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Numerical simulation of tensioned membranes for
kinematic form active structures
Sara Minoodt
Supervisors: Prof. dr. ir. Wim Van Paepegem, Dr. Ali Rezaei
Counsellor: Tien Dung Dinh
Master's dissertation submitted in order to obtain the academic degree of Master of Science in Civil Engineering Department of Materials Science and Engineering Chairman: Prof. dr. ir. Joris Degrieck Faculty of Engineering and Architecture Academic year 2014-2015
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Numerical simulation of tensioned membranes for
kinematic form active structures
Sara Minoodt
Supervisors: Prof. dr. ir. Wim Van Paepegem, Dr. Ali Rezaei
Counsellor: Tien Dung Dinh
Master's dissertation submitted in order to obtain the academic degree of Master of Science in Civil Engineering Department of Materials Science and Engineering Chairman: Prof. dr. ir. Joris Degrieck Faculty of Engineering and Architecture Academic year 2014-2015
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Acknowledgements
A master thesis is the crown on one’s educational career. I would like to thank a number of people who helped
me achieving this work.
Prof. dr. ir. Wim Van Paepegem, for introducing me to the interesting subject of numerical simulations. Prof. dr.
ir. Marijke Mollaert, for letting me be a part of this interesting research in membrane architecture.
Furthermore, I would like to thank M. Sc. Tien Dung Dinh, who thought me all about numerical simulations. Thank
you very much for being a helpful and enthusiastic mentor.
Maarten Van Craenenbroeck and Silke Puystiens, for answering my tsunami of questions soon and without a
problem.
My parents, for believing in me throughout my six years of studies, even when I didn’t believe in myself. Thank
you for giving me the opportunity for chasing my dream to become a civil engineer.
And finally, my friends, who stood by me.
Sara Minoodt, May 22, 2015
Permission for consultation
The author gives permission to make this master dissertation available for consultation and to copy parts of this
master dissertation for personal use.
In the case of any other use, the copyright terms have to be respected, in particular with regard to the obligation
to state expressly the source when quoting results from this master dissertation.
May 22, 2015
vii
Numerical simulation of tensioned membranes for kinematic form active structures
Sara Minoodt
Supervisors: Prof. dr. ir. Wim Van Paepegem, Dr. Ali Rezaei Counsellor: Tien Dung Dinh Master’s dissertation submitted in order to obtain the academic degree of Master of Science in Civil
Engineering.
Department of Materials Science and Engineering Chairman: Prof. dr. ir. Joris Degrieck Faculty of Engineering and Architecture Academic year 2014-2015
Summary
In the current practice, no clear design guidelines are available. The aim of this master thesis is to
model the three dimensional deployment of tensioned membranes, as they could be applied in
practical applications. From the simulations, a prediction of the strains and stresses could be obtained,
which could then be implemented in the design phase of structures. This would increase the efficiency
of these structures.
A case study is investigated, which was built in the framework of the Contex-T project. A
polyvinylchloride coated polyester fabric was used to cover a structure which can (un)fold in different
positions. An equivalent model analysed using the finite element method software Abaqus. By using
the elasto-plastic material model as a user material to simulate the behaviour of the membrane, the
nonlinear behaviour of the material could be grasped in the simulations.
Different approaches were used and compared in this master thesis. In the end, the simulations using
contact properties in Abaqus/Explicit gave good correlation when the isotropic material model was
used. The CPU time was however large, limiting its practical use. As another approach, the belt is
simulated as a belt-like connector, eliminating the convergence problems encountered with the
contact. The simulations showed promising results, which are discussed thoroughly.
Finally, the influence of the stiffness of the belt is investigated. Also a comparison is made between
several material models.
Keywords
Coated fabric, kinetic architecture, finite element method (FEM), user material
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Numerical simulation of tensioned membranes for kinematic form active structures
Sara Minoodt
Extended abstract
In the current practice, no clear design guidelines are available. The aim of this master thesis is to
model the three dimensional deployment of tensioned membranes, as they could be applied in
practical applications. From the simulations, a prediction of the strains and stresses could be obtained,
which could then be implemented in the design phase of structures. This would increase the efficiency
of these structures.
This master thesis is situated within the research on design methods for kinetic tensile architecture. It
presents a method for simulating the three dimensional deployment of tensioned membrane coated
fabrics, on a scale that resembles practical use.
In a first part of this thesis, literature was studied to gain more knowledge in the field of membrane
architecture. The membrane material shows significantly different behaviour in warp and fill direction,
due to the weaving process. The fill direction is less stiff. The different steps in the design of membrane
structures were investigated as well: the principles for the form-finding, patterning, design analysis
and fabrication process are discussed. Furthermore, different types of kinetic architecture were
explored as well and illustrated with some examples.
Next, the necessary tools for simulating 3D models in Abaqus were explored. The Abaqus/Explicit
analysis showed a lot of promise, however, the results are not as reliable as the Abaqus/Standard
analysis.
In order to validate the analysis performed, an experiment was executed at the Free University in
Brussels, where a PVC-coated polyester membrane fabric was deployed three dimensionally. The
membrane was opened from a 50° opening angle towards 90° and then closed back to 10°. Strains
were measured for the intermediate opening angles as well. This was executed by using a Digital Image
Correlation (DIC) technique. In this way, contour plots of the warp and fill strains were obtained, as
well as detailed values along three paths. The opening angle of 90° showed the largest deviations over
the membrane surface. The overall strains measured were low: in the order of 0.1% for the warp
strains and 1% for the fill and shear strains. This indicates that the largest part of the deployment of
the membrane is rigid body motion and does not influence the pretension present in the membrane.
However, it should be noted that difficulties with the reference state were encountered. If the
reference state is taken after first pretensioning is applied, the initial strains are disregarded from the
results. Since the membrane material is history dependent, it is very important that this reference
state is taken at the beginning of the experiment
Different approaches were used to model this deployment. In the end, the simulations using contact
properties in Abaqus/Explicit gave good correlation when the isotropic material model was used.
However, the strains were overestimated due to the large stiffness and no wrinkling or crimp
interchange could be modelled. Especially the 90° opening angle gave very small error (about 10%).
For the closing operation towards 10° however, fluctuations are noticeable and inertia effects are still
too large. Overall, this method shows promising results. The only disadvantage is the CPU time: this is
even with a simplified model very large. Therefore, its practical use is limited.
ix
The central belt can however also be modeled with a belt-like connector. In this way, the difficulties
regarding the simulation of contact are excluded from the model. The concentration of strains is less
pronounced in the simulation as it is in the experiment. The lack of friction plays an important role on
the results: the fill strains are larger. The overal tendency however is seen in the simulations as well.
The strains for the 10° opening angle show better comparison along the paths than for the 90° opening
angle, indicating that the friction has a larger impact on the closing operation than on the opening
operation. From the shear strain, it could be concluded that the 90° opening angle shows differences
with the experiment. This can be due to the properties of the UMAT model: detailed information is
missing on the behaviour of material under load ratios different from 1:1. It can be concluded that the
use of these connectors show a lot of promise, especially regarding the CPU time. However, in order
to obtain results with the inclusion of friction, the model in Abaqus/Explicit has to be elaborated more.
Overall it can be stated that the order of magnitude of the strains obtained from the simulations are
the same as for the experiment. The errors found are quite high, however, it must be noticed that the
strains are quite low, especially in the larger context of the building industry. A lot of parameters affect
the discrepancies between both, where the largest influence will come from a mismatch in paths.
Since the paths could not be exactly located but had to be approximated as close as possible, the error
might become quite high.
The model with belt-like connectors shows promising results. Therefore, the model in Abaqus/Explicit
should be elaborated more so that friction can be included in the simulation. Also, by performing a
parametric study for the friction coefficient, the influence of this parameter can be investigated.
Additionally, a model can be investigated with the correct cutting pattern in such a way that the
influence of this discrepancy can be quantified.
A comparison was made between three material models: the isotropic, orthotropic and elasto-plastic
material model. Finally, from a simulation with a softer belt, it could be concluded that the material
model used to model the belt will influence the strains as well. Especially the strains in the top region
show different results for different stiffnesses.
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Table of Contents Acknowledgements ................................................................................................................................................. vi
Permission for consultation .................................................................................................................................... vi
Numerical simulation of tensioned membranes for kinematic form active structures ........................................ vii
Numerical simulation of tensioned membranes for kinematic form active structures ....................................... viii
Chapter 1: Introduction........................................................................................................................................... 1
1.1 Objectives .............................................................................................................................................. 1
1.2 The FWO project .................................................................................................................................... 1
Chapter 2: Membrane engineering ......................................................................................................................... 2
2.1.1 Materialisation of membranes .......................................................................................................... 2
2.1.2 Coated technical textiles ................................................................................................................... 2
2.1.3 The coating ........................................................................................................................................ 3
2.1.4 Mechanical properties ...................................................................................................................... 4
2.2 Structural behaviour ............................................................................................................................ 10
2.3 Form-finding ........................................................................................................................................ 10
2.3.1 Shape of the membrane .................................................................................................................. 10
2.3.2 Warp and fill directions ................................................................................................................... 11
2.3.3 Edge geometry ................................................................................................................................ 11
2.3.4 The numerical implementation ....................................................................................................... 12
2.4 Patterning ............................................................................................................................................ 12
2.4.1 Development ................................................................................................................................... 12
2.4.2 Compensation ................................................................................................................................. 13
2.4.3 Patterning calculation criteria ......................................................................................................... 14
2.5 Load analysis ........................................................................................................................................ 14
2.5.1 The significance of material properties ........................................................................................... 15
2.5.2 Failure modes .................................................................................................................................. 15
2.5.3 Design guidelines............................................................................................................................. 16
2.6 The fabrication process ....................................................................................................................... 16
Chapter 3: Kinetic architecture ............................................................................................................................. 18
1.1 The evolution of kinetic architecture .................................................................................................. 18
1.2 Kinetic architecture ............................................................................................................................. 20
1.2.1 Morphology ..................................................................................................................................... 20
1.2.2 Kinematic form active structures .................................................................................................... 25
Chapter 4: Analysis methods in ABAQUS software ............................................................................................... 26
4.1 The implicit solver ................................................................................................................................ 26
4.1.1 Convergence .................................................................................................................................... 26
4.1.2 Automatic incrementation control.................................................................................................. 27
4.1.3 Automatic stabilisation of unstable problems ................................................................................ 28
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4.2 The explicit solver ................................................................................................................................ 31
4.2.1 Stable time increment ..................................................................................................................... 31
4.2.2 Computational cost ......................................................................................................................... 31
4.2.3 Comparison of the implicit and explicit results ............................................................................... 35
4.3 The membrane’s element type ........................................................................................................... 36
4.3.1 Shell elements ................................................................................................................................. 36
4.3.2 Membrane elements ....................................................................................................................... 37
4.3.3 Idealised shell elements .................................................................................................................. 37
4.4 Conclusion ........................................................................................................................................... 39
Chapter 5: The experiment ................................................................................................................................... 40
5.1 The Contex-T project ........................................................................................................................... 40
5.2 The experimental set-up...................................................................................................................... 41
5.2.1 The membrane ................................................................................................................................ 41
5.2.2 The belts .......................................................................................................................................... 43
5.2.3 The kinematics ................................................................................................................................ 43
5.2.4 The connections .............................................................................................................................. 44
5.2.5 Execution of the experiment ........................................................................................................... 44
5.2.6 Measurements ................................................................................................................................ 45
5.2.7 Results ............................................................................................................................................. 46
Chapter 6: Numerical deployment of a membrane tensioned by contact ........................................................... 47
6.1 The course of the simulation ............................................................................................................... 47
6.2 The initial geometry ............................................................................................................................. 47
6.3 The implicit analysis in Abaqus/Standard ............................................................................................ 48
6.3.1 Step 1: Pretensioning the membrane ............................................................................................. 48
6.3.2 Step 2: Opening the membrane ...................................................................................................... 55
6.3.3 Results ............................................................................................................................................. 56
6.4 Explicit analysis in Abaqus/Explicit ...................................................................................................... 57
6.4.1 Differences in the model ................................................................................................................. 57
6.4.2 Results of the Abaqus/Explicit analysis ........................................................................................... 57
6.5 Conclusion ........................................................................................................................................... 68
R
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Chapter 7: Numerical deployment of a tensioned membrane with SLIPRING connectors................................... 69
7.1 Including SLIPRING connectors into the model ................................................................................... 69
7.2 Frictionless contact .............................................................................................................................. 70
7.2.1 Optimisation of the mesh ................................................................................................................ 70
7.2.2 The elasto-plastic material model UMAT ........................................................................................ 72
7.2.3 Influence of the material model ...................................................................................................... 82
7.2.4 Influence of the belt stiffness .......................................................................................................... 88
7.3 The inclusion of friction in the SLIPRING connector ............................................................................ 94
7.3.1 Validation of the model ................................................................................................................... 94
7.3.2 Conclusion ..................................................................................................................................... 100
Chapter 8: Conclusions and recommendations .................................................................................................. 101
8.1 General conclusions ........................................................................................................................... 101
8.2 Recommendations for future works .................................................................................................. 102
References……………………………..……………………………..……………………………..……………………………..…..……………………103
1
Chapter 1: Introduction 1.1 Objectives In the current practice of membrane architecture, no clear design guidelines are available. The aim of this master
thesis is to model the three dimensional deployment of tensioned membranes, as they could be applied in
practical applications. From the simulations, a prediction of the strains and stresses could be obtained, which
could then be implemented in the design phase of structures. This would increase the efficiency of these
structures.
A case study is investigated, which was built in the framework of the Contex-T project. A polyvinylchloride coated
polyester fabric was used to cover a structure which can (un)fold in different positions. An equivalent model
analysed using the finite element method software Abaqus. By using the elasto-plastic material model as a user
material to simulate the behaviour of the membrane, the nonlinear behaviour of the material could be grasped
in the simulations.
Different approaches were used and compared in this master thesis. In the end, the simulations using contact
properties in Abaqus/Explicit gave good correlation when the isotropic material model was used. The CPU time
was however large, limiting its practical use. As another approach, the belt is simulated as a belt-like connector,
eliminating the convergence problems encountered with the contact. The simulations showed promising results,
which are discussed thoroughly.
Finally, the influence of the stiffness of the belt is investigated. Also a comparison is made between several
material models.
1.2 The FWO project This master dissertation is part of the FWO project ‘Integrated analysis and experimental verification of Kinematic
Form Active Structures (KFAS) for architectural applications’. It is a collaboration between the department
Mechanics of Materials and Constructions at the Free University of Brussels (VUB) and the department Materials
Science and Engineering at Ghent University. In this project, two case studies are investigated, of which the first
is a case study about the folding of a tension membrane unit about one axis. An integrated nonlinear study is
conducted, involving a numerical form-finding, design of a reference configuration and finite element analysis,
adjustment of the pretension, load analysis, optimisation of the reconfigurable system in different positions, an
experimental set-up and the measurement of the structural behaviour. This will validate the proposed
methodology and elaborate the design and use of KFAS structures.
Usually separate simulations are performed for the reconfigurable frame geometry, the form-finding of the
textile and the analysis of the pretensioned system. Several assumptions are made for the different models,
resulting in several limitations:
The textile behaviour is simplified, although it has a very complex orthotropic behaviour.
The structural analysis is mostly restricted to the evaluation of the final positions of the subsystems,
where the interaction of the different systems is taken into account by transferring reaction forces.
However, since large displacements are involved, there is a need for numerical tools that model the
complex material behaviour, large displacements, deployment, friction, interaction etc. The numerical
models have in fact never been validated.
The intermediate stages and the adjustment of the pretension have never been investigated before.
Three objectives are aimed in this research project. First, a detailed experiment is performed at the VUB to
validate the parameter choices for the numerical simulations, such as element types, material behaviour and
friction. The deployment system needs to be designed as well. Secondly, an advanced nonlinear finite element
model is developed to simulate all deployment stages of the whole KFAS structure. This is performed in this
dissertation. Finally, the stability analysis of KFAS is investigated. A nonlinear finite element study can evaluate
the buckling behaviour of the membrane and the slender supporting structure.
2
Chapter 2: Membrane engineering The goal in the design of membrane structures is to find an equilibrium form for the membrane, free of wrinkling
and with the required performance under the applied load, without neglecting the functional and aesthetic
requirements [1]. Four steps are encountered in the design process: the form-finding, patterning, load analysis
and fabrication process (Figure 2-1). During the form-finding step, the geometry of the textile is determined,
which is then translated into a cutting pattern. The final geometry is then verified in a load analysis. These three
steps influence each other and it is probable that they need to be repeated several times to obtain the desired
structure. Finally, the different steps of the fabrication process are analysed. Each step is described in more detail
in this chapter, but first a description of the membrane material and the general structural behaviour of
membrane structures is given.
Figure 2-1: The different steps in the design process: form-finding, patterning and load-analysis
2.1.1 Materialisation of membranes Since a membrane structure is a lightweight surface structure, it is ideal to cover a large area. Membranes are
very thin, often called technical membranes, which can be technical textiles (coated and uncoated fabrics) and
technical plastics (extruded films) [2].
Technical textiles consist of membrane parts with a large area, joined together, transferring load exclusively in
tension. These textile can be coated or uncoated, where only the first type find practical use in architecture. The
coated textile fabrics are composite materials. On an atomic scale, these textiles consist of manufactured plastics
formed by linking or modification of molecules. Due to their appropriate composition for large spans the
resistance against the effects of load, time and temperature, these are the most common type encountered.
Therefore, these will be discussed further in detail.
Less frequently used are the technical plastics, which are made of fluorothermoplastics in the form of a film. They
are nearly transparent, which explains their increasing popularity amongst architects. However, their strength is
lower when compared to the coated fabrics, so they are only used for smaller spans.
2.1.2 Coated technical textiles Textile fabrics form a grid of woven yarns, oriented orthogonally when unstressed. The yarns consist of single
threads, which are parallel or twisted together. These threads are composed of natural or chemical fibres.
Natural fibres have a round cross section, a finite length and are bound up in strands. They are called spun fibres.
Chemical fibres on the other hand have an infinite length, a free and larger cross sectional shape. These are called
filaments [3].
The raw fabric is coated with a special paste to protect the surface. The most commonly used material
combinations are polyester fabrics with polyvinyl chloride (PVC) coating and glass fibre high-strength fabric with
a polytetrafluoroethylene (PTFE) coating [2], [4].
PVC-coated polyester fabrics
The first type is chosen for its low cost and resistance to damage during fabrication and erection. It has a high
tensile strength and sufficient elasticity. The material undergoes considerable elongation before yielding, so it
makes small corrections during installation possible. Due to its low shear stiffness, only little risk of wrinkling in
the double curvature shapes exists. It is water-resistant over the full lifespan of the structure. Under fire
conditions the material is self-venting, retreating itself from a flame. Disadvantages however, compared to the
Fabrication
3
latter type, are the high creep levels, resulting in a loss of pretension, a lower resistance to dirt build-up and a
shorter lifespan [3].
PFTE-coated glass fibre fabric
The PTFE-coated glass is made out of spun glass threads, who have a certain bending stiffness. It has a high tensile
strength, but it remains brittle and has low elastic strain [3]. Due to its brittleness there is an increasing risk of
folding damage. The tensile strength will decrease when the fabric is subjected to moisture.
The raw fabric is made by weaving threads in two orthogonal directions, called the warp and weft or fill
directions. During the weaving process, the warp threads are tensioned in the weaving machine, making them
initially straight. The fill threads go around them in alternate patterns, by pulling the warp threads up- or
downwards and shuttling a fill thread in between them. This waviness of the threads is also referred to as crimp
[5]. The fill threads show more crimping in the bedded coating. This will result in a different material behaviour
between the two directions. Figure 2-2 shows a microscopic image of the cross-sections in both directions.
Figure 2-2: Microscopic image a PVC-coated polyester fabric: (a) the curved fill threads and (b) straight warp threads [5]
The method of crossing threads will lead to specific crimp of the three-dimensional fabric. Two types are
characteristic in membrane construction: plain weave and basket weave, which is a modified plain weave (Figure
2-3). In the first type the fill threads pass the warp threads alternatively above and underneath. It is the simplest
and closest weave form. The basket weave is a weave operation with more than one thread (two in this case) at
a time. Using multiple yarns results in an improved mechanical behaviour.
Figure 2-3: Two different weave types: the plain weave and the basket weave [2]
2.1.3 The coating The load bearing function is fulfilled by the fabric, whereas the coating protects the fabric from damage and
deterioration. Additionally, it provides waterproofing and shear stiffness. The quality of the fabric decreases over
time due to:
the interaction of alternating loads
creep with load-independent factors such as ageing, climatic and atmospheric effects
a
b
4
The textile membranes need to be resistant against chemical and biological influences and should be non-
flammable. Several measures are taken during the production to obtain sufficient resistance. By applying a
surface treatment, namely the top coating and finishing, water and air tightness can be assured. Important for
the long term mechanical properties is the coating thickness of the thread ridge. The thicker the coating, the
better the protection. The usual thickness varies from 0.08 mm to 0.25 mm.
PVC-coated polyester fabrics are coated with PVC-P (Polyvinylchloride-Plastisol with plasticizer and additives).
These coatings reduce the escape of plasticizer from the PVC-coating, avoiding that the PVC-coating becomes
brittle. The surface treatment also hinders the formation of sedimentation and microbes, resulting in undesired
optical properties.
At last, a surface treatment of the coating is applied, known as the top coating or finishing. This provides an
additional protection against dirt or moisture and delays the loss of plasticizer from the coating. Two processes
are available: the lamination of films and paints.
2.1.4 Mechanical properties Due to the weaving and coating process, the fabrics behaves differently from conventional building materials
such as concrete, steel, wood or masonry. They are nonlinear, anisotropic and inelastic materials [3]. The shear
stiffness requires special attention as well.
2.1.4.1 Nonlinear behaviour Unlike many traditional materials, a nonlinear relation exists between the applied load and the resulting strains
in an uniaxial tension test (Figure 2-4). The increase in deformation is not proportional to the increase in loading.
Figure 2-4: Typical nonlinear behaviour stress-strain-diagram [3]
2.1.4.2 Anisotropy When several strips aligned in warp and fill direction are tested in a uniaxial tension test, stress-strain diagrams
such as in Figure 2-5 are typically obtained. This is due to the weaving process and coating. Tensioning the initially
straight warp threads will lead to only little deformation. However if the fill threads are loaded, these will
straighten first, resulting in a much larger initial deformation and a less stiff behaviour.
Figure 2-5: The stress-strain curves in the uniaxial tensile test for a continuously increased load [5]
F
F
F
F
5
The yarns in these main directions also interact with each other at the crossing points. Straining the weaker fill
threads results in stretching this thread, so the fabric elongates in fill direction. However, the straight warp
threads will now curve, resulting in a shortening of the fabric in warp direction. This results in negative strain
(Figure 2-6). If the warp direction is also loaded in tension, there is resistance when the fill direction is stretched.
The crimp in warp and fill directions are thus complementary, called crimp interchange [5]. This interaction will
lead to a complex, nonlinear biaxial stress-strain behaviour, as described above. After applying the first loading
cycle, this crimp interchange will lead to a permanent strain called the constructional strain [5]. The alternation
of the crimp depends on the applied load, the load ratio between warp and fill and the extension [2], leading to
a highly variable stiffness and Poisson’s ratio. For isotropic, homogeneous solids, this Poisson’s ratio cannot be
higher than 0.5, however, for tensile membranes higher values are used to model the large negative strains [6].
Figure 2-6: Crimp interchange [6]
2.1.4.3 Inelastic behaviour The inelastic property or hysteresis can be examined by applying a cyclic loading on the fabric. Figure 2-7 shows
that the unloading curve is different from the loading curve. When this loading cycle is repeated, all loading and
unloading curves are different. This indicates that the textile undergoes permanent strains, depending on the
loading history. The extent, speed and duration of the loading as well as the number of cycles influence the fabric
deformation [2]. Increasing the duration and the extent will increase the resistance against deformation.
