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Structures 67 (2024) 106960
2352-0124/© 2024 The Author(s). Published by Elsevier Ltd on behalf of Institution of Structural Engineers. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
Conceptual study of an innovative friction damper for the seismic retrot of
precast RC structures with poor connections
Eleonora Grossi
a
,
*
, Matteo Zerbin
a
, Alessandra Aprile
a
, Raffaele De Risi
b
, Flavia De Luca
b
a
University of Ferrara, Ferrara, Italy
b
University of Bristol, Bristol, UK
ARTICLE INFO
Keywords:
Precast RC structure
Friction damper
Conceptualisation
Seismic performance
Importance analysis
ABSTRACT
Precast RC structures have been widely used in industrial and commercial buildings since the 60 s in the most
developed areas. However, during those decades of economic growth, most buildings were constructed without
seismic design criteria, accounting for gravity loads only. For this reason, this structural typology often faces a
signicant seismic risk in earthquake-prone areas due to the lack of effective connection between structural
elements. As a result, the seismic retrot of precast RC structures is essential to prolong their service life and
mitigate seismic losses. The present work shows the conceptualisation study of an innovative seismic protection
device called Bidirectional Rotational Friction Damper (BRFD) for precast RC structures that behave simulta-
neously as a beam-to-column joint and damper. This device unies the concepts of rotational friction dampers
and a movable plate system, producing a damping effect along two main directions. Furthermore, the device’s
ability to dissipate energy through friction enables it to remain undamaged during multiple seismic events while
maintaining its damping capacity. After dening a simplied analytical model, to evaluate the inuence of the
BRFD on a structure’s behaviour during a seismic event, a case study was conducted on a single-story, single-bay
precast reinforced concrete structure made of plane parallel frames, i.e. that lacks secondary frames. Quasi-static
and nonlinear time history analyses were performed to evaluate the BRFD efcacy in reducing seismic forces and
displacements, and an importance analysis was carried out using a multi-criteria decision-making (MCDM)
approach to identify the optimal conguration of the BRFD for the case study. The main results highlight that
introducing the BRFD positively inuences the dynamic performance of the structure, producing a signicant
reduction of interstorey drift and total base shear and preventing structural and non-structural damage.
1. Introduction
Precast RC structures have been widely used in industrial and com-
mercial buildings since the 60 s in the most developed countries. These
structural systems consist of modular, quickly installable, mass-
produced elements that cover large spans. During those decades of
economic growth, buildings were constructed without considering
seismic design criteria, resulting in designs that only accounted for
gravity loads.
Precast RC structures often face a signicant seismic risk in
earthquake-prone areas due to the lack of effective connection between
structural elements. This issue was highlighted in the 2000 FIB report,
which emphasised the combined effect of high seismic vulnerability and
exposure in such structures [1]. This was conrmed during the 2012
Emilia Earthquake [2–7], which occurred in one of the most important
industrial areas in Italy. Thousands of buildings in the region were
designed without seismic detailing and collapsed during the seismic
event, resulting in 28 casualties and hundreds of injured and displaced
people, affecting the regional economy with enormous losses. It is
evident that seismic losses resulting from buildings’structural damage
impact society, the environment, and the industrial and commercial
activities within the epicentral area [8]. The seismic retrot of precast
RC structures is essential to prolong their service life and mitigate
seismic losses. The most common solutions involve traditional and
passive control techniques based on energy dissipation.
Traditional retrotting techniques, like concrete or steel jacketing
and bre-reinforced polymer (FRP) wrapping, are usually adopted to
increase the structural elements’strength, especially the columns [9]. In
* Corresponding author.
E-mail addresses: eleonora.grossi@unife.it (E. Grossi), matteo.zerbin@unife.it (M. Zerbin), alessandra.aprile@unife.it (A. Aprile), raffaele.derisi@bristol.ac.uk
(R. De Risi), avia.deluca@bristol.ac.uk (F. De Luca).
Contents lists available at ScienceDirect
Structures
journal homepage: www.elsevier.com/locate/structures
https://doi.org/10.1016/j.istruc.2024.106960
Received 29 November 2023; Received in revised form 10 May 2024; Accepted 19 July 2024
Structures 67 (2024) 106960
2
addition, external bracings are added to increase global stiffness [10,11]
and steel connections are placed between structural elements to improve
their connectivity [9]. Adopting traditional techniques requires the
execution of additional works at the foundation level and, in case of
signicant seismic events, can still substantially damage the structure.
Passive control techniques primarily rely on the yielding or friction
properties of metallic materials to dissipate seismic energy [12].
Implementing dissipative devices has proven to be an effective and
affordable solution compared to traditional retrotting methods, pre-
venting damage to structural and non-structural elements [13]. During
the last decade, several authors have focused on the seismic rehabili-
tation of precast RC structures, developing devices that are able to
dissipate energy and simultaneously improve the connection between
structural elements. Most of these solutions are intended to be installed
as beam-to-column joints in the main frames, and the main damping
principle relies on the friction and yielding properties of metallic ma-
terials. Eldin et al. [14] introduced a beam-to-column connection that
dissipates energy using the linear sliding mechanism of a friction
damper (FD). The studies of Huang et al. [15] focused on a bolted web
FD and, to increase the stiffness of the system and the associated energy
dissipation, the authors suggested the adoption of friction pads with
added grooves. Valente [16] worked on a Rotational Friction Damper
(RFD) that dissipates energy thanks to the subsequent opening and
closing of the gap between the beam and the column during the seismic
event. Colajanni et al. [17] conducted additional studies on RFDs by
incorporating a bolted web FD into the solution suggested by Valente
[16] to improve the damping capability of the system. Martinelli and
Mulas [18] revised the layout of beam-to-column joints, and their so-
lution was improved by Belleri et al. [19] by introducing a re-centring
component. Pollini et al. [20] and Huang et al. [21] worked on de-
vices whose damping mechanism is based on the yielding of steel tubes
with added connement elements to avoid instabilities during the
yielding process. Bressanelli et al. [22] focused on the design and opti-
misation of a crescent moon steel element.
