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2003 ABAQUS Users Conference 1

Prediction of Collapse Behavior of Confined

Masonry Members with ABAQUS

Alberto Mandara and Domenico Scognamiglio

Second University of Naples, School of Engineering, Department of Civil Engineering

Abstract: The behavior of masonry members confined with steel tied plates and subjected to axial

compression is examined in detail in this paper. The problem is investigated by means of the

F.E.M. ABAQUS code, whose results are calibrated on the basis of experimental tests available inthe technical literature. The numerical analysis, which exploits the advanced non -linear features

of ABAQUS, is used to set up an analytical approach able to provide an useful tool for practicalcalculations. The theoretical procedure allows the evaluation of the effect of confinement,

accounting for the inelastic behavior of both masonry and steel. In spite of the reduced number of

factors the proposed model proves to be reliable in the prediction of the main behavioral aspectsof confined masonry.

1. Introduction

Confinement of vertical load-bearing masonry or stone elements for recovering existing damage or

for increasing their ultimate capacity is a practice started long ago, and widely applied still today.

The basic principle of confinement is the application of a compressive action in one or more

directions transverse to that of the applied load, so to achieve conditions of multi-axial

compressive stress. This leads not only to an increase of the member compressive strength as

respect to the case of uniaxial stress, but also to a remarkable improvement of the ductility

properties at failure. These features are very useful in all cases when higher load bearing

capabilities are demanded to the building, for example when a new use or a seismic upgrading arerequested, or when damage resulting from cracking or excess of axial deformation exists in the

members (Mandara, 2002).

From the practical point of view, transverse confinement can be applied in several ways. The most

common practice is to use metal elements working in tension, such as tie-bars or tie-beams,

fastened to the masonry by means of contrasting end-plates (Figure 1). To this purpose, most

adopted material in such interventions is steel, both mild, low carbon and high strength grade,

even though in the last years innovative materials such as titanium alloys and Shape Memory

alloys have started to be applied with interesting results (Mazzolani, 2002). Tie tension can ariseeither as a consequence of member lateral expansion, or due to external prestressing. In this view,confinement can be defined passive or active, respectively. As an alternative, a combined

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2 2003 ABAQUS Users Conference

procedure can be also used, in particular when a long-term adjustment of the confining force is

required. Confinement interventions can be applied not only to single structural elements, like

columns or walls, but also to a greater extent, by encircling the whole building with a suitable

system of tensioned members. In all cases, a quite rational structural system is obtained, in which

in-situmaterials are exploited in the most rational and effective way. In addition, when externally

fastened plates are used, the intervention can be arranged in such a way to be easily controlled or

removed, if necessary. This aspect reflects the commonly adopted policy, aiming to prevent

existing buildings, in particular when they possess monumental value, from irreversible,inappropriate restoration operations (Mazzolani, 2002).

Even though commonly recognized as an effective practice and traditionally adopted in current

applications, such provision is surprisingly not yet adequately supported neither by a convenient

theoretical assessment nor by procedures able to give a reliable prediction of the load bearing

capacity of confined masonry members. Also, aspects related to the use of materials other than

steel still deserves further investigation. This paper deals with the numerical calibration of a

theoretical procedure for the prediction of the effect of confinement. A F.E.M. model running on

the ABAQUS code is used to this purpose, which accounts for the actual inelastic behavior of bothmasonry and steel. The simulation analysis is preliminarily fitted on the basis of some

experimental results carried out on masonry specimens coming from the collapsed Civic Tower of

Pavia, Italy (Ballio,1993), arranged in such a way to provide all relevant collapse types, namely

collapse by yielding of steel bars, collapse by crushing (punching) of the masonry in the confined

area and collapse by shear-tension failure of masonry in the unconfined areas. The results coming

from ABAQUS simulation are then used to fit the parameters of the theoretical model. In spite of

some simplifications introduced into the analytical developments, the method proposed can be

considered as a first attempt to the direct evaluation of the load bearing capacity of confined

masonry. As a conclusion of the study, some general indications about the prediction of failure

type are also given.

