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Proceedings of the IASS Annual Symposium 2017
“Interfaces: architecture.engineering.science”
25 - 28th September, 2017, Hamburg, Germany
Annette Bögle, Manfred Grohmann (eds.)
Copyright ©
2017 by Pierre Marquis, Amandine Cersosimo, Cyril Douthe
Published by the International Association for Shell and Spatial Structures (IASS) with
permission.
Building elastic gridshells from patches
Pierre MARQUIS*, Amandine CERSOSIMOa, Cyril DOUTHEb
*, a, b Laboratoire Navier – UMR 8205 – Ecole des Ponts Paris-Tech, IFSTTAR, CNRS
6-8 avenue Blaise Pascal – Champs-sur-Marne – 77455 Marne la vallée CEDEX 2
pierre.marquis@eleves.enpc.fr
Abstract
To be able to build elastic gridshells, it is necessary to define and build the flat grids that will be
deformed into the final 3D shapes. For complex topologies, this flattening is not possible without
separating the grid into independent sub-grids which will subsequently be assembled. The purpose of
the proposed paper is therefore the investigation of such construction from patches. A systematic study
of various strategies for patch cutting by grid offset is presented on a toroidal form. The different
propositions are analyzed in terms of their influence on the final form, on the forces in the connectors,
and on the equivalent flat area of the patches. A discussion on the way to deal with short beams
generated by the method closes the study.
Keywords: Elastic gridshell, structural morphology, structural analysis, construction method, cutting pattern, bending active
structures
1. Introduction
This paper is based on the design of a pavilion project developed by the laboratoire Navier (see
figure 1). This full scale pavilion shall illustrate the potential of the recent shape generation method
developed by Douthe et al [1] for elastic gridshells which can be naturally covered by quadrangular
planar panels. These shapes, although limited, allow for the construction of complex topologies
through the introduction of closed forms or singularities. Yet to be able to build those shapes, it is
necessary to define and build the flat which is to be deformed into those shapes. For complex
topologies, such flattening is not possible without first separating the grid into sub-grids that will be
assembled afterwards. Note that, even for simple topologies, the footprint of the grid (its flat
configuration) is always larger than its in-shape configuration, so that when the size of the building
site is limited, it might be interesting to separate the grid into sub-grids to limit the obstruction of the
building site. To be efficient, one must strive to define sub-grids that will form independent stable
modules once in-shape in order to limit scaffolding.
In this exploratory study, we define in this way a reference geometry close to the geometry of the
pavilion project, which is derived in four variants, following two typologies of interesting cuts for the
gridshells: either along a beam of the main frame, or along a bracing member. We present in this paper
the global mechanical behavior of these four variants in comparison to the reference geometry, as well
as the local behavior of the nodes of the beams affected by what within the context of the present
study is called the seam line of the modules. Finally, the patch cutting of a gridshell has to deal with
short beams, which can complicate the construction of elastic gridshells. It would therefore be
necessary to consider these construction characteristics already during the design phase.
Proceedings of the IASS Annual Symposium 2017
Interfaces: architecture.engineering.science
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Figure 1: Rendering of the Ecole des Ponts toroidal gridshell clad in greenery.
2. Reference geometry
To conduct this study, we first choose to model and analyze the behavior of a reference elastic
gridshell made of a grid with cut. This reference geometry takes the form of a half-torus of 13 meters
in length by 10 meters in width which approximates the shape of the pavilion. A torus is a surface of
genus 1 closing on itself, and cannot be meshed by the method of the compass, developed by the
architect Frei Otto in the work IL10 [2]. To obtain our mesh or equilateral network shown in Figure 2,
we have taken our lead from research carried out by Douthe et al. [1], to cover elastic gridshells by
quadrangular planes from an isoradial mesh. It is by using the complementarity between an isoradial
mesh and its Tchebycheff dual network that we succeed in creating a mesh, whilst controlling its
generation process. The main grid of the gridshell consists of two sets of beams, D1 and D2, following
the directions of the lines of the Tchebycheff network, at a bay spacing of 600 mm. Those lines cross
the principal curvature line at approximately 45°, making them optimal for the grid members whose
bending pre-stress is thereby minimized. The third layer of beams, whose role of bracing gives the
gridshell a shell-like behavior, is obtained by recovering the vertical lines of the previous isoradial
mesh (lines which actually are one family of principal curvature of the surface).
