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1
A DIRECT METHOD TO COMPUTE THE INELASTIC
DISPLACEMENT DEMAND OF MASONRY STRUCTURES
Francesco GRAZIOTTI1, Guido MAGENES
2 and Andrea PENNA
3
ABSTRACT
Estimating the maximum inelastic deformation demands for a given earthquake excitation is essential
for the evaluation of the seismic vulnerability of existing structures. There is not a general consensus
on a specific, easy and reliable method to compute the displacement demand for low-period
unreinforced masonry structures. In some of the available methods, the maximum inelastic
displacement is directly calculated starting from the spectral displacement at initial period, others
consider an equivalent linearization in which the maximum deformation is estimated by means of an
equivalent linear elastic system with lower lateral stiffness and higher damping coefficient. In order to
validate existing formulations and propose new ones, a technique for the definition of single degree of
freedom (SDOF) nonlinear systems representing the global behaviour of multi degree of freedom
(MDOF) structural models has been developed to perform fast dynamic analyses. Both SDOF and
MDOF systems are based on a nonlinear macro-element model able to reproduce the in-plane shear
and flexural cyclic behaviour of pier and spandrel elements. The comparison of the results in terms of
maximum displacement obtained for the SDOF system and for the MDOF system proves the
feasibility and reliability of the proposed approach. In this paper, parametric analyses were run on the
SDOF and used to develop and verify a non-iterative method based on secant stiffness specific for
masonry structures. The result of the study is a trustworthy formulation able to predict the ductility
demand for masonry structures with combined flexural and shear behaviour.
INTRODUCTION
The quantification of displacement demands for a structure is a crucial step in performance-based
assessment. Currently, a simple, yet reliable method to compute displacements in masonry structures
is lacking within the literature. The absence of such a procedure is rooted in two key limitations: first
of all, the complex cyclic behaviour of masonry structures does not allow for simple hysteretic rules to
be assigned to them and, secondly, the extremely short periods typical of masonry structures does not
allow the use of the “equal displacement rule” utilised in numerous simplified procedures for other
structural typologies. Nonlinear dynamic analysis can be used to simulate the dynamic response of a
masonry structure, provided that the model is capable of capturing the principal failure modes and
cyclic decay. In many cases, the global displacement response can be modelled though the
implementation of an equivalent frame technique; allowing for straightforward interpretation of the
expected damage pattern (Penna et al., 2014).
One of the objectives of this work is to develop a single degree of freedom (SDOF) model that
allows for the estimation of the seismic response of a masonry structure in a reliable, yet efficient
1 Postdoctoral Researcher, EUCENTRE, Pavia, francesco.graziotti@eucentre.it
2 Associate Professor, University of Pavia, EUCENTRE, guido.magenes@unipv.it
3 Assistant Professor, University of Pavia, EUCENTRE, andrea.penna@unipv.it
2
manner. The reduced computational complexity of simplified SDOF analysis methods greatly
enhances the feasibility of conducting parametric studies using nonlinear dynamic analyses. The
SDOF analyses can be used to study the correlations between the displacement demand and various
seismic intensity measures (e.g. Shome et al., 1998), perform simplified incremental dynamic analyses
(e.g. Azarbakht and Dolšek, 2011), calculate limit state-dependent fragility curves (e.g. Luco et al.,
2004; Mouyiannou et al. 2014) or to create simplified methods that can estimate the maximum
displacement demands on masonry structures (similar to what was done for general structures in Vidic
et al., 1994 and by Graziotti et al., 2014 for masonry ones). Due to the lack of studies in the latter
mentioned field of application, this work utilises the SDOF system to derive a corrected strength and
displacement reduction factors (ηin-R) relationship by means of a parametric study. The results
provided a simplified formulation for determining the inelastic displacement of a masonry structure
initiating from an idealized push-over and an elastic response spectrum, trying to bridge the evident
gap in this particular field of study. The proposed procedure is based on an equivalent period Teff that
was proposed by Lin and Miranda (2004) in their “optimal stiffness” method.