2. Description and calibration of the F.E.M. model

2.1 Review of experimental tests

Results of 10 confining tests on ancient masonry wall specimens coming from the collapsed Civic

Tower of Pavia, Italy, (Ballio, 1993) have been assumed as a benchmark for the set up of the

numerical procedure. These specimens have been chosen for being very significant, in that theyhave been purposely conceived in such a way to achieve all possible collapse types (Figure 2),

namely: 1) collapse by yielding of steel bars; 2) collapse by crushing (punching) of the masonry in

the confined area; 3) collapse by shear-tension failure of masonry in unconfined areas. According

to tests, the main geometrical magnitudes influencing the collapse mechanism were found to be

the plate-to-bar cross section ratio (Ap/As) and the ratio between the unconfined wall length and the

wall thickness (i/t). As expected, the use of small plates (L= 40mm) gave place to failure by

punching of the masonry, whereas greater plates (L= 80mm) involved collapse by shear-tension in

the unconfined area. Comparatively high values ofAp/As, instead, involved failure by bar yielding,

which also the collapse type providing the best performance in terms of ductility. The

experimental set-up is shown in Figure 3, where 4-plate and 9-plate specimens are illustrated. The

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research pointed out the influence of confinement on both the ultimate strength and ductil ity of

compressed masonry, showing all the effectiveness of this practice and also giving some hints to

obtain a given failure mechanism.

2.2 The ABAQUS F.E.M. model

The non-linear features of F.E.M. code ABAQUS release 6.2.1 have been thoroughly exploited for

the assessment of the theoretical model proposed in the next section. As a first step, an accurate

calibration of the F.E.M. model has been made, starting from experimental data available in

(Ballio, 1993). The F.E. model is shown in Figure 4, for both 4-plate and 9-plate specimens.

Eight-node reduced integration C3D8R elements have been used. The standard material model

*CONCRETE embedded in ABAQUS has been used for the representation of masonry behavior.Because of its good accuracy in the reproduction of the progressive development of cracking, the

consequent tension stiffening effect and the actual shape of the interaction domain at failure, it can

be also used for interpreting the behavior of masonry with an acceptable degree of approximation.

In true, such behavior is much more complex than concrete due to the strong non-isotropic

behavior of masonry, as well as to the block-to-mortar interaction. The latter effect would bedominant in case of large block stone masonry, that is where block size is potentially larger than

that of confining plates, or when the quality or mortar adopted is very poor. As this would involve

a failure mechanism strongly different from that predicted assuming material isotropy, t he validity

of this calibration should be limited to relatively small sized blocks, with mortar of good quality,

so as to approach the isotropic behavior as much as possible. In this case, in fact, the main trend of

the mechanical behavior of the material is well interpreted, as also demonstrated by many studies

carried out on this subject. The behavior of steel bars has been described by means of *ELASTIC

and *PLASTIC material keys, by means of which an accurate elastic-perfectly plastic -law hasbeen reproduced, allowing for a small hardening in order to reduce numerical convergence

problems. The contact problems between the steel confining plates and the underlying masonry

have been properly taken into account, in order to consider possible slip phenomena. To this

purpose the *SURFACE INTERACTION facilities have been used, assuming *PRESSURE

OVERCLOSURE = HARD. Due to the post-critical softening in the compressive responseinvolved by any collapse mechanism, the *RIKS algorithm implemented in ABAQUS has been

used. Also, allowance for second order effects (*NLGEOM) has been made.

Material data have been inserted in ABAQUS according to the experimental measurements

reported in (Ballio, 1993). As specimens came from ancient, inhomogeneous masonry, a certain

scattering of the results was observed (Figure 5). In addition, as specimens were tested unconfined

not up to failure but at the first cracking only, a complete description of the load-displacement

curve of plain masonry was non possible, in particular in the softening branch beyond the

attainment of the ultimate load. For this reason, the effect of reloading in confined specimens was

not thoroughly quantified. Also, the relatively small size of specimens, as well as the effect of

restraints and other sub-experimental problems, added additional uncertainty to the input data to

be used in the simulation analysis. All of this involved a certain degree of inaccuracy in the

simulation of both confined and unconfined behavior, already pointed out in (Mandara, 1998). In

any case, the reproduction of the load-displacement curve is quite satisfying, with a faithfulinterpretation of the actual collapse mechanism in all cases considered. Comparison between

experimental tests and corresponding simulation analyses is shown in Figure 6. Eventually, the

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wall deformed shape predicted by ABAQUS for the three collapse mechanisms highlighted in the

tests is depicted in Figure 7, together with the contour of equivalent Von Misesstress and

horizontal displacement of the wall. Based on the distribution of such displacements, it is easy to

distinguish each collapse mechanism from each other.

