Parametric Analysis of a Spiraled Shell: Learning from Nature’S Adaptable Structures

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Article Parametric Analysis of a Spiraled Shell: Learning from Nature’s Adaptable Structures

Diana A. Chen 1,* , Brandon E. Ross 2 and Leidy E. Klotz 3

1 Department of Integrated Engineering, University of San Diego, 5998 Alcala Park, San Diego, CA 92110, USA 2 Glenn Department of Civil Engineering, Clemson University, 310A Lowry Hall Box 340911, Clemson, SC 29634, USA; [email protected] 3 Departments of Architecture, Civil and Environmental Engineering, University of Virginia, Thornton Hall d202, P.O. Box 400259, Charlottesville, VA 22904, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-619-260-4622

Received: 27 August 2018; Accepted: 8 November 2018; Published: 13 November 2018

Abstract: In our current building design philosophy, structural design is based on static predictions of the loads a building will need to withstand and the services it will need to provide. However, one study found that 60% of all buildings are demolished due to obsolescence. To combat our obsolescence-demolition culture, we turn to Nature for lessons about adaptable structural design. In this paper, we investigate the structural adaptability of the T. terebra spiraled turret shell through ﬁnite element modeling and parametric studies. The shell is able to change its structure over time to meet changing performance demands—a feat of adaptability that could transform our current infrastructure design. Modeling the shell’s growth process is an early and simple attempt at characterizing adaptability. As the mollusk deposits material overtime, its shell wall thickness changes, and its number of whorls increases. We designed parametric studies around these two modes of growth and investigated their effect on structural integrity and living convenience for the mollusk. By drawing parallels between the shell structure and human structures, we demonstrate connections between engineering challenges and Nature’s solutions. We encourage readers to consider biomimicry as a source of inspiration for their own quantitative studies for a more sustainable world.

Keywords: biomimicry; bioinspired design; natural systems; structures; adaptability; modeling; parametric analysis

1. Introduction With the rise of powerful information technology developed in the past few decades, obsolescence is a problem that is well-known to most people, evident every time the software of a smartphone outgrows its hardware. Obsolescence, the lack of suitability for desired use, describes the condition of objects, services, or practices when they are no longer able to meet changing requirements. These requirements range from new safety standards to users’ desires, but the cause of abandonment is largely irrelevant to its effect. Contrastingly, adaptability is the ability of an object to change to improve its functionality or suitability for a purpose. Viewed together, adaptable design can be leveraged as a tool for sustainability by reducing the occurrences of obsolescence. Adaptability has great potential for application in the built environment, where the “demolition culture” found in the United States encourages a build-break philosophy. As noted by Lemer [1], obsolescence poses a heavy burden on the owners and users of civil infrastructure. A study in Minnesota found that about 60% of all building demolitions are due to some type of obsolescence rather than reduced structural integrity [2]. While this study was contained in St. Paul, there is no reason

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Designs 2018, 2, 46 2 of 17 rather than reduced structural integrity [2]. While this study was contained in St. Paul, there is no reason to believe that results would be different across the U.S; similar trends have been observed in otherto believe developed that results countries would such be as different the U.K. across [3] and the Japan U.S; similar [4]. This trends high havepercentage been observed of obsolescence in other- baseddeveloped demolitions countries suggests such as that the the U.K. current [3] and way Japan that [4 ].our This structures high percentage are designed of obsolescence-based is inadequate for meetingdemolitions our suggests long-term that service the current needs. way Structures that our structuresare designed are designed to last for is hundreds inadequate of for years meeting but inhabitantsour long-term cycle service through needs. stru Structuresctures are about designed every to 30 last years for hundreds [5], causing of years more but inhabitantsdemolition cycle and construction.through structures about every 30 years [5], causing more demolition and construction. While accurately predicting predicting (let (let alone alone planning) planning) for for the the future future is is difficult, difﬁcult, adaptability can can be be a a tooltool for for maintainin maintainingg relevance. relevance. We We can circumvent our unpreparedness by designing structures structures that are able to adapt to changing demands rather than continuing to build path-dependentpath-dependent structures based on static predictions.

1.1. Adaptability Adaptability in in Structures Structures Adaptability in in the the built built environment environment can be can implemented be implemented on many on levels many to combat levels obsolescence. to combat obsolescence.In his book, How In hisBuildings book, LearnHow , Buildings Stewart Brand Learn, describes Stewart six Brand shearing describes layers six of shearing a building, layers based of on a building,the expected based lifetime on the of expected each layer lifetime [5]. These of each “six layer S’s” are [5]. shown These in“six Figure S’s”1 are with shown their typicalin Figure lifetimes. 1 with theirTreating typical different lifetimes. layers Treating as separate different systems layers issigniﬁcant as separate for systems enabling is adaptability significant forin buildings, enabling adaptabilitysince this allows in buildings, us to address since different this allows components us to address of a building different on components its appropriate of a time building scale. Leton its us appropriateconsider one time ludicrous scale. example:Let us consider if the skinone ludicr layer wereous example: inseparable if the from skin the layer structural were inseparable layer, a building from thewould structural need to layer, be torn a building down every would time need its paintto be colortorn down was changed. every time its paint color was changed.

Figure 1. 1. SixSix shearing shearing layers layers of a structure of a structure and their and lifetimes their lifetimes (adapted (adapted from Reference from Reference [5], Penguin [5],, 1995Penguin,). 1995).

A well-planned well-planned example example of ofadaptability adaptability in structures in structures is illustrated is illustrated by the 2012 by the London 2012 Olympics, London Olympics,where the where designers the designers of the stadium of the stadium arenas tookarenas possible took possible future future needs needs and uses and ofuses the of site the intosite intoconsideration consideration [6]. The[6]. designersThe designers understood understood that thethat Games the Games were were a passing a passing event event and could and could result resultin an enormousin an enormous waste waste of infrastructure of infrastructure and investment.and investment. Their Their plan plan was was to convert to convert the 80,000-seatthe 80,000- seatstadium stadium into into a smaller-sized a smaller-sized arena arena after after the the Games, Games, and and this this was was reﬂected reflected in in thethe useuse of temporary steelsteel structures structures which which allowed allowed for for partial partial disassembly. disassembly. To To allow for deconstruction, the stadium had a simple design with only two parts: a a concrete concrete bowl, bowl, and and a a lightweight steel steel truss structure that comprisedcomprised the the upper tiers and supported the roof membrane. Even Even details details as as small as connections were taking into consideration for component reuse and adaptability adaptability—bolts—bolts were were used rather than welded steel connections. The The foresight foresight of of the the London London Olympics Olympics designers designers in in creating structures adaptable to changing demands ensured a prolonged life for both the site site and and the the structures structures by by taking taking thethe lo longstandingngstanding population’s needs into account.

