The technique to construct the unmixing matrix employs first a whitening step and then an approximate simultaneous diagonalisation of several time-delayed second order correlation matrices. Its efficiency and stability are demonstrated for linear artificial mixtures with 17 sources. Algorithm Computer science Correlation Blind signal separation Matrix mathematics Artificial intelligence Data analysis Pattern recognition Source separation.
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Frank C. A wooden formwork was employed to support the main arch during construction Figure 9c. The construction of the main central arch was preceded by the construction of its foundation at both ends together with the construction of the abutments that were raised up to a certain height in order to resist the thrust of the central arch. The construction of the main central arch was followed in many cases with the construction of a secondary central arch on top of the main central arch Figures 9d and 10a — f. Finally, the mandrel walls were constructed above the abutments in order to form together with the arches the main passage deck at the top of the bridge.
In certain cases, this passage is protected at both sides at the deck level by an in-built continuous stone parapet that rises approximately 0. In the case of the Kontodimou Bridge, this parapet is formed by individual stones in-built at intervals of approximately 1. The thickness of the primary and secondary arches of the main span varies considerably. The primary arch for the Konitsa Bridge with a clear span of 40 m has a thickness of 1. The primary arch of the Plaka Bridge again with a clear span of 40 m has a thickness of 0.
The primary arch of the Kokorou Bridge with a clear span of The primary arch of the Tsipiani Bridge with a clear span of 26 m has a thickness of 0. Finally, the primary arch of the Kontodimou Bridge with a clear span of These thickness values are approximate and correspond to the arch thickness at the maximum rise; in some cases, the primary and the secondary arch thicknesses vary having an increased thickness in the areas where these arches join the abutments.
The construction of both the primary and secondary main central arches as well as the rest of the arches was constructed with stones that were shaped in a very regular prismatic shape. In this way, the mortar joints of the masonry construction for these arches are relatively very small. The same holds for the foundation and the abutments up to a certain height. For these structural parts, according to oral tradition, special attention was paid for the quality of the stone and mortar to be employed.
On the contrary, neither the shape nor the quality of the stones or the mortar was of equal importance for the mandrel walls. As can be seen in Figure 10g that depicts the remaining part of the Plaka Bridge, these mandrel walls were internally constructed with some form of rubble. However, in order to protect these parts from the weather conditions, the mandrel walls were also encased within facades of good-quality stone masonry Figure 10g.
Because the primary and secondary main arches were constructed at different construction stages, there is a continuous cylindrical joint that lies between them see Figure 10a — f. As revealed by the remains of the collapsed Plaka Bridge, wooden beams with iron inserts were employed to connect the primary and secondary arches at certain intervals. Iron ties were also used to connect the two opposite faces of the primary arch in many bridges.
These iron ties are visible in the photos of the main central arch of the Plaka Bridge before its collapse and they are still in place at the parts of the arch that were salvaged after its collapse Figures 11 and The iron ties were also used to connect the opposite faces of the primary arch of the main span in Tsipianis Bridge Figure 13 and in Voidomatis bridge at Klidonia Figure Connection with wooden beams and iron inserts between the primary and secondary arches of the Plaka Bridge.
In measuring the dynamic response of four stone bridges, two types of excitation were mobilized. The first, namely ambient excitation, mobilized the wind, despite the variation of the wind velocity in amplitude and orientation during the various tests. Due to the topography of the areas where these stone bridges are located, usually a relatively narrow gorge, the orientation of the wind resulted in a considerable component perpendicular to the longitudinal bridge axis Figures 15a , 17a , 18a and 19a.
Application of Independent Component Analysis for Evaluation of Ashlar Masonry Walls
For this purpose, the employed SysCom triaxial velocity sensors had a sensitivity of 0. All the obtained data were subsequently studied in the frequency domain through available fast Fourier transform FFT software [ 4 , 5 ]. This wind orientation relative to the geometry of each bridge structure coupled with the bridge stiffness properties could excite mainly the first symmetric out-of-plane eigen-mode, as can be seen in Figure 16c for the Konitsa Bridge. The variability of the wind orientation could also excite, although to a lesser extent, some of the other in-plane and out-of-plane eigen-modes see Figure 16c for the Konitsa Bridge.