Figure 2-7: The stress-strain curves in the uniaxial tensile test for a cyclic load [5]
2.1.4.4 Shear stiffness For PVC-coated polyester fabric, 1/20 of the tensile stiffness can be taken as a good estimation of the shear
stiffness [7]. Due to this low stiffness, a detailed analysis of the shear behaviour is often neglected [4]. For small
shear angles, the shear stiffness is low, since the yarns are free to rotate. The coating and inter-yarn friction
provide the resistance. However, at a critical interlocking angle, the yarns have side-by-side contact, resulting in
a sharp increase in shear stiffness. The weave type and applied coating determine the extent of the angle of
rotation [2]. Typically, PTFE-coated glass fibre fabrics have a tighter weave and a higher crimp level, resulting in
a lower lock-up angle than PVC-polyester. Research aims to quantify the relation of this lock-up angle in function
F
F
6
of warp and fill stresses, and for different types of fabrics. Shear testing nowadays is provided by the uniaxial
extension of bias cut strips (cut at 45° to the yarn directions). This test is explained in 2.1.4.5(a).
2.1.4.5 Material model The different interactions described above result in a very complex material behaviour. Although the yarns are
oriented orthogonally, the elastic moduli, shear stiffness and Poisson’s ratios are independent and do not satisfy
the laws for orthotropic materials. A full quantification of the response of these fabrics to biaxial and shear
loading is however time consuming and expensive and has in fact not yet been achieved [8]. Nevertheless, even
if detailed biaxial testing would be carried out, the techniques to interpret this data are limited.
The most frequently applied standardised method is described in the Commentary to the Membrane Structures
Association of Japan (MSAJ) Testing Method for Elastic Constants of Membrane Materials [7]. In this document,
the method for defining mechanical constants from biaxial and uniaxial tensile tests are described for plain weave
membranes. Since every membrane analysis software is based on elastic parameters, an anisotropic elastic
material model for the membrane is assumed. This allows a simplified quantification of the mechanical constants
from a biaxial tensile test. However, there is a need for a more sophisticated material model which can grasp the
complex material and requires not too much computational time during numerical simulations. This has been
developed at Ghent University. Frequently encountered material models are described hereunder.
ANISOTROPIC ELASTIC MATERIAL MODEL
Due to the weaving process, the mechanical properties differ with changing alignment in orthogonal directions,
which is typical for anisotropic materials. However, due to material nonlinearities such as the crimp interchange,
the material behaviour cannot be described in all its aspects by this material model. An elastic material model is
however still preferred, since calculation methods are based on the elastic material theory. The membrane is
thus often assumed to have an orthogonal anisotropic elastic material model.
In fabric research, the uniaxial and biaxial tensile test are frequently carried out. These tests were also conducted
at the VUB for the PVC-coated fabric former to the 3D deployment experiment. The results of these test will be
used to illustrate the different testing techniques.
(I) UNIAXIAL TENSILE TEST
In a uniaxial tensile test, rectangular shaped specimens are tested, some aligned in warp direction and others in
fill direction. Two types of loading are applied: once with a load that increases to 25% of the ultimate tensile
strength of the fabric (Figure 2-5) and once with a cyclic loading (Figure 2-7). The results show a different
behaviour for the warp and fill direction as expected. Also the nonlinear behaviour shows under a cyclic loading.
The tensile stiffness is obtained from the load and strain in the tensile direction. The Poisson’s ratio is obtained
from strain in the load direction and shrinkage strain perpendicular to it. However, since the crimp interchange
has not been tested, the Poisson’s ratio cannot be accurately determined. A biaxial tensile test solves this
problem.
(II) BIAXIAL TENSILE TEST
During a biaxial tensile test, a cruciform specimen is loaded with different load ratios in warp and fill direction
(Figure 2-8). In this way, the interaction between the orthogonal yarns can be quantified. By applying different
load ratios between warp and weft directions, the initial and stabilised behaviour at limited load levels can be
investigated. The load in warp and fill direction are measured at the arms of the specimen, whereas a Digital
Image Correlation (DIC) allows to measure the strains in both directions in the centre of the specimen. The
obtained stress-strain curves from the VUB experiments are shown in Figure 2-9. It can be concluded that
different slopes are obtained for different load ratios and that passing on to the next load ratio introduces an
additional permanent strain.
7
Figure 2-8: A biaxial tensile test as performed at the VUB [5]
Figure 2-9: Stress-strain curves from the biaxial tensile test for successive load ratios 1:1, 2:1, 1:2, 1:0 and 0:1 [5]
(III) UNIAXIAL BIAS EXTENSION TEST
The stress development at the centre of the specimen is affected by the shear stiffness in a biaxial tensile test.
The development ratio for the load increases with decreasing shear stiffness [7]. The in-plane shear stiffness is
tested at the VUB with a uniaxial bias extension test. Again a rectangular specimen is tested, but it is now cut at
45° from the warp and weft direction. Several loading and unloading cycles were applied. Figure 2-10 shows the
stress-strain curve obtained from the test. The same general trend is obtained as for the uniaxial test in warp
and fill direction, however, the permanent strains are higher and the low stiffness zone is larger.
Figure 2-10: Stress-strain curve from the uniaxial bias extension test [5]
In this testing arrangement, it is difficult to quantify the ratio of the direct stresses and shear stresses. Therefore,
further research on testing methods is necessary [4].
(IV) DETERMINING THE MECHANICAL CONSTANTS
Research showed that for orthotropic textiles under in-plane loading, one single biaxial tensile test is sufficient
to determine the elastic properties of the fabric and estimating the material parameters of the constitutive model
F
F
8
[9]. The stress-strain relationship in orthotropic materials under in-plane loading can be described for low stress
ranges as:
[
𝜖11
𝜖22
2𝜖12 ] =
[
1
𝐸11
−𝜈21
𝐸22
0
−𝜈12
𝐸11
1
𝐸22
0
0 01
𝐺21]
[
𝜎11
𝜎22
𝜎12 ]
where 𝐸11 and 𝐸22 are the elastic moduli along the material symmetry directions, 𝜈12 and 𝜈21 are the Poisson’s
ratios and 𝐺12 is the shear modulus. The stiffness matrix should be symmetric, resulting in the following
reciprocal relationship:
𝜈12
𝐸11
=𝜈21
𝐸22
These equations can be rewritten in function of the measured parameters for low stress ranges during the biaxial
tensile test, namely the Poisson’s ratios and the load in both directions 𝑁11 and 𝑁22:
𝑁11 =𝐸11𝑡
1 − 𝜈11𝜈22
(𝜖11 + 𝜈22𝜖22)
𝑁22 =𝐸22𝑡
1 − 𝜈11𝜈22
(𝜖11𝜈11 + 𝜖22)
Four values are unknown: the tensile stiffness 𝐸11𝑡 and 𝐸22𝑡 and the Poisson’s ratios. Due to the reciprocal
relation, in fact only three constants are independent. All the unknowns can thus be determined from a single
biaxial tensile test. This is the main reason for using this material model, together with its simplicity.
If an uniaxial test is conducted, the unknown Poisson’s ratios can be determined from the obtained tensile
stiffness in warp and fill directions [7]. Two methods are possible: the ratios of tensile stiffness can be set to
known values from the uniaxial test or further biaxial tensile testing can be performed for different load ratios.
An approximate solution can be obtained by for example the least-squares method. If an uniaxial bias extension
test is combined with the uniaxial test, all parameters can be directly derived from the equations above.
The simplification in the material model has several consequences. In elastic materials, the Poisson’s ratio cannot
exceed 0.5. For a membrane material it can even be larger than 1, since the extension and shrinkage are
associated with the crimp interchange (see 2.1.4.2). Therefore, an “apparent” Poisson’s ratio is determined.
Additionally, the reciprocal relation is not often satisfied in the experiment although it is a very important aspect
of the material model.
These issues are taken into account in the calculation process by several measures. The initial slack is excluded
from the experimental data, so the mechanical constants are based on the stabilised behaviour [5]. Nevertheless,
even for load values close to zero, the effects of the viscoelasticity of the coating resin and the slack between the
orthogonal yarns are large. It is thus difficult to obtain accurate elastic constants. Therefore, the measured load-
strain curves are replaced with curves that satisfy the reciprocal relationship by using the least-squares method
or best approximation method in terms of load or strain [7]. The most adequate combination of elastic constants
is retained.
AN ELASTO-PLASTIC MATERIAL MODEL FROM GHENT UNIVERSITY
The anisotropic elastic material model does not at all describe the real material of the membrane. Therefore, a
new material model is proposed for PVC-coated polyester fabrics [5]. Uniaxial and biaxial tests were conducted
to validate the proposed material model.
9
Figure 2-11: The proposed material model in (a) warp and (b) fill direction [5]
The linear hardening elasto-plastic models were separately proposed for warp and fill directions (Figure 2-11). In
warp direction, stress and strain are linearly correlated (𝐸𝑤1) as long as the stress is lower than the yield stress.
Once this stress is reached, the linear hardening occurs with the plastic modulus 𝐻𝑤. If the textile is unloaded,
again the relation is linear, however with a different Young’s modulus 𝐸𝑤3. This fill direction shows similar
behaviour, however, the relation is nonlinear in the elastic region. In the plastic behaviour, there is a piecewise
linear hardening with two different plastic moduli 𝐻𝑓1 and 𝐻𝑓2. For unloading, a linear relation is assumed (𝐸𝑓4).
The material parameters mentioned until now can be determined by uniaxial test. For the membrane used in the
experiment, following values apply for the warp direction:
𝐸𝑤1 = 1364 𝑀𝑃𝑎 if 𝜎𝑤 < 15 MPa
𝐸𝑤2 if 𝜎𝑤 < 15 MPa and Δ𝜎𝑤 > 0
𝐸𝑤3 = 1130 𝑀𝑃𝑎 otherwise
For the fill direction:
𝐸𝑓1 = 104 (14.1309(𝜖𝑓)2+ 0.04𝜖𝑓 + 0.0115) 𝑀𝑃𝑎 if 𝜎𝑤 < 5 MPa
𝐸𝑓2 if 5 MPa ≤ 𝜎𝑓 < 15 MPa and Δ𝜎𝑓 ≥ 0
𝐸𝑓3 if 𝜎𝑓 ≥ 15 MPa and Δ𝜎𝑓 ≥ 0
𝐸𝑓4 = 825 𝑀𝑃𝑎 otherwise
𝐻𝑓1 = 593 𝑀𝑃𝑎 if 5 MPa ≤ 𝜎𝑓 < 15 MPa
𝐻𝑓2 = 236.82 𝑀𝑃𝑎 if 𝜎𝑓 ≥ 15 MPa
The interaction between warp and fill threads is characterised by two Poisson’s ratios 𝜈12 = 0.09076 and 𝜈21 =
0.46923. The load ratio dependency is captured by introducing 𝐸𝑤4 = 852.241 𝑀𝑃𝑎 and 𝐸𝑓5 = 618.402 𝑀𝑃𝑎
when the next load ratio is applied onto the fabric. Biaxial tests allow the determination of these parameters.
This model can be implemented in the ABAQUS software as a user material subroutine (UMAT) by using the
implicit return mapping method.
The proposed material model can capture the nonlinearities in both warp and fill directions as well as the
orthotropic effect, the permanent strains and the load ratio dependence. However, the model should be
additionally validated for different load ratios. This requires additional biaxial tensile experiments. Also nonlinear
shear responses and time dependent behaviour are not considered. It still needs validation for other types of
fabrics as well.
10
2.2 Structural behaviour A membrane fabric is very thin and flexible. It has in-plane stiffness but only low shear and no flexural stiffness
or compression stiffness. The out-of-plane stiffness is achieved by prestress and geometry [10]. If the external
loads eliminate the prestress in the membrane, wrinkling will occur. This is an important material nonlinearity
which should be taken into account during the design and analysis of structures.
The prestress is mechanically introduced by stretching the membrane using boundary elements and prestressed
cables attached to the end and inside the surface. Due to the fabric’s creep behaviour, of which there is little
knowledge so far, the conservation of this prestress over the lifetime of a structure is a major issue. After some
time, slack areas are formed in the structure, which can lead to ponding of rain water. This overloads the
structure and excessive movement (flapping) of the under-stressed fabric can occur, with noise and fatigue
damage in the structure and edge details as a consequence [4]. This issue may be overcome by introducing re-
tensioning capabilities in the support details. Nonetheless, tests still need to be carried out to investigate strain
failure of fabrics and the reduction in failure stress over the lifespan of the structure.
One important aspect of building with flexible membranes is the deformation behaviour of the materials used.
Structures which are loaded in tension, transfer the loads by axial forces, resulting in a more efficient material
use and lightweight structures. In the load-bearing elements, yielding under load to some extent is very
advantageous. If the deformations are acceptable, a successful structure is designed. When deformations occur
in flexible elements, peaks of stress are relaxed through the material. A larger deformability of the textile results
thus in ductile behaviour, lowering the overall deformations. This contrast with traditional materials is due to the
elastic behaviour of the synthetic plastics. The polymer’s chemical composition and physical structure result in
elastic and viscous-plastic deformations. Therefore, the mechanical properties depend on loading, temperature,
time, duration of loading and the speed of loading [2].
2.3 Form-finding Unlike other structures, the shape of the membrane cannot be prescribed. It must be determined based on the
supporting structure. This design process is called shape-finding or form-finding. This design problem is also
called ‘the initial equilibrium problem’ to account for the fact that in many cases the stresses are also unknown
[11]. The final form of the membrane has to be visually appealing while having a sufficiently good engineering
performance and functional suitability [1]. These functional requirements can be sufficient head room or
avoiding ponds under snow load. Intensive cooperation of the architect and the engineer from this very first step
is thus indispensable for obtaining a good design.
2.3.1 Shape of the membrane The shape of a membrane is determined by:
the geometry of the edges
the cable layout
the prestress distribution over the surface
the ratio of that prestress to that in the cables
The boundary geometry can be a fixed connection, like a solid structure, or flexible but relatively stiff, like a cable.
When cables are used along the edges, the geometry of the cable depends on the prestress of the membrane as
well. Thus also the layout of the cables and the prestress within will have an influence on the shape. The surface
shape is defined rather by the ratio of the warp and fill stress, than the absolute values.
From a geometrical point of view, the structure needs to be designed with sufficient curvature so that the applied
loads are resisted by tension forces in the plane of the membrane [4]. The type of surface is determined by the
Gaussian curvature 𝜅 [12]:
𝜅 =1
𝑟1∙
1
𝑟2
11
where 𝑟1 and 𝑟2 are the principal radii of curvature. If 𝜅 is positive, the radii of curvature are on the same side of
the surface, resulting in a synclastic surface. If it is negative, the radii are on the other side and the shape is called
anticlastic. A surface with a single curvature leads to a zero Gaussian curvature (one radii is infinite). To resist
both upward and downward forces, the surface of the structure must be anticlastic since the membrane has only
tensile strength. Often used shapes are the conic and saddle shapes (Figure 2-12).
Figure 2-12: Basic anticlastic surfaces: the conic, barrel vault and hypar shape [4]
Ideally, a uniform stress distribution is reached within the membrane surface. This is obtained by a minimal
surface form. This surface joins the boundary points with the smallest possible membrane area and has uniform
in-plane tensile stresses in both warp and fill directions [8]. For early tensile architecture, this form was obtained
by making models on soap bubbles (Figure 2-13). The curvature of the surface can be synclastic or anticlastic.
Once such a surface is loaded, it will have to get larger to redistribute the impact to the boundaries. In this way,
a local concentration of reaction force and damage to the lightweight structure can be avoided [13].
However, true minimal surfaces cannot be formed between all boundary conditions [4]. In that case, a pseudo-
minimal surface can be developed by increasing stresses in the region where the soap bubble would have failed.
This allows more freedom in shape, however the structure becomes less efficient due to increased stress
variations.
Figure 2-13: Soap bubble experiment by Frei Otto to find the minimal surface [14]
2.3.2 Warp and fill directions The long-term behaviour of the material requires a uniform and smoothly varying stress in warp and fill direction,
so that wrinkling or local stress concentrations are minimised. Ideally, the warp and fill directions coincide with
the principal curvature of the surface. This will optimise the stiffness, reduce the overall deflections and allow
easier initial stressing of the membrane [1]. The practical choice for the warp and fill direction depends upon the
strength, installation strategy and fabrication economics. In a complex model, it is possible that the directions
vary over the surface.
2.3.3 Edge geometry During form-finding, the edge details and forces on the edge element will determine the geometry of the surface
[2]. To stabilise surfaces in the final form, forces are introduced through linear bearing elements at the edges.
The geometry of the edges determines the bearing and deformation behaviour of the membrane. The edge can
be stiff or flexible. The differences lie within the method of load transfer:
A flexible edge carries the tensile forces as tension. These elements will resist the deformation of the
membrane under external loading as deformation. The edge element relaxes, resulting in a relocation
of the supports. This again results in a deformation of the edge element and reduces the maximum
stress in the membrane. This will thus reduce the tension forces in the edge element and support, which
results in reduction of material.
12
In a stiff edge, the tensile forces are taken by compression and bending moments. The peaks of stress
cannot be relaxed, resulting in a heavier edge element and a higher strength requirement for the
membrane. However, by increasing the geometrical stiffness of the membrane, higher forces are
introduced in the membrane, which will alter the surface curvature.
The forces in the flexible edge element, the sag of the edge element and the membrane stress will influence the
surface geometry [2]. By increasing the membrane stress, the edge force is increased as well. If the sag is
increased, the total membrane surface area is reduced so the curvature of the membrane is increased. This
results again in increased forces in the edge elements.
2.3.4 The numerical implementation The form-finding analysis methodologies are beyond the normal scope of finite element codes. It requires
nonlinear finite element analysis. The introduction of the computer has led to the development of several
methods. Discrete cable-net structures were earlier applied and later on extended to surface elements for
membrane structures. The methods can be divided into three large categories [11]:
Stiffness matrix methods are the oldest methods, using standard elastic and geometric stiffness
matrices and including material properties. This category is the least harmonised, with no agreement
on name or principal sources. Disadvantages may be the unnecessary high computational cost due to
the inclusion of material properties.
Geometric stiffness methods have only a geometric stiffness and are based on the force density method.
Later extensions prescribe forces rather than force densities. These methods however often produce
unrealistic solutions, limiting their use during the preliminary design. Additional iterations are
necessary.
The dynamic equilibrium methods solve the dynamic equilibrium to obtain a steady-state solution.
Downsides are the requirement of many parameters such as time step, control stability and
convergence.
The initial geometry of the form-finding analysis can be quite arbitrary, although convenient initial stress is
required in order to guarantee a positive defined stiffness matrix. When the surface has no double curvature for
example, no feasible equilibrium shape can be determined. Initial conditions are assigned to the membrane
elements: the warp and fill prestresses, the prestresss of each cable and the layout of the cables. Also the
boundary conditions are specified. From this, the equilibrium shape is calculated. The amount of contribution of
an element to neighbouring node forces are function of the specified prestress and geometry. During the
iterative search for an equilibrium form, these are continually updated as the geometry changes. Veenendaal
[11] mentions two common formulations for nonlinear, large displacement analysis: the total Lagrangian
formulation and the updated Lagrangian formulation. Based on these formulations, the forces and stiffnesses at
each increment are calculated. There is also a mixed formulation of both, called homotopy mapping. If it is
impossible to find equilibrium with the defined element stresses, no stable solution can be found.
Some nodes might be fixed due to boundary conditions. During form-finding, the remaining node positions are
iteratively adjusted until equilibrium with the internal forces is reached. The final solution is thus material
independent and finds a balance between the internal stresses and the node geometry.
2.4 Patterning After the proper equilibrium shape has been determined, it is divided into 2D strips by projecting the shape onto
a plane. These strips can then be cut from a material roll that is generally up to about 5 m wide. Several strips
are connected according geometrical and structural design requirements. The main influencing factors are the
form, bearing behaviour and erection process.
2.4.1 Development The processing of 2D cutting patterns from the 3D form is called development. This development fixes the layout
of the strips on the roll. The different material axes have to be properly aligned. There are two manners of
13
projecting the geometry on a plane: the true-length projection for woven sheets (Figure 2-14) and the true-angle
projection for foils [2].
When shear deformation is applied on fabrics, the surface can be projected onto a plane with correct lengths.
Since woven fabrics have a low shear stiffness, this deformation, that moves one edge parallel to the opposite
edge, should induce no, or low, stresses in the material. Tests and calculations are performed to determine the
relationship between the force and shear deformation and evaluate the influence of the deformation between
warp and fill threads. The shear modulus is the most important parameter for the production of fabrics.
Figure 2-14: True length projection [2]
2.4.2 Compensation Geometrical surfaces can only be mathematically developed into two-dimensional surfaces when the Gaussian
curvature at every point is zero [2]. However, curved tensile surfaces that found their own form, always have a
negative Gaussian curvature so they cannot be developed. In order to produce these pieces, the different lengths
which result from mathematical-geometrical constraints should be compensated through a patterning
calculation based on material properties that can predict behaviour under deformation.
The crimp interchange is dominant in the first load cycle: it leads to an initial slack. In practice, this phenomenon
is not included in the material model and is therefore compensated in the patterning stage. Additionally, the
creep behaviour of the fabric results in permanent strains over the lifespan of the structure. The prestress will
disappear and the structural requirements will no longer be fulfilled. Therefore, the geometrical dimensions are
corrected. This is called compensation. When negative strains would occur, material should be added, called
decompensation. However, this method of compensation is not well documented [5].
Since the stretch properties are different for warp and fill directions, the deformations are different as well. In
fact, the stiffer warp direction will be compensated less than the fill direction. To determine the compensation
values, data about the creep strain and the reduction of prestress over the lifespan is important. This can be
retracted from biaxial tensile tests.
The aim of compensation is that after assembling the different strips, the intended geometry is obtained.
Therefore, also the curvature of the membrane is important: a synclastic surface will result in convex edges,
whereas a anticlastic surface results in concave edges (Figure 2-15).
F
α α − γ
𝛾
14
Figure 2-15: Compensation depends on the curvature of the surface: synclastic (top) and anticlastic (bottom) [2]
2.4.3 Patterning calculation criteria Different criteria influence the patterning [2]: the optical, topological, practical erection and processing criteria.
The optical criteria treat the appearance of the membrane. The arrangement of the seams and edge appearance
will influence the overall appearance tremendously. The number of seams should be as low as possible. The other
criteria are briefly discussed in the following paragraphs.
2.4.3.1 Typological criteria The major influencing parameter is obviously the shape of the surface. First, the warp and fill directions need to
be determined relative to the main direction of the curvature. From the stresses and resulting strains, the final
strip layout can be determined. The strips are carried out in the main loading direction, which is the direction of
the greatest strain. Since the warp direction is the stiffest direction, it is mostly aligned to this direction so that
the deformations are reduced. This will result in an optimal levelling of the surface stresses through
compensation. Another advantage is that the weaker seams do not bear the highest possible stress in transverse
direction. Since the deformation is also reduced when a stiffer material is used, the material will have an
important role as well. Other important parameters are the surface curvature and the edge shape.
2.4.3.2 Erection criteria During the erection process, the membrane will be tensioned until its final pretensioned state. During this
process the mechanical effects, the time scale, temperature, space requirement and sequence of tensioning play
an important role. A good choice of material, cutting patterns, surface shape and edging type will result in an
economical tensioning scheme. The cost of the tensioning procedure is determined mainly by the arrangement
of the individual strips. All erection measures like the tensioning equipment, stabilising measures for the primary
structure and the scaffolding depends upon the orientation of the strips.
2.4.3.3 Processing criteria In order to be efficient, the cutting out waste should be as low as possible. Therefore the width of the material
roll is an essential parameter in the cutting-out calculation. This is more problematic for smaller than larger
membrane areas. The length distortion by projecting the latter is relatively low for each individual strip. Other
processing criteria are the cutting and connection equipment, the edge detail and the preparation for transport.
2.5 Load analysis After finding the equilibrium shape, the material properties are introduced in the model and the behaviour of
the structure is analysed under snow and wind loads. To evaluate the structure’s behaviour properly, a nonlinear
analysis is necessary to account for the geometrical, material and boundary nonlinearity. The stiffness equations
are solved iteratively, where the element stiffnesses are updated when the geometry is changed. Additional on-
off nonlinearities such as wrinkling and cable slacking effects need to be taken into account as well. Numerically,
an updated Lagrangian formulation is used in the load analysis since it includes large displacements of the
membrane. It is preferred over a total Lagrangian formulation due to the existence of initial stress [1]. It is also
important to include the wrinkling effect.
15
2.5.1 The significance of material properties The three typical fabric forms, the barrel vault, hypar and conical shape, have been analysed in order to establish
rules of thumb for the safe and efficient design of tensile fabric structures [8]. Additionally, the sensibility to
material properties was investigated. The barrel vault proved the value of highly curved surfaces: they are
efficient, robust and result in low stresses and deflections. Additionally, there is a minimal variation for a wide
range of material stiffness values and the larger the curvature of the edge cables, the lower the cable force. This
relation is highly nonlinear. If the dip to span ratio is smaller than 0.1, there is a large increase in the cable force,
resulting in larger cable cross sections and heavier connections.
