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Form-finding bending-active structures with temporary ultra-elastic contraction elements

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FORMFINDING BENDING-ACTIVE
STRUCTURES WITH TEMPORARY
ULTRA-ELASTIC CONTRACTION
ELEMENTS
LIENHARD J.1,2, LA MAGNA R. 1, KNIPPERS J. 1
1Institute of Building Structures and Structural Design, University of
Stuttgart, Germany
2str.ucture GmbH, Stuttgart, Germany
Abstract
Bending-active structures as basis of a structural form defined by elastic
deformation, pose both numeric as well as strategic problems to a computational
form-finding process. As the geometry of a bending-active structure becomes
more complex in a coupled system, so do the equilibrium paths in the
deformation process. Here, it is no longer possible to simply deform a structure
into its elastically deformed shape by a number of linear support displacements.
As a new approach for form-finding coupled bending-active systems in FEM, the
authors developed a strategy using contracting cable elements to pull associated
points from an initially planar system into an elastically deformed configuration.
This paper will discuss the strategy of form-finding bending-active structures
with ultra-elastic contraction elements by looking at basic samples of coupled
bending-active structures, complemented with some considerations of the
numerical background. In addition, samples of complex applications in built
prototype structures are presented to highlight the effectiveness of this approach.
Keywords: Bending-active, form finding
1 Introduction
Especially in the field of flexible structures, it is often impossible to determine
beforehand the geometric description of an equilibrium system. This is well
known in the field of tensile structures, as the output geometry is the result of a
tight interaction between form, forces, material characteristics and boundary
conditions. Thus we speak of form-finding rather than designing form. The
indeterminacy of the form-finding process often poses a complicated problem
from a simulation point of view, as it is non-trivial to identify the kinematics and
motion paths of key elements of the structural system. The proposed approach
greatly simplifies the setup of the simulation model, freeing the analyst from the
difficult and in some cases impossible task of precisely defining the paths of
nodal displacement needed to perform the form-finding of bending-active
structures. The main difference between a simple action type like support
displacement and the proposed approach which employs contracting cables is
exemplified in Figure 1: In the first case the motion path of the supports is
completely determined beforehand. In the latter, where the bending-active
elements is constrained by coupling it in space, the position of the displacements
are unknown; here, a definition of the bending shape via support displacements
is practically not possible. The employed cable elements work with a temporary
reduction of elastic stiffness which enables large deformations under constant
pre-stress. The cable element thereby becomes a free load defined by the
magnitude of pre-tension and the rigidity of the elements in between which it is
contracting.
Figure 1: example of a basic bending-active component form-found via
support displacements (a) and contracting cable elements (b)
2 Finite-Element Approach
The necessity for simulation of large elastic deformations in order to form-find
bending-active structures poses no problem to modern nonlinear finite element
analysis. However, since Finite-Element programs do not serve well as a design
environment, the input data for the pre-processing of the simulation has to be
generated with either physical form-finding or behaviour-based computational
modelling techniques. The necessity and advantage of FEM lies in the possibility
of a complete mechanical description of the system. Provided that form-finding
solvers are included in the software, the possibility of freely combining shell,
beam, cable, coupling and spring elements enables FEM to simulate the exact
physical properties of the system in an uninterrupted mechanical description.
These include: mechanical material properties, asymmetrical and varying cross-
sections, eccentricities, coupling and interaction of individual components,
nonlinear stress-stiffening effects, nonlinear simulation of stresses and
deflections under external loads (e.g. wind and snow), patterning and
compensation.
It is often said that dynamic relaxation has a computation time advantage over
FEM for systems of large deflection; however, this is rapidly diminished with an
increasing number of elements per beam and large deflections. Comparative tests
by [1] showed that computation time for the form-finding of an elastic gridshell
was faster with FEM when the element length was set sufficiently small. For
larger elements, computation time increased significantly or no convergence was
reached at all in FEM, while dynamic relaxation converges quicker with
increasing element size. The use of step size adaptive incremental deformation
additionally drastically reduces the iteration steps and time needed in a FEM
simulation. A disadvantage of FEM based form-finding of bending-active
structures is the necessity to always start the simulation from a planar
configuration in order to track the residual bending stresses in the system. This
problem led to the development of the elastic cable approach, through which
large deformations of complex bending-active structures can be solved using the
Finite Element Method.