Figure 2: Mesh of the reference geometry, top view
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Interfaces: architecture.engineering.science
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The mesh is then released by the dynamic relaxation method, allowing to take into account
the pre-stress, linked to the geometrical and mechanical characteristics of our structure. We
have to specify that we wish to take into account the influence of the flexural stiffness of the
bracing layer on the equilibrium shape from the beginning of the design. Therefore, we first
carry out a form finding process by dynamic relaxation, with extendible bracing. After that,
the lengths of the bracing members are defined and a second dynamic relaxation step is
carried out using bracing of actual mechanical properties, in order to find the mechanical
equilibrium under self-weight. The material used is a fiberglass composite material of
Young's modulus (E) equal to 40 GPa with an ultimate strength of σ = 900 MPa, to which a
safety coefficient of 3 is applied. The beams have a circular and solid section of 18 mm
diameter (r = 9 mm). From these characteristics, one can define the value of the limiting
bending moment which is not to be exceeded.
𝑀max <!ult
!
∙𝑊
!" =171.7!Nm (1)
The structure is studied under the weight load case. The materials having a density of 2.05, a
downward vertical point charge of 2.0 daN is applied to each node of the structure.
Figure 3: Nodal displacements (left) with umax = 112,3 mm; bending moments (right) with Mmax = 162.5 N.m
FIG. 3 shows the deformation under self-weight (left) and resulting bending stresses (right). The
smallest displacements, as well as the least deflected beams are shown in green and the largest
displacements and the most deflected beams in red. The maximum allowable stress of 300 MPa is not
reached by good margin in that load case. The members that are under the most bending are the
bracing elements of the gridshell. The displacements are uniform over the whole structure and
relatively small in view of the section and the bearing surfaces considered. The maximum
displacement is 112 mm.
3. Cutting the patches
Two ways of patching a gridshell have been imposed on us spontaneously: along a beam of the main
frame, or along a bracing element. For the former, cutting along a beam, a mesh is generated following
the method used for the reference geometry, over a portion of this surface. In FIG. 4 (left), we first
mesh the left half. We note that there are two beams of the same array of parallel beams which
constitute the two free edges of the new gridshell. All the beams of the assembly D1 are supported at
their ends. The second set of beams D2 is the only one which is truly cut, supported on the ground on
one side and free of the other at the seam, protruding by half a bay each of the beams D1 forming the
two free edges.
Proceedings of the IASS Annual Symposium 2017
Interfaces: architecture.engineering.science
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The second patch is then generated by a quasi-symmetry of the first patch (FIG. 4, center), shifting the
whole grid by half a step. Finally, the two patches are assembled (FIG. 4, right). The beams of the
network D1 are continuous, and those of the network D2 are discontinuous, arranged in a staggered
manner on either side of the seam created.
Figure 4: Test 1: meshes of a non-relaxed patch (left) and its opposite (center); assembly and bracing (right)
This arrangement makes it possible, on the one hand, to avoid the use of sleeves and multiple
connection types, and on the other hand to allow the gridshell to be braced with continuous elements
(blue lines on FIG. 4, right), thanks to the half-step shift. In addition, the half-step method of shifting
makes it possible to densify the mesh at the seam where one could assume a general structural
weakness.
Following the same method, we make other patches of smaller sizes by dividing our reference
geometry into four parts. In the example of FIG. 5, it should be noted that the free edges of the patches
are derived from the array of beams D1 as well as from the array of beams D2. This does not change
the design process, the assembly of the modules allows in equal amount for it to be braced by
continuous elements.
Figure 5: Test 2: meshes of a non-relaxed patch (left) and the others (center); assembly and bracing (right)
The second patch cutting method involves sewing along a bracing member, or rather along its future
location after mounting. The principle remains the same as with the first method. The beams of both
networks D1 and D2 have free edges at one end and are supported on the ground at the other end. The
mirror patch is designed in the same way, with the same shift of half a bay, thus enabling the mesh to
be densified along the seam. (FIG. 6, left). We also carry out a division of the pavilion into four
patches, following the braces, in order to be able to compare with the first two tests (FIG. 6, right).
Proceedings of the IASS Annual Symposium 2017
Interfaces: architecture.engineering.science
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Figure 6: Test 3 (left) and Test 4 (right)
4. Global Behavior
Once these geometries are determined, it is necessary to analyze their overall behavior i.e. the
configuration of the assembled patches, in order to study how the assembly of "pieces" of a gridshell
influences the final shape of the structure. The analysis of the different self-weight gridshells follows
the same process as described above. The results presented in FIG. 7 shows the maximum
displacements of the test geometries, which are all relaxed from the single and even reference non-
relaxed geometry. It is observed that they are very close to the maximum displacement of the
reference gridshell (in orange in FIG. 7). This applies both to patches with seams that either follow a
beam or a brace of the reference gridshell. Similarly, for the distribution of the bending stresses, the
elements under the highest amount of flexure remain the braces and the values of maximum bending
moment remain very close to that of the reference. It can thus be concluded that regardless of the
number of patches as well as their typology, when they are assembled, the “cutting pattern” has little
influence on the distribution of stresses and displacements within the gridshell. It would be useful,
however, to study the buckling patterns of each of the structures to complement this result, an analysis
not dealt with in the present study.