3. Set-up of a theoretical model for confined masonry

3.1 Basic assumptions

The model discussed herein was initially proposed in (Mandara, 1998), concerning the case of

masonry uniformly confined along one transverse direction by means of tied end-plates (Figure

8a), and is further refined in this paper in order to examine also collapse mechanisms different

from bar yielding. In case of uniformly confined masonry, in fact, the collapse can occur due to

bar yielding only. As the model was purposely conceived for design applications, it has been

based on the assumption of a homogeneous continuum. Hence, in case of masonries with complex

texture, it requires a former application of suitable homogenization criteria in order to get theequivalent masonry properties to be introduced into the calculation procedure (Nemat-Nasser,

1999). The main assumptions remain the same as in (Mandara, 1998), namely:

the behavior of masonry is assumed isotropic;

the behavior of steel bars is elastic -perfectly plastic;

the steel confining plates are rigid, which involves the confining force to be evenlydistributed across the wall side surface; as a consequence, the steel bars can be assumed

as concentrated;

pseudo-elastic relationship between the applied stress mand the confining stresses c,xand c,yhold in both elastic and post-elastic range, which involves the use ofNavier-likeequations written in terms of secant modulusEm,s;

the behavior of masonry in compression, both confined and unconfined, and hence thesecant modulusEm,s, are described by means of an appropriate non-linear -uniaxiallaw, with experimentally fitted parameters.

The model is presented hereafter referring to unidirectional confinement only, namely the case of

masonry walls confined in the transverse direction. Its generalization to the bi-directional case,

even though quite similar conceptually is not reported.

3.2 Constitutive relationships for confined masonry

Referring to Figure 8a, the equilibrium equation along the wall transverse direction must be

satisfied:

xcpss AA , = (1)

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The global strain in the masonry in both load and transverse direction is expressed by means of

Navier-like equations written in terms of secant modulusEm,s, which means that both the elastic

and plastic part of the global deformation are taken into account at the same time:

( )( )ycxcm

sm

mE

,,

,

1 += (2)

( )( )ycmxc

sm

xcE

,,

,

,

1 += (3)

( )( )xcmyc

sm

ycE

,,

,

,

1 += (4)

whereEm,sis the secant modulus of compressed masonry. Owing to symmetry, in case ofunidirectional confinement, it may be assumed c,y= 0. In addition, when steel bars are in elasticrange, c,x= s= s/Es. Substituting c,yfrom Equation 4 into Equations 2 and 3 and consideringEquation 1 yields:

( )( )( )spssm

mxc

AAEE ,2

2

,1 +

+=

(5)

( ) ( )( )( )spssmmsm

m

AAEE

E

,

2222

,

11 ++=

(6)

whereEsis the elastic modulus of steel bars. Assuming that bothEm,sandare a function of thecompressive strain m, then Equation 6 may be considered as the -law of the confined masonry.The above equations hold until the stress in the tensioned bars does not exceed the steel yield

stressfy, When this occurs, from Equation 1 it results:

psyxc AAf /, = (7)

where the steel yield stressfyhas to be taken negative. By substituting Equation 7 into Equation 2

and considering that c,y= 0, the stress in the load direction becomes:

( )( ) ( )22, 1/ += psymsmm AAfE (8)

At the same time, from Equation 7, the stress acting in steel bars must fulfil the condition:

yspxcs fAA == , (9)

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where c,xis given by Equation 5. Equations 8 and 9 allow the wall behaviour in the steel post-

elastic range to be easily described.