2.1 Elastic Cable Approach
Based on Finite Element technology, the contracting elastic cable approach has
been proven to be a reliable technique for the form-finding of flexible structures.
Following, the model setup and simulation details will be further examined. The
model is mainly composed of two parts: the geometry to simulate, and the
contraction elements which enable the form-finding process. In the first place the
geometry is designed and pre-processed as in any CAE environment. The model
is then equipped with a set of linear cables which link together all the pair of
nodes that have to be joined during the form-finding. The contracting cables
have to be defined as one single element which does not contain further internal
nodes to avoid sagging and other collateral phenomena associated to the
simulation of cable elements. Moreover, this reduces the computational
complexity of the model as fewer nodes are present in the system, thus cutting
down the total amount of degrees of freedom for which the system has to be
solved. Having defined the whole geometrical and mechanical setup, a pre-stress
value is then assigned to the cables, which is later translated into a pre-stressing
load acting onto the elements. Due to the assigned pre-stress the cable elements
will start contracting accordingly.
The cable elements work with a temporary reduction of elastic stiffness which
enables large deformations under constant pre-stress. This method was originally
developed for the form-finding of tensile membrane structures using, for
example, the transient or modified stiffness method. For further information on
this method, see [2] and [3]. For the form-finding of coupled bending-active
systems, the great advantage is that the cables allow complete freedom of the
equilibrium paths that are followed during the deformation process. The pre-
stress that is independent of the change in element length also allows the
simultaneous use of several cable elements in the different positions of the
system. Once the cables are contracted to nearly zero length they can be replaced
by various types of coupling elements for the following structural analysis.
The value of nodal displacement can be deduced by multiplying the axial
geometric stiffness of the element with the applied load. Referring to the basic
equilibrium equation of a cable element in nonlinear FEM we can derive
equation (1) for the pre-stressing cable used here:

11
1 1 +
11
1 1 
=
(1)
If Young’s modulus E is temporarily set to zero, the elastic stiffness would
disappear and the tangential stiffness would be entirely based on the ratio pre-
tension to length. Material stiffness therefore has no influence on the behaviour
of the elastic cable which is thereby only constrained by the elements it is
attached to. As such, the cable has become a directionless load defined by the
magnitude of pre-tension and the rigidity of the elements in between which it is
contracting.
Figure 2: Sample system with a spring attached to a contracting cable
element; increasing the stiffness results in smaller nodal
displacements per iteration (dashed line)
This artifice presents two advantages: in the first place the element becomes
independent from its mechanical properties, thus creating a material-less device
which serves only the purpose of pulling the system into shape; on the other
hand reducing the stiffness speeds up the contracting process, as the element
exerts less resistance to the pre-stressing, thus shortening its length more rapidly.
It is worth noting that for each calculation step the amount of contraction is not
constant or generally predictable during the form-finding process. Figure 2
displays a simple system where a spring of given stiffness is attached to a
contracting cable. The graph illustrates the nodal displacement of the attached
spring, which varies according to the stiffness of the spring itself. From the graph
is becomes clear that the total displacement per step is reduced, as the overall
resistance of the system has increased. It is therefore generally complicated to
evaluate the number of contracting steps needed to fulfil the form-finding
process, as a stiffer system will necessitate more iterations to reach the final
configuration with the cable fully contracted.
The advantage of form-finding complex coupled equilibrium systems is shown
by the example in Figure 3. This system is composed of two differently sized
masts with fixed supports which are connected at the top and an additional inner
point. Both cables of the system are simultaneously pre-stressed incrementally.
The graph in Figure 3 shows the reciprocal dependency of cable pre-stress and
length for the two cables in relation to the iterations of the linear incremental
loop. Even though both cables are assigned with the same pre-stress increments,
their actual pre-stress differs depending on the deformation of the system. Also,
the moment of total contraction of the cable elements is reached at different
iterations and still, the calculation continues after one cable has already
contracted to a length close to zero. This astonishingly stable convergence
behaviour enables the form-finding of far more complex systems introduced be
the case studies below.