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While designing for adaptability may be more sustainable than succumbing to obsolescence, learning how to build adaptable structures is still a largely uncharted path. There have only been a handful of architectural designs and structural studies that use biomimicry [7–11], and even fewer have examined adaptability in structures. Here is where we can turn to biomimicry for insight. Biomimicry, a design method that mimics Nature’s time-tested patterns and strategies, has potential for lessons in how to incorporate adaptability into structural design for a sustainable built environment.

1.2. Turning to Biomimicry and Engineering Systems for Adaptability Solutions Biomimicry is a design method that draws inspiration from Nature for sustainable solutions to human challenges. Nature has 3.8 billion years’ worth of time-tested patterns and strategies, which engineers can and have learned from and applied. Janine Benyus, the modern popularizer of biomimicry, describes ten of “Nature’s principles”, referring to rules of thumb that natural systems follow [12]. One principle, form follows function, stands out as particularly relevant for adaptable infrastructure design. While this phrase is widely recognized as originating from modern, industrial architecture (20th century), this concept draws from the theory of evolution (e.g., as described by the overlap in Lamarck and Darwin’s theories in the 16th–17th centuries). Nature shapes its structural forms to help meet functional requirements, rather than adding more material and energy to produce similar outcomes [12]. Natural structures are honed and polished through natural selection, resulting in systems that are effective, efficient, and multifunctional to meet each organism’s performance demands [13]. This research considers Nature’s principle of form follows function through an engineering lens by exploring biological systems through an engineering systems approach to better understand the mathematical functions underpinning natural forms [14]. To describe the behavior of a system, a mathematical model called a governing equation is often used [15]. Governing equations are frequently taught as differential equations obtained by substituting a system’s constitutive relationships into more general laws of physics, such as Newton’s Second Law for mechanical systems and Kirchhoff’s Voltage and Current Laws for electrical systems. In classical mechanics, the dynamic motion of a system is characterized as a harmonic oscillator using three simple elements: a spring, a mass, and a damper. In electrical systems, capacitors, inductors, and resistors are directly analogous to the parameters for mechanical systems. Governing equations are also used in fluid flow and heat transfer, among other topics. The goal in discretizing various system types into fundamental parameters is to create simple models of complex systems, helping us to understand how systems’ inputs affect behavioral responses (e.g., system functions transform the inputs into behavioral outputs—in this case, structural form). This research asks biomimetic engineers, “If we’ve determined governing equations for mechanical systems, and governing equations for electrical systems, can we determine a governing equation (or at least the model elements) for adaptable systems?” Users of biomimicry often function on the border of different disciplines, tools, and databases for connecting ideas for inspiration and application. As such, biomimicry is often a two-way street between biology and an engineering field. Biomimicry can be problem-driven, where engineers turn to biology in a search for sustainable solutions, or solution-driven, where scientists search for applications of their interesting discoveries [16,17]. In this paper, engineering mechanics is used to parameterize a problem-driven example—the adaptable growth of a spiraled seashell—to explore how biomimicry can be used for application to the design of adaptable structures.

2. Materials and Methods This paper describes a biomimetic exploration of adaptability in a natural structure—the Turritella terebra, a spiraled turret shell that adapts its structure (its shell) to meet changing performance demands (mollusk growth). While “enlarging” is a simplistic form of growth, this project is an early attempt at characterizing adaptability in natural systems. The T. terebra grows by adding a layer of calcium carbonate to its newest edge, slowly incrementing its shell wall thickness and its overall helical length, Designs 2018, 2, 46 4 of 17

Designs 2018, 2, x 4 of 17 which affects structural strength and behavior. The T. terebra was selected as the specimen for this studyselected due as to the its specimen simplicity, for as wellthis study as its economicaldue to its simplicity, availability as in well large as quantities its economical for an availability experimental in study.large quantities This structural for an engineering experimental project study. investigates This structural three parameters engineering that project may investigates shed light on three the adaptableparameters growth that may of the shed shell: light (1) on the the ratio adaptable of volumetric growth of capacity the shell: to mass—dependent (1) the ratio of volumetric on wall thicknesscapacity to determined mass—dependent empirically; on (2)wall the thickness number of determined whorls (full empirically; revolutions of(2) the the shell)—investigated number of whorls theoretically;(full revolutions and of (3) the the shell) load capacity—investigated of the shelltheoretically; under three-point and (3) the bending load capacity through of the the development shell under ofthree a ﬁnite-point element bending model through of the theT. development terebra. of a finite element model of the T. terebra.

2.1.2.1. ShellShell GeometryGeometry andand ModelModel TheThe systematic systematic study study of ofseashells seashells is dated is dated as far as back far as back 1838, as when 1838, Moseley when used Moseley a logarithmic used a logarithmicspiral to describe spiral theirto describe geometric their form geometric [18]. Since form Moseley, [18]. Since many Moseley, others havemany improved others have the improvedmathematical the mathematical characterization characterization of shell geometry of shell geometrythrough mathematical through mathematical biology biology[19], digital [19], digitalconstruction construction [20], enhanced [20], enhanced orthogonality orthogonality of curves of [21] curves, computational [21], computational shell generation shell generation [22], and [22 the], anddefinition the deﬁnition of three ofadditional three additional angles for angles more for reliable more modeling reliable modeling [23]. The [ 23spiraled]. The spiraledgastropod gastropod shell can shellbe characterized can be characterized throughout throughoutits taxonomic its class taxonomic using a classset of usinggeometric a set variables. of geometric These variables.variables Thesedetermine variables the form determine of the shell the and form include of the parameters shell and include such as parametersvertical distance such from as vertical the apex, distance radius fromof the the helico apex,-spiral, radius shape of the function helico-spiral, of the shapeaperture, function and various of the aperture, expansion and rates, various among expansion others. rates,Basic amongshell terminology others. Basic is shellillustrated terminology in Figure is illustrated2. in Figure2.

Figure 2.2. Terminology ofof thethe TurritellaTurritella terebraterebrashell shell features. features.

TheThe gastropodgastropod shellshell isis aa compositecomposite materialmaterial ofof calciumcalcium carbonate carbonate (in (in the the form form of of calcite calcite and/or and/or aragonite)aragonite) andand organic organic macromolecules macromolecules (e.g., (e.g., proteins) proteins) [24 [24,25],25]. As. As a reference a reference point, point, Table Table1 shows 1 shows the rangethe range of values of val forues mechanicalfor mechanical properties properties for thefor theOtala Otala lactea lactea(the (the Spanish Spanish snail, snail, a related a related mollusk) mollusk) as summarizedas summarized by by Rajabi Rajabi et al.et al. [26 [26]]. Calcium. Calcium carbonate carbonate has has similar similar strengths strengths to to human-made human-made buildingbuilding materialsmaterials [[13]13] andand cancan bebe consideredconsidered inin futurefuture studiesstudies ofof biomimicrybiomimicry toto developdevelop buildingbuilding materialsmaterials withwith improvedimproved strength-to-weightstrength-to-weight ratios. ratios.

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Table 1. Mechanical properties of O. lactea (summarized from Reference [26]).

