**7. Simplified numerical investigation of the seismic behaviour of the studied stone-masonry bridges**

## **7.1. Simplified dynamic spectral numerical simulation of the seismic behaviour of the Konitsa Bridge**

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. Initially, use is made of the current definition of the seismic forces by EURO-Code 8 [12]. 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 0.16 g (g is the acceleration of gravity) for the location of the Konitsa Bridge. 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 (1.0); the damping ratio is considered equal to 5% and the behaviour factor is equal to 1.5 (unreinforced masonry). 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 1996 at the city of Konitsa located at a distance of approximately 1.5 km from the site of the bridge [15]. In **Figure 26a** and **b**, the eigen-period range of the first 12 eigenmodes 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**).

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. The value of Young's modulus that was adopted for the masonry in these numerical simulations is listed at the bottom of **Table 8** and at the captions of **Figures 24** and **25** that depict the numerical eigen-mode and eigen-frequency numerical results together with the measured *in situ* eigen-frequency values for each bridge. Thus, for the Konitsa (**Figure 24**), Kokorou and Tsipianis Bridges, 4 GPa was adopted for the masonry Young's modulus and 2 GPa for the contact surface. For the Kontodimou Bridge, these values were 1.6 GPa for both the masonry and the contact surface. 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 3.0 and 1.75, respectively. 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.

**studied stone-masonry bridges**

**Konitsa Bridge**

98 Structural Bridge Engineering

**7. Simplified numerical investigation of the seismic behaviour of the**

**7.1. Simplified dynamic spectral numerical simulation of the seismic behaviour of the**

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. Initially, use is made of the current definition of the seismic forces by EURO-Code 8 [12]. 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 0.16 g (g is the acceleration of gravity) for the location

**Figure 26.** (a) Horizontal spectral curves for the 1996-Konitsa earthquake and the type-1 Euro-Code and (b) vertical spectral curves for the 1996-Konitsa earthquake and the type-1 Euro-Code.

As can be seen in **Figure 26a**, the Euro-Code horizontal acceleration spectral curves compare well with the horizontal component-3 of 1996 Earthquake spectral curves for the period range of interest. The Euro-Code vertical acceleration spectral curves, depicted in **Figure 26b**, are approximately 100% larger than the vertical component-2 of 1996 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 1996 earthquake sequence.

In **Table 10**, the base reactions are listed (*FX*, *FY* and *FZ*) 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 1996-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 eigenmodes listed in **Table 9** were employed.


**Table 9.** Modal participating mass ratios for Konitsa Bridge (see **Figure 24**).

As can be seen in **Table 9**, these eigen-modes have modal mass participation ratios that result in sums smaller than 90%. That is, Sum*Ux* = 64.9%, Sum*Uy* = 50.2% and Sum*Uz* = 36.6% of the total mass for the direction of motion in the *Ux*, *Uy* and *Uz* axes, respectively. This was accounted for in the subsequent load combinations where the dead load is combined with the horizontal and vertical spectral curves (rows 5 and 6 of **Table 10**, Combination 1, 1996 earthquake horizontal + vertical spectral curves and rows 10 and 11 of **Table 10**, Combination 7 Euro-Code horizontal + vertical spectral curves). 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 amplification factor is equal to 1/Sum*Ux* for the dynamic analyses employing the out-of-plane *x*-*x* horizontal eigen-modal ratio, to 1/ Sum*Uy* for the in-plane *y*-*y* horizontal eigen-modal ratios and to 1/Sum*Uz* for the in-plane vertical eigen-modal ratios [11]. This becomes evident when one compares the base reaction values without and with these amplification factor values in **Table 10**.