The second type of excitation that was employed, namely vertical in-plane excitation, was produced from a sudden drop of a weight on the deck of each stone-masonry bridge [ 6 , 7 ]. This weight was of the order of approximately 2. Again, the level of this second type of excitation was capable of producing mainly vertical vibrations and exciting the in-plane eigen-modes of each structure that could be captured by the employed SysCom triaxial velocity sensors with a sensitivity of 0.
All the obtained data were subsequently studied in the frequency domain through available FFT software. In Figure 15c , the velocity measurements are depicted along the three axes x - x horizontal out-of-plane, y - y horizontal in-plane and z - z vertical as they were recorded during a typical sampling with the wind excitation. In Figure 15d , the velocity measurements are again depicted along the three axes x - x horizontal out-of-plane, y - y horizontal in-plane and z - z vertical as they were recorded during a typical sampling with the drop weight excitation.
As can be seen, the drop weight excitation could produce at the dominant frequencies vibrations at least one order of magnitude larger than the wind excitation. From these measurements, an attempt was also made to obtain an estimate of the damping ratio for the dominant in-plane and out-of-plane frequencies. As is depicted in Figure 16a for the wind excitation, the main symmetric out-of-plane vibration that is excited by the wind has a dominant period of 2.
Similarly, as is depicted in Figure 16b for the drop weight excitation, the main symmetric in-plane vibration that is excited by the drop weight has a dominant period of 7. This increase in the damping ratio value for this latter dominant frequency must be attributed to the relatively larger amplitudes of vibration that are produced from the drop weight excitation than from the wind excitation, as already underlined. All vibration measurements of the dynamic response of the Konitsa Bridge for either type of excitation were utilized to extract the eigen-frequencies depicted in Figure 16c together with the approximate shape of the corresponding eigen-modes.
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The same process was followed for measuring the dynamic characteristics of another three stone-masonry bridges Kokorou, Tsipianis and Kontodimou using both the wind and the drop weight excitations, as shown in Figures 17 — 19 where the position of the employed velocity sensors is indicated.
Next, by utilizing all these vibration measurements of the dynamic response of each of these studied bridges for either type of excitation, it was possible to extract the relevant eigen-frequencies that are listed in Table 1. At least measurements of three repetitive sampling sequences for each type of excitation, either wind or drop weight, for each bridge Konitsa, Kokorou, Tsipianis and Kontodimou were measured. The eigen-frequency values listed in Table 1 are values representing an average from corresponding values that were obtained by analysing the measured response from all tests.
To gain more confidence in the in situ measurements presented in Section 4. This additional in situ campaign was conducted during the end of October Figure This almost coincides with the in situ campaign described in Section 4. Moreover, for the Konitsa Bridge the measurements presented in Section 4. Based on this timing and the constant weather conditions prevailing during this period, no influence is expected to arise from environmental conditions to all these measurements.
Further data post-processing was performed using a set of additional FFT processing software.
Shown in Figure 21 are the post-wind gust bridge response vertical acceleration and the corresponding power spectrum associated with the trace segment between 12 and 16 s of the record [ 4 ]. The power spectrum associated with the decay segment clearly delineates a the symmetric vertical mode 7. Decay segment of acceleration trace vertical and the corresponding power spectrum 7. Horizontal out-of-plane acceleration time histories recorded simultaneously at two locations on the Konitsa Bridge deck.
Dominant frequency of 2.
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Figure 22 depicts the horizontal out-of-plane acceleration time histories recorded simultaneously at two locations on the Konitsa Bridge deck and their corresponding power spectrum with dominant frequency of 2. Comparing the eigen-frequency values obtained for the in situ experiments, reported in Section 4. Figure 22c and b depict the FFT-averaged Fourier spectral curves that formed the base together with the coherence plot of Figure 22d to identify with confidence the eigen-frequency values [ 9 ]. A laboratory testing sequence was performed having as an objective to study in a preliminary way the mechanical characteristics of the basic materials representative of the materials employed to build the studied stone-masonry bridges [ 4 ].