THE SINGLE DEGREE OF FREEDOM MODEL
Concept of the model
In order to perform simplified nonlinear dynamic analyses, a single degree of freedom model (SDOF)
was created. This model is able to synthetically interpret the seismic response of a multi degree of
freedom (MDOF) model representing a complete masonry building (if the MDOF is governed by a
single dominating mode of deformation, typical in regular buildings). It consists of two macro-
elements in parallel characterised by nonlinear behaviours typical of masonry panels. The two
elements are connected by a top rigid link. Such approach arises from the aim of completely
decoupling the shear and flexural/rocking mechanisms (or behaviours) consequently facilitating an
independent calibration of the parameters. The variables that govern the model are the geometry of the
elements, their axial compression, the mechanical characteristics (related to the flexural behaviour in
one and shear behaviour in the other), and the inertial mass. The single degree of freedom is the top
horizontal displacement of the two elements where the mass is concentrated; the rotation of the top
edge of the elements is restrained. Figure 1 reports a simplified illustration of the SDOF model. The
model was created in the TREMURI computer program (Lagomarsino et al., 2013; Penna et al.,
2014), a nonlinear analysis program capable of performing monotonic and cyclic pushover analyses as
well as time-history analyses of masonry buildings.
M
Axial rigid link
F1 F2
Fh
Figure 1. Scheme of the simplified SDOF model obtained assembling two pier elements, one governed by
flexural and one by shear behaviour.
Reference experimental campaign
An experimental campaign was used to calibrate the MDOF three-dimensional numerical model, it
consisted of a cyclic quasi-static test on an unreinforced masonry structure. The full-scale bricks
Δs
F. Graziotti, G. Magenes and A. Penna 3
structure was tested at the laboratory of Department of Structural Mechanics of University of Pavia in
1994 (Magenes et al., 1995).
Figure 2. Plan and views of the building specimen used to calibrate the MDOF three-dimensional numerical
model (Magenes et al., 1995).
It is important to notice that façade D is decoupled from the walls A, C and that the pushing
forces were equal at the two floor levels. Thanks to the above mentioned test it was possible to
determine the parameters of the MDOF model that allowed a fair simulation of the experimental
results. In particular the calibrated masonry mechanical properties are: elastic modulus E, shear
modulus G, compression strength fm, the cohesion c and friction coefficient μf. Values are reported in
Table 1.
Table 1. Calibrated masonry mechanical model.
E [MPa] G [MPa] fm [MPa] c [MPa] μf [-]
3000 500 2.8 0.14 0.15
The so calibrated macro-element model was used to create two larger symmetrical models that
represent realistic two story buildings with rigid floors. One is characterized by a flexural dominated
cyclic behaviour and the other by a shear one. The flexural model was created imaging a symmetrical
structure composed by two mirrored parts with geometrical characteristics equal to the ones of the lab-
tested structure. On the other hand, in order to create a model that exhibits a shear related failure it was
chosen to create a squatter pier by removing one of the two doors in wall D in one of the models.
Figure 3 shows the plan views and the equivalent frame idealization of the two models.
The parameters related to the flexural dominated structure (F) are indicated with superscript
“f” while those related to the shear dominated structure (S) with “s”. The total masses of the structures
are Mf=224.5 t and M
S=228.2 t, the first modal elastic periods are 0.21 s and 0.17 s and the
participating masses for the first vibration mode are 185 t (82.4%) and 188 t (82.3%) respectively.
Figure 4 plots the nonlinear pushover analyses used to define an equivalent bilinear capacity curve.
These analyses are run considering a limit shear drift equal to 5‰, i.e. the lateral strength and stiffness
of elements exceeding such drift value is set to zero.
4
E68 E69
E70 E71
72 73 74
75 76 77
n37
n38
n39
N13
N14
N15
N16
N17
N18
t112 t113
t114 t115
E68
E70
72 74
75 77
N13
N14
N15
N16
N17
N18
t112
t114
Figure 3. F Building (top) and S Building (bottom): plan view, three-dimensional models and equivalent frame
idealization (red, piers; green, spandrels; blue, rigid nodes)..