**Table 10.** Base reactions from the dynamic spectral analyses, Konitsa Bridge.

**Output case Text**

(first OOP Symmetric)

100 Structural Bridge Engineering

(first IP asymmetric)

(second IP symmetric)

(third OOP symmetric)

(third IP symmetric)

(fourth OOP asymmetric)

(fourth IP asymmetric)

(fifth OOP symmetric)

(fifth IP symmetric)

(sixth IP asymmetric)

(sixth OOP asymmetric)

**Table 9.** Modal participating mass ratios for Konitsa Bridge (see **Figure 24**).

(second OOP asymmetric)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

Mode 9

Mode 10

Mode 11

Mode 12

**Period s**

**Frequency Hz**

*UX* **Unitless** *UY* **Unitless**

0.396 2.5264 0.3300 0 0 0.33002 0 0

0.210 4.7588 0.0004 0 0 0.33043 0 0

0.149 6.7333 0.0 0.08498 0.00055 0.33043 0.08498 0.00055

0.141 7.0777 0.0 0.00104 0.12769 0.33043 0.08602 0.12825

0.130 7.7062 0.2271 0 0 0.55753 0.08602 0.12825

0.099 10.0851 0.0 0.0007 0.15393 0.55753 0.08672 0.28218

0.086 11.6697 0.0 0 0 0.55754 0.08672 0.28218

0.074 13.4317 0.0 0.17916 0.0023 0.55754 0.26588 0.28447

0.065 15.2999 0.08851 0 0 0.64605 0.26588 0.28447

0.062 16.0400 0.0 0.23382 0.000014 0.64605 0.49971 0.28449

0.0565 17.6794 0.0 0.0026 0.08175 0.64605 0.5023 0.36624

0.0522 19.1439 0.00262 0 0 0.64868 0.5023 0.36624

As can be seen in **Table 9**, these eigen-modes have modal mass participation ratios that result in sums smaller than 90%. That is, Sum*Ux* = 64.9%, Sum*Uy* = 50.2% and Sum*Uz* = 36.6% of the total mass for the direction of motion in the *Ux*, *Uy* and *Uz* axes, respectively. This was accounted for in the subsequent load combinations where the dead load is combined with the horizontal and vertical spectral curves (rows 5 and 6 of **Table 10**, Combination 1, 1996 earthquake horizontal + vertical spectral curves and rows 10 and 11 of **Table 10**, Combination 7 Euro-Code horizontal + vertical spectral curves). 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 amplification factor is equal to 1/Sum*Ux* for

*UZ* **Unitless** **Sum***UX* **Unitless** **Sum***UY* **Unitless** **Sum***UZ* **Unitless**

> 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-ofplane 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 30.6 mm. In **Figure 28a–h**, the numerically predicted state of stress (max/min S11, max/min S22), which develops at Konitsa Bridge for load combinations 1 and 7, is depicted. 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 (3.46 MPa, **Figure 28b**) develop at the bottom fibre of the crown of the arch. 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 (1.5 1 MPa, **Figure 28f**) develops at the area where the primary arch joins the foundation block. 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. On the contrary, the largest value of compression stress equal to S11 = −4.3 MPa (**Figure 28d**, for combination 7) is expected to be easily met by the compression capacity of the stone masonry for this bridge [16].

**Figure 27.** (a) Deformations of Konitsa Bridge. For loads Dead + 1996 EQ RS (*u*1+ *u*2 + *u*3). Comb 1. At crown *u*1 = 24.643 mm, *u*2 = 1.209 mm, *u*3 = −10.593 mm. (b) Deformations of Konitsa Bridge. For loads Dead + Euro-Code RS (*u*1+ *u*2 + *u*3). Comb 7. At crown *u*1 = 30.573 mm, *u*2 = 1.348 mm, *u*3 = −15.804 mm.

**Figure 28.** State of stress through the distribution of stresses S11 and S22 for Konitsa Bridge.

## **7.2. Dynamic elastic time-history numerical simulation of the seismic behaviour of the Konitsa Bridge**

An additional linear numerical simulation was performed. This time, apart from the dead load (D, row 1, **Table 11**), the Konitsa Bridge was subjected to the horizontal component (Comp3) and/or the vertical component [17] of the 1996-Konitsa earthquake record (**Figure 29**) in the following way. 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 11**). 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].

**Figure 27.** (a) Deformations of Konitsa Bridge. For loads Dead + 1996 EQ RS (*u*1+ *u*2 + *u*3). Comb 1. At crown *u*1 = 24.643 mm, *u*2 = 1.209 mm, *u*3 = −10.593 mm. (b) Deformations of Konitsa Bridge. For loads Dead + Euro-Code RS (*u*1+

*u*2 + *u*3). Comb 7. At crown *u*1 = 30.573 mm, *u*2 = 1.348 mm, *u*3 = −15.804 mm.

