Dynamic response of reinforced masonry columns in classical monuments

Construction and Building Materials 25 (2011) 4325–4337

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Construction and Building Materials

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Dynamic response of reinforced masonry columns in classical monuments

Ioannis Stefanou a,⇑, Ioannis Psycharis b, Ioannis-Orestis Georgopoulos a

a School of Applied Mathematics and Physics, Section of Mechanics, National Technical University of Athens, 9 Heroon Polytechniou Str., Polytechnic Campus, Athens, Greeceb Laboratory for Earthquake Engineering, School of Civil Engineering, National Technical University of Athens, 9 Heroon Polytechniou Str., Polytechnic Campus, Athens, Greece

a r t i c l e i n f o

Article history:Available online 18 February 2011

Keywords:Masonry3D dynamic analysisMulti-drum columnsFractured drumsRockingWobblingDistinct Element Method (DEM)Restoration processMarble–cement interfaceDirect shear tests

0950-0618/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.conbuildmat.2010.12.042

⇑ Corresponding author. Address: School of ApplieSection of Mechanics, National Technical University oniou Str., P.O. BOX GR15780, Athens, Greece. Tel.: +3067 22 873.

E-mail addresses: [email protected] (I.ntua.gr (I. Psycharis), [email protected]

URLs: http://lee.civil.ntua.gr/ (I. Psycharis), h(I. Stefanou, I.-O. Georgopoulos).

a b s t r a c t

From the mechanical point of view, the particularity of masonry structures stems from the fact that thestructural system is hard to be modeled by the classical Continuum Mechanics approach. The problemgets more complicated when imperfections, such as cracks are present. An example of a single multi-drum column, with fractured drums, is studied herein, using the Distinct Element Method (DEM). Thepurpose of the research is the investigation of the impact of the fractures to the overall stability of thestructure. The 3D DEM numerical results are explained on the basis of simple 2D analytical consider-ations. The shear and normal crack deformation is monitored and the minimum required strength ofthe crack interface is quantified. An experimental program of direct shear tests is set in order to estimatethe strength of the marble–cement interface. The experimental values are compared to the minimumrequired from the numerical analysis.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

During the last decades an increased interest in the dynamic re-sponse of ancient classic monuments has arisen. Typical examplesof such cultural heritage assets are the classical Hellenistic, Greekand early Roman temples (Fig. 1). In certain cases this kind ofstructures may undergo intense earthquake actions without col-lapsing. The particularity of these classical monuments is that theyare constituted by many discrete bulgy stone blocks, which formdry masonry walls and multi-drum columns. The various bulgydiscrete blocks were not systematically connected together inancient times, and when they were, special metal or wooden con-nectors were used. A special element of these structures is themulti-drum column which is made of (sometimes astoundingly)perfectly fitted stone drums placed on top of each other avoidingthe use of cement (mortar) [1]. It is worth noticing that the seismicresponse of these articulated discrete structures has very little incommon with the dynamic response of modern structures, whichexhibit ‘tensegrity’ (tension + integrity) in the sense that they canbear tensile stresses and keep their integrity. The stability and

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Stefanou), ipsych@central.(I.-O. Georgopoulos).ttp://geolab.mechan.ntua.gr/

resistance of modern structures subjected to axial lateral loadsand moments is attributed to the development of internal tensileforces, while the stability and resistance of classical structures isattributed primarily to friction and is affected by their geometriccharacteristics [3,2]. This fundamental difference makes inapplica-ble most of the available structural theories and classical computa-tional tools.

Despite the lack of inter-connection among the stone elements,classical monuments are not, in general, vulnerable to ‘typical’earthquake motions, if they are in their intact condition [4]. Theirlarge ‘apparent’ period and large dimensions make them vulnera-ble only to earthquakes that are characterized by increased valuesof spectral acceleration in the long period regime. The energy dis-sipation, caused by wobbling, rocking and sliding, has also a bene-ficial effect. This good seismic behavior has been proved inpractice, since many classical monuments are standing for morethan 2000 years, although they are located in regions of extensiveseismic activity, as Greece and Italy.

The vulnerability of classical monuments to earthquakes de-pends on two main parameters: the predominant period of theground motion and the size of the structure [4]. The former signif-icantly affects the response and the possibility of collapse withlow-frequency earthquakes being much more dangerous thanhigh-frequency ones. In the first case, the response is characterizedby intensive rocking; in the latter, significant sliding of the drumsoccurs, especially close to the upper part of the structure, whilerocking is usually restricted to small values. This good seismicbehavior may be attributed to their large ‘apparent’ period, which

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Fig. 1. Poseidon’s temple at Cape Sounion, Greece.

Fig. 3. Top displacement of a model of the Parthenon Pronaos column with andwithout imperfections, for the Aigion, Greece, 1995 earthquake, scaled to severalvalues of PGA (numerical results, Psycharis et al. [5]).

4326 I. Stefanou et al. / Construction and Building Materials 25 (2011) 4325–4337

increases with the amount of rocking. The size of the structure isanother important parameter, with bulkier structures being muchmore stable than smaller ones of dimensions with the same aspectratio.

Unfortunately, imperfections are present in many monuments.They are caused by various factors as previous earthquake events,foundation failure, material deterioration or man interventions, asfire and vandalism. The most common imperfections are cut-offdrum corners, displaced drums, inclined columns and broken ele-ment stones (Fig. 2). Previous analyses [4,5] show that such imper-fections reduce significantly the stability of the structure and canlead to collapse even for middle-size earthquakes. An example ofthe significant reduction of the stability, produced by imperfec-tions, is shown in Fig. 3. In restoration practice, when brokenpieces are missing, new pieces are usually constructed by newmaterial to complete the ancient member. These new additionsfrom new material have to respect the geometry of the authenticmember and to fit ‘exactly’ to the failure surface. In this way theintervention at the ancient member is minimal. In order to jointhe additions with the ancient member, dowels and cement pasteare used, as for example, at the Acropolis of Athens and the Epidau-rus restoration sites [6], where dowels made of titanium and whitecement are used to join the additions with the ancient members.The number of reinforcing dowels used, has to be as small as pos-sible in order to preserve and not destroy the ancient member withexcessive drilling, while at the same time has to be strong enoughto bear the external actions exerted to the ancient member. The de-sign of the reinforcement has to account for shear, tension, com-pression, bending and torsion of the addition to the failure

Fig. 2. Photo of the bottom cracked drum of a column at Olympieion temple inAthens.

surface of the ancient member. This problem is a complex contactproblem, which has not been yet systematically addressed.