The hypar structure on the other hand did not obtain straightforward results. The patterning direction proved
critical for the membrane behaviour and reaction forces. Extremely high curvature should be avoided since the
hypar displacements are then very sensitive to changes in the tensile stiffness. So, in order to avoid ponding, a
large range of material properties should be investigated. The stress values showed the same pattern as for the
barrel vault: low curvature means that there will be high stress levels. If the shape is highly curved with an
orthogonal patterning then the stress will be very sensitive to variation in shear stiffness.
Finally, for the conic structure, only a limited range of forms can be achieved. Within the range of efficient shapes,
there is only modest dependence on the material properties. If the stiffness increases, the stress increases. If
asymmetric prestress ratios are applied, more forms are feasible, although the efficiency will decrease since the
form differentiates more from the minimal surface.
From these observations, it can be concluded that approximating the material parameters in the design stage
can have unwanted consequences. For most projects, the only biaxial fabric testing carried out is under prestress
loading to determine compensation. This is generally insufficient for the structural design since the
environmental loads on the membrane will be much higher than the prestress. Therefore, equivalent linear
elastic material properties are used for analysis and design. A more detailed discussion about the used material
model can be found in Section 2.1.4.5.
2.5.2 Failure modes The critical failure modes important for membranes are [3]:
Failure of the membrane within the design lifetime of the structure (Figure 2-16)
Failure of a seam or connection to the supporting structure
Tear failure during installation or because of vandalism
The safety factors used upon the ultimate strength of the membrane material will influence the first failure mode.
These are discussed in the next section. The second failure mode is determined by the width of the seam. By
testing it at the appropriate temperature, this mode can be avoided. When raising the temperature, the strength
of the seam gets considerably lower. Tear propagation often occurs during the installation, starting at the edge
or a hole in the membrane. Therefore, it is critical that the fabric panels are continuously contained with a high
precision at all edges.
Figure 2-16: Membrane failure of the Metrodome roof in Minnesota under extreme snow load (2010) [15]
16
2.5.3 Design guidelines Up to now, there is no standard or European code for the design of fabric structures. Only limited design guidance
is provided. Therefore, the design of membrane structures is based on experience engineering judgement and
pragmatism [8]. Recently, CEN Technical Committee 250 Working Group 5 has started a draft for Eurocode 10 for
membrane structures. It will include guidance on analysis methodologies, reflecting the current practice.
TensiNet Analysis and Materials Working Group is an European collaborative group that is aiming to produce a
draft design guide that will allow good practise and recommendations. For structures with strong geometric
nonlinearity a limit state approach is not feasible since the geometry of the structure depends on the magnitude
and distribution of the loading. Therefore, the two main performance criteria are stress and deflection [8].
In standard practice, a permissible stress approach is used rather than the conventional limit state approach with
safety factors for loading and material strengths. In structural fabrics, ‘stress’ is defined as force per unit width,
since the thickness is not consistent over the membrane area. The maximum stresses will mostly occur in the
warp and fill directions, since the fabric has only low shear stiffness [8]. The maximum design strength is
compared to the fabric’s strip ultimate tensile strength, divided with a composite safety factor.
For fabric membranes, these safety factors are high. Values between 5 to 10 are used for the fabric, whereas
lower factors (2.3 to 3) are applied to the structure elements. Several factors such as material variation, accuracy
of material properties, load variation, environmental degradation, loading uncertainty and tear propagation
result in such a high safety. Tear propagation is the governing factor. The design is elaborated in such a way that
tears are not propagated under the applied load. A maximum tear length of 40 mm is often chosen, which is the
length that is visible from a distance of 15 m. It could be stated that this is somewhat an arbitrary value. Smaller
tears may not be visible, so the fabric must not fail under this probably undetected damage. The final choice of
safety factor is however based on experience and rules of thumb, due to lacking detailed information. More
knowledge about the fabric behaviour would enable to reduce these factors, however, the commonly accepted
minimum safety factor is 4 [4]. The large magnitude of the stress factors can give the designer the false comfort
that there is a large margin of safety, covering other uncertainties that have not been investigated as well.
In contrast to conventional building structures, strict deflection limits do not exist but follow from the need to
avoid ponding or clashes between the fabric and other objects or the supporting structure [8]. Ponding is avoided
by assuring positive drainage. Due to the geometric nonlinearity, no factors are applied to the loads in this
analysis, excluding any safety from the calculation. Additionally, inversion of the fabric should be avoided, since
this leads to flapping and fatigue damage.
This design approach results in highly conservative structures. It is aimed to obtain a reliability based analyses,
which allows a more comprehensive analysis of the stresses in the membrane. By combining results from tear
tests with data from the influence of the fabric on biaxial tests, structural tolerances and probabilities of load
intensity and duration from loading codes, a probability of failure can determined for a structure. This can then
be translated into a safety factor, correlated to a given probability of failure (usually 5%), as in the current practise
for steel and concrete structural design. The probability of failure is then based on both the stress levels and the
loaded membrane area. For example, the probability of failure would be the same if there is one small but very
highly stressed area or if there is a large area with a moderate stress level. This will result in more accurate
loading information for the supporting structure.
That harmonisation of the current analysis methods is necessary was concluded from a round robin exercise [6].
Several form-finding and load analysis techniques were compared, leading to the understanding that further
harmonisation will provide a full picture of the variability inherent in the current design practice.
2.6 The fabrication process During fabrication, strips are prepared from roll material and joint together by bonding or a mechanical
connection. The fabricator makes permanent surface joints, such as welded seams, combination seams, sewn
seams and glued seams. During construction, reusable or temporary joints are made as well, such as clamping
plates [2].
17
The forces out of the load bearing threads in the fabric are transferred to the threads of the adjacent fabric
through seams. Due to differences between the fabric and seam stiffness, irregularities can occur in the overall
shape of the membrane. For the design of the joint, the different yarn directions are very important due to the
anisotropic behaviour. Also the adhesion strength of the coating, the seam widths and seaming or welding
process are important.
Welded seams are the most commonly used type. These can be executed by overlap or with a butt joint and a
cover strip. The strength of the seam depends on the welding process and temperature. The forces are
transferred through shear loading of the coating. Since plastics do not conduct heating, thin fabrics are easier to
weld. Due to this heating, there is shrinkage of the welded seam in the longitudinal direction, called thermal
crumple. This should be compensated by pretensioning the fabric or by compensation.
18
Chapter 3: Kinetic architecture Fabric structures come in two forms: fabric tension structures and air-supported fabric structures. In the first
type, the membrane is form-active, whereas in the latter, the form-active membrane is carried by overpressure.
Fabric tension structures are suited for covering large spaces due to their lightweight. Other advantages are the
low maintenance cost, no need for airtightness and the choice from various shapes with doubly-curved surfaces
compared to air-supported structures [10]. This master dissertation focusses on fabric tension structures, used
in kinetic structures. In this chapter, a brief overview is given of the history of membrane structures and kinetic
architecture. Different types of kinetic architecture are investigated and finally, kinetic form active structures are
discussed.
1.1 The evolution of kinetic architecture The evolution of kinetic architecture is not at all straightforward, but is a result of complex socio-cultural
circumstances, technological developments, changes in physical environment and human needs [16]. Historical
precedents are the nomadic tents, such as North American tepees. These structures were not built to remain
standing, but needed to be transportable. With the onset of sedentism, architecture became more permanent.
From that point on, kinetic structures were installed within these buildings more as a secondary, common
feature. In Roman times for example, antiquity atria and spectators seats in amphitheatres were covered with
retractable awnings, called velum, to protect spectators from the sun (Figure 3-1). With the rising of the industrial
revolution in the 19th century, other kinetic devices such as lifts, rolling blinds or foldable canopies became very
popular.
Figure 3-1: A retractable roof over the Colosseum in Roman times [17]
The small alleys and bazaars in Andalusia, Arabia and North-Africa are covered with shadings made out of cloth,
for protection against solar radiation. As a portable mean, a parasol can be used, or an umbrella when it’s raining.
These portable membrane structures were already used at 500 BC. Tents, awnings and umbrellas have
similarities in their working mechanism: it is a combination of stabilising network-like structures with a textile for
covering.
In the eighteenth century, tents as garden buildings became very popular. For the design of the Versailles Palace
long rectangular and round tents were designed. From the mid-nineteenth century illustrations were found from
tents from festivities in England [1].
In the 1920s-30s suspension bridges were successfully built, mostly in the USA and France. A famous example is
the Golden Gate bridge in San Francisco, built in 1933. These structures created a technical challenge, inspiring
the tensile architecture. The combination of concrete structures and tensile architecture let to a high structural
efficiency. In the 1950s two kinds of tensile architecture were developed: the engineer’s tension, which was
technically motivated, and the architect’s tension, which was an exploitation of novelty of new tension forms.
This led to a first wave of tensile buildings in that time [18]. However, in these years the technology did not yet
match the architecture’s needs.
In 1954 Frei Otto published “Das Hängende Dach” (The hanging roof) as a first documentation of his tensile
structure investigations. In the same year, he founded the institute of Lightweight Structures at the University of
Stuttgart. He is considered a pioneer in membrane structures due to its systemic research. The tent was used as
19
a prototype of modern architecture where the tensile form and technology were developed concurrently. Otto
relied on models to examine the tensile behaviour, not on complex mathematical analysis.
The German Pavilion at the World Fair Montreal in 1967 is considered as the first modern tent, where form and
construction means were properly matched (Figure 3-2). The Pavilion consists of a tent-like roof, where a net of
cables rests on 7 supports of different heights. Its low volume and light-weight structure results in a very
economical structure. A full scale mock-up building in Germany and comprehensive studies proceeded the
structure. From test model measurements and wind tunnel tests the pattern of the net and membrane was
determined. The structures from Frei Otto and his team showed the architectural appeal and possibilities of
these lightweight structures.
Figure 3-2: The German Pavilion at Expo ‘67 [1]
In the sixties and seventies, membrane structures became landmarks in urban planning all over the world. Later
on, especially World Expo fairs like Vancouver (1986), Seville (1992) and Hanover (2000) provided technological
developments which made new uses for these structures possible. A lot of these structures were kept
permanently. The modern membrane buildings are ideal for sport arenas: a large area needs to be covered
without intermediate supports. The standard solution became a shell steel structure, covered with a translucent
membrane. An example is the ice skating hall on the Olympia site in Munich (Achermann and Schlaich, 1983). It
is a prestressed cable net construction, covered with a PVC-coated polyester membrane. A curved three-chord
steel beam supports the roof. The efficient spoked-wheel principle is applied, which is briefly explained in Section
1.2.1.3(b).
Figure 3-3: Ice skating hall, Olympia site, Munich [19]
After the work of Frei Otto, highly curved membrane surfaces became very popular. However, there has been a
significant move towards flatter forms due to changing aesthetic criteria. Additionally, the designers want to
separate these new structures from the existing structures [8]. The ice skating hall (Figure 3-3) is an example of
this phenomenon. However, flat membrane panels have a different structural action and performance, asking
for care during the design.
20
Although kinetic architecture has been around for centuries, the academic research on this type of architecture
is fairly recent. The theoretical framework for this type of structures was set with early modernist architecture.
The conceptual work wanted to combine human happiness, mobility and environment. In 1970 William Zuk and
Roger Clark described, as the first ones in the academic field, kinetic architecture as ‘one that has the capability
of adapting to change through kinetics’ [16]. From that point, kinetic architecture was considered as a new type
of architecture, divergent from the traditional static architecture. At the end of the 20th century, there was a
boom in architectural projects with mainly kinetic components due to the evolution in technology and new
materials.
1.2 Kinetic architecture The idea of adapting the roof to weather conditions has a long tradition. However, due to the lack of sufficiently
developed technologies, only in the mid-20th century large spaces for cultural and sporting events could be
provided with opening and closing roofs. However, this idea should not be restricted to roofs only. By introducing
movable parts into a structure, a sense of transforming space can be developed and kinetic architecture is born.
Due to the growing knowledge in computer systems and material technology, kinetic architecture’s popularity is
increasing rapidly. This type of structures differ fundamentally from the static architecture everyone is familiar
with. Kinetic is derived from the Greek word kinesis which indicates motion, movement or the act of moving [16].
In this section, the different morphological types are discussed and illustrated with several clear examples.
1.2.1 Morphology From the book of Zuk and Clark, considerable classifications of kinetic architecture have been derived based on
deployment technology, morphological aspects and structural characteristics of the mechanisms. An example is
Frei Otto’s classification of convertible roofs. However, kinetic architecture can be considered as a much broader
field. Unfortunately, there is not yet a clear definition of the boundaries of kinetic architecture and no uniformity
in vocabulary. Rodriguez [16] focusses on the transformation and connection between the different elements.
From the nature of the components and the patterns, an overview of the most common transformation
mechanisms is given.
1.2.1.1 The nature of elements In kinetic structures, modularity is the key. Movement needs to be transmitted between different elements
where the physical motion is influenced by materials and connections. There are three main types of materials:
rigid, flexible and smart or intelligent materials.
Rigid elements can move through mechanisms, for example truss elements pinned to hinges. These materials
are used in kinetic structures such as foldable, retractable plates. A successful example is the Hoberman Arch in
front of the stage at 2002 Winter-Olympic-Medals Plaza in Salt Lake City (Figure 3-4). The structure had 96
interlinked angulated-scissor modules, consisting of aluminium elements. By simultaneous operating electric
motors, the structure could be opened and closed. The entirety was covered with a translucent fibre-reinforced
cover.
Figure 3-4: The Hoberman Arch [20]
The movement of flexible materials is characterised by folding, creasing, bending, stretching and/or inflating,
resulting in foldable membranes and/or deformable pneumatic structures. These can be combined with rigid
elements, forming for example a deployable tensegrity system.
21
Smart or intelligent materials move by changing their physical properties. These materials were recently
developed by means of nanotechnology and bio-mimicry. Properties, structure, composition and/or functions
can be changed in a controlled way. An example is the non-structural open columns developed at the Department
of Architecture in Buffalo, USA (Figure 3-5). These collapsible structures are made of composite urethane
elastomers and are programmed to drop when the CO2 level is too high. In that way, people disperse into smaller
groups. If the CO2 level is decreasing, the columns respond by going up, inviting more people to the space [21].
Figure 3-5: Open columns in initial (left) and dropped position (right) [21]
The connection between the parts and the actuators, which bring the system in motion, are in principal relation
with the systems of moving bodies. The methods to realise the transformation strategy are translated into
systems which induce, operate and control the movement within kinetic devices. These methods can be manual,
natural (wind or solar power), mechanical, etc. Digitalising these methods even leads to intelligent control
structures.
1.2.1.2 Patterns The position in space and the interconnection between elements can be described in two or three dimensions,
resulting in many possible shapes and patterns. Two spatial themes dominate architecture: the centric pattern
and the linear pattern [16]. These can then be expanded to a grid.
(A) CENTRIC
The centric configuration holds every pattern that can be inscribed in a circular or spherical shape. The centre is
the focal point of space. In most kinematic structures, the elements radiate from or towards the centre point.
Two main configurations can be distinguished in kinetic architecture: pivotal and peripheral.
In a pivotal pattern, the principal supporting element is placed at the centre, from which the kinetic devices can
move back and forth between the centre and the perimeter, in a radial pattern. In 1960, for example, Roger
Tallibert asked Frei Otto for the design of the membrane geometry for a retractable roof of a swimming pool on
Boulevard Carnot in Paris. The membrane was gathered at the central point by motorised winches on radial
supporting cables. This central point was a tall mast, transferring the vertical force to the ground.
An interesting pivotal configuration is the umbrella structure. Only one mechanism results in the entire
movement of the roof: by moving vertically along the mast, the membrane opens in a radial movement. When
opened, the membrane is beneath the opening structure, as a funnel-shape with a number of fixings. The
intermediate positions need to be investigated properly. Based on this mechanism the two courtyards of the
Prophet’s Holy Mosque in Madinah (SL Rasch, finished in 1994) can be transferred into an enclosed hall [22]. Six
umbrellas, each with a 24 m span, form the convertible shading of the two courts (Figure 3-6). In summer, the
umbrellas provide shading and keep the coolness of the night in the court. At night, they are closed. In winter,
the effect is reversed: opening the umbrellas at night will keep the warmth of the day inside the marble court.
The struts connecting the umbrella with the masts are hinged to hydraulic cylinders, giving the struts the up and
down movement.
22
Figure 3-6: The umbrella structure on the courts of the Prophet's Holy Mosque, Madinah [22]
In the peripheral configuration, the supporting elements are installed along the perimeter, usually equidistant
from the centre. A special type is the spoked-wheel form of construction. This combination has the advantage
that the light weight and flexibility allows the membrane to be moved and positioned easily, while the spoked-
wheel structure provides a lightweight support system for the membrane. A large area can thus be covered with
a small amount of material. Due to the light profiles, the overall roof structure will have a large transparency.
Research showed however that gathering the membrane at the perimeter of the roof has several advantages.
There are no shadows on the playing field and it creates an open space, favourable from a visual point of view
[17] An example is given in Section 1.2.1.3(b).
(B) LINEAR
When forms are organised along an axis, straight or curved, a linear configuration is obtained. Typically the length
dimension is much larger than the width. Intermittently placed modules, which span the width direction, are
geometrically linked at their edges or vertices in such a way that movement is possible. The physical motion will
give a sense of progression and movement.
(C) GRID
By overlapping linear systems or tilting centric systems, a grid can be formed. This has a great relevance in the
process of scaling and changing the proportions of the structure. A grid provides a platform for positioning the
connections during these transformations.
1.2.1.3 Transformation methods Kinetic architecture contains more than incorporating movable elements in the structure, it is the creation of
spatial transforming experiences. When designing a transformation method, a range of variables are set up which
will change during a period of time. These variables are:
Size: proportions and scale are modified
Shape: geometrical patterns and modularity is changed
Geographical positions: elements are relocated by rotation, translation and deformation
Constitution: material properties, such as colour, brightness, texture are changed
Transformations which modify the first three variables, can be categorized as deform, fold, deploy, retract, slide
and revolve [16]. In one structure, these can be combined. For each of these mechanisms, an example will be
given.
In kinetic structures, movement and time are correlated, resulting in structures which live on rhythms, sequences
and progressions. Due to the newest developments in material technology, the structure can even embody
responsiveness. The structure can gain awareness, spatial experience, social engagement and even an interaction
between the building and user.
(A) DEFORM
When a structure deforms, the form is changed in an independent way and it can be reshaped back in its original
position. A successful example is the Floating Theatre (Murata and Kawaguchi) at the expo 1970 in Osaka (Figure
3-7). The structure consists out of three pressurized inflated tubes, connected by a membrane and supported by
buoyancy bags which allow automatic adjustment to audience load and movements.
23
Figure 3-7: The Floating Theatre (Expo Osaka 1970) [16]
(B) FOLD
The fold strategy uses a flexible material that can crinkle or crease until it is very compact. An arena in Saragossa,
Spain, was extended with a spoked-wheel foldable roof in 1990 (Schlaich Bergermann und Partner, Figure 3-8).
By incorporating a foldable roof, the arena was made available for events throughout the whole year. The entire
roof consisted of two parts: a fixed roof and an opening roof over the central sandy arena. A total diameter of
88 m could be obtained without intermediate supports. The horizontal spoked-wheel structure proved very
efficient. A hollow hub with a diameter of 36 m supported the structure for the inner opening roof. It consisted
of two ring cables above each other, with compression rods in between. The inner spoked-wheel had a central
hub connected to a centre node with radially arranged cables, the “spokes”. By tying the upper and lower radial
cables together with a suspension member, a truss structure was obtained which allows greater spans and loads.
In particular the stability is important in this type of roofs: the inner and the entire roof has to be prestressed
when closed. This was obtained by installing a spindle in the central node, operated by electric motors. When
the roof was completely open, it was almost planar, with a slight slope towards the outer edge to drain rainwater.
Figure 3-8: Folding roof in Saragossa [1]
(C) DEPLOY
The deploy strategy is comprised by mechanisms where rigid elements are connected with pivoting joints. In this
way the structure can be compressed or extended when necessary. Inspiration can be found in the traditional
origami art. By using a parallel sliding, the roof can be (un)folded. The bays of the membrane are folded between
the girders, which constitute lines of support. It is important that the membrane fixed between these girders
stays tensioned in every configuration. The structure investigated in this dissertation is in fact deployed.
Pneumatic cushions or ridge-and-valley lay-outs are most decent. Moving arch structures are possible as well. It
is important that a saddle-shaped layout is implemented between the arches.
Figure 3-9: Itinerant Theatre [16]
24
Emilio Perez Piñero was a pioneer for this type of kinetic structure. He presented his Itinerant Theatre at the
International Union of Architects in 1961 (Figure 3-9). Throughout his career, he designed several reticulated
foldable space grids. The basic folding mechanism is a scissor mechanism, forming a 3D folding mechanism [23].
(D) RETRACT
When planar rigid elements are pulled back or in, a retractable kinetic structure arises. The different parts are
retracted on top of each other. Many observatory roofs and stadium roofs are built based on this principle. Figure
3-10 shows the design of doors for an airship hangar built in Brand, Germany (SIAT GmbH and Arup). The
structure consists out of a horizontal half cylinder with a half dome at each end. The cylinder is formed by five
steel arched trusses, protected by a translucent membrane. At both ends of the hall, giant sliding doors are
installed. Each door consists of six movable segments, describing a semicircle in plan. At their ridge points they
are connected to a king pin, while their bottom points run radially on tracks, allowing the segments to slide
behind two fixed segments. The membrane is a PVC-coated polyester fabric. It is pretensioned and consists out
of four layers, forming two air cushions.
Figure 3-10: Airship Hangar, Germany [1]
(E) SLIDE
In a sliding configuration, the structure moves from side to side while in continuous contact with a surface. A
remarkable example is the Sliding House in Suffolk, UK, (dRMM Architects) completed in 2009 (Figure 3-11). The
house has a heavy sliding steel encasement which moves along rails in the ground. The encasement is filled with
insulation. By sliding the encasement over the house and glasshouse, several combinations of enclosure, open-
air living and views are obtained, depending on its position. It controls lighting conditions and creates openness
and enclosure inside the house. The movement is powered by four electric motors, installed inside the wall’s
cavity.
Figure 3-11: Sliding House, UK [24]
(F) REVOLVE
Finally, in the revolve strategy the elements are turned, rotated and/or orbited on an axis. An example is the
Flare System by WHITEvoid (Figure 3-12). An amount of tiltable metal flake bodies are controlled by pneumatic
cylinders. These flakes can be attached on a building’s surface in a certain pattern and animated by a computer
to interact with the context of the structure.
25
Figure 3-12: Revolving elements: Flare system [25]
1.2.2 Kinematic form active structures Form-active structures (FAS) are structures which carry external loads by their shape. They have a high degree
of structural efficiency, resulting in distinctive shapes. Membrane structures as well as structural shells or tensile
cable networks fall within this category. Mostly, these structures are a combination of compressive and tensile
parts. FAS are used for large building envelopes, so the acting loads are distributed rather than concentrated.
Therefore, curved shapes are preferable [26].
There are two main types of FAS: tensile and compressive envelopes. Membrane structures are tensile
envelopes, which carry the applied loads by tensile forces. Due to the lack of rigidity, their shape is adapted for
the acting load, independently of their initial geometry. The detailing of the membrane should be careful, since
deviations in the shape result in a non form-active geometry. The shape will then be forced in the active-form
under external load, resulting in folds, wrinkling and stress concentrations. Compressive form active structure on
the other hand transfer the external loads as compressive loads (Figure 3-13). Heinz Isler was a pioneer in
developing slender shell structures. Since the bending stress is never totally eliminated from the structure, these
are less efficient structures.
Figure 3-13: Compressive FAS: Wyss Garden Center [27]
If FAS are reconfigurable, then they are called kinematic form active structures (KFAS). Usually, these are
lightweight structures covered with technical textiles. The shape of these structures can be changed into
different configurations. KFAS structures are rather new in the building industry and request a complex design
and analysis.