These devices proved to be very efcient in terms of energy dissi-
pation and base shear reduction without limiting the buildings’usable
surface area, which is an important aspect when considering the typical
use of those buildings. As these devices are monodirectional, additional
devices are essential on the plane orthogonal to the main frames to get
energy dissipation in all the structure’s directions as required by the
seismic actions. However, additional devices cannot be inserted when
orthogonal beams do not efciently connect the parallel main frames.
Passive control techniques are known to have less environmental
impact when compared to traditional techniques [23,24], especially if
the adopted devices are easily restorable and can survive several shocks
without damage. This is conrmed in the recent work of Cavalieri et al.
[25], which computed a signicant reduction of CO
2
emissions when
retrotting precast RC structures using innovative solutions instead of
traditional ones.
The need for a dissipative device that can improve the overall seismic
performance of a precast RC structure and has a low architectonical and
environmental impact led the authors to develop a bidirectional
damping device installed as a beam-to-column joint. This device unies
the concepts of RFDs and a movable plate system, obtaining a so-called
Bidirectional Rotational Friction Damper (BRFD) that produces a
damping effect along two main directions [26]. Furthermore, the de-
vice’s ability to dissipate energy through friction enables it to remain
undamaged during multiple seismic events while maintaining its
damping capacity.
The present work shows the conceptualisation study of the BRFD to
assess the feasibility and effectiveness of this innovative solution by a
multi-criteria decision-making (MCDM) approach [27]. Conceptualisa-
tion studies are often used as proof-of-concept tools when introducing
innovative solutions and are usually characterised by a simplied
analytical and numerical characterisation of the object of study [28].
This paper is divided into six sections. Section 2 explains the layout
of the BRFD and how it is expected to impact the structure, while Section
3provides a simplied analytical model of the device. In Section 4, the
properties of a precast RC structure are presented as a case study, and
the inuence of the BRFD on the frame’s structural behaviour is high-
lighted. A sensitivity analysis is carried out in Section 5, and an
importance analysis using a multi-criteria decision-making (MCDM)
approach is conducted to identify the optimal BRFD conguration for
the case study. Finally, Section 6 summarises the main ndings and
provides concluding remarks.
2. The BRFD layout and its structural impact
The BRFD is obtained by assembling layered steel plates, combining
RFDs and a movable plate system to achieve a bidirectional behaviour.
Fig. 1(a) shows an axonometric view of the BRFD, while Fig. 1(b) shows
an example of its installation within one of the main portal frames at a
45◦inclination angle from the existing beam as a beam-to-column joint.
In the following, lowercase letters indicate the local reference system of
the BRFD and, in general, of an element, while uppercase letters indicate
the global reference system of the structure. It is worth noting that the
plane generated by the BRFD’s local x- and y-axis is inclined at 45◦and
perpendicular to the global XZ plane. Furthermore, the connection
braces are hinged to the existing beam and column to allow the rotations
around the global Yaxis only and the deformations along the BRFD local
xaxis.
Core plates are the primary components of BRFD and are held
together with four preload stud bolts. In each core plate’s ends, friction
pads are interlocked and kept in contact by the pressure generated by
the stud bolts; as the device activates, the core plates’ends rotate around
the stud bolts, and the friction pads slide on each other, dissipating
energy and obtaining the deformed shapes of Fig. 2. A general
displacement can be decomposed into a longitudinal component Δxand
a transversal component Δy. Consequently, the BRFD’s activation force
has two components: a longitudinal component Fact,x(local x-direction)
and a transversal component Fact,y(local y-direction). Two alignment
guides with slotted holes are placed at the top and bottom of the device
to keep preload bolts aligned with the central bolt, ensuring controlled
displacements.
When introducing the BRFD inside a precast RC frame, the static
scheme of the columns is expected to change from cantilever (Figs. 3(a)
and 3(c)) into double-xed element (Figs. 3(b) and 3(d)), resulting in a
structural stiffness increment in both in-plane and out-of-plane frame’s
directions (see Fig. 3).
Three main conditions drive the BRFD design:
1. BRFD’s steel elements shall not enter the plastic domain;
2. RC beams and columns shall not slide on each other during seismic
motion, especially when a mechanical fastening is not installed (like
in many pre-seismic code constructions);
3. RC beams and columns shear demand shall not exceed the capacity,
while columns shall not yield.
3. The BRFD simplied analytical model
The analytical model here implemented aims to describe the BRFD
behaviour using a simplied approach. As previously mentioned, the
energy dissipation results from the relative rotation of the friction sur-
faces around the stud bolts, which maintain them coupled with each
other. Assuming the circular contact area of Fig. 4, the relative motion
between two surfaces begins when the interface sliding moment Ms,iis
reached:
Ms,i=Re
ρ
=Ri2
π
θ=0
ρ
2
μ
FP
π
Re2−Ri2d
ρ
dθ=2
3
μ
FP
Re3−Ri3
Re2−Ri2(1)
E. Grossi et al.
Structures 67 (2024) 106960
3
where
μ
is the interface friction coefcient (here assumed constant to
keep the model simple), FPis the stud bolts axial load, Reand Riare outer
and inner radius, respectively, and
ρ
and θare the radial and angular
coordinates [29].
The BRFD motion starts when the bending moment at nodes A,B,A
′
and B
′
of Fig. 4 equals the total sliding moment Msexpressed by Eq. (2),
where Ms,iis given by Eq. (1) and nis the number of sliding interfaces of
the device’s contact areas, depending on the plate’s number.
Ms=nMs,i(2)
As a result, nodes A,B,A
′
and B
′
can be considered as rigid nodes
when bending moment values are lower than Ms, and as hinged nodes
for values equal to Ms.
This behaviour is statically reproduced using a rotational spring with
a stiffness KRthat is highly stiff for bending moment values lower than
Ms, and it is highly soft for bending moment values equal to Ms.Fig. 5(a)
shows the static scheme associated with the BRFD core obtained by
introducing the rotational springs. Nodes A,B,A
′
and B
′
are hinged and
connected with beams of length l=L−b(see Fig. 4), which represents
the stud bolt’s distance. Additionally, each node is equipped with a
rotational spring of stiffness KR, which replicates the rotational friction
Fig. 1. BRFD (a) axonometric view and (b) example of installation as a beam-to-column joint.
