Seismic Assessment of Six Typologies of Existing RC Bridges

: Over the last few decades, the attention on the safety of existing reinforced concrete (RC) structures has signiﬁcantly increased. RC bridges, in particular, are highly relevant for strategic importance. In the Italian context, several of these bridges were built around 1960, when engineering practice commonly ignored or underestimated the presence of seismic actions. Therefore, it is fundamental to quantify as accurately as possible their seismic safety level with state-of-the-art analysis techniques. In this paper, an e ﬃ cient procedure based on the multi-modal pushover analysis approach is proposed for the risk evaluation of several bridges of the Italian highway network. This procedure, tailored for portfolio level assessment, takes into account the non-linear behavior and the complex dynamic response this type of structure with limited computational e ﬀ ort. Three fundamental aspects are deﬁned for the structural modelling of bridges, i.e., materials’ constitutive law, ﬁnite element type and nonlinear hinge models. Flexural and shear nonlinearities of piers are included to account for ductile and brittle damage potential. The standardized procedure guarantees consistent comparisons among di ﬀ erent bridges of the same network in the form of risk indexes.


Introduction
During last years, several collapse phenomena have affected existing reinforced concrete (RC) bridges that led to considerable interest on the evaluation of the residual capacity of these structures under static and dynamic loads [1,2]. For this reason, several researches have presented different types of approaches in order to evaluate the safety level of the existing bridges or of other types of strategic infrastructures [3][4][5][6][7][8][9][10]. In the Italian Highway Network, the majority of these bridges were built in the 1960s and 1970s. Nowadays they require several maintenance operations in order to ensure standard safety levels. Moreover, according to new design codes, particular attention is given to the seismic capacity evaluation assessment of these structures. For instance, the Italian Civil Protection requires highway managing operators to collect relevant data of each asset in their portfolios and to conduct nonlinear seismic assessments for emergency planning and investment prioritization purposes [11].
Several issues remain arises when evaluating the seismic response of existing reinforced concrete (RC) bridges with nonlinear techniques [3]. On one hand, standard nonlinear pushover analysis, which is nowadays widely used in structural engineering firms, e.g., [12], fails to address the complex dynamic response of bridges that are not characterized by a predominant vibration mode. On the other hand, nonlinear time history analyses involve several challenges for professional engineers such as ground motion selection, modeling of strength/stiffness degradation, high computational cost, etc. [13]. An alternative solution presented in the literature is the Modal Pushover Analysis Infrastructures 2020, 5, 52 2 of 14 (MPA) that was initially developed by Chopra and Goel [14,15] to assess the seismic response of unsymmetrical-plan buildings. The MPA consists of an extension of the Response Spectrum Analysis (RSA), particularly effective for irregular structures that do not exhibit a principal mode shape with high participating mass. The methodology was furtherly extended to the case of bridges thanks to the work of Kappos et al. [16,17].
In this paper, an efficient procedure is proposed in order to evaluate the seismic vulnerability of bridges taking into consideration the above-mentioned aspects. This procedure is based on the implementation of Finite Element Models (FEM) where the non-linear behavior of the piers is represented with concentrated plastic hinges that consistently reduce the computational effort allowing the execution of MPA analyses. The result of the assessment is expressed in terms of Risk Index, i.e., the ratio between the maximum Peak Ground Acceleration (PGA) that the bridge can withstand (capacity) and the PGA of the site asset at stake for the given location (demand). The procedure is applied to six representative case studies of a bridge portfolio characterized by cantilever and frame type piers. The results are discussed highlighting the critical aspects of each typology with respect to their seismic behavior.

Multi-Modal Pushover Approach
Pushover analysis is commonly used for the evaluation of the non-linear behavior of an existing structure or infrastructure subjected to an incremental horizontal load. Three base concepts regulate the application of the pushover analysis [18]: (a) the capacity curve, (b) the demand spectrum and (c) the performance point.
