**6. Numerical simulation of dynamic characteristics**

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]. The various main parts of these stone-masonry bridges, that is, the primary and the secondary arches, the abutments, the deck, the mandrel walls and the parapets, were simulated in such a way that narrow contact surfaces could be introduced between them, representing in this way a different 'softer' medium. 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. Young's modulus of the stone samples in axial compression has a value exceeding 40 GPa, whereas they yield a much less stiff behaviour in flexure. 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.

**Code name of sample Width**

94 Structural Bridge Engineering

**Cross section (mm2 )**

**Code name of sample**

**(mm)**

**Height (mm)**

**Height (mm)**

**Table 5.** Flexure tests (18 December 2015) with stone samples taken at Plaka Bridge.

**Maximum load**

**Table 6.** Compression tests (28 January 2016) with mortar samples taken at Plaka Bridge.

**6. Numerical simulation of dynamic characteristics**

**(kN)**

**Span (mm)**

Specimen A 52.0 52.0 180.0 14.75 18.88 33,330 Specimen B 52.0 52.0 180.0 12.13 15.52 36,360 **Plaka Bridge stone specimens Average tensile flex. strength = 17.20 MPa,** *E***<sup>2</sup> = 34,845 MPa**

Specimen 1 27.5 × 57.0 66.0 3.228 2.06 1.562/0.91 1.875 **Plaka Bridge stone specimens Compressive strength = 1.875 MPa,** *E***<sup>1</sup> = 2500 MPa,** *ν* **= 0.35**

**Maximum vertical load (kN)**

**Compressive strength (MPa)**

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]. The various main parts of these stone-masonry bridges, that is, the primary and the secondary arches, the abutments, the deck, the mandrel walls and the parapets, were simulated in such a way that narrow contact surfaces could be introduced between them, representing in this way a different 'softer' medium. 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. Young's modulus of the stone samples in axial compression has a value exceeding 40 GPa, whereas they yield a much less stiff behaviour in flexure. 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

**Tensile strength (MPa)**

**Slenderness ratio\***

**correction coefficient** **/**

**Compressive strength (MPa) with correction due to slenderness\***

**Young's Modulus from flexure (MPa) (S.G.)**


**Table 7.** Comparison of *measured/predicted* eigen-frequencies for four stone-masonry bridges (*pinned* boundary conditions).


*Konitsa Bridge*. Emasonry = 4000 MPa, Econtact = 2000 MPa. Bending Stiffness Modifiers = 3.0. *Kokorou Bridge*. Emasonry = 4000 MPa, Econtact = 2000 MPa. Bending Stiffness Modifiers = 1.75. *Tsipianis Bridge*. Emasonry = 4000 MPa, Econtact = 2000 MPa. Bending Stiffness Modifiers = 1.0. *Kontodimou Bridge*. Emasonry = 1600 MPa, Econtact = 1600 MPa. Bending Stiffness Modifiers = 1.0.

**Table 8.** Comparison of *measured/predicted* eigen-frequencies for four stone-masonry bridges (*fixed* boundary conditions).

**Figure 24.** Numerical and observed eigen-values for the *Konitsa Bridge*. Emasonry =4000MPa, Econtact =2000MPa. Bending Stiffness Modifiers = 3.0.

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. This is denoted in the predicted eigen-frequency values in **Tables 7** and **8** and **Figures 24** and **25** with the subscript '*Fixed Numer*'. 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. This is denoted in the predicted eigen-frequency values in **Tables 7** and **8** and **Figures 24** and **25** with the subscript '*Pinned Numer*'. 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

**Figure 25.** Numerical eigen-values for the Plaka Bridge. *Plaka Bridge*. Emasonry = 4000 MPa, Econtact = 2000 MPa. Bending stiffness modifiers = 3.0.

**Figure 24.** Numerical and observed eigen-values for the *Konitsa Bridge*. Emasonry =4000MPa, Econtact =2000MPa. Bending

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

Stiffness Modifiers = 3.0.

96 Structural Bridge Engineering

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.