26
Chapter 4: Analysis methods in ABAQUS software In this chapter, different aspects of the Abaqus software are investigated, in order to obtain tools to for the
simulations of the 3D deployment of a tensioned membrane. A hypar structure is studied in Abaqus/Standard as
well as with the explicit solver in Abaqus/Explicit. A square membrane is deformed by bringing two opposite
corners 1 m up and 1 m down. This is shown in Figure 4-1.
Figure 4-1: Case study: the hypar structure
The analysis conducted for membrane structures is nonlinear, due to geometrical nonlinearities such as the
wrinkling. For this case study, a linear elastic orthotropic material model is applied for the membrane. The
instability of the analysis can be solved by using following stabilisation methods: a dynamic analysis or an addition
of artificial damping [28, Ch. 7]. The latter is investigated in the first section, whereas the dynamic analysis is
performed in Section 4.2. Finally, the use of idealised shell elements are investigated. These elements prove to
be a good alternative for the membrane elements.
4.1 The implicit solver Unlike linear problems, nonlinear problems are solved in Abaqus/Standard with incremental and iterative
procedures. The load is specified as a function of time. The solution is found by incrementing the time during the
analysis. In this way, the nonlinear response can be obtained. The simulation is thus divided into number of time
increments where an equilibrium configuration at the end of each time increment is obtained. The Newton-
Raphson method is used to solve the equations [29, Ch. 8].
4.1.1 Convergence For a body in equilibrium, the total force acting on every node is zero. Therefore, the internal forces 𝐼 and external
forces 𝑃 should be equal: 𝑃 − 𝐼 = 0. The nonlinear response of the membrane to a small load increment Δ𝑃 is
shown in Figure 4-2. Abaqus/Standard uses the structure’s tangent stiffness 𝐾0, based on the configuration at
𝑢0, and Δ𝑃 to calculate a displacement correction 𝑐𝑎 for the structure. From 𝑐𝑎 the configuration of the structure
is updated to 𝑢𝑎. In this new configuration, the internal forces 𝐼𝑎 are calculated. The force residual for the
iteration can finally be calculated as: 𝑅𝑎 = 𝑃 − 𝐼𝑎. If 𝑅𝑎 would be 0, the configuration would be in equilibrium.
However, for a nonlinear problem, this residual will never be exactly zero. It is accepted as a solution in
equilibrium if it is smaller than a tolerance value. By default, this is 0.5% of an average force in the structure,
averaged over time. Additionally, also 𝑐𝑎 should be small enough (1% by default) compared to the incremental
displacement.
27
Figure 4-2: First iteration of an increment [29, Ch. 8]
If the solution from an iteration did not converge, another iteration will be performed (Figure 4-3).
Abaqus/Standard forms a new stiffness 𝐾𝑎 based on the updated configuration 𝑢𝑎. Together with 𝑅𝑎, this
determines another displacement correction 𝑐𝑏, bringing the system closer to the equilibrium. A new force
residual 𝑅𝑏 is calculated and again this is compared to the tolerance value. Additionally, the displacement
correction 𝑐𝑏 is compared to the increment of displacement Δ𝑢𝑏. If necessary, further iterations are conducted.
Figure 4-3: Second iteration [29, Ch. 8]
4.1.2 Automatic incrementation control For every iteration, the stiffness matrix of the model is formed and the equations are solved. The computational
time of each iteration will thus be the determining factor in the computational cost of the complete analysis. The
first increment needs to be suggested by the user, whereas Abaqus/Standard chooses the subsequent. This leads
to an easy and sufficient analysis. It is important that the initial increment size is reasonable, in order not to waste
CPU time.
For every increment, there can be several iterations. An iteration is an attempt of finding an equilibrium solution.
It is said to be divergent when subsequent iterations move away from the equilibrium state. In that case, the
process is terminated and a new attempt for finding a solution is started with a smaller increment size. The
number of iterations needed depends on the degree of nonlinearity. However, when more than 16 iterations are
needed or the solution diverges, the increment is stopped and restarted with an increment size that is 25% of its
previous. Then the analysis is started again. This process is continued until the solution is found. If the time
increment becomes smaller than the minimum specified by the user, the analysis is stopped and the solution
diverged. By default, five cutbacks of increment size are allowed before the analysis is terminated [29, Ch. 8].
The maximum number of increments allowed can also be specified, the default number is 100. If it needs more
increments to finalise the step, the analysis will be stopped as well.
28
When inertial effects and rate-dependent behaviour of the membrane are important in a nonlinear analysis, the
step time will have a physical meaning. The step time 𝑇𝑠𝑡𝑒𝑝 and the ratios of the initial time increment Δ𝑇𝑖𝑛𝑖𝑡𝑖𝑎𝑙 ,
minimum time increment Δ𝑇𝑚𝑖𝑛 and maximum time increment Δ𝑇𝑚𝑎𝑥 to this step time are specified. This
determines the initial load increment as [29, Ch. 8]:
Δ𝑇𝑖𝑛𝑖𝑡𝑖𝑎𝑙
𝑇𝑠𝑡𝑒𝑝
𝑥 𝑙𝑜𝑎𝑑 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒
The choice of the initial time increment can thus be critical and for most analyses, it is 5% to 10% of the total
step time. The size of the subsequent increments is chosen automatically by the implicit solver.
In this case study, the step time is kept as 1 s, the ratios Δ𝑇𝑖𝑛𝑖𝑡𝑖𝑎𝑙/𝑇𝑠𝑡𝑒𝑝 and Δ𝑇𝑚𝑎𝑥/𝑇𝑠𝑡𝑒𝑝 is 1 and the minimum
time increment Δ𝑇𝑚𝑖𝑛/𝑇𝑠𝑡𝑒𝑝 is 1E-5 s. The maximum number of increments is the default value of 100.
4.1.3 Automatic stabilisation of unstable problems To get convergence in nonlinear problems, Abaqus/Standard has an automatic mechanism which introduces
volume-proportional damping to the model. Viscous forces 𝐹𝑣 are added to the global equilibrium equations [28,
Ch. 7]:
𝑃 − 𝐼 − 𝐹𝑣 = 0 with 𝐹𝑣 = 𝑐M∗𝑣
where 𝑀∗ is an artificial mass matrix calculated with unity density, 𝑐 is a damping factor, 𝑣 = 𝛥𝑢/𝛥𝑇 is the nodal
velocity vector and Δ𝑡 is the time increment. Two parameters are adapted to improve convergence in
Abaqus/Standard:
The damping factor 𝑐: By increasing the damping factor, the viscous forces 𝐹𝑣 are increased. This improves
convergence but also increases the dissipation energy. If this energy becomes too high, the influence of the
damping forces is no longer negligible. The applied damping factors can be constant over the duration of a
step or they can vary with time.
The step time: If the ratio Δ𝑇𝑖𝑛𝑖𝑡𝑖𝑎𝑙/𝑇𝑠𝑡𝑒𝑝 is kept constant, reducing the step time 𝑇𝑠𝑡𝑒𝑝 will reduce the initial
time increment Δ𝑇𝑖𝑛𝑖𝑡𝑖𝑎𝑙 , thus increase the nodal velocity 𝑣 and thus increase the viscous forces 𝐹𝑣.
To ensure that accurate solutions were found in the analysis, the viscous damping energy (ALLSD) should always
be small compared to the total strain energy (ALLIE). A rule of thumb states that the ALLSD should be only 5 or
10% of the ALLIE. The viscous damping may be large if the structure undergoes rigid body motion.
4.1.3.1 Constant damping factor In the automatic stabilization with a constant damping factor, an automatic stabilisation is included in any
nonlinear quasi-static procedure. The damping factor can be calculated based on the dissipated energy fraction
or it can be specified directly by the user.
DIRECTLY SPECIFIED DAMPING FACTOR
The influence of the damping factor is investigated for the case study. In a first attempt, a constant damping
factor of 1E-2 is directly specified. The energy history plot is given in Figure 4-4. It is clear that the dissipated
energy is much too large compared to the strain energy at the end of the step. The energy ratio ALLSD/ALLIE
equals 328.3%. The damping will influence the solution significantly, as can be seen from the deformation in
Figure 4-5. The deformation is localised in the corners, indicating that the damping is too high and the chosen
value is not optimal.
29
Figure 4-4: Energy history plot (Damping factor 1E-2)
Figure 4-5: LE11 contour plot for a 1E-2 damping factor
A suitable factor can be determined by using a self-programmed script (See Annex). For every factor the
dissipated energy (ALLSD) and the total strain energy (ALLIE) are compared at the end of the step. By using the
script, Abaqus/Standard can find the damping factor for which the ratio is smaller than 5%. The results from the
different steps are given in Figure 4-6.
Figure 4-6: The energy ratio in function of the damping factor
0
50
100
150
200
250
300
350
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
ALL
SD/A
LLIE
[%
]
log Damping factor [-]
30
Figure 4-7: Comparison of LE11 along a vertical path for different damping factors
When a factor of 1E-8 is used, the dissipated energy is only 1.65% of the total strain energy. The strains LE11
converge towards the values for 1E-8, proving that this is an appropriate damping factor (Figure 4-7). The results
for the implicit solver with a damping factor of 1E-8 are given in Figure 4-8.
Figure 4-8: LE11 strains [-] (Damping factor 1E-8)
There are some disadvantages to this method. When using a constant damping factor, there is no guarantee that
the chosen factor is optimal or suitable [28, Ch. 7]. Therefore, obtaining an optimal value for the damping factor
is a trial-and-error method until the solution converges and the dissipated stabilisation energy is sufficiently
small.
A DAMPING FACTOR BASED ON THE DISSIPATED ENERGY FRACTION
It is assumed that the problem is stable at the beginning of the step, therefore, the viscous energy dissipated is
very small and the artificial damping has no effect. However, if a local region becomes unstable, the local
velocities will increase and part of the strain energy then released is dissipated by the applied damping [28, Ch.
7]. When the first step is stable without applying damping, the damping factor is determined so that the
dissipated energy for an increment is a small fraction of the extrapolated strain energy. This fraction is the
dissipated energy fraction (default value: 2E-4). In order to ensure that the obtained solution is accurate, the
damping factor at the end of the first increment should be reasonable.
For this case study, if the ratio for the dissipated to the strain energy is specified to be 5%, the solution diverges.
The history plot of the energy ratio from the analysis with a constant damping factor of 1E-8 is plotted in Figure
-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Stra
in L
E11
[-]
True distance along path [mm]
1E-5 1E-7 1E-8
31
4-9. The ratio is very high due in the beginning of the analysis due to rigid body motion. Therefore, this method
cannot reach a converged solution and a constant damping factor is used instead.
Figure 4-9: The ratio ALLSD/ALLIE [-] in function of step time [s] (Damping factor: 1E-8)
ADAPTIVE AUTOMATIC STABILISATION SCHEME
In the adaptive automatic stabilisation scheme, the damping factor can vary spatially and with time, which is
generally preferred over the constant damping method [28, Ch. 8]. The damping factor depends on the
convergence history and the ratio of the dissipated energy of the viscous damping and the total strain energy. If
instabilities or rigid body motion occur, the damping factor is automatically increased. However, since the ratio
of the energies is too high at the beginning of the step, this method cannot be used as well.
4.2 The explicit solver When solving quasi-static problems with very complex contact, convergence problems with the implicit solver
may arise. A possible solution might be to use the explicit solver. Abaqus/Explicit performs a large amount of
small increments efficiently during the analysis.
An explicit central-difference time integration rule is used, which does not require solving a set of simultaneous
equations. Therefore, each increment is relatively inexpensive. Abaqus/Explicit solves problems with the
dynamic equilibrium equation: 𝑃 − 𝐼 = 𝑀 ü [29, Ch. 8]. These equations are satisfied at the beginning of the
increment 𝑡. The accelerations calculated at that time are used to advance the velocity and displacement solution
to 𝑡 + Δ𝑡/2. The key to the computational efficiency is the use of a diagonal element mass matrices, a so-called
lumped mass matric. The explicit analysis requires no iterations and no tangent stiffness matrix [28, Ch. 6].
4.2.1 Stable time increment Abaqus/Explicit calculates through time by using many small time increments. The stable limit for the operator
when no damping is applied to the model can be estimated by the following equation:
Δ𝑡𝑠𝑡𝑎𝑏𝑙𝑒 ≈𝐿𝑚𝑖𝑛
𝑐𝑑
where 𝐿𝑚𝑖𝑛 is the smallest element dimension and 𝑐𝑑 is the dilatational wave speed. In conventional shell or
membranes, the element thickness does not account for the smallest element dimension. Mostly not all
elements are small, so only a few elements will determine the stable time increment for the whole model. This
can be optimised by changing the mesh or by using appropriate mass scaling.
If the time incrementation is larger than the stability limit, then there is a failure to use enough time increment.
This results in an unstable solution, where for example the deformations are oscillating in time. In a nonlinear
analysis, the highest frequency of model continually changes, so the stability limit changes. In general, there are
two strategies for incrementation control: a fully automatic or fixed time incrementation.
4.2.2 Computational cost The time for running a simulation in Abaqus/Explicit depends on the time period of the event. For quasi-static
problems, it is mostly opportune to speed up the process hence reduce the number of increments 𝑛 required.
32
Two methods can be combined: the increased loading rate and mass scaling. In the first approach, the time of
the process is increased, so that the period of the event is reduced. For the latter, the density of the material is
artificially increased so that the ratio of the event time to the time for wave propagation across an element is
reduced [28, Ch. 6]. The process is however still examined in its natural time. Both of them are explored further
in the following paragraphs.
4.2.2.1 Increasing the loading rate For the increased loading rate approach, the time of the process is increased so that the period of the event 𝑇 is
reduced. An increase with a factor 𝑓 results in a reduction of the period time to 𝑇/𝑓. It is however important
that the dynamic effects remain insignificant. The aim is to get the shortest time of the process where the inertia
forces are still insignificant [29, Ch. 13]. Otherwise, these forces will change the response and cause a wave
propagation. This can be investigated by comparing the kinetic energy (ALLKE) to the internal energy (ALLIE) for
the whole model. If the kinetic energy is too high or turbulent, the inertia has too much influence. [28, Ch. 6].
In the hypar case study, the corners are translated with smooth amplitude curves in order to get efficient and
accurate results. A sudden change in boundary conditions leads to divergence. A time period of 1 s is chosen as
a first attempt, thus a very high loading rate. Figure 4-10 indicates that the kinetic energy ALLKE is much too high.
The time period should thus be increased.
From several analyses with different loading rates, it is concluded that by increasing the time period and thus
decreasing the loading rate, the kinetic energy becomes smaller in general, as well as compared to the internal
strain energy. Unfortunately, it makes the kinetic energy more oscillating as well. The result from the analysis
with a 500 s time period is given in Figure 4-11. The deformations are compared for both rates (Figure 4-12). For
the smaller time period, the deformation is more located in the corners and the strains are higher. However, by
further increasing the loading rate, the oscillations of the kinetic energy become too high. To improve the
analysis, this method can be combined with mass scaling.
Figure 4-10: Energy history plot for time period of 1 s: a. ALLIE and ALLKE and b. ALLKE
Figure 4-11: Kinetic energy history plot for a time period of a. 1 s and b. 500 s
33
Figure 4-12: Comparison of the deformations for a. 1 s and b. 500 s
4.2.2.2 Mass scaling For the mass scaling approach, the density of the material is artificially increased so that the ratio of the event
time to the time for wave propagation across an element is reduced [28, Ch. 6]. The dilatational wave speed 𝑐𝑑
can be approximated as [29, Ch. 13]:
𝑐𝑑 = √𝐸
𝜌 (𝑖𝑓 𝜈 = 0) ⟹ Δ𝑡𝑠𝑡𝑎𝑏𝑙𝑒 ≃
𝐿𝑚𝑖𝑛
√𝐸/𝜌
where 𝜌 is the material density and 𝐸 is the elasticity modulus. The expression of the wave speed is only valid
for a material with a zero Poisson’s ratio. This expression can be combined with the estimation for the stable
time increment Δ𝑡𝑠𝑡𝑎𝑏𝑙𝑒. Increasing the density with a factor 𝑓2 will decrease the wave speed and consequently
increasing the stable increment time with a factor 𝑓. The number of increments reduce to 𝑛/𝑓. Mass scaling
performs thus an economic analysis while the natural time of the process is not changed.
Mass scaling can be performed on the entire model, however, it might be useful to scale different parts
separately if the size of the elements are different. In this case, there is a homogeneous mesh, so the same mass
scaling is applied for the whole model. Two types of mass scaling allow more flexibility for modifying the densities
of the materials: fixed mass scaling and variable mass scaling.
Fixed mass scaling is performed at the beginning of the step. The mass scaling factor can be defined
directly or a desired minimum stable increment can be determined. For the latter, Abaqus/Explicit
defines the mass scaling factors. This will allow to modify the masses of only a few small elements,
therefore, fixed mass scaling is computationally efficient [28, Ch. 6].
Variable mass scaling on the other hand scales the mass of elements also periodically during that step.
The mass scaling factors are calculated automatically based on the defined minimum stable time.
The influence of the mass scaling factor is determined by comparing several mass scaling factors, for which two
examples are showed here. The energy history plots are given in Figure 4-13 for a mass scale factor of 10, which
indicates that the kinetic energy is too large. This can be seen from the deformation in Figure 4-14. Also a factor
2 leads to the same conclusions. Mass scaling alone can thus not be applied to obtain a stable and adequate
solution.
34
Figure 4-13: Energy history plots for a mass scale factor of 100, 10 and 5
Figure 4-14: Results for a mass scaling factor of a. 10 and b. 2
4.2.2.3 Combining loading rate and mass scaling Both methods will be combined in order to obtain satisfactory results. The mass scaling factor and the time
period parameters are varied and an adequate combination is determined by trial and error. Finally, a mass scale
factor of 2 and a time period of 25 s proves to be a good combination. The time history plot of the internal energy
and the kinetic energy is given in Figure 4-15.
The LE11 strain plot is given in Figure 4-16, as well as the deformations. When compared to the implicit solver,
the values are approximately the same and the same pattern appears. Locally some distortions show however
they are small in amplitude.
0.0E+00
2.0E+06
4.0E+06
6.0E+06
8.0E+06
1.0E+07
1.2E+07
1.4E+07
1.6E+07
1.8E+07
0 0.2 0.4 0.6 0.8 1
Ene
rgy
[mJ]
Time [s]
Factor 10-ALLIE Factor 10-ALLKE
35
Figure 4-15: Time history plot final explicit solution
Figure 4-16: LE11 strains (explicit)
4.2.3 Comparison of the implicit and explicit results A comparison can be made between the strains along path 1 (Figure 4-17). The overall trend is the same for both
methods. The values show only small differences at the middle zone (order of 0.01%), so the explicit solver is an
adequate tool for solving these kind of problems. However, it demands a lot of work to determine appropriate
parameters. The assessment of these parameters is also subjective. So, using an implicit solver is still favourable.
Figure 4-17: Comparison of LE11 along path 1
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Stra
in L
E11
[-]
True distance along path [mm]
Implicit Explicit
36
4.3 The membrane’s element type The membrane fabric used in the experiment has a very small bending stiffness. However, due to the out-of-
plane deformation caused by the (un)folding, it will be difficult or even impossible to get convergence with
membrane elements. Shell elements could be chosen, however, due to the coupling of the bending and in-plane
stiffness, the elements will be too stiff. Therefore, another option is explored. Shell elements can be idealised by
choosing the ‘membrane only’ option. The three element types are compared for a simple case where a square
membrane (side of 1 m) is clamped at one edge and moved upwards 0.1 m at the opposite edge (Figure 4-18).
An isotropic material model is used to simplify the calculations.
Figure 4-18: Simple case study to compare element types
4.3.1 Shell elements Shell elements are used to model structures in which one direction is smaller compared to the other two [28, Ch.
29]. Abaqus/Standard has a large library of shell elements, where one family is the general-purpose shell
elements. These elements provide accurate solutions for thin and thick shell problems [30, Ch. 3.6]. In this case,
an S4R element is used, which is a general-purpose element. The deformation and strains (E11 and E22) are
displayed in Figure 4-19. The edges A and B show a parabolic upward deformation. This is a consequence of the
bending stiffness of the elements, which counteracts the deformation In x direction (LE11) there is tension over
the whole surface, whereas in the y direction (LE22), the middle zone is in tension and the outside zone is in
compression.
Figure 4-19: The deformation, LE11 and LE22 strains for shell elements
In order to investigate this behaviour in more detail, the mesh is refined to 30 x 30 elements (Figure 4-20). For
the E11 strain, the behaviour is comparable. In the y-direction (E22), almost the whole surface is under tension,
whereas only a small zone at the edge is under compression. This is due to the influence of the clamped edge,
where every degree of freedom is restrained. The differences in strains are however very small.
Figure 4-20: Shell elements: a finer mesh
Edge A
Edge B
37
4.3.2 Membrane elements Membrane elements are planar elements which can carry membrane force but lack bending stiffness. They are
typically used for thin surfaces in space. These elements are in plane stress, so the stress components
perpendicular to the middle surface are zero [28, Ch. 29]. A general membrane element M3D4R is used and a
damping factor of 1E-8 is introduced to obtain convergence. The results are shown in Figure 4-21. A first
difference with the shell elements is that edges A and B are straight. LE11 shows tension over the full membrane
surface, with local negligible differences. The transverse strains however are compressive, with a middle zone
where these are slightly higher (0.03%). This is a consequence of the Poisson’s effect. By tensioning the
membrane longitudinally, there is a transverse contraction.
Again more insight can be gained from refining the mesh (Figure 4-22). The edge is straight, with a very small
wrinkle occurring closer to the clamped edge. For the LE11 strains, two disturbance zones can be noticed,
however of a very small magnitude (0.001%). For the LE22 strains, also some disturbances can be noticed,
however of the same small magnitude. The overall strain field can thus be considered as uniform.
Figure 4-21: Deformations, LE11 and LE22 for membrane elements
Figure 4-22: Membrane elements: a finer mesh
4.3.3 Idealised shell elements Homogeneous shell sections can be idealised to membrane action in this case since the predominant response
of the fabric is in-plane stretching. These idealisations modify the shell general stiffness coefficients after they
are computed normally. The bending stiffness terms are eliminated from the shell stiffness. By retaining only the
membrane stiffness, all membrane-bending coupling terms as well as the off-diagonal terms of the bending
submatrix are set to zero. The diagonal bending terms are set to 1E-6 times the largest diagonal membrane
coefficient [28, Ch. 9]. Again the S4R element is used and no damping is necessary. The results are shown in
Figure 4-23. Again, a straight edge can be recognised as for the membrane elements. Additionally, the LE11
strains show tension and LE22 compression, with only small local differences.
In order to obtain results for the refined mesh, a damping factor of 1E-8 was applied (Figure 4-24). The edges are
again straight, however, some very small wrinkling shows. Additionally, for the E11 strains, a few wrinkling zones
show of negligible differences in tensile strains. In the transverse direction, the whole surface is in compression,
whereas negligible disturbance zones show (an order of magnitude of 0.006%).
38
Figure 4-23: Deformations, E11 and E22 strains for idealised shell elements
Figure 4-24: Idealised shell elements: a finer mesh
The displacement along edge A is given for the three element types in Figure 4-25Error! Reference source not
found.. As can be expected, the shell elements behave differently from the membrane and idealised shell
elements. The shell elements has an out-of-plane bending stiffness, resulting in a parabolic displacement. The
membrane elements on the other hand have no such bending stiffness and will deform along a straight line, with
the occurrence of wrinkling. The occurrence of wrinkling for the idealised shell elements indicates that the
elements indeed have no bending stiffness.
Figure 4-25: Comparison deformation for different element types along edge A
A final comparison is given in Table 4-1. Simulating with idealised shell elements takes some extra CPU time. The
output is namely first calculated as if it were shell elements and the bending stiffness is then removed at the end
of each increment. However, from the findings above, it can be concluded that idealised shell elements are a
good alternative for membrane elements in case convergence would be difficult.
0
10
20
30
40
50
60
70
80
90
100
0 200 400 600 800 1000
U3
[m
m]
x [mm]
Shell Membrane Idealised shell
39
Table 4-1: Comparison between different elements for the fine mesh
Shell elements Membrane elements Idealised shell elements
Damping factor [-] None 1E-8 1E-8
Total CPU time [s] 9.7 59.7 84.9
4.4 Conclusion In this chapter, the case study of the hypar structure was investigated to compare the implicit and explicit
analysis. In order to stabilise the Abaqus/Standard simulation, a constant damping factor is an adequate tool. In
Abaqus/Explicit, the loading rate was decreased in order to minimise the inertia effects. On the other hand, mass
scaling could be applied to increase the total calculation time.