In the present paper we deal with the seismic response of amulti-drum column considering several cases of fractured stonedrums. The aforementioned cases are simplified representationsof flaws that are often displayed in classical monuments. The ad-vance of the present work on previous approaches is that it focusesmainly on the developed stresses at the crack interfaces and that itcorrelates the strength of the latter to the global stability of thesystem. In particular, the objective of the current analysis is theinvestigation of the effect of several parameters to the possibilityof failure of the system, such as the mechanical properties of thefailure surfaces and the position and the inclination of the cracks.More specifically, in Section 2, we firstly highlight the three dimen-sional dynamic behavior of masonry multi-drum columns, whensubjected to seismic actions. The Distinct Element Method (DEM)is a keen and powerful tool which may, when critically used, pro-vide useful results in the understanding of the behavior of suchstructures. Further, in Section 3, the numerical analysis of acracked multi-drum column is presented in detail, focusing onthe effect of different crack types to the stability of the system. Asimplified 2D analytical model is presented in order to approxi-mate the internal stresses developed in the cracked drums andthe results from the numerical and the analytical models are crit-ically compared. Numerical analyses allow for the monitoring ofthe shear and normal crack deformation. Moreover, they makepossible to derive a rough estimate of the minimum required cohe-sion and tensile strength of the crack interface. If these minimumrequired values are not assured then the structure may becomeunstable and collapse. The various failure modes of the structureare critically discussed in the same section and connected to theorientation and the position of the cracks. Finally, in Section 4, anexperimental program of direct shear tests is set, in order to esti-mate the strength of the marble–cement interface. The experimen-tal results for the mechanical properties of the marble–cementinterface are used to qualitatively and quantitatively validate therequired calculated values of cohesion and tensile strength pro-vided by the DEM analyses.

2. Modeling of multi-drum columns

2.1. Three dimensional dynamic response

The complexity of the dynamic behavior of multi-drum col-umns originates from the fact that the structure continuouslymoves from one ‘mode’ of vibration to another; different jointsare opened and different poles of rotation apply for each mode.The term ‘mode’ is used here to denote different patterns of the re-sponse (for an example we refer to Fig. 4) and does not refer to the

MODE 1 MODE 2 MODE 3 MODE 4

Fig. 4. Modes of deformation for two-block assemblies.

Fig. 5. Relative drum displacements and rotations at Poseidon’s temple, CapeSounion, Greece.

1 The method proposed by Cundall in late ‘70s is also called Molecular Dynamics(MD) or Soft Particle Distinct Element Method. Today, the term Distinct (or Discrete)Element Method (DEM) refers to a family of numerical methods that account fordiscontinuous systems of interacting deformable (or not) particles. Except theaforementioned method other DE methods are the non-smooth Contact Dynamics(CD) [64,63,65] and the Discontinuous Deformation Analysis (DDA) [62].

I. Stefanou et al. / Construction and Building Materials 25 (2011) 4325–4337 4327

eigenmodes of the system, since spinal structures do not possessnatural modes in the classical sense and the period of free vibra-tions is amplitude dependent. The dynamic response is dominatedby the ‘‘spinal’’ form of the construction and is governed by thesliding, the rocking and the wobbling of the individual, practicallyrigid stone-drums, independently or in groups.

During rocking, the pole of rotation of each block moves fromone corner of the base to the other. This transition produces impactphenomena among adjacent structural elements and energy dissi-pation, causing a sudden decrease in the angular velocities. Duringwobbling the motion is not limited in the vertical plane of rocking,but it is rather evolving out-of-plane, in the three dimensionalspace. Specifically, the drum rolls on its edge under small inclina-tion angles and, contrary to rocking, the pole of rotation does notchange abruptly from one to the other corner of its base, but it fol-lows a smooth translation along it. A quite simple and interestingexample of the aforementioned three dimensional motion of wob-bling is a coin wobbling on a desk. The significant nonlinearitiesthat appear in the mathematical modeling of such systems havechallenged many scientists and engineers until now, since theyare apparent even in the case of single-degree freedom systems(e.g. the inverted pendulum). The latter, in spite of its apparentsimplicity, is a complicated problem, which attracted the interestof researchers since the end of the 19th century [7,8]. The first at-tempt in two dimensions for the analytical treatment of the dy-namic response was presented by Housner [9] in 1963 whofocused on the rocking of a single rigid block. In the followingyears, many investigators examined the same two dimensionalproblem both analytically and experimentally producing animpressive amount of research on this subject, e.g. [4,3,10–12],which continues up to date. In three dimensions the dynamic re-sponse of a single drum gets even more complicated. Althoughworking in a different context, earlier than Housner, in 1868, Routhwas the first to formulate mathematically the non-holonomicproblem of a symmetric body by revolution that rolls on a planarsurface. Since then a significant number of papers appeared on thisthree dimensional problem, e.g. [13–18] and recently rocking wasshown by Linear Stability Analysis to be an unconditionally unsta-ble degenerated case of the general three dimensional motion ofwobbling [19]. Experimental evidence [20] and observation onmonuments (cf. Fig. 5 and [6]) corroborate the aforementioned re-sult of predominant wobbling. According to Mouzakis et al. [20]experiments on multi-drum columns under seismic excitations:

‘‘[. . .] even for purely plane excitations, significant out-of-plane dis-placements occurred. In some cases, the predominant residualdeformation of the column after the test was in a direction differentfrom the direction of the excitation. This behavior is attributed tothe round cross section of the drums, which can easily roll alongthe perimeter of their base.’’

Finally, it should be mentioned that the energy dissipationmechanism at wobbling differs from that of rocking. As well as atrocking the energy is dissipated at the impact of the drums, atwobbling the dissipation of energy is attributed also (a) to fric-tional torsion [21] at the instant of the impact and (b) to the fric-tional forces at the joints during the stick and slip motion of thedrums [19].

