chamfered bldg corner); adjacent perimeter columns
interconnected at floor level by deep spandrel plates, typically 70cm long and 1312mm deep
WTC: the frame tube structure

WTC: the perimeter walls
column-tree
prefabricated panels

perimeter columns: built up by welding four plates together to form a 356x356 mm square sect.
the 4 plates had different thickness, also varying along the height between 6.35 and 101.6 mm
The perimeter columns

uniform exterior column geometry maintained over most of the height of the 100 st. bldg by
varying the thickness of the plates and the steel grade
The perimeter columns

North side: column cross section areas
reduce from lower to higher floors – as expected
within the same floor, areas nearly constant in central zone of the side (panel #112 to #148)
then areas reduce in the end zones
Column Section Area
Exam of the perimeter structure
130

gravity axial forces and stresses - perimeter cols.
equalize gravity load
stresses and shortening
of very tall steel columns

North side: steel yield stress:
upper levels, steel strength larger than in lower levels
within a single floor side, distribution approximately parabolic
Exam of the perimeter structure
Column Steel Grade – yield stress
14 grades of steel

0
0,2
0,4
0,6
0,8
1
livello 10
tassi
di
lavoro
grav vento (N) vento (M)
stress DCR (s/sy) in base columns due to gravity and wind load almost uniform
columns of North side (web frame): around 80%
columns of West side (flange frame): around 40%
web
EW Wind
north side
DCR under gravity + wind load – “web” north side cols.
0
0,2
0,4
0,6
0,8
1
livello 10
tassi
di
lavoro
grav vento (N) vento (M)
flange
EW Wind
west side

exam of the perimeter structure
credit Guy Nordenson
«Il risultato netto di questa progettazione, raffinatissima
nell’uso dell’acciaio, è una struttura caratterizzata da
compressione uniforme per carichi verticali in tutte le colonne
ad ogni piano, e da un tasso di lavoro (rapporto sollecitazione
su resistenza) altresì uniforme per carichi orizzontali.
Pertanto, le differenze di sollecitazione dovute allo scarico
dei solai e soprattutto al vento, agente sulle varie facciate
dell’edificio e proveniente da diverse direzioni, in diverse
condizioni orografiche e di rugosità, vengono soppresse grazie
a un lavoro quasi artigianale di ingegneria, e sublimate nella
rigida uniformità visiva della struttura e dell'architettura.
Come osservato da Guy Nordenson, l’invisibile creatività del
progetto strutturale di Robertson è, specialmente dopo la
distruzione delle torri, intensa e memorabile come una
grande opera di arte concettuale»
Mele et al. Costruzioni Metalliche n.4 2021
this is optimization!

Lateral drift
dtop = 144 cm
dtop/H = 1/290 >> 1/500
Limitation of building drift under wind loads was not part of original WTC design criteria
dtop = 141.5 cm

• idea of tube → actually a revolution in structural conception
• adoption of traditional rectangular pattern (FT) strongly
undermined the closeness of the idea to its concrete realization
44
limits of Framed Tube efficiency
Khan limit for FT: 80 st.
WTC: 110 st.

tube
diagonalization

the essential character of the structure is created by the continuity of the diagonal X-bracing,
intersecting each other and with columns at tube corners (to obtain fully 3D response)
• withstand the shear produced by horizontal forces
• act as inclined columns to transmit gravity loads to ground
from the Framed Tube to the Braced Tube
BT: 100 st.
shear lag can be virtually eliminated
nearly pure flexural behavior: diagonals
together with spandrel beams provide a
wall-like rigidity against lateral loads
very efficient up to 100 st.

deformation modes: BF
flexural deflection often dominates the deflection characteristics
<<
Shear racking Bending def.
cols
diags
beams
tot.
shear def. bending def.

bracing the tube: diagonalization
fine diagonal mesh
DIAGRID
super diagonals with only corner columns
MEGA SPATIAL TRUSS
tubular variety
BRACED TUBE

50
the braced tube
stacking modules
• X megadiagonals
• main ties at module edges (restraining
horizontal spread of the X form)
• secondary ties at each column-diagonal
intersection (for channeling loads into columns)
mega-diagonals spanning the façade virtually cancel out the bending
deformation in beams and columns, thus almost eliminating shear racking
contribution to the global building behavior

improvement in terms of lateral stiffness and
shear lag reduction
also reflects in the building architecture,
strongly connoted by the clear and disciplined
structure, the honesty of structure
B. Graham
Braced Tube: the John Hancock Center
almost uniform stress distribution on flange face and
approximate linear decrease of stress across the web
→ predominant cantilever mode (80% of total sway)
and very little shear lag effect

mega spatial truss
diagrid braced tube
bracing the tube: diagonalization

the most remarkable innovation of the 21st century
perimeter structural configurations characterized by a
narrow grid of diagonal steel members which are involved
both in gravity and in lateral load resistance
… not new
- steel – not a new material
- diagonalization – not a new solution
- diagrid – earlier use of diagrid is dated back to the ’60s
diagrid structures IBM building, Pittsburgh

