Bubble frame: assessment of a new structural typology starting from the Water Cube

Maurizio Toreno 1; Massimiliano Fraldi 2, Aldo Giordano3 Elena Mele 4; Antonello De Luca 5

Abstract

In this paper a discussion on a new steel structural typology of strong aesthetic impact and efficient use of

structural material is presented. Starting from the Beijing National Aquatics Center “Water Cube”, the authors

focus to the peculiarities of bubble frame structures and to the potentials of their application in large span roofs.

The structure of the Water Cube has a complex, confused and casual appearance; on the contrary, it is obtained

through a few simple groups of many identical elements, that are assembled and organized on the basis of a

rigorous geometry, which, in turn, is derived from a physical theory, the Wheire-Phelan foam theory.

The Wheire-Phelan foam is mathematically formulated and completely repetitive. Starting from this observation, a

new structural type can be defined, i.e. the bubble frame structure, and its application can be proposed by

adopting the same generation process based on the mathematical geometrical model.

However the apparently random configuration of the frame members and nodes of the bubble structure is

prohibitive to be assessed through the traditional approach of structural engineering, since it cannot identified any

hierarchies and orders in the structure elements: in the water cube there is no differentiation between principal

and secondary elements, horizontal roof elements and vertical wall elements. Furthermore, since the directions of

the frame members in the space are highly variable, it is practically impossible identifying a clear load path.

Therefore the need for simplified modelling tools to be used in a preliminary design phase do exist.

In this paper a homogenisation criterion for porous elastic solids, firstly elaborated for describing the mechanical

behaviour of human bones, is used for defining the equivalent mechanical properties of bubble frame solids.

The equivalence between the continuous model and the original bubble frame is based on the parameter volume

fraction of a representative volume element (RVE). A sensitivity analysis is presented by varying dimensions and

position of the RVE in the bubble frame structure, as well as by varying the base unit (the bubble size) used in the

generation process of the bubble frame. Some design guide lines and simplified tools are finally suggested. Keywords Bubble truss, steel, water cube, foam structures, large roofing. Theme Buildings – construction – non conventional architecture.

1 Ph.D.Student, Department of Structural Engineering. University of Naples/Italy maurizio.toreno@unina.it 2 Assistant Professor, Department of Structural Engineering. University of Naples/Italy fraldi@unina.it 3 Ph.D. Eng, Department of Structural Engineering. University of Naples/Italy algiorda@unina.it 4 Professor, Department of Structural Engineering. University of Naples/Italy elenmele@unina.it 5 Professor, Department of Structural Engineering. University of Naples/Italy adeluca@unina.it

1 Introduction

Motivated by a growing interest of material science and engineering in polymers and biological tissues,

micromechanics and physical properties of heterogeneous media have been intensely studied in the last decades

[1] [2]

Nature offers a large number of examples of heterogeneous materials: if observed at different scale

levels many biological tissues exhibit indeed a hierarchical microstructure whose architectural elements are

constituted by collagen fibres, contractile elements, filaments and cell units, differently arranged to form structural

networks aimed to guarantee optimal mechanical performances with respect to specific functions. On the other

hand, synthetic materials such as composites, polymers, foams and porous media have been recently designed

and developed for applications in civil, mechanical and aerospace engineering, and special attention of the

Industry has been also registered for the manufacturing processes, design strategies and optimization techniques

employed to create high-performance man-made products. In this framework how macroscopic effective physical

properties of heterogeneous materials can be derived from their microstructure becomes decisive.

From a mechanical standpoint two main continuum theory-based approaches may be adopted for

analyzing complex microstructures: statistical and deterministic. Both engage averaging techniques and require

the choice of the Representative Volume Element (RVE), that is the smallest statistically homogeneous material

volume to which macroscopic constitutive relationships must be referred [2]

The statistical approach correlates the microstructure of the materials to their overall physical properties

by introducing the statistical n-points probability function, that provides a mathematical representation of

distribution and morphology of phases in heterogeneous materials. It is possible to show that the lowest order

(one-point) probability function represents the volume fraction of the constituents, higher order probability

functions being instead devoted to capture morphology features. Although a statistical approach may be

successfully used for predicting physical properties of heterogeneous materials as well as in material design and

processing optimization problems, the construction of reliable probability functions strongly depends on the

possibility of exploring the material microstructure in detail. This entails to process a large amount of information,

sometimes leading to overwhelming computational costs and intolerable calculation times.

