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: valdes@ugto.mx

aPh.D. Professor

E-mail: adgarcia@ugto.mx bPh.D. Professor

E-mail: michele@cimne.upc.edu cPh.D. Professor

E-mail: alejandro.hernandez@ugto.mx

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

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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

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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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