**8. Non-linear numerical simulation of the seismic behaviour of the Konitsa stone-masonry bridge**

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 general finite element software True-Grid (meshing) and LS-DYNA (static, modal, earthquake analyses) software were employed [20]. The developed three-dimensional model incorporated interface conditions between distinct parts of the structure (i.e. lower and upper stone arches, arches and abutments, etc.) in an effort to capture the interaction between the structural sections as well as differentiate between the building techniques and details that were introduced during the construction of the bridge and thus differentiate between the different failure criteria and mechanisms that may govern the different parts. A modelling approach where the elements (stones) of the arch are represented by solid elements with 'hybrid' behaviour was adopted and used throughout. The detailed model developed for this study included 4009 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 72,540 elements. As noted above, the different structural components are in 'contact' governed by contact interface conditions. The two arches have been modelled with solid 'hybrid' elements (stone-mortar behaviour) that capture the 'non-linearity' or failure rather than the pure contact between stones, an approach that is closer to the actual conditions in the structure. Specifically, it has been assumed that the 'hybrid' element representing mortar and stone behaves as one with the weakness attributed to the mortar part (Modulus and Poisson ratio represent the entire element but critical stresses are dependent on mortar).

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. Based on experience and data for similar structures, the first attempt in establishing the static and dynamic (modal) behaviour of the Konitsa Bridge utilized isotropic material properties for the mortar-stone material with Young's modulus *E* = 17 GPa, compressive strength of 30 MPa, Poisson's ratio of 0.21 and density of 2.69 g/cc. 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 20**).

**Figure 35.** (a) Depiction of sections considered in modelling: the Konitsa stone arch bridge including the partial steel mesh over the intrados placed during restoration work prior to 1996 earthquake and (b) finite element model and details of Konitsa stone bridge.

**Figure 36.** (a) Konitsa Bridge RHS foundation modelling details and (b) Konitsa Bridge parapet modelling details.

#### **8.1. Modal and static analyses**

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 2015) that caused the collapse of this bridge become available.

**8. Non-linear numerical simulation of the seismic behaviour of the Konitsa**

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 general finite element software True-Grid (meshing) and LS-DYNA (static, modal, earthquake analyses) software were employed [20]. The developed three-dimensional model incorporated interface conditions between distinct parts of the structure (i.e. lower and upper stone arches, arches and abutments, etc.) in an effort to capture the interaction between the structural sections as well as differentiate between the building techniques and details that were introduced during the construction of the bridge and thus differentiate between the different failure criteria and mechanisms that may govern the different parts. A modelling approach where the elements (stones) of the arch are represented by solid elements with 'hybrid' behaviour was adopted and used throughout. The detailed model developed for this study included 4009 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 72,540 elements. As noted above, the different structural components are in 'contact' governed by contact interface conditions. The two arches have been modelled with solid 'hybrid' elements (stone-mortar behaviour) that capture the 'non-linearity' or failure rather than the pure contact between stones, an approach that is closer to the actual conditions in the structure. Specifically, it has been assumed that the 'hybrid' element representing mortar and stone behaves as one with the weakness attributed to the mortar part (Modulus and Poisson ratio

represent the entire element but critical stresses are dependent on mortar).

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. Based on experience and data for similar structures, the first attempt in establishing the static and dynamic (modal) behaviour of the Konitsa Bridge utilized isotropic material properties for the mortar-stone material with Young's modulus *E* = 17 GPa, compressive strength of 30 MPa, Poisson's ratio of 0.21 and density of 2.69 g/cc. 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

**stone-masonry bridge**

110 Structural Bridge Engineering

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**. This was done using isotropic elastic material behaviour throughout the numerical model with material properties *ρ* = 2.69 g/cc, *E* = 17 GPa, *ν* = 0.21 for the two arches and similar values for the abutment and mandrel walls. 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 37**. 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.

**Figure 37.** Comparison between numerically predicted eigen-frequencies with measured values.

#### **8.2. Non-linear earthquake analysis and damage criteria**

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.

The mortar-stone material was assumed to behave like 'pseudo-concrete' according to the Winfrith model. 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 0.25–2.1 MPa. The fracture energy assumed in the analysis dissipated in the opening of a tension crack assumed as 80 N/m. 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 1.1 MPa.

## **8.3. Seismic analysis of the Konitsa Bridge**

orthotropic properties for the materials employed in both the linear numerical simulations can

**Figure 37.** Comparison between numerically predicted eigen-frequencies with measured values.

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

The mortar-stone material was assumed to behave like 'pseudo-concrete' according to the Winfrith model. It is controlled by compressive and tensile strength as well as fracture energy

**8.2. Non-linear earthquake analysis and damage criteria**

behaviour was adopted in this study.

and aggregate size.

correct up to a point these discrepancies.