According to the abovementioned fundamental characteristicsof the dynamic response of multi-drum columns, the two dimen-sional (2D) analyses that are often performed in the modeling ofsuch systems should be considered under caution cf. also [22].The by hypothesis a priori omission of the out-of-plane compo-nents of motion may be misleading and inaccurate. Relativelyfew three dimensional analyses are present in literature concern-ing the dynamic response of multi-drum columns. For a literatureoverview we refer to Papantonopoulos et al. [22] by adding the re-cent announcements of Dasiou et al. [23–25] and Toumbakari et al.[26]. This is mainly due to the growing calculation cost of thenumerical analysis as the number of blocks increases. Notice thatanalytical solutions may be obtained only in simple cases, as forexample for two dimensional two-block assemblies [27]. If manyblocks are involved, it seems that the response can be calculatedonly by numerical approaches. In this analysis, the Distinct (or Dis-crete) Element Method is employed [28–30].

2.2. The Distinct Element Method

As previously stated, the deformation and failure of classicaltemples is governed by the relative movement of the blocks. Forsuch structures, ‘‘discrete type’’ analysis may be used (cf. also mi-cro-modeling [31]). In such models the structure is considered as ablock assemblage and the joints are represented explicitly. The Dis-tinct (or Discrete) Element Method1 proposed by Cundall in the ‘70s

1.65m

10.08m

(Drum) No.1:

(Drum) No.12:

Fig. 6. Multi-drum masonry column considered for the numerical model (3DEC).The geometry is inspired by the columns of Parthenon Pronaos of Acropolis Hill inAthens, Greece [5].


in the context of geomechanics [28] and later extended to threedimensional problems [32,33] leading to the code 3DEC [29], is usedin the present study. This code provides the means to apply the con-ceptual model of a masonry structure as a system of blocks, eitherrigid or deformable. Block deformability may be taken into accountby internal discretization of blocks into finite elements. However,in the present study only rigid blocks were used, as they were foundto provide a sufficient approximation and reduce substantially therun times. The system deformation is concentrated at the joints(soft-contacts), where frictional sliding and/or complete separationmay take place (dislocations and/or disclinations).

The efficiency of the Distinct Element Method and particularlyof 3DEC to capture the seismic response of classical structureshas been examined by juxtaposing the numerical results withexperimental data [22,23]. The experimental data were obtainedfrom the shaking table response of a 1:3 scale model of a columnof the Parthenon [20]. The numerical results were found in goodagreement with the experimental data, in spite of the sensitivityof the system to trivial changes of the seismic excitation and/orto the governing parameters of the system (geometry, mechanicalparameters). All the main features of the response, as the ampli-tude, the period and the residual displacements were reproducedwith sufficient accuracy by the numerical model.

It is worth noticing that the mechanical behavior of multi-drummasonry columns, differs from the behavior of typical masonrystructures, which are consisted of numerous blocks (bricks) thatare usually bonded by mortar, forming composite materials, e.g.[34–37]. Therefore, different modeling techniques are requiredfor the analysis of the present system. This is pronounced by thefact that the multi-drum structure shows a quite rich dynamic re-sponse, which is hard to be captured by conventional approaches.This also holds for the modeling in the frame of ContinuumMechanics, through homogenization (cf. also macro-modeling[31]). The ad hoc consideration of such systems as Boltzmann con-tinua is not obvious. On the contrary, the framework of Higher Or-der continuum media as, for instance, the Cosserat continuum,seems to form a basis for a more consistent description of themechanical, dynamic response of these structures under homoge-nization [38]. Moreover, the use of averaging that is inherent inmany homogenization methods is questionable for the presentapplication. The reason is that, especially in the dynamic regime,important characteristics of the discrete system may not be trans-ferred to the continuum approximation cf. [39–41]. Consequently,DEM may not be the only choice for the modeling of the discretesystem at hand, but, as mentioned above, it forms an efficientand validated manner for the study of the dynamic behavior of ma-sonry columns in classical monuments.

3. Numerical analysis of a cracked multi-drum column

Focusing on the improvement of our understanding towards theinternal stresses developed at the crack interface of a fractureddrum of a multi-drum masonry column, under seismic excitation,and the stability of the system due to a potential drum failure,numerical analysis is performed using the Distinct Element Meth-od. The three dimensional DEM model and the results obtainedfrom the numerical analysis using 3DEC software are presentedin the next paragraphs and are explained on the basis of qualitativeand quantitative considerations.

3.1. DEM model

The geometry considered for the present numerical study is in-spired by the columns of the Parthenon Pronaos of the AcropolisHill in Athens [5]. The considered column is composed by 12 stone

drums and a capital. According to the geometry of the real struc-ture, the height of the drums in the numerical model is not con-stant. Moreover, the flutes of the drums are approximated by apolygonal cross section (Fig. 6) and the cracks of the drums are as-sumed to be planar. Eight different cases (types) of cracks areexamined for each drum. Each type of crack is characterized by adifferent inclination and origin (Fig. 7).

After having defined the geometry of the system, a quite impor-tant factor for the numerical analysis is the selection of the appro-priate constitutive laws that govern the mechanical behavior of thejoints of the model. In the present paper we make use of a Coulombtype failure criterion. In Table 1 we define the friction angle, thecohesion, the ultimate tensile strength and the stiffness of thejoints. Notice that the stiffness affects significantly the results ofthe numerical analysis [26]. Therefore, the values presented in Ta-ble 1 are selected according to Psycharis et al. [5], where an exper-imental investigation was performed in order to assess the exactnumerical values of the aforementioned parameters. Finally, weshould mention that the presence of the wooden connectors be-tween the drums, the so-called ‘empolia’, was not considered inthe numerical model, as their impact in the dynamic behavior ofthe system is usually quite weak and insignificant.

The elastic normal and shear stiffness, used to describe themechanical behavior of the crack joints, are the same with the val-ues used for the drum joints. As it will be shown in the next para-graphs, two cases are distinguished concerning the ultimateresistance of the crack joints. In the first case, the joints are as-sumed to be purely elastic (infinite shear and normal strength),while in the second case the joints are assumed to be frictional,i.e. no tensile strength or cohesion is defined (see Table 1). This willallow, in the first case, a rough estimate of the minimum requiredcohesion and tensile strength of the crack joints, when, in the sec-ond case, will allow the investigation of the impact of the fracturesto the overall stability of the structure.