Fig. 2. BRFD bidirectional deformation shape in (a) tension ( +Δ
xand +Δy) and (b) compression ( −Δxand −Δy) congurations.
Fig. 3. Main frame deformation in Xdirection (a) without and (b) with BRFD, and deformation in Ydirection (c) without and (d) with BRFD.
Fig. 4. BRFD core with circular contact areas highlighted in red colour.
E. Grossi et al.
Structures 67 (2024) 106960
4
behaviour.
To compute the relationship between the sliding moment Msand the
BRFD’s activation forces, the conguration before sliding of Fig. 5(b) is
considered in the following. As previously mentioned, nodes A,B,A
′
and
B
′
can be considered as rigid nodes before sliding occurs while the
alignment guides maintain the central symmetry around node O,
allowing the static scheme of Fig. 5(a) to be transformed into the one of
Fig. 5(b), analysing only the upper half. It is worth noting that, because
of the central symmetry, when analysing the half BRFD, the displace-
ments are halves with respect to the overall BRFD displacements, while
the forces are equal to the overall BRFD forces.
When a generic displacement Δ
2
→=Δx
2
→+Δy
2
→is applied to node Ain
the before sliding conguration of Fig. 5(b), the reaction forces HO,VO,
HA,VAand MAare generated on nodes Oand A, respectively. More
precisely, the reaction forces HA,VAand MAcoincide with the forces
transferred by the BRFD into the existing structures. If Δs/2 is the
generic displacement that generates the moment Mson nodes Aand/or
B, then HAand VAare equal to the longitudinal and transversal
components (Fact,xand Fact,y, respectively) of the activation force of the
BRFD.
To better understand the mechanical behaviour of the BRFD, longi-
tudinal (Δx/2) and transversal (Δy/2) displacements are studied sepa-
rately. This allows to focus on the relationship between the sliding
moment Msand the activation forces and initial stiffness of the BRFD. In
addition, to maintain the analytical model simple, only exural defor-
mation is considered in the analysis.
3.1. Longitudinal behaviour
When the BRFD is subjected to longitudinal displacements (Δx/2),
the static scheme of Fig. 5(b) can be simplied into the one of Fig. 6(a). A
is a rigid node that can slide along x-directions, while Ois a hinged node.
When a longitudinal force Fxis applied to node A, the displacement Δx/2
occurs and the reaction forces HO,VO,VAand MAare generated. It is
worth noting that in the half BRFD analysed, the displacements are
halves with respect to the overall BRFD displacements, while the forces
are equal to the overall BRFD forces.
Fig. 5. BRFD core simplied static scheme (a) with rotational springs and (b) in the before sliding conguration.
Fig. 6. BRFD’s longitudinal component (a) static scheme, (b) bending moment, (c) shear and (d) axial forces diagrams.
E. Grossi et al.
Structures 67 (2024) 106960
5
The static scheme of Fig. 6(a) is one-time hyperstatic and has been
analytically resolved by applying the Principle of Virtual Works (PVW).
Figs. 6(b), 6(c) and 6(d) report, respectively, the obtained diagrams of
bending moment, shear and axial forces. Node Breaches the higher
value of bending moment (MB), consequently when longitudinal
displacement occurs, the BRFD activates when MB=Ms. The relation-
ship between Msand the longitudinal activation force Fact,xis described
as follows:
Fact,x=2Ms
l(3)
The displacement Δx/2 is computed by applying the PVW and the
initial longitudinal stiffness Kxis obtained from the ratio between Fxand
Δx:
Kx=Fx
Δx=E
l31
12IOB +1
8IBA−1
(4)
where IOB and IBA are the moments of inertia of OB and BA elements (see
Fig. 5(b)), which represent the core central and core side plates,
respectively. IOB and IBA are computed by considering the whole cross-
section of the elements; however, when designing the BRFD, the ca-
pacity of its elements is evaluated by considering the presence of the
holes to avoid any risk of exceeding the elastic limit. To maintain the
analytical model as general as possible, IOB and IBA are kept independent
to each other: in the real device, core central and core side plates have a
different stratication, and they can have a different geometry, which
affects IOB and IBA inertia. However, when core central and core side
plates have the same section, Eq. (4) can be written as follows:
Kx=
48
7
EI
l3for n=2→IOB =I;IBA =2I
12 EI
l3for n=4→IOB =2I;IBA =3I
(5)
From Eq. (5) it can be observed that when incrementing the number
of friction interfaces and, consequently, the number of core’s plates, the
initial stiffness Kxincrements as well.
3.2. Transversal behaviour
When the BRFD is subjected to transversal displacements (Δy/2), the
static scheme of Fig. 5(b) can be simplied into the one of Fig. 7(a). Ais a
rigid node that can slide along y-directions, while Ois a hinged node.
When the tansversal force Fyis applied to node A, the displacement Δy/2
occurs and the reaction forces HO,VO,HAand MAare generated. It is
worth noting that in the half BRFD is analysed, the registered dis-
placements are halves with respect to the overall BRFD displacements,
while the registered forces are equal to the overall BRFD forces.
The static scheme of Fig. 7(a) is one-time statically indetermined and
has been analytically resolved by applying the PVW. Figs. 7(b), 7(c) and
7(d) report, respectively, the obtained diagrams of bending moment,
shear and axial forces. Node Areaches the higher value of bending
moment (MA), consequently, when transversal displacement occurs, the
BRFD activates when MA=Ms. The relationship between Msand the
transversal activation force Fact,yis described as follows:
Fact,y=2
3
√
3
Ms
l(6)
The displacement Δy/2 is computed by applying the PVW and the
initial longitudinal stiffness Kyis obtained from the ratio between Fyand
Δy:
Ky=Fy
Δy=E
l31
16k2IOB +2
IBA 1
4k−1
22−1
(7)
where IOB and IBA are the moments of inertia of OB and BA elements (see
Fig. 5(b)), which represent core central and core side plates respectively,
and k=IBA/(2IOB )+1 is an non-dimensional parameter introduced to
compact the formulation. IOB and IBA are computed by considering the
whole cross-section of the elements; however, when designing the
BRFD, the capacity of its elements is evaluated by considering the
Fig. 7. BRFD’s transversal component (a) static scheme, (b) bending moment, (c) shear and (d) axial forces diagrams.