102 Structural Bridge Engineering

**Figure 28.** State of stress through the distribution of stresses S11 and S22 for Konitsa Bridge.

The solution this time was obtained through a step-by-step time integration scheme assuming a damping ratio equal to 5% of critical. In these analyses, only the first most intense 6 s of this 1996-Konitsa earthquake record were used [15]. In **Table 11**, the base shear values in the *x*-*x* (*FX*, *u*1, horizontal out-of-plane), *y*-*y* (*FY*, *u*2, horizontal in-plane) and *z*-*z* (*FZ*, *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. Limit (maximum or minimum) base shear *FX*, *FY*, *FZ* values are also listed in rows 8–13 when these seismic excitations (Ex, Ey and Ez) are combined within themselves and the dead load as is shown in the third column of **Table 11** to produce load combinations encoded as COMB9, COMB10 and COMB11. By comparing these base shear values with the ones listed in **Table 10** where the response spectral curves of either the 1996- Konitsa record or the Euro-Code were employed, it can be seen that the limit (max/min) base shear amplitudes in both tables are very similar. **Figure 30a** shows the horizontal (*ux*, out-ofplane) and the vertical (*uz*, 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 1996 earthquake record and the vertical in-plane response when the structure is subjected to vertical component (Comp2) of the Konitsa 1996 earthquake record (**Figure 29**). **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 1996 earthquake record (Comp3) in the out-of-plane (*ux*) direction or the vertical component of the Konitsa 1996 earthquake record (Comp2) in the vertical (*uz*) 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 *ux* displacement and S11 stress response produced by the horizontal out-of-plane excitation are larger than the corresponding response vertical *uz* 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 eigenfrequency values (**Figures 24**, **26a** and **b**), the vertical *uz* displacement and S11 stress response, produced by the vertical in-plane excitation, are of higher frequency content than the horizontal *ux* displacement and S11 stress response produced by the horizontal out-of-plane excitation.


**Table 11.** Base reactions from time-history analyses: Konitsa Bridge.

**Figure 29.** The first eight (8) most intense seconds of the 1996 earthquake record (ITSAK).

**Loading case description**

104 Structural Bridge Engineering

**Loading type description Type**

1 DEAD Dead Load (D) Linear Static 0 0 35,853

8 COMB9 Dead + Ex + Ez Max 5254.6 128.6 38,504.2 9 COMB9 Dead + Ex + Ez Min −5786.6 −140.2 33,472.5 10 COMB10 Dead + Ey + Ez Max 0 4737.3 38,592.4 11 COMB10 Dead + Ey + Ez Min 0 -6596.9 33,390.7 12 COMB11 Dead + Ex + Ey + Ez Max 5254.6 4737.3 38,592.4 13 COMB11 Dead + Ex + Ey + Ez Min −5786.6 −6596.9 33,390.7

2 Comp 2 TH Ver *u*3 IP Konitsa 1996 Comp 2 THist. Ver *u*3 In-Plane (Ez)

3 Comp 2 TH Ver *u*3 IP Konitsa 1996 Comp 2 THist. Ver *u*3 In-Plane (Ez)

4 Comp 3 TH Hor *u*1 OP Konitsa 1996 Comp 3 THist. Hor *u*1

5 Comp 3 TH Hor *u*1 OP Konitsa 1996 Comp 3 THist. Hor *u*1

6 Comp 3 TH Hor *u*2 IP Konitsa 1996 Comp 3 THist. Hor *u*2 In-Plane (Ey)

7 Comp 3 TH Hor *u*2 IP Konitsa 1996 Comp 3 THist. Hor *u*2 In-Plane (Ey)

**Table 11.** Base reactions from time-history analyses: Konitsa Bridge.

**Figure 29.** The first eight (8) most intense seconds of the 1996 earthquake record (ITSAK).

Out-of-Plane (Ex)

Out-of-Plane (Ex)

**limit**

**Global** *FX* **(kN)**

**Global** *FY* **(kN)**

Max 0 128.6 2651.2

Min 0 −140.2 −2380.5

Max 5254.6 0 0

Min −5786.6 0 0

Max 0 4608.7 88.2

Min 0 −6456.7 −81.9

**Global** *FZ* **(kN)**

**Figure 30.** (a) Displacement (Hor. or Ver.) response at the crown of the Konitsa Bridge when subjected to either the horizontal or the vertical component of the Konitsa 1996 earthquake. (b) S11 stress response at the bottom of crown of the Konitsa Bridge when subjected to either the horizontal or the vertical component of the Konitsa 1996 earthquake.