0
100
200
300
400
500
600
0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8
Ba
se s
he
ar [k
N]
Top displacement [cm]
dMU
FMax
70%FMax
FMy
dMy0
100
200
300
400
500
600
700
0 0.2 0.4 0.6 0.8 1 1.2
Ba
se s
he
ar
[kN
]
Top displacement [cm]
dMU
FMax
70%FMax
FMy
dMy
Figure 4. F Building (left) and S Building (right) pushover curve and equivalent bilinearization.
The equivalent bilinear curve is defined according to the recommendations reported in the
Commentary to the Italian Building Code (MIT, 2008 and 2009). The modal participation factors is
equal to Γf= Γ
s =1.2 in both cases and the masses of the equivalent SDOF are Ms
f=154 t and Ms
s=157 t,
respectively. The periods of vibration of the equivalent bilinear system are calculated as:
2S My
My
M dT
F
(1)
where the meaning of the quantities are explained in Figure 4.
F building
S building
F. Graziotti, G. Magenes and A. Penna 5
The periods of vibration are Tf=0.21 s and T
s=0.17 s, respectively.
Calibration of the model
The SDOF model needs to be calibrated by a two-step procedure described in (Graziotti, 2013). The
first calibration to be performed involves the use of nonlinear static analysis. The goal of this
calibration is to obtain a SDOF model with a cyclic behaviour similar to the global one of the
reference MDOF model.
It is useful to repeat more than one cycle for each displacement level, the macro-element
implemented in the TREMURI model is able to take in account mechanical property degradation. A
SDOF cyclic pushover is run at the same maximum displacements of the one run for the MDOF. In
particular the pushover run for the MDOF model was adaptive, with a force distribution related to the
current deformed shape (Galasco et al., 2006). The parameters that characterize the system were
calibrated comparing the static responses of the two analyses. In particular, the variables that have
been checked are: the maximum base shear, the initial stiffness, the unloading and reloading stiffness,
the shear deformation and stiffness and the area of the hysteresis loops. The properties to be calibrated
on the SDOF are: the geometry of the elements, their mechanical characteristics, the load axial forces
and model parameters controlling post-peak softening and nonlinear shear deformation.
Figure 5 shows the comparison between the MDOF and SDOF hysteresis cycles in terms of
base shear/displacement. It is possible to notice how the SDOF models are able to interpret the static
behaviour of the MDOF systems. The shear dominated MDOF model has a non-symmetrical
force/deformation relationship due to the geometry of the walls D. This obviously could not be
simulated by the simplified SDOF models.
In order to create a dynamically calibrated model there is the need to assign a mass MS and the
Rayleigh damping parameter αS (second step of the calibration). The mass MS is assigned using the
equation of the modal analysis:
T
SM φ M1 (2)
where the vector φ is the mode shape normalized for dMax=1 and M is the mass matrix of the MDOF
system. In this way, the mass of the SDOF dynamic system is chosen considering the first modal
elastic vibration mode.
Concerning the Rayleigh damping to assign to the model, Eq. 3 gives the damping matrix for a
general MDOF system according to Rayleigh hypothesis:
M M C M K (3)
where αM and βM are constants with units of s−1
and s, respectively, and K is the linear stiffness matrix
of the structure when the initial tangent stiffness is used. Thus, C consists of a mass-proportional term
and a stiffness-proportional term. The MDOF models have Rayleigh damping parameters αM=0.43 and
βM=7.8.10
-4, which gives to the models a nearly constant damping of 2% between the first elastic
period and the secant period at collapse (minimum value 1.8%). The damping coefficient of the
simplified system is not necessarily equal to the ones used in the MDOF analyses, so a calibration is
needed.
6
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-600
-400
-200
0
200
400
600
Top displacement [cm]
Bas
e s
hear
[kN
]
-1.5 -1 -0.5 0 0.5 1 1.5
-600
-400
-200
0
200
400
600
Top displacement [cm]
Bas
e sh
ear
[kN
]
Figure 5. Hysteretic response of MDOF (red) and SDOF (black) model of the flexural (top) and shear (bottom)
dominated building.
