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THE INFLUENCE OF INTERFACES ON THE MECHANICAL BEHAVIOUR OF RESTORED EPISTYLES

Stavros K. Kourkoulis∗, Vasiliki Panagiotopoulou, Evangelia Ganniari-Papageorgiou School of Applied Mathematical and Physical Sciences, Department of Mechanics,

National Technical University of Athens Theocaris Building, Zografou Campus, 157-73 Athens, Greece

Keywords: Epistyles, Dionysos Marble, Titanium, Bending, Finite Element Method ABSTRACT

A numerical analysis based on the Finite Element Method is presented aiming to the study of the influence of interfaces on the mechanical behavior of fragmented epistyles made from marble and restored with titanium reinforcing bars. The method was introduced a few years ago by the scientists working for the restoration of the Parthenon Temple on the Acropolis of Athens and is still under development. The numerical model was con-sidered centrally fractured into two marble pieces joined together with a single threaded titanium bar and a cementitious material interposed between marble and titanium. Attention is focused to the influence of the inclination of the fracture plane, since a series of previous numerical analyses indicated that the maximum stress and strain values appear in this region and especially in the vicinity of the reinforcing bar around the central cross section. In addition, the contact properties of the marble-titanium, marble-cement and cement- titanium interfaces were also examined. The analysis revealed that the slope of the fracture influences dra-stically the stress intensity in the vicinity of the reinforcing bar even for relatively small angles. INTRODUCTION

For the needs of the structural restoration of the Acropolis of Athens monuments an innovative methodology has been developed from the early seventies by the scientists working for the project, aiming to joining together fractured structural elements. The method is based on the combined use of bolted titanium reinforcing bars with a suitable cementitious material [1-3]. This technique is nowadays used widely for the restoration of multi-fragmented epistyle of the Parthenon Temple, which are subjected to bending under an almost uniformly distributed load over their total length (their own weight, and the weight of the superimposed structural elements).

The motive of the present study was the restoration of a specific multi-fragmented epistyle of the Parthenon Temple, i.e. the fifth external one of the north colonnade. The specific epistyle is frac-tured in eight pieces of arbitrary shape. The problem has been studied already both experimentally and numerically. The experimental results revealed some deviations from the theoretical ones, especially concerning the maximum expected failure load [4, 5]. These deviations were attributed to the assumption that the architrave is equivalent to a statically determined structural element (in the form of a simply supported beam or a two ways projection one), but also to the improper simulation of the uniform load by linearly distributed concentrated forces.

Following the above conclusions a systematic numerical study was carried out with the aid of the Finite Element Method considering either intact or fractured and restored prismatic marble episty-les resting on the supporting abacuses. A series of numerical models were constructed considering that the fragmented epistyles were centrally and vertically cracked into two symmetric marble parts joined together with a single titanium bar. The various numerical analyses [6-9] indicated that the mechanical behaviour of these fragmented architraves is influenced by several parameters,

∗ To whom all correspondence should be addressed.

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such as the load application mode, the geometric features of the reinforcing titanium bars, the existence of the cementitious material interposed between the marble and the bar, the mechanical properties of this cementitious material and the contact properties of the marble-titanium, marble-cement and cement-titanium interfaces. The conclusions of the above analyses can be shortly summarized as follows:

• High stress concentrations appear around the central cross section area of the titanium bar, at the upper side of the architrave close to the cross section area and at the corners of the supporting abacuses. The last one is strongly supported by a thorough in-situ investigation of the architraves of the Parthenon Temple [10]. Such a conclusion dictates that these corners should be rounded in order to prevent local exfoliation and cracking.

• The variation of the axial strain along the height of the architrave is not in accordance to the simplified Bernoulli-Euler technical bending theory (linear distribution). The variation was of sigmoid nature, which is in agreement with earlier experimental results [11] and recent theo-retical predictions (higher order bending theories) [12]. In addition, the neutral axis of the bent architrave is displaced downwards (towards the bottom side of the architrave).

