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Ottimizzazione Strutturale 2013. Lezione della Prof.ssa Elena Mele del 1° febbraio 2023

Franco BontempiSuivre

Ph.D., P.E., Professor of Structural Analysis and Design, School of Engineering, University of Rome "La Sapienza" à University of Rome "La Sapienza"Publicité

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- 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 CORSO EDIFICI ALTI: PRINCIPI DI PROGETTAZIONE, SCELTA DI SOLUZIONI COSTRUTTIVE, ASPETTI DI SISTENIBILITA’ 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

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