Earthquakes and Structures, Vol. 20, No. 2 (2021) 161-173 DOI: https://doi.org/10.12989/eas.2021.20.2.161 161
Copyright © 2021 Techno-Press, Ltd. http://www.techno-press.com/journals/was&subpage=7 ISSN: 2092-7614 (Print), 2092-7622 (Online)
1. Introduction
Tensile fabric structures (TFSs) are used in construction
for spanning large lengths while allowing impressive
architectonical designs. However, they are challenging
structures for engineering design, since their behavior is highly non-linear, their material properties may change in
different directions, they exhibit wrinkling when subjected
to compressive loads (i.e., they only have tensile stresses),
among other issues. Conceptually, the beginnings of TFSs
are referred to shapes of equilibrium unattainable with
computational methods; so, authors as Gaudi from
Barcelona, Heinz Isler from Switzerland and Frei Otto from
Germany are cited in the literature for prestressed and
hanging structures before the advent of computer-based
methods (Linkwitz 1999, Bletzinger and Ramm 1999).
Finding the shape of a TFS can be considered as an inverse
engineering problem (Bletzinger and Ramm 1999), in
Corresponding author, Ph.D. Professor E-mail: [email protected]
aPh.D. Professor
E-mail: [email protected] bPh.D. Professor
E-mail: [email protected] cPh.D. Professor
E-mail: [email protected]
which the usual methodology to find a stress field from a
given section under prescribed loads is inverted, i.e., for a
given stress field the geometry of a TFS is found, so that it
satisfies the prescribed stresses; however, not all the shapes
are executable in practice or desirable for architectonical
interests. Currently, computer-aid methods are used to find the desired shapes of TFSs with the so-called form-finding
approach. Form-finding procedures can be divided in three
main families (Veenendaal and Block 2012), which are
force density methods, dynamic relaxation methods and
stiffness matrix methods. The development of the force
density method is linked to the project for the Olympic
Stadium of Munich 1972 in the literature (e.g., Linkwitz
and Scheck 1971, Scheck 1974, Linkwitz 1999). Dynamic
relaxation methods are listed by Wüchner and Bletzinger
(2005) and include the studies by Barnes (1999), Wakefield
(1999) and Wood (2002). Veenendaal and Block (2012)
refer to the stiffness matrix methods of Haug and Powell (1972), Argyris et al. (1974) and Tabarrok and Qin (1992).
The stiffness matrix method, which is also implemented
in commercial software and recommended for analysis and
design (Huntington 2013), is used in the present study. It is
noted that when mechanical approaches based on the virtual
work of surface stress fields are considered, at the stage of
form-finding the material specifications are not important,
since the problem is to find a geometry which results in
equilibrium for a given stress field, regardless of how such
field was generated (Bletzinger and Ramm 1999); in the
Time history analysis of a tensile fabric structure subjected to different seismic recordings
Jesús G. Valdés-Vázquez1, Adrián D. García-Soto1a, Michele Chiumenti2b and Alejandro Hernández-Martínez1c
1Department of Civil and Environmental Engineering, Universidad de Guanajuato,
Av. Juárez 77, Colonia Centro, C.P. 36000, Guanajuato, GTO., México 2International Center for Numerical Methods in Engineering (CIMNE), Univeridad Politécnica de Cataluña,
Campus Norte UPC, 08034, Barcelona, Spain.
(Received March 28, 2020, Revised January 15, 2021, Accepted January 27, 2021)
Abstract. The structural behavior of a tensile fabric structure, known as hypar, is investigated. Seismic-induced stresses in the fabric and axial forces in masts and cables are obtained using accelerograms recorded at different regions of the world. Time-
history analysis using each recording are performed for the hypar by using finite element simulation. It is found that while the
seismic stresses in the fabric are not critical for design, the seismic tensile forces in cables and the seismic compressive forces in masts should not be disregarded by designers. This is important, because the seismic design is usually not considered so
relevant, as compared for instance with wind design, for these types of structures. The most relevant findings of this study are: 1) dynamic axial forces can have an increase of up to twice the static loading when the TFS is subjected to seismic demands, 2)
large peak ground accelerations seem to be the key parameter for significant seismic-induced axial forces, but not clear trend is found to relate such forces with earthquakes and site characteristics and, 3) the inclusion or exclusion of the form-finding in the
analysis procedure importantly affects results of seismic stresses in the fabric, but not in the frame.