• A contact loss is observed between the two marble parts of the architrave and specifically at the regions just above and below the immediate vicinity of the reinforcing bar as well as at the regions of the beam resting on the abacuses, where the architrave tends to lift up due to the bending load. This observation is very important and may lead to undesired results, since these areas are susceptible to dust or humidity.

• The use of threaded titanium bars in comparison to the use of unbolted (cylindrical) ones yields more severe stress and strain fields due to the existence of singularities at the corner and the roots of the threads. In addition, these stresses are more severe at the roots of the threads rather than at their respective peaks.

• The intensity of the stress field along the reinforcing bar decreases rapidly and is almost zero after the 50th (out of 200) thread. This conclusion is very important indicating that the empiric-al formula for the anchoring length of the reinforcing bars should be reconsidered in the direction of minimizing the intervention on the authentic elements, in accordance, also, to the Venice Chart directions.

• The insertion of the cement layer between the marble and the bolted titanium bar reduces drastically the high stress and strain discontinuities appearing in the vicinity of the reinforcing bar at the central cross section.

• The appropriate choice of the composition of this cementitious material is very important, since a change in its constitutive behavior influences dramatically the stress and strain fields all over the model. More specifically the composition that could produce a multi-linear layer, of slightly ductile nature, rather than a brittle linear one, can be considered as the most successful, since it creates relatively smooth stress and strain distributions.

The present study approaches the problem in a more systematic way and enhanced the previous numerical analyses by increasing the number of parameters influencing the stress and strain fields. In this direction a series of numerical models were constructed considering centrally cracked architraves with an inclined fracture plane along their total height. Therefore an addi-tional parameter must be taken into account, namely the inclination of the fracture plane. In the present study the value of this angle varied in a relatively narrow range of values between 0o (fracture plane parallel to the external transverse load) and 30o. It is obvious that for crack inclination angles exceeding 30o it is mostly unlikely for a single horizontal reinforcing bar to be used. The values of the angle of the inclined crack examined were 0o, 10o, 20o and 30o. It is to be mentioned that for the needs of the numerical analysis the positive direction was chosen to be the clockwise one.

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NUMERICAL ANALYSIS

The geometry of the numerical models

A numerical analysis is carried out with the aid of the Finite Element Method in an effort to study in a parametric manner the mechanical behaviour of a typical prismatic marble architrave of rec-tangular cross section of the north colonnade of the Parthenon Temple. The architrave was assumed to be cracked centrally with an inclined crack plane along its total height and therefore it was simulated considering two equal marble parts in simple contact joined together with a single threaded titanium reinforcing bar and a thin layer of a cementitious material interposed between marble and titanium, as it is shown schematically in Figure 1. The abacuses (capitals) of the columns of the temple, on which the architrave rests, were also taken into consideration and therefore they were simulated as cubic marble blocks for simplicity reasons. In addition only half of the configu-ration was modelled, taking advantage of the plane of vertical symmetry containing the longitud-inal axis of the reinforcing bar.

R1 5.85 mm

R2 6.35 mm

p 2 mm

t 0.5 mm

a 25o

Table 1: The geometry of the reinforcing bar

Figure 2: Schematic representation of the geometrical characteristics

of the titanium bar

p

t

R1 R2 a

y

x

z

abacuses

architrave

cementitious material

titanium bar

Figure 1: (a) An overall view of the model and (b) a detailed view of the titanium bar and the layer of the cementitious material at the position of the inclined crack

(a) (b)

inclined crack

φ

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The model was constructed under an 1:3 scale (taking into account recent conclusions concerning the influence of the size effect and also recent experimental studies of the problem [5]). Therefore the overall dimensions of the epistyle modeled were: length x=1.43 m, height y=0.45 m and thick-ness z=0.18 m. The abacuses were of cubic shape with an edge length equal to 0.63 m. The archi-trave was placed on the abacuses in such a way that its free span was 0.8 m. For the simulation of the reinforcing bar the exact geometrical characteristics of the titanium bars used in the restoration of the Parthenon Temple [13, 14] were taken into account. The reinforcing bar was placed at a distance ht=0.305 m from the upper side of the epistyle (following recent studies by Ioannidou and Pashalides [13] and Mentzini [14]). The anchoring length in each one of the two marble fragments was set equal to 0.2 m. A schematic representation of the bar is shown in Figure 2, while its geome-trical features are recapitulated in Table 1. In addition, the thickness of the cement layer, inter-posed between the marble and the titanium bar, was chosen equal to 2 mm, in accordance to the practice usually followed by the technicians working nowadays for the restoration of the Parthe-non Temple.

