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RESEARCH ARTICLE
Floor spectra for bare and infilled reinforced concrete
frames designed according to Eurocodes
Mariano Di Domenico Paolo Ricci Gerardo M. Verderame
Department of Structures for Engineering
and Architecture, University of Naples
Federico II, Naples, Italy
Correspondence
Mariano Di Domenico, Department of
Structures for Engineering and Architec-
ture, University of Naples FedericoII, Via
Claudio , Naples Italy.
Email: mariano.didomenico@unina.it
Abstract
In this study, nonlinear time-history analyses are performed to assess the floor
response spectra of bare and infilled reinforced concrete framed buildings with
different number of stories and designed according to Eurocode provisions for
different intensity levels of the seismic action. Infill walls are modeled by neglect-
ing and by accounting for the effects of their out-of-plane response and of the in-
plane/out-of-plane interaction. To this aim, a recent out-of-plane response model
is updated and improved.
The results of the numerical analyses are compared in order to assess, first,
the different floor spectra obtained for bare and infilled buildings and, second,
the effect of the in-plane/out-of-plane interaction on the results obtained for
infilled buildings. The main parameters influencing the shape and the ampli-
tude of floor response spectra are investigated, namely higher vibration modes,
structural nonlinearity, and damping of the secondary element. This is also per-
formed with the support of the discussion and application of current code and
literature formulations.
Based on the results of the numerical analyses, a simplified code-oriented for-
mulation for the assessment of floor response spectra in bare and infilled rein-
forced concrete framed structures is proposed. The proposed formulation may be
a useful tool for the seismic assessment and safety check of acceleration-sensitive
nonstructural components.
KEYWORDS
floor response spectrum, formulation, infill wall, nonlinear analysis, out-of-plane, reinforced
concrete building
1 INTRODUCTION
In the framework of performance-based earthquake engineering, the seismic assessment of nonstructural components
is a paramount issue: in fact, most of the earthquake-induced damage to buildings and the consequent economic losses
is related to them. Nonetheless, the heavy damage of certain nonstructural components, such as infill walls, may even
threaten human life safety. For these reasons, there is need for the development of robust and reliable models for the
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the
original work is properly cited.
© The Authors. Earthquake Engineering & Structural Dynamics published by John Wiley & Sons Ltd.
Earthquake Engng Struct Dyn. ;–. wileyonlinelibrary.com/journal/eqe 1
2DI DOMENICO .
GROUND MOTION ACCELERATION TIME HISTORY
a
g
(t)
Sa(T, )
T
PGA
t
THIRD FLOOR ACCELERATION TIME HISTORY
a
3
(t)
t
PSA(T
a
,
a
)
T
a
PFA
Response spectrum for a SDOF supported by
the ground and subjected to the ground
motion acceleration time history
Response spectrum for a SDOF supported by
the third floor and subjected to the third floor
acceleration time history
FIGURE 1 Concept of floor response spectrum
assessment of capacity and demand for their seismic safety assessment. This work is focused on the definition of the
acceleration demand acting on them, that is, on floor response spectra.
As well known, a structure (hereinafter “primary structure”) subjected at ground to a certain acceleration time-history
with a certain Peak Ground Acceleration (PGA) will experience a certain pseudo-spectral acceleration demand (Sa)dif-
ferent from PGA. The pseudo-spectral acceleration demand can be calculated from the response spectrum of the ground
motion based on the structure modal (ie, the period T) and damping (ie, the damping ratio ξ) properties. At the same
time, a certain floor of the structure will experience a certain acceleration time-history with a certain Peak Floor Acceler-
ation (PFA). Consider now a nonstructural component (hereinafter “secondary element”) as a Single Degree of Freedom
(SDOF) system—with vibration period Taand damping ratio ξa—supported by a certain floor of the primary structure.
The maximum pseudo-spectral acceleration, herein identified as PSA, acting on it can be calculated through the pseudo-
acceleration response spectrum of the acceleration time-history of the supporting floor. This is acceptable if the dynamic
interaction between the secondary element and the primary structure is negligible. This pseudo-acceleration response
spectrum is known as floor response spectrum. These concepts are summarized in Figure .
So, based on the above discussion, the seismic demand acting on a certain nonstructural component supported by a
certain floor of the structure can be known if the PFA of that floor and the PSA spectral shape are known. In addition,
based on the above discussion, it is possible to define some parameters that, intuitively, are expected to have an influence
on the PFA and on the PSA spectral shape. Regarding the PFA, one can expect its dependence on the PGA value (the
higher the PGA, the higher the PFA) and, due to the dynamic amplification of the fundamental vibration mode, on the
height of the specific floor considered, z, potentially normalized with respect to the total building height, H. Regarding
the spectral shape, one can expect that the PSA is amplified or de-amplified with respect to the PFA, with the maximum
amplification expected due to resonance of the secondary element and of the primary structure, that is, when Tais closer to
the fundamental vibration period(s) of the supporting structure. Also, the damping of the secondary element is expected to
influence PSA maximum values. As will be shown later in this paper, further parameters may have a significant influence
on both the PFA distribution along the building height and the floor spectral shape.
The study presented in this paper is aimed at defining code-oriented and simplified formulations for the assessment of
floor response spectra in reinforced concrete framed buildings. First, a recall of current code and literature formulations is
presented, mainly to define a systematic review of the geometric and mechanical structural parameters which are expected
to influence the PFA distribution along the building height and the floor spectral shape. Second, nonlinear models of bare
and infilled case-study reinforced concrete framed buildings with , , , and stories designed for different PGA levels
(., ., ., and . g) according to Eurocodes are defined. Then, nonlinear time history analyses are performed on
the case-study buildings to evaluate floor response spectra. For the sake of brevity, the results will be shown in this paper
only for the extreme PGA levels (. and . g) but all the results obtained are used for discussion and proposal purposes.
The numerical outcomes are compared to analyze the effect of the presence of infill walls on floor response spec-
tra considering and not considering their out-of-plane response and the mutual interaction between their in-plane
DI DOMENICO . 3
and out-of-plane responses (ie, the so-called in-plane/out-of-plane interaction). To this aim, a recent out-of-plane and
in-plane/out-of-plane interaction model proposed by the Authors of this study is updated to account for the potential
softening and cyclic degradation of the out-of-plane response of infill walls. Note that modeling the out-of-plane response
of infills and the in-plane/out-of-plane interaction is a novelty element for studies dedicated to floor spectra assessment,
which rarely have been dedicated to infilled buildings in general. Finally, a selection of the outcomes of the analyses is
compared with those expected based on code and literature formulations. This is done to assess the influence of different
parameters on the analyses’ results and is preliminary to the proposal of a code-based simplified formulation for the evalu-
ation of floor response spectra that may be useful for the safety check of acceleration-sensitive nonstructural components.
2 EXISTING CODE AND LITERATURE PROPOSALS
Several studies have been dedicated to floor response spectra in past and recent times. Comprehensive state-of-the-art
reviews can be found in Rodriguez et al,Vukobratović, Degli Abbati et al,Wang et al.In this section, for the sake of
conciseness, the discussion is focused on recent practice-oriented proposals available in the literature, as well as on main
code prescriptions regarding this topic.
2.1 Code proposals
Eurocode ,in Section .., proposes an expression, whose theoretical derivation is not clearly stated, for the PSA as
shown, with some manipulation in the nomenclature, in Equation .
PSA = PGA 3(1+𝑧∕𝐻
)
1+(1−𝑇
𝑎∕𝑇1)2−0.5
.()
In Equation ,Tais the nonstructural element vibration period and Tis the fundamental vibration period of the structure.
If Taequals zero, a linear PFA distribution along the building height is obtained at varying z/Hratio. PFA ranges from PGA
(at zequal to zero) to .PGA (at zequal to H). On the other hand, the maximum PSA is obtained if Taequals T, even at the
ground floor, where the maximum PSA should depend on ground motion characteristics. If Taequals T, the maximum
PSA ranges from .PGA (at zequal to zero) to .PGA (at zequal to H). It should be noted that this formulation returns
a maximum PSA/PFA ratio different for each floor. This will be observed also from the numerical analyses performed in
this study.
ASCE/SEI -,in Section .., proposes an expression for the PSA as shown, with some manipulation in the nomen-
clature, in Equation .