Apart from the friction at the contacts of the blocks, there areseveral additional factors that are responsible for the dissipationof the mechanical energy, but are not considered in the presentDiscrete Element model. Such mechanisms are the impact of thedrums, the mechanical degradation of the materials during theseismic response (e.g. local breakage of the flutes, exfoliations[42]), viscous behavior (cf. Rice and Tse [43]), the air resistanceby the environment etc. An indirect, but artificial way to capturethe various dissipation mechanisms is damping. In the present

θ=22,5° θ=45° θ=67,5°

θ=22,5° θ=45° θ=67,5°

vertical crack

diagonal crack

C3.3C3.2C3.1

C1.3C1.2C1.1

C2.2C2.1

Fig. 7. Crack types (Cn m) at the drums of the masonry column considered in the numerical investigation. The cracks are considered planar.

Table 1Constitutive parameters for the Coulomb elastoplasticmodel considered for the mechanical behavior of thejoints of the numerical model.

Normal stiffness 1 GPa/mShear stiffness 1 GPa/mFriction angle 37�Cohesion 0 MPaTensile strength 0 MPa


model a mass-proportional Rayleigh damping of amplitude 10%and 20% at 0.3 Hz was introduced in the numerical model 10 and30 s respectively after the end of the strong motion part of the seis-mic excitation. During the seismic excitation no damping was ap-plied to the system. In this way the free vibrations of the systemdecay and its residual deformation may be accurately obtained[22]. Notice, that the use of stiffness-proportional damping usuallyappears to be more realistic regarding the rocking motion of thedrums, than the mass-proportional damping that was appliedherein [44]. However, the former requires extra computational ef-fort, which would be unnecessary for the purpose of the currentresearch.

The seismic action of Lefkada, Greece (2003) earthquake [45]was applied to the base of the numerical model by prescribingthe two horizontal components of the motion. Lefkada is a Greekisland situated at Ionian Sea, which suffered from the aforemen-tioned quite strong earthquake (damages of properties, rock falls,landslides, soil liquefaction, subsidence, ground cracks etc.). The

Fig. 8. Accelerogram and elastic response spectra of the lon

epicenter was located in the Ionian Sea, about 30 km east-north-east of the town of Lefkada. The records show a relatively longduration of the seismic event with many cycles (Figs. 8 and 9),which is the reason for using this earthquake for the numericalanalyses. The magnitude of the earthquake was Mw = 6.2 with apeak ground acceleration of PGA = 0.42 g.

3.2. Simplified analysis of a cracked drum

In this paragraph we present a simplified pseudo-dynamic twodimensional analysis of a three dimensional cracked drum sub-jected to in plane (x, y) actions (Fig. 10). The crack plane passesthrough point (2) with inclination angle h. Let us assume that thedrum is a cylinder with diameter D = 2R and height H. The crosssectional area A of the crack and the moment of inertia Izz are pre-sented in Fig. 11 as functions of the inclination h, for the case ofD = 1.6 m and H = D/2 = 0.8 m.

As it is shown in Fig. 11, the cross sectional area A and the mo-ment of inertia Izz of the cracked drum increase with angle h up tothe critical value of tan hcr = H/D = 0.5) hcr ’ 27�. At hcr the crosssection area and the moment of inertia Izz reach maximum values.As the angle h increases, the plane of the crack tends to becomevertical and in the limiting case of hmax = 90� the cross section van-ishes. In the simplified 2D analysis, two forces are exerted on theextreme edge of the drum (upper right corner, Fig. 10). This is aconservative simplification in the current analysis due to the wob-bling and/or rocking motion of the drums. The vertical force, Fv, is

gitudinal component of the Lefkada 2003 earthquake.

Fig. 9. Accelerogram and elastic response spectra of the transverse component of the Lefkada 2003 earthquake.

θ

VF

HF

(2)

(1)

2D R=

H

x

y

z

Fig. 10. Sketch of a fractured column drum, separated in two blocks. The crack isconsidered planar. The vertical and horizontal forces are acting at the extreme edgeof the drum in order to account for the wobbling and rocking motion of the column.

Fig. 11. Cross sectional area A and moment of inertia Izz of the crack cross section asfunction of the crack inclination h. D = 1.6 m and H = D/2 = 0.8 m.

Fig. 12. Normal stresses developed on the crack interface at points (1), (2) andmean normal stress as functions of the inclination of the crack plane calculated (a)analytically by the 2D simplified pseudo-dynamic approach (lines) and (b)numerically using 3D DEM for Drum No. 2 of column shown in Fig. 6.

Fig. 13. Mean shear stress developed on the crack interface as function of theinclination of the crack plane calculated (a) analytically by the 2D simplifiedpseudo-dynamic approach (solid line) and (b) numerically using 3D DEM for DrumNo. 2 of column shown in Fig. 6 (points).


considered to be equal to the total weightP

Wi of the drums, lay-ing above the reference drum, which is a reasonable approximationin the case of static loading. Notice that in dynamic conditions, thevertical load may be exceeded. Assuming a Coulomb type frictionlaw for the contact, the horizontal force is limited by friction:Fmax

h ¼ lFv ’ 0:75Fv (see Table 1). In this configuration the out of(x, y) plane forces and moments/torsions are neglected. Assuminglinear stress distribution at the crack interface, we derive analyti-cally the expressions of the normal stresses at points (1) and (2)and the mean normal stress (see Fig. 10). In Fig. 12 we presentthe variation of the normal stresses r1, r2 and the mean normalstress rm of the cross section as a function of h, for Fv = 353.5 kNand Fh = 265.1 kN that correspond to Drum No. 2, according toFig. 6. In the same figure, numerical results from 3D DEM analysesare also plotted (elastic crack interface behavior) for the seismic

event of Lefkada 2003 and the maximum and minimum values ofnormal stresses are plotted against the angle h. Regarding the shearstresses developed at the crack, we assume, in a first approxima-tion, a uniform distribution along the cross section. This approxi-mation may be justified considering rigid blocks and excludingtorsion at the interface. However, the numerical analysis showsthat the shear stresses developed at the extreme vertices of thecrack can exceed the aforementioned mean shear stress. Fig. 13presents the analytical and the numerical results for Drum No. 2as far as shear stresses are concerned.

