MODULAR DESIGN-CONCEPT, USING HYPAR-SHELLS MADE OF FIBRE REINFORCED COMPOSITES

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1 1st International Conference Textile Reinforced Concrete (ICTRC) 389 MODULAR DESIGN-CONCEPT, USING HYPAR-SHELLS MADE OF FIBRE REINFORCED COMPOSITES E. De Bolster, H. Cuypers, J. Wastiels and P. De Wilde, Department of Mechanics of Materials and Constructions, Vrije Universiteit Brussel, Belgium ABSTRACT: In contemporary architecture, one can notice the tendency towards modular lightweight structures. Another tendency that can be observed is the frequent use of freeforms. These designs are usually lightweight as well as aesthetically pleasing. The designs that combine the advantages of modular structures and freeform structures are usually covered with textiles. However, composites (TRC-composites for example) could present a good alternative; especially when used as faces in a sandwich construction. In this paper, a designconcept that uses sandwich panels (shaped as hyperbolic paraboloids) with reinforced Inorganic Phosphate Cement-faces as building stones of a modular structure is studied and applied in a cantilever design. 1 INTRODUCTION Composites can be manipulated where the material directions are concerned. When a freeform structure is designed, the material directions should be able to follow the directions of the principal stresses. Composites can do that without loosing their lightweight characteristic: arbitrary lay-ups can be manufactured fairly easily. Since materials are more efficiently used in a structural element that is mainly subjected to axial forces and not to bending moments, a sandwich construction can be advantageous. A sandwich construction places its main material further away from its neutral axis, so bending moments are better captured, but without a serious increase in weight. TRC-composites could be used as faces of a sandwich construction. They have the advantage that fire safety is not endangered, unlike most classical constructions. Since glass fibres can easily be introduced in the form of textiles in order to obtain high fibre volume fractions, are not too expensive and are rather reliable in their properties, they represent an attractive choice as fibre reinforcement for the TRC-composites. The economical and widely available E-glass fibres however, are attacked by the alkaline environment of cements. To overcome this shortcoming of E-glass fibres as reinforcement in an alkaline matrix, one is bound to modify the cementitious matrix in order to improve the resistance of the embedded fibres. An example of such a low alkaline matrix is Inorganic Phosphate Cement. IPC has been developed at the Vrije Universiteit Brussel (VUB) and is a two-component mixture (a mixture of a calcium silicate mineral in powder form and a phosphoric acid based aqueous solution of inorganic metal oxides) that represents a non-alkaline environment after hardening. Cement matrix composites under compressive loading show a linear behaviour in the stressstrain curve. On the other hand, the stress-strain behaviour of these cement matrix composites under tension loading is highly non-linear and usually defined by the ACK-theory ([Ave71)] or the stochastic cracking theory ([Cur99] and [Cuy02]), which is derived from it. Looking at a typical experimental curve of a random glass fibre reinforced IPC-composite (Fig. 1.1.),

2 390 DEBOLSTER ET AL: Modular design-concept, using hypar-shells made of fibre reinforced composites however, it is seen that the nonlinear curve could - on a macroscale - be approximated by a bilinear curve with a bifurcation point at a stress of 7MPa. The composite in the first stage of the curve is assumed to be crack-free and therefore behaves linear elastic. The composite in the second stage of the curve is assumed to be fully cracked, resulting in a rather significant reduction of stiffness. Previous study has shown that fibre reinforced IPC loaded up to the fully cracked stage experiences a serious stiffness reduction due to cyclic loading and can even lead to fatigue failure ([Cuy02]). When initially designing a structure, it is Stress (MPa) spanning (MPa) MPa 5 Fig Stage I, E c1 Transition stress Stage II, E c rek (%) 0.042% Strain (%) Typical stress-strain-curve of 2D random E-glass fibre reinforced IPC with a fibre volume fraction of about 12%, approximated by a bilinear curve. therefore advisable to limit the tensile stress-strain behaviour of the composite faces to the bifurcation point shown in Fig In what follows, sandwich panels with a core of polyurethane (PUR) and 2D-random E-glass fibre reinforced IPC-faces will be used in designing cantilever constructions. Relevant material properties are given in Table 1.1. The minimum thickness of a random reinforced IPC-laminate is 1mm, while the thickness of the PUR-core will be considered to change in steps of 10mm. Table 1.1. Relevant material properties Reinforced IPC Tensile strength (MPa) Compressive strength (MPa) Shear strength (MPa) E-modulus (GPa) G-modulus (GPa) Density (kg/m³) (*) PUR (*) (*) (*) irrelevant for the performed calculations 2 DESIGN-CONCEPT 2.1 Design-concept When designing a modular system, two aspects need to be considered: the aspect of one building stone and the aspect of the overall structure. Since a modular system is only as strong as its components (building stones and the connection between them), it is important to choose them right. A hyperbolic paraboloid (also called: hypar) building stone (see Fig. 2.1.) has many advantages over other types of building stones. Firstly, a hypar roof can be applied