PSA = PGA 1+2𝑧
𝐻𝑎𝑝.()
In Equation ,apis a factor accounting for the amplification of acceleration due to the deformability of the nonstructural
element reported in Table .- of the code. It is worth noting that the American approach, with this factor, simplifies
the calculation of the seismic demand acting on a nonstructural element, as there is no need of a more or less detailed
dynamic characterization to determine its period, Ta, which enters Eurocode formulation.
Also according to the ASCE/SEI -approach, the PFA varies linearly along the building height. It ranges from
PGA (at zequal to zero) to PGA (at zequal to H). The maximum value of apreported in Table .- of the
code is equal to .. Hence, the maximum possible PSA value is always equal to .PFA, independently on the floor
considered.
The New Zealand code NZSEE,in section C.., refers to the loading code NZS .for the calculation of
floor response spectra. NZS .,in Section .., provides the formulation reported, with some manipulation in the
nomenclature, in Equation .
PSA = PGA𝐶𝐻𝑖𝐶𝑖(𝑇𝑎).()
4DI DOMENICO .
CHi is calculated through the formulations reported in Equation and defines the PFA distribution along the building
height which is, in this case, multilinear. The PFA ranges from PGA (at zequal to zero) to PGA (at zequal to H).
𝐶𝐻𝑖 =1+𝑧
6for all 𝑧 < 12m, (a)
𝐶𝐻𝑖 =1+10𝑧
𝐻for 𝑧 < 0.2H, (b)
𝐶𝐻𝑖 =3.0 for𝑧≥0.2H. (c)
In Equation ,Ci(Ta) is the spectral shape coefficient. Ci(Ta) is expressed as a function of the period Taand ranges from
., for Tahigher than . seconds, to ., for Talower than . seconds. Hence, the maximum possible PSA value is
always equal to PFA, independently on the floor considered. It is worth noting that, in this case, the maximum PSA value
depends on the “absolute” value of Ta, that is, it does not depend on Ta/Tratio.
As above shown, the previous code prescriptions generally relate the PFA linear or multilinear distribution along the
building height to the floor height normalized with respect to the building height (ie, the higher z/H, the higher the PFA)
and the PSA spectral shape to the ratio between the Taand T. From both these features, it can be assumed that they relate
the floor response spectra to the elastic response of the structure to its first vibration mode.
The commentary (Circolare)to the current Italian building code proposes both a “rigorous” and simplified
approaches for the assessment of floor response spectra. Both account for multimodal contributions and for the effect
on floor response spectra of structural nonlinearity. In fact, structural nonlinearity limits the maximum force acting on a
structure; hence, it also may limit the maximum acceleration acting at its floors. Both topics have been also highlighted
in recent literature, as will be shown in the next subsection.
The rigorous approach accounts for multimodal contributions based on simple considerations related to structural
dynamics. In fact, the PFAij acting at the jth floor of the primary structure associated with its ith vibration mode is deter-
mined by means of Equation .
PFA𝑖𝑗 =Γ
𝑖𝜑𝑖𝑗𝑆𝑎(𝑇𝑖).()
In Equation ,Γ
iis the modal participation factor of the ith vibration mode, φij is the modal displacement of the jth
story for the ith vibration mode, Sa(Ti) is the spectral acceleration of the structure associated with its ith vibration period,
potentially reduced by means of the structure behavior factor. The PSA is obtained, for each floor and for each vibration
mode contribution, by amplifying or de-amplifying the PFA through an Rij factor (which equals PSAij/PFAij)calculated
by means of Equation .
𝑅𝑖𝑗 =PSA𝑖𝑗
PFA𝑖𝑗
=2𝜉𝑎
𝑇𝑎
𝑇𝑖2
+1−𝑇𝑎
𝑇𝑖2
−1
()
Equation returns a PSAij value equal to /(ξa)timesPFA
ij for Ta=Ti. The value of the maximum amplification is a
classical result of the dynamic of the damped SDOF system, even if obtained with a slightly different and less rigorous
formulation.
For each story, the floor spectrum can be obtained by combining multimodal contributions through the Square Root
of Sum of Squares (SRSS) rule, thus resulting in a spectral shape with multiple peaks corresponding to the number of
significant modes considered. Implicitly, also the PFA at the jth floor can be obtained through the SRSS combination of the
different modal contributions, thus resulting in a potentially nonmonotonic distribution along the building height. Both
circumstances will be observed also from the analyses results presented in this study. Regarding the PFA distribution, it is
worth noting that Equation may yield to a de-amplification of the PFA with respect to the PGA value, especially in bottom
floors of high-rise structures. This circumstance has been observed by some authors,,, especially when analyzing elastic
models, but not by other authors and never in the analyses performed in this study. A comment to this issue will be given
in Section ..
DI DOMENICO . 5
Regarding Equation , it may appear quite surprising that a theoretical formulation referring to a harmonic motion
appears in a code. Also, due to its nature, Equation is expected to provide significantly overestimated values of the real
maximum PSA. In the authors’ opinion, this should be interpreted as a very conservative tool provided by the code if no
one of the proposed simplified formulations (one specifically dedicated to reinforced concrete framed structures, the other
validated for masonry structures and borrowed by the work by Degli Abbati et al.) is applicable by the designer.
More specifically, the simplified approach proposed by the Italian regulation for framed reinforced concrete structures
(which are the topic of this study) is borrowed by the work by Petrone et al. Since this proposal is based on numerical
analyses on nonlinear models of reinforced concrete framed structures, it implicitly account for both multimodal con-
tributions and structural nonlinearity effects. The proposed formulations are reported, with some manipulation in the
nomenclature, in Equation .
PSA =
PGA 1+ 𝑧
𝐻
𝑎𝑝
1+(𝑎𝑝−1)1− 𝑇𝑎
𝑎𝑇12
≥PGA for 𝑇𝑎<𝑎𝑇
1,
PGA 1+ 𝑧
𝐻𝑎𝑝for 𝑎𝑇1≤𝑇𝑎<𝑏𝑇
1,
PGA 1+ 𝑧
𝐻
𝑎𝑝
1+(𝑎𝑝−1)1− 𝑇𝑎
𝑏𝑇12
≥PGA for 𝑇𝑎≥𝑏𝑇1.
()
In Equation ,a,b,andapare coefficients depending on Tthat account for higher modes effects (the coefficient a),
resonance period elongation due to nonlinearity (the coefficient b) and effect of nonlinearity on the maximum PSA value
(the coefficient ap). If Taequals zero, a linear PFA distribution along the building height is obtained. PFA ranges from
PGA (at zequal to zero) to PGA (at zequal to H). On the other hand, the maximum PSA is obtained if Tais between aT
and bT. In this case, the floor spectral acceleration ranges from .PFA to PFA, dependently on apvalue.
Circolarealso reports a second simplified formulation borrowed by the work by Degli Abbati et al.This proposal has
rigorous basis and has been validated against the experimental results obtained for a mixed reinforced concrete/masonry
structure as well as against the results of numerical analyses carried out on an existing masonry structure. Given that a
specific formulation is provided in Circolarefor reinforced concrete structures, the second simplified formulation is not
further investigated in this study.
It is worth noting that none of the proposals discussed in this subsection explicitly refers to infilled buildings, as if the
presence of infills may be considered only through Tvalue, that is, as if the presence of infills only affects the resonance
period at which the maximum PSA is attained, except for the simplified proposal of the Italian regulation, in which also
the maximum PSA value depends on Tand attains it maximum potential value for small values of the period (ie, ap=
when Tis lower than . seconds). In other words, based on this approach, an infilled building is expected to experience
an equal or higher maximum PSA value, at a certain floor, with respect to an identical but bare structure. This is also
generally observed in the numerical analyses shown in this paper.
2.2 Literature proposals
Calvi and Sullivan propose an approach for the assessment of floor response spectra for SDOF systems extended to
Multidegree of Freedom (MDOF) systems and validated against the results of numerical time-history analyses on rein-
forced concrete wall structures. The PSA value for each floor and for each modal contribution is calculated by means of
Equation .
PSA𝑖𝑗 =Γ
𝑖𝜑𝑖𝑗𝑎𝑚,𝑖 .()
In Equation ,am,iis a spectral shape function given by Equation .
𝑎𝑚,𝑖 (𝑇𝑎)=
𝑇𝑎
𝑇𝑖𝑎max,𝑖 (DAFmax −1
)+𝑎
max,𝑖 for 𝑇𝑎<𝑇
𝑖
𝑎max,𝑖 DAFmax for 𝑇𝑖≤𝑇𝑎<𝑇
𝑖,𝑒𝑓𝑓
𝑎max,𝑖 DAF𝑖for 𝑇𝑎≥𝑇𝑖,𝑒𝑓𝑓
.()
6DI DOMENICO .
