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EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2009; 38:243267Published online 6 November 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.860
In-plane strength of unreinforced masonry piers
Chiara Calderini,, Serena Cattari and Sergio Lagomarsino
Department of Civil, Environmental and Architectural Engineering, University of Genoa,
Via Montallegro 1, 16145 Genoa, Italy
SUMMARY
The definition of adequate simplified models to assess the in-plane load-bearing capacity of masonry
piers, in terms of both strength and displacement, plays a fundamental role in the seismic verification ofmasonry buildings. In this paper, a critical review of the most widespread strength criteria present in theliterature and codes to interpret the failure modes of piers ( rocking,crushing,bed joint sliding or diagonalcracking) are proposed. Models are usually based on an approximate evaluation of the stress state producedby the external forces in a few points/sections and on its assessment with reference to a limit strengthdomain. The aim of the review is to assess their reliability by discussing the hypotheses, which they arebased on (assumed stress states; choice of reference points/sections on which to assess the pier strength;characteristics of the limit strength domain) and to verify the conditions for their proper use in practice,in terms of both stress fields (depending on the geometry of the pier, boundary conditions and appliedloads) and types of masonry (i.e. regular brick masonry vs rubble stone masonry). In order to achievethese objectives, parametric nonlinear finite element analyses are performed and different experimentaldata available in the literature are analysed and compared. Copyright q 2008 John Wiley & Sons, Ltd.
Received 6 June 2008; Revised 3 September 2008; Accepted 4 September 2008
KEY WORDS: masonry; pier strength; in-plane behaviour; seismic capacity; failure mechanisms; simpli-fied models
1. INTRODUCTION
In the last decade, the achievement of performance-based earthquake engineering concepts has led
to an increasing utilization of nonlinear static procedures in the evaluation of the seismic perfor-
mance of masonry buildings (Coefficient Method[1], Capacity Spectrum Method[2], N2/method
[3]). As widely known, these procedures are based on a comparison between the displacement
capacity of the structure and the displacement demand of the predicted earthquake. Definition of thedisplacement capacity requires the evaluation of a forcedisplacement curve (pushover curve)
Correspondence to: Chiara Calderini, Department of Civil, Environmental and Architectural Engineering, Universityof Genoa, Via Montallegro 1, 16145 Genoa, Italy.
E-mail: [email protected]
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244 C. CALDERINI, S. CATTARI AND S. LAGOMARSINO
Idealized vertical stress
distribution at the base
Pi
Pj
Vi
Vj
Mj
Mi
Piers
Rigid connectionsSpandrels
(a) (b)
Figure 1. Different approaches in modelling of masonry buildings: (a) FEM model
and (b) equivalent frame idealization.
able to describe the overall inelastic response of the structure 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 of increasing
magnitude (describing seismic forces). To this aim, different strategies may be pursued.
A first approach consists of discretizing the masonry continuum in a number of finite elements
(Figure 1(a)), in adopting a suitable nonlinear constitutive law, and, finally, in performing a
nonlinear incremental analysis. Although this approach may provide quite an accurate description
of the structure and of its material, it requires a high computational effort, which is unsustainable
for wide application in engineering practice. Moreover, it poses some problems in correlatingthe displacement capacity of the structure to predefined limit states. In fact, the limit states are
commonly related to the drift parameter, whose reference values are conventionally defined for
single panels on an experimental basis. Since in the finite element method (FEM) the structure is
modelled as a continuum, the identification of the elements on which to monitor this parameter
might be ambiguous, and may imply repeated average operations performed ex post.
For these reasons, a second approach, particularly suitable for the analysis of standard masonry
buildings, made up of well-connected walls with a rather regular pattern of openings, is often
adopted. It is based on the idealization of the structure through an equivalent frame (Figure 1(b)),
in which each resistant wall is discretized by a set of masonry panels in which the nonlinear
response is concentrated [4 6]. Two types of panels are distinguished: piers, which are the
principal vertical resistant elements for both dead and seismic loads; and spandrels, which are
secondary horizontal elements, coupling piers in the case of seismic loads. Only in-plane resistantmechanisms are considered. Actually, an exhaustive seismic verification would require to also
take into account the possible occurrence of out-of-plane mechanisms; however, if the attention is
focused on the overall seismic behaviour of the structure, common practice is to neglect this class
of mechanisms. In fact, they usually involve parts of the structure without significantly affecting
its global response; as a consequence they are usually verified apart, referring only to the involved
portion. Moreover it is worth pointing out that, in existing buildings, in most of the cases they
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IN-PLANE STRENGTH OF UNREINFORCED MASONRY PIERS 245
can be inhibited through specific seismic retrofitting interventions (tie rods, effective connection
between wall and floor).
In the context of this second approach, the prediction of the in-plane load-bearing capacity of
masonry piers, in terms of both displacement and strength, is a fundamental issue.
The load-bearing capacity of masonry piers may be provided by performing experimental teststhat are able to simulate reality as closely as possible, in terms of boundary conditions and acting
forces. From these tests it is possible to define, for a pier of given slenderness and given masonry
type, a limit strength domain in the space of applied forces. Such an approach, though quite accurate
and reliable, is costly and time consuming since it requires a large number of tests to be performed.
Moreover, in most cases, it is technically inapplicable to existing buildings due to its highly
destructive nature. For these reasons, simplified theoretical models are needed to be developed.
Referring to the strength only, different simplified models have been proposed in the literature
and in seismic codes in recent decades. Following a typical engineering approach, they are generally
based on the approximate evaluation of the stress state produced in piers by the external forces and
on the assessment of its admissibility with reference to a limit strength domain. The application
of this approach poses the following issues:
Masonry is an anisotropic material. Also considering only plane homogenous stress states, it is
characterized by many different failure modes and strengths (Figure 2(a)) [716]. This property
strongly affects the response of piers subjected to in-plane seismic forces (Figure 2(b)),
as confirmed by both the observation of seismic damage and experimental laboratory tests
[13, 1730]. It is therefore necessary to schematize their limit strength domain by proper
simplified assumptions.
P
y
x
y
900 45
P
x
y
P
V
A
B1
C
B2
y0
fm fm0
V A =
y P A =A
B1
B2
C
(a) (b)
Figure 2. Failure modes and limit domains of masonry: (a) scale of the material and (b) scale of the pier.
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246 C. CALDERINI, S. CATTARI AND S. LAGOMARSINO
Forces acting on piers produce strongly non-homogeneous stress states. This is true both in
the linear range, due to the boundary conditions and applied forces, and in the nonlinear
range, due to the stress redistribution derived from the damaging process of the material.