Keywords: tensile fabric structure; time-history analysis; finite element simulation; form-finding; seismic-induced
forces
Jesús G. Valdés-Vázquez, Adrián D. García-Soto, Michele Chiumenti and Alejandro Hernández-Martínez
Fig. 1 Curvilinear coordinates for the membrane
Fig. 2 Initial geometry without form finding analysis (m)
form determination stage, the response of the structure is
not very dependent on employed stress-strain functions,
Huntington (2013).
The impact of the form-finding procedure in the seismic
analysis of TFSs is not reported in the literature. In fact, in
general there are scarce studies of TFSs under earthquake-
induced loads; nevertheless, Valdés-Vázquez et al. (2019)
report a parabolic hyperboloid (hypar) formed of prestressed fabric and cables, over a supporting frame of
cables and masts, subjected to an Italian seismic record. To
inspect the seismic response of other TFSs is desirable,
especially if different recordings worldwide are used, so
that possible differences can be investigated. This can be
achieved by using an adequate finite element method
(FEM) which allows the time-history non-linear analysis of
TFSs.
The main objective of the present study is to investigate
the structural behavior of a TFS (a hypar) on a supporting
frame when subjected to seismic loading by using time-history dynamic analysis with the aid of finite element
simulation. The effect of the form finding, as well as that of
the different recordings on the results, is reported.
2. Theory
The finite element simulation is based on the theoretical
background briefly described in the following. The
membrane theory used in this work is based on vectors and
tensors using a curvilinear coordinate system, where Greek
indices on membrane middle surface take on values of 1
and 2 in a plane stress state in Euclidean space, as shown in
Fig. 1. The position vector X on the membrane in the
reference configuration Ω0 is defined by
𝐗 = 𝐗(𝜉1, 𝜉2) (1)
while its corresponding position vector x in the current
configuration Ω is given by
𝐱 = 𝐱(𝜉1, 𝜉2) (2)
With these position vectors, the covariant base vectors
on Ω0 and Ω are defined respectively as
𝐆𝛼 =𝜕𝑿
𝜕𝜉𝛼 (3)
and
𝐠𝛼 =𝜕𝐱
𝜕𝜉𝛼 (4)
The covariant components of the metric tensors are
expressed as
𝐺𝛼𝛽 = 𝐆𝛼 ∙ 𝐆𝛽 (5)
and
𝑔𝛼𝛽 = 𝐠𝛼 ∙ 𝐠𝛽 (6)
Using the above definitions, components of the Green-
Lagrange strain tensor can be found using
𝐸𝛼𝛽 =1
2(𝑔𝛼𝛽 − 𝐺𝛼𝛽 ) (7)
Finally, with appropriate constitutive equations, the
second Poila-Kirchhoff components 𝑆𝛼𝛽 are obtained and
virtual internal work 𝛿𝑊 int in curvilinear coordinates is
𝛿𝑊int = ∫ 𝛿𝐸𝛼𝛽Ω0
𝑆𝛼𝛽𝑑Ω0 (8)
where 𝛿𝐸𝛼𝛽 stands for the virtual strain tensor. From the
virtual internal work, the internal forces 𝐅int can be
obtained and the dynamic equation to be solved yields
𝐅int + 𝐌𝐚 = 𝐅ext (9)
where M is the mass matrix of the system, a is the
acceleration vector and 𝐅ext are the external forces given by the prestressed forces and seismic records; the
recordings used in this study are described in detail later.