In Equation ,amax,iis the minimum between Sa(Ti) and the Sacorresponding to global yielding of the structure (Say,
potentially determined by means of a nonlinear static analysis), DAF is given by Equation ,withDAF
max obtained for
Ta=Ti,eff.
DAF𝑖=1− 𝑇𝑎
𝑇𝑖,𝑒𝑓𝑓 2
+𝜉
𝑎
−1
.()
In Equations and ,Ti,eff is the effective period of the structure (which is assumed potentially different from the elastic
vibration period only for the first vibration mode), potentially evaluated through a nonlinear static analysis.
This approach accounts for structural nonlinearity effects in terms of resonance period elongation and limitation of the
maximum PSA. The values of PSAij determined at a certain jth floor for each ith vibration mode by means of Equation
can be combined through the SRSS rule to obtain a unique PSAjvalue associated with the jth floor accounting for the
contributions of the significant vibration modes. Once PSAjhas been obtained, by assuming Taequal to zero, PFAjcan
be determined. Hence, also PFAjaccounts for multiple vibration modes contributions. Note that the results may be influ-
enced by the combination rule adopted. Also in this case, as highlighted when discussing the Italian regulation “rigorous”
approach, the PFA value may result lower than PGA. Since this is only seldom observed from numerical analyses, Calvi
and Sullivan suggest that the PFA value should never be assumed lower than PGA at bottom floors and that, at the same
floors, the floor response spectrum should never return PSA values lower than those associated with the ground motion
response spectrum.
Vukobratović and Fajfar, propose a theoretical-based (except for the determination of the maximum PSA value,
which is empirical) approach for the assessment of floor response spectra validated against the results of numerical time-
history analyses on reinforced concrete frames. The PSA value for each floor and for each modal contribution is calculated
by means of Equation .
PSA𝑖𝑗 =Γ𝑖𝜑𝑖𝑗
𝑇𝑎
𝑇𝑖,𝑒𝑓𝑓 2
−1
𝑆𝑎𝑇𝑖,𝑒𝑓𝑓
𝑅𝜇,𝑖 2
+𝑇𝑎
𝑇𝑖,𝑒𝑓𝑓 2
𝑆𝑎(𝑇𝑎)2
≤AMP𝑖Γ𝑖𝜑𝑖𝑗
𝑆𝑎𝑇𝑖,𝑒𝑓𝑓
𝑅𝜇,𝑖
.()
In Equation ,Ti,eff is different from the elastic period only for the first vibration mode and can be calculated by means
of a nonlinear static analysis; Rμ,iis the reduction factor due to the nonlinear behavior of the structure and is equal to
the maximum between and the ratio between Sa(T,eff)andSay (potentially determined by means of a nonlinear static
analysis) for the first vibration mode, while it is equal to for higher vibration modes; AMPiis the maximum PSAij/PFAij
value and is expressed as a function of the damping ratio similarly to the approach proposed by Calvi and Sullivan. The
effects of multiple modes are combined by applying the SRSS rule, except in the postresonance region in which modal
effects are combined by algebraic summation. Also in this case, the PFA can be calculated by evaluating the PSA resulting
from modal combination at Taequal to zero. Also in this case, PFA may result lower than PGA. In Vukobratović and
Ruggieri, based on physics, it is suggested to assume PFA never lower than PGA at bottom floors.
Surana et al propose an empirical approach (based on numerical analyses on infilled reinforced concrete structures)
for the assessment of floor response spectra. This approach accounts for the effects of the first two vibration modes as well
as for the effect of structural nonlinearity, which is associated, differently from the previous approaches, also to the second
vibration mode. This is observed also from the structural analyses carried out for this study. The formulations proposed
by Surana et al present a quite complex form; moreover, their results, differently from what occurs when dealing with
the other code and literature models, are significantly different from those obtained by means of the numerical analyses
herein presented. Hence, for the sake of conciseness, these formulations are not shown in detail in this section.
2.3 Final remarks
In the previous subsections, it has been observed that, generally, floor response spectra are determined by two parts: a dis-
tribution of the PFA along the building height and a spectral shape function amplifying or deamplifying the PFA to obtain
DI DOMENICO . 7
TABLE 1 Summary of code and literature proposals
Proposal PFA profile shape
Floor-
dependent
spectral shape
Higher
modes
Structural
nonlinearity
Eurocode Linear ✓
ASCE-SEI /Linear
NZSEE Multilinear
Circolare“rigorous” Combination of mode shapes ✓ ✓ ✓
Circolaresimplified/Petrone et al Linear ✓✓
Circolaresimplified/Degli Abbati et alCombination of mode shapes ✓ ✓ ✓
Calvi and Sullivan Combination of mode shapes ✓✓✓
Vukobratović and Fajfar , Combination of mode shapes ✓ ✓ ✓
Surana et al Multilinear ✓✓✓
the PSA. This spectral shape can be fixed or different, in terms of maximum amplification and resonance period, from floor
to floor. The maximum PSA value, attained in case of resonance, is usually determined empirically. There are different
significant parameters influencing both the PFA and the spectral shape function, above all the effect of higher vibration
mode and structural nonlinearity. Not all the above listed proposals account for all these parameters, as summarized in
Table .
3NONLINEAR ANALYSIS PROCEDURE
3.1 Design and modeling of RC frames
The -, -, -, and -story case-study reinforced concrete moment resisting frames have regular rectangular plan defined
by five and three bays in the longitudinal and transverse direction, respectively. Beams are . m long; columns are . m
high. The buildings were designed with a Response Spectrum Analysis (RSA) according to Eurocode and Eurocode
for four different design PGA values at Life Safety Limit State (., ., ., and . g) in “High” Ductility Class (DCH).
The materials used for design are class C/ concrete with characteristic compressive strength of the cylinder equal to
N/mmand steel rebars with characteristic yielding stress equal to N/mm. It is assumed that floor slabs have a
diaphragmatic behavior. Further details on the design of the case-study reinforced concrete frames are available in Di
Domenico. For the analyses, the reinforced concrete elements’ nonlinearity is modeled by adopting a lumped-plasticity
approach in OpenSees by using ModIMKPeakOriented Material with response parameters determined according to
Haselton et al and with the introduction of the cracking point. In the modeling process, average material properties are
used, namely a compressive strength for concrete equal to N/mmdetermined according to Eurocode , and a steel
yielding stress equal to . N/mmdetermined according to Fardis et al.
3.2 Modeling of infill walls
Two different unreinforced masonry infill layouts are considered. The first is constituted by a two-leaf (thickness:
+ mm) “weak” infill wall (weak layout, WL), the second is constituted by a one-leaf (thickness: mm) “strong”
infill wall (strong layout, SL). Note that the two-leaf infills are constituted by independent noninteracting panels. In other
words, the in-plane and the out-of-plane responses of each leaf are determined (and modeled) independently on those
associated with the other leaf.
The mechanical properties of infills are those calculated for the masonry wallets tested by Calvi and Bolognini for the
WL and those by Guidi et al for the SL. The mass of infill panels is obtained by multiplying the panel nominal volume
times the density of masonry proposed by the Italian regulation, equal to kg/m. Each infill leaf is introduced in the
structural model by using a couple of equivalent no-tension struts.
8DI DOMENICO .
FIGURE 2 Updating of the out-of-plane response model: (a) new response envelope and (b) calibration of Pinching Material hysteretic
parameters based on the cyclic test performed at DIST-UNINA
The in-plane nonlinear behavior is modeled, separately for each leaf, based on the proposal by Panagiotakos and
Fardis. The two layouts analyzed are characterized by similar elastic in-plane stiffness but by different in-plane and
out-of-plane strength capacity. In other words, buildings with WL and SL infills have similar elastic period, but those
with WL infills are more likely to experience nonlinearity due to infills’ cracking. Further details regarding infill walls’
modeling are available in Ricci et al.
Regarding the out-of-plane response, the modeling strategy proposed by Ricci et al is adopted in order to account for
the in-plane/out-of-plane interaction effects, that is, the degradation of the out-of-plane strength and stiffness due to in-
plane damage and of the in-plane strength and stiffness due to out-of-plane damage. The equations adopted for modeling
the out-of-plane response are those proposed in Ricci et al for infills in which the out-of-plane response is governed by
two-way arching strength mechanism, except for those associated with some response parameters. In fact, for this study,
the out-of-plane response model is improved and updated.
