evaluation of design parameters for deployable … of design parameters for deployable planar...

Evaluation of design parameters for deployable planar scissor arches

L. Alegria Mira1, R. Filomeno Coelho2, N. De Temmerman1 & A. P. Thrall3 1Department of Architectural Engineering, Vrije Universiteit Brussel, Belgium 2Building, Architecture and Town planning Department, Université libre de Bruxelles, Belgium 3Department of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, USA

Abstract

Deployable scissor structures can transform from a compact bundle of elements to a fully expanded configuration. Due to this transformational capacity there is a mutual relation between the geometry, the kinematics and the structural response of the scissor system, resulting in a relatively complex design process. In order to understand how geometrical parameters influence the structural performance, it is beneficial to evaluate these structures at a pre-design stage. To reach this goal, we developed an integrated framework for pre-design evaluation using the parametric finite element tool Karamba in combination with Matlab. By doing so an automatic and immediate preliminary structural evaluation can be established that guides the designer in making efficient geometrical design decisions. This paper evaluates design parameters for deployable scissor arches using this framework. More specifically, the influence of the most important geometrical inputs – scissor type, number of scissor units, structural thickness and height of the arch – on structural performance (i.e. stress, deflection and mass) is determined. Results indicate the sensitivity of the considered parameters and their importance in the design of scissor arches. With this knowledge in an early design stage, the subsequent design optimisation, detailed analysis and realisation are enhanced. Keywords: deployable structures, 2D scissor arches, evaluation of design parameters.

Mobile and Rapidly Assembled Structures IV 183

doi:10.2495/MAR140151

www.witpress.com, ISSN 1743-3509 (on-line) WIT Transactions on The Built Environment, Vol 136, © 2 014 WIT Press

1 Introduction

Deployable structures are prefabricated space frames which can be transformed from a compact bundle of components into an expanded, load bearing structural system. Because of this transformational capacity they are adaptable to changing needs or circumstances, offering significant advantages compared to conventional, static structures. They are used for a wide spectrum of applications ranging from temporary and mobile structures or covers (emergency shelters, exhibition and recreational structures), bridge systems, deployable roofs for sports stadia, to the aerospace industry (solar arrays) [1–3]. A specific subgroup of deployable structures is formed by scissor systems [4]. Besides being transportable, they have the great advantage of speed and ease of erection and dismantling, while offering a large volume expansion and high deployment reliability [5]. Scissor structures consist of units comprised of two beams connected through an intermediate hinge allowing a relative rotation. By connecting such scissor units at their end nodes by hinges, a deployable grid structure is formed. Finally, by adding constraints, the mechanism goes from the deployment phase to the service phase, in which it can bear loads and where a membrane attached to the structure offers weather protection. Depending on the location of the intermediate hinge and the shape of the beams, three general unit types can be distinguished: translational, polar and angulated units [1, 3, 5] (Figure 1). From the 1960s through today, many have contributed greatly to the field of deployable scissor structures: Piñero, Escrig, Gantes, Hoberman, Pellegrino, You, among others. A great deal of research has been done in determining the geometric principles for different scissor configurations, in proposing new types of scissor units and in structurally analysing these systems to enhance their performance (e.g. [1, 3, 5-12]). The goal is generally the same: enlarging the practical use of deployable scissor structures in full-scale realisations. Pursuing this aim, De Temmerman and Alegria Mira [13] recently introduced the Universal Scissor Component, USC (Figure 2). This is a single component that can be reconfigured in a variety of geometrical configurations allowing component re-use and multi-use of scissor structures for different applications.

Figure 1: The design parameters are indicated on the arches, from left to right: translational, polar and angulated scissor type.

184 Mobile and Rapidly Assembled Structures IV


Figure 2: The Universal Scissor Component (left) [13], USC, can be composed in an arch configuration in an angulated way (scissor hinge in strut; middle) or in a polar way (scissor hinge in mast; right).