This implies that the limit strength domain of masonry piers is not coincident with the limit
strength domain of the masonry itself (Figure 2). A rigorous approach would require that,given a strength domain of masonry, the admissibility of the stress state would be assessed
in all the points of the pier. However, the equivalent frame idealization imposes assessing
the stress state only in a limited number of points or sections.
Due to the complexity of the material considered, the experimental evaluation of the parameters
required by models is not an easy task. On the one hand, the interpretation of tests is not
always clear. On the other hand, practical or technical reasons do not always allow one to
perform the experimental tests required for a given model.
The paper proposes a critical review of the simplified models present in the literature and codes
for the prediction of the strength of masonry piers. The analysis of the models has the following
main objectives: discussing the reliability of the hypotheses on which they are based (assumed
stress states, choice of points/sections on which to assess the stress state, characteristics of the limitstrength domain); verifying the conditions for their proper use in practice, in terms of both stress
fields (depending on the geometry of the pier, boundary conditions and applied loads) and types
of masonry (i.e. regular brick masonry vs rubble stone masonry). The research has been carried
out both by performing parametric nonlinear finite element analyses and by analysing available
experimental data.
2. CLASSIFICATION OF OBSERVED SEISMIC FAILURE MODES
Observation of seismic damage to complex masonry walls, as well as laboratory experimental tests,
showed that masonry piers subjected to in-plane loading may have two typical types of behaviours,with local cracks according to Figure 2(b), with which different failure modes are associated:
Flexural behaviourThis may involve two different modes of failure. If the applied vertical
load is low with respect to compressive strength, the horizontal load produces tensile flexural
cracking at the corners (Figure 2(b)A), and the pier begins to behave as a nearly rigid
body rotating about the toe (rocking). If no significant flexural cracking occurs, due to a
high applied vertical load, the pier is progressively characterized by a widespread damage
pattern, with sub-vertical cracks oriented towards the more compressed corners (crushing).
In both cases, the ultimate limit state is obtained by failure at the compressed corners
(Figure 2(b)C).
Shear behaviourThis may produce two different modes of failure. In sliding shear failure,
the development of flexural cracking at the tense corners reduces the resisting section; failureis attained with sliding on a horizontal bed joint plane, usually located at one of the extremities
of the pier. In diagonal cracking, failure is attained with the formation of a diagonal crack,
which usually develops at the centre of the pier and then propagates towards the corners. The
crack may pass prevailingly through mortar joints (assuming the shape of a stair-stepped
path in the case of a regular masonry pattern, Figure 2(b)B 1) or also through the blocks
(Figure 2(b)B2).
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IN-PLANE STRENGTH OF UNREINFORCED MASONRY PIERS 247
Tensile
flexural
cracking
Sub-vertical
cracksTensile
flexural
cracking
Sub-vertical
cracks
Sliding on a horizontal planeSliding on a horizontal plane
Diagonal
crack
Diagonal
crack
(a) (b) (c)
Figure 3. Typical failure modes of masonry piers: (a) rocking; (b) sliding shearfailure; and (c) diagonal cracking.
Figure 3 shows three typical damage patterns associated with the above-described main failuremodes.
The occurrence of different failure modes depends on several parameters: the geometry of the
pier; the boundary conditions; the acting axial load; the mechanical characteristics of the masonry
constituents (mortar, blocks and interfaces); the masonry geometrical characteristics (block aspect
ratio, in-plane and cross-section masonry pattern). In the past, many experimental tests have
attempted to analyse the influence of these parameters on the failure mode of masonry piers. In
general, it has been assessed that rocking tends to prevail in slender piers, while bed joint sliding
tends to occur only in very squat piers[2126]. In moderately slender piers,diagonal crackingtends
to prevail over rocking and bed joint sliding for increasing levels of vertical compression [17, 27, 30].
Diagonal crackingpropagating through blocks tends to prevail over diagonal crackingpropagating
through mortar joints for increasing levels of vertical compression [1719, 25, 2729] and for
increasing ratios between mortar and block strengths [7, 8, 18, 19, 27, 31]. Increasing interlockingof blocks (block aspect ratio plus masonry pattern) may induce a transition from diagonal cracking
through mortar joints torocking[30, 32], todiagonal cracking through blocks [31] or to bed joint
sliding [33]. Crushing, in general, occurs for high levels of vertical compression (related to the
compressive strength of the material).
It is worth pointing out that it is not always easy to distinguish the occurrence of a specific type
of mechanism, since many interactions may occur between them.
3. INTERPRETATION OF FAILURE MODES THROUGH AVAILABLE MODELS
In the following paragraphs, the most common simplified models present in the literature for theprediction of the strength of masonry piers are discussed. In order to help the reader to understand
the notation used, a few clarifications should be made. The simplified models analysed are based
on the choice of a reference stress (either shear, normal or principal stress) and of a reference
point or section on which it should be calculated. Its admissibility is assessed by comparison with
a proper stress domain for masonry. In what follows, in order to underline this common approach,
the reference stress is named c.
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248 C. CALDERINI, S. CATTARI AND S. LAGOMARSINO
3.1. Flexural behaviour
Models considering the flexural behaviour of piers usually choose the base section of the pier to
assess the stress state. The maximum normal stress acting on the bed joints plane is assumed as the
reference one(c). It is calculated on the basis of the beam theory, neglecting the tensile strengthof the material and assuming an appropriate normal stress distribution at the compressed toe. In
both rocking and crushing, failure is associated with attainment of the compressive strength of
masonry normal to bed joint plane (vertical directiony, with reference to Figure 2(b)). Neglecting
the dead load of the pier, equilibrium leads to the following general expression:
c =y
k2r(12k1r)fm (1)
where k1r is a coefficient taking into account the slenderness and the boundary conditions of the
pier; k2r is a coefficient, which takes into account the assumed normal stress distribution at the
compressed toe; is the ratio between the horizontal force (V) and the vertical force applied
(P);y = P/DTis the mean vertical stress acting on the section (D and Tare the pier width and
thickness, respectively); fm is the compressive strength of the masonry.The parameter k1r is the shear ratio, computed as the effective pier height H0 over the width
D (H0 is equivalent to the distance from zero moment). The H0 parameter is determined by the
boundary conditions: for piers fixed against rotation at their top and base H0=H/2, whereas for
piers fixed at one end and free to rotate at the other H0=H (Hbeing the height of the pier).
The parameter k2rdepends on the constitutive law assumed for the material. It allows to define
an equivalent stress block, reducing the compressive strength in order to take into account the
actual stress distribution (which is not rectangular). The assumption of an infinite ductility of
the material under compression implies that k2r= 1. Considering a finite ductility of masonry in
compression, under the above-mentioned hypotheses for masonry, the coefficient k2r assumes the
following form:
k2r=(21)2
4(2+1/3)(2)
where is the ductility of the material. It can be observed that the value ofk2r tends quickly to
unity, also for rather low values of the assigned ductility; many codes assume k2r= 0.85, which
corresponds to a ductility =1.18. It should be pointed out that Equation (2) is valid only when
the neutral axis cuts the cross-section (rockingfailure); in the case ofcrushing failure, the use of
k2r from Equation (2) is on the safe side.