First, the empirical equation for the assessment of the out-of-plane first macro-cracking force, Fcrack,isupdatedto
account for recent experimental evidences proposed by De Risi et al andbyDiDomenicoetal.
The new formulation
is reported in Equation .
𝐹crack = 5.90𝑓𝑚𝑣0.11 𝑡0.83
ℎ1.44 𝑤ℎ. ()
In Equation , forces are expressed in Newtons and lengths are expressed in millimeters. In addition, the formulations
proposed by Di Domenico et al are used to model the in-plane/out-of-plane interaction effects. In these formulations,
the reduction of the out-of-plane strength and stiffness capacity is related not only to the in-plane interstory drift ratio
demand and to the infill wall vertical slenderness (ie, the h/tratio), but also to the infill wall aspect ratio w/h.
However, the most important and significant improvement of the out-of-plane response model is the introduction of
the softening branch in the out-of-plane response backbone, as shown in Figure A, and the assessment of the cyclic
degradation of the response envelope. As shown in Figure A, the out-of-plane response backbone adopted in Ricci et al
was trilinear with a plastic branch after the peak load point up to the attainment of the displacement, equal to . times
the infill thickness, corresponding, on average and based on experimental tests, to the % degradation of the maximum
out-of-plane strength. In this study, the “conventional” ultimate point is retained at a displacement equal to . times the
infill thickness. However, a bilinear softening branch is modeled. The first part of the softening branch goes from the peak
load point to the “conventional” ultimate point. The second part goes from the “conventional” ultimate point to the “col-
lapse displacement” point corresponding to zero out-of-plane strength capacity, that is, at vanishing of arching strength
mechanism. As demonstrated in Angel et al and further explained in Di Domenico et al, vanishing of both vertical and
horizontal arching effect is expected, based on geometrical considerations, at an out-of-plane central displacement of the
infill wall equal to . times the infill thickness. So, the out-of-plane collapse displacement is set, in this study, to this
value.
The quadrilinear response envelope shown in Figure A is modeled in OpenSees by adopting Pinching Material. Pinch-
ing Material also allows modeling the hysteretic degradation of strength, unloading and reloading stiffness, as well as the
so-called “pinching” effect. Further details on the nature and meaning of the hysteretic parameters governing Pinching
DI DOMENICO . 9
TABLE 2 Selected Ground Motions for nonlinear time-history analyses. The indicated PGA values are referred to the original records
before matching and scaling
# ESM ID Country Date MwRepi [km]
PGA – NS
[g]
PGA – EW
[g]
ME-- Montenegro // . . . .
IT-- Italy // . . . .
IT-- Italy // . . . .
IT-- Italy // . . . .
IT-- Italy // . . . .
SI-- Italy // . . . .
IT-- Italy // . . . .
GR-- Greece // . . . .
TK-- Turkey // . . . .
IT-- Italy // . . . .
Material response are available in Lowes et al. A preliminary calibration of these parameters has been performed by
carrying out at the laboratory of the Department of Structures for Engineering and Architecture of University of Naples
Federico II (DIST-UNINA) a cyclic out-of-plane test on an unreinforced masonry infill with thickness equal to mm
and poor mechanical properties (ie, nominally identical to those monotonically tested in Di Domenico et al, whose out-
of-plane force (FOOP)-displacement (dOOP )responseisshowninFigureB. Further details on this experimental test are
available in Di Domenico.
The Pinching Material hysteretic parameters have been calibrated by setting damage degradation type to “energy”
and to minimize the distance between the experimental and the predicted energy dissipation history during the entire
experimental test. They have been calibrated by neglecting the force cyclic degradation, since the limited number of com-
parisons between monotonic and cyclic experimental tests available, shows that no significant strength degradation is
observed when comparing monotonic and cyclic out-of-plane tests. The calibrated parameters are gK=., gK=.;
gK=.; gK=; gD=.; gD=.; gD=.; gD=.; gF=gF=gF=gF=; gKLim =; gDLim =;
gFLim =; gE =; rDispP =rDispN =–.; rForceP =rForceN =–.; uForceP =uForceN =–.. As already high-
lighted, the number of cyclic out-of-plane experimental tests is very limited. Further experimentation is needed to achieve
a more robust and reliable calibration of the hysteretic parameters of the out-of-plane response of unreinforced masonry
infills, also characterized by different geometric and mechanical properties.
3.3 Analysis procedure
The nonlinear time-history analyses have been performed by applying to each case-study structure ten bidirectional
ground motions selected in the European Strong Motion (ESM) Database. Some properties of the selected records,
which were registered on stiff and horizontal soils (ie, type A soils according to Eurocode )are reported in Table . Both
components of the selected records were simultaneously matched (to ensure the condition of spectrum-compatibility)
to Eurocode %-damped design spectrum at Life Safety Limit State by using the RspMatchBi software. Then, for each
case-study building, each record was scaled in order to provide it with a PGA equal to the design PGA of the considered
case-study building. The matched and scaled record component registered in the NS direction was applied along the lon-
gitudinal global direction, while the matched and scaled component registered in the EW direction was applied along the
transverse global direction. Further details on record selection are available in Refs. ()and().
Nonlinear time-history analyses were performed on the Bare Frame (BF) models, on the infilled frame models with WL
infills and SL infills but modeling only their in-plane response (WL and SL models), and on the infilled frame models with
WL infills and SL infills with modeling both the out-of-plane response and the in-plane/out-of-plane interaction effects
(WLOOP and SLOOP models). In the following, each case-study building is identified using an acronym, such as XPY_Z,
in which Xis the number of stories, Ythe design PGA at Life Safety Limit State expressed in g/, Zthe model identifier
(BF, WL, SL, WLOOP, and SLOOP).
10 DI DOMENICO .
FIGURE 3 Average level of nonlinear demand experienced by the case-study building during nonlinear time history analyses shown on
static pushover curves. The first element yielding point is determined based on SPO analyses and is not shown on the curves if the
corresponding displacement normalized with respect to ∆TOP,TH ,average is higher than
Consistently with the design process, a % damping ratio was used to calculate damping forces during the analyses.
More specifically, mass- and tangent stiffness-proportional Rayleigh damping model was adopted.
3.4 Expected nonlinear demand on the case-study buildings
As above stated, in this study floor spectra will be determined also by accounting for the effect on them of the nonlin-
ear response of the supporting reinforced concrete structure. Since the case-study buildings were designed with an RSA
for a certain PGA level, and since they are analyzed with nonlinear time-history analyses for the same PGA demand
level, it is not expected that structural members will yield during nonlinear time-history analyses. Hence, nonlinearity is
expected to be due only to elements’ cracking (remember that reinforced concrete elements are modeled by account-
ing also for the cracking point in the assigned moment-chord rotation response envelope) and to infills’ (if present)
cracking.
This is confirmed by Figure , in which the pushover curves obtained by applying a first-mode-shaped lateral load pat-
tern to both bare and infilled structures are shown. By normalizing the top displacement (∆TOP) applied during static
pushover (SPO) analyses with respect to the average top displacement demand registered during time-history analyses
(TH), it is observed that this displacement demand is not able to produce, on average, elements’ yielding, but only tangent
stiffness reduction due to members’/infills’ (if present) cracking. It should also be noted that the first significant nonlin-
earity in the pushover curve occurs for a ∆TOP,SPO /∆
TOP,TH ,average value which is lower for buildings designed for higher
PGA (ie, the stiffer and stronger ones). That being considered, Figure shows that buildings designed and analyzed for
higher PGA level experience a higher nonlinearity demand. All these outcomes will have an influence on the results in
terms of floor response spectra that will be shown in the next section.
4 TIME-HISTORY ANALYSIS RESULTS
In this section, the results of the time-history analyses are presented and discussed. Namely, the outcomes in terms of
PFA/PGA profiles and PSA/PFA spectral shapes are shown. The spectral shapes are calculated by assuming a code-
consistent value of the damping ratio equal to %. For each case-study building, the results shown are obtained as the
average of the results obtained from the ten time-history analyses performed. For the sake of brevity, only some signifi-
cant results are shown. These results can be considered representative of the general trends observed from the outcomes
of all the numerical analyses. Then, these outcomes are compared with the results of the application of code and literature
proposals described in Section .