The Rayleigh parameters of the two SDOF models were calculated by mean of 10 nonlinear
dynamic analyses run on each MDOF model. In the SDOF system the viscous damping matrix C is a
1-by-1 (scalar). Note that the stiffness-proportional part of C does not contribute to the total damping
force in SDOF systems. The SDOF damping parameters that are able to best replicate the nonlinear
analyses output are αSf=1.87 and αS
s=4.19 (which means ξ
F=3% at elastic period for F structure and
ξS=5% at elastic period for S structure).
The slight increase of damping is likely due to the contribution of higher modes that are not
taken in account by the SDOF model and the choice to use the elastic modal participation factor Γ.
Validation of the method
In order to validate the method, a large database of natural accelerograms was used. The database
employed is SIMBAD (Smerzini et al., 2013). Nonlinear dynamic analyses were run using both
models (complete and simplified). Maximum displacements obtained by means of two models were
plotted as a function of different earthquake intensity measures. The equivalent frame modelling has
computational times lower than detailed finite element models, but the time histories analyses still take
more than one hour to be completed (standard structures run on a normal PC). Conversely, the
analyses for the SDOF equivalent models take only a few seconds to be completed.
The multiple nonlinear dynamic analyses allowed the comparison of displacement demands of
the two systems. Figure 6 shows the correlation between the top displacements of the SDOF and
MDOF systems. The graphs do not consider displacements higher than the collapse point dfMC=2.5 cm
and dsMC= 1.3 cm. Lines for mean and ± mean one standard deviation are annotated on the graphs. In
particular, all the Figures below refer to top displacements of the building. It is noticeable how the
SDOF model is able to interpret the response of the MDOF one. The correlation coefficients between
the vectors of maximum MDOF and SDOF displacements are 0.97 and 0.98 respectively for the
flexural and the shear dominated models.
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
Max displacement m-d.o.f. [cm]
Max
dis
pla
cem
ent
s-d
.o.f
. [c
m]
0 0.2 0.4 0.6 0.8 1 1.20
0.2
0.4
0.6
0.8
1
1.2
Max displacement m-d.o.f. [cm]
Max
dis
pla
cem
ent
s-d
.o.f
. [c
m]
Figure 6. Correlation between top displacements obtained with the flexural (first) and shear (second) dominated
SDOF and MDOF models.
F. Graziotti, G. Magenes and A. Penna 7
The use of un-scaled natural accelerograms allows for the correlation of the structural
responses with different spectral intensity measures (Graziotti et al., 2013a; 2013b; Graziotti, 2013).
SIMPLIFIED PREDICTION OF THE INELASTIC DISPLACEMENT DEMAND: AN
OPTIMUM STIFFNESS METHOD
Models combination
Starting from the two calibrated structures (Section 2), five others with intermediate behaviours
(mixed shear and flexural response) were created. This allowed to have a bunch of seven structures, to
represent the nonlinear dynamic behaviour of a variety of masonry buildings. The idea was to combine
in different ways the two structures in order to create more than one system with an intermediate
behaviour. The easiest way to obtain these models is to parallelize the two calibrated systems
weighting them to obtain different configurations. In particular, 7 systems were created from the shear
dominated one (T=0.165 s) to the flexural dominated one (T=0.212 s). Figure 7 (left) shows a
schematic representation of one of the models created while Figure 7 (right) plots in one graph the
cyclic push overs of all the models in terms of equivalent acceleration and displacement. It could be
noticed that the more flexural the response is, the thinner the hysteretic loops are (solid black line).
The Jacobsen (1960) method was used in order to characterize the 7 SDOF structures by
comparing their cyclic dissipated energy at ultimate displacement. The hysteretic damping ξhyst varies
almost linearly from a value of 19.9 % for the shear governed structure to a value of 13.8% for the
flexure dominated one.