Mechanical properties

The transversely isotropic nature of Dionysos marble was ignored since attention is paid for the epistyles to be loaded normally to the material layers of their structure. In addition the slight non-linearity and the bimodularity of the σ-ε curve were ignored in a first approximation. There-fore the material was considered as linearly elastic and isotropic with mechanical properties coin-ciding with those of the strong anisotropy direction [15] (Young's modulus Em=70 GPa, Poisson's ratio vm=0.3). The density of Dionysos marble was set equal to ρm=2.78 gr/cm3 and the coefficient of static friction between the marble architrave and the marble abacuses was set equal to µ=0.7. The mechanical properties of the titanium bar were: Young's modulus Et=105 GPa, Poisson's ratio vt=0.32 and density ρt=4.51 gr/cm3. Finally the mechanical properties of the cement mortar have been determined through a series of uniaxial compression tests [16, 17]. As a first step, the cement layer was considered linearly elastic with Young's modulus Ec=15.4 GPa and Poisson's ratio vc=0.26. The density was set equal to ρc=1.7 gr/cm3 and the coefficient of static friction between marble and cement was assumed equal to 0.5.

Meshing

After the construction of the solid model an appropriate meshing operation was necessary to be carried out. This step is one of the most important of the entire analysis, since the decisions made at this stage of the model development will seriously affect the accuracy and economy of the analysis. The density (element size and shape) as well as the type (free or mapped) of the mesh are two of the most important factors influencing the accuracy of the results. The ideal solution would be the creation of a uniform and relatively fine mesh all over the model. However for the specific numerical model this choice couldn’t be used due to its complicated geometry (threaded titanium bar and inclined fracture plane). A huge number of elements (too fine mesh) were created making the model too large to run on ordinary computer systems.

In order to avoid such inconvenience, a uniform and fine mesh was created only at the regions of highest interest (the vicinity of the titanium bar, the layer of the cementitious material and at the region around these two materials) choosing the free meshing technique (Figure 3b). For the re-maining part of the model a coarser and relatively free uniform mesh was adopted (Figure 3a). The element used was the SOLID187, a higher order 3-D structural solid element, defined by the orthotropic material properties and by 10 nodes having three degrees of freedom per node: translations in the nodal x, y, and z directions (Figure 3c). The finally adopted model (after the appropriate convergence analysis) consisted of 487051 (!) such elements.

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Contact options and boundary conditions

The next step of the numerical analysis was the identification of the contact options of the problem. In this direction, four couples of 3-D contact elements were created as it is shown in Figure 4. These couples represented the simple contact between:

1. the epistyle and the supporting abacuses (marble-to-marble contact, coefficient of friction µ=0.7)

2. the inclined interface of the two equal marble parts of the epistyle which are connected with the titanium bar (marble-to-marble contact, coefficient of friction µ=0.7)

3. the architrave and the layer of cementitious material (marble-to-cement contact, coefficient of friction µ=0.5)

4. the reinforcing bar and the layer of cementitious material (titanium-to-cement contact, coefficient of friction µ=0.5)

The elements used for each of these couples were the “CONTA174” and “TARGE170”, since they are applicable to 3-D structural analyses (Figure 5). “CONTA174” is defined by eight nodes (four of them are mid-side) and it is the most appropriate for the present analysis, since the underlying solid element (SOLID187) has mid-side nodes also. “TARGE170” was chosen since it can be paired with 3-D surface-to-surface contact elements, such as “CONTA174”.