3 1st International Conference Textile Reinforced Concrete (ICTRC) 391 over any foundation shape (rectangular, triangular, circular, ). Secondly, a hypar surface is a curved surface and has therefore better bearing capacities than flat surfaces, because the forces can more easily be introduced in an efficient way (i.e. through higher normal forces and lower bending moments). Due to its anticlastic doubly curved shape, instabilities are less likely to occur: the convex curve will stiffen the behaviour of the concave curve and vice versa (see Fig. 2.1.). Even the fabrication of a hypar surface is not as difficult as one would expect from a three dimensional shape: a hypar surface is a doubly ruled surface Fig Hyperbolic paraboloid and can therefore be formed by two distinct independent families of straight lines (see Fig. 2.1.) which makes the formwork much easier. In the design-concept under study, a grid is used as an integration system (through hinged line connections) of the individual hypar surfaces in the overall roof -design. A two-dimensional grid defines the horizontal coordinates of the four corner points of each hypar surface. The vertical coordinate of the corner points depends on the formal design of the overall structure and is thus insofar independent of the grid. Once the position of four non-coplanar corner points is chosen, a hypar surface can be created. With this systematic method, a variety of shapes can be made modular in five steps: 1. Choose a mathematical function z=f(x,y), representing the formal aesthetics you - as designer - wish. 2. Choose/note the dimensions of the horizontal rectangular base that should be covered. 3. Divide this base in a number of elements (each element having the same dimensions in the x- and y-direction). 4. Calculate the x- and y-coordinates of each corner point (on the base). 5. Use these coordinates and the mathematical equation of the three-dimensional function to calculate the z-coordinate of each corner point. Fig Generating a modular system, based on a grid The two-dimensional horizontal grid not only defines the overall design, but also facilitates the prefabrication of each hypar surface. In Fig. 2.2., the choice has been made to work with a single regular grid to which each building stone is related. Since any mathematical function can be chosen as initial aesthetic design, a variety of freeform constructions can be created without the fear of losing oneself in an infinite amount of choices. Due to the grid and the specific shape of the basic building stone, one is better capable of handling the complexity of freeform design: a framework has been set that still leaves room for ones own creativity and originality.

