Engineering Structures 56 (2013) 1787–1799
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Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
TREMURI program: An equivalent frame model for the nonlinear seismic
analysis of masonry buildings
Sergio Lagomarsino a,⇑, Andrea Penna b, Alessandro Galasco b, Serena Cattari a
a
b
Dept. of Civil, Environmental and Chemical Engineering, University of Genoa, Italy
Dept. of Civil Engineering and Architecture, University of Pavia, Italy
a r t i c l e
i n f o
Article history:
Received 18 February 2013
Revised 30 July 2013
Accepted 2 August 2013
Available online xxxx
Keywords:
Masonry buildings
Seismic assessment
Equivalent frame model
Pushover analysis
Mixed masonry–r.c. constructions
a b s t r a c t
The seismic analysis of masonry buildings requires reliable nonlinear models as effective tools for both
design of new buildings and assessment and retrofitting of existing ones. Performance based assessment
is now mainly oriented to the use of nonlinear analysis methods, thus their capability to simulate the
nonlinear response is crucial, in particular in case of masonry buildings. Among the different modelling
strategies proposed in literature, the equivalent frame approach seems particularly attractive since it
allows the analysis of complete 3D buildings with a reasonable computational effort, suitable also for
practice engineering aims. Moreover, it is also expressly recommended in several national and international codes. Within this context, the paper presents the solutions adopted for the implementation of
the equivalent frame model in the TREMURI program for the nonlinear seismic analysis of masonry
buildings.
Ó 2013 Published by Elsevier Ltd.
1. Introduction
The large population of existing and historical unreinforced masonry buildings all over the world and their potential high vulnerability to earthquake require to improve the knowledge of their
seismic behaviour, setting up analytical and numerical models
for their structural assessment. Actually, the reliability of models
represents one of the most important issues involved in both the
design of new buildings and, in particular, in the assessment and
strengthening of existing ones.
In this paper, the attention is focused only on the global response of masonry buildings, that is assuming that proper connections prevent the activation of local failure modes mainly
associated with the out-of-plane response of walls. Within this
context, the global seismic response is strictly related both to the
in-plane capacity of walls and to the connection and load transfer
effects due to the floor and roof diaphragms. Thus, in most of the
cases, it is necessary to refer to methods of global analysis and
three-dimensional models. As regards the analysis methods for
the seismic assessment, in the last decades, the performance-based
earthquake engineering concepts have led to an increasing use of
nonlinear static analyses (pushover). As it is well known, these
simplified procedures based on pushover analysis [1,2] result in
the comparison between displacement capacity of the structure
(identified for different performance limit states) and the displace⇑ Corresponding author. Tel.: +39 0103532521; fax: +39 0103532546.
E-mail address: sergio.lagomarsino@unige.it (S. Lagomarsino).
0141-0296/$ - see front matter Ó 2013 Published by Elsevier Ltd.
http://dx.doi.org/10.1016/j.engstruct.2013.08.002
ment demand, which depends both on structure and earthquake
characteristics. The definition of the displacement capacity for significant limit states requires the evaluation of a force–displacement curve (‘‘pushover’’ curve), able to describe the overall
inelastic response of the structure under horizontal seismic loadings and to provide essential information to idealize its behaviour
in terms of stiffness, overall strength and ultimate displacement
capacity. This curve can be obtained by a nonlinear incremental
static (pushover) analysis, i.e. by subjecting the structure, idealized
through an adequate model, to a static lateral load pattern (simulating seismic inertial forces), increasing the total force and/or the
displacements, with possible updating of the force distribution
(adaptive pushover).
Among the possible modelling strategies proposed in literature
and codes, this work is focused on the equivalent frame modelling
strategy [3,4]. According to this approach, each resistant masonry
wall is subdivided into a set of deformable masonry panels, in
which the deformation and the nonlinear response are concentrated, and rigid portions, which connect the deformable ones. This
approach, which is also suggested in some seismic codes [5,6], requires a limited number of degrees of freedom, with a reasonable
computational effort, allowing the analysis of complex threedimensional models of URM structures, obtained by assembling
walls and floors, mainly referring to their in-plane strength and
stiffness contributions. Moreover the idealization as an equivalent
frame easily allows to introduce other structural elements, such as
reinforced concrete beams or columns, together with the masonry
ones. Thus, it appears particularly versatile to model also mixed
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structures (e.g. mixed masonry and reinforced concrete structures
which are quite common in existing buildings).
The solutions adopted for the implementation of the equivalent
frame model in the TREMURI computer program are presented and
discussed in the following. TREMURI was originally developed and
gradually improved at the University of Genoa, starting from 2001
[7,8], and subsequently also implemented in the commercial
software 3Muri [9]. Starting from the presentation of some general
issues on the possible strategies for the idealization of the masonry
wall in an equivalent frame (Sections 2 and 3), the solutions
adopted in TREMURI for the different structural elements
(Section 4), the assembling of complete 3D models and the seismic
analysis by means of nonlinear static analysis procedures
(Section 5) are illustrated in the following sections. Regarding the
formulations of structural elements (both masonry and reinforced
concrete ones) the attention is only focused on the nonlinear beam
model with lumped inelasticity idealizations. Finally, some
examples of applications are illustrated in Section 6 to show the
capability of the proposed model of assessing the seismic response
of masonry buildings.
2. Equivalent frame modelling of URM walls
Structural element modelling strategies are based on the identification of macroscopic structural elements, defined from a geometrical and kinematic point of view through finite elements
(solid, shell or frame) and described from a static point of view
through their internal generalized forces. In the field of structural
element models, the ‘‘equivalent frame’’ ones are the most widely
diffused. They consider the walls as an idealized frame, in which
deformable elements (where the nonlinear response is concentrated) connect rigid nodes (parts of the wall which are not usually
subjected to damage). Focusing on the in-plane response of complex masonry walls with openings, usually two main structural
components may be identified: piers and spandrels. This idealization starts from the earthquake damage observation that shows
as usually cracks and failure modes are concentrated in such elements (Fig. 1). Piers are the main vertical resistant elements carrying both vertical and lateral loads; spandrel elements, which are
intended to be those parts of walls between two vertically aligned
openings, are secondary horizontal elements (for what concerns
vertical loads), which couple the response of adjacent piers in the
case of lateral loads. It is worth noting that, although ‘‘secondary
elements’’, spandrels significantly affect the boundary conditions
of piers (by allowing or restraining end rotations) with significant
influence on the wall lateral capacity.