Concerning the boundary conditions it was assumed that the lower bases of the supporting abacuses are rigidly clamped. In addition the whole model was restricted along the direction of its thickness since, as it has already been mentioned, advantage was taken of the vertical plane of symmetry including the axis of the architrave (only half of the configuration was modelled). Finally, a uniformly distributed bending load equal to 65 kN was applied at the upper side of the architrave all along its free span. This choice was dictated by the results of a recent numerical analysis, according to which for the specific configuration (architrave resting on two abacuses) this loading type corresponds to the most severe stress fields [4].

(c)

(b)

(a)

Figure 3: (a) An overall view of the meshing model, (b) a detailed view of the mesh where the titanium bar and the cement layer are placed and (c) the element SOLID187

titanium

cement layer

1

1 3

3

2

2

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RESULTS AND DISCUSSION

An overall view of the variation of the axial stress is plotted in Figures 6-9 for each model (models with crack angles 00, 100, 200 and 300). It is clearly seen that from a qualitative point of view all figures are of similar nature. High stress concentrations appear in three regions:

1. at the corners if the supporting abacuses, 2. at the cross section of the titanium bar close to the crack, and 3. at the upper side of the architrave close to the region where the two parts of the architrave

are in contact.

(a) (b)

Figure 5: (a) The element CONTA174 and (b) the element TARGE170

Figure 4: The identification of the four couples of contact elements created for the needs of the numerical simulation

1st couple

2nd couple

3rd couple

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These stresses are compressive in regions 1 and 3 and tensile in region 2. Additionally, a contact loss of the two parts of the architrave is observed at the regions both above and below the connecting bar and as one approaches the lower base of the architrave the crack opening displacement takes its highest value. Finally, it is observed that in all cases part of the beam resting on the abacuses (1st couple of contact elements) tends to lift up losing contact. This point should be seriously considered since it may be the origin of various problems such as penetration of external matter (dust, humidity etc). Figure 7: The variation of the axial stress in the restored architrave for crack angle φ=10o.

The embedded figure shows a detail of the central section around the titanium bar

Figure 6: The variation of the axial stress in the restored architrave for crack angle φ=0o. The embedded figure shows a detail of the central section around the titanium bar

Region 1 Region 1

Region 3

Region 2

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In an effort to study quantitatively the results of the numerical analysis a series of comparative diagrams were constructed including the data of the four numerical models. As a first step, the vertical displacement of the lowest fibber of the architrave (line 1-1, see Figure 3a) was plotted all over its length (Figure 10). Point (0,0) corresponds to the leftmost point of the architrave while point (1,0) to the rightmost one. It is clearly seen that the maximum deflection is observed at a position very close to the mid-point of the architrave’s length. Especially for the model with the vertical crack (φ=0o) the maximum deflection is observed exactly at the mid-point due to its sym-

Figure 8: The variation of the axial stress in the restored architrave for crack angle φ=20o. The embedded figure shows a detail of the central section around the titanium bar

Figure 9: The variation of the axial stress in the restored architrave for crack angle φ=30o. The embedded figure shows a detail of the central section around the titanium bar

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metry while the respective values for the other three models (models with crack angles 10o, 20o and 30o) appear to be displaced slightly to the left since the symmetry is lost due to the inclined crack. However the maximum vertical displacement is almost the same for all models reaching a value of about 24 µm. In addition negative values are observed at the ends of the epistyle indicating that at these points the epistyle had lifted up losing its contact with the supporting abacuses as it had already been observed in the previous figures. It is to be mentioned that this lift is inversely pro-portional to the crack angle: the lift of the epistyle decreases as the crack angle increases.

The variation of the axial stress along the same line (line 1-1) is plotted in Figure 11. It is interesting to observe that all models yield almost identical results: The axial stress starts from almost zero values due to the contact loss between the epistyle and the abacuses at this region. However, as ones moves towards the corners of the supporting abacuses the axial stress obtains negative (com-pressive) values of the order of about 2 MPa for all models. From this point on the stress starts increasing in an almost parabolic form until it becomes zero at the mid-span of the beam, where the contact between the two constituent parts is lost. The similarity of these variations for all models indicates that the stress and strain fields at the bottom base of the member are almost inde-pendent from the inclination of the central fracture plane (at least for the φ-values studied here). It is also to be mentioned that previous parametric numerical analyses indicated that the stress and strain fields at the bottom base of the architrave are also independent from the geometrical details of the titanium reinforcing bar [6].