Similar results are obtained for Drum No. 5 and Drum No. 10and are presented accordingly in Figs. 14–17.

Fig. 14. Normal stresses developed on the crack interface at points (1), (2) andmean normal stress as functions of the inclination of the crack plane calculated (a)analytically by the 2D simplified pseudo-dynamic approach (lines) and (b)numerically using 3D DEM for Drum No. 5 of column shown in Fig. 6 (points).


Fig. 16. Normal stresses developed on the crack interface at points (1), (2) andmean normal stress as functions of the inclination of the crack plane calculated (a)analytically by the 2D simplified pseudo-dynamic approach (lines) and (b)numerically using 3D DEM for Drum No. 10 of column shown in Fig. 6 (points).



It is worth pointing out that the above results demonstrate thatboth simplified 2D pseudo-dynamic analytical and 3D dynamicnumerical analyses are important for the estimation of the stressesdeveloped at the crack interface. The simplified 2D analytical solu-tion allows for a rough estimate of the stress distribution along thecrack plane. Nevertheless, it may underestimate the normal andshear stresses developed at the crack plane due to dynamic and

three dimensional effects (e.g. torsion). On the other hand, 3DDEM numerical analyses provide more accurate and detailed re-sults, but require a greater range of input data and parameters,which are not always possible to calibrate.

3.3. Required strength at crack interface

From the structural point of view, it is quite interesting to eval-uate the minimum required strength of the crack interface in orderto avoid breakage. For that reason additional DEM analyses areperformed considering the crack interface to behave in the elasticregime (tensile and/or shear failure are prevented), which permitsthe monitoring of the maximum tensile and shear stresses devel-oped at the crack.

For the numerical analysis, Drum No. 2, No. 5 and No. 10 are se-lected as they sustain different levels of axial static loading, i.e. thevertical load applied on Drum No. 2 is Fð2Þv ¼ 353:5 kN, on Drum No.5 is Fð5Þv ¼ 240:6 kN and on Drum No. 6 is Fð10Þ

v ¼ 94:7 kN. In thecontext of a Coulomb type frictional crack interface (frequent con-stitutive behavior for such type of crack joints), it is possible todetermine the minimum required cohesion, creq, in order to avoidshear crack failure. In Fig. 25 we present the calculated requiredcohesion, by assuming marble to marble friction angle (Table 1)and considering the Lefkada 2003 earthquake as case study.Fig. 18 shows that shear failure always would occur without cohe-sion. As it is intuitively expected the required cohesion increasesfor the lower drums. Considering the tensile crack opening, a min-imum tensile resistance, rt,req, is also required, which is equal tothe maximum tensile stress developed at the crack plane duringthe 3D DEM analysis. In Fig. 19 the abovementioned minimum re-quired tensile strength is given for the example of Lefkada earth-quake for all eight different modes of crack.

According to the numerical results a minimum cohesion,creq � 0.6 MPa, and tensile strength, rt,req � 0.5 MPa, is needed inorder to avoid crack opening and/or shear failure for the case studyat hand. Notice, that the reinforcement of the fractured drumshould be able to sustain these values of cohesion and tensilestrength. Otherwise, there is a potential risk of loss of structuralstability, which is examined in the next paragraph.

3.4. Loss of stability related to crack failure

The overall stability of the structure is directly influenced by thepresence of imperfections. Cut-off drum corners, displaced drums,inclined columns or broken element stones are common imperfec-tions that may lead to a local or global loss of stability and collapseof the structure (Fig. 3 [5]). In this paper we focus on the dynamic

Fig. 18. Required cohesion calculated by 3D DEM simulations for the eight cases ofcrack orientation, considered herein, and for Drums Nos. 2, 5 and 10.

Fig. 19. Required tensile strength calculated by 3D DEM simulations for the eightcases of crack orientation, considered herein, and for Drums Nos. 2, 5 and 10.


behavior of multi-drum masonry columns, which include fracturedblocks. One of the objectives of the present research is the investi-gation of the impact of the cracks to the overall stability of thestructure. For this reason, the crack opening and sliding is moni-tored for the seismic event of Lefkada 2003, which was selectedas case study.

As described in the previous sections, eight types of cracks wereintroduced into the system (Fig. 7) at different elevations. In Partic-ular, Drums Nos. 1, 5 and 11 were chosen as reference drums. Inthe numerical model the joint crack was allowed to open and slide

FM-I FM-III FM

Fig. 20. Examples of failure modes (FM) due to cracks introduced at different orientatiopresented in this figure as the excessive crack opening is of several centimeters and can

under friction (Table 1). No tensile strength and cohesion wereconsidered in the calculations leading to either collapse of the sys-tem, significant relative displacements of the blocks, or completedetachment of them from the column (Fig. 20). Five types of failuremodes (FM) are distinguished. These are FM-I, which correspondsto the detachment of blocks from the column and has a local char-acter as it does not necessarily provoke the collapse of the system;FM-II and FM-III, which are characterized by quite large relativedisplacements of the blocks forming the fractured drum, leadingto excessive opening (FM-II) and sliding (FM-III) of the crack joint;FM-IV in which the column obtains a significant inclination makingit potentially unstable in future seismic events and finally FM-V,which corresponds to the collapse of the column.

In Table 2 we present the results from the numerical analysis. Acareful examination of the numerical results reveals that not all ofthe cases of cracks lead to failure and that the failure modes areconnected to the type of the crack that cuts the drum. Three factorsare of particular importance regarding the stability of the system:(a) the inclination of the crack relative to the friction angle of thecrack interface, (b) the extent of the reduction of the area of thebase of the drum situated on top of the fractured drum, due to apotential failure of the latter and (c) the ability of the blocks ofthe fractured drums to displace (keyblocks). For instance, forDrums Nos. 5 and 11, the wedges defined by cracks C3.2 andC3.3 fail under self weight (detachment) due to the fact that theinclination of the cracks exceed the friction angle of the crack inter-face. However, for Drum No. 1 and for the same cases, the wedgesare blocked by the horizontal base and failure is avoided. But evenif the wedge detaches as in the case of Drum No. 5 and C3.3, thisdoes not necessarily lead to global loss of stability, i.e. collapse,as the reduction of the area of the base of the above placed drumis limited (for the selected geometry and seismic excitation).