The explicit analysis has several advantages. First, the use of small increments, governed by the stability limit,
allows the analysis to proceed without no iterations or no tangent stiffness matrix. Also the contact interaction
is simplified. Secondly, the results are not automatically checked for accuracy as in Abaqus/Standard since
Abaqus/Explicit uses the half-increment residual. This check is however mostly not necessary since the time
increments are small, so the solution changes only slightly in one increment. This simplifies the incremental
calculations tremendously. The analysis is thus economical, since the one increment is inexpensive.
As a secondary investigation, membrane elements were compared with the idealised shell elements. These latter
are calculated as if they were shell elements, however, at the end of the increment all bending related factors
are eliminated from the equations. This delivers results with a good agreement. Although the CPU time is larger,
these elements can be used as an alternative for membrane elements.
40
Chapter 5: The experiment Between 2006 and 2010 a project was carried out in the EU for the investigation and development of membrane
structures, called the Contex-T project. As a case study, a retractable dome was built. However, the demonstrator
did not allow a detailed analysis at that time. VUB developed an experimental set-up for further investigation.
Ghent University joined the VUB in this project for the investigation of the material behaviour [31], resulting in a
new, more accurate material model [5]. While the VUB experimentally investigated the demonstrator, Ghent
University was responsible for the numerical validation of these experiments with the new elasto-plastic material
model. The research of the demonstrator consists out of two phases: the pretensioning of the membrane in
plane [32] and finally the total deployment in 3D, investigated in this master thesis. This chapter describes the
case as well as the experiment and obtained results.
5.1 The Contex-T project The Contex-T project (2006-2010) was a project within the EU between different actors of the building industry.
The three main objectives were [33]:
a. to develop a new lightweight building with a lifespan up to 60 years, fulfilling the requirements on
noise reduction, acoustic absorption, thermal insulation, light transmission and reflection
b. the development of safety, healthy and economic buildings
c. easy and fast construction
This could be achieved by introducing high technology materials and finding new approaches for the membrane
structures. A case study within this project was the retractable dome structure, of which a demonstrator has
been built (Figure 5-1).
It is a 360° dome, half a sphere, with a radius of 4.25 m. It consists of an angulated beam structure. The dome
can be opened and closed by (un)folding the beam structure. The space underneath the dome is covered by the
use of a membrane, consisting of flat triangular patterns and connected to the beam structure with belts at the
corners. The aim is to test the kinetic behaviour of the supporting system. If one quarter of the dome is fixed
(three points fixed at ground level) then the top point remains fixed as well and the other parts can transform by
rolling over a circular path at ground level. Since the top point of all parts is fixed, every rhombus has one degree
of freedom.
Figure 5-1: The demonstrator from the Contex-T project [34]
Unfortunately, the biaxial tensions in the different configurations could not be measured in the demonstrator.
Additionally, the pretension of every rhombus changed between configurations. Low tension is necessary in the
unfolded reference position, so once the dome was folded, the tension was insufficient. In order to solve these
issues, the VUB made a separate experimental set-up. Due to symmetry, only one rhombus needs to be
examined. The study of different opening angles for the same membrane structure is a new approach [34]. The
proposed research questions were:
a. Will the configuration be stable in intermediate configurations?
b. Can it carry snow and wind loading?
c. How to control and adjust pretension for different angles?
41
5.2 The experimental set-up As stated above, one rhombus of the demonstrator is examined in a 3D experiment. In advance, a parameter
study and numerical simulation are performed with the software program EASY. By executing a parameter study,
the cutting pattern, necessary pretension and path of deployment was determined [35]. The membrane was
modelled as a cable net and the boundaries of the membrane were modelled as cables as well. The numerical
simulation was performed using a nonlinear analysis but a linear material model for the membrane.
5.2.1 The membrane 5.2.1.1 Geometry In Figure 5-2 the membrane pattern of one rhombus is shown. It consists of two triangular panels, welded
together at the bottom with an overlap of 5 cm. The used membrane is a PVC-coated polyester fabric (T2103
from Sioen) with a strength of 4 kN/5 cm in both warp and fill direction and a self-weight of 1050 g/m². The
membrane is 0.83 mm thick.
Figure 5-2: Compensated cutting pattern of the rhombus [mm]
The pattern is cut in such a way that the warp threads are aligned horizontally and the fill vertically. Every edge
of the membrane has a sag of 5%. In the corners, a double layer of membrane is applied in order to transfer the
load more uniformly from the belts to the membrane. The cutting pattern is derived from numerical form-finding
performed in EASY. The 50° opening angle proved to be the most suitable for obtaining a constant prestress in
all different unfolding stages [35]. Figure 5-3 gives an overview of the experimental set-up.
42
Figure 5-3: Overview of the experimental set-up and detail of the central belt
5.2.1.2 Material parameters Prior to the 3D deployment test, several tests were executed in order to define the material elastic parameters.
The method described in the Japanese standard MSAJ/M-02-1995 [7] was followed. The prestress loads equal
2.5% of the ultimate tensile stress (UTS). The maximal load incorporates a safety factor of 4, resulting in 25% of
the UTS. During one biaxial test, different load ratios of warp and fill direction are applied on the membrane:
1/1, 1/2, 2/1, 0/1, 1/0. The forces and corresponding strains in the membrane are measured.
The recommendations for the prestress and loading ratios should be applied during testing in case the loading
of the structure is unknown. However, in this project a kinematic structure is designed where the kinematic
movement will induce stress distributions in the membrane, even without taking into account external loads. A
sequence of load ratios can thus be determined which correlates to the folding and unfolding of the structure.
Additionally, the numerical model showed that during the (un)folding, the membrane will not be subjected to
25% of the UTS. Therefore, a tailored biaxial test was conducted, loading the membrane with more realistic load
ratios. Instead of the MSAJ ratios, an alternation of 1/2 and 2/1 is simulated, starting from a 2 to 1 kN/m prestress
state. From this test, a compensation of 0% in warp and 1.7% in fill direction is derived (Figure 5-4). This
compensation takes into account the pretension as well as the initial permanent deformations of the membrane.
Figure 5-4: Tailored biaxial test: Protocol (top) and results (bottom) [35]
Apex points
Bottom
Opening link
Fixed point
Compensation
43
Table 5-1 shows the set of material properties obtained from each test. A large difference can be noticed, proving
the influence of the material testing protocols. Note that for both tests the initial large deformation is not taken
into account for the calculation of the different parameters. The shear modulus is determined through a bias
extension test so it is the same for both cases. The parameters of the tailored test were used in the model in
EASY.
Table 5-1: Material properties determined by the MSAJ test and tailored test [35]
𝐸𝑤 [𝑘𝑁/𝑚] 𝐸𝑓 [𝑘𝑁/𝑚] 𝜈𝑤𝑓 [−] 𝜈𝑓𝑤 [−] 𝐺 [𝑘𝑁/𝑚]
MSAJ test 766 659 0.37 0.32 0.22 Tailored test 1037 346 1.08 0.36 0.22
5.2.2 The belts Polyester belts (Loadlok 1506) are used in the experiment to pretension the membrane. The lower valley belt is
laid over the weld, allowing sliding over the membrane. The circumference of the membrane is reinforced by
stitching the belts on the membrane. The belts are 5 cm wide and have a maximum strength of 60 kN. By using
belts instead of cables, enough flexibility is foreseen and lower reaction forces are needed to achieve the
necessary prestress. From the parametric study in EASY the initial pretension was determined as 2 kN/m in warp
direction and 1 kN/m in fill direction.
5.2.3 The kinematics The membrane deploys by means of a square truss frame. (Figure 5-5) It consists of steel bars with a length of
1.4 m. By changing the distance between two opposite corners, the membrane will open or close. In order to
control this distance, a link is introduced. By screwing the wheels at the end, the length of the link is changed. In
this way, the opening angle can be determined in a very precise way. The link is a steel rod with a diameter of 16
mm. The (un)folding of the membrane is thus achieved by a combination of length and position measurements,
together with load control in each of the connections. In Table 5-2 the length of the link is given in function of
the opening angle.
Table 5-2: Relation between opening angle and distance between top points
Opening angle [°]
Distance apex points [m]
Length link [mm]
10 0.35 1389 30 0.93 1320 50 1.46 1195 70 1.94 1010 90 2.34 769
Figure 5-5: The superstructure used to open the membrane
44
5.2.4 The connections Figure 5-6 shows the connections of the membrane. The apex corners are connected to the opening frame by
stitching the belts into a loop and using steel U-shackles. An overlap of belts and membrane is made to provide
a loop and to reinforce the membrane at a location where concentrated forces are introduced. This results in a
layer-up of belt/membrane/belt of 17 cm, followed by a double belt of 25 cm and then a loop of 15 cm.
It is very important that the centrelines of all the belts intersect in the same bottom point. This can be achieved
by connecting the side belts to a main ring by using metal rings. The connection with these smaller rings is again
provided by making a loop. This results in a stitched overlap of 25 cm belt/membrane/belt and a loop of 10 cm.
The central belt has a much smaller loop to connect it to the main ring. This ring is then connected to a turnbuckle,
connected to an anchored steel beam. The bottom connections allow measuring equipment to be installed,
whereas the turnbuckle allows changing the length of the belts.
Figure 5-6: The connections of the membrane (a. apex points, b. connection at the bottom)
5.2.5 Execution of the experiment The total deployment of the membrane consisted of different steps. First, the central belt was pretensioned
separately from the rest of the set-up. By using a ratchet, the belt was lengthened from 5.72 m, the length
extracted from the EASY model, to 6 m, the length between the two bottom points. It is concluded from previous
tests that setting the valley belt to its correct length first has a major influence on the results, as it proved
impossible to tension the belt sufficiently once the membrane was in place. Next the membrane was brought
under the belt and pulled upwards by the means of a crane, in such a way that the apex points were positioned
in their correct geometrical position.
The overall deployment of the membrane was a geometrically controlled process: first, the length of the link was
adjusted after which the position of the apex points were adapted by moving the crane up or down. However,
manual measurements showed that the central belt‘s curvature was lower than in the theoretical model.
Therefore, in a second step, the apex points were brought above their theoretical position, matching the central
belt’s curvature to the theoretical model.
Then the membrane was opened to 90° and the process was repeated. The membrane was then again closed to
50°, 30° and 10°. A view on the set-up for different opening angles is given in Figure 5-7.
Figure 5-7: Opening the membrane during the experiment (50° - 90° - 10°) [36]
45
At the end of the test, the membrane was brought to a fully closed, slightly tensioned configuration. This state
was considered the reference state of the strain field. As stated before, the membrane’s material is highly
depending on its loading history. By folding and unfolding the membrane, permanent strains are introduced. So
if the reference state is taken at the end of the experiment, these permanent strains are neglected from the
results although they are critical in the material behaviour. As a result, the strain output will give incorrect results.
5.2.6 Measurements The opening angle was measured in steps of 20° and for every opening angle four states were measured: the
state with the apex points in the right position and one where the curvature of the central belt was correct. For
every position, measurements were also taken after 15 minutes. In this way some relaxation of the pretension
could be measured as well. Measurements were taken for the geometry (the position of the markers), the sag at
the borderlines, the forces in the anchorage points and the biaxial strains by the means of a DIC.
Figure 5-8: Position of the markers
On the membrane, markers were attached (Figure 5-8). This allowed to measure the geometry during the
simulation. The three markers at the bottom are fixed points, specifying the coordinate system for the markers.
Manual measurements were performed with a laser meter to verify whether the set-up corresponds to the
geometry analysed during the design and optimisation stage.
The reaction forces are measured by custom-made load cells (Figure 5-9). They are small and are placed in line
with the belts. For the side belts, the maximum capacity of the cells is 20 kN. For the central belt, this capacity is
increased to 30 kN. The total vertical force is measured by a commercial load cell (MTS System Corporation
661.21B.03) in the crane, with a maximum capacity of 100 kN.
Figure 5-9: The load cells [36]
Finally, the strains in the membrane and displacements of the overall prototype could be measured by using a
stereoscopic Digital Image Correlation (DIC). By using discrete spatial measurements of marked points on the
prototype, the displacements of the prototype could be obtained as well. This can verify the manual
measurements and give additional information. The fabric strains were measured by a full-field DIC analysis of a
randomised speckle pattern on the fabric.
46
5.2.7 Results From a visual inspection, the initial state showed some wrinkling and bulge in the apex corner, suggesting that
the prestress in fill direction disappeared. In the fully opened position, this wrinkling became even more
pronounced, whereas it reduced when closing the membrane (Figure 5-10).
Figure 5-10: Wrinkling in 50° - 10° - 90° opening angle [36]
From the measurements of the reaction forces, it could be concluded that these reduce when the structure is
unfolding. Overall, the measured reaction forces are 40% lower than the theoretical values. In general, the state
where the central belt is in the correct position corresponds more to the trend from the theoretical model. This
case was further analysed. The measured strains are small, between 0.3% and -1.3% in warp direction and 0.5%
and -0.3% in fill direction. Since the localised effects and complex straining behaviour were not incorporated in
the material model used in EASY and the initial reference state was taken at the end of the experiment, in an
underestimation of the strains in magnitude and uniformity over the membrane was obtained.
The reference state should be taken at the beginning of the experiment. In order to validate the numerical model,
a new set of DIC’s is generated with reference to the 50° opening angle where the apex points are in a correct
position. It is thus already pretensioned, so it is not fully strain-free, however, the impact of the (un)folding will
be more realistic.
47
Chapter 6: Numerical deployment of a membrane tensioned by contact 6.1 The course of the simulation In order to introduce the same strains in the numerical model as in the experiment, the measured reaction forces
have to be applied on the membrane. The initial geometry is the 50° opening angle. In order to validate the
model, the membrane is first brought in the same reference state: the 50° opening angle. Next, the reaction
forces and upper force are adjusted so that the strains can be compared to the DIC output for the 50° opening
angle after relaxation. From that point, the membrane is unfolded to the 70° angle by adjusting the length of the
link. Simultaneously, the forces in the belts and the upper force are adjusted until the forces from the experiment
are reached again. Next, the membrane is opened to 90° in the same way, then closed back to 50°, 30° and finally
to 10°.
6.2 The initial geometry The initial geometrical model is built in a 3D modelling software. As stated in Section 5.2, the 50° opening angle
proved to be the optimal opening angle for maintaining sufficient pretension during the full deployment. From
the form-finding shape, a flat cutting pattern was extracted. This cutting pattern is used to form flat membrane
panels, positioned in a 50° angle. From the experiment, the distance between the apex points is known as well
(Table 5-2). Finally, the connection between the two membrane patterns is obtained by lofting a valley curve in
between the two panels. This valley curve has a small transverse radius and a large longitudinal radius (Figure
6-1). The transverse radius is determined in such a way that a smooth transition between the valley and the two
panels is obtained and that the circumference is close to 5 cm. For the longitudinal curvature, the curvature of
the weld is followed. By pretensioning the membrane with the central belt, the curvature from the experiment
should be obtained.
Figure 6-1: Comparison of the central belt in 50° opening angle between experiment and initial geometry
However, by introducing this curved connection, the membrane area would be enlarged. However, from a
kinematic point of view, it is very important that the central belt intersects with the intersection points of the
side belts. Therefore, the valley of the membrane is brought up and the membrane panels are cut so that the
three belts intersect at the bottom point (Figure 6-2). This brings the membrane area to 8.01 m², compared to
7.69 m² in the experiment. The discrepancy can be related to the use of the wrong cutting pattern (see 6.3.1.3(a)).
48
Figure 6-2: Development of the valley of the belt
The different names used in this dissertation are indicated in Figure 6-3. The whole 3D set-up is symmetrical to
two vertical planes, so only one quarter will be modelled. This will reduce the computational time of the model
tremendously. One vertical plane lies through the chains of the superstructure, whereas the other is vertically
through the middle of the central belt.
Since the final model is very complex, it was built by gradually adding complexity. First, the frame for opening
and closing of the membrane was simulated to finalise the kinematics of the structure. In a next step, the contact
between the central belt and the membrane was simulated separately. Later on, the stitched edge belts were
added as well as the reinforcement in the corners.
Figure 6-3: Names of the different parts used in this master dissertation
6.3 The implicit analysis in Abaqus/Standard 6.3.1 Step 1: Pretensioning the membrane The experiment from the VUB starts by pretensioning the central belt, after which the membrane is brought up,
matching the geometrical position that was determined in Easy. In the Abaqus model, these two actions will be
performed simultaneously during the first step. By pretensioning the central belt, the valley of the membrane
will shape itself around the central belt, simulating what happens during the experiment. Simultaneously, the
upper force is applied as well, ensuring the right position of the membrane. The different components of the
numerical model in Abaqus/Standard are discussed in this section.
6.3.1.1 Side belts The belts have a width of 5 cm, a thickness of 2.3 mm and are modelled with an isotropic material model. These
belts are however also woven material, indicating that two orthogonal material directions exist with different
properties. The material model is thus an approximation, made because only results from an uniaxial test were
Removed membrane area
Added membrane valley
Top
Apex points
Central belt
Link
Side belt
Bottom
Bottom
Chains
Fixed point
Fixed point
Point B
49
received (Table 6-1). The elastic modulus is based on the stiffness between 50 MPa and 290 MPa. However,
further testing showed that the stiffness changes during cyclic loading and that the applied elastic modulus may
be too high. Therefore, the influence of the stiffness will be investigated in Chapter 8.
Table 6-1: Mechanical properties of the belts
𝐸 [𝑁/𝑚𝑚²] 𝜈 [−] 𝜌 [𝑘𝑔/𝑚³]
Belts 3492 0.3 1400
The element type chosen is a linear shell element S4R. It is a general purpose 4-node element, often used to
model doubly curved thin shells. These type of elements have a bending stiffness, which will result in a stiff
behaviour compared to the experiment. However, if membrane elements are chosen, the stability of the overall
set-up is endangered during the simulation. Therefore, the shell elements are a good approximation.
The four belts are stitched to the membrane so that sufficient pretension can be obtained in the membrane. This
stitching will be simulated as a tie constraint between the membrane and each belt. All degrees of freedom are
tied, so that the two surfaces act together as a whole. A partition is made on the membrane and belts in order
to restrict the zone for the tie constraint. The belts are the master surface, where the membrane is the slave
surface. This implicates that the mesh of the belts has to be coarser than the membrane’s. The initial position of
the surfaces is adapted so that they overlap perfectly.
A double layer of belt material is provided at the top and the bottom of the belt in order to:
provide loops in the side belts, making it possible to connect the belts to the superstructure and bottom
rings
add additional reinforcement at the location where a large concentrated force is introduced in the
membrane
By using an extra belt layer as slave surface and tying it to the master belt surface, this reinforcement can be
simulated in the model (Figure 6-4).
Figure 6-4: Different layers in the model
Initially, the belts are stitched on the membrane with a circular form due to the sag. If the belts are pretensioned,
the belts will be uniformly loaded due to the contact with the membrane (Figure 6-5). Where there is no contact,
the belts are simply straight. The pretensioning force will thus always be tangent to the belt. The same principle
holds for the central belt.
Belt (master)
Reinforcement belt (slave)
Membrane (slave/master)
Reinforcement membrane (slave)
50
Figure 6-5: Forces on edge belts [1]
The use of the turnbuckles to adapt the pretension is simulated as an axial displacement between the centre of
the belt and the bottom point (Figure 6-6). A connector parallel to the edge of the belt with a ‘slot and align’ type
simulates this behaviour properly. This connector only allows axial displacement between two points. The axial
displacement direction is specified by introducing a local coordinate system that has its x-axis between the centre
of the belt and the bottom point. The pretension can then be introduced as a connector force. In this way, a
virtual load cell is modelled. The variation of the force is given in Figure 6-7.
Figure 6-6: The use of a 'slot and align' connector to tension the belts
To ensure that the edge of the belt follows the movement of the centre, the edge of the belt is coupled to the
centre. All degrees of freedom are coupled since the shell elements have a bending stiffness. Finally, mechanical
properties can be assigned to the belt connector. The connector receives a spring stiffness, reflecting its material
stiffness. The spring constant can be calculated as: 𝐷11 = 𝐸𝐴.
Table 6-2: Mechanical properties of the belt connector
𝐴 [𝑚𝑚²] 𝐷11 [𝑁/𝑚𝑚]
Belt connector 201.06 449,650
Slot+align
Slot+align Bottom
membrane
Pretensioned belt
51
Figure 6-7: Variation of the forces during the simulation
6.3.1.2 Central belt The central belt is the same belt as the side belts and is therefore also modelled with an isotropic material model.
Pretension is introduced in the same way as for the side belts: by using a ‘slot and align’ connector and a coupling
constraint. Since a quarter of the model is simulated, only half of the measured force is applied. The variation of
this force is also given in Figure 6-7.
The membrane will curve around the central belt during pretensioning. This is thus a type of form-finding, where
the membrane reaches its equilibrium state based on the pretension, upper force and position of the fixed
boundary (see 6.3.1.4). In the experiment, the belt has only little bending stiffness, so it will curve as well (Figure
6-1). To improve convergence as much as possible, the curve of the belt and membrane are exactly the same in
the initial geometry. Membrane elements were used for the central belt, so that the central belt could be curved
by the membrane during pretensioning. However, this resulted in divergence of the simulation. Therefore, again
S4R shell elements are used, leading to satisfactory results as well.
It was noticed that the central belt and the membrane valley penetrated in the initial geometry, due to
approximations of the curve when importing the model in Abaqus (Figure 6-8). The two surfaces did not
recognize each other, so no contact was introduced between the two, resulting in penetration. To solve this
issue, the overclosure was adapted in the contact properties. In this way, the position of the nodes is adapted to
eliminate penetration. However, this did not solve the issue. Therefore, the central belt is translated 3 mm
upwards, so that it does not make contact at the beginning of the simulation. By applying the pretension force,
the central belt will straighten and make contact.
Figure 6-8: Approximations of the curvature of the central belt and membrane result in penetration
0
2,000
4,000
6,000
8,000
10,000
12,000
0 50 °Reference
50° 70° 90° 50° 30° 10°
F [
N]
Step
Central belt Belt Upper
52
It is very important that the discretisation of the surfaces is sufficiently small so that the curvature is
approximated as close as possible. In Figure 6-9, the difference is showed: the central belt has only two elements
to form the valley curvature, whereas the membrane has much more. This leads to a difference in curvature
between the belt and the membrane during simulation: gaps form and the central belt penetrates the
membrane. By switching the master/slave relation so that the membrane becomes the master surface, the
central belt’s mesh can be refined. It is namely a rule of thumb that the master surface has to have a coarser
mesh than the slave surface in order to obtain convergence. From different combinations, it appears that setting
the mesh size equal to 5 mm along the transverse curvature is the most efficient way to get convergence.
Additionally, quadratic elements are chosen. These elements have three integration points along an edge,
whereas linear elements have only two. This allows the elements to approximate a curve better.
Figure 6-9: Discretisation of the membrane vs. the central belt
The central belt can slide over the membrane, allowing adjustment of the membrane’s geometry to pretension
it. In the model, the interaction of the two surfaces is modelled by using a contact property with an isotropic
friction coefficient of 0.15 in a penalty friction formulation. The value of the coefficient was however not
measured, but obtained from previous observations. The normal behaviour is described by hard contact. This
minimizes the penetration of the slave surface into the master surface and does not allow tensile stress across
the surface. If the surfaces are in contact, any contact pressure is transmitted. If the contact pressure is zero, the
surfaces separate and the constraint is removed [2]. The discretisation method is the surface-to-surface method.
6.3.1.3 The membrane The membrane is the main point of interest in this simulation. It is reinforced at the corners with an extra layer
of membrane material. These will reduce the stresses from the concentrated force in those zones (Figure 6-4).
The two layers are tied together, where the membrane is the master surface. The two panels of membrane are
welded in the valley. This results in an overlapping strip of membrane material of 5 cm wide. In the simulation,
an additional strip is tied as the slave surface to the membrane.
THE CUTTING PATTERN
The cutting pattern is shown in Figure 5-2. Unfortunately, for the numerical simulations an old cutting pattern
was used, which is 2 cm larger in both warp and fill direction. Since the forces are in that case distributed over a
larger area, the strains will be lower for the larger pattern. However, since the difference is only 2 cm for a length
of 6 m horizontally and 1.7 m vertically, the influence will be rather small. It is however recommended to rerun
the analysis in order to quantify the influence of the geometry on the results.