13-story structure’s exterior truss load-bearing wall, made of
welded steel in a diamond-patterned grid, is a radical break
from post-and-beam construction
early diagrid: IBM bldg. Pittsburgh (1960’s)
(now known as the United Steel Workers Building)
“one of the most unique office structures”
1962, Progressive Architecture
"the only fine and daring piece of new architecture in the downtown area”
1963, AIA Journal

early diagrid: IBM bldg. Pittsburgh
use of different steel grades – “the 4th dimension of steel construction”: it
allows, in addition to 3 geometrical dimensions, the adjustment of the steel
strength in order to optimize the working conditions of structural members
during construction, the three different types of steel in the exterior walls
were painted red, white, or blue to designate their different structural roles
the building uses steels of varying strengths,
max at supports, reducing along elevation
upper storeys: ordinary carbon steel 36.000 lb/in2
(248 Mpa) yield strength
first 6 storeys: low-alloy high-strength steel 50.000
lb/in2 (345 MPa)
areas immediately above the supports: heat-treated
alloy steel - yield strength 100.000 lb/in2 (690 Mpa),
nearly 3 times the strength of normal steel

Humana HQ Louisville
Kentucky
design competition
Norman Foster, 1982
Norman Foster and the diagrid

efficiency: mechanical concept and structural pattern
efficiency: accomplishment of performance targets by means of minimum employment of
structural material
efficiency of TB structural system
• mechanical concept behind the global configuration → tube
• arrangement of the structural elements, i.e. geometry and connectivity of the
structural pattern
while the idea of tube was actually a revolution in the structural conception of tall
buildings, the adoption of the traditional rectangular pattern in the framed tube, made of
the orthogonal arrangement of beams and columns, strongly undermined the closeness of
the idea to its concrete realization, due to racking deformation and shear lag effect
path towards tube efficiency → optimization of façade structural pattern

58
diagrid - the latest mutation of tube structures
the underlying idea is quite trivial
preserving the spatial configuration of the pure tube,
while changing the structural pattern in façade
from rectangular to triangular,
thus moving from a bending dominated pattern to a
stretching dominated pattern
an improved pattern for tube configuration

the mega-diagonal members are diffusely spread over
the façade
giving rise to closely spaced diagonal elements
allowing for the complete elimination of the
conventional vertical columns
i.e. the frame rectangular grid, which is the major
source of shear racking and shear lag effects
diagrid - the latest mutation of tube structures
diagrid as the evolution of the braced tube

triangle tasselation and module
like in the BT, also here the façade is vertically organized in modules, usually
extending over multiple floors
quite intuitively, the triangle unit, and consequently the overall system, has a
structural behaviour mainly depending on geometry, with mainly axial force
and axial def. modes in the façade grid members → great efficiency
diagrid – the role of geometry

NG,mod
NdG
NcG
NdG
2







=
2
cos
2
mod
,

G
dG
N
N







=
2
sin

dG
cG N
N
NdM
NcM
NdM
2
NW,mod

NW,mod
NdM
NcM
NdM
2

NdV NdV.
2
VW,mod






=
2
cos
2
mod
,

w
dM
N
N







=
2
sin

dM
cM N
N






=
2
sin
2
mod
,

w
dV
V
N
Horizontal force due
to inclination of base
diagrid module
H
NcH
NcH
H
Horizontal force due
to inclination of base
diagrid module
H
NcH
NcH
H
gravity
load
lateral load
overturning moment
lateral load
shear force
effect of geometry
basic structural unit – the triangle module
61
NG,mod
NdG
NcG
NdG
H
2 H
NG NG,mod
NG NG,mod


62
( )
d
w
,
d
w
T
L
cos
E
A
n
K

2
2



=
( ) 

2
2
2
sen
L
E
A
B
n
K
d
f
,
d
f 



=
the module stiffness capacity shear and bending stiffness provided by diagrid
on façades (KT and K)→ grid geometry, diagonal section properties, material
h
V
K *
m
*
T

=
 h
M
K *
m
*

=


known the shear force (Vm) and the overturning moment (Mm) at basis of module due to wind action→
design values of shear and bending stiffness (KT* and K*)



 ,
design values
500
2
2
H
H
H
uH =
+

=


Ad,w

 2
2 cos
h
E
n
L
V
A *
w
d
m
w
,
d






=

 2
2
2
sen
h
E
B
n
L
M
A *
f
d
m
f
,
d







=
lateral stiffness of the module
module stiffness demand
module stiffness capacity
Ad,f