Property Value References Currey, 1976; Young’s modulus 30–68 GPa Currey and Taylor, 1974; Kearney, 2006 Designs 2018, 2, xShear modulus 19–39 GPa Kearney, 2006 5 of 17

BulkTable modulus 1. Mechanical properties 47–69of O. lactea GPa (summarized fromKearney, Reference 2006[26]). Currey, 1976; Tensile strengthProperty 25–60Value MPa References Currey,Currey 1976; and Taylor, 1974 Young’s modulus 30–68 GPa Currey and Taylor,Currey, 1974; 1976; Compressive strength 198–336 MPa Kearney,Liang 2006 et al. 2008 Shear modulus 19–39 GPa Kearney, 2006 Shear strengthBulk modulus 22–3647–69 MPaGPa Kearney, Kearney,2006 2006 Currey, 1976; Fracture energyTensile strength 157.725–60 N/mMPa Ashby et al., 1995 Currey and Taylor, 1974 3 Density 2800 kg/m Currey,Ashby 1976; et al., 1995 Compressive strength 198–336 MPa Poisson’s ratio 0.3Liang et Karmat al. 2008 et al., 2004 Shear strength 22–36 MPa Kearney, 2006 Fracture energy 157.7 N/m Ashby et al., 1995 This section describes howDensity a finite element2800 kg/m^3 (FE) modelAshby was et createdal., 1995 in order to run parametric Poisson’s ratio 0.3 Karmat et al., 2004 studies on the T. terebra. First, a 3D geometry of the shell was constructed in MATLAB to represent a shell specimenThis section with describes approximately how a finite 19.75 element whorls, (FE) model a height was created of 5.4 in order inches, to run and parametric base width of 1.4 inchesstudies [27]. on Thethe T. 3D terebra. geometry First, a 3D of geometry the shell of isthe dependent shell was constructed on two in angles MATLAB (α and to representβ, as showna in shell specimen with approximately 19.75 whorls, a height of 5.4 inches, and base width of 1.4 inches Figure3), an initial apex length R 0, and the “generating curve” of the shell aperture. The generating curve of[27] a particular. The 3D geometry specimen of the can shell be is extracteddependent on from two aangles cross-sectional (훼 and 훽, asimage shown in of Figure the shell, 3), an and this initial apex length R0, and the “generating curve” of the shell aperture. The generating curve of a 2D shapeparticular is then specim useden as can the be template extracted from for creatinga cross-sectional whorls image as they of the revolve shell, and around this 2D theshape columnella is axis [26,then28,29 used]. Toas the extract template the for outline, creating awhorls simple as they code revolve uses around basic the image columnella processing axis [26,28,29] techniques. on a manuallyTo extract isolated the outline, aperture a simple to determine code uses basic the image curve-fit processing function techniques (see Reference on a manually [27 ]isolated for full code). The extractedaperture generating to determine curve the from curve the-fit shell function specimen (see Reference and its accompanying[27] for full code). curve-fit The extracted used in the FE model aregenerating shown curve in Figure from4 the. The shell resulting specimen MATLAB and its accompanyin geometryg (combiningcurve-fit used the in the generating FE model curveare with shown◦ in Figure◦ 4. The resulting MATLAB geometry (combining the generating curve with 훼 = α = 88.63 , β = 3.3 , R0 = 6.8 mm) is shown in Figure5 as a progression from its cross-sectional scan 88.63°, 훽 = 3.3°, 푅0 = 6.8 푚푚) is shown in Figure 5 as a progression from its cross-sectional scan to to its overlayits overlay with with each each geometrically geometrically modeled half. half.

FigureFigure 3. Diagram 3. Diagram of shell of shell variables. variables Reproduced. Reproduced with with permission permission from from [29], Elsevier, [29], Elsevier, 2012. 2012.

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Figure 4. Manually extracted generating curve outline from cross cross-sectional-sectional scan of shell (left) and curve-ﬁtcurve-fit function generated fromfrom outlineoutline ((rightright).).

Figure 5. 5.Cross-sectional Cross-sectional scan scan of shell of shell halves halves (left); ( MATLABleft); MATLAB generated generated the model the using model the specimen’s using the parametersspecimen’s (parameterscenter); and (center digital); overlaysand digital of eachoverlays half (ofright each). half The hand(right symbols). The hand represent symbols the represent opening ofthe the opening shell into of the two shell halves. into two halves. The shell geometry, developed using Rajabi’s code [26,29], was then imported into ANSYS ﬁnite The shell geometry, developed using Rajabi’s code [26,29], was then imported into ANSYS finite element software through direct generation to create nodes and shell elements [30]. This model element software through direct generation to create nodes and shell elements [30]. This model uses uses SHELL181 elements, the simplest version of the four-node shell element, with six degrees of SHELL181 elements, the simplest version of the four-node shell element, with six degrees of freedom freedom at each node. The structural FE model uses a Young’s modulus and Poisson’s ratio typical at each node. The structural FE model uses a Young’s modulus and Poisson’s ratio typical of mollusk of mollusk shells, E = 50 GPa, ν = 0.31 [26,29,31,32]. The structural model was then subjected shells, 퐸 = 50 퐺푃푎, 휈 = 0.31 [26,29,31,32]. The structural model was then subjected to simple three- to simple three-point bending as the base approach for investigating adaptability through stress point bending as the base approach for investigating adaptability through stress as a failure criterion. as a failure criterion. (Interested readers are directed to Reference [27] for details of the FE model, (Interested readers are directed to Reference [27] for details of the FE model, such as constraints, such as constraints, boundary conditions, mesh convergence, comparison with experimental results, boundary conditions, mesh convergence, comparison with experimental results, and treatment of and treatment of numerical error.) Figure6 shows the typical deformation of the shell model in ANSYS numerical error.) Figure 6 shows the typical deformation of the shell model in ANSYS as it undergoes as it undergoes three-point bending. three-point bending.