More details of the formulation and discretization with the
finite element method can be found in Valdes et al. (2009),
where, for a particular direction i and node I, the internal
forces in tensorial form are
𝑓𝑖𝐼𝑖𝑛𝑡 = ∫ 𝐵𝛼𝛽𝑖𝐼 𝑆𝛼𝛽 𝑑Ω0
Ω0
(10)
with the second Piola-Kirchhoff components 𝑆𝛼𝛽 obtained using proper constitutive equations, and the fourth-order
strain-displacement tensor given by
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Time history analysis of a tensile fabric structure subjected to different seismic recordings
𝐵𝛼𝛽𝑖𝐼 =1
2(
𝜕𝑁𝐼(𝛏)
𝜕𝜉𝛼 𝑥𝑖,𝛽
ℎ + 𝜕𝑁𝐼(𝛏)
𝜕𝜉𝛽 𝑥𝑖,𝛼
ℎ ) (11)
where 𝑁𝐼(𝛏) are the element shape functions and
𝑥𝑖,𝛼ℎ = ∑
𝜕𝑁𝐽(𝛏)
𝜕𝜉𝛼 𝑥𝑖𝐽
𝑛𝑛𝑜𝑑𝑒
𝐽=1
(12)
is the discretized form for the derivative of the shape
functions times the element nodal coordinates.
3. Geometry definition The geometry used in this study is defined by a
hyperbolic-paraboloid (hypar) of 8 meters of side in a plan
view, as shown in Fig. 2, with a difference between its
upper U and lower L corners of 3 meters in height. In this
figure, the upper corner nodes of the membrane fabric
structure are shown with filled black circles, while the
lower corner nodes are depicted with empty circles. The
lower fabric nodes L in Fig. 2 are 4.5 meters high from the
ground level, while nodes H for the frame have a height of
9 meters from ground level. Projected dimensions of the
supporting structure in a plan view are also depicted in Fig. 2 (i.e., cables in blue and masts in red in Fig. 2 and Fig. 4).
It is point out that the geometry in Fig. 2 corresponds to the
TFS before the so-called form-finding procedure is
performed.
The manufacture warp direction of the membrane is
defined along the diagonal between upper nodes U of the
fabric, while the fill direction is perpendicular to the warp
direction (i.e., along the diagonal going from one L corner
to the other L corner). The fabric is surrounded by a cable
(yellow lines in Fig. 2). The membrane fabric and the
surrounding cable are prestressed. As mentioned before, the
hypar frame consists of cables and masts, whose material properties, together with those of rest of the TFS, are given
in Table 1, where Ew=Ef is the tensile stiffness for warp and
fill respectively, 𝜈 is the Poisson ratio, t is the fabric
thickness, 𝜌 is the density, Esc is the elastic modulus for
the surrounding cable with dsc as its diameter, Em is the
elastic modulus for the mast with a cross section defined by
Am and Ec is the elastic modulus for the frame cable with a
diameter given by dc.
3.1 Form-finding
In this work the direct stiffness method, also known as
matrix stiffness method (Tabarrok and Qin 1992) was used
for the form-finding. Since the minimum surface of the
structure must be independent of the material properties, the
assumed material properties for the membrane and
surrounding cable are given in Table 2; the fictitious values
of some properties in Table 2 (e.g., a modified modulus of
elasticity, after Tabarrok and Qin, 1992) are used simply for
convenience, so that the direct stiffness method can be
applied in the form-finding stage, which does not impact on the finding of the minimum surface (Bletzinger and Ramm
1999, Huntington 2013). Material properties for the frame
Table 1 Material properties
Membrane Fabric Surrounding
Cable Mast Frame Cable Frame
Ew=Ef 800
kN/m Esc
210
GPa Em 210 GPa Ec
210
GPa
𝜈 0.4 𝜈 --- 𝜈 --- 𝜈 ---
T 1 mm dsc 12 mm Am 0.018387 m2 dc 12 mm
Prestress
warp
and fill
4 kN/m Prestress
cable 40 kN
Prestress
mast ---
Prestress
cable ---
𝜌 1800
kg/m3 𝜌sc
7800
kg/m3 𝜌m 7800 kg/m3 𝜌c
7800
kg/m3
Table 2 Form finding material roperties
Membrane Fabric Surrounding Cable
Ew = Ef 0.008 kN/m Esc 0.008 kN/m2
𝜈 0.4 𝜈 ---
t 1 mm dsc 12 mm
Prestress warp
and fill 4 kN/m Prestress cable 40 kN
𝜌 1800 kg/m3 𝜌sc 7800 kg/m3
g 9.806 m/s2 g 9.806 m/s2
Fig. 3 Membrane norm of displacements after form finding
analysis
remain unchanged. Note that the self-weight of the structure
is added in the form-finding.