E. Grossi et al.
Structures 67 (2024) 106960
6
presence of the holes to avoid any risk of exceeding the elastic limit. To
maintain the analytical model as general as possible, IOB and IBA are kept
independent to each other: in the real device, core central and core side
plates have a different stratication and they can have a different ge-
ometry, which affect their inertia. However, when core central and core
side plates have the same section, Eq. (7) can be written as follows:
Ky=
32
5
EI
l3for n=2→IOB =I;IBA =2I;k=2
21
2
EI
l3for n=4→IOB =2I;IBA =3I;k=7/4
(8)
From Eq. (8) it can be observed that when incrementing the number
of friction interfaces and, consequently, the number of core plates, the
initial stiffness Kyincrements as well.
3.3. Bidirectional behaviour
When the BRFD is subjected to a generic displacement Δ
2
→=
±Δx
2
→±Δy
2
→, the reaction forces and the internal forces diagrams of
Fig. 5(b) are computed by combinig the solutions of §3.1 and §3.2. More
precisely, Eq. (9) represent the reaction forces when Δxand Δyare
applied with the same and opposite sign, while Eq. (10) represent the
condition that Fxand Fyshall satisfy to reach the sliding moment Msat
nodes Aand Bat the same time.
HA=HO= ±Fx±
3
√
2kFy
VA=VO= ±
3
√
6Fx±Fy
forΔ
2
→= ± Δx
2
→±Δy
2
→(9)
MB= ±l
2Fx±
3
√
4klFy= ±Ms
MA= ∓l
4Fx∓
3
√
2lFy= ∓Ms
forΔ
2
→= ± Δx
2
→±Δy
2
→(10)
If Fxand Fysatisfy the condition of Eq. (10), the dimensionless
interaction domain of Fig. 8 is obtained. The blue dots indicate a mono-
directional loading condition, the orange dots indicate a loading con-
dition with Δxand Δyapplied with the same sign, and the green dots
indicate a loading condition with Δxand Δyapplied with the opposite
sign.
The interaction domain of Fig. 8 is rectangular and indicates that the
activation forces of the BRFD remain constant and equal to the mono-
directional conditions of §3.1 and §3.2, independently from the
loading direction. This behaviour permits considering the longitudinal
and transversal components of the BRFD as unrelated, allowing the
denition of a simplied uncoupled hysteretic behaviour.
3.4. The simplied hysteresis cycles
One of the main hypotheses of the simplied analytical model for the
BRFD is that the nodes associated with the dissipative areas can be
considered as hinged nodes once the device activates (see Fig. 4). As a
result, when the BRFD is subjected to the longitudinal force Fact,x, which
activates the kinematic along the local x-direction, the rigid node B
becomes a hinged node (see Fig. 9(a)). Similarly, when the BRFD is
subjected to the transversal force Fact,y, which activates the kinematic
along the local y-direction, the rigid node Abecomes a hinged node (see
Fig. 9(b).
When longitudinal displacement occurs, the bending moment
around node Bequals Msand, because of the friction mechanism, it
cannot increase. The relationship between Msand the longitudinal
activation force Fact,xis described as follows:
Fact,x(Δβ) = 2Ms
l•cosΔβ(11)
where Δβis the angle OB element generates with its original position
when moving. The distance between nodes Band Ochanges when the
BRFD activates along the longitudinal direction, becoming l/2cosΔβ.
When transversal displacement occurs, the bending moment of node
Aequals Msand, because of the friction mechanism, it cannot increase.
Moreover, the relationship between Msand Fact,yis given by Eq. (6),
since the distance between nodes Aand Okeeps equal to
3
√/2lwhen the
BRFD activates along the transversal direction.
Fig. 10 shows the shape of the two hysteresis cycles associated with
BRFD’s longitudinal (blue line) and transversal (red line) components
(x- and y-direction, respectively). The shapes are obtained by combining
the Eqs. (4), (11) and (5) for the longitudinal component, and Eqs. (6)
and (7) for the transversal component.
The BRFD’s hysteresis cycle associated with its longitudinal
component exhibits a force increment when incrementing the
displacement. This increment is a function of cosΔβ, but for small
displacement Δx, typical of seismic devices real use condition, the
increment can be assumed as linear. The BRFD’s hysteresis cycle asso-
ciated with its transversal component exhibits a force constancy when
incrementing the displacement. It is worth noting that the two compo-
nents are unrelated and independent of each other. This behaviour
simplies the numerical implementation of the BRFD, as it can be
dened as a link with two separated hysteresis laws, one for each
component.
The presented analytical model assumes a constant friction coef-
cient
μ
, according to Coulomb’s law. For instance, the shapes of Fig. 10
can be signicantly affected by the real
μ
behaviour during the real use
conditions. This topic may be critical and will be the focus of further
research after the execution of mechanical testing. In fact, the here
presented simplied analytical model is useful to understand the BRFD’s
inuence when implemented inside a precast RC structure, which is the
main goal of the present work.
4. Case study and BRFD inuence
To evaluate the inuence of the BRFD on a structure’s behaviour
during a seismic event, a case study is conducted on a single-story,
single-bay precast reinforced concrete structure, whose geometry is
represented in Fig. 11. Main frames (shown in Fig. 11(b)) consist of 7 m
height square columns and 15 m length prestressed I-shaped beams
connected by three 15 m length prestressed PI-shaped slabs. The total
weight of the case study is 1393 kN, which averages a mass of 14 tons. It
is worth noting that the prestressed PI-shaped slabs cannot be consid-
ered a proper connection between the two parallel main frames (Xdi-
rection); as a result, the case study lacks secondary frames (Ydirection).