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Figure 6. Typical deformation of the shell model as subjected to three-point bending. The pin (bottom left), roller (bottom right), and loading (top left at highest point) boundary conditions are shown. FigureFigure 6. Typical 6. Typical deformation deformation of of the the shell shell model model as subjected to to three three-point-point bending. bending. The Thepin ( pinbottom (bottom left),left roller), roller (bottom (bottom right right), and), and loading loading ( top(top left left atat highest point) point) boundary boundary conditions conditions are areshown. shown. 2.2. Parametric Studies 2.2. Parametric2.2. Parametric Studies Studies The structural model was developed as a means to explore parametric effects of growth on the shell’sThe adaptability. structuralThe structur model alAs model shell was wasgrowth developed developed was the asas a aprimary meansmeans to toform explore explore of adaptabilityparametric parametric effects investigated, effects of growth of growth onparametric the on the shell’sstudiesshell’s adaptability. were adaptability. designed As Asaround shell shell growth growththe method was was the the in primary primarywhich theform form shell of of adaptability adaptabilityadded material investigated, investigated, to its structure. parametric parametric Here, studies were designed around the method in which the shell added material to its structure. Here, studiesthree outcomes were designed due to the around addition the methodof material in whichare discussed: the shell number added materialof whorls to as its a structure.characterization Here, three outcomes due to the addition of material are discussed: number of whorls as a characterization three outcomes due to the addition of material are discussed: number of whorls as a characterization of of shellof shell maturity maturity; optimization; optimization of of living living convenience convenience for for the the mollusk mollusk; and; and impact impact on the on tensile the tensile shellstrengthstrength maturity; of the of optimizationtheshell. shell. of living convenience for the mollusk; and impact on the tensile strength of the shell. 2.2.1.2.2.1. Numbe Number ofr Whorls of Whorls 2.2.1. Number of Whorls As theAs molluskthe mollusk grows, grows, it itsecretes secretes a a substance substance whichwhich solidifies solidifies at atthe the edge edge of the of shellthe shell aperture aperture [33]. As[33]This the. This gradual mollusk gradual deposition grows, deposition it secretes of of primarily primarily a substance calciumcalcium which carbonatecarbonate solidiﬁes sutures sutures at the the edgethe current current ofthe revolution shellrevolution aperture to the to [ 33the]. Thisprevious gradualprevious shell depositionshell section, section, creating of creating primarily a abuild build calcium-up-up ofof carbonate material land and sutures lengthening lengthening the current the the spire revolution spire over over time. to time. As the such, previousAs such, the number of whorls increases with a shell’s maturity and was our proxy for characterizing shell shellthe number section, of creating whorls a build-upincreases ofwith material a shell’s and maturity lengthening and the was spire our over proxy time. for Ascharacterizing such, the number shell age. ofage. whorls increases with a shell’s maturity and was our proxy for characterizing shell age. In experimental studies conducted on approximately 200 T. terebra specimens, a small collection In experimental studies conducted on approximately 200 T. terebra specimens, a small collection (Inn =experimental 24) were cut into studies two longitudinal conducted halveson approximately and their wall 200thicknesses T. terebra were specimens, measured a as small a function collection (n = 24) were cut into two longitudinal halves and their wall thicknesses were measured as a function (n = 24)of position were cut along into the two columnella longitudinal [27]. halvesThis simple and studytheir wallcorroborated thicknesses that materialwere measured is only deposited as a function ofof positionpositionat the shell alongalong aperture, thethe columnellacolumnella as smaller [[27]shells27].. ThisThis did not simplesimple have studystudy thinner corroboratedcorroborated walls than their thatthat larger materialmaterial counterparts isis onlyonly deposited(bothdeposited atat thethein shellshell terms aperture,aperture, of whorl ascountas smallersmaller or absolute shellsshells shell diddid notlength).not havehave thinnerthinner wallswalls thanthan theirtheir largerlarger counterpartscounterparts (both(both inin termsterms ofofAs whorlwhorl each shell countcount specimen oror absoluteabsolute has unique shellshell length).geometriclength). variables, a control study where all constants but AstheAs eachnumbereach shellshell of specimen whorlsspecimen are has identicalhas unique unique is geometricunfeasible. geometric variables,Instead, variables, the a controlnumba controler studyof studywhorls where where was all parameterized constantsall constants but but the numberthe numberin the of model whorls of whorls by are simulating identical are identical three is unfeasible.-point is unfeasible. bending Instead, on Instead, a section the number the of thenumb ofshell whorlser spire,of whorls was as shown parameterized was in parameterized Figure 7. in the It should be noted that the effects of the cantilever base were negligible, as decreasing the material modelin the model by simulating by simulating three-point three bending-point bending on a section on a ofsection the shell of the spire, shell as shownspire, as in shown Figure 7in. ItFigure should 7. thickness of the cantilever to effectively zero (negating its stiffness and ensuring no force beIt should noted that be noted the effects that ofthe the effects cantilever of the basecantilever were negligible, base were asnegligible, decreasing as the decreasing material the thickness material of propagation) produced the same result as leaving it intact [27], as per St. Venant’s principle regarding the cantilever to effectively zero (negating its stiffness and ensuring no force propagation) produced thicknessstresses of distant the canfromtilever points of to interest. effectively zero (negating its stiffness and ensuring no force thepropagation) same result produced as leaving the it same intact result [27], as as leaving per St. Venant’sit intact [27] principle, as per regardingSt. Venant’s stresses principle distant regarding from pointsstresses of distant interest. from points of interest.

Figure 7. Three-point bending model with pin-roller connections acting upon a simulation of one shell at three different stages of growth.

FigureFigure 7. ThreeThree-point-point bending model with pin-rollerpin-roller connectionsconnections actingacting uponupon aa simulationsimulation ofof oneone shell atat threethree differentdifferent stagesstages ofof growth.growth.

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2.2.2.2.2.2. LivingLiving ConvenienceConvenience TheThe thickness thickness measurements measurements described described above above did, did, however, however, increase increase as as the the shell’s shell’s diameter diameter increased.increased. Considering Considering wall wall thickness thickness from from the the mollusk’s mollusk’s perspective, perspective, we posit we posit that the that mollusk the mollusk likely trieslikely to tries maximize to maximize internal internal volume volume (the snail’s (the functional snail’s functional living space, living with space, the with shell the being shell a physical being a constraintphysical constraint on its body), on its while body), minimizing while minimizing shell weight. shell Wall weight. thickness Wall thickness is inversely is inversely related to re internallated to capacity,internal andcapacity, directly and related directly to therelated mass to of the the mass shell. of Thus, the theshell. ratio Thus, of capacity the ratio to of mass, capacity renamed to mass, here asrenamed “living here convenience”, as “living isconvenience”, used as one variableis used as in one parametric variable analyses in parametric as thickness analyses is systematicallyas thickness is incremented.systematically The incremented. internal living The space internal is approximated living space is as approximated a sphere with aas diameter a sphere half with the a widthdiameter of thehalf shell’s the width widest of diameter, the shell’s less widestits wall diameter,thickness (see less Figure its wall8). Mass thickness was determined (see Figure empirically 8). Mass was in andetermined experiment empirically similar to in wall an thicknessexperiment described similar to above: wall shellsthickness of various described sizes above: (n = 22) shells were of cutvarious into segmentssizes (n = 22) along were their cut spire, into segments and these along segments their spire, were weighedand these individually segments were to determineweighed individually a curve-ﬁt equationto determine for cumulative a curve-fit massequation as a for function cumulative of length. mass as a function of length.