For this purpose, stone samples were selected from the neighbourhood of the collapsed Plaka Bridge as well as from a quarry near the Kontodimou and Kokorou Bridges. Moreover, stone samples were also taken from the river bed of the Kontodimou Bridge. Furthermore, it was possible to take a mortar sample from the collapsed Plaka Bridge. From both the stone and mortar samples collected in situ , it was possible to form specimens of regular prismatic geometry.
These specimens were subjected to either axial compression or four-point bending tests. For the compression tests, the loaded surfaces of the prisms were properly cupped. Figure 23a and c depict typical loading arrangements employed for the compression stone and mortar specimens tests, whereas Figure 23b depicts the loading arrangement employed for the four-point bending tests. The applied load was measured through a load cell and the deformation of the tested specimens was measured employing a combination of displacement sensors as well as a number of strain gauges.
These measurements were continuously recorded with a sampling frequency of 10 Hz. The obtained values of these mechanical parameters are listed in Tables 2 — 6. In this section, the dynamic characteristics of the four studied stone-masonry bridges will be predicted through a numerical simulation process. Initially, this numerical simulation will be based on elastic behaviour, assuming the stone masonry as an orthotropic continuous medium and limiting these numerical models at approximately the interface between the end abutments and the rocky river banks, thus introducing boundaries at these locations [ 10 ].
For simplicity purposes, the bulk of these numerical simulations are made in the 3-D domain representing these bridge structures with their mid-surface employing thick-shell finite elements [ 11 ]. All available information, measured during the in situ campaign, on the geometry of each one of these parts for every bridge was used in building up these numerical simulations. The mechanical property values obtained from the stone and mortar sample tests, which were presented in Section 5, indicate the following main points. It is well known that the complex triaxial behaviour of masonry cannot be easily approximated from the mechanical behaviour of its constituents.
For the studied stone-masonry bridges, this becomes even more difficult considering the various construction stages that were discussed in Section 3, the variability of the materials employed to form the distinct parts during these construction stages and the interconnection and contact conditions between the various parts formed during these construction stages abutments, primary and secondary arches, deck, parapets, mandrel walls.
Moreover, there is important information that is needed in order to form with some realism the boundary conditions at the river bed and banks [ 11 ]. The lack of specific studies towards clarifying in a systematic way all these uncertainties represents a serious limitation in the numerical simulation process. Konitsa Bridge. Kokorou Bridge.
Tsipianis Bridge. Kontodimou Bridge. Numerical and observed eigen-values for the Konitsa Bridge. Numerical eigen-values for the Plaka Bridge. Plaka Bridge. The approximation adopted in this study is a process of back simulation [ 6 , 7 ]. That is, adopting values for these unknown mechanical stone-masonry properties, respecting at the same time all the measured geometric details, which result in reasonably good agreement between the measured and predicted in this way eigen-frequency values.
Following this approximate process, two distinct cases of boundary conditions were introduced. In one series of numerical simulations, all the boundaries, either at the river bed or at the river banks, were considered as being fixed in these 3-D numerical simulations for all studied bridges. Alternatively, the rotational degrees of freedom were released all along the locations where the abutments are supported at the river banks thus excluding the footings.
It is shown from this sensitivity analysis that this variation in the boundary conditions approximation influences, as expected, the out-of-plane and not the in-plane stiffness of the studied stone-masonry bridges. This out-of-plane stiffness variation is more pronounced for the relatively small dimensions Kontodimou Bridge rather than for the relatively large Konitsa Bridge and Plaka Bridge.
Moreover, for the Tsipianis Bridge whereby the main central arch is supported at the North end in adjacent arches rather than on the rocky bank, this variation of the boundary conditions, as expected, has again a less pronounced influence. For the Kontodimou Bridge, these values were 1. A partial explanation is that the mortar joints and contact surface between the various bridge parts in the Kontodimou Bridge Figure 10e and f were wider than in other bridges and the mortar was in some cases washed out at some depth.
In order to approximate the in-plane and the out-of-plane stiffness of the studied stone-masonry bridges, which directly influences the corresponding numerical eigen-frequency values, listed in Tables 7 and 8 and depicted in Figures 24 and 25 , a flexural stiffness amplifier was introduced for the Konitsa Bridge and the Kokorou Bridge equal to 3. From the comparison of the results of these numerical simulations in terms of eigen-frequencies and eigen-modes, listed in Tables 7 and 8 and depicted in Figures 24 and 25 , it can be seen that in most cases the predicted eigen-frequency values are in reasonably good agreement with the measured values.
