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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.

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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.

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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.

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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

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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

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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

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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).

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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.

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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.

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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

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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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