Abstract—Origami, the craft of folding paper, has been a source

of inspiration for developable systems in various engineering disciplines, as it results in lightweight and stiff structures. In this design study, a single-crease arc is employed as driver for the design of a 56m carbonfiber-foam composite footbridge system. The design concept of the origami-inspired footbridge is enhanced with aperture patterns in the walls of the cross-section. Apertures provide ventilation, natural lighting and views, while achieving an aesthetic design intent. Aperture patterns are created parametrically by varying the number of cutouts, their shape, orientation and size, whether aperture rows are stacked or staggered, and in case of polygonal cutouts, the number of vertices and the corner-fillet radius. Six patterns, selected to align with conventional truss typologies (Pratt, Town lattice, Vierendeel), are investigated using Finite Element analysis. This selection ensures structural soundness, and helps follow recommendations regarding aesthetics in bridge design discussed by bridge designers Christian Menn and Fritz Leonhardt. The structural impact of the aperture patterns is evaluated according to AASHTO footbridge design-code. The curved-crease origami footbridge with the Pratt truss-like aperture pattern is found to meet the controlling AASHTO deflection limit with the lowest volume of material, suggesting higher structural efficiency.

Keywords— origami, curved crease, footbridge, design, aperture patterns.

I. INTRODUCTION RIGAMI can be used to design developable three dimensional shapes and stiffened structures due to folding

planes. Curved-crease origami was first explored for artistic purposes during the Bauhaus (1919-1933, Germany) by Josef Albers [1]. In his preliminary course on Paper Study in 1927-28, he and his students created sculptures by folding paper along curves instead. Since then several artists used curved crease origami to create sculptures from paper and other sheet materials. Similar to traditional straight-crease origami, curved-crease origami creates three dimensional shapes made of developable surfaces. However, curved-crease folding reflects a hybrid of both folding and bending of its surfaces [2-4].

The goal of this paper is to investigate a curved-crease Luca Nagy was with Princeton University, Princeton NJ, USA and is now

with the University of Toronto, Toronto ON, Canada (email: luca.nagy@mail.utoronto.ca). Landolf Rhode-Barbarigos, was with Princeton University, Princeton NJ, USA and is now with the University of Miami, Coral Gables FL, USA (email: landolfrb@miami.edu). Sigrid Adriaenssens is with Princeton University, Princeton NJ, USA (phone: 609-258-4661; e-mail: sadriaen@princeton.edu).

origami-inspired footbridge concept from a structural and aesthetic perspective, using cutout patterns in the walls of the closed cross-section. To achieve this goal, Section 2 presents the design of the curved-crease origami-inspired footbridge. Section 3 describes the method of aperture pattern generation, and the aesthetic and structural impact of selected patterns. Finally, Section 4 presents the conclusions of this study.

II. DESIGN OF A CURVED-CREASE ORIGAMI-INSPIRED FOOTBRIDGE STRUCTURE

A. Description of the Footbridge Structure The proposed footbridge system is composed of two single

curved 40 m radius arcs, approximated with a segmented design. The two arcs connect in plane forming an S-shaped 56 m span beam. The S-shape geometry of the beam enhances its structural behavior by increasing the horizontal stiffness against wind and other lateral loads. Moreover, a change in length due to temperature results in a slight change of the S-shape curvature instead of longitudinal stresses in the entire beam. Fig. 1 shows a schematic drawing, while Fig. 2 shows technical drawings of the footbridge. The S-shaped beam is assumed fixed at each end and a deck width of 4 m complying with footbridge design codes [5] is assumed.

Fig. 1: Schematic drawing of the curved-crease origami-inspired movable

footbridge design. In cross-sectional view, the curved-crease origami-inspired

footbridge has an innovative tubular profile with creases implemented on its top and bottom flanges. Two single-crease plates, inclined at a 15° angle, form the flanges of the girder section (see Fig. 1). Avoiding a flat geometry increases in-plane flexural stiffness of the flange supporting the deck

Curved-Crease Origami-Inspired Footbridge: The Structural and Aesthetic Influence of Aperture Patterns

Luca Nagy, Landolf Rhode-Barbarigos, and Sigrid Adriaenssens

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structure. Flanges are connected with 5 m tall planar vertical webs. Webs and flanges are divided into the same number of quadrilateral segments. This cross-section with a single curved-crease in its flanges has substantial flexural and torsional stiffness.

Fig. 2: Longitudinal cross-section and plan of the footbridge design.