The resulting deformed shape of the membrane fabric
and surrounding cable is shown in Fig. 3, after the form-finding analysis is carried out. This deformed structure is
taken as the initial geometry to perform the dynamic
seismic analysis in the next section and it is shown in Fig. 4
in a 3D view.
4. Seismic records Once the form finding analysis is performed, a dynamic
time history analysis using accelerations of four recordings
is carried out for the hypar in Fig. 4. The four seismic records to be used in the dynamic analysis are listed in
Table 3, where the earthquake characteristics are also
included. These records are briefly discussed and results of
the dynamic analysis are described in the next sections, first
for a record from an earthquake in Alaska, then for the
remaining events.
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Jesús G. Valdés-Vázquez, Adrián D. García-Soto, Michele Chiumenti and Alejandro Hernández-Martínez
The first considered record was obtained during the last
of a sequence of upper crustal normal-faulting earthquakes
which struck the Apenines in Central Italy (D’Amico et al.
2019, Civico et al. 2018). It was the Norcia October 30th,
2016 earthquake, with a moment magnitude, Mw 6.5, a
focal depth of 9.2 km, and an epicentral distances, Repi, ranging from 4.6 to 66.6 km (D’Amico et al. 2019). It was
recorded at 60 sites, including at the station known as Forca
Canapine located at rock site class A* (EC8 code, Luzi et
al. 2016) and Repi=11 km; A* means that site classification
is not based on the shear wave velocity at 30 m, Vs30,
(Clementi et al. 2020). We have selected the processed
record from Forca Canapine retrieved from
https://strongmotioncenter.org/ because of the maximum
PGA recorded at the site. Note that according to a
preliminary report (ReLUIS-INGV Workgroup 2016), the
records from Forca Canapine were under technical revisions at the time but deemed adequate and, if confirmed, the
strongest records ever recorded in Italy (close to a
maximum peak ground acceleration, PGA, of 1 g). More
recently, this record seems to be included to investigate the
fling effects from near-source strong-motion records
(D’Amico et al. 2019) and it is certainly included in a study
to assess the seismic structural response of ancient Italian
churches (Clementi et al. 2020). We include this seismic
record in the present study as input for the time-history
analysis of the TFS previously described, since this record
was also used in a previous study, Valdés-Vázquez et al.
(2019), resulting in large seismic demands for some elements. Other large PGAs recorded during the Norcia
Mw6.5 earthquake are reported for the vertical component
being 869 and 782 cm/s2 (absolute values) at stations
known as T1213 and CLO, respectively (Luzi et al. 2017).
It would be interesting to inspect the vertical response of
TFSs using these records, as well as the simultaneous effect
of two or three orthogonal components, which is
recommended in future studies. The second record listed in Table 3 was generated by an
instraslab normal-faulting event, the November 30th, 2016,
Mw7.1 Anchorage earthquake, in Alaska (West et al. 2019), at a depth of 47 km. The record corresponds to station
known as Chugach Park located over till and glacial
deposits with bedrock at 34 m; the horizontal maximum
PGA of approximately 2 g at the site is partly attributed to
topographic and radiation effects (Cramer and Jambo 2020).
The processed record was obtained from
https://strongmotioncenter.org/, where a distance to the
epicentre for Chugach Park station of approximately 27 km
is given. The inclusion of the record is based on previous
evidence of important stresses in the supporting frame of a
TFS when large PGAs are used (Valdés-Vázquez et al.
2019). The third listed accelerogram in Table 3 was recorded
during the September 16th, 2015 Mw8.3 Illapel earthquake
in Chile. This was a megathrust subduction earthquake
recorded at several stations, including C11O station, whose
record is used in the present study. Seismic intensities, over
the median plus one standard deviation, were recorded for
structural periods under 0.4s at this station, which was
attributed to possible site amplification (Candia et al. 2017).