63

64

Swiss Re tower
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
(G+W) Fem
(G+W) Hand
calculation
Wind
direction
Diagonals stress level [Gravity + wind] z=0
at the base
gravity + wind – diagonal DCR
65
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Diagonals stresslevel [Gravity load]
(G) Fem
(G) Hand calculation
[Level]
[N/Nb] gravity – diagonal DCR
diagridmodule
DCR
DCR ≈ 1 drift/drift limit ≈ 1
importance of local (diagonal)
strength design criteria
0
40
80
120
160
200
240
280
320
360
400
440
480
0.00% 0.10% 0.20% 0.30% 0.40% 0.50%
Normalizedlateral displacement[Windforce]
HearstTower
G.W. Tower
Swiss Re
Height[m]
dtop/H [%]
dlim = 1/500
design issues: stiffness vs. strength

diagrid – already optimized pattern ?
solution for reducing both shear flexibility and shear lag
in this perspective, the diagrid seems an “already optimized pattern” for tube
configurations, since it eliminates “a priori” the major source of the tube
inefficiency, i.e. the beam-column grid
however, the diagrid-like structures can be further improved by means of
optimization processes, that can be carried out at the member level (sizing
optimization), at the pattern level (topology optimization), or both
further, in more general terms, diagrid-like optimized patterns can be
rationally derived on the basis of the mechanical behavior of the hollow
cantilever beam describing the tube configuration

diagrid varieties: not only regular
regular variable angle variable density
regular diagrids reflect the traditional approach for
accommodating the stiffness and strength demands
along the building elevation, which is based on the
assumption:
fixed grid, variable member capacity (cross-section)
this design approach preserves the structure geometry
(i.e. column spacing in frame tubes, mega-diagonals
length in braced tubes, module scale/size and density
in diagrids), and adjusts the cross section properties
and/or the steel strength of the structural members
however, in the case of diagrid, design strategies based
on variation of the grid geometry can also be adopted,
resulting in diverse geometrical patterns characterized
by density, size, scale, angle and/or depth of the base
unit varying along the building façades.

68
diagrid …not only regular triangle tasselation
…variable-geometry strategies can be adopted according to specific structural rationales resulting in diverse
geometrical patterns characterized by density, size, scale, angle and/or depth of the base unit varying along
the building façades…
exploring this further potential of diagrids
Strategy 1: Regular diagrid
angle constant along the height
constant module size
uniform diagonal density
cross sections varying according demand
Strategy 2: Variable angle (VA)
variable angle along the height
variable module size
uniform diagonal density
Strategy 3: Variable density (VD)
constant angle
variable module size along the height
non-uniform diagonal density
Strategy 4: ISO diagrid-like pattern
concept of principal stress trajectories, employed to obtain a
pattern with both angle and density varying along elevation
this pattern is expected to be highly efficient
and visually appealing, since it expresses the
flow of forces arising in the building façade

69
an optimized structural pattern for the building with beam-like
behaviour should follow and accommodate variation of stresses along
height and across width
therefore, placing the structural members on the façade according to
direction and magnitude of internal forces, would improve the local
strength and stiffness, and, in turn, the global structural efficiency
pursuing this objective in structural systems characterised by
members mainly stressed by axial forces, naturally leads to the study
of the stress lines theory and the principal stress directions
principal stress lines: the optimal structural pattern

principal stress lines: the optimal structural pattern
70
structural efficiency – pattern optimization and stress lines
isostatic inspiration for the rib
patterns of Nervi’s floor systems

CORSO EDIFICI ALTI: PRINCIPI DI PROGETTAZIONE, SCELTA DI SOLUZIONI COSTRUTTIVE, ASPETTI DI SOSTENIBILITA’
Transbay Tower Competition,
San Francisco
Al Sharq Tower, Dubai
CITIC Towers, Guohua China
principal stress lines: inspiration for structural patterns
optimization and stress lines

72
principal stress lines: inspiration for structural patterns
CITIC
Towers,
Guohua
Financial
Center,
China
SOM

73
principal stress lines of 3D cantilever beam equivalent to the building as structural pattern
the idea is to model the building as a 3D
vertical cantilever beam with tubular section
under horizontal load, and to derive the
principal stress lines
optimization and stress lines

74
principal stress directions
biaxial stress state
the problem can be treated using Mohr’s circles
𝜎𝑥 − 𝜎𝑦
𝜎𝑦
𝜎𝑥
𝜎1, 𝜎2 =
൯
ሺ𝜎𝑥 + 𝜎𝑦
2
±
൯
ሺ𝜎𝑥 − 𝜎𝑦
2
2
+ 𝜏𝑥𝑦
2
𝑡𝑎𝑛2𝜃p =
2𝜏𝑥𝑦
𝜎𝑥 − 𝜎𝑦
principal stress magnitudes
by subdividing the solid body into
infinitely small elements, for each
of them it is possible to calculate
the principal directions

generation of the PDI pattern
H/B 6,67
H/B 5
H/B 3

77
generation of the PDI pattern
For each H/B, the 3D cantilever beam structure with hollow cross section is created by assembling four
2D shell elements representing the building façades.
Then, the cantilever faces are meshed using the Delaunay triangulation algorithm.
The centroids of mesh elements are selected as the points where principal stress values σ1 and σ2, and
relevant directions should be derived. For this purpose, the structural model is transferred in Karamba.
H/B 6,67
H/B 5
H/B 3
definition of the design domain