112 Structural Bridge Engineering

The most recent earthquake in the proximity of the Konitsa Bridge occurred in August 1996 [21]. The epicentre of the 6th August earthquake (*M* = 5.7) with 8-km depth was about 15 km to the South West (SW) of the bridge. 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 0.39 g [22]. A similar recording on rock (~1 km away and on the same rock formation to that supporting the left-hand side (LHS) buttress of the Konitsa Bridge) indicated a peak ground acceleration of 0.19 g. During the 1996 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. A considerable number (16%) of the checked 925 buildings of the town of Konitsa, located at close proximity to the stone masonry bridge, developed structural damage typical to Greek construction [24]. 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 1996 earthquake analysis, the in-plane and out-of-plane horizontal components were identical and reflected the recorded horizontal acceleration trace of **Figure 29**. The vertical excitation component was the one also shown in **Figure 29**. 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.e. global damping of 1.6%, **Figure 38b**). **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).

**Figure 38.** (a) Static state of stresses of Konitsa Bridge at the start of seismic analysis. The arrow indicates the start of the dynamic (earthquake analysis) following the gravity load analysis stabilization. (b) Two percent response spectra of the 1996-Konitsa earthquake recorded on rock.

**Figure 39.** (a) Principal stress profile of Konitsa Bridge under dead load and (b) compressive stress concentration at the foot of the main arch.

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). **Figure 40b** depicts the horizontal (in-plane and out-of-plane) and vertical crown displacement seismic response of the Konitsa Bridge predicted using the non-linear numerical analysis. As can be seen in **Figure 40b**, the maximum predicted out-of-plane horizontal crown displacement is somewhat larger than the maximum value predicted by the linear time-history analysis in Section 7.1 (**Figure 30a**). The maximum predicted in-plane vertical crown displacement (**Figure 40b**) predicted by this nonlinear earthquake analysis is significantly larger (approximately four times) than the maximum value predicted by the linear time-history analysis in Section 7.1 (**Figure 30a**). This must be attributed to the fact that the linear analysis performed in Section 7.1 is three-dimensional but employing a numerical model of the bridge that represents its mid-surface, whereas the 3-D non-linear simulation utilizes a model where the bridge is simulated with its actual thickness (compare **Figure 15** with **Figures 35** and **36**). Thus, the vertical displacement at the crown (see **Figure 40a**) predicted by the 3-D non-linear analysis represents the vertical displacement at the façade of the crown cross section of the bridge, which includes a contribution from the outof-plane response, and not the vertical displacement of the crown at mid-surface, as is the case for the simplified analysis of Section 7.1 (**Figure 30a**). The in-plane horizontal displacement predicted by both the linear and the non-linear earthquake analyses has relatively very small amplitude. As discussed before, this clearly demonstrates the much larger stiffness of the bridge structure along the horizontal in-plane direction than along the out-of-plane direction. In **Figure 41a** and **b**, the absolute velocity response at four locations of the Konitsa Bridge as well as at its base is depicted in the horizontal in-plane or out-of-plane direction, respectively. As can be seen again in these figures, the stiffness of the bridge combined with the applied seismic motion results in very small amplification of this velocity response in the in-plane direction than in the out-of-plane direction between the base and the four Konitsa Bridge locations (Crown, Loc-3, Loc-2, Loc1). This crown/base velocity response amplification factor in the out-of-plane direction has a value approximately equal to 2.

(**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

**Figure 38.** (a) Static state of stresses of Konitsa Bridge at the start of seismic analysis. The arrow indicates the start of the dynamic (earthquake analysis) following the gravity load analysis stabilization. (b) Two percent response spectra

**Figure 39.** (a) Principal stress profile of Konitsa Bridge under dead load and (b) compressive stress concentration at the

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). **Figure 40b** depicts the horizontal (in-plane and out-of-plane) and vertical crown displacement seismic response of the Konitsa Bridge predicted using the non-linear numerical analysis. As can be seen in **Figure 40b**, the

stabilization for the start of earthquake analysis).

114 Structural Bridge Engineering

of the 1996-Konitsa earthquake recorded on rock.

foot of the main arch.