The capacity curve defines the nonlinear response of a structure subjected to a predefined lateral load distribution. The curve usually consists in a top displacement versus base shear diagram. The shape of the lateral load profile is usually proportional to a mode shape: where: M is the mass matrix of the structure, φ n is the n-th eigenvector, s n * is the loading vector applied to the structure during the analysis. The obtained curve can be converted in the spectral displacement (S d ) versus spectral acceleration (S e ) plane through these fundamental relations: where: V bn is the base shear for the n-th vibration mode, u rn is the top displacement value of the control point for the n-th vibration mode, M n * is the modal mass of the n-th mode, Γ n is the modal participation factor and φ rn is the control point component of the n-th eigenvector. The seismic demand curve can be represented in the Acceleration Displacement Response Spectrum (ADRS) format, obtained from the horizontal acceleration response spectrum using the formula below: The performance point is obtained by intersecting the capacity curve with the demand curve and represents how the structure would behave under the specific seismic action (Figure 1a). Several techniques have been proposed in order to evaluate the performance point. An extended state-of-the-art review is reported by Causevic and Mitrovic [19]. Through a bilinearization of the capacity curve (Figure 1b), a demand reduction coefficient is obtained. This can be evaluated in terms of ductility factor or equivalent damping and takes into account the energy dissipation of the postelastic phase. The intersection between the reduced demand spectrum and the capacity curve identifies the performance point. In this work, the Capacity Spectrum Method (CSM) technique is adopted [18,20,21] where these fundamental steps are executed: 1. Definition of the seismic demand in the ADRS form; 2. Selection of the first iteration point api, dpi on the capacity; 3. Bilinearization of the capacity curve with KI as elastic stiffness followed by a hardening branch. The hardening branch is defined by applying the equal energy rule between the capacity curve and its bilinear idealization ( Figure 1b); 4. Scaling of the ADRS according to the effective damping coefficient. This takes into consideration both the hysteretic damping (referred to the cyclic plastic deformations) and the inherent damping (equal to 5% in the case of concrete structures), Figure 1c; 5. Evaluation of the performance point by intersecting the capacity curve and the scaled demand spectrum through an iterative process. The selection of the horizontal load profile for the pushover analysis is not univocal and can decisively influence the results. As discussed in the Introduction, there has been consistent research on the topic, e.g., [13]. In this work, mode-shape load profiles are adopted as for Chopra and Goel [14]. Operationally, N capacity curves are determined, one for each significant vibration mode. For each capacity curve, the performance point is evaluated with reference to the same seismic demand spectrum. Lastly, relevant internal-forces/displacements at the performance configuration are extracted and combined with the classical modal combination rules (e.g., CQC).

Structural Modelling
A fundamental step in assessing the seismic vulnerability of existing structures is the determination of the actual materials' characteristics through the execution of laboratory/in-situ tests. It is worth mentioning that, within the same structure, the variability of mechanical properties can be high. Therefore, the Italian Building Code [11,22] requires the use of an appropriate confidence factor that is related to the level of knowledge obtained through the survey campaign. Consequently, materials' strength is reduced for structural verifications. In absence of specific laboratory investigations, concrete and steel mechanical properties are taken from technical-scientific studies [23,32].
The materials' constitutive laws should take into account the mechanical phenomena that occur at both element and cross section levels. The concrete behavior is significantly influenced by the confining effect of transverse reinforcement. In this work, the concrete model developed by Kent and Park [24] was chosen, considering only the compressive behavior. The Kent and Park concrete model Several techniques have been proposed in order to evaluate the performance point. An extended state-of-the-art review is reported by Causevic and Mitrovic [19]. Through a bilinearization of the capacity curve (Figure 1b), a demand reduction coefficient is obtained. This can be evaluated in terms of ductility factor or equivalent damping and takes into account the energy dissipation of the post-elastic phase. The intersection between the reduced demand spectrum and the capacity curve identifies the performance point. In this work, the Capacity Spectrum Method (CSM) technique is adopted [18,20,21] where these fundamental steps are executed:

1.
Definition of the seismic demand in the ADRS form; 2.
Selection of the first iteration point a pi , d pi on the capacity; 3.
Bilinearization of the capacity curve with K I as elastic stiffness followed by a hardening branch. The hardening branch is defined by applying the equal energy rule between the capacity curve and its bilinear idealization ( Figure 1b); 4.