Moreover, the order of the out-of-plane and the in-plane eigen-modes predicted by the numerical simulation is in agreement with the observed response. An exception is the first asymmetric in-plane eigen-mode for the Konitsa Bridge Figure 24 and Kokorou Bridge Tables 7 and 8 that indicates a corresponding measured stiffness smaller than the predicted one.
On the basis of this comparison, an additional numerical simulation was performed for the Plaka Bridge Figure 25 , despite the lack of measured response in this case, adopting the same assumptions that were described before specifically for the Konitsa Bridge. As can be seen by comparing the numerical eigen-frequency values of the Konitsa Bridge Figure 24 with those of the Plaka Bridge Figure 25 , the latter, as expected, is more flexible both in the in-plane and in the out-of-plane direction.
This section includes results of a series of numerical simulations of the Konitsa Bridge when it is subjected to a combination of actions that include the dead weight D combined with seismic forces. The seismic forces will be defined in various ways, as will be described in what follows. Towards this, horizontal and vertical design spectral curves are derived based on the horizontal design ground acceleration.
This value, as defined by the zoning map of the current Seismic Code of Greece [ 13 , 14 ], is equal to 0. Furthermore, it is assumed that the soil conditions belong to category A because of the rocky site where this bridge is founded, that the importance and foundation coefficients have values equal to one 1. The design acceleration spectral curves obtained in this way are depicted in Figure 26a and b for the horizontal and vertical direction, respectively. In the same figures, the corresponding elastic acceleration spectral curves are also shown derived from the ground acceleration recorded during the main event of the earthquake sequence of 5 August at the city of Konitsa located at a distance of approximately 1.
In Figure 26a and b , the eigen-period range of the first 12 eigen-modes is also indicated ranging between the low and the high modal period. For the vertical response spectra, this is done for only the in-plane eigen-modes see also Table 9.
As can be seen in Figure 26a , the Euro-Code horizontal acceleration spectral curves compare well with the horizontal component-3 of Earthquake spectral curves for the period range of interest. Based on these plots, it can be concluded that this bridge sustained a ground motion that in the horizontal direction was approximately comparable to the design earthquake; however, the design earthquake in the vertical direction is shown to be more severe than the one this stone-masonry bridge experienced during the earthquake sequence.
In Table 10 , the base reactions are listed F X , F Y and F Z in the x - x u 1, out-of-plane , the y - y u 2, in-plane and z - z u 3, in-plane directions see Figure 15a and b from the various load cases, which were considered in this numerical study. Apart from the dead load D, row 1 in rows 2—4 of Table 10 , the base reaction values listed are obtained from dynamic spectral analyses employing the horizontal and vertical response spectral curves of the Konitsa earthquake event Figure 26a and b.
In rows 7—9 of Table 10 , the base reaction values are again obtained from dynamic spectral analyses employing this time the Euro-Code horizontal and vertical design spectral curves of Figure 26a and b. In all these dynamic spectral analyses, the 12 eigen-modes listed in Table 9 were employed. Modal participating mass ratios for Konitsa Bridge see Figure Towards this end, the dynamic spectral analysis results were multiplied by an amplification factor equal to the reverse of the relevant ratio values before superimposing the dead load results.
This becomes evident when one compares the base reaction values without and with these amplification factor values in Table In Figure 27a and b , the numerically predicted deformation patterns of Konitsa Bridge are depicted for load combination 1 and 7, respectively. As can be seen, this stone-masonry bridge develops under these combinations of dead load and seismic forces relatively large out-of-plane displacements at the top of the main arch. As expected, the deformations for the Euro-Code design spectra reach the largest values attaining at the crown of the arch a maximum value equal to Again, as expected, the most demanding state of stress results for the load combination 7 that includes seismic forces provided by Euro-Code [ 12 ].