In **Figure 43**, the contours of the effective von-Mises stresses are depicted for the Konitsa Bridge subjected to the previously described 1996-Konitsa earthquake record. As can be seen in this figure, tensile distress is indicated at the right and left ends of the primary and secondary arches where they join the foundation blocks. This is also shown in some detail in **Figure 42a** and **b** in terms of von-Mises and vertical stress response in this location. The time-history plot of the vertical stress at the foundation block (A) at the arch-to-foundation block interface (B) and at the primary arch (C) clearly indicates that the tensile stress at location C reaches, as expected, the largest value, which is in excess of the tensile capacity of the bridge construction material (see also Section 7.1. and **Figure 28e**). By comparing the results of the displacement and stress response of the Konitsa Bridge, as obtained by the present 3-D non-linear analysis, with the corresponding time-history results of the simplified linear analysis of Section 7.1, it can be concluded that the 1996-Konitsa ground motion employed in both cases was of such an intensity and frequency content that very limited non-linearities developed at this 3-D advanced non-linear model of the structure. This conclusion is in line with the observed performance of this bridge during the 1996 main event. As already mentioned before, limited damage was experienced by this bridge in this 1996 earthquake in the form of (a) spalling of the protective cement layer in the bridge intrados that was introduced following upgrades performed few years earlier and (b) loss of parapet sections (**Figure 43**).

**Figure 40.** (a) 3-D model of Konitsa Bridge together with the locations of input (excitation) and predicted seismic response and (b) horizontal (in-plane and out-of-plane) and vertical crown displacement seismic response of the Konitsa Bridge predicted using the non-linear numerical analysis.

**Figure 41.** Earthquake response of Konitsa Bridge when subjected to the 3-D 1996-Konitsa earthquake: (a) in-plane horizontal velocities and (b) out-of-plane horizontal velocities.

**Figure 42.** Vertical stress response at the right end of the primary and secondary arches where they join the foundation blocks.

**Figure 43.** Tensile stress concentration at the foot of the primary and secondary arches of the Konitsa Bridge.

#### **8.4. Seismic vulnerability assessment and code guidance effects**

**Figure 40.** (a) 3-D model of Konitsa Bridge together with the locations of input (excitation) and predicted seismic response and (b) horizontal (in-plane and out-of-plane) and vertical crown displacement seismic response of the Konitsa

**Figure 41.** Earthquake response of Konitsa Bridge when subjected to the 3-D 1996-Konitsa earthquake: (a) in-plane hor-

Bridge predicted using the non-linear numerical analysis.

116 Structural Bridge Engineering

izontal velocities and (b) out-of-plane horizontal velocities.

In order to examine the capabilities of the 3-D non-linear numerical simulation performed in the previous section and in an effort to understand the potential influence of the time structure and period content of the exciting earthquake which may be missed when utilizing envelope code spectra (i.e. Euro-Code), the Konitsa Bridge was subjected to two (2) additional earthquakes that represent distinct classes, namely near-field (impulsive-type) and far-field earthquakes. Specifically, the NS component observed at Shiofukizaki site in the 1989 Ito-Oki earthquake of moment earthquake magnitude 5.3, epicentral distance of 3 km and the depth of the seismic source of 5 km. The record was observed at the surface of basalt rock and has a maximum acceleration of 0.189 g. It has been characterized as a near-field earthquake and it exhibits remarkable similarity to the Konitsa 1996 earthquake (**Figure 44**, top).

**Figure 44.** Acceleration time histories of Ito-Oki 1989 and El-Centro1940 PGA-adjusted earthquakes.

The second earthquake is the 1940 El-Centro normalized to 0.19 g (**Figure 44**, bottom) allowing for direct comparison with the similar PGA Konitsa-1996 and Ito-Oki near-field earthquakes. The direct comparison of the response spectra of the three earthquakes (Konitsa-1996, Ito-Oki and normalized 1940 El-Centro) is shown in **Figure 45**. The objective of subjecting the Konitsa Bridge to the same PGA but different spectral content earthquakes is to directly compare the damageability potential based on the non-linear response of the bridge and shed some light on sensitivities to the type of earthquake these type of structures (masonry stone bridges) exhibit. This ultimately will aid in the modification/updating of the seismic codes to capture the unique structural design and response characteristics of large span arch masonry bridges in their provisions. While for the Konitsa-1996 earthquake the actual vertical acceleration was used, for the Ito-Oki and modified 1940 El-Centro the vertical component was assumed as 75% of the employed horizontal component. The results drawn from the three (3) non-linear analyses (Konitsa-1996, Ito-Oki and 1940 El-Centro) and the comparative damageability potential are very revealing. Specifically, very similar response and bridge damage are observed for the two impulsive-type earthquakes, *M* =5.7 Konitsa-1996 and *M* = 5.3 Ito-Oki earthquakes, which are similar PGA and time structure. Their damage potential is quite limited and it confirms the observations made post 1996-Konitsa earthquake of the bridge. **Figure 46a** and **b** depict the Konitsa Bridge out-of-plane displacement and stress response, respectively. On this basis and by comparing these maximum response values with the corresponding maximum values obtained utilizing the 1996-Konitsa earthquake record as input motion (**Figures 30a** and **b**, **40b**, **41a** and **b**), it can be concluded that the potential damage vulnerability from the Ito-Oki earthquake resembles that of the Konitsa-1996 earthquake.