78
generation of the PDI pattern
By applying lateral loads and constraints, the design domain is analyzed, thus
magnitude and directions of principal stresses are defined in every centroid point
of triangulated mesh; since principal directions are tangent to the principal stress
lines, the latter can be easily evaluated and plotted as a contour diagram
According to symmetry conditions imposed to patterns, attention is focused on one half of web façade, where the principal
stress vector field is plotted

inclinazione delle stress-line in B/2 al variare dell’altezza
• per yi/H → 0 e xi/(B/2) → 0 θσpi → 90°
• per ogni yi/H e xi/(B/2) → 1 θσpi → 45°
• per yi/H → 1 e xi/(B/2) → 0 θσpi → 45°
x
y
generation of the PDI pattern
procedura di surface fitting → equazione della
superficie parametrica che fornisce il valore di θ
𝜃 = 𝑎𝑥2 + 𝑏𝑦2 + 𝑐𝑥𝑦 + 𝑑𝑥 + 𝑒𝑦 + 𝑓 • 𝑥 = ൗ
𝑝
ሺ
𝐵
2
)
• 𝑦 = Τ
𝑧
𝐻
B
p
z
H
• 𝑥 = Τ
𝑝 Τ
𝐵 2
• 𝑦 = Τ
𝑧 𝐻

generazione di ISO-patterns
H/B=6,67 θ = −4.12x2
− 10.48y2
+ 32.83xy − 32.95x − 29.68y + 86.33
H/B=5 θ = −0.32x2
− 5.59y2
+ 34.29xy − 35.02x − 32.45y + 83.15
H/B=3 θ = 2.16x2
− 1.65y2
+ 31.54xy − 30.87x − 29.51y + 74.38
1
1 1
procedura di surface fitting → coefficienti

1
81
𝜃𝜎𝑝,𝑖 = 𝑎𝑥𝑖
2
+ 𝑏𝑦𝑖
2
+ 𝑐𝑥𝑖𝑦𝑖 + 𝑑𝑥𝑖 + 𝑒𝑦𝑖 + 𝑓
1 1
yi/H → 0
xi/(B/2) → 0
yi/H → 0
xi/(B/2) → 1
generation of the PDI pattern

1
82
𝜃𝜎𝑝,𝑖 = 𝑎𝑥𝑖
2
+ 𝑏𝑦𝑖
2
+ 𝑐𝑥𝑖𝑦𝑖 + 𝑑𝑥𝑖 + 𝑒𝑦𝑖 + 𝑓
1 1
yi/H → 1
xi/(B/2) → 0 or 1
yi/H → 0.5
xi/(B/2) → 0.5
generation of the PDI pattern

83
generation of the PDI pattern
x
y
𝐵/2
𝐵
𝐻
Macro-Module
(MM)
Module (M)
Pi (xi, yi)
𝜃𝑑
,
𝑖
,
𝑀
≈ 𝜃𝜎𝑝
,
𝑖
,
𝑀
slope of the
diagonals
slope of the
stress lines
Pi (xi, yi)
the primary condition to define the PDI Pattern is to guarantee:

84
generation of the PDI pattern
However, the value θσp,i,M depends on Pi, which is the diagonal mid-point, which varies as the
diagonal inclination θd,i,M changes
𝜃𝑑
,
𝑖
,
𝑀
= 𝑓ሺ𝑃𝑖
) 𝑃𝑖
= 𝑓 𝜃𝑑
,
𝑖
,
𝑀
and
the search of θd,i,M is a non-linear problem
∆𝜃 = ෍
𝑖=1
Τ
𝑛 2
ቁ
ሺ𝜃𝜎𝑝,𝑖,1 − 𝜃𝑑,𝑖,1
2
+ ⋯ + ෍
𝑖=1
Τ
𝑛 2
ቁ
ሺ𝜃𝜎𝑝,𝑖,𝑛𝑀
− 𝜃𝑑,𝑖,𝑛𝑀
2
solved by a search function, representing the minimization of the value Δθ, defined by:

CORSO EDIFICI ALTI: PRINCIPI DI PROGETTAZIONE, SCELTA DI SOLUZIONI COSTRUTTIVE, ASPETTI DI SISTENIBILITA’
x
y
𝐵/2
𝐵
𝐻
Macro-Module
(MM)
Module (M)
𝑃𝑖 = 𝑥𝑖; 𝑦𝑖 =
𝑏𝑖,𝑀
2
+ ෍
𝑖=1
Τ
𝑛 2
𝑏𝑖−1,𝑀 ; ൗ
ℎ𝑀
4
∆𝜃 = ෍
𝑖=1
Τ
𝑛 2
ቁ
ሺ𝜃𝜎𝑝,𝑖,1 − 𝜃𝑑,𝑖,1
2
+ ⋯ + ෍
𝑖=1
Τ
𝑛 2
ቁ
ሺ𝜃𝜎𝑝,𝑖,𝑛𝑀
− 𝜃𝑑,𝑖,𝑛𝑀
2
𝜃𝑑,𝑖,𝑀 = arctan
ൗ
ℎ𝑀
2
𝑏𝑖,𝑀
values of bi,M : unknown to be obtained by minimizing Δθ
M=1
ൗ
ℎ𝑀
2
generation of the PDI pattern