4 392 DEBOLSTER ET AL: Modular design-concept, using hypar-shells made of fibre reinforced composites 2.2 Fabrication method Due to the anticlastic nature of the basic building stone in the modular design-concept, it is possible to create it with two sets of cables. Ordinary cable nets usually have fixed nodes and the cable-lengths need to be carefully calculated before assembly. After assembly - in a pretensioned state - only one hypar surface can be generated. If the fixed character of the nodes could be removed and if the length of the cables could be varied within certain limits, it would be possible to generate a larger range of hypar surfaces with just one cable net. The ADAPTENT -concept gives us the possibility to do just that ([Heb01]). The fabrication method used to generate the hypar sandwich panels with IPC-composite faces is largely based on the abovementioned ADAPTENT -concept, which is a generating system for temporary, adaptable and re-usable cable nets. Although the concept was created in order to generate tensile architecture, it could very well be used as a reconfigurable mould, resulting in an easy, fast, re-usable and economical methodology in fabricating the desired hypar sandwich panels. In Fig on the right, it is shown how a specific configuration of the adaptable cable net has been used to laminate a hypar sandwich panel with reinforced IPCfaces. Fig A pretensioned cable net used as mould to create a hyperbolic paraboloid-sandwich panel with E-glass fibre reinforced IPC-faces ([Heb01]) 2.3 Design attitude When designing according to European standards, two limit states need to be considered: the serviceability limit state (SLS) and the ultimate limit state (ULS). In the SLS, the limitation of deformations of the structure should be assured (amongst others) and in the ULS, the structure should not fail. In both limit states, certain design parameters should be checked. Although the other parameters cannot totally be omitted, the tensile stresses in the face in tension (ULS) and the maximal deflection of the structure (SLS) will in most cases - probably be the two designing parameters. In a previous study, it has been confirmed that the following design procedure, where only a few iterations are needed, leads to a lightweight design with the least amount of weight: designing in SLS and performing a consequent check-up in ULS ([Deb06]). A starting design with an arbitrary core-thickness and arbitrary face-thicknesses is chosen. The maximum deflection will then be controlled by dimensioning the core-thickness and the

5 1st International Conference Textile Reinforced Concrete (ICTRC) 393 strength will mainly be limited by manipulating the face-thicknesses. The design procedure itself starts at gradually changing the core-thickness of the whole structure until the maximum deflection in SLS stays below the limit value of L/250. Next, the tensile stresses in the faces are controlled in ULS. The thickness of the faces of the hypar sandwich panels are augmented when the tensile stresses in those faces exceed the stress of 7MPa. The change in thickness is therefore done per hypar surface: only the faces of the sandwich panels that are stressed beyond 7MPa will be re-dimensioned. Although the stress of the face in compression of these sandwich panels will not have exceeded the compressive strength of 50MPa, its thickness will also be increased in order to avoid the use of asymmetrical sandwich panels. 3 CASE-STUDY: CANTILEVER CONSTRUCTION 3.1 Cantilevers Cantilever structures are one of the more difficult structures to build: deflections are large and the remaining supports are highly solicited. However they do allow a base-area to be covered and protected against wind, snow and rain without the need of extra supporting structures that can interrupt the mainstream of people walking under it. Cantilever structures are therefore very popular for building entrances. Most cantilever constructions apply the principle of a counteracting load or a fixed support. In spatial cantilever structures, however, this is not a necessity. As can be seen in Fig. 3.1., stability can be acquired through form. When each beam-like strip of the plate (or shell) causes a rotation around a different axis, the strips tend to stabilize each other and the use of a fixed end connection is no longer stringent. COLLAPSES CAN BE STABLE Fig Stability through formdesign, rather than through fixed end supports. Another possibility to omit the fixed end connections is shown in Fig. 3.2.: the connection of two neighbouring edges to the outside world. This effect, however, becomes less and less noticeable when the strip is situated further away from the structural edges and is less efficient than the design method explained in Fig. 3.1.

6 394 DEBOLSTER ET AL: Modular design-concept, using hypar-shells made of fibre reinforced composites COLLAPSES CAN BE STABLE Fig Cantilever plate without the need of fixed end supports, due to two neighbouring line connections. The design-concept under study (including the hinged line connections) can thus be used to cover the area in front of building entrances. Especially when both, the design methods of Fig and Fig are implemented, lightweight and freeform cantilever constructions can be generated. Moreover, it is possible to assemble the whole structure without the need for extra resurrectionsupports because the structure can be made self-stabilizing through application of design methods, explained in Fig and Fig Case-study In what follows, a case-study with an arbitrary chosen aesthetical appearance will be studied. The form finding of the structure is done according to the rules of the design-concept explained in paragraph 2.1. and the design methods shown in Fig and Fig Surface design The proposed design covers a base of 12m² with 12 different hypar sandwich panels, numbered as can be seen in Fig (on the left). The Cartesian coordinate system is easily read from the same figure. The X-, Y- and Z-coordinates of the midplane of the hypar surfaces are tabulated in Table 3.1. The Z-coordinate is written relative to the lowest point of the structure. Each hypar surface contains 4 corner points. They will be listed as follows: lower left point (LLP), lower right point (LRP), upper right point (URP) and upper left point (ULP).