3.3 Application of the method

For the above method to be applied, appropriate functions forEm,sandhave to be assigned. Thesecant modulusEm,scan be obtained starting from a suitable -relationship for plain masonry. Anumber of -relationships susceptible to be used exist in the technical literature, derived fromexperimental tests on either plain concrete, or fitted directly on masonry specimens. In this

procedure, a model derived from the Saenzs law for concrete has been proposed (Sargin, 1971):

( )

( )

+

+

=z

um

m

um

m

umum

um

m

umum

um

m

E

Ek

,,,,

,,,

,

2/1

/

(10)

whereE, m,uand m,uare the initial elastic modulus, the ultimate compressive stress and thecorresponding strain, respectively. Such a model represents a slight variation of that proposed in

(Mandara,1998) and is able to provide a good description of both pre- and post-collapse

compressive behavior. There are some differences in the model presented h erein, compared to the

original formulation of Saenzs law: 1) a strength enhancement factor kdue to confinement has

been introduced in order to take into account the increase of masonry resistance produced by the

combined state of stress; 2) the coefficientzhave been introduced instead of a numerical factor

equal to 2, in order to have a more accurate reproduction of the softening branch of the -relationship. Assuming values ofzother than 2 causes the actual maximum of the -curve to beslightly different from m,u, but gives a much better approximation of the material post-collapsebehavior. As shown in the next section, both kandzhave been found being rather dependent onboth mechanical and geometrical properties of the masonry wall. An appropriate expression for k

can be put into the form:

mycmxck ,,1 ++= (11)

where is a numerical coefficient to be fitted experimentally, which depends on the masonryfeatures. The c,x/mratio can be evaluated from Equation 5 - or from Equations 7 and 8 after thebar has yielded - as a function of,Em,s,Es,ApandAs. Similarly, from Equation 4 it is easy toshow that:

( )mxcmyc ,, 1+= (12)

The expression of secant modulus can be easily deducted from Equation 10 remembering thatEm,s

= m/m:

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z

um

m

um

m

umum

m

msm

cE

EkE

+

+

==

,,,,

,

1

(13)

In the evaluation of c,x/mand c,y/ma tr ial-and-error procedure would be necessary for theircalculation. In fact, since c,xdepends onEm,sthrough m, its value depends on kitself, too.Because of Equation 12 this happens for c,yas well. Nevertheless, it can be observed that, in theview of design calculations, the terms of Equation 11 may be computed in an approximate way

assuming k= 1 in the evaluation ofEm,s, independently of the bar yielding. The related inaccuracy

can then be covered by an appropriate value of the parameter .

Together with , in the calculation of ma suitable function for thePoissons modulus(m) hasto be assigned in order to evaluate the -relationship of confined masonry. An accurateestimation of is extremely important for the accuracy of the model, that is for a correctevaluation of the confinement effect, as the confinement mainly depends onitself.Unfortunately, the meaning of thePoissons modulus in masonry is not exactly the same as in an

elastic continuum, in particular when the collapse load is approached. The transverse expansion of

the masonry is, in fact, strongly influenced by the onset of cracks along the load direction. As a

consequence, when the masonry wall is regarded as a whole, specific allowance should be made

for cracks in the evaluation of the expansion ratio. Also, the actual masonry texture, that is the

block size and configuration as well as the mortar properties, should be considered in the

assumption of a(m) function. In this view, it is clear that a direct evaluation of the transverseexpansion ratioleads to a so-called apparentPoissons modulus, whose mechanical meaningis far different from the one of an elastic, isotropic contin uum. Some existing tests (Faella, 1993),

in fact, indicate apparent values of the coefficientat the ultimate strength equal to or higher than1.5 2, depending on the masonry features.

Such values, clearly incompatible with the physical meaning of, can not be assumed in case ofconfined masonry, as the development of cracks is greatly contrasted by confining ties. In this

case, cracks occur as well, but with a rather different aspect from the case of unconfined masonry.

In general, depending on the value of confining stress, the concept of apparent expansion drops

most of its meaning, being downsized by an unknown extent. In absence of reliable data onunder combined stress conditions, assuming that in such a case a reduction of the void volume due

to the local crushing of the masonry could take place, it seems more appropriate to assign a law for

reaching values not higher than 0.5 (no volume change) in the large displacement range. This ismore complying with the assumption of isotropic continuum made for deriving the -law. At thesame time, the condition that= 0 for m= 0 should be fulfilled, this corresponding to whatcommonly observed in tested specimens. As being stated, a possible law forcan be put into theform:

( ) ( ) ( )( )2

,2

,, //// ummummumm ba += (14)

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The coefficients , aand bhave to be fitted in order to adequately reproduce the results comingfrom either numerical simulation or direct experimentation. Since thePoissons ratio isresponsible for the lateral expansion of the masonry and, consequently, for the confining pressure

of the steel plates, the coefficients aand bcan be fitted in such a way the bar yielding occurs at the

same point as in the reality. On the contrary the factor, taking into account the effect ofconfinement on the masonry strength, can be evaluated by considering the actual value of ultimate

load bearing capacity, as deducted by experimentation or by F.E.M. analysis.