By comparing the overall strength of the pier obtained through Equation (1) with the limit value
obtained with the rigid block assumption, it can be observed that, for moderate axial loads (related
to the compressive strength of the material), the value of the compressive strength fm has a limited
influence. Considering that in complex masonry buildings piers are subjected to axial loads usually
far from the limit value of the compressive strength, it can be stated that uncertainties, also quite
wide, regarding the estimate of fm (due to large scattering of experimental data results) do not
significantly affect the resistance prediction.
3.2. Shear behaviour
Different models have been developed to describe the failure associated with shear behaviour. It
is possible to recognize two main types of models: models describing masonry as a composite
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IN-PLANE STRENGTH OF UNREINFORCED MASONRY PIERS 249
Table I. Meaning of the parameter in shear failure modes.
Failure mode k1d k1s c
Bed joint sliding 1 Function of the assumed
constitutive law c Diagonal cracking Function of the
(through joints) slenderness () 1 c1
1+
1
1+
Considering a no-tension material and linear distribution of stresses, k1s=3(0.5), being a parametertaking into account the boundary conditions of the pier.
In the following, it will be assumed: k1d=, with 1.0
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250 C. CALDERINI, S. CATTARI AND S. LAGOMARSINO
On the basis of the same mechanical hypotheses adopted for the description ofdiagonal cracking
through joints, Mann and Muller [19] also developed a criterion for the cracking of blocks. Since
no shear stresses can be transferred through head joints, they assumed that an approximately double
shear force must be transferred through the blocks. The criterion adopts the maximum principal
stress acting in the centre of a block as reference stress c; it must not exceed the tensile strengthof the block itself fbt. The criterion may therefore be expressed in the following form:
c =y
2+
(k1dk2d)2+
y
2
2fbt (4)
wherek2d is the ratio between the mean shear stress applied on the block and the local shear stress
at its centre. It has been demonstrated that k2d=2.3 for standard masonry where =0.5.
Among the models that consider indistinctly the development of a crack along a principal stress
direction, the most widely used one was originally proposed by Borchelt [34] and Turnsek and
Cacovic [35]. Borchelt based his formulation on a square panel tested in diagonal compression,
while the formulation of Turnsek and Cacovic was developed by racking tests on cantilever piers.They both consider as reference stress c the maximum principal stress acting at the centre of the
panelI. Masonry is assumed to be an isotropic material. It is assumed that the reference principal
stress c must not exceed a reference tensile strength of masonry ft:
c =I=y
2+
(k1d)2+
y
2
2ft (5)
Borchelt assumed the shear stress at the centre of the panel as coincident with the mean stress
acting on the transversal section; thus, in its formulationk1d=1. As demonstrated by other authors
[36], this assumption implies a strong approximation of the actual stress field. In their original
work [35], Turnsek and Cacovic assumed that k1d=1.5. Later, other authors proposed more
detailed expressions of this parameter. A common criterion for design practice was proposed byBenedetti and Tomazevic[37]as: k1d=, with 1
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IN-PLANE STRENGTH OF UNREINFORCED MASONRY PIERS 251
Table II. Criteria adopted by principal codes.
Code EC6/EC8 EC8 Part III ACI 530-05 FEMA 356/306 DIN 1053-100
Flexural behaviour Equation (1) X X X
Other X X X XShear behaviour Equation (3) X X X X X
Equation (4) XEquation (5) X Other X X X
Reference is made to generic strength limitations, not related to specific mechanical interpretations.
Table III. Meaning of the parameter in codes.
Flexural behaviour Shear behaviour
Code k1r k2r k1d k1s c k2d
EC6/EC8 1 Note Note 0.4
EC8Part III 1 0.85 1 Note Note 0.4
ACI 530-05 Note 1.5 1 0.255 MPA 0.45
FEMA 356/306 H0/D 0.7 Note 1 Note 0.75
DIN 1053-100 Note 0.85 Note 1 Note 1
1+ 2.3
Depends on the constitutive law adopted.Obtained by the triplet test or from a table as a function of the block type (hollow %) and of the mortar type.To be calculated, taking into account the tensile strength of masonry.Bed joint sliding: 1.5; diagonal cracking: 1.5 for =2,1 for 1, linear interpolation elsewhere. From experimental tests, multiplied by the coefficient 0.56.
To be calculated, depending on the slenderness and boundary conditions.1.5 for =2,1 for 1, linear interpolation elsewhere.From a table, as a function of the mortar type.
not considered null; by adopting a linear stress distribution (with a fragile behaviour in tension),
an independent verification of compressive and tensile stress is proposed.
As far as shear behaviour is concerned, the majority of codes assumes the MohrCoulomb-type
verification criterion expressed in a general form in Equation (3). However, it can be observed
that:
In some cases, it is not declared whetherbed joint slidingor diagonal crackingthrough joints
is considered. This is the case of Eurocodes 6 and 8[3, 38]. On the one hand, it considers only
the compressed part of the cross section and assumes that k1d=1.0; regarding the strengthparameters, is considered as a constant value and c is assumed to be dependent on the
masonry typology, but neither of them depends explicitly on the interlocking or overlapping
of blocks. The latter assumptions seem to be consistent with bed joint sliding. On the other
hand, the constant value 0.4 for seems too conservative to represent the local friction
coefficient of the mortar joints, while it appears more appropriate if referred to Mann and
Mullers theory. Indeed, it can be obtained by assuming a local friction =0.6 and an
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252 C. CALDERINI, S. CATTARI AND S. LAGOMARSINO
interlocking coefficient=0.83 (corresponding to a ratiob/hequal to 2.4, typical of standard
masonry). The latter observation induces one to bring back the criterion to diagonal cracking.
In other cases, the values attributed to some coefficients seem to be inconsistent with the
failure mode considered. FEMA 306 [41] considers a criterion like that of Equation (3),
declaring that it may be used either for bed joint sliding (on the central or the base section)or for diagonal cracking through joints. However, a coefficient k1d=1.5 and the whole cross
section of the pier is indistinctly considered. The German code [40] is clearly based on
Mann and Mullers theory and, thus, refers to diagonal cracking through joint failure mode.
However, only the compressed part of the cross section is considered, an assumption that
usually leads to consider the end section as reference one. This latter choice may provide an
excessively conservative estimation of the strength, especially if the cracking starts from the
centre of the pier.