Figure 7: Comparison of the displacements (left) and the bending moments (right) of the tests with the reference
5. Local Behavior
We have seen that the seams do not affect, or not significantly, the overall behavior of the structure
considering the reference structure. However, we wish to know more precisely the seams’ influence on
the local behavior, especially the stresses in the connections between the patches. To this end, we go
back to how forces are calculated from the geometry at local scale in the algorithm of dynamic
relaxation [3, 4]. In this model, internal forces are calculated for each member and the connectivity is
Proceedings of the IASS Annual Symposium 2017
Interfaces: architecture.engineering.science
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achieved by ensuring that the nodes are at the same location in the starting configuration and always
submitted to the same force which is equal to the sum of internal forces in each member plus the
external load. To have access to forces in the connection in the final equilibrium configuration, one
can therewith calculate the internal forces in the member and, knowing that each node is in
equilibrium, deduce that the contribution of connection is the missing force (the force that does not
come from internal stress) at each node.
The graph of FIG. 8 superimposes the total forces of each of the nodes of a beam serving as seam of
the test 1 in blue with those of the same nodes of the corresponding or dual beam of the reference
geometry in orange. On the abscissa, one reads the numbering of the nodes. The total force is
expressed in Newton, for each node, on each of the columns of the histogram.
Figure 8: Total forces, in Newton S.I., in the test 1 nodes (left); tearing and shear for the same nodes (right)
If the nodes 0 and 31, corresponding to the supports, are considered separately (their values
correspond to reaction forces rather than connection forces, and is very much influenced by the
presence of the short beams next to supports), the difference between test 1 configuration and the
reference geometry is small. 50 % on the ordinate shows the equality of the forces.
By projecting the total forces onto the local coordinate system of each connection of the test geometry
(FIG. 8, right), it is interesting to note that the stress for this sewing is essentially linked to shear.
Moreover, what the superposition histogram does not present is that one observes the staggered
arrangement of the beams underlined by the Z-chart. This seems to be due to an alternation of short
and long beams on both sides of the seam near its ends. Conversely, near the center of the curve, or the
"key of the vault", the beams on both sides of the seam have substantially the same length, hence a
shift of the rhythm of the shear peaks around the Nodes 13, 14, 17, 18 and 20. Although it is not
constant, the removal does not appear to be affected by the arrangement of the beams.
Considering then the geometry of test 2, the orange columns of the histogram are mostly larger than
the blue ones. This suggests that the seam which densifies the mesh slightly lightens the forces in the
nodes (FIG. 9, left). In the local reference frame of each of the nodes, the tearing is here more
important and sometimes exceeds the shear, as can be seen for nodes 7 and 8, for example (FIG. 9,
right).
Proceedings of the IASS Annual Symposium 2017
Interfaces: architecture.engineering.science
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Figure 9: Total forces, in Newton S.I., in the test 2 nodes (left); tearing and shear (right)
In the case of cutting along the bracing, no significant difference is observed between test 3 and the
reference geometry. However, in the case of test 4, the forces in the nodes increase considerably, up to
6.5 times the total stress of the node of the reference geometry for node 10, where the maximum
bending forces are concentrated (FIGS. 10, left and right).
Figure 10: Total forces, in Newton S.I., in the test 4 nodes (left); tearing and shear (right)
As a first conclusion, we note that the proposed methodology of building a gridshell by connecting
patches seems efficient but that the sewing of highly utilized beams has to be avoided. Furthermore,
we observe that the seam generates short beams that are too stiff, and which therefore induce
considerable stresses in the connections (see section 7 for more details).
6. Individual behavior of patches
Considering the problem of assembling such structures, the study carried out up to this point is
extended to the analysis of the behavior of the individual patches which corresponds to the stages of
construction before assembly. We therefore consider pieces of gridshells of different sizes, some of
which are supported on the ground (this is the case for all bracing) and some which are waiting to be
“sown” together with the adjacent patch.