THE DISCRETISATION
As mentioned above, the discretisation in the valley of the membrane is very important when the contact is
implemented in the model. Since the membrane is the master surface in the contact relation, it has to have a
coarser mesh compared to the belt. The element size along the transverse curve is the same as for the central
belt (5 mm), whereas in longitudinal direction it is larger (10 mm).
For the membrane panels, it is important that the zones with the largest stress and strain variations are
discretised fine enough. This is for example the zone at the top of the membrane, since a lot of wrinkling occurs
there. On the other hand, the zone in the middle of the membrane shows less variation in the experiment and
can thus be meshed coarser. Since the membrane is the slave surface in the tie with the belts, it has to be meshed
53
coarser than the belt. On the other hand, the membrane reinforcement in the corners will be meshed finer. The
optimisation of the mesh is described in Chapter 8.
The quadratic membrane element M3D8 is used, which has 8 integration nodes. As stated before, by using
quadratic elements, the curvature in the valley of the belt will be more easily followed. The same goes for the
weld and reinforcement.
THE MATERIAL MODEL
The aim of this thesis is to compare the impact of three material models on the analysis: the isotropic, linear
elastic orthotropic and elasto-plastic UMAT model. Since the isotropic model is the most simplified, it is most
likely to obtain convergence. Therefore, it is used during the development of the model. The mechanical
properties are given in Table 6-3.
Table 6-3: Mechanical properties of the isotropic material model [3]
𝐸 [𝑁/𝑚𝑚²] 𝜈 [−] 𝜌 [𝑘𝑔/𝑚³] Isotropic 1270 0.313 1416
The properties of the orthotropic material model were obtained from a biaxial test (Table 6-4) [4]. In the
simulations, a plane-stress material model is used, which will simplify the calculations. For the Poisson’s
coefficient, the average value of 𝜈𝑤𝑓 and 𝜈𝑓𝑤 is taken. The stiffness is decreased in comparison with the isotropic
material, therefore, it is expected that the strains will increase. The orthotropic material model demands the
assignment of two orthotropic material directions, which are in this case aligned with the horizontal and vertical
direction.
Table 6-4: Mechanical properties of the linear elastic orthotropic material model [4]
𝐸𝑤 [𝑁/𝑚𝑚²] 𝐸𝑓 [𝑁/𝑚𝑚²] 𝜈𝑤𝑓 [−] 𝐺𝑤𝑓 [𝑁/𝑚𝑚²] 𝜌 [𝑘𝑔/𝑚³]
Orthotropic 975.6 716.1 0.2 87.714 1416
Finally, the simulations will also run with the more complex material model UMAT. The values from the tailored
test will be used in a user material subroutine (Section 2.1.4.5(b)).
6.3.1.4 Boundary conditions Since only a quarter of the membrane will be simulated, boundary conditions have to be assigned along the
symmetry planes. At the transverse edge of the membrane and central belt an XSYMM condition is applied,
constraining the translation in x-direction as well as the rotation around the y- and z-axis (Figure 6-10). Along the
longitudinal edge of the central belt and valley of the membrane, a ZSYMM condition is applied (restraining Uz,
URx and URy).
Figure 6-10: Boundary conditions on the membrane due to symmetry: XSYMM and ZSYMM
54
By tensioning the belts, the apex point will move inwards, which changes the opening angle. This can be solved
by constraining the z-translation at this apex point. This boundary condition becomes inactive once the
membrane is opened. The upper force at the top point is introduced, where the vertical translation is free.
The belts are connected to the bottom point, which represents the large ring in the experiment. This point can
move freely, but is connected with a turnbuckle to a fixed point. The mechanism of the turnbuckle is however
translated into ‘slot and align’ connectors which are loaded with the reaction forces measured in the load cells.
The simulation is thus force controlled, whereas the experiment is geometrically controlled. The connection
between the bottom and fixed point can be represented by a ‘link’ connector, which constrains only the axial
displacement between the two points, allowing the bottom point to move freely. At the fixed point, the
translation (Ux, Uy and Uz) is fixed. Due to symmetry, the z-translation is constrained at the bottom point (Figure
6-12).
Figure 6-11: A link connector [2]
Figure 6-12: Uz constraint bottom point due to symmetry
One major issue is the position of this fixed point. The coordinates of this point are known with respect to the
positions of the markers in the reference state, however, not in the initial geometry state from which the
simulation starts. However, in order to obtain the position of the markers, pretensioning has to be applied,
hence, the position of the fixed point has to be known. This will require a series of iterative simulations.
Additionally, the position of this fixed point will influence the vertical translation of the membrane. This is
illustrated in Figure 6-13. The force in the link is in horizontal and vertical equilibrium with the forces in the
central belt, side belt and upper force. For simplicity, the influence of the latter is neglected. If for example the
force of the central belt increases faster than the force in the belt, then the force in the link will increase as well
and change direction. The direction of the link will thus change as well. Since the fixed point is fixed in translation,
the membrane will move downwards. However, if the fixed point would be located at a different y-position, the
translation of the membrane would be different as well. The position of the link and the vertical translation of
the membrane will thus influence each other.
Figure 6-13: Forces in the connection between the belts, bottom and fixed point
Bottom
Fixed
𝐹𝐶𝐵 𝐹𝐶𝐵 ↑
𝐹𝐵
𝐹𝑙𝑖𝑛𝑘 𝐹𝑙𝑖𝑛𝑘
Bottom Fixed
55
6.3.2 Step 2: Opening the membrane The frame consists out of a planar truss system and two chains which connect it to the upper point. These chains
are modelled as truss elements as well. An isotropic material model is assumed, with mechanical properties as
given in Table 6-5.
Table 6-5: Mechanical properties of the steel members
𝐸 [𝑁/𝑚𝑚²] 𝜈 [−] 𝜌 [𝑘𝑔/𝑚³] 𝐴 [𝑚𝑚2]
Steel 21,000 0.3 7850 736 As stated above, the opening of the frame is performed by adjusting the length of the link between two nodes.
The link is introduced in the model as a connector with a ‘slot and align’ constraint, so that only axial displacement
is allowed between the two nodes. The connector receives again a spring stiffness. Since it is a steel rod with
diameter 16 mm, the spring constant can be calculated as: 𝐷11 = 𝐸𝐴 = 𝐸𝜋𝑑2/4.
Table 6-6: Mechanical properties of the link connector
𝐴 [𝑚𝑚²] 𝐷11 [𝑁/𝑚𝑚]
Link connector 201.06 42,223,005
The length is adjusted continuously by using a tabular amplitude over the steps (Figure 6-14). The opening of the
membrane is the result of a shortening of the link (which is positive on the graph), as can be seen in Figure 6-15.
Figure 6-14: Variation of the length of the opening link during the simulation
Figure 6-15: The opening structure in 50° - 90° - 10°
During the experiment, the position of the upper point is adjusted by moving the crane vertically. Since this would
be difficult to implement continuously in the simulation, it is the upper force that is changed over the different
-500
-400
-300
-200
-100
0
100
200
300
0 50 °Reference
50° 70° 90° 50° 30° 10°
l
[mm
]
Step
Link Truss
Chains
56
steps. This approach should give the same results. The kinematics of the opening frame were tested by using a
simplified membrane (Figure 6-16). When this worked, it was translated into only a quarter of the frame.
In fact, the force measured in the experiment includes the total weight of the structure. Since gravity is excluded
from the simulations, the weight has to be subtracted from measurements. The total mass of the superstructure
can be measured in Abaqus as 54.3 kg. The weight of the steel superstructure is thus:
𝑊𝑠𝑢𝑝𝑒𝑟 = 𝑚 ⋅ 𝑔 = 532.7 𝑁 ⟹ 𝐹𝑢𝑝𝑝𝑒𝑟 = 𝐹𝑢𝑝𝑝𝑒𝑟,𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 − 𝑊𝑠𝑢𝑝𝑒𝑟
The membrane is a very lightweight material, so the weight is neglected. The total upper force is about 16.5 kN,
so the weight of the superstructure is only small in comparison (about 3%). Due to symmetry, only a quarter of
the top force 𝐹𝑢𝑝𝑝𝑒𝑟 is applied on the model.
Figure 6-16: Model for testing the kinematics of the truss system and final superstructure
The centre of the frame is introduced so that the link can be simulated. Several boundary conditions are applied
at the different points of the superstructure:
Centre: The point can move vertically. Therefore, the translation along and the rotation around the
x- and z-direction is constrained.
Point A: The z-translation is constrained due to symmetry. This will prevent rotation of the link in
the xz-plane.
Apex: The translation in x-direction is constrained due to symmetry. To connect the superstructure
to the belts, a tie constraint is used between the apex and the belt edge. The apex point is the
master node, whereas the edge is the slave surface. No adjustment of the initial position is allowed.
Top: It can move vertically. The rotational degrees of freedom are not active. The z-translation is
constrained due to symmetry.
Finally, the opening frame should move in the same horizontal plane. This is achieved by using an ‘equation’-
constraint for the apex and point B:
1 ∙ 𝑢𝑦,𝑎𝑝𝑒𝑥 + (−1) ∙ 𝑢𝑦,𝐵 = 0
6.3.3 Results Divergence occurs very early in the analysis. Several measures were tried to improve convergence:
Applying a directly specified damping factor: with low damping factors, convergence cannot be reached
whereas with high factors the stabilisation energy is too high.
Changing the time increment controls as described in Section 4.1.
Using idealised shell elements S4R: this improves convergence slightly.
B
Centre Apex
Top
57
However, divergence occurs after 0.01 s in the first step. It should also be remarked that even if convergence
would be reached, the computational time would be much too large to have a practical use. The implicit analysis
is therefore not suited for the analysis.
6.4 Explicit analysis in Abaqus/Explicit 6.4.1 Differences in the model Due to convergence problems and a large CPU time in Abaqus/Standard, the analysis is performed in
Abaqus/Explicit as well. The model is based on the same geometry and principles as described above. Some
aspects of the analysis are however different:
Element types
The membrane elements are changed into explicit elements. No quadratic elements can be chosen. An idealised
shell element cannot be used since no strain or stress output is available which can be compared to the
experimental values.
Amplitudes
Important in an explicit analysis is that no load or boundary conditions has an abrupt change in value. Therefore,
smooth step amplitudes are introduced for the reaction forces in the belts, the upper force and the change in
length of the link. Also an amplitude is used for the z-translation constraint of the apex point during the
pretensioning state.
Boundary conditions
In the Abaqus/Explicit analysis, the rotation constraints have to be added on the centre point and point B on the
frame (Figure 6-16), since these points have no rotary inertia.
Contact definition
The penalty contact method is preserved in the Abaqus/Explicit analysis, where finite sliding of the surfaces is
allowed.
Mass scaling and loading rate
As discussed in Section 4.2, the computational costs can be decreased by using mass scaling and decreasing the
loading rate. From several observations, it could be concluded that adding mass scaling results quickly in dynamic
effects. This is due to low mass of the membrane material. Therefore, mass scaling is not applied. From a trial-
and-error method, adequate step times could be chosen.
6.4.2 Results of the Abaqus/Explicit analysis 6.4.2.1 General method of comparing results First, the general method of comparing the simulation with the experiment is discussed. Several properties were
measured during the experiment which can be compared with the output of the analysis:
The overall geometry of the membrane
During the experiment, the geometry and position of the membrane was measured by the means of markers.
The position of these markers can be compared with the numerical model as well to obtain information about
the deformation as well as the sag of the central belt.
Contour plots for the strains in warp and weft direction
It was shortly mentioned in Section 5.2, that the actual reference state was measured at the end of the
deployment. The reference state was a fully closed, slightly tensioned state which showed a lot of wrinkling.
Since the material is history dependent, this reference state will exclude a large amount of the permanent strains,
resulting in very small strains during the deployment. Additionally, this state cannot be simulated. Therefore,
58
another reference state is chosen: the 50° opening angle where the apex points are in the correct position.
Contour plots of 𝑒𝑥𝑥 and 𝑒𝑦𝑦 were then obtained for the 50°/70°/90°/50°/30°/10° after relaxation. Since the
strains are small, the comparison will focus on the extreme opening angles of 90° and 10°. When comparison is
made, both contour plots are plotted on the same colour scale.
Values of the strains 𝜖𝑥𝑥, 𝜖𝑦𝑦 and 𝜖𝑥𝑦
Exact values of the strains were extracted along three paths (Figure 6-17):
Path 1: A vertical path, starting under marker 11 and ending above the central belt
Path 2: A horizontal path starting next to marker 10
Path 3: An oblique path with the starting point under the starting point of path 2 and the end point
as the middle of the membrane at the top.
Figure 6-17: Three paths in the experiment
Unfortunately, the software to post-process the DIC does not use the same coordinate system as was used to
subtract the position of the markers. It chooses the best fitting plane. Therefore, the coordinates of the markers
and the points along the paths are in a different coordinate system, which are related by an unknown translation
and 3D rotation. Since the markers lie outside the DIC speckle pattern, their position had to be estimated in the
coordinate system of the paths.
Marker 11 lies at the centre of the belt and at the top of the membrane, which makes it easy to find its position
in the numerical model. A translation in x- and z-direction between the two positions will then allow to find the
coordinates of the points on the paths as well.
The definition of the three paths is not straightforward. From the coordinates of marker 11 in the experiment
and in the numerical model, an x- and y-transformation can be applied on the coordinates of the paths, resulting
in coordinates in the model. The three paths are given in Figure 6-18. Path 1 could be defined from the
transformation and gives good correlation with the coordinates from the experiment. Path 2 was defined in the
same way, however, it was noticed that the starting point was not as close to the belt as in the experiment.
Therefore, the path was extended with along a straight horizontal line. Finally, path 3 had to be determined
manually, since the coordinate transformation resulted in a starting point outside the membrane. This is due to
the different cutting pattern that was used as well as different deformations obtained in the simulation.
1
2
3
Marker 11
Marker 10
Marker 6
59
Figure 6-18: Determination of the three paths
It should be noted that although the x- and y-coordinates correlate within some extend (the maximum deviation
is 4 mm), the z coordinate did not correspond well at all. This could indicate that the opening angle is different
between the experiment and the model. However, the length of the link in the simulation corresponds to
theoretical values obtained from the EASY-model (Figure 6-19). During the experiment, the angles were
approximated as close as possible however, imperfections are inevitable. Additionally, the DIC cannot measure
the z-coordinate perfectly as well. Also the influence of the cutting pattern will play a role as well.
Figure 6-19: Variation of the length of the link
Relative strain deviation
In order to get an idea of the error between the simulation and the experiment, relative strain deviation can be
calculated along the path as:
Δ𝜖 =𝜖𝑒𝑥𝑝 − 𝜖𝑠𝑖𝑚
𝜖𝑒𝑥𝑝
If the value is positive, it means that the simulation underestimates the strains. If it is negative, the strains
obtained from the simulation are larger than in the experiment.
1000
1500
2000
2500
3000
0 1 2 3 4 5 6 7 8
Len
gth
of
the
lin
k [m
m]
Path 1: from transformation Path 2: from transformation
Path 2: adapted Path 3: manually
60
6.4.2.2 Results of the explicit analysis Convergence was obtained for the explicit simulation with an isotropic material model. No mass scaling was
introduced and the step time was increased to reduce the loading rate. This minimises the inertia effects. The
final combination is given in Table 6-7.
Table 6-7: Step time in the Abaqus/Explicit simulation
Step Step time [s]
50° reference state 10 50° after relaxation 10
Open to 70° 20 Open to 90° 20 Close to 50° 40 Close to 30° 20 Close to 10° 20
GEOMETRY
By pretensioning the central belt, the membrane finds its equilibrium shape around the belt. This can be seen in
Figure 6-20. The initial flat panels are now curved, and slight wrinkling is introduced around the belt. It should be
remarked that this stage showed wave propagation, indicating that the inertia effects are still too high.
Figure 6-20: Form-finding: the difference between the initial state and the reference state
An overview of the deformation for different opening angles is given in Figure 6-21. After pretensioning, wrinkling
is introduced on the top of the membrane and in the bottom corners. A bulge is also formed at the top. This was
observed in the experiment as well. Additionally, wrinkling occurs in the middle of the membrane, which was not
encountered in the experiment. After opening the membrane to 90°, the wrinkling becomes less whereas in the
10° opening angle, the wrinkling is worse and occurs over the full membrane area. In the simulations, small wave
propagation is observed, indicating that the inertia effects might still be too high.
Figure 6-21: Overview of the deformations: initial geometry - 50° reference state (top) - 90° - 10° (bottom)
61
CONTOUR PLOTS OF THE STRAINS
In Figure 6-22 the contour plots for the 90° opening angle are given. The overall strains in warp direction are
positive, indicating that the membrane is in tension. The strains are however lower than in the experiment. This
can be explained by the use of an isotropic material model: the stiffness is too large, resulting in smaller strains.
At the top zone, a compression zone is noticeable in the numerical results, however, in the experiment, a
concentration of compressive strains is found in the corner whereas a tensile zone is found in the middle. This
indicates that the material shows wrinkling. Although the geometry shows wrinkling, the variations are very small
and cannot be noticed on the contour plots. Finally, the results from the simulation show a very ‘speckled’
pattern, indicating that the explicit analysis results are not uniform over the membrane.
The strains in warp direction for the 10° opening angle (Figure 6-23) show a highly speckled pattern. The overall
strain values are however in the same order as for the experiment: in the lower part of the membrane tensile
strains are present. In the bottom corner, the strains are higher. In the top zone, again the concentration of
strains under the corner is missing, due to the isotropic material model.
The strains in fill direction are much more uniform for both opening angles. The values are also overestimated
by the analysis, indicating that the stiffness in the material model is too high.
Overall it can be stated that the explicit analysis with an isotropic material model results in a small overestimation
of the strains and lacks the possibility of modelling wrinkling. Additionally, the analysis shows a speckled pattern,
more for the warp strains then for the fill strains.
Figure 6-22: Contour plots for the 90° opening angle - 𝝐𝒙𝒙 and 𝝐𝒚𝒚
62
Figure 6-23: Contour plots for the 10° opening angle - 𝝐𝒙𝒙 and 𝝐𝒚𝒚
COMPARISON OF THE WARP STRAINS 𝜖𝑥𝑥
The values for the warp strains along path 1 are found in Figure 6-24. For both opening angles, the strains are
smaller in the simulation than in the experiment. This indicates that the isotropic material is too stiff to model to
real behaviour of the membrane. For the 10° opening angle, the zone at the top (which is the start of path 1)
shows large deviations. This can be explained by the inertia influence which is too larger. The 90° opening angle
shows large compressive strains along the path, indicating the wrinkle that was visible on the contour plots. This
is not visible in the simulations.
For the strains along path 2, the same trends can be observed (Figure 6-25). The disturbed zone for the 10°
opening angle is however more extended and located in the central zone of the membrane. The strains along
path 3 show very similar behaviour (Figure 6-26). Due to a miscorrelation of the coordinates, the paths in the
simulation are shorter. However, the overall trend is again visible: the strains are smaller and smoother in the
90° opening angle, whereas more disturbance is noticeable in the 10° opening angle.
The conclusions are confirmed by the relative strain deviation (Figure 6-27). The deviations are larger for the 10°
opening angle, with an average error of 200%. For the 90° opening angle, this is only 10%. This indicates that the
closing operation is more susceptible to the inertia effects.
It should however be mentioned that the strains measured and modelled are very small: in the order of 0.1%.
Therefore, even though the deviations seem large, these are still very small when the larger context of the
building industry is considered.
Overall it can be stated that the explicit analysis gives adequate trends but large fluctuations are found, especially
for the 10° opening angle. This might indicate that the inertia effects are still too large in the closing operation.
The use of the isotropic material model leads to an underestimation of the strains and does not make it possible
to simulate wrinkling.
63
Figure 6-24: Values of the warp strains along path 1
Figure 6-25: Values of the warp strains along path 2
Figure 6-26: Values of the warp strains along path 3
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0 200 400 600 800 1000 1200St
rain
[-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10°
-0.015
-0.01
-0.005
0
0.005
0.01
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10°
-0.015
-0.01
-0.005
0
0.005
0.01
0 500 1000 1500 2000 2500
Stra
in [
-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10°
64
Figure 6-27: Relative strain deviation for the warp strains
COMPARISON OF THE FILL STRAINS The values along the three paths are given in Figure 6-28, Figure 6-29 and Figure 6-30. The overall strains are
slightly larger than the strains in the warp direction (0.8% instead of 0.5%). The same trends are observed: the
strains are underestimated due to the isotropic material model and the angle of 10° shows a lot of fluctuation,
due to the inertia effect.
The relative strain deviation is given in Figure 6-31. Again, the 90° opening angle shows less deviation, however,
for all paths outliers are recorded. Path 1 results a larger error. The fault for 10° is again about 50%.
Again it is noticed that adequate trends are found from the simulations, but large fluctuations show, especially
for the 10° opening angle.
Figure 6-28: Values of the fill strains along path 1
-2
-1
0
1
2
3
4
5
6
90°Path 1
10°Path 1
90°Path 2
10°Path 2
90°Path 3
10°Path 3
Re
lati
ve s
trai
n d
evi
atio
n [
-]
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10°
65
Figure 6-29: Values of the fill strains along path 2
Figure 6-30: Values of the fill strains along path 3
-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
0 200 400 600 800 1000 1200Stra
in [
-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10°
-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
0 200 400 600 800 1000 1200Stra
in [
-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10°
66
Figure 6-31: Relative strain deviation of the fill strains
COMPARISON OF THE SHEAR STRAIN
Finally, also the shear strain values can be analysed (Figure 6-32, Figure 6-33 and Figure 6-34). The shear strains
are much larger than the warp or fill strains (order of 1%). Additionally, they show more variation and range
between 3% and -0.1%. The extreme values are reached for the 90° opening angle. Again the fluctuations are
visible for the 10° opening angle. The values for path 2 are almost 0 in the simulations, and slightly higher in the
experiment. This can be explained by the high value of the shear modulus for an isotropic material model. This
modulus can be calculated as:
𝐺 =𝐸
2(1 + 𝜈)= 483.625 𝑀𝑃𝑎
This explains the very small values for the strains along the three paths. The relative strain deviation is given in
Figure 6-35. Again the error is much lower for the 90° opening angle as for the 10° opening angle. Path 1 is here
an exception: this is due to the fact that the wrinkling cannot be simulated by the isotropic material model. The
lack of wrinkling makes the strain output very different in the simulation.
Figure 6-32: Values of the shear strains along path 1
-1
-0.5
0
0.5
1
1.5
2
2.5
90°Path 1
10°Path 1
90°Path 2
10°Path 2
90°Path 3
10°Path 3
Re
lati
ve s
trai
n d
evi
atio
n [
-]
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10°
67
Figure 6-33: Values of the shear strains along path 2
Figure 6-34: Values of the shear strains along path 3
Figure 6-35: Relative strain deviation of the shear strains
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10°
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10°
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
90°Path 1
10°Path 1
90°Path 2
10°Path 2
90°Path 3
10°Path 3
Re
lati
ve s
trai
n d
eiv
atio
n [
-]
68
6.5 Conclusion The experiment conducted at the VUB in the framework of the Contex-T project was translated into a numerical
model in Abaqus/Standard. The different aspects were discussed in this chapter. The main point of interest was
the pretensioning of the membrane by the means of a central belt. A type of form-finding analysis was
performed, where the membrane was brought to its equilibrium state under the application of the reaction
forces.
The analysis in Abaqus/Standard did however not result in a trustworthy, efficient and converging solution.
Therefore, the model was transferred to an Abaqus/Explicit model by adapting some properties. This resulted in
a converging solution, where the inertia forces were toned down by reducing the loading rate.
Overall, it can be concluded that analysis shows the tendency of the strains. However, for the 10° opening angle
a lot of wrinkling in the membrane area and flutter in the results is observed, indicating that the inertia effects
are still too high. The loading rate should thus be decreased further to obtain more uniform results. Also the
influence of the material model could be further investigated.
It should however be mentioned that the Abaqus/Explicit model asks a lot of CPU time at this point. It is expected
that adding the complex material model and decreasing the loading rate will further increase this time
tremendously. This can endanger the efficiency of the analysis.
69
Chapter 7: Numerical deployment of a tensioned membrane with SLIPRING connectors Due to the difficult convergence and large CPU time encountered with the Abaqus/Standard and Abaqus/Explicit
simulations, another approach for modelling the central belt was explored. In the EASY simulations performed
at the VUB, the belts were simulated as cable elements, disregarding the sliding interaction between the belts
and the membrane. This interaction can however be introduced by using SLIPRING connectors. Additionally, the
uncertainties about the curvature of the central belt as well as the wrinkling at the bottom during the
pretensioning stage are eliminated.