Figure 3: Sample system composed of two differently sized masts with fixed
supports. b: Behaviour of the elastic cable approach for the system
2.2 Incremental Load Steps
The large deformations that beam and cable elements undergo during the form-
finding of bending-active structures cannot be simulated in a single load step. In
order to still take advantage of a civil engineering FEM environment (in the
presented cases Sofistik®), the authors developed custom programmed routines
which enable incremental-iterative calculation.
The basic calculation procedure for incremental load steps is based on the option
of primary load cases. Here, the deformations and strains of an already solved
load case may be referred to by defining it as a primary load case for the
calculation of a current load step. Therefore, deformations and stresses from a
prior load step may be linked to another load case, thus reducing the calculation
time.
An aspect to be considered is the exit criteria of the form-finding process. As the
cable elements start contracting, they will continue doing so indefinitely, as the
amount of displacement is proportional to the flexibility of the cable (i.e. the
inverse of its stiffness) which is in turn proportional to its length. By successive
contractions, the cable element shortens its length diminishing the total amount
of length reduction that occurs at each pre-stressing load step. This entails that
the element will never reach a zero-length configuration, but will rather converge
asymptotically to zero as the graph in Figure 2 and 3 displays. Given this fact, it
is necessary to define a threshold length under which the simulation can be
deemed acceptable. This can be achieved by retrieving the length of each
contracting element, checking whether all have reached the selected threshold,
and finally halting the process if this is the case.
The asymptotical behaviour is reflected also in the global form-finding
procedure, as the amount of total nodal displacement decreases as the cables tend
to reach zero. This problem can be leveraged by introducing an adaptive
framework that increases the pre-stressing load at each step depending on the
number of iterations needed to find the equilibrium solution of the nonlinear
problem.
3 Case Studies
The following two case studies are briefly introduced to exhibit the strength of
the elastic cable approach in form-finding complex bending-active structures.
3.1 Research Pavilion ICD/ITKE
At the end of July 2010 the Institute of Computer based Design (ICD) and the
Institute of building structures and structural design (ITKE) at the University
Stuttgart realised a temporary research pavilion made of plywood. The design of
the pavilion was the result of a student workshop which focused on the
integration of physical experiments and computational design tools to develop
bending-active structures. The project was discussed in detail in [4].
The Pavilion structure is based on a radial arrangement and interconnection of
the self-equilibrating arch system made of 6.5 mm ply wood. Due to the reduced
structural height, the connection points locally weaken the coupled arch system.
In order to prevent these local points from reducing the structural capacity of the
entire pavilion, the locations of the connection points between the strips need to
change along the structure, resulting in 80 different strip patterns constructed
from more than 500 geometrically unique parts.
Figure 4: Form-finding of the pavilion using the elastic cable (red lines)
approach in three subsequent form-finding steps.
Given the unrolled geometry and connection points of the coupled arches, it was
possible to form-find the pavilion structure with the elastic cable approach. The
form-finding was separated into three subsequent steps (Figure 4). In a first step,
80 contracting cables simultaneously pull the flat plywood strips into the simply
curved half torus shape. Thereafter, two sets of 205 contracting cables
simultaneously form the coupled arch systems and then couple the arc pairs to a
continuous torus shell. The nodal positions of the final equilibrium shape are
entirely defined by the inner constraints. Such a system could not have been
form found in FEM without the elastic cable approach. In fact, it was this project
that ignited the research and development of this approach, and therefore laid the
basis for the later case study structures.
3.2 M1
The Textile Hybrid M1 at La Tour de l’Architecte showcases the research on
hybrid form- and bending-active structure systems by the Institute for
Computational Design (ICD) and Institute for Building Structures and Structural
Design (ITKE) with students of the University of Stuttgart. The scientific goal of
the project was the exploration of formal and functional possibilities in highly
integrated equilibrium systems of bending-active elements and multi-
dimensional form-active membranes (termed “Deep Surfaces”). The resulting
multi-layered membrane surfaces allowed not only for structural integration, but
also served as a functional integration by differentiating the geometry and
orientation of the membrane surfaces. The project was discussed in detail in [5].