H/B=6,67
Inclinazione diagonali
(dalla prima in mezzeria)
Modulo
Tipo
3
θ1 [°] 60
θ2 [°] 60
θ3 [°] 60
θ4 [°] 60
hi [m] 32
Modulo
Tipo
2
θ1 [°] 76
θ2 [°] 73
θ3 [°] 66
θ4 [°] 55
hi [m] 40
Modulo
Tipo
1
θ1 [°] 81
θ2 [°] 76
θ3 [°] 73
θ4 [°] 57
hi [m] 48
H/B=5
Inclinazione diagonali
(dalla prima in mezzeria)
Modulo
Tipo
3
θ1 [°] 53
θ2 [°] 53
θ3 [°] 49
θ4 [°] 45
hi [m] 16
Modulo
Tipo
2
θ1 [°] 67
θ2 [°] 67
θ3 [°] 60
θ4 [°] 50
hi [m] 24
Modulo
Tipo
1
θ1 [°] 73
θ2 [°] 73
θ3 [°] 69
θ4 [°] 55
hi [m] 32
H/B=3
Inclinazione diagonali
(dalla prima in mezzeria)
Modulo
Tipo
3
θ1 [°] 53
θ2 [°] 49
θ3 [°] 49
θ4 [°] 49
hi [m] 16
Modulo
Tipo
2
θ1 [°] 58
θ2 [°] 49
θ3 [°] 49
θ4 [°] 45
hi [m] 16
Modulo
Tipo
1
θ1 [°] 67
θ2 [°] 67
θ3 [°] 60
θ4 [°] 50
hi [m] 24
risultati

verifiche della procedura
ISO-Pattern
Efficacia
Efficienza

verifiche della procedura - efficacia
ISO-Pattern
pattern ad angolo costante e variabile in altezza,
suggeriti dalla letteratura
H/B=6,67
H/B=5
H/B=3
Efficacia

?
?
?
ISO-Pattern risultati ottimizzazione
topologica
pattern ad angolo costante e variabile in altezza,
suggeriti dalla letteratura
H/B=6,67
H/B=5
H/B=3
Efficacia
Efficienza
verifiche della procedura - efficienza

θ = 76/70/66 °
θ = 70° θ = 60° θ = 66/60/50°
ISO-Pattern
verifiche della procedura – efficacia: ISO pattern vs. diagrid - H/B=6,67

verifiche della procedura – efficacia: ISO pattern vs. diagrid - H/B=6,67
θ = 70° θ = 60°
ISO-pattern
spostamenti
laterali
DCR
ISO

?
?
?
iso-pattern
risultato
di
ottimizzazione
topologica
valutare l’efficienza del metodo di definizione degli iso-pattern
mediante
confronto con risultati di un’ottimizzazione topologica
obiettivo
metodologia per la ricerca topologica basata sulla creazione di
shape grammar, all’interno del quale è ricercato l’ottimo topologico
mediante l’uso di algoritmi genetici
procedimento
verifiche della procedura – efficienza – ottimizzazione topologica

Computational Design (CD) → for exploring structural solutions and developing design ideas
Generative design (GD) is a CD approach that employs algorithmic or ruled-based processes to
generates multiple and complex solutions
able to seek for uncommon solutions that are not within standard and defined set of
shapes/topology, and cannot be anticipated.
so-called “happy accidents” - in product design, mechanical engineering, and architecture
rule-based or grammar-based strategies show highly versatility and effectiveness in generating
novel and unexpected solutions
shape grammar Stiny and Gips for analyzing, interpreting and formulating design styles and
solutions both in art and architecture; generalised and imported in engineering design by
coupling shape generation and performance assessment (e.g. FEA or graphic statics methods)
computational design, generative design, shape grammar

generative design, shape grammar, structural grammar
Generative design (GD) è un approccio che utilizza processi algoritmici o basati su regole per
generare soluzioni multiple e complesse
shape grammar definisce le regole progettuali per la creazione della geometria del pattern
un numero teoricamente infinito di geometrie viene generato e considerato potenzialmente
realizzabile
→ lo shape grammar esplora solo le diverse soluzioni in termini di geometria, quindi è
completamente agnostico sul loro comportamento strutturale
le soluzioni progettuali generate attraverso le regole grammaticali vengono poi esaminate e
classificate nella fase di analisi strutturale e ottimizzazione. Sia la generazione di alternative
progettuali che la scelta della soluzione ottimale sono demandate agli algoritmi di ottimizzazione
→ structural grammar definito dall’insieme di shape grammar, analisi strutturale e processo di
ottimizzazione

95
Flow chart of the structural grammar
The flow chart is divided into four blocks (a, b, c, d):
generation of geometry with
rule-based shape grammar
a
creation of the
structural model
b
structural analysis, cross-section sizing
and output processing
c
optimization with genetic algorithms,
Topology Optimization (TO) patterns
d
optimal solution

ruled-based algorithm
structural grammar

1a
4a
1c
3c
2c
2b
1b
3b
4b
5b
2a
3a
3a
2a
2a
a) Geometry
b) Structural model
c) Analyses and results
subdivision of the algorithm in blocks and sub-blocks
Structural grammar

98
Design Rules of the shape grammar
definition of the
design domain
Two types of rules are defined in the block a:
definition of the
geometrical parameters
definition and subdivision
of design domain
1.1 – 1.2
definition of distances between
intersection points of diagonals aij
2.1 – 2.4 3.1 – 3.4 4.1
transition belt corners chamfering
DGD pattern
DGD module
a
discretization of the
design domain