Since scissor structures are characterised by a dual functionality as either kinetic mechanisms (during deployment) or load-bearing skeletal structures (after deployment), it is crucial to understand that there is a direct and mutual relation between the geometry, the kinematics, and the structural response. This results in a relatively complex design process. Moreover, the designer has a large freedom when choosing design parameters for a scissor structure: scissor type, number of scissor units U, structural thickness t when fully deployed, etc. (Figure 1). With more structural insight in geometrical aspects of the scissor system the designer can make legitimate decisions related to these parameters in the design process. The earlier this knowledge is available, the more beneficial for the further design, analysis and realisation. To evaluate these structures at an early design stage, the authors have developed an integrated framework for pre-design evaluation using the parametric finite element tool Karamba in combination with Matlab [14, 15]. Within this framework we performed a sensitivity analysis of design parameters for planar scissor arches. This paper presents these results which indicate the influence of design parameters and their importance in the design of scissor arches. The investigation is performed on two-dimensional scissor arches since the insights retrieved on this level can be extrapolated to the corresponding three-dimensional shape (e.g. barrel vault shelters composed of arches). Also, at this pre-design stage, design details of the scissor structure (e.g. joints, membrane covering) are not considered. These aspects will surely influence the structural behaviour, but minimally affect the influence of the geometrical design parameters on the performance. This paper begins with a review of the integrated framework. Afterwards, the sensitivities of inputs X on outputs Y are calculated. In this paper X = {scissor type, number of units, thickness-span ratio and height-span ratio of the arch} and Y = {deflection of the arch, stress value in the scissor beams and mass of the arch}. It includes the new scissor type component, the USC, in order to compare it with the traditional scissor types. The results are presented and the design parameters are evaluated. Finally, conclusions are drawn in respect to the next research step: an exhaustive parameter analysis leading to knowledge on appropriate choices for the design parameters.



2 Integrated framework for pre-design

The ultimate purpose of this research is to guide the designer in making efficient design decisions and facilitate the design process of scissor structures. For this, an evaluation methodology has previously been proposed [14]. In this prior paper the goal and advantage of this methodology is presented: investigating the structural performance of some scissor arch case studies in an early design stage. This opens the possibility to an automatic and immediate evaluation framework [15]. This integrated framework consists of two main parts which form a loop: parametric finite element (FE) simulations with Karamba [16], controlled by Matlab (Figure 3). This integrated framework has been used to perform the sensitivity analysis presented in this paper and will be briefly reviewed here.

Figure 3: The framework, developed for a preliminary evaluation methodology, allows an automatic and immediate generation of structural properties for a set of geometrical inputs.

Karamba is a finite element program within Grasshopper [17], the parametric geometric plug-in for Rhinoceros [18] a commercial computer aided design package. Karamba, in the case of the research presented here, interactively calculates the linear-elastic response of beam structures which are parametrically modelled in Grasshopper. It is not another FE tool for detailed engineering analysis. Instead it is meant to aid designers in principal decisions during early project phases where design flexibility transcends depth of detailing. Although the actual behaviour of scissor structures involves geometric nonlinearities and is sensitive to effects such as friction in joints, a linear-elastic analysis can generally be considered acceptable for a preliminary design phase. An advantage of Karamba is the fast bi-directionality between the geometrical and structural data, allowing for an immediate update of the structural data when geometrical



parameters are changed. This is possible since the developers made deliberate choices in terms of analysis in order to reduce the calculation time: e.g. linear-elastic analysis, Euler-Bernoulli beam theory, use of hermitian finite elements. In the integrated framework this parametric finite element environment is combined with Matlab. Files with input parameters are generated and read in at the level of the parametric model in Grasshopper. Immediately afterwards, the structural output data, from Karamba, are fed back into Matlab (Figure 3). The flexibility of an automatic and immediate generation of different structural output values, based on the desired set of geometric inputs, is an important asset of the integrated framework and methodology. It allows a fast investigation of a large design space. The latter is demonstrated for the first time in [15] through a preliminary sensitivity analysis. For more information concerning these tools and the integrated framework, the reader is referred to prior research [14, 15]. This paper implements this framework to rigorously investigate the influence of geometrical inputs on structural properties: more design parameters and structural properties are examined leading to valuable insights for further analysis steps.

3 Description of the analysis

Within the integrated framework we evaluated the design parameters for the case of planar scissor arches. The outcome lets us examine which geometric parameters significantly influence the structural properties of a scissor system, and to what extent. With this knowledge the designer can explore the parametric space efficiently, considering the relevant geometric variables, in order to eventually create competitive scissor structures. Through finite element simulations on the parametric model of the scissor arches the influence of inputs X on outputs Y are calculated. Here, X = {scissor type; number of scissor units; thickness-span ratio; height-span ratio} and Y = {deflection; stress value; mass}. The design parameters X are indicated in Figure 1. The deflection is the maximum vertical displacement of a beam node in the overall arch. The stress value is the axial stress always extracted from the right outer support beam, as indicated in red in Figure 1 (which in most cases is the heaviest loaded). The mass indicates the total mass of the arch. In case of the thickness-span ratio, t/S, and the height-span ratio, 2H/S, which are continuous variables, we can calculate the influence as sensitivity values. The normalised relative sensitivities are calculated empirically:

= .