B. Finite Element Analysis of the Curved-Crease Origami-Inspired Movable Footbridge

In the finite element (FE) analysis, the panels composing the curved-crease origami-inspired footbridge are assumed to be made of a carbon fiber-foam composite material. Carbon fiber-foam has been successfully used before in civil engineering applications [6-7] offering high strength and low weight compared to traditional engineering materials [8-12]. The selected composite is a 65 mm thick sandwich material, with two layers of carbon-fiber on the outside and a layer of foam in the middle. Carbon-fiber layers carry the bending forces, while the foam layer is inserted in between to increase the second moment of inertia for the material. A linear elastic behavior is assumed in this study. Material properties are based on information provided by commercially available products from DragonPlate [13] and are presented in Table 1. Internal stiffeners (created by changing the ratio of materials in the carbonfiber-foam composite) are assumed so that local plate buckling does not occur. Static analyses and sizing are performed according to AASHTO codes [5, 14-15]. Linear static analysis is performed in the FE software AXIS [16]. The dynamic behavior of the structure is outside the scope of this research.

Table 1: Material properties for the carbon-fiber and foam composite

The analysis of the structure is conducted with a pedestrian

load of 4.3 kN/m² (90 psf) in addition to the self-weight of 1.15 kN/ m2 (24 psf, calculated per plan area), and the weight of the carbon-fiber sandwich material deck of 0.5 kN/m2 (10.5 psf). Loads and load combinations are assembled according to AASHTO guidelines [15]. Wind loads and temperature are also considered, however, combinations involving these loads do not control design. Snow and other loads are not considered in this preliminary design.

The design of the curved-crease origami-inspired footbridge is deflection controlled as maximum stresses in the carbon-fiber sandwich panels do not exceed 150 N/mm2 or 60% of the design yield stress of the carbon-fiber sandwich material in any load combination. For the evaluation of deflection, the L / 360 = 155 mm limit [5] is employed. A camber equal to the deflection from dead load is assumed. Therefore, the deflection limit applies to the deflection resulting from service loads. The 5 m overall height of the cross-section and the 65 mm wall thickness are both determined by this deflection limit.

III. AESTHETIC CONSIDERATIONS AND THEIR INFLUENCE ON THE DESIGN

The design of the curved-crease origami-inspired footbridge is enhanced by aesthetic considerations, staying true to its origins in art. The closed cross sectional profile requires the implementation of openings to allow air and light inside in the structure, and to improve the experience of pedestrians walking across the bridge. Cutouts in the walls provide thus an opportunity for the implementation of aesthetic considerations in the design process.

Fig. 3: Preliminary Finite Element study of cutout shapes.

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Preliminary FE analyses on selected cutouts, such as triangles and squares, revealed that the location and shape of the cutouts have little effect on the maximum midspan deflection (Δ) under self-weight and uniform pedestrian load (see Fig. 3). The determining factor of the maximum deflection is the cumulative aperture area (A) for each segment. Therefore, apertures can be designed freely regarding shape, following aesthetic considerations.

In this study, apertures have multiple functions. First, they serve for ventilation and natural lighting. This goal is the easiest one to meet, as the number of cutouts employed is assumed to be sufficient. Second, the selected pattern should express some aspect of how the bridge functions. Third, apertures should have an overall aesthetically pleasing effect. The second and third goals are discussed in further detail.

To meet the second goal, aperture patterns are designed to reflect the way the curved-crease origami-inspired footbridge functions. Since material is removed from the walls only, the relevant internal forces are the shear forces (see Fig. 4). Shear forces resulting from a bending moment have a value of zero at midspan, and increase towards the two fixed supports of the structure. Therefore, to express the main forces the walls of the structure carry, the aperture pattern progresses similarly, with more removed area towards midspan, and less at the supports.

Fig. 4: Shear force diagrams.

The third goal of creating an aesthetically pleasing structure

is defined with the help of Fritz Leonhardt, a German engineer who wrote on aesthetics in design. He claimed that the truism that ‘there is no accounting for taste’ is only valid if aesthetics are not studied [17]. Once they are, several guidelines emerge that can guide an engineer to fulfil the sometimes ignored functions of structures to be “comfortable”, “beautiful”, or “cozy”, in addition to safe and economical. In addition to Leonhardt, Christian Menn [18] also outlined aesthetic guidelines for bridge design. The main points that both agree on are slenderness, lightness or transparency, order or regularity (including symmetry, and aesthetic homogeneity), and integration into the environment.