**Figure 45.** Acceleration response spectra (2% damping) of the similar PGA but different type (near- vs. far-field) earthquakes utilized in the study.

**Figure 44.** Acceleration time histories of Ito-Oki 1989 and El-Centro1940 PGA-adjusted earthquakes.

118 Structural Bridge Engineering

from the Ito-Oki earthquake resembles that of the Konitsa-1996 earthquake.

The second earthquake is the 1940 El-Centro normalized to 0.19 g (**Figure 44**, bottom) allowing for direct comparison with the similar PGA Konitsa-1996 and Ito-Oki near-field earthquakes. The direct comparison of the response spectra of the three earthquakes (Konitsa-1996, Ito-Oki and normalized 1940 El-Centro) is shown in **Figure 45**. The objective of subjecting the Konitsa Bridge to the same PGA but different spectral content earthquakes is to directly compare the damageability potential based on the non-linear response of the bridge and shed some light on sensitivities to the type of earthquake these type of structures (masonry stone bridges) exhibit. This ultimately will aid in the modification/updating of the seismic codes to capture the unique structural design and response characteristics of large span arch masonry bridges in their provisions. While for the Konitsa-1996 earthquake the actual vertical acceleration was used, for the Ito-Oki and modified 1940 El-Centro the vertical component was assumed as 75% of the employed horizontal component. The results drawn from the three (3) non-linear analyses (Konitsa-1996, Ito-Oki and 1940 El-Centro) and the comparative damageability potential are very revealing. Specifically, very similar response and bridge damage are observed for the two impulsive-type earthquakes, *M* =5.7 Konitsa-1996 and *M* = 5.3 Ito-Oki earthquakes, which are similar PGA and time structure. Their damage potential is quite limited and it confirms the observations made post 1996-Konitsa earthquake of the bridge. **Figure 46a** and **b** depict the Konitsa Bridge out-of-plane displacement and stress response, respectively. On this basis and by comparing these maximum response values with the corresponding maximum values obtained utilizing the 1996-Konitsa earthquake record as input motion (**Figures 30a** and **b**, **40b**, **41a** and **b**), it can be concluded that the potential damage vulnerability

**Figure 46.** Konitsa Bridge response to the *M* = 5.3 Ito-Oki (0.19g PGA) near-field (impulsive) earthquake. (a) Out-ofplane displacement response and (b) tensile stress response.

**Figure 47.** Konitsa Bridge out-of-plane displacement response to PGA-adjusted (0.19 g) 1940 El-Centro earthquake.

A strikingly different bridge response and damage potential are observed when Konitsa Bridge is subjected to an excitation with the 0.19-g normalized 1940 El-Centro earthquake, which represents a different type (far-field) of seismic event lacking that characteristic dominant velocity pulse (**Figure 29**). **Figures 47** and **48** clearly demonstrate the different damage potential of this type of earthquake on such relatively long-span stone-masonry bridges. **Figure 47** depicts the Konitsa Bridge out-of-plane displacement response when subjected to PGAadjusted (0.19-g) 1940 El-Centro earthquake. **Figure 48** depicts the variation of tensile stresses together with relevant non-linear deformations of large amplitude at critical locations of the main arch during certain time windows of the response when these deformations are maxi-

**Figure 48.** Evolution of damage resulting from the El-Centro (0.19-g) far-field-type earthquake. (a) Time = 2.32 s, (b) 3.4 s, (c) 3.6 s and (d) 3.8 s.

mized, thus indicating the collapse potential of that portion of the main arch. This prediction of the main arch performance is in agreement with a similar conclusion reached by the simplified analyses of Section 7.1 when Konitsa Bridge (**Figures 27b**, **28b** and **d**) and Plaka Bridge (**Figures 33** and **34**) were subjected to the design spectra as defined employing provisions of Euro-Code 8. This large variation in the damageability potential, therefore, should be accounted for in establishing seismic code guidelines for relatively fragile old cultural heritage structures (as the old stone-masonry bridges studied in this chapter) as they apply to these non-typical structures. It should be noted that similar conclusions regarding the damageability of near-field-type earthquakes, as compared to their far-field counterparts based on which seismic codes for nuclear structures were deduced, were reached following an International Atomic Energy Agency (IAEA)-launched coordinated research project (CRP) experimental study [23] augmented with numerical analysis and response/damage predictions conducted by an international participation.