Fig. 8. Activation forces dimensionless domain.
E. Grossi et al.
Structures 67 (2024) 106960
7
Fig. 12 shows the reinforcement detailing of columns and beams,
while the material properties were assessed by on-site sampling: the
concrete has a cylindrical compressive strength of 38 MPa and a Young
modulus of 33 GPa, reinforcing steel bars have a yielding stress of
544 MPa and a Young modulus of 200 GPa, prestressed steel bars have a
yielding stress of 1670 MPa and a Young modulus of 200 GPa.
Given the absence of proper out-of-plane frames in the Ydirection,
the BRFD is only inserted inside the main frames in the Xdirection, as
shown in Fig. 1(b).
The case study was numerically implemented in Opensees [30] using
the scheme of Fig. 13 and STKO [31] as pre- and post-processor. I-shaped
beams are modelled as elastic beams hinged at the ends, while square
columns are bre sections with a xed base. To validate the numerical
implementation of the columns, the relationship between the bending
moment (M) and the curvature (
χ
) was analytically computed and
compared with the numerical one in Fig. 14(a). The comparison high-
lights the good matching between analytical and numerical previsions,
validating the adopted modelling approach. The roof is simulated with a
diaphragm constrain type in X-Y plane using a control node at the centre
of the roof.
The BRFDs are modelled using ZeroLength links with two different
Steel01 materials along their local xand ydirections to describe the two
components of the BRFD hysteretic laws described in Section 3. More
precisely, the BRFD longitudinal component xaligns with the axial
Fig. 9. BRFD core simplied kinematics when (a) longitudinal and (b) transversal displacements occur.
Fig. 10. Longitudinal and transversal components of the BRFD’s simplied
analytical hysteresis cycles.
Fig. 11. Precast RC structure used as a case study: a) top view, b) A-A view and c) B-B view. Dimensions in m.
E. Grossi et al.
Structures 67 (2024) 106960
8
direction of the ZeroLength links, and the BRFD transversal component y
aligns with the global Ydirection. Figs. 14(b) and 14(c) compare the
analytical (A tag) and the numerical (N tag) previsions of the BRFD
longitudinal and transversal components, respectively. The comparison
highlights the good matching between analytical and numerical pre-
visions in both longitudinal (x) and transversal (y) directions, validating
the adopted modelling approach.
4.1. Quasi-static performance without and with BRFD
A quasi-static analysis was performed to preliminarily evaluate the
inuence of the BRFD when installed inside the case study. The control
node located at the top of the structure (see Fig. 13) was subjected to a
sinusoidal displacement law with an amplitude of ±300 mm, a fre-
quency of 0.05 Hz for three cycles and a time step increment of 0.001 s
Three different frames have been modelled: F0, which represents the
case study without the BRFDs, F1, which represents the case study that
implemented a beam-to-column connection with the elastic properties
of the BRFDs, and F2, which represents the case study that implemented
a beam-to-column connection with the hysteretic properties of the
BRFDs. It is worth noting that F1 resembles a traditional retrot
executed by introducing a steel connection between the structural ele-
ments and it is used, together with F0, as a reference for a better un-
derstanding of the nonlinear behaviour of F2.
Table 1 describes the properties of the implemented frames in terms
of linear elastic period along directions Xand Y,T0,Xand T0,Yrespec-
tively, and BRFD’s properties in terms of activation forces along its local
x- and y-direction, Fact,xand Fact,yrespectively, and initial stiffness along
its local x- and y-direction, Kxand Kyrespectively. The values of the
BRFD properties reported in Table 1 come from the results of §5.2 and
are here used as an example. It is worth noting that the presence of the
BRFD decreases the elastic period of the case study by about 45 % in
both Xand Ydirections. This is because the BRFD elastic stiffness in-
creases the rigidity of the beam-to-column node, causing the columns
Fig. 12. Elements detailing: a) column base (Sections 1–1), b) beam end (Sections 2–2) and c) beam centre (Sections 3–3). Dimensions in mm.
Fig. 13. Numerical implementation scheme of the case study.
Fig. 14. Comparison between analytical (A) and numerical (N) previsions in terms of (a) columns M−
χ
relationship and BRFD F−Δrelationship along its (b)
longitudinal xand (c) transversal ydirections.
Table 1
Description of the three implemented frames and BRFD’s properties.
Frame Description T0,X
[s]
T0,Y
[s]
Fact,x
[kN]
Fact,y
[kN]
Kx
[kN/
m]
Ky
[kN/
m]
F0 No BRFD 0.77 0.77 - - - -
F1 BRFD elastic
properties
0.44 0.41 - - 1⋅10
6
9⋅10
5
F2 BRFD
hysteretic
properties
0.44 0.41 125 72 1⋅10
6
9⋅10
5
E. Grossi et al.
Structures 67 (2024) 106960
9
static scheme to change from cantilever into double-xed elements. This
is quite a general outcome, whenever the beam-to-column joints are
strengthened using additional steel plates. However, the stiffening effect
may be different depending on the specic structural layout.
Fig. 15 shows the results of the quasi-static analysis in terms of top
displacement and total base shear relationship in directions Xand Y,
Figs. 15(a) and 15(b) respectively. F0 exhibits a similar hysteresis cycle
in the Xand Ydirection due to the steel yielding at the square columns
footings; this similarity is due to the frame’s square plant and the sym-
metric detailing of the columns (see Fig. 12(a)). F1 exhibits a higher
stiffness than F0, with higher forces and smaller displacement associated
with the columns yielding. Furthermore, the hysteresis cycle in the X
direction differs from the one in the Ydirection; this difference is caused
by the dissimilar changes in the frame’s static scheme that the BRFDs
induce in the two global directions. More precisely, the column goes
from a cantilever to a xed scheme along the Xdirection and from a
cantilever to a semi-xed scheme along the Ydirection. In fact, the
columns yield simultaneously at the base and the top sections in the X
direction, while in the Ydirection, the columns yield rstly at the base
and then at the top section. F2 combines the behaviours of F0 and F1: F2
exhibits the same stiffness as F1 until the BRFD activates, then behaves
more similarly to F0, registering a similar hysteresis cycle in the Xand Y
direction.