FigureFigure 8.8. DiagramDiagram showingshowing thethe differentdifferent measurementsmeasurements usedused forfor basebase diameterdiameter andand livingliving space.space. 2.2.3. Structural Capacity 2.2.3. Structural Capacity Lastly, the parametric analysis examines how the shell’s addition of material affects its strength. Lastly, the parametric analysis examines how the shell’s addition of material affects its strength. In structural engineering, the phrase form follows function implies that the architecture (form) of a In structural engineering, the phrase form follows function implies that the architecture (form) of a structure should support rather than detract from its structural integrity—the ability to keep a structure structure should support rather than detract from its structural integrity—the ability to keep a from failing in a crisis (function). One simple assessment of integrity is to examine the structure’s structure from failing in a crisis (function). One simple assessment of integrity is to examine the behavior under stress. structure’s behavior under stress. In three-point bending, brittle materials such as calcium carbonate typically experience tensile In three-point bending, brittle materials such as calcium carbonate typically experience tensile failure; hence, tensile strength was the failure criteria of interest for the shell FE model. The tensile failure; hence, tensile strength was the failure criteria of interest for the shell FE model. The tensile strength typical of mollusk shells is 40.3 MPa (5845 psi) [31], and when this value was exceeded at any strength typical of mollusk shells is 40.3 MPa (5845 psi) [31], and when this value was exceeded at location in the model, the shell was predicted to have fractured in real life. Typically, more material any location in the model, the shell was predicted to have fractured in real life. Typically, more (e.g., thicker walls) enables structures to withstand higher loads, and this was found to be the case for material (e.g., thicker walls) enables structures to withstand higher loads, and this was found to be the shell as wall thickness was incremented in the parametric analysis. (For details on how varying the case for the shell as wall thickness was incremented in the parametric analysis. (For details on thickness was implemented in the FE model, see Reference [27].) Figure9 illustrates the bottom surface how varying thickness was implemented in the FE model, see Reference [27].) Figure 9 illustrates the of the shell under deformation (bending out of the page) and highlights the suture where fracture is bottom surface of the shell under deformation (bending out of the page) and highlights the suture predicted. This location matched the failure observed in physical tests of shells [27]. where fracture is predicted. This location matched the failure observed in physical tests of shells [27]. As turret shells develop over time, they adapt and grow by depositing calcium carbonate to strengthen and lengthen their structure. The addition of material is considered through parametric analyses in two ways: (a) deposition at the aperture opening, slowly lengthening the spire as the shell matures; and (b) deposition in terms of wall thickness, which affects the shell’s volume, mass, and mechanical strength. Variations in percent thickness of the shell wall inﬂuence living convenience and load capacity (b), and these changes can be examined over time with shell maturity (a). Thus, it was determined that wall thickness is a signiﬁcant variable in the shell’s adaptability. Incrementing thickness captures the shell’s changes in 3D space, as shown in Figure 10.

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Figure 9. Tensile failure occurs at a suture on the bottom surface of the shell. The color scale ranges from 0 psi to 5845 psi, the tensile strength of calcium carbonate. The gray sliver in the cutout indicates that the tensile capacity is exceeded and the location of the failure.

As turret shells develop over time, they adapt and grow by depositing calcium carbonate to strengthen and lengthen their structure. The addition of material is considered through parametric analyses in two ways: (a) deposition at the aperture opening, slowly lengthening the spire as the shell matures; and (b) deposition in terms of wall thickness, which affects the shell’s volume, mass, and mechanical strength. Variations in percent thickness of the shell wall influence living conven ience and load capacity (b), and these changes can be examined over time with shell maturity (a). Thus, it FiFiguregure 9. Tensile failure occurs at a suture on the bottom surface of the shell. The color scale ranges was determined that wall thickness is a significant variable in the shell’s adaptability. Incrementing from 0 psi to 5845 psi, psi, the the tensile tensile strength strength of calcium carbonate. The The gray gray sliver sliver in in the the cutout cutout indicates indicates thickness captures the shell’s changes in 3D space, as shown in Figure 10. that the tensile capacity is exceeded and the locationlocation of the failure.

As turret shells develop over time, they adapt and grow by depositing calcium carbonate to strengthen and lengthen their structure. The addition of material is considered through parametric analyses in two ways: (a) deposition at the aperture opening, slowly lengthening the spire as the shell matures; and (b) deposition in terms of wall thickness, which affects the shell’s volume, mass, and mechanical strength. Variations in percent thickness of the shell wall influence living convenience and load capacity (b), and these changes can be examined over time with shell maturity (a). Thus, it was determined that wall thickness is a significant variable in the shell’s adaptability. Incrementing thickness captures the shell’s changes in 3D space, as shown in Figure 10. FigureFigure 10.10. Three-dimensionalThree-dimensional representation of the effects ofof varyingvarying shellshell wallwall thicknessthickness onon livingliving convenienceconvenience andand loadload capacity,capacity, andand naturalnatural additionaddition ofof materialmaterial on on shell shell maturity. maturity.

TheThe linearlinear equationequation determineddetermined toto bestbest modelmodel wallwall thicknessthickness asas aa functionfunction ofof lengthlength waswas

t(푡x()푥)==0.00240.0024××d(푑x()푥+) +0.029,0.029, (1)(1) where t(x) is the wall thickness, and d(x) is the diameter of the shell, in inches at any position x along where t(x) is the wall thickness, and d(x) is the diameter of the shell, in inches at any position x along the columella. Equation 1 was developed from a series of measurements of T. terebra shell segments the columella. Equation 1 was developed from a series of measurements of T. terebra shell segments in experimental studies [27]. Incrementing wall thickness in parametric studies was performed by in experimental studies [27]. Incrementing wall thickness in parametric studies was performed by scaling the entire thickness equation by a factor in 5% increments, up to ±20%. Because the study scalingFigure the entire10. Three thickness-dimensional equation representation by a factor of in the 5% effects increments, of varying up toshell±20% wall. thickness Because theon living study was interested convenience in overall and load trends, capacity, increments and natural of 5% addition were of deemed material to on provide shell maturity. sufﬁcient resolution for the parametric study; smaller increments would not have provided signiﬁcant information to add to the overallThe trend. linear These equation varying determined thicknesses to best were model implemented wall thickness in the as FEa function model, andof length the results was of the model provided the corresponding load at which the shell fractured, as discussed in the next section. 푡(푥) = 0.0024 × 푑(푥) + 0.029, (1) where t(x) is the wall thickness, and d(x) is the diameter of the shell, in inches at any position x along the columella. Equation 1 was developed from a series of measurements of T. terebra shell segments in experimental studies [27]. Incrementing wall thickness in parametric studies was performed by scaling the entire thickness equation by a factor in 5% increments, up to ±20%. Because the study