The deterministic approach is instead based on a continuum theory where the micro-structural properties

are incorporated in prescribed architectural variables, whose choice is generally suggested by analytical results or

based on engineering assumptions that invoke phenomenological or heuristic arguments. Furthermore, in order to

obey to some thermodynamic restrictions, these variables are usually introduced in the form of scalars or even-

order tensors. Obviously, physically consistent and mathematically well-posed models constitute the premise for a

good engineering analysis, but do not exhaust the subject of their practical applicability to real problems. In this

sense, difficulties in the actual measurability of tensor variables representing the RVE architecture, unshared

definitions of some morphological features and ambiguous geometrical interpretations of the RVE microstructure

contribute to limit the applications of the deterministic approach.

Among heterogeneous materials great attention is reserved in Literature to depleted media that well

interpret many different biological and man-made material behaviors.

Depleted media are materials whose overall stiffness and strength are reduced as a consequence of the

presence of cavities in the solid matrix. Nevertheless advantages can be traced in possible high stiffness/weight

ratios due to optimal distribution of voids. In general, two limit structural configurations can be identified in

depleted media. The first one is characterized by media exhibiting High Volume Fraction (HVF), for example solid

matrices containing n dilute voids. This class is often met in studying porous materials, damage or elastic solids

weakened by non interacting penny-shaped micro-cracks. Complementarily, the second limit configuration is

represented by materials possessing Low Volume Fraction (LVF), where a structural ordered or randomly-

arranged network of thin elements constituted by beams and/or plates can be generally observed at the micro-

scale level. Typical examples of this situation is offered by trabecular bone tissues, honeycomb materials or truss

structures. In particular, spatially organized truss structures can be also interpreted as overall macroscopically

arranged depleted media and the micromechanical approaches can be thus employed for analyzing and

designing structures in civil engineering.

Overall mechanical properties of depleted media depend on the elasticity and strength of the solid matrix

and upon some selected architectural variables inferred from the RVE micro-structural features. In the perspective

of rational thermodynamics, by making use of the theory of homogenization or by invoking micromechanical

approaches that exploit heuristic and phenomenological arguments, suitable architectural variables are

established in the form of scalars or even-order tensors, in order to guarantee mechanical consistency.

A physical-geometrical interpretation of the architecture variables related to the mechanical properties of

both HVF and LVF depleted media allows to classify them by means of three main RVE microstructure features,

namely topology, density, and orientation.

RVE topology refers about the micro-structural architecture of the material trough two topological

properties: connection and integrity. Connection reveals the “skeleton” of the RVE solid matrix and then

constitutes a measure of the amount of internal links which virtually form a dormant network, directly responsible

of the effective stiffness as well as of the strength of the material. A scalar measure of connectivity can be

represented by topological indexes of connectivity, such as the so-called Euler Characteristic, .

A more complex tensor measures may be however obtained by averaging over the number of solid paths

that interconnect any pair of arbitrary points belonging to the RVE matrix. Depleted media with RVEs

characterized by solid skeletons in the form of three-dimensional truss structures constitute examples of micro-

and macro-structures where the connection, say the number and the arrangement of bars, plays a crucial role in

the overall stiffness of a selected cell. Integrity is the other important topological property of a depleted material. It

refers the structural integrity of the solid network and then ascribes to the presence of interfaces (e.g., possible

broken skeleton sections) the responsibility of a reducing in the overall RVE stiffness or strength. Truss structures

where a geometrical interface is generated by interrupting the physical continuity between some links or puzzle-

like materials where the presence of discontinuities at the interface between the tesserae significantly modifies the

mechanical response of the ensemble, are all examples in which the RVE stiffness is reduced for a lack of

integrity.

The main stereological measure of a depleted media is constituted by the well-known (apparent) density,

whose architectural variable is usually identified with a dimensionless parameter named volume fraction,

]0,1] .

Finally, eventual RVE mechanical anisotropies are related to the orientation of the microstructure, whose

conventional variable is usually identified with second order so-called Fabric Tensors. However, how to detect and

measure the microstructure orientation of a depleted RVE is still a controversial matter. This depends on both

theoretical and practical reasons. In fact, an explicit expression of the Fabric parameters is often not available as a

consequence of the absence of a widely shared and univocal interpretation of the meaning of the microstructure

orientation. Moreover, in its full formulation, Fabric would request too expensive measures investigations for the

number of parameters that for each RVE should be identified. It must be also considered that, depending on the

material nature to be explored, the cavities which weaken depleted media can be found arranged inside the

material domain so that the sole weapons of the direct observation result often inappropriate and therefore the

Fabric should match the data furnished by the specific above mentioned techniques.

Connection and integrity – however – can be neglected if topology features are preliminarily specified.