The BRFD positively inuences the case study’s structural behaviour,
unaltering the top displacement associated with the column yielding and
slightly increasing the related total base shear. Moreover, when the
BRFD activates, it restores the original static scheme of the structure,
allowing the columns to behave as cantilevers.
5. Dynamic performance
To investigate the benets of the BRFD on the case study during a
seismic event, Nonlinear Time History Analyses (NLTHAs) were per-
formed on frames F0, F1 and F2 using natural ground motions. Seven
pairs of ground motions were selected from the European Strong Motion
database [32]; the ground motions are spectrum-compatible for a high
seismic area in Italy (i.e. L
′
Aquila [33]) and are scaled to match the
Life-Safety limit state assuming soil type B and Importance Class III
according to Eurocode 8–1[34].Table 2 summarises the properties of
the selected ground motions, including the original Peak Ground Ac-
celerations (PGA) ad Scale Factors (SF) in x and y directions.
NLTHA were performed in Opensees [30] using STKO [31] and
Matlab [35] as pre- and post-processing tools. The results are assessed
considering the total base shear (Fb), interstorey drift (IDR), columns’
base rotation and shear utilisation factors (
ρ
C,θand
ρ
C,V), and equivalent
damping (ξeq). The columns’utilization factors are a-dimensional and
are dened as follows:
ρ
C,θ=θEd
θy
(12)
ρ
C,V=VEd
VRd
(13)
Columns’utilisation factors are computed as a ratio between demand
and capacity: values lower than 1 indicate the satisfaction of the col-
umns’safety checks. The rotation capacity was set as the rotation
associated with the base column yielding θycomputed according to
Eurocode 8–3[36], while the shear capacity as the shear associated with
the shear failure VRd computed according to Eurocode 8–3[36]. It is
worth noting that when dening
ρ
C,θ, the choice to associate the capacity
rotation with the yielding rotation of the column was driven by one of
the main objectives of this work, which is to avoid structural damage to
the building.
The equivalent damping ξeq is computed from the acceleration
spectrum in ADRS (Acceleration-Displacement Response Spectrum)
format [37] once the top displacement and total base shear demand are
determined.
5.1. Sensitivity analysis
To investigate the effects of the BRFD’s activation forces (Fact,xand
Fact,y) and initial stiffnesses (Kxand Ky) on the case study during a
seismic event, the F2 frame was subjected to a sensitivity analysis.
The analytical model developed in Section 3 allows to describe Fact,y
as a function of Fact,xand Kyas a function of Kxas follow:
Fact,y=2
3
√Ms
3l=
3
√
3Fact,x(14)
Ky=
32
5
EI
l3=14
15Kx≅0.93Kx≈0.9Kxfor n=2
21
2
EI
l3=7
8Kx≅0.88Kx≈0.9Kxfor n=4
(15)
Eqs. (14) and (15) are obtained by merging Eq. (3) with (6) and Eq.
(5) with (8), respectively.
When performing the sensitivity analysis, Fact,xand Kxvaried as a
function of the BRFD feasibility while Fact,yand Kywere computed ac-
cording to Eqs. (14) and (15). Considering the BRFD feasibility, Fact,xwas
set to range between 25 and 250 kN, while Kxbetween 10
5
and 10
6
kN/
m. The main objective of this sensitivity analysis is to identify the
Fact,x-Kxcouple which better improves the case study performance under
the selected seismic action.
Fig. 16 summarises the sensitivity analysis results in terms of
ρ
C,θ,
ρ
C,V, IDR and ξeq as mean values of the NLTHA with seven ground mo-
tions for each Fact,x-Kxcouple. The response levels are expressed using
different colours, as shown on the scaling bar of each graph. In the
graphs associated with
ρ
C,θand
ρ
C,V(Figs. 16(a) and 16(b) respectively),
green shades are associated with the satisfaction of the columns’safety
checks, while red shades corresponds to unsatisfactory columns’safety
checks. In the graph associated with IDR (Fig. 16(c)), the darker blue
Fig. 15. Quasi-static performance using a cyclic displacement law along (a) Xand (b) Ydirection.
E. Grossi et al.
Structures 67 (2024) 106960
10
colour shades indicate lower interstorey drift values, which are usually
associated with a lower damage level [38]. In the graph associated with
ξeq (Fig. 16(d)), the lighter yellow colour shades indicate higher equiv-
alent damping values, which are associated with a better overall per-
formance of the BRFD.
When incrementing Fact,xvalues, F2 performance improves and ex-
hibits a decrement in column base rotations and interstorey drifts: for
Fact,xvalues higher than 100 kN,
ρ
C,θare lower than 0.50, and IDR are
lower than 1 %. However, when incrementing Fact,xvalues, F2 exhibits
an increment of total base shear, which may affect the columns’shear
check and the equivalent damping. This behaviour is typical of addi-
tional damping systems and has been observed by several authors
[39–42]. It is worth noting that the columns’shear failure VRd is 196 kN,
which is quite a high value and leads to
ρ
C,Vlower than 0.50 for all the
investigated Fact,x-Kxcombinations. As a result, in this case, the incre-
ment of total base shear is insignicant, but it is a critical aspect to be
considered, especially when the columns’detailing is poor for the shear
capacity.
Table 2
Properties of the selected ground motions.
TH n◦Earthquake event Station ID Magnitude
(Richter scale)
EC8 soil class Original PGA x [g] Original PGA y [g] SF x SF y
1 Central Italy
(24/08/2016)
PZI1 6.0 B 0.05 0.05 8.65 8.49
2 Central Italy
(30/10/2016)
MZ24 6.6 C 0.76 1.02 0.38 0.51
3 Central Italy
(26/10/2016)
NOR 5.9 B 0.21 0.12 1.82 3.24
4 Friuli 1st shock
(06/05/1976)
TLM1 6.4 B 0.32 0.35 1.24 1.11
5 Emilia 2nd shock
(29/05/2012)
T0819 5.5 C 0.26 0.25 1.52 1.56
6 Umbria Marche 2nd shock (26/09/1997) CSA 6.0 C 0.11 0.17 3.70 2.27
7 Emilia 1st shock
(20/05/2012)
MRN 6.1 C 0.26 0.26 1.49 1.48
Average values 6.1 0.28 0.32 2.68 2.67
Fig. 16. Sensitivity analysis results in terms of (a) column rotation and (b) shear factors, (c) interstorey drifts and (d) equivalent damping.