Designs 2018, 2, x 10 of 17

was interested in overall trends, increments of 5% were deemed to provide sufficient resolution for the parametric study; smaller increments would not have provided significant information to add to Designsthe overall2018, 2, 46trend. These varying thicknesses were implemented in the FE model, and the results10 of of 17 the model provided the corresponding load at which the shell fractured, as discussed in the next section. 3. Results 3. Results Results from the parametric studies are summarized in three ﬁgures below by presenting data as three separateResults planes,from the each parametric with a pair studies of axes are from summarized Figure 10 in. While three thefigures data below in each by graph presenting are primarily data governedas three byseparate the x- andplanes,y-axis, each the with third a pair parameter of axes isfrom illustrated Figure 10 as. aWhile color the spectrum data in distributedeach graph acrossare thatprimarily variable’s gov fullerned range. by the Furthermore, x- and y-axi thes, the incrementation third parameter of thickness is illustrated is superimposed as a color spectrum on each graph.distributed The increments across that of ± variable’s5% wall thickness full range. are indicatedFurthermore, as a thespectrum incrementation that follows of the thickness curve of is the superimposed on each graph. The increments of ±5% wall thickness are indicated as a spectrum that baseline thickness; percent changes are shown with red circular indicators, with the X’s representing follows the curve of the baseline thickness; percent changes are shown with red circular indicators, the +20% change to help orient the reader. For example, Figure 11 shows living convenience on the with the X’s representing the +20% change to help orient the reader. For example, Figure 11 shows y-axis, age (number of whorls) on the x-axis, and failure load shown in color. The baseline thickness living convenience on the y-axis, age (number of whorls) on the x-axis, and failure load shown in (0% change) is shown as a red line and can be seen in the center of the vertical spectrum as an asterisk. color. The baseline thickness (0% change) is shown as a red line and can be seen in the center of the Figurevertical 11 showsspectrum that as as an thickness asterisk. Figure increases 11 shows (moving that towards as thickness red X),increases failure (moving load increases towards and red livingX), conveniencefailure load decreases increases whileand living age isconvenience held constant. decreases while age is held constant.

FigureFigure 11. 11.Living Living convenience convenience increasesincreases withwith age. Contour Contour levels levels represent represent the the fail failureure load load (lbs). (lbs).

FigureFigure 11 11 shows shows that that a a mollusk’s mollusk’s livingliving convenienceconvenience increases increases as as it itgrows grows—a—a shell shell with with more more whorlswhorls is moreis more efﬁcient efficient at at increasing increasing its its living living space space while while keeping keeping massmass low.low. TheThe coloredcolored contourscontours in Figurein Figure 11 illustrate 11 illustrate the the magnitude magnitude of of the the load load that that causes causes failurefailure ( (aa darker darker shade shade indicates indicates a smaller a smaller loadload magnitude). magnitude). It It should should be be noted noted thatthat thethe contour lines lines follow follow the the load load magnitudes, magnitudes, not not the the percent percent changechange in in thickness. thickness. The The shell shell is is able able toto withstandwithstand larger loads loads as as it it gets gets older, older, which which is issimply simply a result a result ofof the the boundary boundary and and loading loading conditions conditions usedused in this model. As As expected, expected, by by theoretically theoretically increasing increasing wallwall thickness thickness at at any any age, age, the the shell shell decreasesdecreases itsits chances for for failure, failure, but but marginally marginally loses loses some some living living convenienceconvenience by by doing doing so. so. FigureFigure 12 12 shows shows that that as as a a shell shell addsadds moremore whorls, the the load load at at which which three three-point-point bending bending failure failure occursoccurs generally generally increases. increases. In In otherother words,words, older shells are are more more resistant resistant to to breaking breaking than than younger younger shells are, for the given boundary conditions. In Figure 12, the contours illustrate the living shells are, for the given boundary conditions. In Figure 12, the contours illustrate the living convenience; convenience; the lighter colors indicate higher living convenience (i.e., high volumetric capacity to the lighter colors indicate higher living convenience (i.e., high volumetric capacity to low mass). low mass). As expected, a decrease in thickness would both allow for more volumetric capacity and As expected, a decrease in thickness would both allow for more volumetric capacity and lessen total lessen total mass (thus, increased living convenience), but would also cause failure to occur more mass (thus, increased living convenience), but would also cause failure to occur more easily (at a smaller load magnitude). Interestingly, the failure load increases more steeply in youth and appears to stabilize after around 13–15 whorls, suggesting that immature shells are more intent on increasing their load capacity, and that there may be an upper limit to load capacity. Designs 2018, 2, x 11 of 17 Designs 2018, 2, x 11 of 17 easily (at a smaller load magnitude). Interestingly, the failure load increases more steeply in youth andeasily appears (at a smaller to stabilize load aftermagnitude). around 13Interestingly,–15 whorls, thesuggesting failure load that increasesimmature more shells steeply are more in youthintent Designs 2018, 2, 46 11 of 17 onand increasing appears to their stabilize load capacity,after around and 13that–15 there whorls, may suggesting be an upper that limit immature to load shellscapacity. are more intent on increasing their load capacity, and that there may be an upper limit to load capacity.

Figure 12.12. Shells are lessless likelylikely toto breakbreak withwith maturity.maturity. ContourContour levelslevels representrepresent livingliving convenienceconvenience (in(inFigure33//oz).oz). 12. Shells are less likely to break with maturity. Contour levels represent living convenience (in3/oz). Figure 13 shows thethe relationship ofof loadload andand livingliving convenienceconvenience asas ageage changes.changes. ItIt is to be noted that the theFigure contours contours 13 shows in in this thisthe plot relationship plot line line up up with of with load the the integerand integer living number numberconvenience of whorls. of whorls. as ageIt appears changes. It appears that It aroundis that to be around noted13–15 13–15whorls,that the whorls, contoursthe rate the of in rate change this of plot change drops line dropsup off with dramatically. off the dramatically. integer In number other In other word of whorls. words,s, after It after13 appears–15 13–15 whorls, that whorls, around the shell the 13 shell can–15 canincreasewhorls, increase theits livingrate its livingof convenience change convenience drops but off can but dramatically. only can gain only a gainminimal In other a minimal increase words, increase inafter load 13 capacity in–15 load whorls, capacityfrom the its shellyounger from can its youngerself.increase This its self.is living the This same convenience is trend the same as butseen trend can previously: only as seen gain previously: athe minimal young increase theshell young may in loadbe shell more capacity may intent be from more on itspreventing intent younger on preventingfracture,self. This while is fracture, the a same larger while trend shell a larger asspends seen shell previously:energy spends on energy increasing the young on increasing its shell living may itsconvenience livingbe more convenience intent (suggesting on (suggestingpreventing that the thatshellfracture, themay shellwhile approach may a larger approach a limit shell of spends amaximum limit energy of maximum strength). on increasing strength).As demonstrated its living As demonstrated convenience by the red bysymbols,(suggesting the red a symbols,decrease that the ainshell decrease thickness may approach in corresponds thickness a limit corresponds to ofa lowermaximum load to a capacity lowerstrength). load but As capacity an demonstrated increase but in an volume increaseby the tricred in capacity volumetricsymbols, to a mass. decrease capacity toin mass.thickness corresponds to a lower load capacity but an increase in volumetric capacity to mass.