Fig. 2 reports a sketch aiming at representing the idealization of
a wall with openings as an assemblage of structural elements. Different schemes are illustrated according to very simplified models,
for which the actual modelling of spandrels behaviour is not
requested, and the Equivalent Frame (EF) discretisation that
Fig. 2. URM wall idealization according to simplified and equivalent frame models.
considers both pier and spandrel elements. In particular, the idealization of a ‘‘strong spandrels-weak piers’’ model (SSWP in Fig. 2) is
based on the assumption piers crack first, thus preventing the failure of spandrels which can be then assumed as infinitely stiff portions, assuring a perfect coupling between piers. This corresponds
to assuming a fixed-rotation boundary condition at the piers
extremities and it is also known as ‘‘storey mechanism’’ [10]. On
the contrary, in case of the ‘‘weak spandrels-strong piers’’ (WSSP
in Fig. 2), the hypothesis of both null strength and null stiffness
of spandrels is adopted then assuming the piers as uncoupled (this
corresponds to the cantilever idealization). However, it is worth
noting that in most cases it is correct to assume that horizontal displacement of the vertical structural elements are at least coupled
at the floor levels by the presence of horizontal diaphragms.
Once the choice has been made, according to the assumptions of
these simplified models, since only pier elements are modelled, the
definition of both their effective height and boundary conditions
plays a crucial role for the reliable assessment of the overall capacity of the wall. Usually only preliminary evaluations on the effectiveness of spandrels are requested in order to properly orientate
the choice between these two extreme idealizations. Both SSWP
and WSSP models are expressly suggested by FEMA guidelines
[11,12] and model SSWP is consistent with the POR method [13],
which was largely adopted in Italy after the 1980 Irpinia earthquake [14]. In the Italian Building Code [6] the WSSP hypothesis
Fig. 1. Examples of in-plane failure modes with damage concentration in piers and spandrels (examples from L’Aquila 2009 – left and centre – and Emilia 2012 earthquakes –
right).
S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799
is assumed for the simplest allowed modelling technique (cantilever models), whilst the SSWP hypothesis (storey mechanism) is no
more allowed for the assessment of multi-storey masonry
buildings.
Despite the advantage of adopting very simplified and manageable models, since they are based on an aprioristic choice, the following troublesome issues arise. First of all, it is conceivable that
both of these limiting cases are inappropriate for certain walls,
which may display both types of response in different regions or
which can be involved in a different behaviour with the increase
of nonlinear response. Moreover, it is not at all a foregone conclusion that the presence of certain constructive details (e.g. r.c.
beams coupled to spandrels), not supported by a quantitative evaluation of their effectiveness, is sufficient to assure the achievement
of the hypotheses which these simplified models are based on [15].
On the contrary in case of the EF Model, since both pier and spandrel elements are modelled, the transition through different
boundary conditions is directly a consequence of the progressive
damage of the elements. Actually, in some cases, the use of the
EF Model is regulated in codes, by defining the cases in which masonry spandrels may be taken into account as coupling elements in
the structural model [5]; these provisions mainly concern the
bonding to the adjoining walls, the connection both to the floor
tie beam and to the lintel.
Once having idealised the masonry wall as an assemblage of
structural elements, the reliable prediction of its overall behaviour
mainly depends on the proper representation of characteristics of
each single structural member. In this paper the attention is focused only on the nonlinear beam and lumped inelasticity idealizations. Within this context, several applications may be found in the
literature. Some of them focus on the formulation of nonlinear
beams [16] or programs specifically oriented to the analysis of masonry buildings [17,18]. Others are based on the use of general purpose software packages [19–22].
In the following the attention is focused on the solutions
adopted in TREMURI program developed at the University of Genoa
[7], starting from the formulation of a more refined nonlinear
macro-element model [23–26]. A more simplified nonlinear formulation similar to the one suggested in [17] was then introduced
mainly addressing to engineering practice aims and to performing
pushover analysis (this is also implemented in the commercial release of the program, 3Muri [9]).
The main distinctive features of the TREMURI program, when
compared to the other models mentioned above, are: (a) as it is
specifically oriented to the seismic analysis of masonry structures,
the possiblity to easily implement different formulations for masonry panels (Section 4.1.1) and alternative algorithms for the
pushover analysis (Section 6.1); (b) the explicit modelling of flexible horizontal diaphragms (Section 5.2), which are very common,
particularly in ancient existing buildings; (c) the 3D assembling
of masonry walls, which behave in-plane, and floor/roof diaphragms, drastically reducing the number of degrees of freedom
(Section 5.1).
3. Idealization of the masonry wall in an equivalent frame
model
The first step for modelling of the masonry wall as an equivalent frame is the identification of the main structural components,
previously introduced as piers and spandrels.
For the identification of the geometry of pier and spandrel
elements, conventional criteria are often assumed in literature,
supported by the damage survey after earthquakes and experimental campaigns. However, a systematic parametrical analysis either
numerical or experimental has never been performed in order to
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define rigorous criteria. Despite this, although the identification
of masonry piers and spandrels may result rather trivial and easily
automated in case of perforated walls with regularly distributed
openings, it becomes more difficult and ambiguous when openings
are irregularly arranged (Fig. 3). In the following some possible criteria are examined.
Usually the criteria for the definition of the height of masonry
piers are defined as a function of that of adjacent openings. A commonly adopted criterion conventionally assumes a maximum 30°
inclination of the cracks starting from the opening corners and
consistently provides an increased height for the external piers.
This is also the initial hypothesis proposed in [27] for the definition
of the equivalent height of masonry panels in models based on the
storey mechanism. In [28] it is proposed to define it as the height
over which a compression strut is likely to develop at the steepest
possible angle (i.e. assuming that cracks can develop either horizontally or at 45°). In case of existing buildings, the pattern of
pre-existing cracks should be taken into account in order to properly define the geometry of spandrels and piers.
In the following some criteria that can be easily automated (and
actually already implemented in the 3Muri software) are discussed. Fig. 4 summarizes the main steps of the frame idealization
procedure in a regularly perforated masonry wall: from the identification of spandrels and piers (steps 1 and 2) to that of nodes (step
3). Spandrel elements (step 1) are defined on basis of the vertical
alignment and overlap of openings: the length and the height are
assumed equal to the distance and width (in case of full alignment)
of the adjacent openings, respectively. Pier elements (step 2) are
defined starting from the height of adjacent openings: when these
latter are perfectly aligned, as the case of the internal pier shown in
Fig. 4, the height is assumed equal to that of openings. For the definition of height of the external piers the possible development of
inclined cracks from the opening corners (and/or from the lintel
edges) has to be considered, as previously discussed. As possible
approximated criteria, it can be assumed equal to the height of
the adjacent opening or as the average of the interstorey height
and the height of the opening. The geometry of the rigid nodes
(step 3) comes out directly from the previously defined elements
that are connected to them.
To complete the frame idealization for the whole wall, such a
calculation is done separately for each storey and each wall. It is
worth noting that the application of such a criterion without any
limitation to the cone diffusion angle may induce a significant
overestimation of the effective in-plane aspect ratio of external
piers in case of adjacent openings with a limited height and close
to the wall edge. Actually, in these situations flexural failure
modes are likely predicted for such slender piers, with possible
underestimation of the lateral strength and overestimation of
the deformation capacity. The presence of other structural elements, such as lintels and r.c. tie-beams, can influence the effective height of masonry piers and, in principle, for irregularly
distributed openings it should also vary depending on the direction of analysis.