Fig. 4 Geometry after form finding analysis and frame
elements
Station C11O is located at Monte Patria, over Quaternary
alluvial deposits surrounded by volcanic rocks (Vs30=625
m/s), at a rupture distance RRup of approximately 55 km; an
unexpectedly very high maximum PGA of 0.77 g was
recorded at the site (Fernández et al. 2019). The seismic
signal used in this study for this Chilean record was
retrieved from http://evtdb.csn.uchile.cl.
The last record in Table 3 was obtained during the
November 14th, 2016, Mw 7.8 Kaikõura earthquake in New Zealand; the rupture at 15 km depth, with ground motions
exceeding 1g for the horizontal and vertical components,
from the epicentral region at Station Waiau to Station Ward
at approximately 125 km away from the epicenter (Kaiser et
al., 2017). Although the preliminary study stated that the
failure mechanism was oblique thrust (Kaiser et al. 2017)
other studies describe it as a mechanism which possibly
represents the most complex multifault rupture ever
recorded (Crowell et al. 2018). In other study it is
considered that a simultaneous rupture of the subduction
interface and overlying faults occurred (Wang et al. 2018).
The selected station is in the town of Ward about 6 km from the Kekerengu fault over deep soil (Allstadt et al. 2018,
Kaiser et al. 2017). The processed record used was also
obtained from https://strongmotioncenter.org/, where a
maximum PGA of around 1.2 g is reported.
The records in Table 3 are selected because of their large
PGA; all of them lead to important structural response of
the TFS frame structure. Plots for these seismic records are
shown in Fig. 5, where it can be observed that their
maximum PGA is significant. They can be near-source
recordings, like the Norcia one, but also have source-to-site
distances of up to 125 km, like the New Zealand record. Some recordings with PGAs under 0.5g were also inspected
but did not lead to significant increases in the seismic
demand. Thus, only records in Table 3 are considered for
the time-history analysis. Nonetheless, research should be
extended using accelerograms from different tectonic
environments, different frequency contents, etc. (even if
they do not have large maximum PGAs), to further inspect
this aspect.
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Time history analysis of a tensile fabric structure subjected to different seismic recordings
5. Finite element discretization and results for the Alaska record
The seismic dynamic analysis of the TFS is carried out
by using three types of finite element meshing, first for the
earthquake recorded in Alaska in the x-direction (Fig. 2),
simply because it has the maximum recorded PGA of the
recordings depicted in Fig. 5. The first meshing is denoted
as FF-S-Mesh and refers to a structured mesh from the
form-finding results. The second one is denoted as FF-NS-
Mesh, a non-structured mesh, that is also generated from
the form-finding results. Finally, a third mesh which is
structured, and not based on the form-finding, but in the original geometry denoted as NFF-S-Mesh is considered.
The three types of meshing are shown in Fig. 6. The frame
elements are the same for the three meshing cases (Figs.
6(a), 6(b) and 6(c), respectively).
As shown in Fig. 6, the type of finite element used for
the analysis consists of membrane triangular elements, each one with three nodes. The mass matrix discretization for
each finite element consists in computing the finite element
mass and dividing it with equal parts to its three nodes. In
structural analysis, usually the damping matrix is associated
with the Rayleigh damping that requires two periods to
compute the constants for the mass and stiffness matrices,
in such a way, the vibration modes of the structure need to
be previously calculated. In this work, the eigenperiods
have not been computed and instead a parametric analysis
with different damping coefficients for the mass is studied,
and the coefficient that produces an exponential decay in
the structural response has been chosen once the external loads are withdrawn.