. (1)

Relative-sensitivity functions show which parameters have the greatest effect on the output for a certain percent change in the parameters (dx = 0.01%) [19]. In this paper we consider values for the thickness from 2 to 20% of the span, which is a broad range compared to the 10% t/S value generally used for space truss



structures [3]. When examining the height, the values range from 10 to 100%, meaning from a very shallow (e.g. for a roof configuration) to a semi-circular arch (e.g. for shelter, exhibition spaces). In the investigation of the scissor type and number of scissor units, the influence as sensitivities, calculated as derivatives obtained by as finite differences, cannot be determined since these are discrete numerical variables. For this reason, the actual values of the structural outputs are given instead of the sensitivity values. Each scissor type is considered and the amount of units ranges from 4 to 20. If a parameter is investigated, the other inputs are kept constant: t/S = 10%, 2H/S = 50% and U = 8. The calculation method, as explained here, is applied on several case studies. Five scissor types are considered: translational, polar and angulated (Figure 1), USC as angulated and USC as polar (Figure 2). Also, two different spans are considered: 6 m and 15 m. The beam elements of the scissor arches are modelled with aluminium material properties (EN-AW 6060 T6 grade) and with hollow square profiles 100 x 100 x 5 mm (for 6 m span) and 200 x 200 x 12 mm (for 15 m span). For supports, the four free beam ends of the scissor arches are pin restrained. The arch is loaded with point loads at the lower nodes, representing a snow load of 1 kN/m2 (calculated for the area of the corresponding barrel vault shape with 2 m depth) [20]. In Karamba the scissor hinges are modelled as zero-length springs, which is an easy and effective method for a linear elastic analysis in a first stage. In order to simulate the mechanism behaviour during deployment of these scissor structures, the corresponding degrees-of-freedom (or stiffnesses) of the springs are set.

4 Evaluation of design parameters

4.1 Sensitivity analysis of the height-span ratio (2H/S)

The graphs in Figure 4 depict the sensitivity values of the 2H/S parameter on deflection, stress and mass for the different scissor types. The evolution of the sensitivity curves between 6 m span and 15 m span is quasi identical for all scissor types (Figure 4a and 4b; here only shown for the angulated type). This means that the influence of the variation in height is span independent as one would expect for linear elastic analysis. The angulated scissor type is the least sensitive to the height parameter (sensitivity values < 1.4) compared to the other scissor types. Comparing the structural output for this scissor type, the deflection is more sensitive to changes in height. For the other types, stress becomes more sensitive from a 2H/S value around 50%. For the translational, we can distinguish a high peak for the stress sensitivity around 2H/S = 1. When investigating what happens in that region, we understand that the stress decreases and reaches zero and goes even below zero (meaning that a tension stress is changed to a compression stress). Thus around 2H/S = 1, a small change in height can mean a big change in stress. The stress can evolve in this way because a change in this geometrical parameter changes the orientation



of the beams in the structure (with reference to the loading). If we zoom in on this graph, ignoring the peak value (as shown in Figure 4d), the sensitivity values are comparable over all graphs.

a) b)

c) d)

e) f)

Figure 4: The sensitivity values of the height-span parameter on the deflection, stress and mass are low and comparable over all scissor types if we ignore the peak values around 2H/S=1: a) angulated, 6 m, b) angulated, 15 m, c) polar, 6 m, d) translational, 6 m, e) USC angulated, 6 m, f) USC polar, 6 m.

4.2 Sensitivity analysis of the thickness-span ratio (t/S)

The graphs in Figure 5 depict the sensitivity values of the t/S parameter on deflection, stress and mass for the different scissor types. Like for the 2H/S parameter, the evolution of the sensitivity curves between 6 m span and 15 m span is quasi identical for all scissor types (Figure 5a and 5b; here only shown for the angulated type).



The angulated scissor type is also the least sensitive to the thickness parameter compared to the other scissor types. For all scissor types, the sensitivity values are quite low (< 8) except for the translational type which shows high peak values for the stress for small thickness-span ratios. The reason for this peak value is the same as explained in the previous section (4.1): stress values drops to values near zero. If we zoom in on this graph, ignoring the peak values (as shown in Figure 5d), the sensitivity values are comparable over all graphs. a)

b)

c) d)

e) f)

Figure 5: The sensitivity values of the thickness-span parameter on the deflection, stress and mass are overall comparable over all scissor types: a) angulated, 6 m, b) angulated, 15 m, c) polar, 6 m, d) translational, 6 m, e) USC angulated, 6 m, f) USC polar, 6 m.



4.3 Number of scissor units (U)

Since the number of units is a discrete numerical variable, the sensitivity values, for a small dx, cannot be calculated. Therefore, the evaluation of this parameter is done for all scissor types in case t/S = 10% and 2H/S = 50% considering the actual values of the stress, deflection and mass (Figure 6). a) b)

c) d)

Figure 6: Considering t/S=10% and 2H/S=50% , the deflection, stress and mass generally increase when the number of units increase: a) deflection, 6 m b) deflection, 15 m c) stress d) mass.