The maximum and the residual crack opening and sliding dur-ing and after the dynamic excitation are recorded and presentedin Table 2 in mm, for the cases of cracks where the column didnot collapse. It should be mentioned that the residual values arequite small compared to the opening and sliding of the crack dur-ing the seismic event. However, they cannot be neglected as theyalter the geometrical configuration of the system making it weakerin future seismic events.

4. Application to reinforced drums of marble

Recent techniques [46] in the restoration process of ancientmonuments include the use of cement paste and dowels as rein-

-IV FM-V

ns and positions along the multi-drum masonry column. Failure mode FM-II is notnot be clearly depicted.

Table 2Numerical results of the maximum and residual opening and sliding of the cracksduring and after, respectively, the seismic excitation of the multi-drum masonrycolumn. The failure mode is indicated in the case of failure.

Case Lefkada earthquake

Drum No. Crack type Failure mode Crack opening and sliding (mm)

MaxUn ResUn MaxUt ResUt

11 C1.1 V5 C1.1 III 60.5 0.0 260.1 260.01 C1.1 III 22.1 0.0 114.7 114.6

11 C1.2 V5 C1.2 IV 43.0 5.7 211.5 211.41 C1.2 IV

11 C1.3 – 109.2 13.0 65.2 9.55 C1.3 – 60.3 0.2 60.9 37.61 C1.3 – 19.1 12.1 12.9 7.7

11 C2.1 V5 C2.1 V1 C2.1 IV

11 C2.2 – 30.4 17.9 66.2 11.35 C2.2 – 49.9 40.7 35.4 34.71 C2.2 – 59.5 0.0 22.6 14.5

11 C3.1 III 61.9 10.5 517.8 517.55 C3.1 III 29.9 0.1 147.4 147.31 C3.1 III 26.4 17.1 61.5 61.1

11 C3.2 V5 C3.2 V1 C3.2 II and III 62.6 44.7 57.9 56.3

11 C3.3 N/A N/A N/A N/A N/A5 C3.3 I1 C3.3 – 16.9 16.9 17.0 8.9


forcement elements. Here we focus on the beneficial effect of thecement paste by isolating its contribution to the mechanicalstrength of the reinforced element (cracked column drum, frag-mented beams, architraves, capitals etc.). The use of cement pasteis twofold: On one hand it increases the adhesion (cohesion) of thecracked interface and on the other hand it anchors the dowels(reinforcement) placed between the two joining members. Cementpaste is also favorable in restoration works as a filling material, dueto the fact that is can be relatively easily removed, without causingsignificant damage to the ancient material, compared to the use ofdowels. However, the vulnerability of cement paste due to envi-ronmental actions is an open subject. Moisture diffusion andchemical weathering are quite important degradation factors thatshould be further investigated. For aesthetic reasons, in the major-ity of restoration works performed on classical monuments, whitecement paste is used [47]. The reason is that the many historicallyimportant and famous monuments are made of marble blocks (cf.among others the temples of Parthenon and Olympieion in Athens

5 pla

mm tes

marble

Fig. 21. Typical specimen consisting of two 5 mm Dionysos marble plates cemented tthickness is 2 mm.

and the temple of Poseidon in Cape Sounion). White portland ce-ment or white ordinary portland cement (WOPC) is similar to or-dinary, gray portland cement in all respects except for its highdegree of whiteness.

Typical constituents of portland cement are calcium oxide (CaO)by 61–67%, silicon oxide (SiO2) by 19–23%, aluminum oxide (Al2O3)by 2.5–6%, ferric oxide (Fe2O3) by 0–6% and sulfate (S) by 1.5–4.5%.Cement sets when mixed with water by a way of complex series ofchemical reactions that are still not completely understood. Thestrength of cement is owed to the interlocking of its constituents,which slowly crystallize during the setting process. Carbon dioxideis slowly absorbed to convert the portlandite (Ca(OH)2) into insol-uble calcium carbonate. There are different standards for classifica-tion of Portland cement. The two major standards are the ASTMC150 used primarily in the US and the European EN-197. The Euro-pean EN-197 cement types CEM I, II, III, IV, and V do not correspondto the similarly named US cement types.

In the restoration design process, the knowledge of the mechan-ical parameters of the cement paste is of vital importance. Theoverall strength of the system is much depending on the mechan-ical properties of the material found at the interface. For instance,the resistance of a contact between two pieces of a cracked marbledrum is depending on the dowels used to keep the two parts to-gether, on the cement used for the anchoring of the dowels andon the cement layer placed at the interface. There are many andquite important publications studying the mechanical behavior ofthe cement paste itself [48–52]. However, this is not the case forthe cement–marble interface, whose mechanical properties arenot yet thoroughly investigated, even though it seems to be weakerthan the cement paste [53,46,54]. The failure mode of a crackeddrum, consisting of two pieces connected with cement paste, is ashear plane passing through their interface. Therefore, themechanical behavior of the above system depends primary onthe mechanical properties of the thin shear layer, which is sub-jected to bending, shearing and torsion [21]. Closer investigationof the failure surfaces reveals that the shear plane does not passthrough the thin layer of the cement paste placed between thetwo pieces, but coincides with the surface of the one of the twopieces. In other words, shearing takes place just at the interfaceof the thin cement layer and the one of the two pieces to be con-nected. For the aforementioned reasons, a series of experimentaltests was dedicated to the investigation of the mechanical strengthof the contact interface between the marble and the cement paste.

4.1. Experimental program and procedure

In order investigate experimentally the mechanical behavior ofthe marble–cement paste interface, a set of direct shear tests was

2 pa

mm ste

celaye

mener

t

o each other. The cement paste conforms to CEM I-52.5 N standard and its layer

Shearing plane

N

T

T

~12 mm

~60 mm

T

T~30 mm

N

cement paste layerof 2 mm thickness

Fig. 22. Top and side view schematic of the specimens.


performed. The experimental program focused on the frictionalbehavior of the marble–cement paste interface. The mechanicalstrength of the interface was quantitatively assessed according tothe direct shear tests performed for that reason in the Laboratoryof Geomaterials, at NTU Athens. The specimens used are shownin Fig. 21 and are schematically presented in Fig. 22.