7 1st International Conference Textile Reinforced Concrete (ICTRC) 395 1m 1m 1m Y X m 1m 1m 1m 1m Fig Cantilever design for the entrance of a building. Left: view from above. Right: Three-dimensional impression Table 3.1. Hypar surface X Cantilever design for the entrance of a building. LLP LRP URP ULP Y Z X Y Z X Y Z X Y Z Dimensioning the surface The structure is situated in Belgium with a reference height below 10m and thus is subjected to own weight, snow load (uniformly distributed load of 0.4kN/m²) and wind load (pressure = ( )*633N/m² = 1.6kN/m²). The safety factors for the load are 1.35 and 1.5 for the dead load and mobile loads, respectively. The iterative design calculations are done numerically,

8 396 DEBOLSTER ET AL: Modular design-concept, using hypar-shells made of fibre reinforced composites with the help of Ansys. The finite element SHELL91 is used for which the sandwich option is included. Since cantilevers are expected to result in large deformations, the starting design already has a medium core-thickness (100mm, to be more specifical) and faces that consist of 2 random glass fibre reinforced IPC-lamina of 1mm each. In SLS, the structure needs to be dimensioned so that the maximum deflection does not exceed the limiting value of L/250 (in this case: w limit = 5m/250 = 20mm). The maximum deflection will appear around the lower right point of hypar surface 3 (see Fig. 3.3.) and is for the assumed starting design valued at 16mm, which is allowable. In ULS, however, the tensile stresses in half the hypar sandwich panels exceeds the limiting stress of 7MPa. Overall design: t face = 7mm; t core = 100mm t face = 3mm; t core = 100mm Maximum induced compressive stress (face) = 14.87MPa Maximum induced compressive stress (face) = 5.2MPa Maximum induced shear stress (core): S XZ = 0.09MPa << 0.25MPa w max = 12mm Tensile strength (face) (7MPa) Fig Static design-requirements for the proposed cantilever structure. Although it was mentioned that the faces would only be re-dimensioned locally, the high amount of hypar surfaces that would need thicker faces and the fact that the maximal occurring tensile stress is twice the limiting value, lead to the decision of re-dimensioning all the faces of each hypar surface at once. All hypar sandwich panels are given faces of 3mm, except for hypar surface number 5. Further iterative calculations have shown that this hypar surface needs thicker faces than the rest (namely: a total thickness of 7mm) in order to stay below the defined tensile stress-limit of 7MPa. With this final design the maximum deflection in SLS is once again calculated - for illustrative purposes only: it is reduced to 12mm. Fig shows that the design acquired through limitation of deflection in SLS and limitation of tensile stresses in ULS, also satisfies all the other design-requirements (compressive strength, wrinkling strength, shear strength, etc.). Some starting design values will lead to the same solution (cf. starting design 2/80/2 and 3/60/3 in Table 3.2.), but other starting design values could also lead to another final design, as can be seen in Table 3.2. Since the faces of the hypar surfaces are dimensioned