E. Grossi et al.
Structures 67 (2024) 106960
11
When incrementing Kx, F2 performance improves (especially when
associated with higher Fact,xvalues) and exhibits an increment of
equivalent damping: for Kxhigher than 5⋅10
5
kN/m and Fact,xvalues
between 100 and 200 kN, ξeq is higher than 30 %.
The sensitivity analysis highlights that, for a better improvement of
F2 dynamic performance, Fact,xideal values range between 100 and
200 kN, while Kxideal values are higher than 5⋅10
5
kN/m.
5.2. Importance analysis using a multi-criteria decision-making (MCDM)
approach
To better identify the optimal conguration of the BRFD for the case
study, an importance analysis using a multi-criteria decision-making
(MCDM) approach was conducted. The adopted approach uses the
entropy-right method to consider the multiple-factors effect of columns’
safety checks (
ρ
C,θand
ρ
C,V) and performance parameters (IDR and ξeq)
to identify the most important Fact,x-Kxcouple in terms of F2 overall
performance. The details of the adopted approach can be found in the
work of Guo et al. [27].
The importance indexes are divided into two groups: the rst one
considers the columns’safety checks associated with rotation and shear
factors (
ρ
C,θand
ρ
C,V, respectively), the second group considers the
overall F2 performance in terms of interstorey drift and equivalent
damping (IDR and ξeq respectively). These parameters are rstly
standardised as non-dimensional parameters, then combined according
to the entropy-right method. Each importance index is associated with a
weight that comprehends an objective weight, which is a function of the
entropy, and a subjective expert-based weight λ. In this work, the
selected groups are assumed to have the same importance, which results
in equal λvalues following the hierarchy scheme of Fig. 17.
The comprehensive indexes are nally obtained by combining the
standardised importance indexes with the comprehensive weight: the
Fact,x-Kxcouple with the highest comprehensive index is the most
important and resembles the optimal solution.
Fig. 18 shows the results of the importance analysis in terms of
performance level from “Really good”(white colours) to “Really bad”
(red colours) with an additional area associated with “BRFD unfeasible”
(black region). This additional area is obtained considering BRFDs with
maximum length of 90 cm with
μ
equal to 0.45, according to previous
authors ndings [43–45], and the Fact,xvalues that are effectively
obtainable from the geometry associated with each Kx. In fact, lower
values of Kxare associated with smaller BRFD’s plate width and a
smaller range of available studs’diameters and applicable torque. As a
result, higher values of Fact,xare not reachable for the lower Kxvalues.
F2 exhibits the best performance when Fact,xranges between 100 and
150 kN and Kxis higher than 6⋅10
5
kN/m (see the white area of Fig. 18),
conrming the remarks concerning the sensitivity analysis of §5.1. More
precisely, the MCDM approach suggests a solution with Fact,xequals to
125 kN and Kxequals to 10
6
kN/m as the most important, i.e., the
optimal BRFD conguration.
It is worth noting that, Fact,xvalues have a major impact on the ob-
tained performance level than Kxvalues. In fact, while a change in the
initial stiffness Kxmaintains the performance level mainly unaltered, a
change in activation force Fact,xmay result in a signicant modication
of the performance level. Since the activation force is strictly related to
the axial tension of the preload stud bolts (see Eq. (1) and Fig. 1(a)),
during the actual design of the BRFD this force can be optimised by
varying the torque applied to preload bolts.
The optimal BRFD conguration identied using the MCMD
approach represents the case study when located in the site selected at
the beginning of §5. A change in the structure or the site location can
affect the obtained results and the optimal BRFD conguration.
5.3. Dynamic performance without and with BRFD
To investigate the benets of the BRFD on the dynamic response of
the case study during a seismic event, NLTHA were performed on frames
F0, F1 and F2 using natural ground motions. F1 and F2 frames were
modelled, implementing the characteristics of Table 1, using the BRFD
optimal conguration properties.
Fig. 19 compares the dynamic performance of F0, F1 and F2 in terms
of Fb, IDR,
ρ
C,θand
ρ
C,Vas mean values of the NLTHA with seven ground
motions. Given the frame’s square plant and the symmetric detailing of
the columns, the response of F0 and F2 in the Xdirection resembles the
one in the Ydirection; on the contrary, the response of F1 in the Xdi-
rection differs from the one in the Ydirection, especially in terms of
displacements, conrming what already highlighted in §4.1.
F0 results indicate that while the shear checks are perfectly satised
with
ρ
C,Vvalues lower than 0.5 (see Fig. 19(d)), the damage registered
by the columns is quite signicant. In fact,
ρ
C,θvalues are higher than 1
(see the red line of Fig. 19(c)), and IDR values are higher than 2 %,
usually associated with high damage of structural and non-structural
elements [38] (see the black line of Fig. 19(b)).
F1 results indicate an overall performance improvement, with
ρ
C,θ
and
ρ
C,Vvalues lower than 1, and IDR values lower than 1 %. However,
Fb values exhibit an average increment of 64 % with respect to F0. While
in this case study, the structural response is not affected by this incre-
ment, limiting the total base shear increment is a good practice, espe-
cially for structures with poor shear detailing. Despite the total base
increment, F1 avoids panels and structural damage (red and black lines
of Fig. 19(b)).
F2 exhibits the best overall performance improvement, with
ρ
C,θand
ρ
C,Vvalues lower than 1, and IDR values lower than 1 %. Moreover, Fb
values exhibit an average decrement of 28 % with respect to F0. It can be
Fig. 17. Subjective weights adopted in the importance analysis.