99
• definition of the design domain
Two types of rules are defined in the block a:
the rules 1 identify the volume occupied by the building model
the external surface represents the building façades, where the
diagrid pattern should be applied
the façades are vertically subdivided into stacking macro‐modules,
each containing a certain number of modules
PARAMETERS:
- Base points (b1, b2, b3, b4)
RULE 1.1 RULE 1.2
MODULE
B
H
hs
PARAMETERS:
- Base surface
- hs
- nM
- nmj
B
b1 b2
b4 b3
Macro-module
module
module (m): pattern unit that represents the horizontal fascia covering
the full width of the building façade and spanning multiple floors
Macro-module (M): horizontal fascia covering the full width of the building
façade and spanning multiple modules characterized by the same
geometrical features of the pattern, i.e., the number and slope of diagonals
Design Rules of the shape grammar

Page 100
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RULE 1.1 RULE 1.2
MODULE
MACROMODULE
B
H
- nM
- nmj
B
b1 b2
b4 b3
definition of the base
points b1-b4
RULE 1.1 RULE 1.2
MODULE
MACROMODULE
B
H
- nM
- nmj
B
b1 b2
b4 b3
definition of the
base surface, hs, nM, nmj
nm1 = 5
nm2 = 6
nm3 = 6
PARAMETERS:
- Base points (b1, b2, b3, b4)
RULE 1.1 RULE 1.2
MODULE
B
H
hs
PARAMETERS:
- Base surface
- hs
- nM
- nmj
B
b1 b2
b4 b3
M1
M2
M3
M1
M2
M3
base B = 54 m, altezza interpiano hS = 4 m, altezza (H/B = 3, 5, 6.6)
numero di macromoduli nM=3
numero di moduli nel j-esimo macromodulo nmj (j=1,…,nM)
numero totale di moduli nm = Σjnmj
Es. B=54 m, hS=4 m, H = 272 m (H/B=5)
suddiviso in 3 macromoduli (nM=3), il più basso copre 5 moduli (nm1=5),
l’intermedio e il superiore coprono 6 moduli (nm2=nm3=6)
Design Rules of definition of the design domain

101
• discretization of the design domain
Two types of rules are defined in the block a:
the rules 2, 3, 4 generate the diagrid pattern on the façades, with
distribution and cross‐sections of the diagonals that vary within a
module and from one macro‐module to another
module
in each module,
building façade is covered
by 2 triangle units
M1
M2
M3
a5
a1 a2 a3 a4
𝑎𝑖 =
2
𝑛
𝑖=1
𝑧
RULE 2.3
RULE 2.4
MIRROR
REPLICATE
Design Rules of the shape grammar
the geometrical features of the pattern, i.e. the number and slope of diagonals, are the same for
the modules within a macro-module, though they vary from one macro-module to the other
Ex. 68-storey building, subdivided into 3 macro-modules (nM=3), the lowest one spanning 5
modules (nm1=5), while the intermediate and upper ones both cover 6 modules (nm2=nm3=6)

module
H
YMj-1
Mj
Mj-1
hmj = ns,mj∙hs
macro-module
B
X
Y
hmj
module
x
y
+
hmj/2
B
B/2
mj
hs
HMj = nmj∙hmj
rules for the discretization of the design domain – the module
definizione delle diagonali in ¼ di modulo

macro-module
module
module
global coordinates
local
coordinates
definition of the diagonals in ¼ of module, by
superimposing two triangle units, base to base
from ¼ of module,
the diagonals are
mirrored in the
two sym axes
rules for the discretization of the design domain – the module

macro-module
macro-module
global coordinates
the discretisation of macro‐module is obtained by
simple replication of the module along vertical direction
nm3 = 6
rules for the discretization of the design domain – the macro-module

the algorithm entrusts the geometric definition of the modules to a group of sliders, thanks to which it is possible
to set the spacing between the initial points of the elements
ℎ𝑚
2
a1 a2 a3 a4
a1 a2 a3 a4
y = 0
y = hm/2
Pi (xi, yi)
the sliders indicate the coordinates of the points Pi
ai = xi – xi-1
ℎ𝑚
2
y
x
slope and length of diagonals vary
by varying the abscissa of points Pi
rules for the discretization of the design domain

define the horizontal distance (xij - xj-1,j ) between the end points of
the i‐th diagonal of the j-h macro-module
Parameters a1j , a2j , …, aij
𝜃𝑖𝑗 = tan−1
ℎ 𝑗
2𝑎𝑖𝑗
𝑑,𝑖𝑗 =
𝑎𝑖𝑗
tan−1
ℎ 𝑗
2𝑎𝑖𝑗
𝑎𝑖𝑗 =
ℎ𝑠 × 𝑛𝑠, 𝑗
2 × tan ቁ
ሺ𝜃𝑖𝑗
y = hmj/2
y = 0
local
coordinates
rules for the discretization of the design domain

rules for the discretization of the design domain
define the horizontal distance (xij - xj-1,j ) between the end points of
the i‐th diagonal of the j-h macro-module
Parameters a1j , a2j , …, aij
local
coordinates
the sum of the
distances aij should be
exactly equal to B/2
y = hmj/2
y = 0

rules for the discretization of the design domain
define the horizontal distance (xij - xj-1,j ) between the end points of
the i‐th diagonal of the j-h macro-module
Parameters a1j , a2j , …, aij
local
coordinates
the sum of the
distances aij should be
exactly equal to B/2
double sym
y = hmj/2
y = 0

section 3a: construction of the transition belt for the presence of succesive modules with
different geometric characteristics
rules for the discretization of the design domain: transfer fascia