In case of not perfectly aligned openings, a possible choice is to
conventionally assume a mean value for the height of spandrel elements as a function of the overlapping part between the openings
at the two levels (Fig. 5); when no overlap is present or the opening
lacks at all, it seems more appropriate to assume the portion of masonry as a rigid area (Fig. 5). Further studies, based on both experimental testing and numerical research, should be performed in
order to validate the capability of the presented procedure for different types of opening layout.
Finally, the actual efficiency of masonry panels must be carefully assessed and considered in the equivalent frame modelling
of the wall. For example, infilled openings (as shown in Fig. 3)
are sometimes weak and badly connected and, in this case,
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Fig. 3. Examples of façades with regularly and irregularly distributed openings.
Step 2- Identification of piers
Step 3- Identification of nodes
Equivalent frame
(Hint.+ Hdoor) /2
bs
bs /2
Step 1- Identification of spandrels
Fig. 4. Example of equivalent frame idealization in case of regularly distributed openings.
Equivalent Frame Idealisation
Non-linear beam / macro-element
Barycentric Axis of element
Pier
Spandrel
Rigid node
Fig. 5. Example of equivalent frame idealization in a case of irregularly distributed
openings.
they could be idealized, on the safe side, as openings, hence
neglecting the contribution of added masonry. This can be justified by the difficulty to guarantee a full interlocking with the
adjacent pre-existing masonry portions and the stress redistribution effects, which hardly may reproduce the original configuration without opening. As an alternative, reduced mechanical
properties could be assigned to the corresponding infilled masonry portions.
4. Modelling of structural elements
Once having idealised the masonry wall into an assemblage of
structural elements, the reliable prediction of its overall behaviour
mainly depends on the proper interpretation of the single element
response. As mentioned above, several formulations, characterized
by different degrees of accuracy, may be adopted either for
masonry panels and other structural types. The possibility of
modelling the nonlinear response of structural elements other than
masonry ones, such as reinforced concrete (r.c.), steel or wooden
beams, is particularly useful for the analysis of new and existing
buildings. As an example, from the beginning of the twentieth century, the spreading of r.c. technology has caused the birth of mixed
structural solutions inspired by practical aspects and higher architectural freedom: (a) new mixed masonry–r.c. buildings (e.g. buildings with perimeter masonry walls and internal r.c. frames); (b)
mixed buildings resulting from interventions carried out on existing masonry structures (e.g. replacement of internal masonry walls
by r.c. frames, r.c. walls inserted for supporting lifts and staircases,
additional storeys made of r.c. structure). Indeed, these structural
modifications may turn out in a potential high increase of the seismic vulnerability, as discussed in [29].
In the following, the attention is focused on a simplified formulation based on nonlinear beam elements with lumped inelasticity
idealization (bilinear elastic perfectly plastic behaviour). The element response is directly faced in terms of global stiffness,
strength and ultimate displacement capacity by assuming a proper
force–displacement relationship and appropriate drift limits (or
chord rotation limits in the case of r.c. elements). Despite some
unavoidable approximations of the actual behaviour (e.g. related
to the mechanical description of damage and dissipation mechanisms), this simplified formulation implies the following main
advantages:
– It allows performing nonlinear static analyses with a reasonable
computational effort, suitable also in engineering practice;
– It is based on few mechanical parameters that may be quite
simply defined and related to results of standard tests.
Moreover, it has to be stressed this formulation is consistent
with the recommendations included in several seismic codes
[30,31,6], since strength criteria defined for both bending and
shear failure modes can be easily implemented and adopted to
define the lateral strength of the different structural elements.
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4.1. Masonry elements
The specific characterization of the force–displacement relationship, aiming to describe the masonry panels behaviour, starts
from the knowledge and interpretation of the different failure
modes which may occur.
Observation of seismic damage to complex masonry walls, as
well as laboratory experimental tests, have shown that a masonry
panel subjected to in-plane loading may show two typical types of
behaviour: flexural behaviour, that may be associated to the failure
modes of Rocking (panel starts to behave as a nearly rigid body
rotating about the toe) and Crushing (panel is progressively characterized by a widespread damage pattern, with sub-vertical cracks
oriented towards the compressed corners); shear behaviour, that
may be associated with the failure modes of Diagonal Cracking (panel usually develops cracks at its centre, that after propagate towards the corners) and Shear Sliding (failure is attained with
sliding on a horizontal bed joint plane). Despite this classification,
it is evident that also mixed modes are possible and quite common.
Actually it is worth noting that this classification usually is implicitly referred to the pier element type. In fact, while many experimental researches related to the behaviour of piers have been
carried out in the last decades, tests on spandrels are very limited
and quite recent [32–35]. Indeed, the boundary conditions that
characterize spandrel elements and the orientation of main mortar
joints activated are very different from those of piers: as a consequence, relevant differences may be noticed. In particular in case
of flexural behaviour, due to low values of axial load, which usually
characterize spandrel elements (especially in case of lack of
tie-rods or r.c. beams), Crushing represents a very rare instance.
Moreover, in case of the shear behaviour, due to the interlocking
phenomena, Sliding failure (meant as sliding on a vertical joint
plane at the end-sections) usually cannot occur.
As well known, the occurrence of these different failure modes
depends on several parameters. In case of piers, they may be summarized as follows: the geometry; the boundary conditions; the
axial load; the mechanical characteristics of the masonry constituents (mortar, blocks and interfaces); the masonry characteristics
(block aspect ratio, in-plane and cross-section masonry pattern).
In case of spandrels, as shown in the referenced experimental campaigns, some additional variables can play an important role, like
the interlocking phenomena which can be originated at end-sections with the contiguous masonry portions, the type of lintels
(in particular masonry arches or architraves in stone, timber, steel
or r.c.), the interaction with other structural elements coupled to it
(in particular if tensile resistant such as r.c. beams or steel tierods).
The above introduced failure modes may be interpreted, in
terms of resultant maximum shear, by some simplified strength
criteria, based on mechanical or phenomenological hypotheses,
which are proposed in literature and codes. Usually, they are based
on the approximate evaluation of the local/mean stress state produced by the applied forces on predefined points/sections of the
panel, assessing then its admissibility with reference to the limit
strength domain of the constituent material, usually idealised
through oversimplifications based on few mechanical parameters.
As a function of the current value of the axial force (N) acting on
the element, the minimum value – as predicted by the criteria
adopted to model the flexural and shear responses, respectively –
is usually assumed as reference. In addition, it is worth noting that,
due to the application of horizontal load patterns, aimed to simulate seismic actions, the acting axial load changes from the initial
value consequent to the vertical dead loads; moreover, due to
redistribution phenomena associated with the progressing of nonlinear response, further variations may occur. As a consequence, it
is evident how also the value of the corresponding shear strength
varies in each panel during the nonlinear static analysis. Then, failure of the panel is usually defined through the definition of a maximum drift (du) based on the prevailing failure mechanism
occurred in the panel (e.g. as proposed in both national and international codes [5,6,31,30]). Fig. 6 schematically shows the abovementioned issues.
Further details on the specific criteria and formulations implemented in TREMURI program are illustrated in Section 4.1.1.