This choice produces the possibility of assuming that, once volume fraction and orientation of the RVE are fixed,

changes in topology which do not modify volume fraction and orientation result in unchanged overall mechanical

properties and in elasticity in particular. Nevertheless, truss- like structures or foam-like topologies, contrarily to

intuitive arguments, generally exhibit a connection–independent behavior in terms of homogenized elastic

properties.

The attention of the work is here focused on depleted media where volume fraction and orientation are

the sole microstructure features responsible of the overall RVE mechanical (elastic) response, thus avoiding to

explore the influence of topological properties on stiffness and strength of the material. In particular, with reference

to spatially distributed foam-like macrostructures characterized by interconnected beam elements, the present

work is aimed to examine the possibility of utilizing classical micromechanical-based strategies for the design and

the assessment of civil structures generated by complex polyhedral truss-like assemblies. With reference to a

recently proposed explicit formulation in which volume fraction of the solid matrix and a Fabric tensor constitute

the sole architecture variables to be employed for determining the elastic properties of a porous media, a

simplified model is derived to describe the mechanical behavior of bubble-like shells, finally showing an example

application related to the Beijing National Aquatics Center “Water Cube”.

2 Homogenized elastic model for bubble-like structures: general formulation

With reference to an arbitrary scale level, a porous material constituted by a solid skeleton (or matrix)

and voids imprisoning a mechanically non- collaborating fluid phase can be regarded as a depleted media and its

overall elastic response described through a Fabric Tensor approach in the following linear form [3]

0 0 0 0, ( 2 ) 2

3 3 3

2k k k k k

i i i i j i j i j i j

i=1 i, j=1 i, j=1i j i j

m m m m m M M M M M M MC

(1)

where 0 and 0 are the Lamè moduli of the solid matrix, is the skeleton volum fraction, M represents the

second order Fabric Tensor responsible of the anisotropy of the RVE, i.e.

3 3

i i i i i

i=1 i=1

= m = m M M m m (2)

k being a suitable positive real penalization power and , MC is the homogenized elasticity tensor in the

case of linear stress-strain relationship.

An explicit expression of M has been recently furnished by assuming that the Fabric tensor is writeable

as a linear function of the geometrical inertia of the elastic matrix. However, if the three-dimensional arrangement

of the constitutive solid elements can be approximated as stereological within the RVE, the anisotropy can be

neglected and the sole variable responsible to the heterogeneity of the material becomes the volume fraction, .

Nevertheless, it remains to select the mathematical form of the overall homogenized elastic moduli C .

To make this, Perrella et al.[4] proposed an unified formulation based on the choice of a third-order

polynomial law with respect to the volume fraction, in which the coefficients explicitly depend on the Young

modulus and the Poisson ratios of the elastic matrix, assumed isotropic. In order to find this analytical expression

the Authors invoked the classical homogenization theory and utilized two special limit cases – the Flugge solution

for low volume fractions and the dilute voids solution for low porosities – so obtaining the following relationship:

2 3

1 2 3 , 0,1i i i ik k k k (3)

where 0 0( )ij i ijk k k – no summation – are coefficients depending on the matrix Poisson ratio 0 and 0ik

and ik represent the homogenized and the corresponding matrix elastic moduli, respectively, that is the bulk,

the first Lamè (shear) and the Young moduli. In particular, for the case of interest the homogenized Young

modulus can be written as

2 3

0 01 02 03E E E E E (4)

whose dimensionless parameters 0 0( )jE are

0

01 2

0 0

2 3 4

0 0 0 002 2 3

0 0 0

2 3 4

0 0 0 003 2 3

0 0 0

2 1 2

3 2

22 17 59 9 45

6 14 17 2 5

50 17 91 21 45

6 14 17 2 5

E

E

E

(5)

3 Case study: the Beijing National Aquatics Center

The approach outlined in the previous paragraphs as been applied to an “iconic” project realized for the last

Olympic games in China. Unveiled in 2008, the “Watercube” (as it is commonly referred to) outstands among the

most admired structures ever.

Figure 1– the “Watercube”

3.1 Geometry and discrete analysis model

The building derives its conceptual inspiration from the foam aggregation, as illustrated in the Weaire-Phelan

theory of the bubble clusters. Such clusters are made up of a repetitive core constituted by eight polyhedral

bubbles, namely 2 dodecaedrons and 6 tetracaidecaedrons. The polyhedrons feature respectively pentagonal

faces (dodecaedreons) and a mix of 2 hexagonal and 12 pentagonal faces.