Interesting conclusions can, also, be drawn by plotting the variation of the axial stress along the central vertical line of the cross section (line 2-2) of the epistyle for all models. The results are shown in Figure 12 where point (0,0) corresponds to the geometrical centre of the cross section, while points (0,0.5) and (0,-0.5) to the upper and lower base of the member respectively. It is clearly seen that from a qualitative point of view all models have similar behaviour. High com-pressive stresses are observed at the upper one third of the section and as one approaches the tita-nium bar the situation changes dramatically: The stress is almost zeroed (indicating the position of the neutral axis) since the contact of the two parts of the architrave is lost and only in the immedi-

x/L

-10

-5

0

5

10

15

20

25

0 0,2 0,4 0,6 0,8 1

def

lect

ion

[µ

m]

.

0 degrees10 degrees20 degrees30 degrees

Figure 10: The deflection of the lowest fibber of the architrave all

over its length

ax

ial

stre

ss [

MP

a]

.

-2,5

-2

-1,5

-1

-0,5

0

0,5

0 0,2 0,4 0,6 0,8 1

x/L

0 degrees10 degrees

20 degrees30 degrees

Figure 11: The variation of the axial stress along the bottom central axial

line (line 1-1) of the architrave

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ate vicinity of the reinforcing bar the stress attains high tensile values. From this point on these values become zero since below the reinforcement the contact of the two marble parts is lost again. However, from a quantitative point of view some interesting discrepancies appear. More ana-lytically,

• the compressive stress observed at the upper side of the architrave decreases as the crack angle increases (the highest stress is observed for crack angle φ=00 while the lowest one for crack angle φ=30o)

• the position of the neutral axis is displaced downwards as the crack angle increases (the highest position is observed for crack angle φ=0o and the lowest one for crack angle φ=30o)

• The high tensile stress values observed in the immediate vicinity of the titanium bar in-crease as the crack angle decreases (the highest stress is observed for crack angle φ=0o while the lower one for crack angle φ=30o). However, since the maximum fracture stress of Dionysos marble is about 6-8 MPa [18] it is concluded that even in the region of the titanium bar the member is safe.

As a last step, the variation of the axial stress along the axial lines, positioned at the lower and upper fibers of the titanium bar, is plotted in Figure 13, in an effort to investigate thoroughly the high stress and strain concentrations appearing at the cross section of the reinforcing bar in the im-mediate vicinity of the cracked plane (Region 2, see Figure 6). From Figure 13a, it is observed that only the model with the vertical crack angle (φ=0o) is clearly symmetric exhibiting tensile values all over the length of the bar reaching a maximum at its mid-point. For the remaining three models the symmetry is lost (as it is expected) and sign changes appear around the central cross section of the titanium bar: The initially low tensile stresses are rapidly converted to very high compressive ones. In addition, as one moves close to the central cross section they increase abruptly attaining maximum tensile values, almost double compared to the previous compressive ones. Finally, it is observed that the maximum tensile values near the mid section become higher by increasing the crack angle. A similar behaviour is observed for the upper fibber of the titanium bar (Figure 13b). The only difference is the fact that the maximum stress values appearing near the mid section are compressive, as it was perhaps expected.

-0,5

-0,25

0

0,25

0,5

-2 -1,5 -1 -0,5 0 0,5 1

axial stress [MPa]

y/

h

.

0 degrees10 degrees20 degrees30 degrees

Figure 12: The variation of the axial stress along the central vertical line of the cross section (line 2-2) of the architrave

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CONCLUSIONS

The influence of the inclination of the interface on the mechanical behavior of fragmented marble epistyles restored with a threaded titanium reinforcing bar was studied in the present paper with the aid of the Finite Element Method. The conclusions of this parametric analysis verified the re-sults of previous numerical analyses and revealed some additional critical points. More analytically:

• High stress concentrations appear at the corners if the supporting abacuses, at the cross section of the titanium bar close to the crack, and at the upper side of the architrave close to the region where the two parts of the architrave are in contact.