In Figs. 6(a) and 6(b) it is clearly observed that the
perimeter of the surrounding cable is longer than for the one
in Fig. 6(c), since in the latter the cable is straight leading to
a shorter length. This difference in length affects very
Table 3 Earthquakes and related seismic records characteristics
Record number
Earthquake location
Country Station Mw Date Repi or
RRup (km) Focal
depth (km) Earthquake type/ Focal mechanism
Station Site Classification
1 Norcia Italy Forca
Canapine 6.5
Oct 30th,
2016 11 9.2
Upper crustal
normal-faulting
A* (EC8 Code), rock
site
2 Anchorage Alaska, U.S.A
Chugach Park
7.1 Nov 30th,
2018 27.4 47
Intraslab normal-faulting
On till and glacial deposits with bedrock at
34 m
3 Illapel Chile Monte Patria
8.3 Sep 16th,
2015 55 11
Megathrust subduction earthquake
Vs30=625 m/s (e.g., very dense soil
and soft rock, NEHRP)
4 Amberley New Zealand Ward Fire 7.8 Nov 13th,
2016 125 15
Subduction interface-
overlying faults- multifault rupture
Deep soils
Fig. 5 Acceleration recordings used in the dynamic analysis
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Jesús G. Valdés-Vázquez, Adrián D. García-Soto, Michele Chiumenti and Alejandro Hernández-Martínez
slightly the results because it leads also to a decrease in the
weight for NFF-S-Mesh. In Figs. 7, 8 and 9 the principal stresses at the membrane
center, upper corner and lower corner, respectively, are
presented. Figs. 7(a)-9(a) show the global behavior; Figs.
7b-9b show extreme (maximum and minimum) dynamic
values and the static loading before imposing the seismic
demand (i.e., that due to the self-weigh and the prestress).
Other figures shown later follow the same format. In Figs.
7(a), 8(a) and 9(a), the straight inclined line at the beginning
of the time represents a monotonic increment in the membrane prestress, which reaches a static prestressed state
(the horizontal line), before the hypar is subjected to the
seismic excitations; the dynamic response can be clearly
observed when the stresses fluctuate around the static
prestress as a function of time.
In Fig. 7(b), it is observed that, when the hypar is
analyzed from the form-finding mesh FF-S, the principal
(a) FF-S-Mesh (b) FF-NS-Mesh (c) NFF-S-Mesh
Fig. 6 Finite element meshes and configurations used in the dynamic analysis
(a) Global behavior (b) Static response and extreme values
Fig. 7 Membrane principal stresses at membrane’s center
(a) Global behavior (b) Extreme values
Fig. 8 Membrane principal stresses at top corner
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Time history analysis of a tensile fabric structure subjected to different seismic recordings
stresses are increased approximately 10% with respect to
the static analysis, while for the FF-NS mesh the increase is
approximately of 13%; the order of the stresses is similar
for these two cases. Nonetheless, when the analysis is
performed without considering the form-finding (NF-SS
mesh), the change is the principal stresses is noticeable and
the variation in stresses surpass 160% with respect to the
static prestress values. This means that the stresses at the
membrane center is not realistic without a previous form-
finding procedure.
With regards to Figs. 8 and 9, it is observed that for the
membrane corners behavior (in terms of principal stresses)
similar conclusions can be drawn, but with a lower
percentage. Overall, results presented in this section for the
dynamic response, when the hypar is subjected to the
Alaska recording, indicate that the form-finding is
(a) Global behavior (b) Extreme values
Fig. 9 Membrane principal stresses at lower corner
(a) Global behavior (b) Extreme values
Fig. 10 Principal tensor axial force
(a) Global behavior (b) Extreme values
Fig. 11 Mast axial force
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Jesús G. Valdés-Vázquez, Adrián D. García-Soto, Michele Chiumenti and Alejandro Hernández-Martínez
important to adequately assess the membrane stresses and
should be taken into consideration.
The axial force for one of the principal tensors is shown
in Fig. 10, where the same three mesh cases are shown and
it can be observed that, broadly speaking, the behavior is
very similar for all cases, being the main difference that of the static case; this occurs because the surrounding cable
perimeter is less when the form finding is not accounted for.
However, the difference between the static forces and the
maximum reached dynamic forces is in the order of 5%.
Fig. 11 is analogous to Fig. 10, except that the axial
forces correspond to the compressive forces in one of the
masts. Qualitatively, results in Fig. 11 resembles results in
Fig. 10, but quantitatively an important difference is
observed, since the increment in axial forces with respect to
the static compressive forces is approximately 54%, 53%
and 55% for cases FF-S, FF-NS and NFF-S, respectively. It is point out that this increase is important from a design
standpoint, since it surpasses the 50% in all cases.