RULE 3.1
POINT
IDENTIFICATION
RULE 3.2
JOINING
CORRESPONDENT
POINTS
RULE 3.3
JOINING
SHIFTED
POINTS
RULE 3.4
MIRROR
DIAGONALS
rules for the construction of the transition belt
Example of unfeasible solution
the module geometry of two stacking
macro‐modules is usually different
staggered macro-modules without
continuity of the diagrid structure

111
RULE 4.1
CORNER
TAPERS
RULE 4.1
CORNER
TAPERS
Hearst Tower
New York
Due to the bird mouth profile at the building corners, the generative grammar redefines the
volume of the building and, in turn, the design domain
rules for the construction of chamfered corners
section 4a: construction of the tapered chamfering of the corners (bird mouth
profile), arising due to the triangulated pattern and lack of corner columns

creation of
structural
elements
external restraint conditions
interior constraint
conditions
material and
cross sections
loads
outputs
verification of the procedure – efficiency – topology optimization
analyses
cross section
sizing

113
𝑊 = 𝛾𝑠 ∙ 4 ∙ ෍
𝑗=1
𝑛𝑀
𝑛 𝑗 ∙ 4 ∙ ෍
𝑖=1
𝑛𝑑 Τ
𝑗 2
𝑑,𝑖𝑗 ∙ 𝐴𝑖𝑗
𝑎𝑖𝑗, 𝑖𝑛 ≤ 𝑎𝑖𝑗 ≤ 𝑎𝑖𝑗, 𝑎𝑥, with 𝑖=1
𝑛𝑑𝑗/2
𝑎𝑖𝑗=B/2
i = 1, …, ndj/2 and j = 1, …, nM
Genoma Design variable
• distances between end points of diagonals in the module:
Fitness Objective function
• minimise structural weight:
Subject to Constraints
resolution
𝑎𝑖𝑗 =
ℎ𝑠 ∙ 𝑛𝑠, 𝑗
2 ∙ tan ቁ
ሺ𝜃𝑖𝑗
optimal patterns are generated by means of a Topology Optimization (TO).
cross‐section sizing of the diagonals, fully integrated into the pattern generation process, is carried out
through the Karamba component Optimize Cross‐Sections. The genetic algorithm employed is
implemented in Galapagos component of Grasshopper and it is based on the principle of natural selection.
optimization of pattern and cross‐section sizing

resolution optimization of pattern and cross‐section sizing

MM
1
MM
2
MM
3
𝑎𝑖𝑗
varying 𝑎𝑖𝑗
INITIAL RANDOM
POPULATION
The first generation with random individuals, is obtained
by assigning arbitrary values to the parameters aij
Genetic Algorithms (GA) Genetic Algorithms (GA)

116
computation of the fitness function (minimum weight) for each
individual of the current generation and ranking of the individuals
(patterns) according their fitness value
INITIAL RANDOM
POPULATION
MAINTAIN CROSSOVER
MUTATION
selection of the best individuals (lightest patterns) of the current
generation, which survive in the next generation, defined by the
parameter Maintain (here set to 5%), while the others create
offspring by Mutation or Crossover
creation of offspring by coupling the other individuals of the
current generation (based on the Genetic Distance and governed
by the parameter Inbreeding Factor = 75%)
PARENT
OFFSPRING
definition of the way the parents are genetically combined,
by means of the parameter Coalescence Crossover (half the
genes, i.e., aij values, belonging to one parent and half
belonging to the other parent)
definition of the random genetic changes of the offspring
genome, in order to increase the biodiversity in the
population (controlled by the parameter Mutation, which
mutates a single gene, i.e., the aij value of the individuals)
NEW POPULATION

Macromodule 1 Macromodule 2 Macromodule 3
n 1 2 3 4 5 6 7 n 1 2 3 4 5 6 7 n 1 2 3 4 5 6 7
Macromodule 1 Macromodule 2 Macromodule 3
n 1 2 3 4 5 6 7 n 1 2 3 4 5 6 7 n 1 2 3 4 5 6 7
80% Wav
50% Wav
Generative design: parallel coordinates plot for H/B=6.6
numero e inclinazione delle diagonali nel
singolo modulo di ciascun macromodulo
caratterizzato da diverse soluzioni
tutte le soluzioni
soluzioni caratterizzate da un peso minore
dell’80% e del 50% del valore medio