Scaling of the ADRS according to the effective damping coefficient. This takes into consideration both the hysteretic damping (referred to the cyclic plastic deformations) and the inherent damping (equal to 5% in the case of concrete structures), Figure 1c; 5.
Evaluation of the performance point by intersecting the capacity curve and the scaled demand spectrum through an iterative process.
The selection of the horizontal load profile for the pushover analysis is not univocal and can decisively influence the results. As discussed in the Introduction, there has been consistent research on the topic, e.g., [13]. In this work, mode-shape load profiles are adopted as for Chopra and Goel [14]. Operationally, N capacity curves are determined, one for each significant vibration mode. For each capacity curve, the performance point is evaluated with reference to the same seismic demand spectrum. Lastly, relevant internal-forces/displacements at the performance configuration are extracted and combined with the classical modal combination rules (e.g., CQC).

Structural Modelling
A fundamental step in assessing the seismic vulnerability of existing structures is the determination of the actual materials' characteristics through the execution of laboratory/in-situ tests. It is worth mentioning that, within the same structure, the variability of mechanical properties can be high. Therefore, the Italian Building Code [11,22] requires the use of an appropriate confidence factor that is related to the level of knowledge obtained through the survey campaign. Consequently, materials' strength is reduced for structural verifications. In absence of specific laboratory investigations, concrete and steel mechanical properties are taken from technical-scientific studies [23,24].
The materials' constitutive laws should take into account the mechanical phenomena that occur at both element and cross section levels. The concrete behavior is significantly influenced by the confining effect of transverse reinforcement. In this work, the concrete model developed by Kent and Park [25] was chosen, considering only the compressive behavior. The Kent and Park concrete model takes into account the confining effect of stirrups through the confinement parameter K. The coefficient Z defines the post-peak (softening) response of the material (Figure 2a). For the steel reinforcements, the Park Strain Hardening [26] constitutive law was adopted (Figure 2b).
Infrastructures 2020, 5, 52 4 of 15 takes into account the confining effect of stirrups through the confinement parameter K. The coefficient Z defines the post-peak (softening) response of the material (Figure 2a). For the steel reinforcements, the Park Strain Hardening [25] constitutive law was adopted (Figure 2b).
(a) (b) In this work, the FEM models of the reinforced concrete bridges were developed using the software MIDAS Civil [26]. A simplified approach has been adopted where (i) the deck, the piers and the pier cups are schematized with beam elements (ii) the bearings are modeled using general links with stiffness values calculated as in EN 1337-3:2005 [27]. The connection between beam elements and general links is guaranteed thanks to rigid links. The abutments are represented as restrains located at deck-abutments interface bearings base. Lastly, the piers are assumed fixed into rigid foundations. Stiffness reduction due to cracking is taken into account when assessing the natural frequencies of RC structures. This can be done with a specific reduction coefficient of the cross-section elastic stiffness obtained from the moment-curvature (M-χ) diagram, as for EN 1998-2:2005 [28]. Structural and non-structural masses are considered for eigenvalue analysis while traffic loads are neglected [11].
For the nonlinear response of elements two types of mechanisms that characterize piers have been considered: i) the flexural-ductile mechanism and ii) the shear-brittle mechanism. The ductile mechanism refers to the rotational capacity of the plastic hinges while the brittle mechanism depends on the shear strength. These two collapse mechanisms interact and affect simultaneously different structural elements. As a result, it is quite complex to obtain a reliable estimate of the nonlinear dynamic response. In this work, the bridges' capacity has been assessed by investigating these failure mechanisms separately.
The ductile response is modeled with concentrated plastic hinges. Figure 3 shows the example related to the piers characterized by a cantilever behavior.  In this work, the FEM models of the reinforced concrete bridges were developed using the software MIDAS Civil [27]. A simplified approach has been adopted where (i) the deck, the piers and the pier cups are schematized with beam elements (ii) the bearings are modeled using general links with stiffness values calculated as in EN 1337-3:2005 [28]. The connection between beam elements and general links is guaranteed thanks to rigid links. The abutments are represented as restrains located at deck-abutments interface bearings base. Lastly, the piers are assumed fixed into rigid foundations. Stiffness reduction due to cracking is taken into account when assessing the natural frequencies of RC structures. This can be done with a specific reduction coefficient of the cross-section elastic stiffness obtained from the moment-curvature (M-χ) diagram, as for EN 1998-2:2005 [29]. Structural and non-structural masses are considered for eigenvalue analysis while traffic loads are neglected [11].