4.1.1. Modelling of masonry piers and spandrel
Masonry panels (piers and spandrels) are modelled as 2D elements by assuming a bi-linear relation with cut-off in strength
(without hardening) and stiffness decay in the nonlinear phase
(for non-monotonic action). The kinematic variables and generalized forces aimed to describe the 2D element are summarized in
Fig. 6. It is important to stress that loads are applied only on nodes,
thus no loads act along the element. The initial elastic branch is directly determined by the shear and flexural stiffness, computed on
the basis of the geometric and mechanical properties of panel, as
summarized in the stiffness matrix (Ke, in squared brackets), as
follows:
2 12EJg
g
g
g 3
0 h26EJ
h312EJ
0 h26EJ
8 9
9
h3 ð1þwÞ
ð1þwÞ
ð1þwÞ
ð1þwÞ 8
7> ui >
Vi > 6
>
7>
6
>
>
>
EA
EA
>
>
>
>
0
0
0
0
7>
> Ni >
> 6
h
h
>
>
>
wi >
>
>
>
>
7
6
>
>
>
>
EJ gð4þwÞ
EJ gð2wÞ 7>
6EJ g
6EJ g
>
>
6
<M = 6 2
</ >
=
0
0
2
7
hð1þwÞ
hð1þwÞ
i
h ð1þwÞ
h ð1þwÞ
i
7
¼6
7
6
12EJ
g
6EJ
g
12EJ
g
6EJ
g
>
>
>
Vj >
0
>
>
7> uj >
6 h3 ð1þwÞ 0
>
>
h2 ð1þwÞ
h3 ð1þwÞ
h2 ð1þwÞ 7>
>
> >
>
>
6
>
>
>
> Nj >
>
> 6
7>
w
j>
>
>
>
EA
EA
>
>
>
7
0
h
0
0
0
: >
:
; 6
;
h
5
4
/
Mj
j
EJ gð2wÞ
EJ gð4þwÞ
6EJ g
6EJ g
0
h2 ð1þwÞ 0
hð1þwÞ
hð1þwÞ
h2 ð1þwÞ
ð1Þ
2
2
where the w coefficient is computed as 1.2El /(Gh ); E and G are the
Young and shear moduli, respectively; A and J are the cross-section
and the moment of inertia of the panel, respectively; l and h are
length and height of the panel; g is a stiffness reduction coefficient
aiming at accounting for the panel ‘‘cracked’’ conditions. As regards
the g coefficient, since the progressive degradation of the stiffness is
not actually modelled, a calibration of the initial mechanical properties is necessary. Concerning this point, codes and recommendations [5,6] provide only rough information. Usually, it is proposed
to adopt reduced values of the elastic stiffness properties: unless
more detailed information are available, a reduction of 50% is proposed. Indeed, the results of parametric nonlinear FEM analyses
performed by the authors on panels subjected to static in-plane
loading with different levels of axial loads and slenderness
[36,37], showed that the reduction factor depends on the acting
compressive state (likewise to that proposed in [38] for r.c. elements). Fig. 7 shows the reduction factor g as a function of the compressive stress state (ry), normalized to the masonry compressive
strength (fu), for two different levels of the nonlinear masonry
behaviour.
Rigid end offsets are then used to transfer static and kinematic
variables between element ends and nodes. A nonlinear correction
procedure of the elastic prediction is carried out based on comparison with the limit strength values as defined hereafter; the redistribution of the internal forces is made according to the element
equilibrium.
The ultimate shear and bending strength is computed according
to some simplified criteria that are consistent with the most common ones proposed in the literature and codes for the prediction of
the masonry panel’s strength as a function of the different abovementioned failure modes. Table 1 summarizes the criteria implemented in TREMURI program for URM piers and spandrels,
respectively; as debated in Section 4.1, the program updates, at
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Idealization of the single panel
Nj
Vj
Kinematic variables,
generalized forces and
geometrical properties
h
Mi
Failure criteria
V
(uj , wj , ϕj )
l
Mj
Ni
Flexural strength domain
Shear strength domain
t
Vi
(ui , wi , ϕi )
Bi-linear relationship
V
V u,flexural
V u,shear
V u =min (V u,shear; V u,flexural)
K
N k-1 N k N k+1
N
δu
δ
Influence of the current axial load acting on the panel
Fig. 6. Sketch of the idealization of masonry pier response by adopting simplified strength criteria based on applied axial compression force.
1
0,9
0,8
0,7
For increasing values of axial
and shear compliance
η
0,6
0,5
0,4
0,3
0,2
0,1
0
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
σy/ fu
Fig. 7. Reduction factor of the elastic stiffness properties (as results from nonlinear
FEM analyses in [36]).
each step of the nonlinear analysis, the current ultimate strength
taking into account the axial load variation.
A check for the ultimate compressive strength is also implemented in the TREMURI nonlinear analysis procedure. The maximum element capacity in compression is limited to Nu = 0.85ltfu,
where fu is the masonry compressive strength, l is the length of
the cross-section, t the wall thickness.
When different strength domains are implemented for the same
failure mode (e.g. in case of the shear mode), a choice has to be
made by the user. Recently, a critical review of the use and the
choice among these criteria as a function of the masonry type
has been discussed in [37]. In particular, it has been noticed that
two main parameters may address this choice: the regularity of
the masonry pattern and the ratio between the strength/stiffness
parameters of mortar and blocks.
Moreover, it has to be highlighted how, due to the rather limited and quite recent attention specifically addressed on spandrel
elements, most of these strength criteria have been formulated
and validated only by comparison with experimental results on
pier elements. Thus, common practice is to adopt the same failure
criteria for both element types, assuming spandrel behaviour as
that of a pier rotated by 90°. Indeed, very few specific formulations
are proposed in both literature and codes, as recently discussed in
[44]. For example, the Italian Building Code [6] makes a distinction
in the strength criteria to be adopted for spandrels as a function of
the acting axial load. If it is known from the analysis, the same criteria assumed for piers are adopted. If it is not known (that is the
Table 1
Strength criteria for URM panels implemented in TREMURI program.