• The maximum deflection is almost the same for all models reaching a value of about 24 µm. However, it was concluded that the ends of the epistyle not only lifted up losing their contact with the supporting abacuses but also this lift was inversely proportional to the crack angle (the lift decreased as the crack angle became larger).

• The similarity of the stress and strain variations along the bottom central axial line for all models indicated that the stress and strain fields at the bottom base of the member are almost independent from the inclination of the central crack.

• The compressive stress observed at the upper side of the architrave decreased by increasing the crack angle.

• The position of the neutral axis was displaced downwards as the crack angle increased.

• The maximum tensile values in the immediate vicinity of the titanium bar became higher by increasing the crack angle.

Coming to an end it is emphasized that the stress field developed in a restored structural member (somehow characterizing the quality of restoration) depends also on many other parameters like for example the geometrical features of the thread [16], and obviously the degree of damage of the original material of the monuments. It is therefore clear that the results of the present analysis are only indicative. The final decision concerning the optimum combination of the parameters de-pends mainly on the irreplaceable experience of the scientists and technicians working for a given restoration project since they are the only ones who have an overall view of the specific problem they deal with.

-3

-2

-1

0

1

2

3

0 0,2 0,4 0,6 0,8 1

x/L

ax

ial

stre

ss [

MP

a]

.

0 degrees10 degrees20 degrees30 degrees

-2

-1

0

1

2

3

4

0 0,2 0,4 0,6 0,8 1

x/L

ax

ial

stre

ss [

MP

a]

.

0 degrees10 degrees20 degrees30 degrees

Figure 13: The variation of the axial stress along the axial line positioned at (a) the lower and (b) the upper fiber of the titanium bar

(a) (b)

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REFERENCES

[1] Zambas C., Ioannidou M., Papanikolaou A., (1986), "The Use of Titanium Reinforcements for the

Restoration of Marble Architectural Members of Acropolis Monuments", in Proc. of the IIC Congress on Case Studies in the Conservation of Stone and Wall Paintings, Bologna, Italy, pp.138-143.

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[13] Ioannidou M., Pashalides V., (2002), "Joining Together Beams of the Propyleae of Acropolis Using Titanium", in F. Mallouchou-Tufano (Ed.), Proc. of the 5th International Symposium for the Restoration of the Acropolis Monuments, October 4-6, Athens, Greece, pp.291-300.

[14] Mentzini M., (2002), "Joining Together Architraves and Drums of the Parthenon", in F. Mallou-chou-Tufano (Ed.), Proc. of the 5th International Symposium for the Restoration of the Acropolis Monu-ments, October 4-6, Athens, Greece, pp.233-242.

[15] Vardoulakis I., Kourkoulis S.K., (1997), ''Mechanical Properties of Dionysos Marble: Final Report of the Environment Project EV5V-CT93-0300: Monuments under Seismic Action'', National Technical Univer-sity of Athens, Athens, Greece.

[16] Kourkoulis S. K., Papanikolopoulos S.A., Marinelli A., Vayas I., (2008), ''Restaurierung Antiker Tempel: Experimentelle Untersuchungen zum Ausziererhalten vor Verankerungen im Marmor'', in Bautechnik, Vol.85, No.2, pp.109-119.

[17] Marinelli A., Papanikolopoulos S. A., Kourkoulis S. K., Vayas I., (2008), ''The Pull-Out Problem in Restoring Marble Fragments: A Design Criterion Based on Experimental Results'', in Strain, Vol. 45, No.5, pp.433–444.

[18] Vardoulakis I., Kourkoulis S.K., Exadaktylos G.E., Rosakis A., (2002), "Mechanical Properties and Compatibility of Natural Building Stones of Ancient Monuments: Dionysos Marble", in M. Varti-Mata-ranga and Y. Katsikis (Eds.), IGME Publishing, Proc. of the Interdisciplinary Workshop “The Building Stone in Monuments”, Athens, Greece, pp.187-210.