Like for the thickness and height parameter, the evolution of the output values between an arch of 6 m span and 15 m span is identical for all scissor types (Figure 6a and 6b; here only shown for the deflection value). From the graphs we can conclude that in general, when the number of units increases, also the deflection, stress and mass values increase. Only in case of the translational scissor type the stress decreases again for a large number of units.

4.4 Scissor type

The scissor type is also a discrete variable, resulting in an evaluation considering the actual values of the stress, deflection and mass for all scissor types in case t/S = 10%, 2H/S = 50% and U = 8 (Table 1). Just like the previously investigated design parameters, the evolution of the output values between an arch of 6 m span and 15 m span is identical for all scissor types.



Between the traditional scissor types (angulated, polar and translational), the polar type shows the biggest deflection and stress (both for 6 m and 15 m span), while the translational type shows the smallest deflection, stress and mass. Compared to the traditional scissor types, the USC types show a much lower deflection and stress (both for 6 m and 15 m span). This is due to the triangular shape of the USC which is more beneficial in terms of stress distribution compared to the beams of the traditional scissor types. The mass, however, is much larger since a fixed cross-section for all types is considered. In a next phase the cross-sections will be optimised leading to interesting insights in terms of mass for the different scissor types. Even if the USC requires more mass, this disadvantage is considered to be of less significance than the obtained advantages, related potential to re-use and recycle a mass-producible component.

Table 1: Compared to the traditional scissor types, the USC types show a much lower deflection and stress (both for 6 m and 15 m span).

Angulated Polar Translational USC angulated USC polar

Span 6 m

Deflection (m) 0.06 0.07 0.04 0.01 0.01

Mass (kg) 90.57 89.96 83.02 205.97 205.97

Stress (kN/cm2) 23.81 25.87 18.47 2.14 4.19

Span 15 m

Deflection (m) 0.13 0.16 0.08 0.02 0.02

Mass (kg) 1034.93 1027.99 948.64 2353.58 2353.58

Stress (kN/cm2) 18.09 19.52 13.95 1.13 2.55

5 Conclusions

This paper presented an integrated framework for the evaluation of the influence of geometrical design parameters on structural behaviour in the case of planar scissor arches. The investigated input parameters were the thickness-span ratio (t/S), height-span ratio (2H/S), the number of units (U) and the scissor type – all typical and relevant design parameters for scissor arches. The influence of these parameters was examined in terms of the structural deflection, stress and mass output. We can conclude that the evolution of the investigated parameter values between an arch of 6 m span and 15 m span is quasi identical for all scissor types. Since the analysis is linear-elastic, the influence of parameter variations is span independent. For the thickness (t/S) and height (2H/S) parameters, the sensitivity values range in the same order of magnitude, meaning that one parameter is not more influential than the other. Moreover, the sensitivity curves do not show any discrepancies or unjustifiable peaks. Furthermore, from the evaluation of the



number of units and the scissor type, we can conclude that these design parameters have an important effect on the structural outputs and need to be taken into account in the early design stage. Furthermore, this research has indicated the potential of the use of Universal Scissor Component (USC) since it can reduce the stress and deflection level significantly. An automatic integrated framework for pre-design evaluation helps the designer to make efficient design decisions and facilitates the further design process of scissor structures: going from optimisation to detailed analysis to realisation. More fundamentally, the advantage of performing a sensitivity analysis and evaluation of design parameters in an early stage is that the design space for further analyses is narrowed down by focusing the search on the geometric input parameters that are relevant to consider. From the investigation done in this paper we can formulate some conclusions to this respect. If the height-span ratio is not a fixed constraint set by the designer, it is interesting to examine which scissor type performs better for low or high height-span ratios. The sensitivity graphs indicate interesting values to consider in a next optimisation step, 2H/S taken around 50 and 100%. Concerning the number of scissor units, an increase generally results in an increase in stress, deflection and mass. Since there is little change in structural performance between 15 < U < 20, the unit number range can be limited from 4 to 15. Further, in the optimisation step it is valuable to consider certain structural thickness values for each scissor type, each height-span ratio and number of unit. The output of the sensitivity analysis distinguishes interesting values to consider: t/S between 2 and 12%. Future work will consist of setting up this structural optimisation framework in order to identify good choices for the design parameters and set up preliminary design guidelines for scissor structures.

Acknowledgement

This research is funded by a PhD grant of the Agency for Innovation by Science and Technology (IWT).

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