For the form-finding and analysis of the structure, FEM was utilised. Here, the
parameters of the complex equilibrium system were explored to determine the
exact geometry and evaluate the structural viability. Based on the elastic cable
approach, the beams were initialised as straight elements and gradually deformed
into interconnected curved geometries, finally being reshaped by the inclusion of
pre-stressed membrane surfaces (Figure 5). The geometric data therein was
determined initially by the physical form-finding models in defining the lengths
and association points on the rods for the topology of FE beam elements. Given
the unrolled geometry and connection points of the rods, it was possible to
simulate the erection process and therefore the residual stress in a Finite -
Element based form-finding process.
Figure 5: FEM form-finding of the Textile Hybrid based on the rod topology
given by the physical model.
The form-finding of the beam elements in the M1 was separated into four
subsequent steps. Starting with the straight beams, the loop modules were all
form-found simultaneously by first forming an arc and then droplet shape. Based
on the topological connection points given from the physical model, the loop
elements were successively joined into a coupled system by contracting a final
set of cables between the common nodes of the structure.
By means of automatic mesh generation, the membrane surfaces were added and
a final form-finding of the fully coupled textile hybrid was undertaken. This
form-found structural analysis model allowed verification of the geometrical
shape, including its residual stress, as well as analysing the deformations and
stress levels under external wind loads. Furthermore, the form-found membrane
surfaces could be processed directly by the textile module of the software for
patterning.
4 Conclusion
The use of ultra-elastic contraction elements in FEM has proven to be a powerful
instrument for the form-finding of complex bending-active structures via large
elastic deformations. An approach that can also be applied to surface elements,
which may be used for the contraction of edge conditions rather than distinct
points.
This approach enables the setup of an uninterrupted mechanical description of
bending-active structures in a general purpose FEM environment. The three main
process steps of form-finding, structural analysis and unrolling/patterning can
thereby be represented in a single modelling environment. This ensures that
structurally substantial residual stresses can be traced throughout all stages of
design.
References
[1] Douthe, C., Baverel, O. and Caron, J.-F. (2006) Form-finding of a grid shell
in composites materials. In: Proceedings of the International Association for
Shell and Spatial Structures (IASS) Symposium.
[2] Hartmann, F. and Katz, C. (2004) Structural Analysis with Finite Elements.
Berlin: Springer. p. 507
[3] Lewis, W. J. (2003) Tension structures. Form and behaviour. London:
Thomas Telford. p. 41
[4] Lienhard, J., Schleicher, S. and Knippers, J. (2011) Bending-active
Structures Research Pavilion ICD/ITKE. In: Nethercot, D. et al.
Proceedings of the International Symposium of the IABSE-IASS
Symposium, Taller Longer Lighter, London, UK
[5] Lienhard, J., Ahlquist, S., Menges, A. and Knippers, J. (2013)
Extending the Functional and Formal vocabularyof tensile membrane
structures through the interaction with bending-active elements. In:
TensiNet symposium [RE]THINKING lightweight structures,
Istanbul.
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Structural Analysis with Finite Elements, 2nd Edition provides a solid introduction to the foundation and the application of the finite element method in structural analysis. It offers new theoretical insight and practical advice on why finite element results are 'wrong,' why support reactions are relatively accurate, why stresses at midpoints are more reliable, why averaging the stresses sometimes may not help or why the equilibrium conditions are violated. This second edition contains additional sections on sensitivity analysis, on retrofitting structures, on the Generalized FEM (X-FEM) and on model adaptivity. An additional chapter treats the boundary element method, and related software is available at www.winfem.de. © Springer-Verlag Berlin Heidelberg 2007. All rights are reserved.
Bending-active Structures -Research Pavilion ICD/ITKE
  • J Lienhard
  • S Schleicher
  • J Knippers
  • D Nethercot
Lienhard, J., Schleicher, S. and Knippers, J. (2011) Bending-active Structures -Research Pavilion ICD/ITKE. In: Nethercot, D. et al. Proceedings of the International Symposium of the IABSE-IASS Symposium, Taller Longer Lighter, London, UK