4. Parametric analysis

An extensive parametric analysis has been carried out by means of the F.E.M. model described in

the previous chapter, assuming several material properties and uniformly confined walls (Figure

8b). The analysis has been concerned with the case of both uniformly and partially confined

masonry walls, in order to investigate all possible failure modes. Three masonry types have been

considered, whose main mechanical parameters are shown in Table 1, together with relevant

values ofz, a, band , as found from a best fitting procedure of theoretical curves against F.E.M.results. A corresponding *CONCRETE material model has been used. Values of wall thickness

equal to 300 and 480mm have been considered. Different values ofAs, represented through the bar

diameter , have been assumed, so as to emphasize the effect of confining steel area. A yielding

stressfy= 600 N/mm2has been considered in the analysis.

In general, a good agreement between the proposed method and the numerical simulation is found,

in particular in the estimation of ultimate load bearing capacity (Figure 9). Some minor

discrepancies with respect to the F.E.M. results exist only around the knee point and in the

softening branch of the -curve. This is partially to be related to the -relationship assumed theunconfined material. It can be observed that, owing to the geometrical symmetry of the masonry

panel, a certain amount of confinement in the wall plane (y-direction) does exist even without

plates in thex-direction. This results in the ultimate load and the corresponding strain of the

unconfined wall being higher than those of the plain masonry. This aspect is well caught by theanalytical model when the positionAs= 0 is made.

In order to consider collapse mechanisms other than yielding of steel bars, partial confinement

with uniformly spaced plates has been also considered. The corresponding F.E.M. model, where

due account of symmetry has been taken, is shown in Figure 10. In such a case, depending on both

plate spacing andAs/Ap ratio, collapse by either local crushing or shear-tension may occur. For agivenAs/Apvalue, the corresponding i/tvalue has been assumed as relevant parameter to establish

whether the wall collapse occurs due to bar yielding, masonry punching or shear-tension failure of

masonry between plates. As long as the i/tratio increases, the corresponding wall collapse load

decreases, according to the trend shown in Figure 11, where the ratio between confined and

unconfined strengthRc/Rncis plotted against i/tratio, forAs/Ap= 50, 100 and 400. In general, for i/t

1.5, the effect of confinement vanishes completely. Correspondingly, the failure type movesfrom bar yielding to punching or shear/tension depending on the value ofAs/Apvalue. Both

experimental and numerical results show that the value of i/tandAs/Apratio determine the collapse

to occur by shear-tension or punching, respectively. A synopsis view of all possible failure

conditions is given in Figure 12, where the relevant collapse mechanisms are also indicated.

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Accordingly, a border line between them has been traced, which can give useful indication from

the design point of view. In practice, collapse conditions other than that involved by bar yielding

should be avoided, in that they cause confinement ineffectiveness and/or local crushing of

masonry, and hence, a brittle behavior at collapse for the wall.

Because of the relatively large number of simulation tests referred to in Figure 12, it can be

considered as a helpful tool in orienting design choices. Nevertheless, concerning practical

calculations, an easy tool for the prediction of collapse load for a given value of i/tratio is needed.

As a matter of fact, the solution to this problem would involve the definition of a very complex

mechanical model, taking into account all relevant aspects of the collapse mechanism. Such

difficulty arises due to the fact that in case of bar yielding or local punching the wall collapse

occurs due to masonry crushing between confining plates, whereas in case of shear-tension failure

it is predominantly a matter of local instability of outer masonry leaves. To this purpose, suitable

interaction models should be used to represent the actual collapse phenomenology. As a more

direct approach is needed for design calculations, a simplified procedure is proposed herein, based

on the combined use of results provided by both F.E.M. simulation and theoretical model for

uniformly confined masonry. In this case, results coming from the theoretical model can be usedfor i/t= 0, whereas F.E.M.-based data can be exploited to reproduce the variation of the wall

strength as long as the i/tratio increases. Curves in Figure 11 can be used to this purpose, in order

to fit a simple relationship relatingRc/Rncratio to i/tratio. The following equation is proposed:

( )[ ]( )

11

1.01

67.0

0

0 +

=

ti

ncc

ncc

nc

c

RRRR

R

R (15)

whose results are plotted in Figure 11 as well. Such equation can be used to predict the resistanceof confined masonry for a given value of the i/tratio when the resistance ratio of uniformly

confined (i/t= 0) to unconfined masonry (Rc/Rnc)0 is known. Since a certain degree of confinement

exists in walls also without ties, such value is to be calculated via the procedure illustrated in the

previous section. Information on the possible collapse mechanism is then obtained from Figure 12.

5. Conclusions

The non-linear F.E.M. code ABAQUS has been used to simulate the inelastic response of confined

masonry walls up to collapse. This has led to a thorough understanding of global behavior of

masonry in such loading conditions, highlighting all relevant collapse mechanisms. The results of

the wide parametric analysis carried out in this paper have been used to calibrate a purposely

conceived theoretical model. With respect to existing models, mostly concerned with confined

concrete in compression, the procedure discussed herein is based on a reduced number of

parameters, to be fitted on the basis of either experimentation or numerical simulation. With a

suitable choice of these factors, the model exhibits a satisfying degree of accuracy.

In addition, collapse mechanisms other than that involved by bar yielding have been investigated

by means of numerical simulation, leading to the attainment of some general conclusions on the

range of geometrical and mechanical properties to be adopted in practice. At the same time,

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starting from results obtained through the theoretical model, F.E.M. results have been further

exploited to set a simplified procedure for the evaluation of the wall load bearing capacity as a

function of confinement ratios i/tandAp/A

s.

In conclusion, an exhausting framing of the problem has been reached, at least referring to the case

of wall confinement (one-direction confinement). As a further step of the research, an extension to

the case of two-directional confinement is planned, in order to cover application to columns,

pillars, and other vertical masonry elements. To this purpose, the theoretical model has been

already formulated, even though not reported herein, and F.E.M. simulation is about to start.

6. References

1. ABAQUS, User and Theory Manual, Version 6.2, Hibbitt, Karlsson & Sorensen Inc., 2002.

2. Ballio, G., Calvi, G.M., Strengthening of Masonry Structures by Lateral Confinement, Proc.of IABSE Symp. Structural Preservation of the Architectural Heritage, Rome, 1993.

3. Faella, G., Manfredi G., Realfonzo, R., Stress-Strain Relationships for Tuff Masonry: Ex-perimental Results and Analytical Formulations, Masonry Int., Vol. 7, No. 2, 1993.

4. Mandara A., Strengthening Techniques for Buildings, in Refurbishment of Build ings andBridges (F.M. Mazzolani & M. Ivanyi Eds), Springer Verlag, Wien-New York, 2002.

5. Mandara, A., Mazzolani, F.M., Confining of Masonry Walls with Steel Elements, Proc. OfInt. IABSE Conference Save Buildings in Central and Eastern Europe, Berlin, 1998.

6. Mazzolani, F.M. Strengthening Options in Rehabilitation by means of Steelwork, Proc. ofSSRC International Colloquium on Structural Stability, Rio de Janeiro, 1996.

7. Mazzolani, F.M., Principles and Design Criteria for Consolidation and Rehabilitation, inRefurbishment of Buildings and Bridges (F.M. Mazzolani & M. Ivanyi Eds), Springer Ve r-lag, Wien-New York, 2002.

8. Mazzolani, F.M., Mandara A., Modern Trends in the Use of Special Metals for the Im-provement of Historical and Monumental Structures, Jour. of Struct. Eng. 24, Elsevier, 2002.

9. Nemat-Nasser, S., Hori, M., Micromechanics: Overall Properties of Heterogeneous Materi-als, 2ndEd., Elsevier, Amsterdam, 1999.

10. Sargin, M., Stress-Strain Relationship for Concrete and Analysis of Structural Concrete Sec-tions, Study n. 4, Solid Mechanics Division, University of Waterloo, Canada, 1971.