E. Grossi et al.
Structures 67 (2024) 106960
12
conrmed that F2 results in a particularly effective solution that com-
bines the satisfaction of the columns’safety checks and an average 62 %
decrement of IDR without a signicant alteration of the existing struc-
tural system, avoiding structural and non-structural damage (red and
black lines of Fig. 19(b)).
6. Conclusions
The present work shows the conceptualisation study of an innovative
seismic protection device called Bidirectional Rotational Friction
Damper (BRFD) for precast RC structures with poor connections. The
BRFD behaves as a beam-to-column joint and damper at once; the device
unies the concepts of Rotational Friction Dampers and movable plate
geometry, producing a damping effect along two main directions. This is
a relevant characteristic because, up to date, the existing damping de-
vices produce a damping effect in one direction only. Furthermore, the
device’s ability to dissipate energy through friction enables it to remain
undamaged during multiple seismic events while maintaining its
damping capacity.
The BRFD behaviour is described as a combination of two compo-
nents, a longitudinal (local xdirection) and a transversal (local ydi-
rection) one, identifying an activation force (Fact,xand Fact,y) and an
initial stiffness (Kxand Ky) for each one.
To evaluate the inuence of the BRFD on a structure’s behaviour
during a seismic event, a case study is conducted on a single-story,
single-bay precast RC structure that lacks secondary frames, allowing
the implementation of BRFD inside the main frames only. Quasi-static
and nonlinear time history analyses were performed on the case study,
identifying three different frames: F0, which represents the case study
without BRFDs, F1, which represents the case study with the elastic
properties of the BRFD, and F2, which represents the case study with the
hysteretic properties of the BRFD.
To identify the optimal conguration of the BRFD for the case study,
a sensitivity analysis was carried out on F2, performing an importance
analysis using a multi-criteria decision-making (MCDM) approach.
The main ndings are summarised in the following:
•The simplied analytical model indicates that the two behaviours in
the two directions are unrelated and independent. Moreover, the
hysteresis cycles can be described using a bilinear hysteretic law.
This behaviour simplies the numerical implementation of the
BRFD, as it can be dened as a link with two separated hysteresis
laws, one for each component.
•The quasi-static analysis shows that the BRFDs application positively
inuence the case study’s structural behaviour, unaltering the top
displacement associated with the column yielding and slightly
increasing the related total base shear. Moreover, when the BRFDs
Fig. 18. Performance level map considering only the feasible BRFD congurations.
Fig. 19. F0, F1 and F2 dynamic performance: (a) total base shear, (b) interstorey drift, (c) column rotation and (d) shear factors.
E. Grossi et al.
Structures 67 (2024) 106960
13
activate, the original static scheme of the structure is restored,
allowing the columns to behave as cantilevers.
•The sensitivity analysis performed on F2 highlights that the dynamic
performance better improves when the activation force ranges be-
tween 100 and 200 kN for Fact,xand between 58 and 115 kN for Fact,y,
and when the initial stiffness is higher than 5⋅10
5
kN/m for Kxand
higher than 4⋅10
5
kN/m for Ky.
•The importance analysis performed using the MCDM approach sug-
gests that the optimal BRFD conguration is obtained when Fact,x
=125 kN, Fact,y=72 kN, Kx=10
6
kN/m and Ky=9⋅10
5
kN/m.
•The introduction of the BRFD positively benets the case study’s
dynamic performance, signicantly reducing interstorey drift (62 %)
and a reduction of total base shear (28 %) without altering the
existing structural scheme, avoiding structural and non-structural
damage.
Obviously, the dynamic results of the present work are representa-
tive of the case study when located in the site selected at the beginning of
§5. Since the obtained results are dependent on both the structural
layout and location site hazard, the MCMD approach can be proposed as
an effective design tool for the BRFD sizing.
As a summary, the conceptualisation analysis performed on BRFD
highlights that such a system can effectively improve a precast RC
structure’s behaviour during seismic events. However, further research
is needed to fully develop the potential of the device. More precisely,
future studies will be focused on the execution of bidirectional me-
chanical tests on a real-scale prototype, the development of a numerical
model which includes a friction law that considers the inuence of
sliding velocity, pressure and temperature, and topological studies to
investigate the effects on multi-span and multi-storey buildings.
CRediT authorship contribution statement
Matteo Zerbin: Conceptualization, Methodology, Resources, Su-
pervision, Validation, Writing –review &editing. Alessandra Aprile:
Conceptualization, Funding acquisition, Methodology, Project admin-
istration, Resources, Supervision, Validation, Writing –review &edit-
ing. Eleonora Grossi: Conceptualization, Data curation, Formal
analysis, Investigation, Software, Visualization, Writing –original draft.
Raffaele De Risi: Data curation, Formal analysis, Methodology, Soft-
ware, Writing –review &editing. Flavia De Luca: Methodology, Project
administration, Supervision, Validation, Writing –review &editing.
Declaration of Competing Interest
The authors declare the following nancial interests/personal re-
lationships which may be considered as potential competing interests:
Eleonora Grossi has patent #102020000013738/2021 issued to Italian
Patent and Trademark Oce, Ministry of Economic Development. Ales-
sandra Aprile has patent #102020000013738/2021 issued to Italian
Patent and Trademark Oce, Ministry of Economic Development. Matteo
Zerbin has patent #102020000013738/2021 issued to Italian Patent
and Trademark Oce, Ministry of Economic Development.
Acknowledgements
The authors wish to acknowledge the support provided by the Uni-
versity of Ferrara: “Bidirectional Friction Links for Seismic Retrot of
Existing Precast RC Structures (FAR2078914)”,“A friction damper for
seismic retrot of precast RC structures”(FAR2151302), “Experimental
testing of innovative friction dampers for seismic retrot of precast RC
structures”(FAR22718675). The authors wish also to acknowledge the
support provided by FIRD funding from the Engineering Department of
the University of Ferrara, year 2022: “Mechanical and tribological
testing of a novel damping device for seismic risk mitigation of industrial
buildings”(2022-FAR.L-FIRD_DE_AA_001).
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