It is worth noting that many codes introduce limit values to the shear stress, which seem to take
into account brick failure. In most cases, these limitations are related to a reduced (through a
square root or a small multiplier) compressive strength of masonry[42]or of blocks [3, 38]. Only
the German code [40] provides a strength criterion explicitly referred to the tensile strength ofblocks, based on the theory of Mann and Muller (Equation (4)).
Finally, the model expressed by Equation (5) is considered only in FEMA 356/FEMA 306
[1, 41].
4. DISCUSSION OF THE CRITICAL ISSUES
With reference to the models described above, issues of both intrinsic and extrinsic nature
should be discussed: among the intrinsic ones, the reliability of the hypotheses on which they are
based; among the extrinsic ones, the conditions for their proper use in verification methods.
Regarding the first issue (Section 4.1), two points appear particularly critical: first, to whatamount the actual stress distribution differs from the simplified distribution assumed in the criteria,
considering that a transition from the elastic to the nonlinear range may occur; second, whether
the choice of establishing the strength of the pier referring to only some specific points/sections is
correct, considering that resistance should further increase in relation to stress redistribution. For
this purpose, a set of numerical nonlinear analyses has been performed.
The second issue (Section 4.2) is related to the choice of the most suitable criteria to be adopted
in the verification of piers. Since each resistance criterion provides a mechanical interpretation of a
specific failure mode, its suitability is related in particular to the actual occurrence of the predicted
failure mode. Moreover, its actual employment depends on the technical possibility of evaluating
the parameters required through experimental tests.
4.1. Investigation into the reliability of the hypotheses of the models
The following points will be discussed. First, whether the assumption of the base section or the
point at the centre of the pier as reference ones is correct or not, with reference to the different
failure modes. Second, what is the relationship between the evolution of the stress state in the
reference section/point and the attainment of the strength of the pier. Third, whether the stress
state assumed in the criteria is appropriate or not.
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IN-PLANE STRENGTH OF UNREINFORCED MASONRY PIERS 253
In order to investigate these points, a set of parametrical analyses on piers subjected to static
in-plane loading, with different slenderness and different levels of axial loads, were performed. A
standard brick masonry, characterized by a regular pattern and by lime mortar, was considered.
The same mechanical properties were adopted for all the piers. The latter choice is motivated by
the will to particularly deepen the influence of the geometrical aspects rather than the mechanicalones on the maximum resistance attainment of the pier.
The finite element method, together with a nonlinear constitutive model for masonry [43],
described shortly in Section 4.1.1, was adopted.
4.1.1. The adopted constitutive law. The constitutive law adopted for the nonlinear parametric
analyses has recently been described in[43]. It is based on a micromechanical approach. The plane
stress hypothesis is assumed. Constitutive equations consider the nonlinear stressstrain relation in
terms of mean stresses and mean strains on a unit cell. The latter are produced by an elastic strain
contribution, associated with a homogenized elastic continuum, and by inelastic strain contributions
depending, besides on an overall damage to masonry in compression, on the damage to mortar
joints and blocks in traction and shear. Mortar joints are reduced to equivalent interfaces. Under the
hypothesis of neglecting the mechanical properties of head joints, symmetries due to periodicity of
the masonry pattern assumed (running bond) lead to defining the mean inelastic tensor of mortar
joints as a function of the inelastic strains of only two couples of hemisymmetric bed joints. The
mechanisms of inelasticity are expressed on the basis of the model proposed in [44].
The limit domain associated with the failure of mortar joints is defined by a discrete set of
equations depending on the sign of the normal stresses acting on the two couples of bed joints and
on the opening/closing state of head joints. Table IV summarizes such failure domain together
with that of blocks and masonry in compression. It is worth noting that, in this case, the mean
stressesx ,y and are referred to the unit cell instead of the cross section of the pier, as assumed
elsewhere.
Figure 4 shows different sections of the complete domain in the plane y , for different
values ofx (the same parameters of the parametrical analyses of Section 4.1.2 are adopted). Itcan be observed that Equation (6), describing the case in which the couples of bed joints are both
compressed and head joints are open, is similar to that proposed by Mann and Muller[19](except
for the presence of the x component) and by Alpa and Monetto [45]. Unlike Mann and Mullers
domain, however, Equation (6) is here replaced by Equation (8) for low values of compression.
Although Equation (8) extends within a very moderate range it may significantly influence the
shear strength of masonry under zero compressive stress.
4.1.2. Parametrical analyses. In the parametrical analyses performed, three configurations of piers
were investigated, respectively characterized by slendernesses =0.65 (Pier 1), =1.35 (Pier 2)
and=2 (Pier 3). A fixedfixed boundary condition was imposed. Increasing horizontal displace-
ments at the top (u)under constant axial loads (P)were applied. The range of the axial load was
such as to cause a mean vertical stressy varying between the values 0.05 and 0.8 of the masonrycompressive strength fm . The mechanical properties assumed for masonry correspond to those
of the piers tested by Anthoine et al. [25]: fm=6.2MPa; c=0.23MPa; =0.58; fbt=1.22MPa;
tensile strength of mortar joints 0.04 MPa;=0.5. A detailed simulation of these tests, validating
the reliability of the constitutive model adopted, is reported in [43].
The results will be discussed following two complementary forms of logic. On the one hand, the
evolution of the stress state in the main reference points/sections will be analysed for a fixed value
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254 C. CALDERINI, S. CATTARI AND S. LAGOMARSINO
Table IV. Full set of equations defining the limit domain of [43].
Mortar joints
State of bed Head joint
joint couples condition Equation
Both compressed Open =1
1+(cxy) (6)
Both compressed Closed =cy (7)
One is compressed,one is tense
Open =(x+y)
2(x+y)2(1+2)(2
2x+
2y c
2)
1+2 (8)
Note
One is compressed,one is tense
Closed The following system of equations have to benumerically solved in and
[2(2)2+2]2+2[2(2)+]y
+[22+]2y c2=0
2[(2)+y ]23c2=0
(9)
Note
Blocks and masonry
in compression =
f2m [2H(x )+H(x )]2x[H(y)+H(y)]
2y (10)
Note
is the square of the ratio between the cohesion c and the tensile strength of mortar joints.=/(1+), being the damage variable of the uncracked couple of joints (0
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256 C. CALDERINI, S. CATTARI AND S. LAGOMARSINO
Central cross section
Pier 1
0 0.25 0.5 0.75 1
x/D
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
x
/y
Pier 2
0 0.25 0.5 0.75 1
x/D
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
x
/y
Pier 3
0 0.25 0.5 0.75 1
x/D
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
x
/y
0 0.25 0.5 0.75 1
x/D
0
0.4
0.8
1.2
1.6
/
0 0.25 0.5 0.75 1
x/D
0
0.4
0.8
1.2
1.6
/
0 0.25 0.5 0.75 1
x/D
0
0.4
0.8
1.2
1.6
/
Base cross section
Pier 1
0 0.25 0.5 0.75 1
x/D
-0.8
-0.6
-0.4
-0.2
0
0.2
y
/fm
Pier 2
0 0.25 0.5 0.75 1
x/D
-0.8
-0.6
-0.4
-0.2
0
0.2
y
/fm
Pier 3
0 0.25 0.5 0.75 1
x/D
-0.8
-0.6
-0.4
-0.2
0
0.2
y
/fm
Figure 6. Stress evolution in the central and base cross section, with referenceto the drifts marked in Figure 5.
parametrical analyses have been performed. The results obtained confirm that the softening does
not significantly influence ; in fact, even for a limit condition of a perfectly plastic behaviour,
this ratio never falls below 0.90.