7. Acknowledgements

This research initially started within the project Metal Systems for the Consolidation of Structures

(resp. F.M. Mazzolani), in the framework of Progetto Finalizzato Beni Culturali issued by the Italian

National Research Council (C.N.R.). The ongoing development is now framed within the project

Innovative Metal Materials in the Seismic Strengthening of Masonry Structures (resp. A. Mandara),which is a part of the project Diagnostic and Safeguard of Architectonic Works, sponsored by Italian

National Research Council (C.N.R.) with funds granted by Italian Ministry of University and Research(MIUR) (L. 449/97).

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Table 1. Synopsis of wall mechanical properties assumed in the parametric

analysis and relevant material calibration parameters (z, a, b and ).

Masonrytype

E(MPa)

m,u(MPa)

m,u t(mm)

(mm)z a b a

1 3300 3.5 0.0025 480 8,12,16 1.5 1 2.1 0.2

2 2300 2.5 0.0025 300 8,10,12 1.8 3 2 0.5

3 660 2.5 0.007 300 4,6,8 2 3 1.8 0.5

Figure 1. Examples of masonry members confined by steel tied elements.

a) b) c)

Figure 2. Collapse types: a) bar yielding; b) masonry crushing in the confined

area; c) shear-tension of masonry in unconfined areas.

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Figure 3. Experimental set-up referred to in (Ballio, 1993).

Figure 4. The ABAQUS F.E.M. model of specimens tested in (Ballio, 1993).

0

1

2

3

4

5

6

0.000 0.005 0.010 0.015 0.020 0.025 0.030

e

s (Mpa)

h)

b)

a)

g)

0

1

2

3

4

5

6

0.000 0.005 0.010 0.015 0.020 0.025 0.030

e

s (Mpa)

f)

d)

c)

e)

l)

i)

Figure 5. The masonry -laws of specimens tested in (Ballio, 1993).

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0

1

2

3

4

5

6

0 0.005 0.01 0.015

(Mpa)

h) )

g)

a)

0

1

2

3

4

5

6

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

(Mpa)

l)

f)

d)

c)i)

e)

Figure 6. Calibration of numerical F.E.M. model (dotted line) against tests.

Collapse by bar yielding Collapse by masonry punching Collapse by shear-tension

Figure 7. Contour of Von Misesequivalent stress (top) and horizontal

displacement (bottom) for observed collapse mechanisms.

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Ap

A s

c,x c,x

m

m

t

Figure 8. The mechanical model of uniformly confined masonry (a) and the

corresponding F.E.M. idealization (b).

0.0

0.5

1.0

1.5

2.0

2.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

e m/em,u

s m/s mu

F= 16mm

F= 12mm

F= 8mm

Unconfined

Plainmasonry

Masonry type 1

0.0

0.5

1.0

1.5

2.0

2.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

em/emu

s m/s mu

F= 12mm

F= 10mmF= 8mmUnconfined

Plainmasonry

Masonry type 2

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

e m/emu

s m/s mu

F= 8mm

F= 6mmF= 4mm

Unconfined

Plainmasonry

Masonry type 3

Figure 9. Comparison between theoretical and F.E.M. results (dotted line) for

uniform confinement.

a) b)

8/10/2019 Mandar A

15/15


0

1

2

3

4

5

6

7

8

910

11

12

0 1 2 3 4 5 6 7 8 9 10

i = 100mm

i = 150mm

i = 200mm

i = 250mmi = 300mm

i = 500mm

i = 700mm

s m(MPa)

d(mm)

Ap/A

s= 50

Figure 10. The F.E.M. model in case of partial confinement (left) and the

corresponding force-displacement relationship.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 0.5 1.0 1.5 2.0 2.5

i/ t

Rc/Rnc

Ap/A s = 50

400

100

Masonry type 1

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 0.5 1.0 1.5 2.0 2.5

i/t

Rc/Rnc

Ap/A s = 50

400

100

Masonry type 2

Figure 11. Influence of i /tratio on the resistance of partially confined walls:F.E.M. analysis (dotted line) versus Equation 15 (full line).

0

1

2

3

4

5

0 1 2 3 4

(A p/A s)*(s mu/fy)

i /t COLLAPSE BY BAR YIELDING

COLLAPSE MECHANISM BY SHEAR-TENSION

COLLAPSE MECHANISM BY PUNCHING

Figure 12. Synopsis view of possible collapse conditions.

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