DI DOMENICO . 11
FIGURE 4 PFA/PGA profile along the longitudinal direction of a selection of case-study buildings
4.1 Observed general trends
The PFA/PGA profiles along the longitudinal direction for a selection of case-study buildings are shown in Figure .
Some considerations can be drawn:
(i) The maximum value of PFA/PGA is generally between and , it is registered at top floors and is lower for buildings
designed and analyzed for higher PGA (see point v. for comments on this). Among code and literature formulations,
NZSEEapproach returns a maximum PFA/PGA value equal to at the top floors of buildings; in addition, a
maximum PFA/PGA ratio equal to was expected at top stories according to Uniform Building Code. Hence, the
PFA/PGA maximum values obtained from the analyses can be deemed reasonable.
(ii) The shape of the profiles is roughly linear for -story buildings, while it becomes multilinear for the other buildings
and nonmonotonic for taller (- and -story) buildings. This is an effect of the influence of higher vibration modes,
which, as it is well-known, is more significant in high-rise structures. This is expected based on the more refined code
and literature proposals discussed in Section . The effect of higher modes appears also more visible and significant for
infilled structures than for bare structures, as well as in presence of higher nonlinearity (ie, for higher PGA demand,
see point v. for commentson this). This occurs because nonlinearity mainly affects the structural response to the first
vibration mode, thus making more "visible" the contribution of higher vibration modes on PFA/PGA profiles.
(iii) Generally, at fixed building and at fixed floor, the maximum PFA/PGA value is registered for infilled buildings with-
out modeling of out-of-plane response and in-plane/out-of-plane interaction, except for the -story buildings. This
occurs due to the dependence of the PFA on Sa(T). In fact, in -, -, and -story buildings, Sa(T) is generally higher
for infilled buildings, since Tis lower for infilled buildings and Tfor bare frames is higher than the corner period
TCequal to . seconds; on the contrary, Tfor -story bare structures is lower than TCso, infilled buildings, which
are always characterized by lower Tthan bare buildings, are characterized, in this case, also by lower Sa(T). The
dependency of the PFA values on Sa(T) is expected based on the more refined code and literature proposals discussed
in Section .
(iv) Generally, at fixed building, the PFA/PGA profiles for infilled buildings with out-of-plane response of infills and in-
plane/out-of-plane interaction modeled tend to approach the profile obtained for the bare structures. This is expected,
since when the in-plane/out-of-plane interaction is modeled, the in-plane response of the infill deteriorates, in terms
of both stiffness and strength capacity, due to the out-of-plane displacement demand. So, the general behaviur of the
building appears intermediate between those of the bare and of the infilled (with in-plane response only of infills
modeled) buildings.
(v) Generally, the trends described above are confirmed when passing from the buildings designed and analyzed for PGA
equal to . g to those designed and analyzed for PGA equal to . g. However, they are sometimes disturbed due
12 DI DOMENICO .
FIGURE 5 PSA/PFA spectral shapes along the longitudinal direction of a selection of case-study buildings
to structural nonlinearity, which in general, yields to a reduction of the PFA/PGA demand. This is expected based
on the more refined code and literature proposals discussed in Section . Remember that, as shown in Section .,
the reduction of PFA at increasing demand PGA and, as will be shown in the following, period elongation and PSA
reduction, is mainly due to elements’/infills’ cracking. This has been observed also by Petrone et al.
Based on the above considerations, which are corroborated by the discussion of literature and code proposals presented
in Section , in can be concluded that, with reference to the set of case-study buildings analyzed for this study, the signif-
icant parameters influencing the values of PFA/PGA ratios are the Sa(T) acting on the structure and on the PGA value
itself (which is representative of the level of expected nonlinearity in the structure), both referred to the ground motion.
On the other hand, the shape of the PFA/PGA profile depends on the importance of higher modes, which is higher for
taller buildings, for infilled buildings, and in presence of higher nonlinearity demand (ie, for higher PGA).
The PSA/PFA spectral shapes obtained along the longitudinal direction for a selection of floors of a selection of case-
study buildings are shown in Figure .ThevalueofTused to normalize the period Tais the period of the elastic uncracked
bare frame for WLOOP and SLOOP models: in fact, as previously discussed, due to the in-plane/out-of-plane interaction
effects, the behavior of models in which the out-of-plane response of infills is considered is intermediate between the
behavior of the bare structural model and that of the infilled structural model with only the in-plane response of infills
DI DOMENICO . 13
considered; for bare and WL/SL models, the period adopted for normalization is the one calculated by means of modal
analysis of the elastic uncracked bare models for BF models, of the elastic uncracked infilled models for WL/SL models.
Some considerations can be drawn:
(i) The peak values of PSA/PFA (especially those corresponding to the first vibration mode in high-rise buildings) can
vary from floor to floor, as some of the code and literature models described in Section show. In general, it appears
dependent on the PFA/PGA value: the higher the PFA/PGA, the higher the peak value of PSA/PFA associated with
the first vibration mode. Generally speaking, the maximum observed value of the PSA/PFA ratio is roughly equal to
., which is consistent with the maximum value, equal to , predicted by the proposal by Petrone et al adopted also
by the Italian code.
(ii) The spectral shapes are characterized, in general, by multiple peaks corresponding to resonance with the multiple
vibration modes of the primary structure. This is quite evident in taller buildings, in which the maximum value of
PSA/PFA associated with the higher modes (ie, at Ta/Tsignificantly lower than the unit) may be even higher than
that associated with the first vibration mode (ie, at Tasimilar to T). In -story buildings, the effect of higher modes
is visible only at the first floor, at which, actually, the maximum influence of the second vibration mode is expected.
On the other hand, in the -story buildings, peaks of PSA/PGA associated with higher modes are visible at different
Ta/Tvalues dependently on the floor considered. This occurs because the predominant effect among higher modes
may be associated with different higher modes from floor to floor (eg, the second or the third vibration mode of the
structure).
(iii) In general, infilled buildings are characterized by higher values of the maximum PSA/PFA ratio associated with
the first vibration mode with respect to bare buildings, except for the -story buildings. This occurs, most likely,
because Sa(Ta=T) is higher for infilled buildings with respect to Sa(Ta=T) for bare buildings, since Tof the
infilled building is lower than Tof the bare building, which, in tune, is higher than TC. As already explained
when discussing maximum PFA/PGA trends, this is not true for -story buildings, since in this case Sa(Ta=T)
is lower for infilled buildings than for bare buildings. This occurs because Tof the infilled building is still lower
than Tof the bare building, but Tof the bare building is lower than TC. The dependence of the maximum
PSA/PFA value on Sa(Ta=T) may be also expressed by relating it directly to the period T, as done by Petrone
et al.
(iv) As also shown when discussing the PFA/PGA profiles, also the spectral shapes for WLOOP and SLOOP models
are intermediate between those obtained for the BF models and those obtained for WL and SL models. The fact
that for WLOOP and SLOOP models the peak of PSA/PFA due to resonance with the first vibration mode occurs at
Ta/T< is due to the fact that, in this case, Tais normalized with respect to Tof the bare frame, while the real
resonance period is lower than it, being intermediate between the bare and the infilled structure elastic vibration
period.
(v) Generally, the trends described above are confirmed when passing from the buildings designed and analyzed for PGA
equal to . g to those designed and analyzed for PGA equal to . g. However, the peak PSA/PFA associated with
the first vibration mode of the structure is noticeably reduced due to the effect of structural nonlinearity; in addition,
the resonance period is higher than Tfor the period elongation due to nonlinearity. Remember that, as previously
highlighted, this effect is due to elements’/infills’ cracking. Both effects of structural nonlinearity are expected also by
applying the more refined literature proposals discussed in Section . Note also that, even if with a lower impact, also
the peaks associated with higher vibration modes may be reduced due to structural nonlinearity. This was observed
also by Surana et al.
Based on the above considerations, which are corroborated by the discussion of literature and code proposals presented
in Section , in can be concluded that, with reference to the set of case-study buildings analyzed for this study, the sig-
nificant parameters influencing the values of the maximum PSA/PFA ratios associated with the resonance with the ith
vibration mode at a certain floor are the PFA/PGA value at that floor, the Sa(Ta=Ti) value, potentially substituted directly
by Ti, and the PGA value (which is representative of the level of expected nonlinearity in the structure). On the other hand,
the shape of the PSA/PFA profile is characterized by a peak associated with the resonance of the secondary element with
the structure first vibration mode and with a group of close peaks associated with the resonance with the structure higher
modes.