Figs. 12 and 13 show tensor axial forces for “x” and “y”
tensors (see Figs. 2 and 4), respectively; these tensors are
those used to give the masts support. Tensors behave alike
in both directions, having increases in the axial force with
respect to the static case of 167% and 179% when the form-
finding is considered or not, respectively. Conclusions that
can be drawn are, in one hand, that while the form-finding
has an important effect in adequately computing the
membrane stresses, it does not significantly change the
resulting axial forces and, in the other hand, that such
significant increment in the axial load imposed by the
seismic effects should not be neglected in the design of the frame elements.
In the next section, results when the TFS is subjected to
the remaining records are presented. Figures are like those
reported in this section. Descriptions of results are given,
and a discussion, focused on any possible effect of
considering earthquakes from different tectonic
environments worldwide, is also presented.
6. Results and discussion including Chile, Italy and New Zealand seismic records
In this section a comparison of the structural response
for all the records in Fig. 5 is carried out. Even though in
the previous section no significant difference, whether the
dynamic analysis includes the form-finding or not, was
found for the axial forces (i.e., FF-S-Mesh or FF-NS-Mesh),
in this section the results are compared using only the FF-S-
Mesh (i.e., including the form-finding), since the results are
(a) Global behavior (b) Extreme values
Fig. 12 Tensor x-axis axial force
(a) Global behavior (b) Extreme values
Fig. 13 Tensor y-axis axial force
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Time history analysis of a tensile fabric structure subjected to different seismic recordings
Fig. 14 Mast axial force
Fig. 15 Mast axial force limits
Fig. 16 Tensor x-axis axial force
169
Jesús G. Valdés-Vázquez, Adrián D. García-Soto, Michele Chiumenti and Alejandro Hernández-Martínez
more reliable, especially for the fabric stresses.
In Figs. 14 and 15 the variations in axial force in a mast
are presented. The increases with respect to the static load
(self-weight and prestress) in percentage are shown for the
different seismic events in Figs. 14 and 15, resulting in
values 54%, 30%, 30% and 69% for Alaska, Chile, Italy
and New Zealand earthquakes, respectively. It is
noteworthy that for the Mw6.5 Italy event as well as for the
Mw8.3 New Zealand event the increase is only 30%, while
the other events lead to increases of approximately twice
the static loads.
In Figs. 16 and 17 the increments for the tensile force in
a tensor, after the seismic dynamic loads are applied, are
reported. Values of 156%, 205%, 189% and 143% for
recordings from Alaska, Chile, Italy and New Zealand are
obtained. In this case the maximum increment does
Fig. 18 Principal Tensor axial force
Fig. 17 Tensor x-axis axial force limits
Fig. 19 Principal Tensor axial force limits
170
Time history analysis of a tensile fabric structure subjected to different seismic recordings
coincide with the event with the largest Mw, the Mw8.3
Chile earthquake, leading to an increment of 205%. From
the design standpoint, no element designed for the static
loads would be able to withstand twice the demand as compared to the static design. This proves that the seismic
design of TFS should not be neglected.
Figs. 18 and 19 show the results for the principal tensor.
In this case the increments are not significant, since values
of 5%, 4%, 7% y 5% for the Alaska, Chile, Italy and New
Zealand cases, respectively, are obtained. In this case an
original design from the static loads may cover the
additional axial forces induced by the seismic activity.
Finally, Figs. 20 and 21present resulting stresses for the
upper corner of the fabric. It is observed that also for this
case the increments can be deemed somewhat unimportant, like the case of axial forces for the principal tensor, with
values equal to 8%, 4%, 5% and 12% for the recordings
from the Alaska, Chile, Italy and New Zealand earthquakes.
It should be highlighted tough, that the shown stresses are
principal stresses at the corners (the maximum stresses
occur at the corners), where it is recommended to have
additional strengthening (thus the common use in practice
of double fabric at the corners is justified).