The largest values of tensile stress S11 3. This is a relatively large tensile stress value that is expected to exceed the tensile capacity of the stone masonry of this bridge [ 16 ]. The largest value of tensile stress S22 1. Again, this is a relatively large tensile stress value that is expected to exceed the tensile capacity of the stone masonry of this bridge. Both these remarks indicate locations of distress for this stone-masonry bridge predicting in this way the appearance of structural damage.
Application of Independent Component Analysis for Evaluation of Ashlar Masonry Walls | SpringerLink
Comb 7. An additional linear numerical simulation was performed. The bridge was subjected only to the vertical Comp2-Ez, rows 2 and 3 of Table 11 or only to the horizontal component of this record in the out of-plane direction Comp3-Ex, rows 4 and 5 of Table Alternatively, the bridge was subjected to the horizontal component of this record in the in-plane horizontal direction Comp3-Ey, rows 6 and 7, Table 11 [ 15 ].
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In these analyses, only the first most intense 6 s of this Konitsa earthquake record were used [ 15 ]. In Table 11 , the base shear values in the x - x F X , u 1, horizontal out-of-plane , y - y F Y , u 2, horizontal in-plane and z - z F Z , u 3, vertical directions are listed in terms of limit values maximum or minimum that arose during the 6 s of these time-history analyses. Figure 30a shows the horizontal u x , out-of-plane and the vertical u z , in-plane displacement response at the crown of the Konitsa Bridge, obtained from the time-history numerical analyses. The horizontal response was obtained when the structure was subjected to horizontal component Comp3 of the Konitsa earthquake record and the vertical in-plane response when the structure is subjected to vertical component Comp2 of the Konitsa earthquake record Figure Figure 30b shows the variation of the S11 stress response at the bottom fibre of the crown of the Konitsa Bridge when this structure is subjected to either the horizontal component of the Konitsa earthquake record Comp3 in the out-of-plane u x direction or the vertical component of the Konitsa earthquake record Comp2 in the vertical u z in-plane direction.
The location of the plotted stress is at the bottom fibre at the middle of the arch crown of the Konitsa Bridge. As can be seen in both Figure 30a and b , the horizontal u x displacement and S11 stress response produced by the horizontal out-of-plane excitation are larger than the corresponding response vertical u z displacement and S11 stress response produced by the vertical in-plane excitation.
Moreover, as expected from the relevant response spectral curves depicted and the dominant eigen-frequency values Figures 24 , 26a and b , the vertical u z displacement and S11 stress response, produced by the vertical in-plane excitation, are of higher frequency content than the horizontal u x displacement and S11 stress response produced by the horizontal out-of-plane excitation.
State of stress through the distribution of stresses S11 envelope at the bottom fibre of the crown for Konitsa Bridge. By examining the displacement and stress response, it could be concluded that the application of the horizontal component of the Konitsa in the horizontal u y in-plane direction is of too small amplitude to be of any significance.
This must be attributed to the stiffness properties of this bridge in this direction and the resulting in-plane eigen-frequencies and eigen-modes that combined with the frequency content of this record result in displacement and stress response of relatively small amplitude. As was discussed before, the Euro-Code design spectral curves result in much higher displacement and stress demands for the Konitsa Bridge. From all these numerical analyses, it can be concluded that the most vulnerable part of this stone-masonry bridge is the slender central part of the main arch, composed as described in Section 3 of the primary and secondary arch, when the structure is subjected to seismic forces in the horizontal out-of-plane direction.
The vertical in-plane excitation is expected to be significant when in-phase with the horizontal excitation in a way that it can offset the beneficial effect of the dead weight. This observation is thought to be of a general nature, as it is demonstrated by the numerical analyses of the Plaka Bridge in the following Section 7. This section includes results of a series of numerical simulations of the Plaka Bridge when it is subjected to a combination of actions that include the dead weight D combined with seismic forces.
The seismic forces will be defined as was done in Section 7. This value, as it is defined by the zoning map of the current Seismic Code of Greece, is equal to 0. The design acceleration spectral curves obtained in this way are depicted in Figure 32a and b for the horizontal and vertical direction, respectively. In Figure 32a and b , the eigen-period range of the first 12 eigen-modes is also indicated ranging between the low and the high modal period. For the vertical response spectra, this is done for only the in-plane eigen-modes see also Table By comparing these design spectral acceleration curves of Figure 32a and b for the Plaka Bridge with the corresponding spectral curves for the Konitsa Bridge Figure 26a and b , it becomes apparent that the former represent a more demanding seismic force level than the latter.