For the nonlinear response of elements two types of mechanisms that characterize piers have been considered: (i) the flexural-ductile mechanism and (ii) the shear-brittle mechanism. The ductile mechanism refers to the rotational capacity of the plastic hinges while the brittle mechanism depends on the shear strength. These two collapse mechanisms interact and affect simultaneously different structural elements. As a result, it is quite complex to obtain a reliable estimate of the nonlinear dynamic response. In this work, the bridges' capacity has been assessed by investigating these failure mechanisms separately.
The ductile response is modeled with concentrated plastic hinges. Figure 3 shows the example related to the piers characterized by a cantilever behavior. In this work, the FEM models of the reinforced concrete bridges were developed using the software MIDAS Civil [26]. A simplified approach has been adopted where (i) the deck, the piers and the pier cups are schematized with beam elements (ii) the bearings are modeled using general links with stiffness values calculated as in EN 1337-3:2005 [27]. The connection between beam elements and general links is guaranteed thanks to rigid links. The abutments are represented as restrains located at deck-abutments interface bearings base. Lastly, the piers are assumed fixed into rigid foundations. Stiffness reduction due to cracking is taken into account when assessing the natural frequencies of RC structures. This can be done with a specific reduction coefficient of the cross-section elastic stiffness obtained from the moment-curvature (M-χ) diagram, as for EN 1998-2:2005 [28]. Structural and non-structural masses are considered for eigenvalue analysis while traffic loads are neglected [11].
For the nonlinear response of elements two types of mechanisms that characterize piers have been considered: i) the flexural-ductile mechanism and ii) the shear-brittle mechanism. The ductile mechanism refers to the rotational capacity of the plastic hinges while the brittle mechanism depends on the shear strength. These two collapse mechanisms interact and affect simultaneously different structural elements. As a result, it is quite complex to obtain a reliable estimate of the nonlinear dynamic response. In this work, the bridges' capacity has been assessed by investigating these failure mechanisms separately.
The ductile response is modeled with concentrated plastic hinges. Figure 3 shows the example related to the piers characterized by a cantilever behavior.  ultimate capacity of the member. The curvature ductility µ ϑ is the ratio between the ultimate and yield curvatures: The data extracted from the M-χ curve are incorporated in the moment-rotation diagram by integrating the curvatures over the plastic hinge length (L pl ). For constant distribution of the bending moment over L pl , the yield and ultimate rotations are: The plastic hinge length (L pl ) is calculated according to the EC8 [29]: where: d bl is the diameter of the longitudinal bars, f y is the yield stress of the steel rebars and f c is the concrete compressive strength.
The FEMA 356 [20] reports the definition of nonlinear hinge relationships for pushover analysis.  ultimate capacity of the member. The curvature ductility μϑ is the ratio between the ultimate and yield curvatures: The data extracted from the M-χ curve are incorporated in the moment-rotation diagram by integrating the curvatures over the plastic hinge length (Lpl). For constant distribution of the bending moment over Lpl, the yield and ultimate rotations are: The plastic hinge length (Lpl) is calculated according to the EC8 [29]: where: dbl is the diameter of the longitudinal bars, fy is the yield stress of the steel rebars and fc is the concrete compressive strength.
The FEMA 356 [20] reports the definition of nonlinear hinge relationships for pushover analysis.  The verification criterion corresponding to the Life Safety limit state is assumed equal to ¾ of the ultimate rotation ϑu. The brittle collapse mechanism depends on the shear capacity of the piers. It is worth mentioning that RC bridges were generally designed to resist small lateral loads. Thus, their horizontal bearing capacity (e.g., seismic resistance) is quite low.
In the present work, the shear strength of the piers is assumed according to EC8 [29], where the cyclic shear resistance VR in the plastic hinge region accounts for the contribution of three factors: (i) the axial load, (ii) the concrete strength and (iii) the transversal reinforcement. An elastic-brittle forcedisplacement constitutive law is considered in this work [20]. The verification criterion for the Life Safety limit state is assumed equal to the achievement of VR.