The specimen consists of two marble square plates (60/30 mmlength/width and 5 mm thick) cemented to each other (Fig. 22).The cement paste layer thickness is equal to 2 mm. The water/ce-ment ratio is 0.5 [55]. The cement used is a white portland cementthat conforms to CEM I-52.5 N standard and it is a typical choice inrestoration works. The marble plates were cut from a larger blockof Dionysos marble. Dionysos marble is used as a substitute of thePentelicon marble from which the ancient monuments of theAcropolis Hill were built. For a detailed study of the mechanicalproperties of Dionysos marble we refer to Vardoulakis et al.[56].

Table 3Experimental program.

Specimen Thickness(ts) (mm)

Cross sectional Area(A) (mm2)

Mass(m) (g)

Normal force(N) (kN)

MC1 12.25 1590 51.2 1.90MC4 12.50 1647 52.4 1.00MC5 12.50 1566 49.5 2.50MC6 12.25 1610 50.6 2.50MC7 12.25 1581 50.1 2.20MC10 12.25 1609 50.6 1.60

Fig. 23. Direct shear apparatus in the Laboratory of Geom

Special care is taken during the preparation of the specimens inorder to avoid air from being trapped between the two marbleplates. Guides were used to ensure parallelism between the twomarble plates and constant cement paste layer thickness. Excesscement paste was carefully removed in order to avoid the creationof notches and to minimize the possibility of stress concentrations.The specimens were cured in a humid (100%) environment at con-stant temperature (around 22 �C). Testing took place 28 days afterspecimen preparation. Table 3 summarizes the testing program.

In Fig. 23, the VJT 2760 ShearTest apparatus, used in the currentexperimental program, is depicted. The apparatus is strain con-trolled and applies constant horizontal strain rates, from 0 to10 mm/min. In the present case, a strain rate of 0.5 mm/min wasselected. The horizontal and vertical displacement and forcesdeveloped during the tests were monitored by transducers.

The above described direct shear apparatus applies normal andshear stresses on a specified surface. This surface is pre-defined bythe geometrical dimensions of the apparatus and the specimen.Special care regarding the positioning of the specimens is given,so as the shear forces pass through the cement paste layer. Itshould be mentioned, though, that the way that the shear forcesare applied by the apparatus to the specimen (Fig. 22) may causethe development of undesired bending moments on the shearingsurface, which, eventually, may lead to the tilting of the specimenif the applied normal force is of small extent (rigid-body motion).In general, the eccentricity of the applied shear loading and conse-quently the development of bending moments at the interface areinfluenced by the thickness of the specimens and the applied nor-mal force. In the present case the values of the applied normal

aterials (left) and MC5 specimen arrangement (left).


forces and specimen thickness (Table 3) are properly selected in or-der to minimize bending and to avoid the abovementioned rigid-body motion.

Measurements of shear and normal elastic constants must beperformed in a modified direct shear type apparatus [57], whereon-sample transducers are installed. The reason is that the rigidityof the loading frame must be at least 5–10 times greater than thespecimen’s. Care must be also given to keep the measuring accu-racy of the applied loads and displacements (position and value)high. Notice, that the standard apparati used in Experimental RockMechanics do not always fulfill the desired accuracy needed forthis kind of experimental studies.

4.2. Experimental results

All the specimens are tested up to failure. Qualitatively, themechanical behavior of the marble–cement interface indicates abrittle-like behavior. After failure the marble plates are totally sep-arated from the cement paste (Fig. 24), showing a complete deb-onding. In Fig. 25 we present a magnification of the marblefailure surface, where no cement paste remains. Therefore, theweakest element of the system marble–cement–marble is theinterface of the two different materials. However, this is not anastonishing result and has been noted in previous studies concern-ing pull-out tests [46,54].

It should also be noted that the surface of the marble plates areextremely smooth and smeared due to the specimens’ preparationprocedure. Hence, the adhesion of the cement paste to the marbleis rather attributed to micromechanical and chemical bonding andnot to the macroscopic roughness of the marble surface. Notice,that in restoration practice the marble surfaces are deliberatelyroughened in random directions in order to increase the adhesionbetween the materials (Fig. 25). However, the beneficiary role ofthis artificial macro roughness is neglected here in order to obtain

Fig. 25. On the left: Magnified segment of the marble failure surface. No remaining cemroughened surface during restoration works at the Acropolis Monuments (detailed viewmarble surface.

Fig. 24. Typical failure and debonding of the cement

a lower bound of the shear strength of the marble–cement inter-face. The ultimate normal stress, ru, and shear stress, su, are de-fined as follows:

ru ¼Nu

A; su ¼

Tu

Að1Þ

where Nu, Tu are respectively the normal and shear forces at failureand A is the cross sectional area of the specimen.

In Fig. 26 we present in the deviatoric plane the stress stateswhich correspond to the failure of the marble–cement interfaceunder direct shear. For low normal stresses, it is reasonable to as-sume a Coulomb type failure criterion [50,51]:

su ¼ ru tan uþ c ð2Þ

where / and c are respectively the friction angle and the cohesion ofthe marble–cement interface. Linear regression of the experimentaldata yields / = 37.0� and c = 1.55 MPa. Note that the frictional angle/ is comparable to the frictional angle of the cement paste [50] andof the interface of marble–marble [58,5]. It should be mentioned,though, that the mechanical behavior of cement paste is generallynon-linear [52]. Hence, additional experimental tests under med-ium and higher normal stresses should be performed before estab-lishing design criteria for engineering practice. However, under lowand relatively medium normal stresses, the frictional behavior ofthe interface marble–cement is described by Eq. (2).