For the Plaka Bridge, the modal mass participation ratios and the base reactions are listed in Tables 12 and 13 , respectively. The base reactions are F X , in the x - x u 1, out-of-plane , F Y the y - y u 2, in-plane and F Z in the z - z u 3, in-plane directions see Figures 7d , 25 , 29a and b Apart from the dead load D, row 1 in rows 2—4 of Table 13 , the base reaction values were again obtained from dynamic spectral analyses employing, as was done in Section 7.
In all these dynamic spectral analyses, the 12 eigen-modes listed in Table 12 were again employed. Modal participating mass ratios for Plaka Bridge see Figure In Figure 33a and b , the numerically predicted deformation patterns of Plaka Bridge are depicted for load combination 1. As can be seen, this stone-masonry bridge develops under this combination of dead load and seismic forces relatively large out-of-plane displacements at the top of the main arch.
As expected, the out-of-plane displacement response of the Plaka Bridge, when subjected to Euro-Code design spectra, reaches the largest value at the crown of the arch with a maximum value equal to This maximum out-of-plane value for the Plaka Bridge is almost twice as large as the corresponding value predicted numerically for the Konitsa Bridge. Again, as expected, the most demanding state of stress results is for the load combination 1 that includes seismic forces provided by Euro-Code.
The largest value of tensile stress S11 5. This relatively large tensile stress value [ 11 , 16 ] is exceeding by far the tensile capacity of traditionally built stone masonry. The largest value of tensile stress S22 3. Again, this is a relatively large tensile stress value and is exceeding by far the tensile capacity of traditionally built stone masonry.
Both these remarks indicate locations of distress for the Plaka stone-masonry bridge, as was done for the Konitsa Bridge predicting in this way the appearance of structural damage. The maximum tensile stress values that were numerically predicted for Plaka Bridge are approximately twice as large as the corresponding values obtained for Konitsa Bridge. This is because Plaka Bridge is located in seismic zone II design ground acceleration equal to 0.
Furthermore, although the main central arches of the two bridges are very similar in geometry with the deck of the Plaka Bridge being somewhat wider than the deck of the Konitsa Bridge , the Plaka Bridge has a much larger total length than the Konitsa Bridge due to the construction of a mid-pier and arches adjacent to the main central arch.
Thus, Plaka Bridge is more flexible and has a much larger total mass than the Konitsa Bridge. Based on these remarks, it is reasonable to expect for the Plaka Bridge larger seismic displacement values in the out-of-plane direction and consequently larger tensile stress values, than the corresponding values predicted for the Konitsa Bridge. The final consequence of these remarks is that, according to the results of this simplified numerical approach, the Plaka Bridge has a higher degree of seismic vulnerability than the Konitsa Bridge. A similar simplified numerical study of the performance of the Plaka Bridge could be done when measurements of flow data of the flooding of river Arachthos 31st January that caused the collapse of this bridge become available.
A three-dimensional finite element model of the Konitsa stone bridge was developed and utilized in the linear modal and gravity and non-linear earthquake analyses [ 18 , 19 ]. The developed three-dimensional model incorporated interface conditions between distinct parts of the structure i. The detailed model developed for this study included beam elements that formed the steel mesh in the intrados of the bridge rigidly connected to the stone array.
The bridge was modelled using four different solid materials with 72, elements. The model is assumed to be fixed on competent rock on both sides and no soil-structure interaction SSI effects are considered. Figure 35a and b depict the finite element model that was developed and utilized based on in situ technical information collection, images and other historically available technical data. In developing the finite element method FEM model, special attention to the foundation and abutment details was paid and incorporated. Orthotropic elastic behaviour of the hybrid stone-mortar material was also utilized in the numerical modal analysis during the calibration phase and following the field vibration test.