14 DI DOMENICO .
4.2 Comparison with code and literature formulations
In the following, comparisons between the results of the numerical analyses and the proposals by code and literature
discussed in Section are shown. SL and WL models are neglected, since we refer hereafter to the case-study structures
in which a “complete” modeling of infills has been performed, that is, in which the out-of-plane response of infills
is considered as well as the in-plane/out-of-plane interaction effects. When applying proposals in which structural
nonlinearity is considered, the necessary parameters are determined by means of a nonlinear static analysis of the
case-study buildings; when applying Circolare“rigorous” approach, the Sa(Ti) is calculated by dividing the elastic
spectral acceleration by the behavior factor adopted during the design process, q, equal to .. This may yield to a
significant underestimation of PFA/PGA and PSA/PFA values, since dividing Sa(Ti) by the behavior factor is equivalent
to assuming that the pseudo-spectral acceleration at global yielding of the structure is exactly equal to the design
pseudo-spectral acceleration: this is not true, since different overstrength sources make the case-study structures able to
withstand a pseudo-spectral acceleration higher than Sa(Ti)/qwithout yielding as shown, for the case-study buildings
analyzed in this study, by Ricci et al. and in section ..
The PFA/PGA profiles obtained along the longitudinal direction for a selection of case-study buildings are compared
with those predicted by code and literature formulations in Figure .
As already discussed in Section , some approaches based on the combination of modal contributes combination may
yield to the prediction of a PFA value lower than PGA at bottom stories, especially in tall buildings, a circumstance never
observed from the numerical time-history analyses presented in this study (and rarely observed in past studies). According
to Hadijan, this is due to the fact that the “real” response of bottom stories is strongly influenced by the “stiff” motion of
the dynamic system that is included both in higher modes (which, being typically characterized by very low participating
mass are often neglected in common-practice modal combination rules) and in lower modes (which partially contribute
to the stiff motion, too). Only elaborated modal combination rules can account for these effects. So, some authors, such
as Calvi and Sullivan and Vukobratović and Ruggieri, suggest assuming a PFA value never lower than PGA at bottom
stories of buildings. On the contrary, this is not suggested explicitly by Circolarefor the “rigorous” approach.
It is observed from Figure that among code and literature models able to provide different PFA/PGA profiles for the
different models (ie, for models in which the PFA/PGA profiles do not depend only on geometric parameters), Calvi and
Sullivan approach works quite well for -story buildings but, as most of the other proposals, significantly underestimates
the PFA/PGA profiles for the -story buildings. Regarding the remaining code proposals according to which the PFA/PGA
profile only depends on geometric parameters, it seems that Eurocode formulation works quite well for the -story
building, while NZSEEproposal have a better performance on the -story building. In general, the underestimation
of the numerical results appears quite significant. However, it should also be considered that the capacity of providing good
estimates of floor spectra should be assessed on the absolute PSA values: in this sense, the underestimation of PFA/PGA
provided by literature proposals could be equilibrated by an overestimation of PSA/PFA values. In addition, it should be
remembered that some proposals existing in the literature have been validated against the results obtained on numerical
models in which members’ force-deformation response was modeled by adopting a typical elastic-plastic bilinear curve.
Such proposals should be always applied with caution for comparison with numerical outcomes obtained by adopting
different modeling approaches for reinforced concrete members, as in the present case.
The PSA/PFA spectral shapes obtained along the longitudinal direction for a selection of case-study buildings are com-
pared with those predicted by code and literature formulations in Figure . The comparison with ASCE-SEI /formu-
lation is not reported, since it does not define a spectral shape, as discussed in Section . Note that the application of code
and literature formulation was performed by normalizing Tawith respect to the appropriate effective Tperiod. However,
considering that the normalization period adopted for different formulations may be different in the graphs reported in
Figure and only for representation purposes, Tais always normalized with respect to the elastic period of the bare frame.
It is observed from Figure that the approach proposed by Eurocode fails in predicting the PSA/PFA peaks due to
higher vibration modes, while it works quite well, in terms of “envelope” of different floor response spectra, for buildings in
which a significant level of nonlinearity is expected (ie, those designed and analyzed for PGA equal to . g). This occurs
also for NZSEEapproach. The approach by Vukobratović and Fajfar, shows a good performance, especially when
dealing with buildings designed and analyzed for low PGA, while it appears slightly conservative for buildings designed
and analyzed for high PGA. However, it should be noted that, as already pointed out, the capacity of providing good
estimates of floor spectra should be assessed on the absolute PSA values: in this sense, the overestimation of PSA/PFA
provided by literature proposals is equilibrated by the underestimation of PFA/PGA values. Circolare“rigorous” approach
DI DOMENICO . 15
FIGURE 6 PFA/PGA profile along the longitudinal direction of a selection of case-study buildings (continuous lines) compared with
the outcomes of code and literature formulations (dashed lines)
appears significantly conservative (for the reasons discussed in Section ). A better (even if still conservative, especially at
bottom floors) performance is provided by the simplified model proposed by Circolare.In summary, the best performance
seems to be obtained by applying the approach by Calvi and Sullivan, both in terms of localization of maximum peaks
and in terms of maximum PSA/PFA values corresponding to resonance periods, even if with a slight conservativeness for
high values of Ta/T.
5CODE-ORIENTED PROPOSAL FOR THE ASSESSMENT OF FLOOR RESPONSE
SPECTRA AT DESIGN LEVEL OF SEISMIC DEMAND
Based on the outcomes of the numerical analyses described in the previous section, a new proposal for the assessment of
floor response spectra at the design level of seismic demand for bare and infilled reinforced concrete framed structures
is herein presented. More specifically, a formulation is proposed for the assessment of the PFA/PGA profile as well as a
formulation for the assessment of the PSA/PFA spectral shape. The proposal is based on the basic principles listed below.
16 DI DOMENICO .
FIGURE 7 PSA/PFA spectral shapes along the longitudinal direction of a selection of case-study buildings (continuous lines) compared
with the predictions of code and literature formulations (dashed lines)
(i) The formulations proposed should depend on predictor parameters which can be simply calculated by the practi-
tioner, that is, only by means of a linear analysis of the elastic uncracked structure. For this reason, the effect of
nonlinearity is considered, when necessary, through the value of the demand PGA at Life Safety Limit State, which
is equal, for a practitioner and for the structural analyses carried out in this study, to the design PGA at Life Safety
Limit State. This simplification, which improves the ease-of-use of the proposal, can be deemed acceptable since the
higher the design PGA, the higher is the observed nonlinearity demand in the case-study buildings (ie, design PGA
and nonlinearity demand level are positively correlated). Remember that the nonlinearity demand in the case-study
buildings herein analyzed is due to cracking of members (and infills, if present). This nonlinearity demand level at
the design PGA is not equal for all the case-study buildings (ie, it is not independent on the design PGA) since it is
not controlled during the design process. At PGA higher than the design one (which is typically outside the scope of a
DI DOMENICO . 17
FIGURE 8 Assessment of PFA/PGA profiles: adopted shape function
code-oriented safety assessment of nonstructural components), an even higher nonlinearity demand level is expected
due to members’ yielding and, potentially, softening. This could qualitatively and significantly change the observed
results in terms of both PFA/PGA profiles and PSA/PFA spectral shapes.
(ii) The formulations proposed for infilled buildings are calibrated based on the results obtained for infilled models with
“complete” modeling of infill walls, that is, by accounting for their out-of-plane response and for the in-plane/out-
of-plane interaction.
(iii) The formulations proposed should account for the effect of higher vibration modes and of structural nonlinearity. In
addition, the formulation proposed for the assessment of PSA/PFA spectral shape is referred to an elastic secondary
element with damping ratio equal to %. However, a correction coefficient accounting for the damping ratio of the
secondary element is provided in Section ..
5.1 Assessment of PFA/PGA profiles
Based on the outcomes of the numerical analyses, the shape function suggested for the assessment of PFA/PGA profiles
is reported in Equation and shown in Figure .
PFA
PGA =𝛼
𝑧
𝐻𝛽
≥1. ()
In Equation ,αis the value of PFA/PGA at the top floor of the building, while βis a shape factor ranging from (“bulged”
shape of PFA/PGA profile) to (linear shape of PFA/PGA profile).
As discussed in the previous section, the PFA/PGA value at the top floor, α, is strictly related to the Saacting on the
considered structure; in addition, it may be limited due to the effect of nonlinearity, which is higher at higher PGA demand.