**Figure 31a** and **b** depict the envelop of the limit (maximum/minimum) values of the S11 stress distribution in the Konitsa Bridge for load combination 11 that includes the dead load, the application of Comp3 of the Konitsa earthquake record in both the horizontal in-plane and out-of-plane direction as well as Comp2 of the Konitsa earthquake record in the vertical inplane direction. By examining the displacement and stress response, it could be concluded that the application of the horizontal component of the Konitsa 1996 in the horizontal *uy* 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. By comparing these S11 stress response maximum/minimum values with the ones shown in **Figure 28** where the response spectral curves of either the 1996-Konitsa record or the Euro-Code were employed (Section 7.1.), it can be seen that the limit (max/min) S11 stress maximum/minimum amplitudes is very similar, as expected, to the corresponding values obtained from the dynamic spectral analyses employing the 1996-Konitsa record spectral curves. As was discussed before, the Euro-Code design

**Figure 31.** State of stress through the distribution of stresses S11 (envelope) at the bottom fibre of the crown for Konitsa Bridge.

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 7.3.

#### **7.3. Simplified numerical simulation of the seismic behaviour of the Plaka Bridge**

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 7.1 by making use of the current definition of the seismic forces by EURO-Code 8 [12]. Towards this, horizontal and vertical design spectral curves are derived based on the horizontal design ground acceleration. This value, as it is defined by the zoning map of the current Seismic Code of Greece, is equal to 0.24 g (g is the acceleration of gravity) for the location of the Plaka Bridge [13, 14]. 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 (1.0), the damping ratio is considered equal to 5% and the behaviour factor is equal to 1.5 (unreinforced masonry). 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 11**). 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.

**Figure 32.** (a) Horizontal spectral curves for type-1 Euro-Code to be applied in Plaka bridge and (b) vertical spectral curves for type-1 Euro-Code to be applied in Plaka bridge.

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 *FX*, in the *x*-*x* (*u*1, out-of-plane), *FY* the *yy* (*u*2, in-plane) and *FZ* 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 7.1., the Euro-Code horizontal and vertical design spectral curves of **Figure 32a** and **b**. In all these dynamic spectral analyses, the 12 eigen-modes listed in **Table 12** were again employed. As can be seen in **Table 12**, these eigen-modes have modal mass participation ratios that result in sums that are Sum*Ux* = 67.4%, Sum*Uy* = 58.7% and Sum*Uz* = 39.3% of the total mass for the direction of motion in the *Ux*, *Uy* and *Uz* axes, respectively. In the subsequent load combination 1, where the dead load is combined with the Euro-Code horizontal + vertical spectral curves, the dynamic spectral analysis results were multiplied again by an amplification factor equal to the reverse of the relevant ratio values before superimposing the dead load results. This amplification factor is equal to 1/Sum*Ux* for the dynamic analyses employing the out-of-plane *x*-*x* horizontal eigen-modal ratio, to 1/Sum*Uy* for the in-plane *y*-*y* horizontal eigen-modal ratios and to 1/Sum*Uz* for the in-plane vertical eigen-modal ratios [11]. This becomes evident when one compares the base reaction values without and with these amplification factor values in **Table 13**.

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 dem-

onstrated by the numerical analyses of the Plaka Bridge in the following Section 7.3.

**7.3. Simplified numerical simulation of the seismic behaviour of the Plaka Bridge**

demanding seismic force level than the latter.

106 Structural Bridge Engineering

curves for type-1 Euro-Code to be applied in Plaka bridge.