Examples of patterns obtained from the generative design workflow
P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4
H/B=3 H/B=5 H/B=6.6
RE (60 ) PDI TO RE (70 ) PDI TO RE (70 ) PDI TO
H/B=3 H/B=5 H/B=6.6
TO
H/B = 3
P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4
H/B=3 H/B=5 H/B=6.6
P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4
H/B=3 H/B=5 H/B=6.6
P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4
H/B=3 H/B=5 H/B=6.6
P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4
H/B=3 H/B=5 H/B=6.6
RE (60 ) PDI TO RE (70 ) PDI TO RE (70 ) PDI TO
H/B=3 H/B=5 H/B=6.6
P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4
H/B=3 H/B=5 H/B=6.6
P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4
H/B=3 H/B=5 H/B=6.6
P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4
H/B=3 H/B=5 H/B=6.6
TO
H/B = 5
RE (60 ) PDI TO RE (70 ) PDI TO RE (70 ) PDI TO
H/B=3 H/B=5 H/B=6.6
P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4
H/B=3 H/B=5 H/B=6.6
P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4
H/B=3 H/B=5 H/B=6.6
P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4
H/B=3 H/B=5 H/B=6.6
P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4
H/B=3 H/B=5 H/B=6.6
TO
H/B = 6.6
generazione di diversi pattern → diversità geometrica
il progettista ha ampia scelta nella selezione della soluzione, anche considerando aspetti non strutturali
(es. luce diurna, distanze tra le diagonali per funzioni o ragioni estetiche)
non esplicitamente contemplati nel processo di ottimizzazione ed
eventualmente, dando priorità ad uno di essi a scapito di un consapevole,
lieve sacrificio in termini di efficienza

RE (60 ) PDI TO RE (70 ) PDI TO RE (70 ) PDI TO
H/B=3 H/B=5 H/B=6.6
Structural patterns: (RE) regular, (PDI) principal direction inspired, (TO)
topology optimization
TO TO TO

120
building model and structural solutions
total number of modules nm in the analysed patterns
TO – H/B = 5
nm = 10
7
9
6
7
11 10
9 9
11
0
2
4
6
8
10
12
RE
PDI
TO
RE
PDI
TO
RE
PDI
TO
H/B=3 H/B=5 H/B=6.67
nm
50
60
70
MacroModule
Module
Modules
working
as
transition
belt

3
2
4 4
3
4
3
2
4 4
3
4
3
2
3
2
3 3
0
1
2
3
4
5
PDI
TO
PDI
TO
PDI
TO
H/B=3 H/B=5 H/B=6.67
121
building model and structural solutions
MacroModule
Module
Modules
working
as
transition
belt
number of modules for each macro-module nmj in
the PDI and TO patterns
TO – H/B = 5
nm = 10
nmj
nm1
nm2
nm3
n
m1
=
4
n
m2
=
4
n
m3
=
2

H/B=6.67 H/B=5 H/B=3
OPT ISO OPT ISO OPT ISO
0 20 40 60 80 100
H/B=3
H/B=5
H/B=6.67
Weight [kg/m²]
ISO OPT
verification of the procedure – efficiency – topology optimization

20
30
40
50
60
70
80
90
100
3 4 5 6 7
Weight
(kg/m
2
)
H/B
TO
PDI
RE
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
z
/H
D/Dlim
Unit steel weight Normalised drift along elevation of the analysed patterns
the minor stiffness provided by the TO pattern
testifies that the proposed design strategy
accommodates correctly the demand of both
strength and stiffness, avoiding unnecessary stiffness
at the expense of weight increase

• snellezze alte: ruolo governante del pattern nelle prestazioni in termini di rigidezza laterale
il metodo euristico proposto restituisce configurazioni che rasentano l’ottimo topologico
• snellezze basse: il metodo fornisce rigidezze molto alte, tali da rendere maggiormente
governanti i requisiti di resistenza locali
Efficienza
Efficacia
conclusioni
strategie per ulteriore ottimizzazione del pattern nei problemi governati da resistenza:
• impiego acciaio di grado superiore

125
• la procedura proposta è caratterizzata da un elevato livello di versatilità e può essere anche impiegato
per la generazione di diversi tipi di pattern strutturale (come griglie esagonali o diagrammi Voronoi)
• il quadro concettuale stabilito per trattare le strutture diagrid e, più in generale, i modelli strutturali, può
essere utile per i progettisti strutturali coinvolti nell'esplorazione di soluzioni di modelli alternativi e,
contemporaneamente, può ampliare la libertà e liberare l'inventiva di architetti disposti a sfruttare
potenzialità espressive dei pattern strutturali
• l'approccio proposto genera pattern strutturali caratterizzati da una significativa diversità geometrica,
fornendo così un'ampia scelta di soluzioni ottimizzate, opportunamente classificate in termini di efficienza
strutturale, che consente al progettista di integrare criteri non quantitativi nel processo di selezione
ed eventualmente di dare priorità a uno di essi anche a scapito di un consapevole, lieve sacrificio in termini
di efficienza
• pattern molto simili a quelli ottenuti mappando le linee di tensione (PDI) emergono automaticamente
dall'applicazione della strategia generativa proposta, in particolare per gli edifici snelli. Questo risultato
suggerisce anche l'affascinante possibilità di utilizzare tali strategie come strumento di acquisizione
delle conoscenze progettuali
conclusioni