7.1 Including SLIPRING connectors into the model SLIPRING connectors model material flow and stretching between two pints. No relative motion is constrained
[28, Ch. 31]. The belt material flows and stretches between nodes a and b (Figure 7-1). At point A, the material
flow 𝜓 equals zero, as well as at the bottom point, indicating that no material can leave the SLIPRING segment
at that position. The belt is thus stretched between these two points. The belt’s mechanical properties can be
assigned to the connectors (Table 7-1).
Figure 7-1: SLIPRING connectors: a. principle [1] and b. location in the model
The definition of friction for SLIPRING connectors is related to the angle 𝛼 over which there is contact in point b
(Figure 7-1). In Abaqus/Standard, this angle has to be specified at the beginning of the analysis. Since it changes
constantly during pretensioning, friction cannot be included. This issue can be solved in Abaqus/Explicit. The
angle is computed automatically, allowing adjustment of the contact angle during the simulation.
Table 7-1: SLIPRING connector properties
𝐴 [𝑚𝑚²] 𝐷11 [𝑁/𝑚𝑚] Mass per length [kg/mm] Friction [-]
SLIPRING connector 201.06 449,650 9570 E-6 0.15
Along the symmetrical edge in the valley of the membrane, a connector is introduced between each node of the
mesh. This is automatically performed by using a script (See Annex). The outer point of the membrane is then
connected to the bottom point with another SLIPRING connector. This last connector is loaded with half of the
reaction force measured in the central belt in order to provide pretension.
This model was again built by adding complexity. It is observed that adding the tied reinforcement of the
membrane and belts, the convergence becomes very difficult (time increments in the order of 1E-7 s). Therefore,
a composite section is used instead of a layer-up composition. Partitions are made in the belts and membrane.
The thickness is increased locally.
Material flow = 0
A
SLIPRING
70
7.2 Frictionless contact The model is first analysed in Abaqus/Standard. This implicates that friction between the central belt and the
membrane cannot be included in the model. The analysis will however provide an insight in the mechanism of
SLIPRING connectors. Easy convergence is obtained.
7.2.1 Optimisation of the mesh Since the analysis in Abaqus/Standard with an isotropic material asks low CPU time, it is the ideal model to
optimise the mesh. Due to the reinforcement at the top, there is a sudden change in stiffness and thus in strain
in that area. In order to make this transition less abrupt, the mesh is refined in this upper area. Additionally, also
the mesh in the middle of the membrane is refined. Four different meshes are compared. The influence of the
number of elements on the CPU time is analysed (Figure 7-3) as well as the strain in warp direction E11 along the
vertical path (Figure 7-2 and Figure 7-4). Only the first pretensioning step with an isotropic material is simulated.
Figure 7-2: The position of the vertical path
Figure 7-3: CPU time in function of the number of elements
Figure 7-4: Comparison along the path for different meshes
From Figure 7-3 it can be concluded that by increasing the number of elements, the CPU time is increased as
well. From the comparison along the path in Figure 7-4 it can be concluded that the two last mesh compositions
give a more sudden transition from the double-layered zone to the single layer membrane. The mesh with 9788
elements is chosen to reduce the CPU time as much as possible.
0
5,000
10,000
15,000
20,000
25,000
30,000
0 2,000 4,000 6,000 8,000 10,000 12,000
Tota
l CP
U t
ime
[s]
Number of elements on membrane [-]
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 200 400 600 800 1000 1200
E11
[%
]
True distance along path [mm]
4993 8193 9788 11397
-0.1
-0.08
-0.06
-0.04
-0.02
0
100 150 200
E11
[%
]
True distance along path [mm]
4993 8193
9788 11397
71
Additional verifications are performed. The mesh is a combination of quadrilateral and triangular mesh elements.
Several properties are verified [28]:
The shape factor: this propertie is only calculated for triangular elements as:
𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 =𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑎𝑟𝑒𝑎
𝑜𝑝𝑡𝑖𝑚𝑎𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑎𝑟𝑒𝑎≥ 0.01
The small face corner angle: two edges of the element should meet at an angle larger than 10
(quadrilateral) or 5 (triangular).
The large face corner angle: two edges of the element should meet at an angle smaller than 160
(quadrilateral) or 170 (triangular).
The aspect ratio: the ratio beteen the longest and shortest edge should be smaller than 10 for both
types.
Geometric deviation factor: this property indicates how much the element edge deviates from the
original geometry, by dividing the maximum gap between an element edge and its parent geometric
face by the length of the edge. It should be smaller than 0.2 by default.
These verifications are performed automatically in Abaqus and no error was discovered. The mesh is shown in
Figure 7-5.
Figure 7-5: Final mesh of the membrane with 9788 elements
72
7.2.2 The elasto-plastic material model UMAT In order to obtain convergence of the analysis, a constant damping factor is specified (Chapter The value is again
determined by trial-and-error, where the stabilisation and total internal energy are compared. The parameters
obtained are given in Table 7-2.
Table 7-2: SLIPRING elements: convergence control parameters (UMAT)
Step time [s] Damping factor [-]
50° reference state 1 1E-8 50° after relaxation 1 1E-8
Open to 70° 1 1E-6 Open to 90° 1 1E-6
Close back to 50° 2 1E-6 Close to 30° 1 1E-6 Close to 10° 1 1E-6
The dissipated energy is sufficiently low compared to the total strain energy, as can be seen from Figure 7-6. The
total internal energy reaches its maximum in the 90° opening angle and returns to zero if the membrane is closed
back to the 50°. Over a short period of 0.015 s around 6 s, the damping energy becomes too high due to this
phenomenon. However, the deformation in the simulation does not show localised deformations. During the
first 2 seconds, the membrane is not opened, hence the internal energy is small. From the detail of the graph, it
can be concluded that the stabilisation energy is also during that period smaller than 5% of the internal energy.
The damping will thus not influence the solution.
Figure 7-6: SLIPRING UMAT simulation: the energy plots – Detail of the first 2 s
0
5E+11
1E+12
1.5E+12
2E+12
2.5E+12
3E+12
3.5E+12
4E+12
0 2 4 6 8
Ene
rgy
[mJ]
Time [s]
ALLIE ALLSD
0.0E+00
5.0E+04
1.0E+05
1.5E+05
2.0E+05
2.5E+05
3.0E+05
0 1 2
Ene
rgy
[mJ]
Time [s]
ALLIE ALLSD
73
Geometry The geometry for different opening angles is given in Figure 7-7. For the 50° reference state, a bulge forms as well as wrinkling in the top zone. Also a small wrinkle forms at the bottom points. During the experiment, it was observed that the membrane lost contact with the central belt at these bottom points. Since SLIPRING connectors model the central belt as if it is always in contact with the membrane, the excess of membrane results in wrinkling.
For the 90° opening angle, the bulge disappears and only small wrinkles are present in the top zone. The wrinkling at the bottom is still present.
Finally, if the membrane is closed to 10°, the wrinkling at the top has almost disappeared. A slight bulge shows however.
Overall, the membrane shows less wrinkling in the simulation than in the experiment. This is due to the use of membrane elements: these model wrinkling more arbitrarily in the simulation.
Figure 7-7: The geometry in the different opening angles:
50° (reference state) - 90° - 10°
Comparison of the contour plots (A) THE 90° OPENING ANGLE
In Figure 7-8 the contour plots for the 90° opening angle are given. For the strain in warp direction 𝜖𝑥𝑥, it is
immediately noticed that a high concentration of (negative) compressive strains is visible in the top zone of the
membrane during the experiment. This is due to the fact that the belts are moved towards each other by the
application of the upwards upper force. Additionally, crimp interchange plays a role as well: since the upper force
is in the direction of the fill yarns, the yarns are stretched. This results in compressed warp yarns and thus a
shortening of these yarns. This concentration is however less pronounced in the simulation.
At the bottom, the experiment shows an compression zone, whereas the membrane is in tension in the
simulation. This is due to the lack of friction: friction would counteract the tension force from the belts. In this
case however, the tensioning of the belts results in a horizontal tensioning of the membrane. The concentration
of high tensile strains is noticed in both the simulation and the experiment.
The tensile strains in fill direction 𝜖𝑦𝑦 are larger in the simulation than in the experiment for the largest part of
the membrane. This can again be explained by the lack of friction: the central belt will pull the membrane down
without any resistance, resulting in larger vertical strains. The legend is set uniformly for the contour plot of the
simulation to receive more detailed information (Figure 7-9). The zone with a large strain is located again at the
top, which is a tendency that is similar to the experiment. The influence of the reinforcement is clearly visible:
where the double layer ends, a sudden increase in strain is noticed due to a stiffness that is reduced by half.
74
Figure 7-8: Contour plots for the 90° opening angle - 𝝐𝒙𝒙 and 𝝐𝒚𝒚 [-]
Additionally, where negative warp strains were measured, positive fill strains are obtained. This can be explained
by the crimp interchange.
At the zone in the middle at the top, the membrane is in compression and a bulge is formed. This is again due to
the inward movement of the belts due to the upper force. This phenomenon is more pronounced in the
experiment than in the model. This is again due to the lack of friction: the central belt does not interact
horizontally with the membrane, so the membrane is smoother. However, this high compression zone is rather
localised. For the largest part of the membrane, the strains are smaller than 1%.
Figure 7-9: Detailed contour plot of 𝝐𝒚𝒚 in 90° opening angle [-]
(B) THE 10° OPENING ANGLE
The contour plots for the 10° are given in Figure 7-10. From the strains in warp direction, following comparisons
can be made. Again, the membrane is in compression in a small zone at the top corners due to the movement of
the belts and the crimp interchange. Also for this angle, this is absent in the simulation. The rest of this upper
75
zone is in small tension, which is resembled by the simulation as well. The rest of the membrane shows uniform
compression in the numerical model, which corresponds to the experiment. However, the experiment shows
slight variations at the corners, indicating some slight wrinkling. Again the lack of friction straightens the
membrane area and decreases wrinkling.
The strains in fill direction show similar behaviour between the experiment and the simulation. A concentration
of tension is found in the upper corner of the membrane. The top zone is for both cases in tension, however, this
zone is larger in the simulations than in the experiment. Again, the numerical model shows less variation in
strains. The lack of frictions decreases wrinkling clearly.
Again the crimp interchange shows in both the model as the experiment: negative warp strains result in positive
fill strains and vice versa.
Figure 7-10: Contour plots for the 10° opening angle - 𝝐𝒙𝒙 and 𝝐𝒚𝒚 [-]
In general, it can be concluded that the lack of friction decreases the wrinkling in the membrane. Additionally, it
shows that this influence is more pronounced for the 90° opening angle than for the 10°.
Comparison of the strains along paths (A) THE WARP STRAINS 𝜖𝑥𝑥
The values along the vertical path are given in Figure 7-11. The values for the 90° opening angle are much lower
in the simulation, which was also noticed from the contour plots. The lack of the wrinkle in this area results in
much lower strains. However, the tendency of decreasing strains when going down on the path is resembled in
the simulation as well. The largest difference is observed for the beginning for the top area. This was noticed
from the contour plots as well: the lack of friction decreases variations in strains. An additional influence on
discrepancies between the values can be due to the mismatch in paths: only an estimation of the begin and end
point could be made. Since the upper zone is susceptible to large deviations in strains, a mismatch in paths can
result in different values as well.
For the strain values along the horizontal path (Figure 7-12) the same trends can be observed. The strains are
much lower and more stabilised in the simulation for the 90° than in the experiment. This can be explained by
the lack of friction: The strains for the 10° opening angle are in good comparison.
76
Figure 7-11: Strain values 𝝐𝒙𝒙 along path 1
Figure 7-12: Strain values 𝝐𝒙𝒙 along path 2
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0 200 400 600 800 1000 1200St
rain
[-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10°
-0.016
-0.014
-0.012
-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10°
77
Figure 7-13: Strain values 𝝐𝒙𝒙 along path 3
Figure 7-14: Horizontal stress 𝝈𝒙𝒙 in the 90° opening angle [N/mm²]
In Figure 7-14, the horizontal stress is given at the end of the simulation. Since the stress is larger than 15 MPa
at the bottom of the membrane, plastic deformations will occur in that zone. The zone under the top corner is
very slightly in tension (about 1 MPa), whereas the middle of that zone is in compression. The compressive
stresses are due to the inward movement of the belt when the upper force is applied. Pretension is lost in this
area and the membrane is compressed. The tensile stresses under the corner indicate wrinkling. Additionally,
the influence of the crimp interchange can be seen as well: by tensioning the fill yarns, the warp yarns are
compressed.
The relative deviation of the strain in warp direction is given in Figure 7-15. The error is maximally 100%. The
deviation is largest for the 10° opening angle. For all cases, the simulated strain is smaller than the experimentally
measured strain. For the opening angle of 90°, path 1 and 2 show excessive outliers, which was also noticed from
the graphs.
The values should be considered as a result of several differences between the experiment and the numerical
model. The model is made perfectly symmetric, however, in practise always an error is introduced. The loads are
slightly asymmetric, the geometry is imperfect, there is an error in the DIC output and the material model is not
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0 500 1000 1500 2000 2500Stra
in [
-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10°
78
perfect as well. Additionally, the paths do not correspond exactly, so that strains are compared at slightly
different positions along the paths. This influence is the largest for path 3.
It should however be remarked that the strains are very small, in an order of magnitude of 0.1%. Therefore, a
deviation of 120% is not a dramatic increase when the strains are considered in the larger context of the building
industry.
It can be concluded that the strains in warp direction are lower in the simulations and show less deviations due
to the lack of friction. A maximum error of 100% is encountered.
Figure 7-15: Relative strain deviation of 𝝐𝒙𝒙
(B) COMPARISON OF THE FILL STRAINS 𝜖𝑦𝑦
The values of the fill strains along path 1 are given in Figure 7-16. Overall, the strains are a factor 10 larger than
the horizontal strains. This is due to the upper force, which works directly in the direction of the fill yarns. Initially,
the fill yarns are crimped, which will result in an higher elongation once they are loaded compared to warp yarns.
For both angles, the strains are in the same order of magnitude as in the experiment. The strains for the 10°
opening angle show clearly the same tendency. The deviations along the path are not present in the simulation,
indicating that there is less wrinkling. The influence of the reinforcement is clearly visible in the graph: a sudden
increase in strains is encountered after about 50 mm along the path. Again, large deviations occur at the
beginning and end of the path, which is due to a mismatch in path.
The strains along path 2 are given in Figure 7-17. Again the deviation of the strains is much less for the simulation.
For the 90° opening angle, the strain value is almost constant. This can be explained by the frictionless contact
between the central belt and the membrane: the central belt causes an almost uniform downward pretension in
the middle zone of the membrane. The strains are up to 6 times higher in the simulation. For the 10° opening
angle, the strains approximate the experiment better. The largest difference can be found at the beginning of
the path. Since the vertical stresses are smaller than 5 MPa, the membrane is in the elastic regime, resulting in
small strains (Section 2.1.4.5).
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
90°Path 1
10°Path 1
90°Path 2
10°Path 2
90°Path 3
10°Path 3
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Figure 7-16: Strain values 𝝐𝒚𝒚 along path 1
Figure 7-17: Strain values 𝝐𝒚𝒚 along path 2
Figure 7-18: Strain values 𝝐𝒚𝒚 along path 3
-0.01
-0.005
0
0.005
0.01
0.015
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0 200 400 600 800 1000 1200Stra
in [
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Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0 500 1000 1500 2000 2500Stra
in [
-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10°
80
Finally, the relative strain deviations were also calculated for the fill strains. Again the 10° opening angle shows
the largest deviation, with a maximum error of 150%. This should be put in perspective however: the strains are
from an order of 0.1%, an increase with a factor 15 would still not be very high. It should be noticed that the
error is much larger than for the 𝜖𝑥𝑥 values. This could be due to the effect of the frictionless contact: this affects
the vertical strain more than the horizontal strain.
Figure 7-19: Relative strain deviation of 𝝐𝒚𝒚
(C) COMPARISON OF THE SHEAR STRAIN 𝜖𝑥𝑦
Only output along the three paths is provided for the shear strain, no contour plots. Along path 1, the strains are
larger than the fill strains, especially along path 1. The strains are close to zero in the centre of the membrane,
except for the 90° opening angle in the experiment. When opening the membrane, the upper force is increased,
resulting in a rotation in the belts. This causes shear in the membrane.
From the in-plane loading of a similar triangular membrane, it was concluded that the differences in simulated
results with the UMAT material model increases when the upper force is increasing, due to a change in load ratio
which was not taken into account in the determination of the properties of the model.
Figure 7-20: Strain values 𝝐𝒙𝒚 along path 1
0
5
10
15
20
25
30
90°Path 1
10°Path 1
90°Path 2
10°Path 2
90°Path 3
10°Path 3
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-]
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10
81
Figure 7-21: Strain values 𝝐𝒙𝒚 along path 2
Figure 7-22: Strain values 𝝐𝒙𝒚 along path 3
From the strain deviations given in Figure 7-23, it can be concluded that the error is largest for path 3 (maximum
160%), whereas the both angles show an equal error of about 200% when path 3 in the opening angle of 10° is
neglected. The error is in the same range as for the vertical fill strains.
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
0 200 400 600 800 1000 1200
Stra
in [
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Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0 500 1000 1500 2000 2500
Stra
in [
-]
Distance along path [mm]
Simulation 90° Experiment 90° Simulation 10° Experiment 10°
82
Figure 7-23: Relative strain deviation for 𝝐𝒙𝒚
Conclusion By using SLIPRING connectors, the difficulties regarding the simulation of contact are excluded from the model.
The concentration of strains is less pronounced in the simulation as it is in the experiment. The lack of friction
plays an important role on the results: the fill strains are larger. The overall tendency however is seen in the
simulations as well. The strains for the 10° opening angle show better comparison along the paths than for the
90° opening angle, indicating that the friction has a larger impact on the closing operation than on the opening
operation. From the shear strain, it could be concluded that the 90° opening angle shows differences with the
experiment. This can be due to the properties of the UMAT model: detailed information is missing on the
behaviour of material under load ratios different from 1:1.
7.2.3 Influence of the material model The same analysis is now performed with an isotropic and orthotropic material model. In this way, the influence
of the use of simplified material models can be investigated. The mechanical parameters used are given in Section
5.2.1.2. It is remarked that the damping factor had to be larger for these analyses: 1e-6 was used throughout the
whole simulation. This indicates that the UMAT material model results in a more stable analysis.
The contour plots are again compared to receive a qualitative comparison. For the quantitative comparison, only
the vertical path is analysed. This path takes into account the top of the membrane, where large differences in
strains are present. By choosing this path, the influence of mismatch of coordinates is only limited.
Contour plots The contour plots of the strains in warp direction are given in Figure 7-24, Figure 7-25, Figure 7-26 and Figure
7-27. For the strains in warp direction, an increase in strains in the upper corner is noticed for the 90° opening
angle if an isotropic material model is used. This is in fact conflicting with the expectations for the isotropic
material model. Since it is more stiff, the strains should be smaller. This zone is even more pronounced for the
orthotropic material model. For the orthotropic material model, the stiffness in fill direction is lower than for the
UMAT material model, explaining the increase in strain.
The strains in warp direction of the isotropic material model are higher in the central zone of the membrane
compared to the experiment (and the UMAT model), indicating that the stiffness is too large. In fill direction, the
strains are much lower. This is due to the larger stiffness. For the 10° opening angle, the compressive zone is less
extended in case of the isotropic material model. Also, compressive concentration in the top is not found in the
simulation. The absence of crimp interchange shows clearly at the top of the membrane: in warp and fill
directions, the strains are positive.
0
2
4
6
8
10
12
14
16
18
90°Path 1
10°Path 1
90°Path 2
10°Path 2
90°Path 3
10°Path 3
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Both material models show bad correlation for the central area of the membrane. For example, the warp strains
for the 90° opening angle show a tensile region, whereas a tensile region was measured. This tendency was also
noticeable for the UMAT model, however, the tensile strains were lower. The warp strains also show a lack of
difference between tension and compression in the top zone.
The strains in fill direction show similar behaviour as for the experiment. The tensile zone at the top in the 10°
opening angle is less extended as for the UMAT material model. Especially the 90° opening in the isotropic
material model angle shows better correlation as for the UMAT material model.
In general it can be concluded that the simplified material models give a good indication from a qualitative point
of view. Moreover, the concentration in strains in the top corner is even better simulated with these material
models. A possible explanation can be found in the difference in damping factor between the simulations: by
applying a higher damping factor, the deformations are more localised. It should be noted that the stabilisation
energy is sufficiently low in both cases however. This might be something to investigate further.
Figure 7-24: 90° opening angle: the isotropic material model - 𝝐𝒙𝒙 and 𝝐𝒚𝒚 [-]
84
Figure 7-25: 10° opening angle: the isotropic material model - 𝝐𝒙𝒙 and 𝝐𝒚𝒚 [-]
Figure 7-26: 90° opening angle: the linear elastic orthotropic material model - 𝝐𝒙𝒙 and 𝝐𝒚𝒚 [-]
85
Figure 7-27: 10° opening angle: the linear elastic orthotropic material model - 𝝐𝒙𝒙 and 𝝐𝒚𝒚 [-]
Quantitative comparison
Figure 7-28: Strain values 𝝐𝒙𝒙 along path 1
Figure 7-29: Strain values 𝝐𝒚𝒚 along path 1
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Isotropic Orthotropic UMAT Experiment
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Isotropic Orthotropic UMAT Experiment
86
Figure 7-30: Strain values 𝝐𝒙𝒚 along path 1
The values of the strains along path 1 can be read from Figure 7-28, Figure 7-29 and Figure 7-30. The strains in
warp direction are smaller for the three models. In the UMAT simulation, variations along the path show, which
indicates that wrinkling can be modelled in this model. Since the other material models give a smooth curve,
wrinkling is not modelled. Overall, the strains are much smaller in the simulation. This can be explained by the
lack of friction: the contact only applies a vertical force on the membrane, whereas the sliding of the belt along
the membrane does not affect the membrane horizontally.
For the strains in fill direction, the three material models show somewhat different behaviour. Whereas the
isotropic material model shows only small strains, the orthotropic material shows larger strains. This is due to a
difference in stiffness: the material’s elastic modulus in fill direction is 716.10 MPa compared to 1270 MPa in the
isotropic material model. Since the vertical stresses are smaller than 5 MPa, the membrane is in the elastic
regime, resulting in small strains. However, more variations show along the path; indicating that wrinkling can
be modelled with the UMAT model.
Finally, it can be noted that the shear strains are approximated better by the orthotropic material model than
for the UMAT material model. The difference with the isotropic material model can be explained in the shear
modulus: it is 483.625 MPa for the isotropic material model, whereas it is only 87.714 MPa for the orthotropic
model. The isotropic material model shows thus much less strains. The orthotropic material model shows better
results compared with the UMAT model. A further investigation in the shear strain under different loading ratios
is therefore recommended.
The relative strain deviation is given in Figure 7-31. The UMAT shows significantly lower deviations for the 𝜖𝑥𝑥
and 𝜖𝑥𝑦 strains compared to the other simplified material models. Only the error of the strain along the fill
direction 𝜖𝑦𝑦 lies in the same order of magnitude as for the other material models. A reason can be found in the
properties of the UMAT material: the initial stiffness is only 160 MPa, and once 5 MPa is reached, the plastic
regime is entered with large strains as a consequence. The properties of the UMAT are based on biaxial tests for
only limited loading ratios in warp and fill direction, which are different from the loading rates in the experiment.
The influence of the loading rate should thus be investigated further.
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Isotropic Orthotropic UMAT Experiment
87
Figure 7-31: Relative strain deviation compared for three material models
In order to have a practical use and to be economically efficient, it is important that the simulations do not take
too much time. The necessary CPU time is showed in Table 7-3. The orthotropic material model requires less
calculation time since it is modelled with an in-plane stress material model. This simplifies the matrices used
during calculation, resulting in a more efficient analysis. The use of the more complex material model increases
the calculation time tremendously. This is due to the simulation of wrinkling. It should be remarked that this time
is for simulations with frictionless contact. Including friction in the model will increase the calculation time. From
this comparison, it is clear that quality pays a price: the large CPU time is a disadvantage of the UMAT material
model.