Failure mode and element type
Rocking/crushing
Strength domain
Piers
Mu ¼
Nl
2 ð1
Spandrels
Mu ¼
dHp
2
0
Notes
N
Þ
0:85f u lt
h
H0
1 0:85fp
fu masonry compressive strength, l length of section, t thickness
i
hu dt
M u ¼ f N; fftu ; lc ; lt
hu
ftu ¼ min f2bt ; c þ lrs u
Shear
Bed joint
sliding
Diagonal
cracking
0
Piers
V u;bjs ¼ l tc þ lN 6 V u;blocks
Spandrels
Vu = htc
Piers/
spandrels
V u;dc
1
¼ lt 1:5bso
V u;dc
2
^ NÞ 6 V u;blocks
¼ 1b ðlt~
cþl
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ 1:5Nso lt
H0p is assumed as the maximum value between the axial load N acting on spandrel and Hp
as proposed in [6] by assuming a strut-and-tie mechanism (if a tension member is
present). Hp is the minimum value between the tensile strength of elements coupled to
the spandrel (such as r.c. beam or tie-rod) and 0.4fhudt, where fhu is the compression
strength of masonry in horizontal direction
As proposed in [39] the limit domain is obtained by assuming an elasto-perfectly plastic
constitutive law with limited ductility both in tension (lt) and compression (lc) and an
equivalent tensile strength for spandrel ftu (fbt tensile strength of bricks; l and c friction
coefficient and cohesion of mortar joint, respectively; / interlocking parameter; rs entity
of compressive stresses acting at the end-sections of the spandrel)
Coulomb criterion with: l0 length of compressed part of cross section. A limit value
(Vu,blocks) is imposed to take into account in approximate way the failure modes of blocks
h height of spandrel transversal section (assumed only in case of a strut-and-tie
mechanism may develop)
so masonry shear strength, b stress distribution factor as function of slenderness [40,41]
^ and ^c equivalent cohesion and friction parameters,
Coulomb-type criterion with: l
related to the interlocking due to mortar head and bed joints, as proposed in [42] (with
b = 1). The introduction of b, proposed in this paper, is implicitly justified in [42] by some
comments on the shear stress distribution; a similar corrective factor is proposed in [43]
S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799
case when floors are modelled as infinitely stiff), if the spandrel is
coupled to another tensile resistant element (e.g. steel tie rod or r.c.
beam), a strut-and-tie mechanism is assumed to be developed,
with a maximum compression force in the spandrel equal to the
tension strength in the coupled element. However, as introduced
in Section 4.1, several factors differentiate spandrels from piers.
In particular, regarding the flexural response, due to both the low
values of axial load or the lack of other tensile resistant elements
coupled to spandrel (as usual in case of existing buildings), the criteria proposed in codes lead to very conservative predictions of the
spandrel strength: as a consequence in many cases flexural failure
tends to prevail over shear much more frequently than that observed by earthquake damage assessment in existing buildings or
in experimental campaigns. To overcome this result and, in general, to take into account for the additional factors which may
influence the spandrel behaviour (as discussed in Section 4.1), it
turns out the pressing need of referring to more reliable and corroborated strength criteria for them. As regards to this and with
particular reference to the flexural failure mode, the formulation
proposed in [39], as shown by the comparison with experimental
results discussed in [44], seems to provide rather good results, at
least in the case of clay brick masonry. The formulation is based
on the assumption that, in case of spandrels, masonry may exhibit
an ‘‘equivalent’’ tensile strength ftu in the horizontal direction (parallel to mortar bed joints) in virtue of the interlocking phenomena
occurring with the contiguous masonry portions.
In TREMURI program, the user can choose between the flexural
strength criterion proposed by Cattari and Lagomarsino [39], if one
wants to consider the contribution of ftu, and the criteria proposed
in [6]. In this latter case, the maximum value provided by two
abovementioned hypotheses on the acting axial load is assumed
as reference. This assumption is justified by the observation that
the axial force computed by the program for spandrels represents
an underestimation of the actual one (usually very low, apart from
the case of the presence on tensioned tie-rods): in fact, some phenomena (e.g. the effect of interaction with floors) are modelled
only in an approximate way.
It is worth stressing that expressions summarized in Table 1
only refer to the case of URM masonry, while the TREMURI program also allows modelling reinforced masonry structures. To this
aim, proper strength criteria, consistent with those usually proposed in codes, have been implemented to predict the panel’s
strength.
The panel collapse, is checked by assuming a limit value for the
drift (d), computed as:
d¼
ðuj ui Þ ðuj þ ui Þ
þ
6 du
2
h
ð2Þ
The limit value assumed (du) varies as a function of the prevailing failure mode that occurs in the panel. According to some recommendations proposed in codes [6,30,31], in case of URM
masonry piers, it usually ranges from 0.4% to 0.8%; indeed, in case
of spandrels, the recent experimental campaigns showed generally
greater values. Once collapse is reached, the element becomes a
strut; this assumption is on the safe side, because no residual shear
and bending strengths are considered, while the axial load is still
supported, checking it does not exceed the axial strength Nu.
4.2. Reinforced concrete elements
Nonlinear r.c. elements, modelled as 2D or 3D elements in the
case of beams or columns and walls, respectively, are idealized
by assuming elastic-perfectly plastic hinges concentrated at the
ends of the element. The choice of this simplified concentrated
plasticity model, with respect to more accurate ones like as the fibre approach, is justified by the will to assume a computational
1793
burden comparable to that of masonry elements and a similar level
of accuracy.
The initial elastic branch, similarly to masonry elements, is directly determined by the stiffness contributions in terms of shear
and flexural behaviour by neglecting that offered by reinforcement.
It is computed by means of a stiffness matrix analogous to that
introduced in Eq. (1) (with some necessary modifications in case
of 3D elements). The reduction of stiffness due to cracking phenomena may be taken into account, analogously to masonry elements, by the g coefficient (e.g. assumed as proposed in [38]),
kept constant during the analysis.
Shear and compressive/tensile failures are assumed as brittle
failures while combined axial-bending moment, modelled by plastic hinges at the end of element, are regarded as ductile failure.
Shear strength is computed according to the criteria proposed
in [6,45] in the case of low-medium ductility classes, for different
element types (beam, column and r.c.-wall). Both cases of transverse shear reinforcements present or not are considered; if present, the shear strength criteria adopted are based on an
equivalent truss with the variable strut inclination method.
In the case of combined axial force (N) and bending moment
(M), the interaction M–N domain is computed on the common
hypotheses of: plane-sections; perfect bond between concrete
and steel bars; rectangular stress block distribution.
In case of columns, only the case of symmetrical reinforcements
is considered; in case of r.c. walls, the domain is computed taking
into account the contribution of longitudinal bars in their actual
position. In order to determine the formation of a plastic hinge,
the comparison between the elastic prediction and the limit values
obtained from the M–N interaction domain, is carried out. The case
of r.c. walls and columns is more complex since these elements can
be affected by a biaxial bending–compression behaviour. In this
latter case, the Mx–My–N domain is traced by computing, on the
basis of the axial force acting on the element, the resistant bending
moments separately in each plane (Mx,Rd and My,Rd, respectively)
and, then, by assuming a proper interaction domain (linear or with
more accurate formulations, such as elliptic). It is necessary to
point out that the plastic hinge, once activated, involves both X
and Y planes at the same time.
The ultimate limit state of the section, in the case of ductile
mechanisms, is identified when the chord rotation (computed
referring to the shear span LV) reaches its ultimate value (hu), calculated by widely used expressions [30,46,47], based on an empirical approach starting from a number of experimental data [46].
Once failure is reached, for both ductile and brittle failure
modes, the beam element is converted to a strut, as in the case
of masonry elements. Instability phenomena and second order effects are not considered. This modelling approach has been recently adopted for the assessment of mixed masonry–r.c.
buildings [29].
4.3. Steel and wooden elements
Also steel and wooden beams or tie-rods may be modelled.