Footbridge design tends to have more leeway on aesthetic rules due to its smaller scale, which allows more “play” with design without becoming too uneconomical. Therefore, the footbridge design looks to incorporate some, but not all, of the aesthetic guidelines by these structural designers. The aperture pattern is designed to have an ordered feel with symmetry and proportionality, but occasionally might violate rules such as limiting the direction of lines to no more than three [17], or simplicity [18]. The pattern generation method detailed below allows for the creation of a nearly unlimited number of designs, even with the guidelines described above. The number of possible patterns is reduced by focusing on designs

that resemble to traditional structural designs and thus reflect Leonhardt’s and Menn’s guidelines. These traditional structure types are the Pratt truss, the Town lattice truss, and the Vierendeel truss (Fig. 5).

Fig. 5: Traditional truss types used for the aperture design.

A. Aperture Pattern Generation Patterns are generated using parametric geometry definition

in Grasshopper, and extension of Rhinoceros 5.0 [19-20]. Pattern generation is implemented so that cutout parameters could be varied separately for each panel. The parameters to be varied are the number of cutouts (length of array in the horizontal and vertical direction), the shape and size of cutouts, in case of polygonal cutouts, the number of vertices, the corner fillet radius (used to avoid sharp corners that would create stress concentrations), the orientation of the cutout shapes, and whether the rows are stacked or staggered. For examples of how each parameter can be varied, see Fig. 6-10.

Fig. 6: Panels with different (4x3 versus 4x4) cutout arrays.

Fig. 7: Panels with different cutout sizes.

Fig. 8: Panels without (left) and with (right) filleted corners.

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Fig. 9: Patterns with different rotation angles.

Fig. 10: Patterns with stacked (left) or staggered (right) cutouts.

Fig. 11: Pratt truss-like cutout pattern.

Fig. 12: Lattice truss-like pattern.

Fig. 13: Vierendeel truss-like pattern.

Fig. 14: Pratt to lattice truss pattern.

The six patterns selected in this study are aligned with a traditional type of truss structure - a Pratt truss, a Town lattice truss, or the Vierendeel truss - or their combinations. These patterns also meet the three main design goals detailed earlier. They offer sufficient ventilation and natural light, they express the shear forces in the cross section walls, and they follow the crucial aesthetic guidelines that both Menn and Leonhardt agree on for bridge design. Fig. 11-16 show the side view of one half of the bridge (due to the curved shape of the bridge, panels in a side view appear to slightly shorter towards the middle, even though their dimensions are constant). The cutout pattern is then mirrored.

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Fig. 15: Pratt to Vierendeel truss pattern.

Fig. 16: Lattice to Vierendeel truss pattern.

B. Analysis of Selected Patterns The selected six aperture designs are analyzed using AXIS

Finite Element software [9]. Designs are compared to each other and to the original configuration of the structure without cutouts, using the total volume of carbonfiber-foam composite material required to meet the deflection requirement. These designs are controlled by deflection, as the stresses do not exceed 150 kN/mm2, staying well below the allowed design strength. The design of the structure remains deflection controlled even with added cutouts. The analysis is performed by keeping a fixed thickness (90mm) for the flanges, and modifying wall thickness between the different designs until the deflection limit (L/360) is met (see Table 2). Wall thicknesses are adjusted at 5 mm steps, since a design more exact than that may not be reliably met by the available construction methods. Therefore, net deflection values vary within a range of 6 mm (from 148 mm to 154 mm), but still remain within the deflection limit.

Table 2: Total required carbonfiber volume to meet the 155 mm deflection limit, assuming different flange (deck support and roof) and web (wall) thickness. Net deflection is defined after camber equal to the deflection under dead load.

It is also shown that only the Pratt aperture design (see Fig.

17) performs better than the other patterns, as well as the original design (without apertures). However, using variable wall thicknesses across the segments, or using more advanced optimization techniques, other aperture patterns could lead to a more efficient aperture pattern.

IV. CONCLUSION In this paper, the design of a curved-crease origami-inspired

footbridge is enhanced by adding aperture patterns. Apertures are employed to provide ventilation and natural lighting, while expressing structural function and highlighting the artistic origins of the origami-inspired system. Aperture patterns are inspired by the traditional truss types of the Pratt (diagonal) truss, the Town lattice truss, and the Vierendeel truss, as well as their combinations. Analyses reveal that the Pratt aperture pattern performs better than the original design (without cutouts) and other patterns, minimizing the total volume of carbonfiber-foam composite material volume required to meet the deflection limit. The structural impact of aperture patterns in the walls of the curved, closed cross-section footbridge can be attributed to the total removed area remaining thus largely independent of cutout shape and location. Therefore, the aperture designs can be used for the expression of how the structure functions, or to meet purely aesthetic goals.

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