For these reasons, the predictor parameter s and the upper and lower bound values of α,αmin,andαmax , are defined in
Equations -.
𝑠=1
2(Γ1𝑆𝑎(𝑇1))2+(Γ2𝑆𝑎(𝑇2))2
PGA ,()
𝛼min = 2.50, ()
𝛼max =3.20+4(0.35 − PGA)≥3.20. ()
In Equations and , accelerations are expressed in g units; Γiis calculated by normalizing the mode shape with respect
to the top floor modal displacement. The αvalue can be calculated by means of Equation .
𝛼=𝛼
min +(𝛼max −𝛼
min)(
𝑠−1
)≤𝛼max.()
18 DI DOMENICO .
The shape factor βshould be calculated by accounting, as shown in Section ., for the fact that the PFA/PGA profile
tendstobelinear(βtends to ) for low-rise buildings, for bare buildings and for low level of nonlinear demand; on the
other hand, the same profile tends to “bulge” (βtends to ) for high-rise buildings, for infilled buildings and for high level
of nonlinearity demand. For these reasons, Equations – are suggested for the calculation of this coefficient. Also in
this case, PGA is expressed in g units.
𝛽= 𝑇1
𝑇1𝐿
𝛽min +1− 𝑇1
𝑇1𝐿 𝛽max ≥𝛽min,()
𝛽min = 0.20, ()
𝛽max =0.80−PGA≥0.45, ()
𝑇1𝐿 = 0.80 s for bare buildings
𝑇1𝐿 = 0.40 s for inf illed buildings .()
The comparison between the numerical and predicted PFA/PGA profiles for the longitudinal direction of a selection of
case-study structures is shown in Figure .
To apply Equations and , the elastic vibration periods and modal participation factor of the uncracked structure
must be known. In case of infilled structure, if its elastic model has not been built, Equations and could be applied
by adopting the modal participation factor of the bare structure with no significant error; the first vibration period of the
infilled structure can be calculated by applying Equation by Ricci et al, in which His the height of the building and Lis
the length of the building plan dimension in the direction of interest, both expressed in meters. This formulation predicts
the observed elastic periods of the infilled buildings analyzed in this study with observed-to-predicted ratios equal to .,
median equal to ., and coefficient of variation equal to %. The second vibration period of the infilled structure can be
calculated by applying Equation , which has been derived based on the second elastic vibration period of the buildings
analyzed for this study (when applying Equation , observed-to-predicted ratios have mean and median equal to and
coefficient of variation equal to %).
𝑇1= 0.063 𝐻
𝐿
,()
𝑇2= 0.026 𝐻
𝐿
.()
5.2 Assessment of PSA/PFA spectral shape
Regarding the assessment of PSA/PFA at resonance and of the corresponding periods, it is assumed that two “resonance
regions” exist, as shown in Figure : the first is associated with the first vibration mode of the structure; the second is
associated with higher modes. In the first “resonance region,” PSA/PFA maximum value is named AMP; in the second
“resonance region,” PSA/PFA maximum value is named AMP. Note that AMP is not necessarily associated with the
second vibration mode: as shown in Section ., at bottom stories of high-rise buildings the maximum amplification of
PSA/PFA may occur due to the third (or higher) vibration mode. This choice (ie, adopting a unique “plateau” enveloping
the PSA/PFA peaks due to higher vibration modes) may be not appropriate for tall buildings, for which two (or more) peaks
associated with higher vibration modes can be clearly distinct and visible. However, it has been adopted as a reasonable
compromise between accuracy and ease-of-use of the proposed approach.
First, AMP and AMP values, together with the resonance periods at which they are registered, TRand TR, respec-
tively, were determined for each floor of each case-study structure. These observed variables are related to predictor param-
eters by means of a least square regression analysis. More specifically, the regressions were performed by correlating the
natural logarithm of each observed variable with a set constituted by i(i=,...,n) candidate parameters. First, the full
model (i=n) was selected as the “reference model”. Then, F-tests were performed comparing the reference full model
DI DOMENICO . 19
FIGURE 9 PFA/PGA profile along the longitudinal direction of a selection of case-study buildings (continuous lines) compared with
the outcomes of the proposal presented in Section (dashed lines)
PSA/PFA
T
a
/T
1
1
1T
R1
/T
1
T
R2
/T
1
AMP1
AMP2
Resonance region 1
Resonance region 2
observed proposed
LEGEND
FIGURE 10 Proposed PSA/PFA spectral shape. [Correction added on , Jul after first online publication: the ‘!’ symbol should be
replaced with ‘’ along y axis.]
20 DI DOMENICO .
TABLE 3 Statistics of the observed-to-predicted ratios for AMP, AMP, TR/T,BARE FRAME,andTR/T,BARE FRAME
Observed-to-predicted ratios (bare buildings)
Observed-to-predicted ratios (infilled
buildings)
Observed Mean Median
Coefficient of
variation Mean Median
Coefficient of
variation
AMP . . % . . %
AMP . . % . . %
TR/T,BARE FRAME . . % . . %
TR/T,BARE FRAME . . % . . %
with all reduced models (i=,...,n–) to associate with each reduced model a p-value related to the null hypothesis of
statistical equivalence between the considered reduced model and the reference model. Models with a p-value lower than
the significance level were immediately rejected, because in this case the null hypothesis itself should be rejected. Among
all possible reduced models, the one with a minimum number of parameters and a higher p-value was accepted and is
proposed in the paper. Note that the significance level was set to . to conservatively reduce the risk of a Type II error (ie,
not rejecting a false null hypothesis—in this case, accepting a reduced model with a statistically significant difference from
the reference model). The selected predictors resulting from this procedureare those individuated within the discussion
of the results of numerical analyses presented in Section ..
Namely, AMP is related to the PFA/PGA of the specific floor, to the PGA value (to account for nonlinearity effects)
expressed in g units and to the elastic uncracked vibration period of the bare structure T. AMP is related to z/H(given
that different higher modes may be predominant at different floors) and to the PGA value (to account for nonlinearity
effects). Regarding the resonance periods, TRis related to Tof the elastic uncracked bare frame and to the PGA value (to
account for the effect of nonlinearity). TRis related to Tof the bare frame, to the PGA value (to account for the effect of
nonlinearity) but also to the height of the building Hand to z/Hof the specific floor (given that different higher modes
may be predominant at different floors). Different predictive formulations are provided for bare and infilled buildings, as
reported in Equations –. The statistics of the observed-to-predicted ratios for the predicted variables are reported in
Table .
AMP1 =
min 5.5; 0.99 ⋅ (PFA∕PGA)1.57 ⋅PGA
−0.27 ⋅0.11
𝑇1for bare buildings
min 5.5; 1.25 ⋅ (PFA∕PGA)1.27 ⋅PGA
−0.29 ⋅0.17
𝑇1f or inf illed buildings ,()
AMP2 =
min 5.5; 2.61 ⋅ PGA−0.13 ⋅0.89
𝑧∕𝐻forbarebuildings
min 5.5; 2.52 ⋅ PGA−0.13 ⋅0.86
𝑧∕𝐻f or inf illed buildings ,()
𝑇𝑅1
𝑇1,𝐵𝐴𝑅𝐸 𝐹𝑅𝐴𝑀𝐸
=1.08 ⋅ 1.05PGA for bare buildings
0.75 ⋅ 1.97PGA for inf illed buildings ,()
𝑇𝑅2
𝑇1,𝐵𝐴𝑅𝐸 𝐹𝑅𝐴𝑀𝐸
=2.21 ⋅ 𝐻−0.52 ⋅(𝑧∕𝐻)0.37 ⋅1.20
PGA for bare buildings
0.82 ⋅ 𝐻−0.17 ⋅(𝑧∕𝐻)0.42 ⋅0.84
PGA f or inf illed buildings .()
Although not perfectly rigorous, as explained in section , the formulation proposed by Circolareis adopted to express
the variation of PSA/PFA in the two “resonance regions” as reported in Equations and .
PSA
PFA1
=min1
AMP1
𝑇𝑎
𝑇𝑅1 2
+1− 𝑇𝑎
𝑇𝑅1 2
−1
;AMP1
,()
DI DOMENICO . 21
PSA
PFA2
=min1
AMP2
𝑇𝑎
𝑇𝑅2 2
+1− 𝑇𝑎
𝑇𝑅2 2
−1
;AMP2
.()
The spectral shape is defined by the envelope of the spectral shapes determined in the two “resonance regions,” as reported
in Equation and shown in Figure .