Similarly to r.c. elements, steel and wooden beams are idealized
by assuming elastic-perfectly plastic hinges, concentrated at the
ends of the element. Obviously the strength criteria adopted as reference are modified according to these different materials and
checks on the ultimate deformation capacity are not included.
Tie-rods are idealized as non-compressive spar elements with the
possibility to assign also an initial strain e0 (thus to impose a corresponding pre-stress action equal to EAe0, with E Young modulus
of material and A transversal section of tie-rods). In this case, the
stiffness matrix of the element Ke (Eq. (1)) is updated by resetting
all terms containing the J contribution; thus, the nonlinearity is
kept into account by updating at each step of the analysis the
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S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799
global stiffness K (obtained by assembling those of all elements)
accounting only for the active tie-rods (i.e. those in tension).
5. 3-Dimensional model
Starting from the equivalent frame modelling discussed in the
previous paragraphs for a single wall, complete 3D models may
be assembled.
The 3-dimensional modelling of whole URM buildings starts
from the following basic hypotheses: (a) the construction bearing
structure, both referring to vertical and horizontal loads, is identified with walls and horizontal diaphragms (roofs, floors or vaults);
(b) the walls are the bearing elements, while diaphragms are the
elements governing the sharing of horizontal actions among the
walls; (c) the flexural behaviour of the diaphragms and the wall
out-of-plane response are not computed because they are considered negligible with respect to the global building response, which
is governed by their in-plane behaviour. The global seismic response is possible only if vertical and horizontal elements are
properly connected; then, if necessary, ‘‘local’’ out-of-plane mechanisms have to be verified separately through suitable analytical
methods.
Within this general context, two main issues have to be solved,
in particular related to: (i) the strategy for assembling 2D masonry
walls (discussed in Section 5.1); (ii) the modelling of floors
(Section 5.2).
5.1. 3D assembling of masonry walls
In order to assemble a 3D model, a global Cartesian coordinate
system (X, Y, Z) is defined. The wall vertical planes are identified by
the coordinates of one point and the angle formed with the global X
axis (Fig. 8). In this way, the walls can be modelled as plane frames
in the local coordinate system and internal nodes can still be 2dimensional nodes with 3 d.o.f. At corners and where two or more
walls intersect 3-dimensional nodes are used. They are characterized by 5 degrees of freedom (d.o.f.) in the global coordinate system (uX, uY, uZ, /X, /Y). In fact, the rotational degree of freedom
around vertical Z axis can be neglected because of the membrane
behaviour adopted for walls and floors (Fig. 8). These nodes can
be obtained assembling 2D rigid nodes acting in each wall plane
and projecting the local d.o.f. along global axes. The assemblage
is then obtained condensing the degrees of freedom of two
2-dimensional nodes by assuming the full coupling among the
connected walls. This solution is particularly efficient to reduce
the total number of d.o.f. and perform nonlinear analyses with a
reasonable computational effort also in case of large and complex
building models.
Since the 2D nodes have no d.o.f. along the direction orthogonal
to the wall plane, the nodal mass component related to out-ofplane degrees of freedom is shared to the corresponding d.o.f. of
the two nearest 3D nodes of the same wall and floor according to
the following relations:
lx
l
lx
I
I
M y ¼ M y þ mð1 j sin ajÞ
l
M Ix ¼ M Ix þ mð1 j cos ajÞ
where the meaning of the terms is shown in Fig. 8. This solution
permitted to maintain the adopted simplification hypotheses in
the implementation of static analyses with 3 components of acceleration along the 3 principal directions and 3D dynamic analyses
with 3 simultaneous input components.
5.2. Modelling of diaphragms
A proper assumption on the diaphragm stiffness may significantly affect the overall response. In fact, in the limit case of ‘‘infinitely’’ flexible floors, there would be no load transfer from heavily
damaged walls to still efficient structural elements. On the contrary, in the other limit case of floors assumed as ‘‘infinitely’’ stiff,
this contribution could be overestimated. Although it represents a
crucial feature to be considered, the floor behaviour in 3D modelling is frequently assumed (with a rough approximation) as completely rigid. This hypothesis may be completely unrealistic in
case of existing buildings (e.g. historical masonry structures),
where various ancient constructive technologies (i.e. timber floors
and roofs, structural brick or stone vaults) can be found for floor
and roofing systems; moreover, this is also a major issue in new
masonry buildings with wooden floors and roofs.
In order to simulate the presence of flexible diaphragms, specific floor elements were introduced in the TREMURI model. They
are modelled as 3- or 4-nodes orthotropic membrane finite (plane
stress) elements, with two displacement degrees of freedom at
each node (ux, uy) in the global coordinate system. They are identified by a principal direction (floor spanning orientation), with
Young modulus E1, while E2 is the Young modulus along the perpendicular direction, m is the Poisson ratio and G12 the shear modulus. The moduli of elasticity E1 and E2 represent the normal
stiffness of the membrane along two perpendicular directions, also
accounting for the effect of the degree of connection between walls
and horizontal diaphragm and providing a link between the inplane horizontal displacements of the nodes belonging to the same
wall-to-floor intersection, hence influencing the axial force computed in the spandrels. The most important parameter is G12,
which influences the tangential stiffness of the diaphragm and
the horizontal force transferred among the walls, both in linear
and nonlinear phases. Starting from these entities, the orthotropic
b aiming at correlating strain and stress (in case of 3 nodes
matrix D,
membranes) may be computed as follows:
2
^¼
D
E1
1et2
6 etE1
4 1et2
0
Fig. 8. 3D assembling of masonry walls: classification of 3D and 2D rigid nodes and
out-of-plane mass sharing.
ð3Þ
etE1
1et2
eE1
1et2
0
0
G12
0
3
7
5
ð4Þ
b may be modified (in D)
where e is the ratio E2/E1. Thus, the matrix D
through a proper rotation matrix R in order to take into account for
the actual orientation of the diaphragm. Finally, the stiffness matrix
S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799
1795
simulations, the definition of equivalent stiffness properties has
been proposed for some types of vaults (barrel, cross and cloister
vaults) as a function of different thickness-to-span and rise-tospan ratios, constraints conditions and masonry texture pattern
(parallel, orthogonal and oblique). In case of timber floors and
roofs, some recent experimental works suggest a critical review
of the formulae proposed in the literature and codes [31,49–51].
6. Seismic analysis procedures
Fig. 9. 4-Node membrane element as average of 3-node element meshes.
is assembled starting from D by adopting linear shape functions. For
each node i of the 3-node element, the matrix Bi can be defined as:
2
Bi ¼
1 6
4
2A
yj yk
0
xk xj
0
3
xk xj 7
5
yj yk
ð5Þ
where xj, yj, xk, and yk are the coordinates of nodes j and k, and A is
the area of the triangle. Starting from the matrices Bi and D, the
stiffness matrix of the 3-node membrane element can be assembled
as
2
kii
6
K ¼ 4 kji
kki
kij
kjj
kkj
kik
6.1. The pushover analysis
3
7
kjk 5
kkk
In order to perform nonlinear seismic analyses of masonry
buildings a set of procedures has been implemented in the TREMURI program [8]: incremental static with force or displacement control; 3D pushover analysis with fixed load pattern; 3D time-history
dynamic analysis (Newmark integration method; Rayleigh viscous
damping). In the following, the attention is focused on the numerical algorithm implemented for pushover analyses, which became
more and more popular for seismic structural assessment in the
last decades, in particular in conjunction with the spreading of performance-based earthquake engineering concepts.