Figure 13. Shells fracture more easily with fewer whorls. Contour levels represent age. Load-to-living Figure 13. Shells fracture more easily with fewer whorls. Contour levels represent age. Load-to-living convenienceFigure 13. Shells rate fractureappears more to stabilize easily withafter fewer13–15 whorls. Contour levels represent age. Load-to-living convenience rate appearsappears toto stabilizestabilize afterafter 13–1513–15 whorls.whorls.

Designs 2018, 2, 46 12 of 17

4. Discussion This project examines a simpliﬁcation of the T. terebra’s structural adaptability to growth over time. While direct parallels drawn to human structures are certainly fantastical (i.e., by no means do we suggest that buildings should be shaped like spiraled shells), the purpose of the comparisons made below is to demonstrate that connections can be found in Nature for engineering solutions. This study also provides a basic example of quantitative biomimicry through the use of structural modeling outside of its usual context.

4.1. Analysis of Shell Adaptability As shown in Figure 11, the mollusk may become more efficient at increasing living space while minimizing mass as it grows, but a simple thought experiment suggests that humans are not as efficient with their usable space. A common measurement of usable living space in real estate is the floor area ratio, which the total floor area of a building (usable space across multiple stories) as a percent of its footprint. While the mollusk is able to capitalize on its internal capacity (volume approximated as a sphere), only floor area is usable for humans. In real estate, the floor area ratio does not increase as ceilings are raised. Take a skyscraper for which every floor adds area A and mass m: its living space-to-mass ratio is essentially constant (A/m), unlike the shell. If we consider volume, instead—in a direct comparison to the shell—we would find a similar increase in the living space-to-mass ratio as the height (number of floors) increases. However, as much of our vertical space is unused, this is best characterized as the space that requires heating or cooling [34]. In contrast to the shell, this is an increasing relationship that we would like to avoid, as it necessitates an increase in energy costs. Figure 12 shows that older shells (more whorls) are able to resist larger loads than younger shells (fewer whorls); however, this is the opposite of what we observe in human structures. The geometries and structures of shells versus buildings are obviously very different, but the fact remains that our older structures are usually in a worse condition than newer ones. This is often due to poor upkeep or not meeting current standards, but ultimately structural health does not tend to improve with age. With regards to the shell’s length as a factor in resisting lateral loads, the simple analogy of building height can be made. Lateral loads on buildings (like wind) affect the behavior of tall skyscrapers more than shorter structures. The effect of changing wall thickness translates easily to buildings. Engineers could build thinner walls, but they would likely compromise structural integrity in the process. Similarly, walls that are overly thick interfere with internal occupant capacity. In Figure 13, we see that the change in slope of load capacity-to-living convenience alters dramatically after a certain number of whorls. This may suggest that the shell undergoes a transformation after which it spends its energy on different development activities. At a younger age, preventing fracture may be its focus, while after maturing and approaching a strength limit, the shell instead focuses on increasing living convenience. Based on commentary by Etter et al. [35], the importance of size on survival may be separate from that of load capacity on survival, since we see that the shell does not stop growing as it approaches maximum load capacity. According to the Encyclopedia of Tidepools and Rocky Shores:

A larger size and thicker shell reduce the efficacy of shell-crushing predators such as crabs, fishes, and birds. For example, when crabs are provided with both thick- and thin-shelled prey, the thin-shelled morphs suffer much greater mortality because they are easier to break. Size is important because as snails increase in size, fewer predators can feed on them and they may often attain a size refuge, where they are sufficiently large that most predators are unable to consume them. [35]

This change in the need for protection may explain the continuance of growth even after maximum load capacity is approached; increasing overall size may be more important than fracture strength after a certain age. A similar change in purpose can be considered for buildings—does vulnerability Designs 2018, 2, 46 13 of 17 to certain hazards change with size, age, or other properties? If so, can we design for a mid-life transformation in performance demands?

4.2. Implications of Adaptability in Structural Design The takeaway from this shell case study is not that we should design our buildings to look like shells. In order to utilize bio-inspiration as a practical tool, as designers we need to look past surface level aesthetics and examine the functions of the design. The spiraled turret shell was chosen as a simpliﬁed analogy to human structures, which also protect and house their inhabitants. As the shell grows, its adaptability for accommodating its inhabitant is crucial for its survival. The shell’s adaptability is the critical function examined in this paper—what can we learn from biomimetic adaptability to help our obsolescence-prone infrastructure? As described in the introduction, while structural engineers are well-trained in considering structural integrity, designers do not often consider future use and structural adaptability to be a design parameter. This shell study provides some insight to structural designers into what we can consider during the design process for more sustainable infrastructure. Here, we present two ideas that reinforce the fact that planning must happen, even if only planning for the unknown, before construction is set in motion.

4.2.1. Planning Is Crucial for Adaptable Design Shelled gastropods are born with a protoconch, which is the initial part of the shell that develops as a coil in the embryo. This prenatal spiral sets the pattern for the shell’s growth to follow for the rest of its existence. This pattern is simple (defined by just a few parameters, as described in Section 2.1), yet guided, leading to structured but unrestrained growth. This planning in advance to support future growth is not a concept we often consider in human structures. As designers, we need to think critically about how we can design structures to allow for the expansion that they will inevitably need, effectively designing the “prenatal” structure that supports future growth. For example, increasing occupancy numbers while maintaining access to natural light for occupants presents an interesting design challenge. If, in the future, we begin to build ad hoc vertical expansion out of necessity, how might we redesign structural components to be easily modified when an unknown number of floors could be added? These are questions that structural designers need to begin considering in the planning stages in order to set a path for growth, when working with what exists is a constraint, and demolition is no longer an option. While there are on-going studies of adaptable design in human structures, practical implementation is currently rare. Ross et al. conducted a study to examine if there are particular design strategies that professionals perceive to enable adaptability more than others [36]. Eleven strategies (see Figure 14) were identified in the small but growing body of literature calling for an adaptable design paradigm. Fifteen design professionals were asked to rank, in their opinion, the importance of “adaptability enablers” such as the accuracy of drawings, open floor plans, and modular components. These “adaptability enablers”—factors that enable a structure to be adaptable to its occupants’ needs—provide insight into what the construction industry can focus on to prolong the lifetimes of structures. One of the eleven parameters described in Ross’ work is designing for adaptability and deconstruction (DfAD or DfD), a construction approach that emphasizes the importance of the design stage in considering the life cycle of structural components (rather than just their usefulness) [37]. Ross posits that, even though DfD sounds like a promising technique, it may have received a low perceived effectiveness rating due to two factors. First, many of the other enablers overlap with design decisions made in DfD; and second, while some practicing engineers may consider the after-life implications of their designs, they may be unaware of DfD as a design framework, as it is not commonly applied in the United States [36]. Both awareness and implementation of adaptable design are needed in the U.S., Designs 2018, 2, 46 14 of 17

and turning to Nature to learn how to design adaptable structures may provide engineers with tools Designsto better 2018, prevent2, x obsolescence. 14 of 17

FigureFigure 14. 14. RelativeRelative effectiveness effectiveness of ofdesign design-based-based adaptability adaptability enablers enablers.. Data Data from from [36 [36], Elsevier,], Elsevier, 2016 2016..