Referring to Pier 3, by analyzing the evolution of the stress components in the central cross
section, it can be observed that:
x componentAnalogous to Piers 1 and 2, progressively pass from tension to compression,
even if with entity lower than for other piers.
componentRegarding the/ratio, the initial value equal to 1.48 remains almost unvaried;
it is worth highlighting that this value results particularly coherent with the hypothesis of the
De Saint Venant idealization, undoubtedly justifiable for this geometry.
Also in this case, the y distribution stays almost unvaried.
If attention is focused on the central cross section, it is not possible to observe in Pier 3 the
sudden changes in the stress distribution that occurred in Piers 1 and 2 after the attainment of Vu .
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IN-PLANE STRENGTH OF UNREINFORCED MASONRY PIERS 257
0 0.1 0.2 0.3 0.4 0.5
y/fm
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
x
/y
Pier 1
Pier 2
Pier 3
Figure 7. Stress component x occurred at the centre of the piers as a function of the applied load.
y y
x y
-7
-5.5
-4.0
-3.0
-2.0
-1.0
-0.5
0.0
0.05
-0.75
-0.65
-0.55
-0.40
-0.30
-0.20
-0.10
0.0
0.25
-0.25
0.35
0.65
0.95
1.25
1.55
1.85
2.15
2.45
Figure 8. Complete stress-state description for Pier 2, in the post-peakphase (drift=0.3%), for y=0.6MPa.
Since the response is predominated by a rocking mechanism, the analysis of the stress evolution
at the base cross section seems more representative in this case: from Figure 6, it is possible to
observe a progressive reduction in the effective uncracked section length, much more evident in
this pier than in Piers 1 and 2. Moreover, it can be seen that the ratio y/fm in the compressed toe
results far from unity, even for the highest drift value considered. This result should suggest that
the failure condition expressed by Equation (1) is far from being attained. However, if the Vu
curve is analysed, it can be evidenced that, following the tensile flexural cracking at the base of the
pier, relevant increases in drift actually correspond to very low increases in resistance. The strength
predicted by Equation (1) appears, thus, an asymptotic limit to which the pier very slowly tends.
It can be noted that, since in seismic codes limitations are usually posed on the drift (a typical
value is 0.8%), the actual attainment of the failure condition in the compressed toe appears asecondary requirement. It should be considered that, in the analysis presented here, a softening
parameter corresponding to an infinite ductility in compression has been adopted. A sensitivity
analysis on this parameter has been performed. It has been assessed that, even if more localized
damage occurred in toes and greater values of the ratio y/fm were obtained in correspondence
of same drift values for increasing fragility of the material, this parameter does not meaningfully
influence the overall response of the piers.
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258 C. CALDERINI, S. CATTARI AND S. LAGOMARSINO
0 0.2 0.4 0.6 0.8 1
y/fm
0
0.1
0.2
0.3
0.4
/f
m
Pier 1
0 0.2 0.4 0.6 0.8 1
y/fm
0
0.1
0.2
0.3
0.4
/f
m
Pier 2
0 0.2 0.4 0.6 0.8 1
y/f
m
0
0.1
0.2
0.3
0.4
/f
m
Pier 3 Eq. (1) -Rocking (fm= 6.2 MPa)
Eq. (3) -Diagonal Cracking (c= 0.18 MPa; =0.45)
Eq. (4) -Diagonal Cracking (fbt= 1.85 MPa)
Eq. (3) -Bed Joint Sliding (c= 0.23 MPa;= 0.58)
Eq. (5) -Diagonal Cracking (ft= 0.22 MPa)
Num. resultsRockingNum. resultsDiagonal Crackingth. joints
Num. resultsDiagonal Crackingth. blocks
Num. resultsMixed behaviour*
Figure 9. Comparison between numerical and analytical strength domains.
The evolution of the stress distribution leads to the preliminary conclusion that the choice of
the reference points/sections adopted in the criteria examined is reliable.
In what follows, the numerical and analytical strength domains, represented in Figure 9, will
be compared and discussed. In the figure, the points representing the numerical results summarize
two types of information: the value of Vu and the prevailing failure mode that occurred. Thecriteria have been plotted on the basis of the experimental parameters adopted in the numerical
analyses. The parameter ft, for which a direct experimental evaluation lacked, has been derived
from Equation (5) referring to the experimental test performed in Ispra on the pier with =1.35
[25]. In general, a good correlation can be observed from both qualitative (failure mode occurred)
and quantitative (predicted value of Vu ) points of view. For low values ofy , the failure mode
of the piers is generally classified as rocking; however, it has been observed that, for Pier 1, a
mixed rocking/bed joint sliding behaviour actually occurred, without a clear prevalence of one
over the other. For higher values ofy , in the case of Pier 1 and Pier 2, the prevailing mechanism
is diagonal cracking; increasing ofy leads to a transition from diagonal cracking through joints
to diagonal cracking through blocks. In the case of Pier 3, the prevailing mechanism is rocking,
even if for high values ofy the development of diagonal cracking has been noticed starting from
the end sections cracked in flexure (named in Figure 9 as Mixed behaviour). This circumstancehas also been observed in experimental tests [25, 27].