This is described in [ 20 ] as one of the options for elastic materials but with orthotropic behaviour. Figure 36a depicts modelling details of the foundation of the Konitsa Bridge and of the way the primary and secondary arches are joined with the foundation block.
Figure 36b depicts the modelling detail of the parapet and the deck of the Konitsa Bridge see also Figure Before proceeding to the complex non-linear analyses, a modal analysis was performed as a first attempt utilizing the numerical model depicted in Figure 36a and b. The same process was followed, described in the numerical simulation of Section 6, whereby the measured eigen-frequencies, reported for this bridge in Section 4, were taken into account in the best possible way.
This modal analysis led to mode and corresponding frequencies shown in Figure The first five 5 modes include the first two bending modes, the first torsional mode, the first asymmetric vertical mode and the first pure vertical mode, as were also reported in Section 6. In what follows is again a comparison of the modal characteristics of the current 3-D numerical simulation with the results of the 3-D numerical simulation of Section 6 as well as with measured values.
As can be seen from this comparison, the values of the eigen-frequencies for the out-of-plane eigen-modes compare well with the measured values, as was also discussed in Section 3. Moreover, as was also discussed in Section 6, certain discrepancies can be seen for the in-plane eigen-modes. It is believed that the use of orthotropic properties for the materials employed in both the linear numerical simulations can correct up to a point these discrepancies.
For the static analysis and subsequently dynamic earthquake analyses where the bridge structure is expected to exhibit non-linear behaviour and damage, the following material behaviour was adopted in this study. It is controlled by compressive and tensile strength as well as fracture energy and aggregate size. The compressive strength is considered to be controlled by the stone portion of the hybrid element 30 MPa and the tensile strength by that of the mortar. The range of the tensile strength assumed in this study for the different sections of the Konitsa Bridge is 0.
Upon formation of a tension crack, no tensile load can be transferred across the crack faces. An additional failure criterion that controls the detachment of elements from the structure is that of pressure negative in tension. This criterion is used to simulate the failure of mortar in the hybrid element, which is considered to fail when the negative pressure exceeds a critical value.
The pressure threshold assumed in the study was 1. The most recent earthquake in the proximity of the Konitsa Bridge occurred in August [ 21 ]. While no recording at the bridge location is available, the earthquake was recorded at less than a kilometre away on soft soil with maximum acceleration of 0. During the earthquake, limited damage was experienced by the bridge in the form of a spalling of the protective cement layer in the bridge intrados that was introduced following upgrades performed a few years earlier accompanied by the introduction of a steel mesh in the intrados and b loss of parapet sections.
The recorded ground motion see Figure 29 of strong motion acceleration [ 15 ] exhibits the characteristics of an impulse-type or near-field earthquake especially its horizontal component that contains the characteristic pulse. This acceleration record, shown in Figure 29 , is used as bridge base excitation in the non-linear analysis.
Three-dimensional excitation was considered for all the seismic analyses performed. For the Konitsa earthquake analysis, the in-plane and out-of-plane horizontal components were identical and reflected the recorded horizontal acceleration trace of Figure The vertical excitation component was the one also shown in Figure No SSI considerations were introduced at the bottom of the two abutments, which were assumed to be fixed on rock.
Further, for these analyses no differentiation in ground motion between abutment supports was considered despite the fact that one abutment is supported on competent rock and the other in what appears to be weathered rock. The seismic study was conducted in two steps. Specifically, during the first step, the static conditions of the structure were reached by introducing a fictitiously high global damping.
Upon stabilization throughout the structure see yellow arrow in Figure 38a , the earthquake analysis was initiated with the correct damping estimated based on the experimental measurements made during the two campaigns i. Figure 39 depicts the state-of-stress profile throughout the Konitsa Bridge due to gravity load Figure 39a depicts principal deviatoric stress, 39b vertical stress around the right-hand side RHS abutment and 38a vertical stress evolution during the gravity load analysis reaching stabilization for the start of earthquake analysis.
The arrow indicates the start of the dynamic earthquake analysis following the gravity load analysis stabilization. Shown in Figure 40a is the location of the numerical model of Konitsa Bridge where the seismic response is predicted crown, Loc-3, Loc-2, Loc-1 having as input motion the described seismic excitation throughout all the base points Base EQ input.