-4
-3
-2
-1
0
1
2
3
4
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Acc
eler
atio
n [m
/s2
]
Displacement SDOF [cm]
0%
20%
33%
50%
66%
80%
100%
Figure 7. Scheme of one of the combined systems (20% shear) (left) and direct comparison of the cyclic
pushovers of the 7 SDOF systems (right).
Effective period
For existing structures, the value of strength reduction factor (R) is known while that of μ is unknown.
In an assessment process, calculating the ductility demand is the final goal. Lin & Miranda (2004)
proposed a method that defines an effective period Teff and an equivalent damping coefficient ξeq as a
function of R.
The aim of the proposed assessment method is to calculate the displacement demand by means
of a non-iterative procedure that defines the effective period Teff and the displacement reduction factor
ηin as a function of R.
In particular, concerning the equivalent period, Lin & Miranda (2004) proposed:
8
1 ( 1)eff el
h
RT T
R
(4)
where αh is the stain hardening ratio and Tel is the elastic period calculated by Eq. (1). In case of
elasto-plastic approximation the equation becomes:
eff elT T R (5)
ηin–R relations
The seven calibrated SDOF structures described in a previous section were used in order to run several
nonlinear analyses. Natural records of SIMBAD database were used as input. The procedure to
calculate ηin for each analysis is straight forward and it is here summarized:
1. Run the analysis in order to calculate Sd
2. compute the strength reduction factor as: R=m.Sa(Tel)/Fy
3. Calculate Teff=Tel.R
0.5
4. Compute Sde(Teff) as the spectral displacement
5. Compute the displacement reduction factor by mean of Eq. (6).
Eq. (6) was derived based on the results obtained from the nonlinear time history analyses
reported earlier, the following ηin-R expressions are proposed:
1.82(8.94 0.4) 1 1 1
R
in hyst
e
e R
(6)
The proposed equations are plotted together with the numerical results in Figure 8. In
particular, the graphs represent the displacement reduction factor ηin vs. strength reduction factor R for
structures with different Jacobsen damping ξhyst. The black line refers to the mean value and the red
line is the regression line with a shape given by Equation (8). The displacement reduction factor is
smaller than 1 in a range of R between 1 and 3. The minimum value of R is approximately 1.7. The
scatter is almost constant for all the ductility values and for all the structures and increases for higher
R.
Eq. (7) is calculated by regression using the results of the seven structures, it is plotted on
graph in Figure 8. The formulation could be simplified if a cyclic pushover analysis is not available,
considering an average value:
1.821.1 1 1 1
R
in
e
e R
(7)
F. Graziotti, G. Magenes and A. Penna 9
0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
R [-]
in
[-]
0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
R [-]
in
[-]
0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
R [-]
in
[-]
0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
R [-]
in
[-]
0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
R [-]
in
[-]
0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
R [-]
in
[-]
0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
R [-]
in
[-]
Figure 8. Displacement vs. strength reduction factor for structures with different Jacobsen damping (mean, in
black; mean ± 1 standard deviation, in blue; proposed regression, in red).
Displacement prediction by optimal stiffness DBA
In the previous sections all the parameters to use the optimal stiffness displacement based assessment
(DBA) method were calculated in terms of capacity of the structure (dy, Fy) and in terms of spectral
characteristics. With these data it is possible to compute the predicted displacement demand.
ξhyst=19.9%
ξhyst=13.8%
ξhyst=18.5%
ξhyst=17.7% ξhyst=17.1%
ξhyst=15.4% ξhyst=14.9%
10
To perform the displacement-based assessment procedure the following steps must be
sequentially performed:
1. Computation of a force-displacement response by mean of a pushover analysis identifying a
yield displacement dy (bilinear approximation, e.g. Figure 4)
2. Calculation the effective mass m* and the elastic period T
*(=Tel)
6. Calculation of the spectral acceleration at T*
and computation of the strength reduction factor
as: R=m*.Sa(T
*)/Fy
3. Calculation of ηin as a function of R (Figure 8, Eq. (6) or, for simplicity Eq. (7))
4. Calculation of Teff= T.R
0.5
5. Calculation of the spectral displacement at Teff, Sde(Teff)
6. Sd(predicted)= ηin. Sde(Teff)
The predicted displacement (or ductility) was compared with the same quantity calculated by
the nonlinear 7 SDOF systems defined in previous sections. Figure 9 shows the predicted ductility
demand (by mean of optimal stiffness method) vs. the calculated ductility demand (computed by mean
of nonlinear SDOF systems) for the two outermost systems (F and S).