The numerical results find good quantitative agreement with the criteria expressed by Equations
(1), (3) (particular reference is made todiagonal crackingaccording to the interpretation given by
Mann and Muller) and (4). This accordance validates the hypothesis that the limit strength of the
pier can reasonably be predicted on the basis of the attainment of limit strength condition of the
material in only few reference points/sections. The slight overestimation of the resistance provided
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IN-PLANE STRENGTH OF UNREINFORCED MASONRY PIERS 259
by Equation (3) in the case of Pier 1 may be attributed to the adoption ofk1d=1. Although in
general the hypothesis of assuming this coefficient as a function of (with lower and upper bounds
equal to 1 and 1.5, respectively) results consistent enough, it does not seem precautionary in the
case of very squat piers. In fact, by analysing the evolution of the stress state in Pier 1, it has been
ascertained that the coefficientk1ddoes not come down to the value of 1.15.It is worth noting that the good correlation between the numerical analyses and the criterion
proposed by Mann and Muller (Equations (3) and (4)), rather than the one by Turnsek and Cacovic
(Equation (5)), may be mainly related to the good agreement between the hypotheses of Mann and
Muller and those of the constitutive law adopted. Actually, these hypotheses are coherent with the
type of masonry considered here, characterized by a regular texture and by blocks much resistant
and stiffer than mortar joints and, thus, by a clearly anisotropic behaviour. The anisotropy of the
material implies a variation of ftas a function of the stress state at the centre of the pier: for this
reason, the assumption of a constant value of ftadopted by Turnsek and Cacovic leads to strong
underestimations of the strength, in particular for increasing levels of the axial loads.
In what follows, the effect of the stress component x in the diagonal crackingfailure mode is
discussed. It should be noted that in both the constitutive law adopted in the numerical analyses
and in the Mann and Muller criteria, the strength contribution of head joints is neglected. However,
in the constitutive law adopted, the compressive stress component x influences the limit domain
by reducing the local shear stress acting on mortar bed joints (Equation (6)) and by reducing
the strength of blocks (Equation (10)), while in Mann and Mullers criterion the x component
is totally neglected. The good accordance between the numerical and theoretical results can be
explained by observing that, for the actual values ofx that occurred in the piers, the increment
in the limit strength domain associated with the adopted constitutive law of the material is small:
this is evident by observing Figure 4, in which the line associated with x=0.05fm represents an
upper bound for the cases examined (see Figure 7). The observations above lead to the conclusion
that, if the mortar head joint is neglected, the effect of the x component may not be meaningful.
Despite this, the occurrence of a compressive stress component x evidenced by the numerical
analyses, suggests that the contribution of head joints may not always be negligible. Regarding thislatter issue, particularly interesting are the results provided by Magenes and Calvi [21], obtained
by racking test on piers of two different slendernesses (=1.33 and 2) characterized by a masonry
similar to that of the Ispra tests; by analysing the results of two piers subjected to the same axial
load for which a diagonal cracking failure mode occurred, they observed that Mann and Mullers
theory underestimates notably the strength of the squat pier, for which a better prediction could be
obtained by employing the extended version of Mann and Mullers theory developed by Dialer[46]
in order to take into account the contribution of the head joints. This evidence may be explained
by the occurrence of compressive stresses x , much higher for squat piers than for slender ones
as evident from Figure 7. Finally, it is worth noting that also Turnsek and Cacovics theory does
not consider the stress componentx (Equation (5)). This assumption, as noticed also by Magenes
and Calvi [21], may lead to different values of ftas a function of the slenderness.
A final observation regards the scarce occurrence of the bed joint sliding failure mode inthe numerical simulations. The main reasons of this can be deepened by comparing the previ-
sion provided by (Equation (3)) with the one more precautionary between the diagonal cracking
(Equation (3)) and the rocking/crushing(Equation (1), withk1r= 0.5). Figure 10 (top) shows, for
piers of different geometrical and mechanical parameters, the value ofy/fm up to which bed
joint slidingprevails. It can be observed that, in general, bed joint sliding may occur only for low
values of the ratio y/fm . If the maximum ratio between the strength prevision provided by the
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260 C. CALDERINI, S. CATTARI AND S. LAGOMARSINO
(c/fm=0.05; =1) (c/fm=0.05;=0.6) (=0.6; =1)
0.5 0.75 1 1.25 1.5 1.75 2
0
0.1
0.2
y/f
m
=0.4
=0.6
=0.8
0.5 0.75 1 1.25 1.5 1.75 2
0
0.1
0.2
y/fm
=0.5
=1
=2
0.5 0.75 1 1.25 1.5 1.75 2
0
0.1
0.2
y/fm
c/fm=0.01
c/fm=0.05
c/fm=0.1
0.5 0.75 1 1.25 1.5 1.75 2
0
10
20
30
40
50
[%]
=0.4
=0.6
=0.8
0.5 0.75 1 1.25 1.5 1.75 2
0
10
20
30
40
50
[%]
=0.5
=1
=2
0.5 0.75 1 1.25 1.5 1.75 2
0
10
20
30
40
50
[%]
c/fm=0.01
c/fm=0.05
c/fm=0.1
Figure 10. Value of y/fm up to which bed joint sliding prevails on diagonal cracking orrocking/crushing and value of the ratio : effect of the friction coefficient ; effect of the
interlocking; effect of the ratio c/fm .
bed joint sliding (in the range ofy/fm in which it is prevailing) and that more precautionary
provided by the other two criteria considered is calculated (Figure 10, bottom), it can be assessed
that it is meaningful (greater than 20%, in term of percentage) only for very low values of the
slenderness (lower than unity). Thus, it can be concluded that this mechanism has little relevance
with respect to the other ones, at least in common masonry piers. It should also be considered
that, in real masonry buildings, it may be further inhibited by irregularities of the construction,such as, for example, the non-uniformity of the setting up of bed joints and the vertical axial loads
applied.
4.2. Conditions for the proper use of the criteria in the verification methods
The analyses performed highlighted the necessity of defining the limit strength domain of masonry
piers through a set of criteria. In particular, one criterion for each failure mode (rocking/crushing,
bed joint sliding, diagonal cracking) should be considered. However, in the case of diagonal
cracking, two different models have been proposed in the literature [19, 35]. Such models are
founded on strongly different hypotheses and, as previously highlighted, may provide very different
previsions of the strength. Thus, a choice between them should be made case by case, depending
on the consistency of the examined masonry with the hypotheses assumed. In particular, it is worthremarking that Turnsek and Cacovics theory is founded on the hypothesis of homogeneous and
isotropic continuum, whereas Mann and Mullers theory considers masonry as a heterogeneous and
anisotropic material in which rotations in blocks may be induced. It is well evident that a crucial
role is played by the isostropy/anisotropy of masonry. Two main features of the masonry may
influence this property: the chaoticity of the masonry pattern; the ratio between the strength/stiffness
parameters of mortar and blocks. In particular, masonry tends to behave as a homogeneous and
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IN-PLANE STRENGTH OF UNREINFORCED MASONRY PIERS 261
0 0.25 0.5 0.75 1 1.25 1.5
y(MPa)
0
0.1
0.2
0.3
0.4
0.5
(MPa)
0.35 y =
WS
0 0.25 0.5 0.75 1 1.25 1.5
y(MPa)
0
0.1
0.2
0.3
0.4
0.5
(MPa)
0.04 0.3 y = +
WI
0 0.25 0.5 0.75 1 1.25 1.5
y(MPa)
0
0.1
0.2
0.3
0.4
0.5
(MPa)
0.11 0.19 y = +
WR
Experimental results
Eq. (1) -Rocking Eq. (3) -Bed Joint Sliding Eq. (5) -Diagonal CrackingEq. (3) -Diagonal Cracking
Figure 11. Experimental tests by Vasconcelos and Lourenco [30]: comparison between the experimentalresults and the predictions provided by the discussed simplified models.
isotropic material as far as: the chaoticity of the masonry pattern is high (rubble masonry); the
ratio between the strength/stiffness of mortar joints and blocks is high (close to unity or even
greater).In order to identify qualitative rules to choose the most suitable criterion in the case ofdiagonal
cracking, two sets of experimental tests published in the literature will be analysed and compared
in the following.