The Italian Building Code [11] requires quantification of the seismic response of the bridge at a given location in terms of risk index. The risk index is the ratio between capacity (C) of the bridge and the seismic demand (D) expressed in Peak Ground Acceleration, PGA (or return period). The PGAD is directly taken from the seismic hazard map for the given location. The estimation of the PGAC requires an iterative process. The N pushover curves are intersected with an increasing spectrum until the safety limit on at least one structural member is exceeded. According to the "second level vulnerability assessment form" by the Italian Civil Protection, risk indices are defined as follow [30]: The verification criterion corresponding to the Life Safety limit state is assumed equal to 3 /4 of the ultimate rotation ϑ u . The brittle collapse mechanism depends on the shear capacity of the piers. It is worth mentioning that RC bridges were generally designed to resist small lateral loads. Thus, their horizontal bearing capacity (e.g., seismic resistance) is quite low.
In the present work, the shear strength of the piers is assumed according to EC8 [30], where the cyclic shear resistance V R in the plastic hinge region accounts for the contribution of three factors: (i) the axial load, (ii) the concrete strength and (iii) the transversal reinforcement. An elastic-brittle force-displacement constitutive law is considered in this work [20]. The verification criterion for the Life Safety limit state is assumed equal to the achievement of V R .
The Italian Building Code [11] requires quantification of the seismic response of the bridge at a given location in terms of risk index. The risk index is the ratio between capacity (C) of the bridge and the seismic demand (D) expressed in Peak Ground Acceleration, PGA (or return period). The PGA D is directly taken from the seismic hazard map for the given location. The estimation of the PGA C requires an iterative process. The N pushover curves are intersected with an increasing spectrum until the safety limit on at least one structural member is exceeded. According to the "second level vulnerability assessment form" by the Italian Civil Protection, risk indices are defined as follow [31]: • Risk index in acceleration (RI PGA ): is the ratio between capacity (PGA C ) and demand (PGA D ) in terms of peak ground acceleration; • Risk index in return period (RI TR ): is the ratio between capacity (T RC ) and demand (T RD ) in terms of return period of the earthquake, raised to 0.41 [11,31].
Values close to one or larger than one characterizes cases where the risk level is acceptable. On the contrary, values close to zero characterizes high-risk cases.

Case Studies
The procedure, described in Section 3, has been applied to six different bridges, representative of the Italian Highway Network. In all the models, the longitudinal axis of the bridge is represented by the X axis, while the transversal direction is oriented with Y axis of the coordinate system.
The first bridge ( Figure 5) is characterized by the presence of two adjacent and independent carriageways, consisting in a sequence of seventeen simply supported 36 m spans (except for the central span which is 60 m long). The planimetric and altimetric layout is rectilinear. The overall width of the roadway is about 11 m and each span of the bridge is realized by a precast concrete slab of six prestressed U girders. The bridge deck consists in a 20 cm thick concrete slab. Each span of the bridge is supported by 2 × 6 elastomeric bearings placed at the ends of each longitudinal beam.
Infrastructures 2020, 5, 52 6 of 15 • Risk index in acceleration (RIPGA): is the ratio between capacity (PGAC) and demand (PGAD) in terms of peak ground acceleration; • Risk index in return period (RITR): is the ratio between capacity (TRC) and demand (TRD) in terms of return period of the earthquake, raised to 0.41 [11,30]. Values close to one or larger than one characterizes cases where the risk level is acceptable. On the contrary, values close to zero characterizes high-risk cases.

Case Studies
The procedure, described in Section 3, has been applied to six different bridges, representative of the Italian Highway Network. In all the models, the longitudinal axis of the bridge is represented by the X axis, while the transversal direction is oriented with Y axis of the coordinate system.