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 7.1 by making use of the current definition of the seismic forces by EURO-Code 8 [12]. Towards this, horizontal and vertical design spectral curves are derived based on the horizontal design ground acceleration. This value, as it is defined by the zoning map of the current Seismic Code of Greece, is equal to 0.24 g (g is the acceleration of gravity) for the location of the Plaka Bridge [13, 14]. 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 (1.0), the damping ratio is considered equal to 5% and the behaviour factor is equal to 1.5 (unreinforced masonry). 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 11**). 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

**Figure 32.** (a) Horizontal spectral curves for type-1 Euro-Code to be applied in Plaka bridge and (b) vertical spectral


**Table 12.** Modal participating mass ratios for Plaka Bridge (see **Figure 25**).


**Table 13.** Base reactions, Plaka Bridge.

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 52.84 mm. 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.

**Figure 33.** (a) Deformations of Plaka Bridge. For loads Dead + Euro-Code RS (*u*1+ *u*2 + *u*3). Comb 1. At crown *u*1 = −52.84 mm, *u*2 = −21.13 mm, *u*3 = −22.11 mm. (b) Deformations of Plaka Bridge. For loads Dead + Euro-Code RS (*u*1 + *u*2 + *u*3). Comb 1. At crown *u*1 = −52.84 mm, *u*2 = −21.13 mm, *u*3 = −22.11 mm.

In **Figure 34a–d**, the numerically predicted state of stress (max/min S11, max/min S22), which develops at Plaka Bridge for load combination 1, is depicted. 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 (5.73 MPa, **Figure 34a**) develops at the bottom fibre of the crown of the arch. 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 (3.40 MPa, **Figure 34c**) develops at the area where the toes of the primary arch join the foundation block. 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. On the contrary, the largest value of compressive stress equal to S11 = −6.14 MPa (**Figure 34d**, for combination 1) could be met by the compression capacity of the stone masonry for this bridge. 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 due to the seismic forcing levels, which for Plaka Bridge are by 50% higher than those applied for Konitsa ridge. This is because Plaka Bridge is located in seismic zone II (design ground acceleration equal to 0.24 g) whereas Konitsa Bridge is located at seismic zone I (design ground acceleration equal to 0.16 g). 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

**Loading case description**

108 Structural Bridge Engineering

2 Euro-Code RS Hor *u*1 OP

3 Euro-Code RS Hor *u*2 IP

4 Euro-Code RS Ver *u*3 IP

Konitsa Bridge.

**Loading type description**

Linear Resp. Spectral Euro-Code

Linear Resp. Spectral Euro-Code

Linear Resp. Spectral Euro-Code

(*u*1+ *u*2 + *u*3)

(*u*1+ *u*2 + *u*3)

5 Combination 1 Dead + Euro-Code RS

6 Combination 1 Dead + Euro-Code RS

**Table 13.** Base reactions, Plaka Bridge.

1 DEAD (D) Linear Static 0 0 42,544

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 52.84 mm. This maximum out-of-plane value for the Plaka Bridge is almost twice as large as the corresponding value predicted numerically for the

**Figure 33.** (a) Deformations of Plaka Bridge. For loads Dead + Euro-Code RS (*u*1+ *u*2 + *u*3). Comb 1. At crown *u*1 = −52.84 mm, *u*2 = −21.13 mm, *u*3 = −22.11 mm. (b) Deformations of Plaka Bridge. For loads Dead + Euro-Code RS (*u*1 +

In **Figure 34a–d**, the numerically predicted state of stress (max/min S11, max/min S22), which develops at Plaka Bridge for load combination 1, is depicted. Again, as expected, the most demanding state of stress results is for the load combination 1 that includes seismic forces

*u*2 + *u*3). Comb 1. At crown *u*1 = −52.84 mm, *u*2 = −21.13 mm, *u*3 = −22.11 mm.

**Type limit Global** *FX* **(kN) Global** *FY* **(kN) Global** *FZ* **(kN)**

Max 6382 0 0

Max 0 5773 311

Max 0 677 6581

Max 9436 11,555 59,823

Min −9436 −11,555 25,265

**Figure 34.** State of stress through the distribution of stresses S11 and S22 for Plaka Bridge.

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.