By considering the earthquakes and site characteristics
of the employed recordings, which cover an important range of moment magnitude (6.5≤Mw≤8.3), epicentral or
rupture distances (11 km - 125 km), focal depths (9.2 km -
47 km), as well as tectonic environments, failure
mechanisms and site soils classifications, the only clear
conclusion that can be drawn, is that all of them have very
large maximum PGAs (approximately 0.7 g - 2.0 g) and this
is the main feature which can be related to significant
increases in the axial forces imposed by the seismic demand
in TFSs, if compared to the static loading. An attempt was
made to inspect whether higher increments are a function of
increasing maximum PGA, but not clear trend was found. For instance, while the largest seismic demand increment
for the mast compressive force is due to the New Zealand
record, the maximum increase in the tensile force for the
tensor is imposed by the Chile earthquake (whose
maximum PGA is the smallest considered), followed close
by the Norcia Earthquake, which nor has one of the largest
Fig. 20 Membrane top corner principal stress
Fig. 21 Membrane top corner principal stress limits
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Jesús G. Valdés-Vázquez, Adrián D. García-Soto, Michele Chiumenti and Alejandro Hernández-Martínez
maximum PGAs considered, neither a large magnitude (the
smallest indeed), albeit it has the shortest source-to-site
distance. Similar unclear trends are found for the principal
tensor and membrane stresses. Therefore, simple recipes for
design are not recommended. Nevertheless, further research
is required to inspect whether the PGA could be solely the main parameter for design (for instance in probabilistic
rather than in deterministic terms), since it is usually one of
the most relevant parameters of ground motion for design
purposes. The seismic structural response of TFSs seem to
be a complex combination of source, path, and site
characteristics, plus perhaps frequency content of the signal
and even the supporting frame configuration (Valdés-
Vázquez et al. 2019). If seismic risks analysis of TFSs is
pursued in the future, the seismic hazard from different
contributing sources and seismic scenarios, as well as the
vulnerability of different TFS geometries and configurations of their supporting frames, may be
necessary. The finite element formulation, the meshing and
other structural analysis aspects should also be selected
carefully, so that time-history analyses can be performed
adequately in TFSs.
8. Conclusions
The seismic structural response of a tensile fabric
structure, in terms of membrane stresses and axial forces in
the supporting frame, is investigated by considering a
geometrical configuration commonly known as hypar.
Recordings from several parts of the world are used. They
cover wide ranges of moment magnitude (6.5≤Mw≤8.3),
epicentral or rupture distances (11 km - 125 km) and focal
depths (9.2 km - 47 km), and they come from different
tectonic environments with different failure mechanisms, site soil classifications and frequency contents. However,
they have in common large maximum PGAs. Finite element
simulation is employed to perform time-history analyses
using the seismic records. The analyses include the so-
called form-finding procedure.
The main findings of this study are:
• The maximum seismic tensile forces in cables and
compressive forces in masts should be incorporated in
design; especially if considering that seismic loading
(unlike wind loading) is usually not considered relevant
in practice for TFSs. The opposite occurs for the
membrane earthquake-induced stresses, which do not increase importantly under seismic loading.
• Records with large maximum peak ground
accelerations (> 0.7 g) induce axial forces in the masts
and cables up to twice as large as those imposed by the
static case (i.e., self-weight and prestress).
• Although, the maximum PGA is a key parameter
correlated to significant seismic-induced axial forces,
there is not a clear trend to express the latter as a
function of the former.
• The exclusion of the form-finding procedure, before
the dynamic analysis, significantly impacts the resulting fabric stresses; the opposite occurs with respect to the
compressive and tensile forces in masts and tensors,
respectively (i.e., resulting axial forces in the time-
history analysis are relatively unaffected by excluding
the form-finding procedure).
• Other than large axial forces linked to large maximum
PGAs, no clear trends were found between the different
earthquakes and recordings characteristics and the largest obtained axial forces.
It is concluded that simplistic rules and analyses for
designing support frame elements of TFSs subjected to
seismic loads are not recommended. It is also concluded
that the seismic structural response of TFSs is a complex
combination of earthquake failure mechanism, frequency
content and site effects, among other possible aspects.
Seismic hazard from different contributing sources, as well
as the capacity of different TFSs geometries and
configurations, should be considered if seismic risk
analyses are desired in the future. Not every finite element formulation can capture the complex, highly non-linear,
behaviour of TFSs, therefore care should be exercised to
select the adequate tools for the analysis.
Acknowledgments The support from Universidad de Guanajuato is
gratefully acknowledge. We are thankful to two anonymous
reviewers for their comments, suggestions, and constructive criticism.
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