The first bridge ( Figure 5) is characterized by the presence of two adjacent and independent carriageways, consisting in a sequence of seventeen simply supported 36 m spans (except for the central span which is 60 m long). The planimetric and altimetric layout is rectilinear. The overall width of the roadway is about 11 m and each span of the bridge is realized by a precast concrete slab of six prestressed U girders. The bridge deck consists in a 20 cm thick concrete slab. Each span of the bridge is supported by 2 × 6 elastomeric bearings placed at the ends of each longitudinal beam. Each pier is structurally independent from the adjacent one and it has a rectangular tapered section where the base dimension is equal to 7 × 2 m while the top dimension is 8 × 2 m. At the top of every RC pier, there is a hammerhead cap where the elastomeric bearings are located. The piers are made of C25/30 concrete with 74Ø22 longitudinal AQ50 steel rebars confined by Ø10/30 cm stirrups.
The second viaduct ( Figure 6) is also made by two adjacent and independent carriageways. It is constituted by a sequence of five simply supported 22 m length spans, realizing a rectilinear planimetric and altimetric layout. The overall width of the roadway is 9.85 m and each span is realized with a precast concrete girder of four I longitudinal beams and four transverse beams. The viaduct deck consists in a 25 cm thick concrete slab. Each span of the bridge is supported by 2 × 4 elastomeric bearings. In this case, the piers are composed by two independent frames. Each frame has two cylindrical columns (diameter equal to 1 m). The two columns are connected at the top by a trapezoidal beam where the elastomeric bearings are located. The piers' characteristics are: C25/30 concrete, 16Ø20 longitudinal AQ50 steel rebars, Ø8/20 cm spiral stirrups. Each pier is structurally independent from the adjacent one and it has a rectangular tapered section where the base dimension is equal to 7 × 2 m while the top dimension is 8 × 2 m. At the top of every RC pier, there is a hammerhead cap where the elastomeric bearings are located. The piers are made of C25/30 concrete with 74Ø22 longitudinal AQ50 steel rebars confined by Ø10/30 cm stirrups.
The second viaduct ( Figure 6) is also made by two adjacent and independent carriageways. It is constituted by a sequence of five simply supported 22 m length spans, realizing a rectilinear planimetric and altimetric layout.
Infrastructures 2020, 5, 52 6 of 15 • Risk index in acceleration (RIPGA): is the ratio between capacity (PGAC) and demand (PGAD) in terms of peak ground acceleration; • Risk index in return period (RITR): is the ratio between capacity (TRC) and demand (TRD) in terms of return period of the earthquake, raised to 0.41 [11,30]. Values close to one or larger than one characterizes cases where the risk level is acceptable. On the contrary, values close to zero characterizes high-risk cases.

Case Studies
The procedure, described in Section 3, has been applied to six different bridges, representative of the Italian Highway Network. In all the models, the longitudinal axis of the bridge is represented by the X axis, while the transversal direction is oriented with Y axis of the coordinate system.
The first bridge ( Figure 5) is characterized by the presence of two adjacent and independent carriageways, consisting in a sequence of seventeen simply supported 36 m spans (except for the central span which is 60 m long). The planimetric and altimetric layout is rectilinear. The overall width of the roadway is about 11 m and each span of the bridge is realized by a precast concrete slab of six prestressed U girders. The bridge deck consists in a 20 cm thick concrete slab. Each span of the bridge is supported by 2 × 6 elastomeric bearings placed at the ends of each longitudinal beam. Each pier is structurally independent from the adjacent one and it has a rectangular tapered section where the base dimension is equal to 7 × 2 m while the top dimension is 8 × 2 m. At the top of every RC pier, there is a hammerhead cap where the elastomeric bearings are located. The piers are made of C25/30 concrete with 74Ø22 longitudinal AQ50 steel rebars confined by Ø10/30 cm stirrups.