9 1st International Conference Textile Reinforced Concrete (ICTRC) 397 individually, different face thicknesses are possible within one overall design. Therefore, the face thickness of most of the hypar surfaces is mentioned in the third column of Table 3.2, while the exceptional facial dimensions are mentioned in the fourth column. Due to the relatively low ratio (density core)/(density faces), the least-weight design will be attained with the solution with the thickest core and thinnest faces. This however, does not necessarily represent the best final design. Other parameters do influence the choice of the best final design: aesthetics and cost-efficiency for example. Cost-efficient structures are structures with the least amount of material, the least amount of calculation-time and the least amount of rehabilitation- and maintenance-costs. In this casestudy, the amount of rehabilitation- and maintenance-costs are assumed to be equal for every design. The remaining parameters are thus the calculation-time and the amount of material required. It was established that, although the starting values of 2/100/2 (thickness upper face in mm/thickness core in mm/thickness lower face in mm) lead to the least-weight structure, more calculation-iterations need to be performed than in the case of starting values 4/60/4. Nevertheless, it is estimated that since the material costs here are far more important than the additional numerical iteration-cost - the starting design of 2/100/2 will lead to the most economical solution for the chosen surface design and chosen starting design values. Table 3.2. Different cantilever designs for an entrance of a building. STARTING DESIGN FINAL DESIGN t face /t core /t face t face (mm) t core (mm) in (mm/mm/mm) Most hypar surfaces Exceptions 2/80/ (hypar 11) 9 (hypar 5) 2/100/ (hypar 5) 3/60/ (hypar 11) 9 (hypar 5) 4/60/ (hypar 5) 4 CONCLUSIONS A sandwich construction with TRC-faces can be advantageous in efficiently capturing and transferring the applied load. Moreover, it creates lightweight structures for which a minimal amount of materials is required. Due to the non-alkaline environment of Inorganic Phosphate Cement, glass fibre reinforced IPC-faces can be made at relatively low cost. With this lightweight concept of sandwich panels with composite faces in mind, a design-concept has been generated for modular constructions in which hyperbolic paraboloid building stones (preferred for their intrinsic favourable mechanical behaviour due to their double curvature) are connected to each other and to the outside world through hinged line connections. Two designrequirements are important when considering sandwich panels with IPCcompositefaces: the maximal deflection (in SLS) of the overall structure and the tensile (face- ) strength of 7MPa (in ULS). An iterative approach has been adopted in order to come to a final, least-weight design: first adapting the core-thickness of the total structure until the limit

10 398 DEBOLSTER ET AL: Modular design-concept, using hypar-shells made of fibre reinforced composites deflection (SLS) is no longer exceeded and then changing the face-thicknesses of the hypar surfaces that have reached the tensile strength of the faces (ULS). Cantilever structures are difficult to build and design, but they are nevertheless very interesting structures for various building components such as building entrances. In onedimensional cantilever designs, stability can only be acquired through a fixed end connection (or similarly: a counteracting load). Spatial designs offer other possibilities - mainly through form-design so that hinged line connections could be used. A case-study has been chosen to demonstrate that all principals mentioned above can be assembled in a lightweight, freeform modular cantilever design of a building entrance. ACKNOWLEDGMENT Financial support by the Institute for the Promotion and Innovation by Science and Technology in Flanders ( IWT-Vlaanderen ) is gratefully acknowledged. REFERENCES [Ave71] [Cur99] [Cuy02] [Deb06] [Heb01] J. Aveston, G.A. Cooper, A. Kelly, Single and multiple fracture; the properties of fibre composites. Proc. Conf. National Physical Laboratories, IPC Science & Technology Press Ltd., UK, W.A. Curtin, Stochastic Damage Evolution and Failure in Fibre-Reinforced Composites. Advances in Applied Mechanics, vol. 36, pp , H. Cuypers, Analysis and Design of Sandwich Panels with Brittle Matrix Composite Faces for Building Applications. PhD-thesis, Vrije Universiteit Brussel, Belgium, E. De Bolster, H. Cuypers, P.W. De Wilde and J. Wastiels, Use of hypar-shell structures with fibre reinforced cement matrix composites in lightweight constructions. 12 th European Conference on Composite Materials (ECCM 12), France, aug S. Hebbelinck, A generating system for temporary, adaptable and reusable nets and tensile structures. PhD-thesis, Vrije Universiteit Brussel, Belgium, 2001.