0 1 2 3 4 5 6 70
1
2
3
4
5
6
7
Ductility SDOF [-]
Duct
ility
op
tim
um
sti
ffn
ess
met
ho
d [
-]
0 1 2 3 4 5 6 70
1
2
3
4
5
6
7
Ductility SDOF [-]
Duct
ility
op
tim
um
sti
ffn
ess
met
ho
d [
-]
Figure 9. Predicted ductility demand vs. inelastic SDOF ductility (mean, in black; mean ± 1 standard deviation,
in blue; on left 100% shear, ξhyst=19.9%, and on right 0% shear, ξhyst=13.8%).
In particular, the predicted vs. calculated by inelastic SDOF ductility demand and the relative
variation of the prediction are plotted in the graphs, that are repeated for S and F structure (from shear
to flexural behaviour). Graziotti (2013) reported similar graphs for all the analyzed structures. On the
graphs lines for mean and for mean ± one standard deviation are reported. The red dashed line on the
diagonal of the graphs represent the best possible prediction: μ(opt.stiff.)=μSDOF. From the graphs it
is possible to derive some considerations:
- There is a dispersion in the results also for μ<1 (elastic range) because the model used for the
SDOF is elastic nonlinear in this range.
- The method is able to approximate with an acceptable accuracy the mean value of the
displacement demand for all the considered ductility values. A slight overestimation was
noticed in all the structures considered. (The correlation coefficient is 82% for the shear
dominated structure and 85% for the flexural dominated one.)
- The coefficient of variation of the predicted values was calculated. The CoV of the results is
similar remains quasi constant for all the ductility levels. This is very important facilitating the
introduction of a safety factor, as example, in a code formulation. The average CoV of the
predicted ductility is higher for the shear dominated structure (0.43), and lower for the flexural
one (0.36).
F. Graziotti, G. Magenes and A. Penna 11
The optimal stiffness method provide a slight better prediction for flexural dominated
structures. An opposite trend was observed for the corrected N2 method (Graziotti et al., 2014a) which
better approximates the nonlinear behaviour of shear dominated structures. However, good results
were obtained also using the simplified formulation reported in Eq. (7).
CONCLUSIONS
A simplified system to simulate the nonlinear behaviour of masonry building was studied. In particular
an equivalent SDOF system was developed and calibrated statically and dynamically. It synthesized,
with sufficient accuracy, the seismic response of the case study MDOF system under investigation,
while minimizing the computational effort.
The SDOF model was used to evaluate the displacement demand of a masonry structure in a
reliable and fast way proposing a relation between strength reduction factor and displacement
reduction factor. The proposed method is based on secant period spectral displacement. In particular a
non-iterative optimum stiffness (Lin & Miranda, 2004) method was calibrated using the available
nonlinear analysis results. The method based on this simplified formulation is able to give a fair
average prediction, in particular for flexure dominated structures. The encouraging results suggest
continuing the study. For example, nonlinear time history analyses on larger statistical samples
(models) should be performed in order to validate the formulation with a wider structural database.
ACKNOWLEDGMENTS
This work was carried out with the partial financial support of the EUCENTRE Executive Project
2012-2014 e3 “Seismic vulnerability of masonry buildings”, funded by the Italian Department of Civil
Protection. The authors would also like to thank Dr. Alessandro Galasco and Mr. Emanuele Bossi for
their valuable support in the numerical analyses.
REFERENCES
Azarbakht, A., & Dolšek, M. (2011). Progressive Incremental Dynamic Analysis for First-Mode Dominated
Structures. J. Struct. Eng.
Galasco, A., Lagomarsino, S., & Penna, A. (2006). TREMURI Program: Seismic Analyzer of 3D Masonry
Buildings. University of Genova, Italy.