Figure 2 – Geometrical construction of the elementary core

In order to make the necessary comparisons and to assess the applicability of the homogenized approach, we

have first built up the wireframe model of the structure, following the bibliographic data available and taking

advantage of the information provided directly by the structural designers (Arup). Once the vertices have been

defined through the W-P theory, it is relatively simple to realize the solid polyhedrons and to aggregate them in

the elementary grapes, then replicated indefinitely to make a prism. Such prism is first rotated by 60° about one

axys (which gives the structure an apparent randomness) and then cut by suitable planes. The deriving

geometry is then internally hollowed to suite architectonical/functional needs. Finally, only the wireframe is

retained to constitute the real structure, which is exported into a FE program such as SAP2000 (the original

made in OASYS GSA and STRAND7) for analysis and checks.

Figure 3 – From the aggregated prism to the finite element model

The structural members dimensions have been chosen to reproduce the real structure, grouped in three

categories, according to different location within the scheme. In particular, tubular and box sections have been

used, following what available in the relevant description provided by the structural designers of ARUP, as well

as the steel material properties.

3.2 Homogenised model

In order to apply the equivalent material approach, we first have to determine the reference volume element

(RVE). This choice is made by selecting a suitable portion of the structure that can be referred as repetitive in a

statistical sense among the overall structural distribution, as schematically shown in figure 4.

Then, the equivalent material properties are derived through the equations defined in the previous paragraphs,

once the volume fractions have been evaluated.

Figure 4 – Selection of the RVE

With the aim of investigating the sensibility of the results to the variation of the RVE, four different alternatives

have been taken into account, changing its position and dimensions. In particular, in the first choice, labeled in

the following “B”, the RVE is a cube with 7.2 dimension, equal to the roof external thickness and it is positioned

at mid-length of the major span. In the second case, “B2”, the RVE has the same dimension as the previous but

different position since its is translated by 7.2 m. Finally, “B3” is an average RVE with increased overall

dimensions (42.67 x 42.67x7.2m)

3.3 Analyses results and comment

Since the aim of the research is to investigate the applicability of the new approach, only two loading conditions

have been taken into account, with vertical dead and live loads acting.

In figure 5 the deformed shapes of the “real” and homogenized structures are shown. Just as a comparison, on

a normal Core I3 based machine with 4GB RAM, the first model, with features more than 22.000 beams runs in

about 12 minutes to make vibration modes extraction and to solve 10 loads combinations. The same task is

performed for the homogenized model almost in real time.

Figure 5 – Deformed shapes

The following graphs contain the comparison among the “real” and homogenized models in terms of vertical

displacement at mid-length of the main roof span. The label G refers to self weight, while G+Q to the self-weight

+ live loads. It is evidenced how the differences among the models are particularly small, and in any case within

a few percent points. For this reason, it is opinion of the authors that a simple analysis using the homogenized

model might be used for a quick dimensioning of such a structure, basing on deformability limits that can be

derived from code provisions.

0

100

200

300

400

500

600

A B B2 B3

Comparison among vertical displacements

G

G+Q

[mm]

[Model]

Figure 6 – Comparison of the vertical displacement in the different models

Then more refined analyses have to be performed since local checks must be in any case carried out. Of

course, at this stage of development such verifications cannot be done since we do not retain sufficient

information about the organization of the discrete structure. Nevertheless, the authors are presently working on

a dual technique that allows for re-localising after global analyses have been performed.

4 Conclusive remarks

The proposed approach allows for very fast assessment of non-conventional structures made up of thousands

of members, regardless of the complexity. In particular, very good results are obtained with respect to the

deformability, since the scatters among the “real” and “homogenized” models are very low and in any case

within acceptable values for engineering purposes. Certainly, though much work has been done in deriving the

mathematical formulation of the equivalent material, a long way ahead is still to be explored. This is mainly due

to the need of re-localising the effect for local checks. Nevertheless some unique features seem to be very

promising. In particular, since the main aspects that govern the model regard the equivalent density, the

connection degree and the orientation, issues such topological optimization may be treated with relatively small

effort and, above all, retaining a conceptual consistency and capacity of control that is precluded to other

analysis methods.

References

[1] Fraldi M., Cowin, S.C., Inhomogeneous elastostatic problem solutions constructed from stress-associated

homogeneous solutions, Journal of the Mechanics and Physics of Solids, 52, 2207-2233, 2004.

[2] Nemat-Nasser, S. and Hori, M. Micromechanics: overall properties of heterogeneous materials. North-

Holland, Amsterdam, The Netherlands, 1999.

[3] Zysset, P.K., Curnier, A. An alternative model for anisotropic elasticity based on fabric tensors, Mech. of

Materials, 21: 243-250, 1995.

[4] Perrella, G., Esposito, L., Cutolo, A. and Fraldi, M. A Proposal of Unified Elastic Moduli-Matrix Volume

Fraction Law for Porous RVEs, Proceedings of AIMETA Conference, 2009.