As for the assembled gridshells, the distribution of the bending moments under self-weight within the
individual patches remains the same as for the reference geometry. The free beams on hold do not
generate excessive stresses. However, displacements become significant. Indeed, the case illustrated in
FIG. 11 (left) shows a maximum displacement of 500 mm on the edge. However, the addition of point
struts (or props), which are modeled by fixed points, considerably reduce these displacements. FIG. 11
(center) effectively shows that the addition of two props (black nodes on the image) makes it possible
Proceedings of the IASS Annual Symposium 2017
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to reduce the maximum deformation to 177 mm, a value which seems acceptable for a temporary state
of the structure.
Figure 11: Nodal displacements (left) umax = 500 mm; nodal displacements with props (center) umax = 177 mm;
nodal displacements (right) umax = 150 mm
In the case of "bigger" gridshell pieces such as that shown in FIG. 11 (right), the displacement of the
beams remains reasonable (150 mm in this case) and the use of temporary supports during the
construction of these patches is no longer necessary.
7. Short beams treatment
As mentioned earlier, when the grid is anchored to the ground and the gridshell is shaped, the beams
which are joined together by ball joints and hitherto without internal stress, are strongly bent. These
beams, although of small diameters, have an important stiffness and the shorter the beams, the greater
the force needed in order to give them their final curvature. Cutting the gridshell into patches
generates short beams by necessity and we must ask ourselves how these models are realistic from a
construction point of view. In other words, the problem can be posed as follows: what is the minimum
beam length that can be bent "bare-handed"?
If one considers a slender elastic beam, pinned at its supports and shaped by flexure, the latter
undergoes large rotations and its post-buckled geometry can be related to the theoretical curve of the
Euler elastic [4], which can be described by a simple expression. This curve is characterized by a
reduced number of parameters: l the length of the beam, a its chord, f its rise and F an axial
compression force. These parameters are shown in the FIG. 12.
Figure 12: Transversal displacement of a straight bi-articulated beam, with constant cross-section [4]
Under this loading, the beam is subjected to a bending moment Fv, where v(x) is the transverse
displacement. Considering then that the curvature is given by the first derivative of the angle θ made
by the tangent and the chord, the local equilibrium of the beam is (3):
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Interfaces: architecture.engineering.science
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(2)
By deriving this equation with respect to the curvilinear abscissa and integrating the angle θ defined
by sin θ = dv / ds and the angle α designating the starting angle, one finds:
(3)
with:
(4)
This last equation (3) can be integrated numerically in order to relate the bending moment, radius of
curvature, force and beam length. The "bare-hand" force being estimated to 30 kg, the minimum
permissible beam length for a desired curvature radius of 1.20 m is equal to 3.56 m. This short beam
criterion being established, FIG. 13 shows in red the beams whose length is less than 3.56 m for each
case of study. It can be concluded that each configuration studied has about 15% of beams that are so
short that it would be a priori impossible to form with bare hands. One could think about various
methods to mitigate this disadvantage, for example to replace these short beams by longer ones that
will be cut in the deformed configuration once the patches would have been assembled.
Figure 13: Beams of insufficient length (in red) for each of the studies 1 to 4
7. Conclusion
To allow for the fabrication of complex topologies or large elastic gridshells, we have investigated
here various cutting strategies to subdivide the flat grid into independent patches. Two patterns have
been shown: cutting along a beam of the main grid, or cutting along a bracing member. In our
parametric study, we varied the number of "seams" either to increase or reduce the size of the
modules, which leads to an increase or decrease in the ground extension of the flat grid. We also
varied the location of the cut, depending on the curvatures of the project's final configuration. We have
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shown first that none of the types of sewing influences significantly the global behavior of the
structure. Then, at the local scale, we did not notice a major influence either, although the shear rate of
the nodes is quite sensitive to the layout of the beams and their length. However we evidence that the
major difficulty induced by our patching strategy is linked to the generation of "short" beams,
preventing the possibility of those member to be bent by hand power alone. In practical cases, this
subject would need to be addressed specifically but does not seem particularly difficult. We thus
consider the strategy validated and will apply it to the construction of the pavilion.
References
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3. ADRIAENSSENS, S., BARNES, M. and WILLIAMS, C. A new analytic and numerical basis
for the form-finding and analysis of spline and gridshell structures. In: Kumar, B. and Topping,
B. H. V., eds. Computing Developments in Civil and Structural Engineering. Edinburgh: Civil-
Comp Press, 1999, pp. 83-91.
4. DOUTHE, Cyril, BAVEREL, Olivier, CARON, Jean-François. Gridshell in composite materials
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