ð6Þ
where kij ¼ BTi DBj As, with s equivalent thickness assumed for the
membrane element. In the case of 4-nodes elements, the stiffness
matrix is obtained as the averaged contribution of the two possible
meshes of 3-node elements (Fig. 9).
The evaluation of the abovementioned quantities may be rather
simple in case of some floor typologies, ascribing it to the structural role shown by some specific elements. For example, in the
case of a r.c. floor with beams and slab the shear stiffness is mainly
given by the slab whereas the beam axial stiffness leads to the definition of E1. On the contrary, in case of various ancient floor technologies, as the case of vaults, beside thickness and material
properties, the stiffening contribution strongly depends on shape
and geometrical proportion (e.g., rise-to-span ratio). In [48], starting from the results of both linear and nonlinear FEM numerical
The pushover procedure implemented [7,18] transforms the
problem of pushing a structure maintaining constant ratios between the applied forces into an equivalent incremental static
analysis with displacement control at only one d.o.f. In this sense
this procedure is conceptually similar to the one proposed in [52].
In particular, the general formulation of the pushover problem
can be represented by equations:
2
K FF
K Fm
6 KT
4 Fm
kmm
K TFC
T
kmC
9
38 9 8
K FC <
> xF >
< kfF >
= >
=
kmC 7
5 xm ¼ kfm
>
>
: >
:
; >
;
xc
rc
K CC
ð7Þ
where m is the control degree of freedom and fF is the coefficient
vector of the applied load pattern.
The system of equations can be transformed subtracting the
mth row, multiplied by a proper factor, from the first m 1 rows;
the ith equation then becomes:
Fig. 10. Front view and 3D building model (on the top); modal analysis results (on the bottom).
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S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799
Fig. 11. Comparison between the real and numerically simulated damage pattern (from [54]): the experimental one (on the left); that simulated through TREMURI program
(at the centre); that simulated by the finite element mode (on the right, in terms of principal inelastic strain).
fi
fi
K i1 km1 x1 þ þ kim kmm xm þ fm
fm
fi
þ kin kmn xn
fm
¼0
ð8Þ
The new system of equations, with a modified stiffness matrix:
2
e FF
K
6 T
4 K Fm
K TFC
e Fm
K
kmm
T
kmC
9
38 9 8
e FC > xF > > kfF >
K
< = <
=
7
kmC 5 xm ¼ kfm
>
>
: >
:
; >
;
xc
rc
K CC
ð9Þ
is then equivalent to a displacement control one, in which the mth
d.o.f. xm is the imposed one. This formulation was obviously rewritten by introducing the nonlinear contribution in incremental form,
in order to be implemented in the nonlinear procedure.
The algorithm results quite effective and robust, so allowing
pushover analyses on very complex models.
Finally, it has to be stressed that horizontal forces, proportional
to tributary masses and the assumed mode shape, are applied to
each node at the level of each floor. Actually, the application of nodal forces in the pushover analysis represents a crucial issue, in
particular, for a reliable computation of the axial load acting on
spandrel elements. As an example, if in case of flexible floors forces
are applied only on the corner nodes, a wrong axial load on spandrel elements is obtained.
structural behaviour both for linear and nonlinear response has
been carried out, as described in [18]. In particular, modal tests allowed good data for dynamic identification of the 3D model
(Fig. 10): numerical modal analysis results, accordingly to experimental ones, are shown in Fig. 6. MAC index value (Modal Assurance Criteria), which quantify the agreement between modal
shapes from numerical model and experimental data, is 0.96 and
0.94 for the first two modes.
The second example focuses on the model ability to reproduce
the seismic response in nonlinear range. In particular, the in-plane
response of the ‘‘Door-Wall’’ of the full scale two storey brick masonry building prototype, tested at the University of Pavia by Magenes et al. [53], has been analyzed. Experimental results have been
simulated by the TREMURI program (as described in more detail in
[54]) and compared with those obtained from a finite element
model by shell elements, adopting the nonlinear continuum damage model proposed in [55]. The parameters of the models have
been defined on the basis of those directly obtained by experimental tests on the components (mortar and brick) and their assemblies (triplets, masonry prisms, in-plane cyclic shear tests on
piers) carried out on the same brick masonry adopted for the full
scale prototype [56]. No additional calibration of parameters has
been performed. In fact, the will was to simulate a real condition
of seismic assessment on an existing building for which, in
200
7. Validation of the model and examples
160
120
V [kN]
In the following, some examples of applications are illustrated,
showing the capability of TREMURI program (and more in general
of the equivalent frame approach) in describing the seismic response of masonry buildings.
The first example shows how, despite of some unavoidable
approximations consequent to the idealization of a real structure
in an equivalent frame model (e.g. in the proposed model the
deformability of portions idealized as rigid nodes is neglected), it
is able to provide a good simulation of the structural behaviour
in linear range. In particular, the application concerns the case
study of the Hall of the Giuncugnano village in Tuscany, an instrumented building included in the Structure Seismic Observatory
(OSS) program of the Italian Department of Civil Protection.
Fig. 10 shows the front view of the building and the 3D model. This
URM building, like several others all around Italy, now hosts a set
of accelerometers that permanently monitor its dynamic response.
Based on both in situ and laboratory tests, a characterisation of the
80
FEM
Experimental test
40
EF-full stiffness
EF-reduced stiffness
0
0
5
10
15
20
25
u [mm]
Fig. 12. Simulation of the in-plane response of Door Wall: comparison in terms of
pushover curve (from [54]).