Table 7-3: The used CPU time for different material models
Material model CPU time [h]
Isotropic 14.9 Orthotropic 8.5
UMAT 127.7
Conclusion The frictionless simulation has been compared for three material models: the isotropic, linear elastic orthotropic
and elasto-plastic material model.
The simplified material models give a good indication from a qualitative point of view. Moreover, the
concentration in strains in the top corner is even better simulated with these material models. Maybe this is due
to the difference in damping factor, however, this influence should only be small since the dissipative energy is
sufficiently small. By comparing the values along the vertical path, it can be concluded that the orthotropic
material model shows the best results, especially for the shear strain. Variations along the path are however not
present in the simplified material models, indicating that the UMAT model can model wrinkling.
Finally, from a computational point of view, it can be concluded that the orthotropic material model is the most
efficient case: the CPU time is within reasonable limits and the results are comparable/slightly better than for
the complex elasto-plastic material model.
0
1
2
3
4
E11iso
E22iso
E12iso
E11ortho
E22ortho
E12ortho
E11umat
E22umat
E12umat
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7.2.4 Influence of the belt stiffness During the development of this model, the VUB has performed further tests on the stiffness of the belts and the
influence on the prediction of the reaction forces in the experiment. It could be concluded that the stiffness is
different under cyclic loading. Therefore, a new simulation is run where the stiffness of the belt is changed to
𝐸 = 2,500 𝑁/𝑚𝑚². Also the stiffness of the connector is changed to 𝐷11 = 287,500 𝑁/𝑚𝑚. The UMAT material
model is used.
Unfortunately, divergence occurred during when the membrane was closed from 90° to 50°. Due to the limited
time in this master thesis, the simulation could not be rerun with a higher damping factor. However, the results
obtained for the 90° opening angle can be analysed. The damping energy is small enough compared to the total
internal energy.
Geometry In Figure 7-32 the geometry is showed for the
50° reference state and 90° opening angle. In
the 50° reference state, wrinkling is noticeable
in the bottom corner as well as in the top.
Additionally, the bulge that was noticed in the
experiment is simulated as well.
In the 90° opening angle, the bulge disappears
and wrinkling is still present in the corners.
This behaviour in geometry is comparable to
the behaviour with a stiff belt.
Figure 7-32: Geometry of the 50° reference state and 90° opening angle for a lower belt stiffness
Contour plots The contour plots for the 90° opening angle are given in Figure 7-33. The compressive zone at the top corner is
extended vertically and more concentrated under the top corner if lower stiffness of the belts is used. Although
the absolute value of the strain is still too high, the tendency is better approximated. The strains in the bottom
zone are larger, indicating that the softer belts induce a larger horizontal strain in the bottom of the membrane.
Also for a softer belt, the fill strains in the largest part of the membrane are too large. In order to obtain more
detailed information, a more uniform plot is made in Figure 7-34. In comparison with the stiffer belt (Figure 7-9),
the maximum (positive) tensile strains are larger and extended over a larger zone. In the middle of the top zone,
compressive strains are found, indicating the presence of a bulge. The stiffness of the belt has no influence on
the fill strains at the bottom corner.
It can be concluded that the stiffness of the belt influences the strains in the membrane. Moreover, a softer belt
gives results that approximate the output from the experiment better.
89
Figure 7-33: Contour plots for the 90° opening angle - 𝝐𝒙𝒙 and 𝝐𝒚𝒚 [-]
Figure 7-34: The fill strains 𝝐𝒚𝒚 in the 90° opening angle for a softer belt [-]
Comparison of the warp strains 𝜖𝑥𝑥 The comparison for the warp strains along the three paths are given on the following pages for both a stiff and
soft belt. The results along path 3 can only be compared to the experiment if the tendencies are considered,
since the path location and length is different.
Along path 1, the influence of the stiffness is very small. Along path 2 and 3 on the other hand, strains are higher
and become positive. The difference is in the order of magnitude of 0.5% à 1%. By applying the same
pretensioning force, the belts will tend to go outward more if they are softer. This will strain the membrane more
in horizontally direction.
90
Figure 7-35: Strain values 𝝐𝒙𝒙 along path 1
Figure 7-36: Strain values 𝝐𝒙𝒙 along path 2
Figure 7-37: Strain values 𝝐𝒙𝒙 along path 3
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Soft belt Stiff belt Experiment
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0 200 400 600 800 1000 1200Stra
in [
-]
Distance along path [mm]
Soft belt Stiff belt Experiment
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0 200 400 600 800 1000 1200Stra
in [
-]
Distance along path [mm]
Soft belt Stiff belt Experiment
91
Comparison of the fill strains 𝜖𝑦𝑦 The values for the fill strains are given in Figure 7-38, Figure 7-39 and Figure 7-40. Overall, it can be concluded
that the vertical strains are larger in case of a soft belt and show less similarity with the experimental results. The
increase is of the order of magnitude of 0.2%.
Comparison of the shear strains 𝜖𝑥𝑦 Finally, the shear strains are compared for both belt stiffnesses along the three paths (Figure 7-41, Figure 7-42
and Figure 7-43). For path 1, the soft belt approximates the experimental results much betters, especially at the
top region. The side belts will rotate more if their modulus of elasticity is lower. Therefore, the phenomenon of
the rotating side belts at the top is more clear. For path 2 and 3, the strains are higher (order of 1%) and do not
approximate the experiment better.
Conclusion From the qualitative and quantitative comparison of the strains for two different belt stiffnesses, it can be
concluded that this property has an influence on the strains in the membrane. This difference is largest for the
horizontal strains, with a maximum of 1%. For the fill and shear strains, the differences are in an order of
magnitude of 0.2%. It is especially the top region which approximates the strains from the experiment better for
a softer belt. The clear concentration of strains under the top corner is more clear in this case.
Overall, it can be stated that the quantification of the stiffness of the belt needs further investigation. By applying
cyclic loading cycles in uniaxial test and performing biaxial testing, the real behaviour of the belts can be
quantified.
92
Figure 7-38: Strain values 𝝐𝒚𝒚 along path 1
Figure 7-39: Strain values 𝝐𝒚𝒚 along path 2
Figure 7-40: Strain values 𝝐𝒚𝒚 along path 3
-0.002
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Soft belt Stiff belt Experiment
-0.002
0
0.002
0.004
0.006
0.008
0.01
0.012
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Soft belt Stiff belt Experiment 90°
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0 500 1000 1500 2000 2500 3000
Stra
in [
-]
Distance along path [mm]
Soft belt Stiff belt Experiment
93
Figure 7-41: Strain values 𝝐𝒙𝒚 along path 1
Figure 7-42: Strain values 𝝐𝒙𝒚 along path 2
Figure 7-43: Strain values 𝝐𝒙𝒚 along path 3
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Soft belt Stiff belt Experiment
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0 200 400 600 800 1000 1200Stra
in [
-]
Distance along path [mm]
Soft belt Stiff belt Experiment
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0 500 1000 1500 2000 2500Stra
in [
-]
Distance along path [mm]
Soft belt Stiff belt Experiment
94
7.3 The inclusion of friction in the SLIPRING connector When the SLIPRING elements are used in Abaqus/Explicit, the friction angle can be computed automatically. A
validation of the model is made for a frictionless simulation.
7.3.1 Validation of the model The simulation from Abaqus/Standard is compared to a frictionless simulation in Abaqus/Explicit. Both are
performed with an isotropic material model since this reduces the calculation time. The results are discussed in
the following paragraphs. It should be noted that a small error was found in the amplitude of the opening towards
90° so that the angle is not reached. This error will however not affect the comparison.
Contour plots (A) THE REFERENCE STATE
The contour plots for the 50° opening angle reference state are given in Figure 7-44. For the strains in warp
direction 𝑒𝑥𝑥, a concentration of tensile strains in the top corner can be found for both analyses. However, the
strains are larger in the case of the explicit analysis: 3.79% compared to only 0.56% for Abaqus/Standard. Low
compressive strains can be found in the middle of the membrane. For the explicit analysis, a concentration of
strains occurs in the lower corner whereas an increase in strains can be seen around the full area of the central
belt.
Also for the strains in fill direction 𝑒𝑦𝑦, the same tendency can be observed: a concentration of tensile strains in
the upper corner, with compressive strains in the middle at the top. This indicates that wrinkling occurs. Overall,
the strains are larger in the Explicit analysis. A maximum tensile strain of 5.65% is found, compared to only 1.2.%
in the case of the implicit analysis. For the compressive strain, -1.43% is found compared to -0.77% in
Abaqus/Standard.
Overall, the same tendency is obtained from both analyses. The extreme values differ, however, the strains
observed are small. That the strains are larger in Abaqus/Explicit can indicate that the inertia effects are still too
high. Although the kinetic energy is sufficiently low, the loading rate should be further decreased in order to
obtain better results.
(B) THE 90° OPENING ANGLE
The contour plots for the 90° opening angle with respect to the reference state can be found in Figure 7-45. The
overall strain field is more uniform in the case of the Abaqus/Standard analysis. No strain concentration is present
in the bottom corner. A small concentration can be found for 𝑒𝑥𝑥 in the implicit analysis, whereas this zone is
spread over the full top in the Explicit analysis. The strains are much larger: for Abaqus/Standard, the maximum
and minimum strain are respectively 0.89% and -0.23% whereas it is 5.21% and -2.24% in case of an Explicit
analysis.
For the strains in fill direction 𝑒𝑦𝑦, the zone with a concentrated strain in the top of the membrane is also larger
and more irregular. For Abaqus/Standard, the maximum strain of 0.87% is reached at the top and -0.23% is
reached at the bottom corner. For Abaqus/Explicit however, the maximum value of 4.39% is found at the bottom.
The minimum value of -1.46% is found in the zone around the top. Again, the strains are larger in case of the
Abaqus/Explicit analysis, indicating that the kinetic energy might be too large.
95
Abaqus/Standard Abaqus/Explicit
Figure 7-44: Comparison of the reference state 50°- 𝒆𝒙𝒙 and 𝒆𝒚𝒚 [-]
Abaqus/Standard Abaqus/Explicit
Figure 7-45: Comparison of the contour plots for 90° - 𝒆𝒙𝒙 and 𝒆𝒚𝒚 [-]
96
Comparison of the warp strains 𝜖𝑥𝑥 The comparison of the warp strains are given in Figure 7-46 and Figure 7-47. The explicit simulations show large
variations in the 10° opening angle. This indicates that the inertia effects are still too large for this case. From the
simulation, a wave propagation is noticed, which explains these large differences. This discrepancy is largest for
at the top zone of the membrane. In the middle of the membrane, the strains show good correlation. For the 90°
opening angle, the strains become compressive in the middle of the membrane, whereas for the implicit, the
strains are positive overall.
For the horizontal path however, the strains for the 10° opening angle due not match at all. Large deviations and
differences along the path are measured for the explicit case. The 90° opening angle shows the same tendency,
that is: a decrease in strain along the path.
From the relative strain deviation (Figure 7-48), it shows that path 1 shows less error, however it has more
extreme outliers. The large deviation that is noticed from the values along the paths are outliers compared to
the rest of the values. An average error of 230% is found. For path 2, it is maximum 100%. It should however be
noticed that the impact of the mismatch in the path will have a large influence. A small error will result in a large
deviation in shear, since the values vary a lot along the path.
Figure 7-46: Strain values 𝝐𝒙𝒙 along path 1
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Explicit 90° Implicit 90° Explicit 10° Implicit 10°
97
Figure 7-47: Strain values 𝝐𝒙𝒙 along path 2
Figure 7-48: Relative strain deviation for
Comparison of the fill strains 𝜖𝑦𝑦 The values for the fill strains are given in Figure 7-49 and Figure 7-50. Again, the same tendency can be observed:
the explicit simulation shows large deviations in the top zone of the membrane, however, in the middle of the
membrane, the strains are in the same order of magnitude. For path 2, the strains in the opening angle of 10°
vary much about the values from the implicit analysis. In the 90° opening angle, a good correlation can be found.
The relative strain deviation (Figure 7-51) shows that the strains along path 1 for the 10° opening angle has a
very large deviation (up to 200%). The 90° opening angle shows a maximum error of 4, whereas it is much more
for the 10° opening angle. This indicates that the inertia effects are larger in the 10° opening angle than in the
90° opening angle.
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Explicit 90° Implicit 90° Explicit 10° Implicit 10°
-4
-2
0
2
4
6
8
10
12
90°Path 1
10°Path 1
90°Path 2
10°Path 2
Re
lati
ve s
trai
n d
evi
atio
n [
-]
98
Figure 7-49: Strain values 𝝐𝒚𝒚 along path 1
Figure 7-50: Strain values 𝝐𝒚𝒚 along path 2
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0 200 400 600 800 1000 1200Stra
in [
-]
Distance along path [mm]
Explicit 90° Implicit 90° Explicit 10° Implicit 10
-0.02
-0.015
-0.01
-0.005
0
0.005
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Explicit 90° Implicit 90° Explicit 10° Implicit 10
99
Figure 7-51: Relative strain deviation for 𝝐𝒚𝒚
Comparison of the shear strains 𝜖𝑥𝑦 The values for the shear strains are given in Figure 7-52 and Figure 7-53. For path 1, the same trend as for the
warp and fill strains is observed: a good correlation is found between the implicit and explicit analysis, except for
the top zone. This discrepancy is larger for the 10° opening angle.
For path 2, the 10° opening angle is behaves totally different in the Explicit analysis as in Abaqus/Standard. The
90° opening angle shows however good correlation: the same decreasing trend can be observed for both
analyses.
The relative strain deviation in Figure 7-54 shows however a different behaviour: path one shows very large
deviations, where the strains in Abaqus/Standard are larger than in Abaqus/Explicit. Path 2 shows a much lower
error: maximum 2, where it is larger for the 10° opening angle. This was noticed from the graphs as well.
Figure 7-52: Strain values 𝝐𝒙𝒚 along path 1
-15
-10
-5
0
5
10
90°Path 1
10°Path 1
90°Path 2
10°Path 2
Re
lati
ve s
trai
n d
evi
atio
n [
-]
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0 200 400 600 800 1000 1200Stra
in [
-]
Distance along path [mm]
Explicit 90° Implicit 90° Explicit 10° Implicit 10
100
Figure 7-53: Strain values 𝝐𝒙𝒚 along path 2
Figure 7-54: Relative strain deviation of 𝝐𝒙𝒚
7.3.2 Conclusion From the validation of the model, it can be shown that the Abaqus/Explicit analysis shows good promise.
However, up to this point, the inertia effects still influence the solution, especially for the 10° opening angle. It is
recommended to investigate this further. This could however not be investigated further. It is therefore
recommended to vary the step time (and thus loading rate) more than in the model that was used now. Especially
the closing needs to be simulated very slowly.
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0 200 400 600 800 1000 1200
Stra
in [
-]
Distance along path [mm]
Explicit 90° Implicit 90° Explicit 10° Implicit 10
-5
0
5
10
15
20
25
30
35
90°Path 1
10°Path 1
90°Path 2
10°Path 2
Re
lati
ve s
trai
n d
evi
atio
n [
-]
101
Chapter 8: Conclusions and recommendations 8.1 General conclusions This master thesis is situated within the research on design methods for kinetic tensile architecture. It presents
a method for simulating the three dimensional deployment of tensioned membrane coated fabrics, on a scale
that ressembles practical use.
In a first part, literature was studied to gain more knowledge in the field of membrane architecture. The
membrane material shows significantly different behaviour in warp and fill direction, due to the weaving process.
The fill direction is less stiff.
In order to validate the model, an experiment was executed at the Free University in Brussels, where a PVC-
coated membrane fabric was deployed three dimensionally. The membrane was opened and closed, while
strains were measured for the intermediate opening angles as well. This was executed by using a Digital Image
Correlation (DIC) technique. In this way, contour plots of the warp and fill strains were obtained, as well as
detailed values along three paths. The opening angle of 90° showed the largest deviations over the membrane
surface. The overall strains measured were low: in the order of 0.1% for the warp strains and 1% for the fill and
shear strains. This indicates that the largest part of the deployment of the membrane is rigid body motion and
does not influence the pretension present in the membrane. However, it should be noted that difficulties with
the reference state were encountered. If the reference state is taken after first pretensioning is applied, the
initial strains are disregarded from the results. Since the membrane material is history dependent, it is very
important that this reference state is taken at the beginning of the experiment.
The deployment of the membrane was then simulated with the use of finite element modelling software Abaqus.
In a first approach, the contact between the central belt and the membrane was modelled by modelling contact
and sliding between the central belt and membrane. The results from an analysis in Abaqus/Explicit with an
isotropic material model show the same tendency as for the experiment. Especially the 90° opening angle gave
very small error (about 10%). For the closing operation towards 10° however, fluctuations are noticeable and
inertia effects are still too large. Overall, this method shows promising results. The only disadvantage is the CPU
time: this is even with a simplified model very large. Therefore, its practical use is limited.
The central belt can however also be modelled with a belt-like connector. In this way, the difficulties regarding
the simulation of contact are excluded from the model. The concentration of strains is less pronounced in the
simulation as it is in the experiment. The lack of friction plays an important role on the results: the fill strains are
larger. The overall tendency however is seen in the simulations as well. The strains for the 10° opening angle
show better comparison along the paths than for the 90° opening angle, indicating that the friction has a larger
impact on the closing operation than on the opening operation. From the shear strain, it could be concluded that
the 90° opening angle shows differences with the experiment. This can be due to the properties of the UMAT
model: detailed information is missing on the behaviour of material under load ratios different from 1:1. It can
be concluded that the use of these connectors show a lot of promise, especially regarding the CPU time.
However, in order to obtain results with the inclusion of friction, the model in Abaqus/Explicit has to be
elaborated more.
Overall it can be stated that the order of magnitude of the strains obtained from the simulations are the same
as for the experiment. The errors found are quite high, however, it must be noticed that the strains are quite
low, especially in the larger context of the building industry. A lot of parameters affect the discrepancies between
both, where the largest influence will come from a mismatch in paths. Since the paths could not be exactly
located but had to be approximated as close as possible, the error might become quite high.
102
8.2 Recommendations for future works The model with belt-like connectors shows promising results. Therefore, the model in Abaqus/Explicit should be
elaborated more so that friction can be included in the simulation. Also, by performing a parametric study for
the friction coefficient, the influence of this parameter can be investigated. Additionally, a model can be
investigated with the correct cutting pattern in such a way that the influence of this discrepancy can be
quantified. In order to fully quantify the strains in the membrane, the reference state should be taken at the
beginning of the analysis, before pretensioning of the membrane.
69
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#METHOD OF ASSIGNING SLIPRING CONNECTORS ALONG AN EDGE
# -*- coding: mbcs -*-
# Do not delete the following import lines
from abaqus import *
from abaqusConstants import *
import __main__
from abaqus import *
from part import *
from material import *
from section import *
from optimization import *
from assembly import *
from step import *
from interaction import *
from load import *
from mesh import *
from job import *
from sketch import *
#from visualization import *
from connectorBehavior import *
import mesh
EConnect = 449650
openMdb(pathName='C:\Users\saram_000\Dropbox\Thesis\Slip ring\Slip ring
connectors 2.cae')
myAssembly = mdb.models['Model-1'].rootAssembly
mdb.models['Model-1'].rootAssembly.Set(name='myNodes',
nodes=myAssembly.sets['myEdge'].nodes)
myNode = mdb.models['Model-1'].rootAssembly.sets['myEdge'].nodes
tmp=[]
tmpLabel=[]
#f = open('E:\\Work\\ABQwork\\SARA\\num.txt', 'w')
for i in range(0,len(myNode)):
tmp.append([myNode[i].coordinates[0], myNode[i].coordinates[1],
myNode[i].coordinates[2],myNode[i].label])
tmpLabel.append(myNode[i].label)
tmp = sorted(tmp)
# for i in range(len(tmp)):
# tmpNodeArray.append(mdb.models['Model-
1'].parts['Membrane'].Node(coordinates=(tmp[i][0],tmp[i][1],tmp[i][2]))
)
# myMeshNodeArray = mesh.MeshNodeArray(nodes=tmpNodeArray)
# mdb.models['Model-1'].rootAssembly.Set(name='myNodes2',
nodes=myMeshNodeArray)
#f.close()
for i in range(0,len(myNode)-1):
index1 = tmpLabel.index(tmp[i][3])
index2 = tmpLabel.index(tmp[i+1][3])
point1 = myNode[index1].coordinates
point2 = myNode[index2].coordinates
myAssembly.WirePolyLine(mergeWire=OFF, meshable=OFF,
points=((myNode[index1],myNode[index2]), ))
middlePoint = ((point1[0] + point2[0])/2.0,(point1[1] +
point2[1])/2.0,
point1[2])
myAssembly.Set(edges=
myAssembly.edges.findAt((middlePoint,
)), name='SlipRing' + str(i))
mdb.models['Model-1'].ConnectorSection(assembledType=SLIPRING,
contactAngle=
180.0, massPerLength=9.57E-8, name='SLIPRING')
mdb.models['Model-
1'].sections['SLIPRING'].setValues(behaviorOptions=(
ConnectorElasticity(table=((EConnect, ), ),
independentComponents=(),
components=(1, )), ))
myAssembly.SectionAssignment(region=myAssembly.sets['SlipRing' +
str(i)], sectionName='SLIPRING')
#AUTOMATIC SCRIPT FOR DETERMINING AN ADEQUATE DAMPING FACTOR# -*-
coding: mbcs -*-
# Do not delete the following import lines
from abaqus import *
from abaqusConstants import *
import __main__
import section
import regionToolset
import displayGroupMdbToolset as dgm
import part
import material
import assembly
import step
import interaction
import load
import mesh
import optimization
import job
import sketch
import visualization
import xyPlot
import displayGroupOdbToolset as dgo
import connectorBehavior
openMdb('sendToSara implicit.cae')
model = mdb.models['Model-1']
part = model.parts['Part-1']
#step = model.steps['Step-2']
assembly = model.rootAssembly
instance = assembly.instances['Part-1-1']
job = mdb.jobs['check1']
# Initialising constants
t=1
di = 1e-5
ratio = 100
#The parameter study as long as ALLSD is too high.
d = di
while ratio > 5:
print 'damping: %s' % (d)
#Create new step
## a = mdb.models['Model-1'].rootAssembly
## session.viewports['Viewport: 1'].setValues(displayedObject=a)
## session.viewports['Viewport: 1'].assemblyDisplay.setValues(
## adaptiveMeshConstraints=ON, optimizationTasks=OFF,
## geometricRestrictions=OFF, stopConditions=OFF)
## mdb.models['Model-1'].StaticStep(name='Parameter',
previous='Initial', stabilizationMagnitude=0.0002,
##
stabilizationMethod=DAMPING_FACTOR,continueDampingFactors=False,
adaptiveDampingRatio=None, nlgeom=ON)
## session.viewports['Viewport:
1'].assemblyDisplay.setValues(step='Parameter')
mdb.models['Model-1'].steps['Step-
2'].setValues(stabilizationMagnitude=d,
stabilizationMethod=DISSIPATED_ENERGY_FRACTION,
continueDampingFactors=False)
#history output opvragen!
## mdb.models['Model-1'].HistoryOutputRequest(name='Energy',
createStepName='Parameter', variables=('ALLSD', 'ALLIE'))
##
####
## mdb.models['Model-1'].FieldOutputRequest(name='F-Output-2',
createStepName='Parameter', variables=PRESELECT)
# Run the job, then process the results.
job.submit()
job.waitForCompletion()
print 'Completed job for %s damping factor' % (d)
from odbAccess import *
## odb = openOdb(path='viewer_tutorial.odb')
##step2 = odb.steps['Step-2']
##region = step2.historyRegions['Node PART-1-1.1000']
##u2Data = region.historyOutputs['U2'].data
##dispFile = open('disp.dat','w')
##for time, u2Disp in u2Data:
##dispFile.write('%10.4E %10.4E\n' % (time, u2Disp))
##dispFile.close()
odb = openOdb(path='check1.odb')
step2 = odb.steps['Step-2']
region = step2.historyRegions['Assembly ASSEMBLY']
allieData = region.historyOutputs['ALLIE'].data
## print type(allieData)
## print allieData
## print allieData[1]
print 'ALLIE: %s' % (allieData[len(allieData)-1][1])
allsdData = region.historyOutputs['ALLSD'].data
ratio = allsdData[len(allieData)-
1][1]/allieData[len(allieData)-1][1]*100
print 'Ratio: %s procent' % (ratio)
odb.close()
d = d/10
print 'Einde'