PSA
PFA =maxPSA
PFA1
;PSA
PFA2()
This is consistent with the fact that AMP and AMP were not determined by “separating” the contributions of different
modes. In other words, the empirical values of AMP and AMP determined from the numerical results already account
for the contribution of different modes; hence, combining (PSA/PFA)and (PSA/PFA)by means of a modal combination
rule (eg, the SRSS rule) would not be correct.
The comparison between the numerical and predicted PSA/PFA spectral shapes for the longitudinal direction of a
selection of case-study structures is shown in Figure .
It is observed that the proposed approach works quite well on a wide range of case-study structures, even if it may result
overconservative or unconservative in some “extreme” cases (eg, for very low- or very high-rise buildings designed and
analyzed for very low or very high PGA). Of course, this is expected since an empirical approach has been adopted when
evaluating the maximum PSA/PFA ratio in the “resonance regions,” so, a certain prediction error is unavoidable. Within a
code-based approach, it could be appropriate to assume a safety factor to amplify the predicted maximum PSA/PFA value,
in order to have a prediction error always on the side of safety. Future studies may address this issue, also considering
record-to-record variability and the uncertainty related to mechanical properties and modeling assumptions, especially
regarding infill walls.
5.3 Effect of damping and of potential nonlinear behavior of the secondary element
As above stated, the proposed formulations are given by calculating PSA/PFA spectral shapes with damping ratio of the
secondary element, ξa, equal to %. However, nonstructural components may be characterized by different values of ξa.
The specific value of ξacould significantly affect the expected value of PSA/PFA: clearly, secondary elements with low ξa
are expected to experience higher values of PSA/PFA and vice-versa; no effect of ξais expected on PFA, of course.
For each case-study building, for each floor and for each direction, the observed values of AMP, AMP, TR,andTR
were re-evaluated for ξaequal to %, %, %, %, and %. Then, the observed values were normalized with respect
to those predicted by means of Equations – (ie, those predicted for ξa=%). This has been done in order to define
correction coefficients able to account for the effect of damping ratio on the PSA/PFA spectral shape. The trends of the
average values of AMP(ξa)/AMP(ξa=%) and of TR(ξa)/TR(ξa=%) with ξaare shown in Figure .
It is observed that a clear decreasing trend relates both AMP and AMP to ξa, nearly independently on the infill layout.
The correction coefficient ηareported in Equation is proposed. It should be calculated with ξaexpressed as a percentage
and used to increase/reduce both AMP and AMP. The proposed formulation is analogous to the one proposed by Calvi
and Sullivan andbyVukobratovićandFajfar.
,
𝜂𝑎=5∕𝜉𝑎.()
On the other hand, the effect of the damping of the secondary element on TRand TRappears quite negligible. Hence, no
correction is proposed. Also, the PSA/PFA spectral shape could be significantly affected by the nonlinear response of the
secondary element as shown, among others, by Vukobratović and Fajfar., This effect has not been considered in this
study. It is suggested that the PSA values obtained from the proposed formulations are reduced when a nonlinear response
of the secondary element is expected. To select appropriate values of the PSA reduction factor, the reader could refer, for
example, to the values of the behavior factor proposed for nonstructural components by codesor by the literature,for
example, for the out-of-plane assessment of unreinforced masonry infill walls.
22 DI DOMENICO .
FIGURE 11 PSA/PFA spectral shapes along the longitudinal direction of a selection of case-study buildings (continuous lines)
compared with the predictions obtained by applying the proposal presented in Section (dashed lines)
5.4 Applicability range of the proposed formulation. Potential application to existing
reinforced concrete framed buildings
The proposed approach for the assessment of floor response spectra is empirical and based on the results of numerical
analyses. Hence, it is fully applicable on reinforced concrete buildings similar to those analyzed in this study, while every
DI DOMENICO . 23
FIGURE 12 Effect of the damping ratio of the secondary element on (a) AMP and AMP and (b) TRand TR
application on buildings significantly different from those analyzed in this study should be performed with caution (eg, for
irregular structures, for structures with different structural system—for example, reinforced concrete wall structures or
dual structures—for tall structures with more than eight stories, for infilled reinforced concrete structures with irregular
distribution of infill walls—eg, pilotis structures).
The proposed approach has been calibrated based on the assessment of reinforced concrete structures designed
according to Eurocodes. It has been shown, among other effects, that the higher the PGA demand at Life Safety Limit
State, the higher the reduction of maximum PFA and PSA demand due to nonlinearity, mainly due to elements’/infills’
cracking. Most likely, in an existing building not designed according to current seismic provisions (or not designed to
seismic action at all), at equal PGA demand currently associated with the assessment at Life Safety Limit State, an even
higher nonlinearity is expected with respect to a new building, not only due to elements’ cracking, but also due to their
yielding. This could further limit PFA and PSA demand with respect to that predicted according to the approach herein
proposed. In other words, the approach described in this section is expected to be conservative (and then, potentially,
applicable) for existing reinforce concrete framed buildings. However, further investigation is needed regarding this issue.
6CONCLUSIONS
Floor response spectra are a useful tool for the assessment of the seismic demand on nonstructural components, whose
safety check is a paramount issue in performance-based earthquake engineering. This study was dedicated to the assess-
ment via nonlinear time-history analyses of floor response spectra of -, -, -, and -story case-study reinforced concrete
framed structures designed according to Eurocodes for four different levels of design peak ground acceleration at Life
Safety Limit State (., ., ., and . g). The case-study bare structures have also been analyzed by including in
the model two different kinds of uniformly-distributed infill walls. The main novelty of this study is not only the assess-
ment of floor spectra for infilled reinforced concrete buildings, which has been rarely addressed in the past, but above
all the fact that infill walls are modeled by considering their in-plane response, as usual, and their out-of-plane response,
as well as the in-plane/out-of-plane interaction, that is, the degradation of the in-plane response due to the out-of-plane
seismic demand and vice-versa.
First, after an introduction on the concept of floor spectrum, the main code prescriptions and literature proposals have
been presented and discussed. Based on these formulations, the main parameters influencing floor response spectra have
been identified, namely the floor height normalized with respect to the building height, the pseudo-spectral accelerations
acting on the considered building, the vibration periods of the considered building and the shape of its vibration modes,
the building height, the expected PGA demand. Moreover, floor spectra shape is strongly influenced by higher vibration
modes and by the nonlinearity demand experienced by the structure. Also, it was observed that the in-plane/out-of-plane
interaction in infill walls makes the floor response spectra of infilled buildings intermediate between those predicted for
bare structures and those predicted for infilled structures when only the in-plane response of infills is modeled.
The expected influence of these parameters on floor spectra has been confirmed by the results of the numerical analyses
and by comparing these outcomes with the prediction of code prescriptions and literature formulations. In addition, it has
24 DI DOMENICO .
been observed that among the available approaches, the one by New Zealand code NZSEEis the most appropriate for
the assessment of the peak floor acceleration distribution along the building height, while the one by Calvi and Sullivan
is the most appropriate for the assessment of the floor spectra shapes at different floors, together with the simplified
approach proposed by the commentary to the Italian technical regulation (Circolare),which may be, however, quite
conservative.
Based on the above discussion, formulations for the assessment of floor response spectra at design level of seismic
demand have been proposed for both bare and infilled buildings. The proposed formulations, which account for the effect
of higher vibration modes, of structural nonlinearity and of the damping ratio of the secondary element, are characterized
by ease-of-use, since they are applicable by knowing basic geometric and dynamic properties of the bare structures, and
by a satisfactory matching with the outcomes of the numerical analyses.
ACKNOWLEDGMENTS
The out-of-plane cyclic experimental test presented in Section was performed at the laboratory of the Department of
Structures for Engineering and Architecture of University of Naples Federico II under the technical and scientific super-
vision of engineer Giuseppe Campanella. The Authors express to his memory their deepest gratitude for his irreplaceable
support over the last years.
The authors are grateful to the eminent scholars who reviewed this paper. Their challenging comments improved the
quality of the work and inspired potential future developments for the research.
FUNDING
Dipartimento della Protezione Civile, Presidenza del Consiglio dei Ministri—Grant Number EC.
METROPOLIS (Metodologie e tecnologie integrate e sostenibili per l’adattamento e la sicurezza di sistemi urbani—PON
Ricerca e Competitività -)—Grant Number BF.
ORCID
Mariano Di Domenico https://orcid.org/---
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https://doi.org/./eqe.