The first set presented here was carried out by Vasconcelos and Lourenco [30]at the University
of Minho, where stone piers subjected to in-plane quasi-static cyclic loads were tested. The interest
in this experimental campaign derives mainly from the fact that different masonry patterns of
increasing chaoticity were adopted. With reference to Figure 11, the following types of masonry
were considered: dry stone masonry (WS), composed of blocks of regular shape and dimensions
and dry joints; irregular stone masonry (WI) consisting of hand-cut blocks with similar shape
but variable dimensions, assembled with mortar joints; rubble stone masonry (WR) composed of
blocks with variable shape and dimensions, randomly assembled with mortar. The specimens were
1200 mm high (H), 1000 mm wide (D) and 200 mm thick (T); the corresponding slendernesswas =1.2. A fixedfree cantilever scheme was considered. Different axial loads were applied:
y,1=0.5MPa; y,2=0.875MPa; y,3=1.25MPa. The materials used in the construction were
granite stone blocks and low-strength mortar.
From a qualitative point of view, the results may be summarized as follows:
Mainly diagonal cracks developed.
In WS and WI, diagonal stepped cracks through joints first developed at the corners and then
extended towards the centre of the piers. Such cracks, which may be attributed to a mixed
flexural-shear behaviour of the pier, led to a rocking mechanism about the lower corners;
for higher levels of compression and for high displacements levels, toe crushing occurred. In
some cases, for higher levels of compression, in piers WI diagonal cracks began to develop
from the centre of the pier, suggesting a more typical shear behaviour. In WR piers, diagonal cracks first opened in the middle of the piers, and then extended
towards the corners. Some of these cracks passed through the stone blocks. A typical shear
behaviour may be identified.
Figure 11 shows a comparison between the experimental results obtained and the strength
predictions provided by the simplified models illustrated in Sections 3.2 and 3.3.
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262 C. CALDERINI, S. CATTARI AND S. LAGOMARSINO
The experimental results were plotted by calculating the stresses y and as P/DTandVu/DT,
respectively. The simplified models were plotted considering the following mechanical parameters
available from experimental investigations: fm equal to 73 MPa and 18.4 MPa, for WS and WI
piers, respectively (since in the case of WR piers a direct experimental evaluation of fm lacked,
Equation (1) was not plotted); equal to 0.65 and 0.63, for dry and mortar joints, respectively; c=0.36MPa. Moreover, from a visual inspection, it was possible to identify the interlocking coefficient
for WS (=1.5) and WI piers (=2). Turnsek and Cacovics limit domain (Equation (5)) was
plotted considering a different value of ftfor each masonry pattern, calculated in mean considering
the different levels of axial loads. In the figure, a linear interpolation of the experimental data is
also plotted. Referring to experimental tests, Figure 11 shows that the relationship y/ is quite
linear for piers WS and piers WI, whereas it loses linearity for WR piers. Concerning the prediction
provided by the models, it may be observed that:
RockingEquation (1). For all the masonry typologies examined, a good agreement between
this criterion and the experimental results associated with the lower level of axial load applied
(y,1=0.5MPa) can be found; it is worth highlighting that, in these cases, rocking failures
actually occurred. Bed joint slidingEquation (3). In all the cases, it greatly overestimates the actual resistance
of tested piers; actually, this failure mechanism never occurred in this experimental campaign.
Diagonal crackingEquation (3). In general, it leads to reasonable previsions, even if with
some differences depending on masonry typology. In the case of WS piers, a light under-
estimation can be observed. It should be mainly ascribed to the assignment of k1d coef-
ficient; in fact, though the pier slenderness leads to k1d=1.2, different values could also
be justified since the actual activation of the shear failure mechanism starts from the end
sections instead of the centre of the piers. In the case of WI piers, comparing the values
ofc and assigned on the basis of the experimental investigations with the ones obtained
by the linear fitting, a significant underestimation of the mortar cohesion c can be ascer-
tained. This mismatch can be reasonably attributed to the following main factors: first, the
pattern (not perfectly regular) of this masonry can lead to a wide scatter of this param-eter; second, since the experimental points correspondent to y,1=0.5MPa clearly showed a
rocking failure, the linear fitting could be much more coherently performed by considering
only the higher values of axial load. Finally, in the case of WR piers, this criterion leads to
a quite overestimation of the actual experimental resistance. However, it can be noted that
the much more irregular pattern of WR piers could justify the assumption of a lower friction
coefficient.
Diagonal crackingEquation (5). Only the WR masonry shows a good agreement with this
criterion: in fact this masonry typology stresses a marked loss of linearity in its behaviour.
These results confirm the general conclusion that Turnsek and Cacovics criterion seems to be
more reliable for chaotic masonry piers, while Mann and Mullers model appears suitable for more
regular masonry ones.The second experimental campaign considered here was carried out at the University of Ljubljana
[27]. The main peculiarity of this experimental research is to have performed both diagonal
compression and racking tests on various masonry piers, characterized by equal blocks and masonry
pattern, but assembled with different mortars. In what follows, reference is made to tests on solid
clay brick masonry, assembled with three types of mortar: a cement mortar ( mix1), a cementlime
mortar (mix2) and a lime mortar (mix3).
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IN-PLANE STRENGTH OF UNREINFORCED MASONRY PIERS 263
0 0.05 0.1 0.15 0.2 0.25 0.3
y/fm
0
0.025
0.05
0.075
0.1
/fm
Exp. results forMix 1
Exp. results forMix 2
Exp. results forMix 3
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
y/fm
0
0.025
0.05
0.075
0.1
/fm
Eq.(1)
Eq.(3)
Eq.(4)
Eq.(5)
Exp. res. (1st Set)
Exp. res. (2ndSet)
(a) (b)
Figure 12. Experimental tests by Bosiljkov et al. [27]: (a) comparison between the experimental resultsobtained by the first set of racking tests and the prediction of Equation (5) (continuum lines are tracedby the mean value of ft; dashed lines consider its standard deviation; same shades of grey for lines andpoints correspond to equal mix) and (b) comparison between the results of the racking tests for mix 2 and
the predictions provided by the discussed simplified models.