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S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799
0,3
WSSP
SSWP
0,25
3m
EF Model - Case A
EF Model - Case B
V/ W
0,2
3m
EF Model - Case C
0,15
0,1
0,05
3m
0
0
2m
n20
Flexural plastic phase
Shear collapse
5
13
CASE A
Shear plastic phase
4
N3
10
n19
N16
n18
31
15
N7
N15
8
12
N6
2
N14
25
9
n17
N5
N13
N4
n20
n27
n26
26
n25
5
13
N11
22
330
N10
20
n20
N8
5
N3
10
N2
27
n19
n18
7
N3
N6
2
N2
N1
9
n17
n19
1
n18
N7
4
12
11
7
N6
2
9
8
8
N9
3
10
12
11
1
N7
4
15
14
15
14
3
N8
6
6
13
33
29
19
N4
N12
24
32
21
328
8
N1
n28
23
4
11
1
7
Flexural collapse
N8
6
14
3
N2
6
u/uSSWP
N4
Rigid Node
Elastic phase
2
1m 1m
CASE B
1m 1m
N5
N1
n17
N5
Fig. 13. Comparison among simplified models (SSWP and WSSP) and the Equivalent Frame model as a function of different hypotheses assumed for spandrels (from A to C)
for a three-storey URM wall (adapted from [15]).
addition, results of standard tests are available. According to the
masonry type (brick masonry) and configuration (two steel beams
introduced to apply the horizontal forces that worked as ‘‘tie-rods’’
at level of spandrels) of the ‘‘Door Wall‘‘, nonlinear beams in the
TREMURI program have been modelled by adopting the criterion
proposed in [42] for the shear response of masonry elements
and, in particular for spandrels, the criteria proposed in [6] (see Table 1). Figs. 11 and 12 show the comparison of results in terms of
damage pattern and pushover curve (global shear forces V at the
base of the wall versus mean displacement u of the second floor),
respectively. It is worth noting that, differently from the experimental tests, in the numerical simulations the nonlinear static
analysis has been performed monotonically. Results are in fair
agreement and substantially confirm the reliability of the EF modelling approach, in particular for masonry walls characterized by
regular opening patterns. As regards the pushover curve, the moderate overestimation of the global strength given by the FE and EF
models with respect to the experimental one may be explained by
considering that:
– numerical simulation has been performed monotonically;
– a scattering of the material parameters (in particular those of
joints) should characterize the real building;
– parameters have been calibrated on those obtained on tests performed on single panels (for which a higher building care is
expected than the whole structure leading to better
performance).
In the case of the TREMURI model, two hypotheses for the stiffness properties have been adopted (by assuming ftu/fhu – see Table 1 – equal to 1 and 0.5, respectively). The case of ftu/fhu = 0.5
highlights how this reduction of the stiffness parameters is rather
coarse. Actually, in the case examined, the response is driven by
few structural elements: thus, the incorrect assignment of the
deformability parameters strongly influences the overall response
of the structure. Finally, as regards the damage pattern numerically
simulated, it can be observed that different failure modes occur in
the two lateral piers at the ground storey. Although these piers are
characterized by the same slenderness, the flexural behaviour prevails in the left one, while diagonal cracking prevails in the right
one. This is due to the different levels of normal forces they are
subjected to, associated with the global overturning of the wall.
In the central pier, which is the more squat, diagonal cracking occurs. The more symmetric experimental damage pattern is due to
the application of a cyclic load history.
Finally, the third example aims to showing the higher versatility
of the equivalent frame model to simulate the actual behaviour of
masonry building with respect to other more simplified approaches. Fig. 13 illustrates the response of a three-storey URM
wall with two lines of vertically aligned openings [54] in which,
respectively: SSWP and WSSP represent the two extreme idealization proposed in [11,12]; Case A refers to the EF model in which the
same strength criteria have been assumed for piers and spandrels;
Case B to the EF Model in which for the flexural behaviour the criterion proposed in [39] has been assumed for the spandrel by
assuming g as 0.05; Case C to the EF Model in which reinforced
concrete beams have been modelled coupled to spandrel elements.
Results are illustrated in terms of V/W (ratio between base shear
and total weight of the structure) versus u/uSSWP (displacement
of control node located on top of the wall, normalized to the ultimate value obtained in the case of SSWP). SSWP and WSSP define
the range of the possible pushover curves of the structure. It appears too wide in terms of strength, stiffness and ductility definition, all three aspects that play a fundamental role by referring
to the adoption of nonlinear static procedures as tools of verification. Moreover, even if the systematic adoption of WSSP in case
of existing buildings of course should lead to results which are
on the safe side, such a severe underestimation of the actual capacity would not be acceptable, in particular if an unsuccessful assessment would lead to unrealistically heavy retrofitting interventions.
The comparison between SSWP and Case C stresses how the presence of certain constructive details is not in general sufficient to assure the satisfaction of some simplified hypotheses: this is the case
of the presence of reinforced concrete beams (usually associated
with the presence of rigid floors), characterized by a finite stiffness,
which does not correspond with the assumption of fixed rotations
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S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799
at the level of each floor. As a consequence, SSWP provides an
upper bound which operates on the unsafe side. Case A provides
results similar to those of WSSP because resistance criteria assumed for spandrel elements are too cautionary. By assuming the
criterion proposed in [39] for spandrel elements (Case B), both a
significant increase in the overall resistance and a decrease in the
global ductility can be observed with respect to case A. The latter
result can be explained by both the different pattern and sequence
of damage that occur in cases A and B (Fig. 13). In fact in Case A, due
to the moderate axial load acting on the spandrel elements, since
the initial steps of the analysis a Rocking mechanism occurs in almost all spandrels which thus supply a weak coupling for piers.
On the contrary, in case B the following phases are recorded: a first
phase in which only the spandrels located on the top floor show
the activation of a Rocking mechanism (in fact, due to the moderate
compressive stresses acting on the contiguous masonry portions,
they cannot rely much on the interlocking phenomena); a final
phase, in which damage increases in spandrels and also spreads
to piers located on the ground floor. Analogous results and comments on the role of spandrel elements are presented in [3,17].
8. Conclusions
The theoretical bases of TREMURI program are explained in the
paper, with particular reference to its distinctive features:
the creation of an equivalent 3-dimensional frame based on
assembling 2-dimensional (plane) structures (masonry walls
and floor/roof diaphragms), which allows for an effective condensation of the degrees of freedom of the global model, so
reducing the computational burden for the global analysis of
the building;
the implementation of specific elements allowing for the representation of the main characteristics of nonlinear response of
masonry piers and spandrels, as well as other structural members (r.c. members, steel tie-rods, etc.);
the implementation of an orthotropic membrane element, for
the simulation of the in-plane behaviour of flexible diaphragms
(e.g. timber roofs and floors), which can be commonly found in
new and existing masonry structures;
an original and versatile algorithm for the pushover analysis,
suitable for assessing the nonlinear evolution of the lateral
response of 3-dimensional masonry buildings, including the
deterioration of the base shear for increasing lateral displacements after the attainment of peak strength.
Most of the TREMURI features were developed and implemented with the aim of filling some of the existing gaps in the seismic analysis of masonry structures, with a specific target on the
assessment of existing buildings with flexible diaphragms. The
computer program that has been developed in the last decade allows, with a reasonable computational effort, a reliable simulation
of the actual behaviour of traditional masonry structures, provided
that proper connections prevent the activation of local outof-plane mechanisms and favour the development of a global
response, governed by the in-plane behaviour of the different
structural components.
This versatile computer program for the analysis of the seismic
response of historical masonry structures represents a useful tool
for supporting their conservation. In fact, a correct identification
of the most critical walls and related collapse mechanisms (soft
storey/failure in spandrel elements) may suggest different
strengthening strategies, addressed to increase strength or displacement capacity of masonry panels, as well as stiffness of horizontal diaphragms. TREMURI is a framework in continuous
progress for the development and implementation of new advanced nonlinear elements and analysis procedures (static/dynamic), both at research level and engineering practice.
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