Graziotti, F., Magenes, G., Penna, A., & Galasco, A. (2013a). Modello numerico semplificato ad un grado di
libertà per l’interpretazione del comportamento dinamico di strutture in muratura. Proc. XV ANIDIS
Conference. Padova, Italy.
Graziotti, F., Penna, A., & Magenes, G. (2013b). Use of equivalent SDOF systems for the evaluation of the
displacement demand for masonry buildings. Proc. of VEESD , (p. Paper No. 347). Vienna, Austria.
Graziotti, F. (2013) Contributions towards a Displacement-Bases Seismic Assessment of Masonry Structures,
IUSS, Pavia, Ph.D. thesis.
Graziotti, F., Penna, A., Bossi, E., Magenes, G. (2014a). Evaluation of displacement demand for unreinforced
masonry buildings by equivalent SDOF systems. Proc. of EURODYN2014 , Porto, PT.
Jacobsen, L. S. (1960). Damping in composite structures. Proceedings of the 2nd world conference on
earthquake engineering, (p. Vol. 2., pp. 1029-1044).
Lagomarsino, S., Penna, A., Galasco, A., Cattari, S. (2013) TREMURI program: an equivalent frame model for
the nonlinear seismic analysis of masonry buildings, Engineering Structures, 56, 1787-1799.
Lin, Y. Y., & Miranda, E. (2004). Non-iterative capacity spectrum method based on equivalent linearization for
estimating inelastic deformation demands of buildings. Structural Engineering/Earthquake Engineering,
21(2), 113-119.
Luco, N., Bazzurro, P., & Cornell, A. (2004). Dynamic versus Static Computation of the Residual Capacity of a
Mainshock-Damaged Building to withstand an Aftershock. Proc. 13th WCEE. Vancouver, Canada.
Magenes, G., Calvi, G., & Kingsley, G. (1995). Seismic Testing of a Full-Scale, Two-Story Masonry Building:
Test Procedure and Measured Experimental Response. University of Pavia, Italy.
12
MIT (2008) DM 14.01.2008: Norme Tecniche per le Costruzioni, Italian Ministry of Infrastructures and
Transportation: Rome. (in Italian)
MIT (2009) Circ. C.S.Ll.Pp. No. 617 of 2/2 Istruzioni per l’applicazione delle nuove norme tecniche per le
costruzioni di cui al Decreto Ministeriale 14/1/2008. (in Italian)
Mouyiannou, A., Penna, A., Rota, M., Graziotti, F., Magenes, G. (2014) Cumulated Seismic Damage on
Unreinforced Masonry Buildings. Bulletin of the NZ Society for EE (submitted).
Penna, A., Lagomarsino, S., Galasco, A. (2013a) A nonlinear macroelement model for the seismic analysis of
masonry buildings, Earthquake Engineering & Structural Dynamics.
Penna, A., Rota, M., Mouyiannou, A., & Magenes, G. (2014). Issues on the use of time-history analysis for the
design and assessment of masonry structures. Proc. COMPDYN2013, (p. Paper No. 1327). Kos Island,
Greece.
Lagomarsino, S., Penna, A., Galasco, A., Cattari, S. (2013) TREMURI program: an equivalent frame model for
the nonlinear seismic analysis of masonry buildings, Engineering Structures, 56, 1787-1799.
Saiidi, M., Sozen, M.A. Simple nonlinear seismic analysis of R/C structures, Journal of the Structural Division,
107.5: 937-953, 1981.
Shome, N., Cornell, C. A., Bazzurro, P., & Carballo, J. (1998). Earthquakes, Records, and Nonlinear Responses.
Earthquake Spectra, V.14, No. 3.
Smerzini, C., Galasso, C., Iervolino, I., Paolucci, R. (2013) Ground motion record selection based on broadband
spectral compatibility, Earthquake Spectra.
Vidic, T., Fajfar, P., & Fischinger, M. (1994). Consistent inelastic design spectra: strength and displacement.
Earthquake engineering and structural dynamics, Vol.23, 507-521.