4.3. Parameters of the mechanical strength of marble–cement pasteinterface

The above presented experimental program allows for the esti-mation of the cohesion c and angle of friction / of the interface be-tween marble and cement paste. Three dimensional numericalsimulations performed in a cracked multi-drum column result inthe calculation of the minimum required cohesion and tensile

ent paste is observed (MC6 specimen). On the right: Example of the deliberately). The increased roughness contributes to the adhesion of the cement paste to the

paste from the marble plates (MC5 specimen).

Fig. 26. Values of normal and shear stresses at failure. Under low and relativelymedium normal stresses, the frictional behavior of the marble-cement interface isdescribed by a Coulomb failure criterion.


strength of the crack interface. By assuming that the friction coeffi-cient / is constant in the tension regime, we may estimate the ten-sile strength of the marble–cement interface: rt 6 c= tan u’ 2:0 MPa. However, this tensile strength, as roughly calculatedfrom the Coulomb friction type law may not be close to reality. Atension cut-off may be more appropriate in the tensile stress re-gime. Therefore, in a first approximation half of the tensile strengthmay be adopted, i.e. rt � 1.0 MPa. Nhis value seems a reasonableestimation according to van der Pluijm’s quasi-static experiments[59] regarding the tensile strength of the interface between highstrength wire cut clay bricks and thin layer mortar (values rangingfrom 1.57 to 3.06 MPa). Notice that actual peak tensile strength ofthe interface may be inferior to the aforementioned quasi-static va-lue in dynamic loading [60]. However, a detailed experimentalanalysis focusing on the mechanical behavior of the marble–ce-ment interface under combined tension and shear should be per-formed in the future.

Having in mind the above experimental values for the cohesionand tensile strength it seems that the crack would not fail for thecase study of Lefkada earthquake and for drums No. 2, No. 5 andNo. 11 (see Figs. 18 and 19). In a restoration process of a crackedmulti-drum column, this could well mean that the dowels usedas reinforcement are not necessary. The cement paste itself, havingthe above mentioned mechanical properties, may resist to theshear and tensile stresses. However, the mechanical behavior ofthe cement paste due to external actions, such as environmental,physical or chemical degradation along time (moisture diffusion,cement deterioration, temperature variations, crack propagation),still remains an open to research question.

5. Discussion and conclusions

The discrete nature of masonry structures differentiates theirdynamic behavior from modern structures, which exhibit ‘tenseg-rity’. From the modeling point of view, the particularity of masonrystructures and especially of masonry multi-drum columns isattributed to the fact that the structural system cannot be modeledby the classical Continuum Mechanics approach. Higher Order con-tinua are needed in order to model the mechanical behavior ofsuch kind of structures [39,40,61]. The mechanical problembecomes even more complicated when imperfections, such ascut-off drum corners, displaced drums, broken element stonesand fractures are present in the ancient monuments. In these cases‘‘discrete type’’ of analyses are more convenient. A numericalmethod that is keen in modeling relatively large systems ofparticles (blocks), is the Distinct Element Method. DEM has alreadybeen applied for the 3D dynamic modeling of multi-drumcolumns with quite successful and promising results compared

to experimental evidence. Therefore, it is a reasonable choice forthe objectives of the present research and the increased calculationcost of the dynamic analyses.

Based on the geometrical configuration of a column of Parthe-non Pronaos of Acropolis Hill in Athens, several DEM simulationswere performed. In each one of them a different crack was intro-duced to one of the drums of the column. The fundamental param-eters of the cracks that are related to the global stability of thestructure and the internal stresses developed at the crack interface,are the inclination of the crack plane and the position of the frac-ture. During the simulations it was possible to monitor the normaland shear crack deformation and to quantify the minimum re-quired strength of the crack interface. The numerical results wereexplained on the basis of simpler 2D analytical pseudo-dynamicconsiderations. It was then revealed that the dynamic three dimen-sional effects are quite important to be neglected. Actions such astorsion should not be ignored in analytical approximations, as theywould inevitably lead to underestimate the true actual stressesdeveloped at the crack interface. In addition, the dynamic effectsof seismic actions cannot be accurately captured by pseudo-dy-namic loading, because wobbling and rocking may increase theinternal forces at the contacts of the drums.

A parallel result of the numerical analyses was the calculation ofthe minimum required cohesion and tensile strength of the crackinterface. This was possible by assuming elastic interface and mon-itoring the developed shear and normal stresses at the crack. Aminimum cohesion, creq � 0.6 MPa, and tensile strength,rt,req � 0.5 MPa, was found to be necessary in order to avoid crackopening and/or shear failure for the case study that was analyzed.If the mechanical parameters of the interface were inferior to theaforementioned values, then a potential risk for loss of structuralstability would exist. According to Table 2, there are certain caseswhere local or global loss of stability is remarked. Notice, thatthe detachment of blocks or the collapse of the columns (even par-tial) may become very dangerous in touristic places. However, onthe other hand, excessive reinforcement of the ancient membersin order to ensure stability is not a well-established practice as itdamages the ancient material and it is opposed to the principleof reversibility of the interventions.

Cement paste is often used as filling material in the restorationof classical monuments consisted of marble. Yet, the mechanicalproperties of the marble–cement interface are not well explored.For that reason, an experimental program of direct shear testswas set in order to estimate the strength of the marble–cementinterface. In the context of a Coulomb failure criterion, the cohe-sion was determined equal to 1.55 MPa and the frictional angleof the marble–cement paste material was found to be equal withthe one of the marble–marble interface. Therefore, the frictionalstrength of the marble–cement paste interface plays an importantstructural role. The adhesion it offers is not negligible and perhapsit could be considered in order to reduce the number of reinforcingdowels and the excessive boring of the brittle, stone members ofmonuments. However, it still remains an open question, whetherthe additional strength offered by the cement paste, should be ta-ken into account. Future work should focus on the vulnerability ofthe cement paste to physical and chemical actions and to thedevelopment of new materials, such as self-healing materials,which allow a friendlier, to the ancient member, restoration de-sign. The reduction of the reinforcement percentage is quite impor-tant in the case of ancient monuments, where preservation of theancient material is a priority.

Acknowledgements

The research leading to these results was supported by theEuropean Research Council under the European Community’s


Seventh Framework Programme (FP7/2007-2013)/ERC GrantAgreement No. 228051.

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Dynamic response of reinforced masonry columns in classical monuments