4.2.2. Planning Ahead Can Increase Efficiency One of the eleven parameters described in Ross’ work is designing for adaptability and deconstructionCities may(DfAD even or DfD), be aa construction better analogy approach for the that spiraled emphasizes gastropod the importance shell than of the buildings. design stageWhile in buildingsconsidering do notthe usuallylife cycle increase of structural their efficiency components with (rather scale, cities than do. just For their example, usefulness) New York[37]. is Rossthe posits largest that, and, even yet, most though sustainable DfD sounds city inlike the a Unitedpromising States technique, [38]. We actuallyit may have see that received an increase a low in perceivedpopulation effectiveness density increases rating infrastructural due to two factors. efficiency, First, since many the ofphysical the other components enablers of overlap infrastructure with design(e.g., cables,decisions pipes, made subways) in DfD; haveand second, less absolute while areasome to practicing span. On theengineers other hand, may consider the urban the sprawl after- of lifeLos imp Angeles,lications caused of their by designs, poor (or they lack may of) infrastructural be unaware of planning, DfD as a hasdesign stimulated framework, wealth as inequalityit is not commonlybetween theapplied urban in center the United and suburbia States [36 at]. Both the fringe. awareness and implementation of adaptable design are neededThere in is thea long U.S., history, and turning from Mumford to Nature (1961) to learn [39] to how Batty to (2013)design [40 adaptable], of studying structures the form may of providehuman engineers settlements. with Recent tools to work better shows prevent that obsolescence. some fundamental equations can be used to describe human behavior and efficiencies in these cities [41]. For example, as cities grow, they require fewer 4.2.2.resources Planning per Ahead capita andCan produceIncrease more Efficienc innovationsy per capita. While such research derives equations fromCities the behavior may even of be cities, a better this research analogy hints for the at the spiraled potential gastropod to use equations shell than to buildings. inform the While design buildingsof cities. do As not seen usually in Figure increase 11, the their seashell efficiency grows with in scale, a way cities that isdo. efficient For example, for its occupantNew York and is the also largestbecomes and, more yet, most efficient sustainable as it grows. city Well-plannedin the United urbanStates areas[38]. We grow actually in an analogoussee that an manner—as increase in a populationcity’s population density grows, increases higher infrastructural density commercial efficiency, and residential since the structures physical tend components to be constructed, of infrastructureleading to more (e.g., living cables, space pipes, for subway less material,s) have like less the absolute shell achieves. area to span. Studying On the Nature’s other hand, adaptable the urbanstructures sprawl for of hidden Los Angeles, yet practical caused lessons by poor can (or inspire lack usof) toinfrastructural design and help planning, us to plan has forstimulated structures wealthand infrastructure inequality between that support the urban our center growing and population. suburbia at the fringe. There is a long history, from Mumford (1961) [39] to Batty (2013) [40], of studying the form of 5. Conclusions human settlements. Recent work shows that some fundamental equations can be used to describe humanThis behavior research and explores efficiencies biomimicry in these as cities a design [41]. approachFor example, for reimagining as cities grow, our they building require philosophy fewer resourceto disrupts per the capita current and obsolescence-demolitionproduce more innovations culture. per capita. By turningWhile such to Nature, research engineering derives equations systems fromand the quantitative behavior of analysis cities, techniquesthis research can hints lead at us the to potential an understanding to use equations of how complexto inform systems the design work, ofand cities. we As may seen be ablein Figure to discover 11, the fundamental seashell grows model in elementsa way that that is affectefficient adaptable for its occupant behavioral and response. also becomesIn more this paper, efficient the as structural it grows. adaptabilityWell-planned of urban the T. areas terebra growspiraled in an turretanalogous shell manner is investigated—as a city’sthrough popul finiteation element grows, modelinghigher density and parametriccommercial studies. and residential The shell structures is able to tend change to be its constructed, structure over leadingtime to to meet more changing living space performance for less demands—amaterial, like feat the ofshell adaptability achieves. that Studying could transformNature’s adaptable our current structuresinfrastructure for hidden design yet if practical implementable. lessons can Based inspire on the us resultsto design of theand parametric help us to plan studies, for structures two design andapplications infrastructure are provided.that support First, our we growing emphasize population. that the planning stage in the design process is crucial. Current implementation of adaptability in the structural design process is rare; moreover, parallel work 5. inConclusions civil engineering has found that adaptable design is misunderstood and underutilized effectively. Turning to Nature for biomimetic lessons in adaptability may help us better identify the critical design This research explores biomimicry as a design approach for reimagining our building parameters that lead to adaptability. Second, analogies are also drawn between the shell’s adaptability philosophy to disrupt the current obsolescence-demolition culture. By turning to Nature, engineering systems and quantitative analysis techniques can lead us to an understanding of how complex systems work, and we may be able to discover fundamental model elements that affect adaptable behavioral response.

Designs 2018, 2, 46 15 of 17 and city design. Our city planners can learn from the adaptable growth of the shell to more effectively manage population and infrastructural growth. By planning ahead strategically, our cities are capable of increasing infrastructural efficiency as boundaries and populations expand, much like the shell. While shell and human structures are incommensurable, especially given the assumptions, approximations, and simplicity of the shell model, the purpose of these analogies is to demonstrate that the challenges that we face as engineers may resemble those already resolved in Nature. While we may not have the step-by-step answers to how to design adaptable buildings yet, we show readers that adaptable structures are possible, as demonstrated by the shell. Recent work has begun the search for adaptability blueprints, but more effective solutions might be found through further biomimetic studies. We encourage readers to consider biomimicry as a source of inspiration for their own research and hope that this work will incite collaboration between disciplines, as natural forms do not conform to artificial disciplinary boundaries. Some of the largest opportunities for scientific advances lie at the convergence of multiple disciplines [42–44], and applying biomimicry to structural forms can place structural engineers at the cutting edge of these advances.

Author Contributions: Conceptualization, D.A.C.; Methodology, D.A.C.; Software, D.A.C.; Validation, D.A.C., Formal Analysis, D.A.C.; Investigation, D.A.C.; Resources, D.A.C. and B.E.R.; Data Curation, D.A.C.; Writing—Original Draft Preparation, D.A.C.; Writing—Review and Editing, D.A.C., B.E.R., and L.E.K.; Visualization, D.A.C.; Supervision, B.E.R. and L.E.K.; Project Administration, DA.C.; Funding Acquisition, B.E.R. and L.E.K. Funding: This research has been performed in part using funding received from the Department of Education Graduate Assistance in Areas of National Need program (Award number P200A12022). Acknowledgments: The authors thank Hamed Rajabi and his research team for their magnanimous support of this PhD project through their generous contribution of MATLAB code for shell geometry generation. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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