Diagonal compression tests showed a correlation between the failure modes occurred and the
type of mortar: for mix3 a stair-stepped diagonal cracking resulted prevailing in four out of five
piers; formix 1, on four piers tested, in two cases a stair-stepped diagonal crackingoccurred while
in two others a straight path diagonal cracking through blocks developed.
Racking tests were performed on piers of 1400 mm height(H), 950 mm width(D)and 120 mm
thickness(T). A fixedfree cantilever scheme was considered. Two series of tests were carried out:
in the first, all the piers were tested under the same relative axial load (approximately 0.16 of fm );
in the second, piers made of cementlime mortar masonry were subjected to four additional levels
of compression, varying from 0.06 to 0.33 of fm . With regard to the observed failure modes, the
results may be summarized as follows: in the first set, the failure was mainly obtained by diagonal
cracking (even if differences were observed in the first phase of loading as a consequence of the
different mortars employed); in the second set, a transition between a mixed rockingbed jointslidingfailure mode to diagonal cracking was observed for increasing levels of the axial loads.
Figure 12 shows a comparison between the experimental results and the prediction provided
by the simplified models illustrated in Sections 3.2 and 3.3. The parameters adopted, directly
available from experimental investigations, are: mean compressive strength of masonry ( fm,mix 1=
14.98MPa, fm,mix 2=12.51MPa, fm,mix 3=6.93MPa); tensile strength of the blocks (fbt=
1.89MPa); mean reference tensile strength of masonry, as obtained from the diagonal compres-
sion tests ( ft,mix 1=0.46MPa, ft,mix 2=0.38MPa, ft,mix 3=0.1MPa). From a visual inspection, an
interlocking coefficient =0.6 was identified; this value may be assigned on the basis of the
block dimensions without significant uncertainties due to the masonry pattern regularity. It should
be pointed out that, for plotting Equation (3) in Figure 12(b), the value of c=0.25 has been
obtained on the basis of the bond strength tests; the friction coefficient =0.6 has been obtained
by linear regression on the basis of the experimental result corresponding to y =1.5MPa, forwhich a diagonal cracking through mortar joints was observed.
Figure 12(a) summarizes a comparison between the experimental results of the first set of
racking tests and the prediction obtained from Equation (5). In the cases ofmix 1 and mix 2, good
agreement can be noticed; on the contrary, in the case ofmix3, Turnsek and Cacovics criterion
leads to a significant underestimation of the actual maximum resistanceVu of the pier. This result
may be attributed to the fact that, in the case of mix 3, the hypothesis of isotropic continuum
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264 C. CALDERINI, S. CATTARI AND S. LAGOMARSINO
Figure 13. Different types of masonry patterns.
appears less coherent than in the two other cases, because of the strong difference between the
mechanical properties of mortar and blocks. Unfortunately, the experimental data available do not
allow us to certainly verify whether the prediction provided by Mann and Mullers model, which
appears more suitable to describe an anisotropic continuum, would be more reliable.
The results of the second set of racking tests made on mix 2 piers supply further elements to
deepen the suitability of the criteria under discussion. In Figure 12(b), it can be noted that Mannand Mullers criterion (Equations (3) and (4)), even if with uncertainties correlated with the indirect
evaluation of mechanical parameters, leads to a satisfying global interpretation of the experimental
results. Turnsek and Cacovics criterion is not in such good agreement with all the points for
whichdiagonal crackingoccurred. This statement suggests that the good agreement ascertained in
Figure 12(a) could not be considered as conclusive for mix 2. Actually, the mechanical properties
ofmix 2 do not suggest a clear tendency to an isotropic or anisotropic behaviour.
In conclusion, the problem of the choice of the most appropriate criterion describing the diagonal
cracking may be traced back to the evaluation of the degree of anisotropy of the examined type of
masonry (through, for example, the visual analyses of the masonry pattern, the assessment of the
damage pattern and the performing of mechanical tests on the constituent materials). Although in
limit cases such evaluation may appear trivial (see, for example, Figure 13, cases a and c, both
characterized by a weak mortar), in other cases it may raise many doubts (Figure 13, case b).
5. FINAL REMARKS
In this paper, a critical review of the simplified models present in the literature and codes for the
prediction of the in-plane load-bearing capacity of masonry piers subjected to seismic actions has
been proposed.
Parametrical analyses have enabled us to verify the reliability of the hypotheses on which the
models are based. With reference to the stress state assumed in the diagonal cracking, it has been
assessed that, unlike what is hypothesized by most of the criteria, thex component at the centre of
the pier is not null at failure. This is due to nonlinear phenomena and is true, in particular, for rathersquat piers. However, the entity of this stress component does not seem to appreciably influence
the overall strength; thus, the hypothesis of disregarding it appears acceptable. Furthermore, it has
been confirmed that the assessment of the stress state in few points/sections is sufficient to predict
the failure of the entire pier.
The relevance of a correct and aware use of the criteria in the verification of piers has been
stressed. The necessity of defining suitable strength domain of masonry piers through a set of
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criteria, considering one criterion for each failure mode (rocking/crushing, bed joint sliding,
diagonal cracking) has been highlighted. With reference to the diagonal cracking failure mode,
it has been pointed out that two main criteria are usually adopted [19, 35]. They are founded on
very different hypotheses and may provide very different strength previsions, which may lead, in
some cases, to severe underestimations of the strength (acceptable for the design of new buildingsbut not for the assessment of existing ones, due to the invasiveness and the expensiveness of the
resulting retrofitting interventions). For these reasons, a choice between them should be made.
The anisotropy of masonry plays a decisive role in addressing such choice. In fact, coherently
with the hypotheses adopted, Turnsek and Cacovics criterion seems more suitable if masonry
behaves as a homogeneous and isotropic material, whereas Mann and Mullers theory seems
more appropriate if masonry behaves as anisotropic material. It has been noticed that two main
parameters determine these different behaviours: the chaoticity of the masonry pattern; the ratio
between the strength/stiffness parameters of mortar and blocks. Actually, the evaluation of the
degree of anisotropy is not always an easy task, representing, in some cases, an open issue (in
particular for ancient existing buildings).
A last observation, concerning the conditions for proper use of the criteria, regards the meaning
of the mechanical parameters adopted in the criteria. An emblematic case is that of bed joints
slidingand diagonal cracking. As discussed in this paper, such very different failure mechanisms
may be reconducted to the same formal expression (Equation (3)), in which the meaning attributed
to the parameters represents the main distinctive feature.
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