caldin-plane strength of unreinforced masonry piers

7/25/2019 CaldIn-plane strength of unreinforced masonry piers

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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]

Copyright q 2008 John Wiley & Sons, Ltd.


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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

Copyright q 2008 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn.2009; 38:243267

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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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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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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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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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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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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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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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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 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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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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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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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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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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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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(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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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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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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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



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


DOI: 10.1002/eqe


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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


DOI: 10.1002/eqe


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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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caldin-plane strength of unreinforced masonry piers

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