The second viaduct ( Figure 6) is also made by two adjacent and independent carriageways. It is constituted by a sequence of five simply supported 22 m length spans, realizing a rectilinear planimetric and altimetric layout. The overall width of the roadway is 9.85 m and each span is realized with a precast concrete girder of four I longitudinal beams and four transverse beams. The viaduct deck consists in a 25 cm thick concrete slab. Each span of the bridge is supported by 2 × 4 elastomeric bearings. In this case, the piers are composed by two independent frames. Each frame has two cylindrical columns (diameter equal to 1 m). The two columns are connected at the top by a trapezoidal beam where the elastomeric bearings are located. The piers' characteristics are: C25/30 concrete, 16Ø20 longitudinal AQ50 steel rebars, Ø8/20 cm spiral stirrups. The overall width of the roadway is 9.85 m and each span is realized with a precast concrete girder of four I longitudinal beams and four transverse beams. The viaduct deck consists in a 25 cm thick concrete slab. Each span of the bridge is supported by 2 × 4 elastomeric bearings. In this case, the piers are composed by two independent frames. Each frame has two cylindrical columns (diameter equal to 1 m). The two columns are connected at the top by a trapezoidal beam where the elastomeric bearings are located. The piers' characteristics are: C25/30 concrete, 16Ø20 longitudinal AQ50 steel rebars, Ø8/20 cm spiral stirrups.
The third case study (Figure 7), is a long span bridge characterized by a total length equal to 77 m. The overall width of the roadway is about 10 m and the long span is realized by a spiroll prestressed precast concrete slab while the deck consists in a 20 cm thick concrete slab. The long span is supported by 20 elastomeric bearings divided between the two piers and the abutments. The third case study (Figure 7), is a long span bridge characterized by a total length equal to 77 m. The overall width of the roadway is about 10 m and the long span is realized by a spiroll prestressed precast concrete slab while the deck consists in a 20 cm thick concrete slab. The long span is supported by 20 elastomeric bearings divided between the two piers and the abutments. Each pier is characterized by a rectangular tapered section which presents a base dimension equal to 5 × 0.9 m and a top dimension equal to 5.6 × 0.9 m. The elastomeric bearings are placed in correspondence to the top of the pier. The two piers are made of C20/25 concrete with 38Ø18 longitudinal AQ50 steel rebars confined by Ø10 stirrups having a spacing of 25 cm.
The fourth bridge (Figure 8), is characterized by the presence of two adjacent and independent carriageways. It consists in a sequence of eighteen simply supported 29 m length spans. The layout presents a slight curvature. The overall width of the roadway is about 12 m and each span of the bridge is realized by a precast concrete lattice girder formed by four longitudinal beams (three characterized by a I section and one by a U section) and five transverse beams while the deck consists in a 22 cm thick concrete slab. Each span of the viaduct is supported by 2 × 4 elastomeric bearings placed at the end of each longitudinal beam.
The piers are characterized by a frame system consisting of four columns with a rectangular section 0.8 × 2 m. The fourth pier presents two independent columns characterized by a triangular section and by a rectangular section 3 × 2.4 m. The fifth pier is composed by a rectangular 6 × 2.4 m column. The piers are made of C32/40 concrete with 22Ø20 longitudinal FeB44K steel rebars confined by Ø12/20 cm stirrups. The fifth case study is a long span bridge with a total length equal to 65 m (Figure 9). The overall width of the roadway is 11.25 m and the long span is realized by a precast concrete lattice girder with five longitudinal I beams, seven transverse beams and a 20 cm thick concrete deck. The long span is supported by twenty elastomeric bearings divided between the two frame piers and the abutments. Each pier is characterized by a rectangular tapered section which presents a base dimension equal to 5 × 0.9 m and a top dimension equal to 5.6 × 0.9 m. The elastomeric bearings are placed in correspondence to the top of the pier. The two piers are made of C20/25 concrete with 38Ø18 longitudinal AQ50 steel rebars confined by Ø10 stirrups having a spacing of 25 cm.
The fourth bridge (Figure 8), is characterized by the presence of two adjacent and independent carriageways. It consists in a sequence of eighteen simply supported 29 m length spans. The layout presents a slight curvature. The overall width of the roadway is about 12 m and each span of the bridge is realized by a precast concrete lattice girder formed by four longitudinal beams (three characterized by an I section and one by a U section) and five transverse beams while the deck consists in a 22 cm thick concrete slab. Each span of the viaduct is supported by 2 × 4 elastomeric bearings placed at the end of each longitudinal beam.
The third case study (Figure 7), is a long span bridge characterized by a total length equal to 77 m. The overall width of the roadway is about 10 m and the long span is realized by a spiroll prestressed precast concrete slab while the deck consists in a 20 cm thick concrete slab. The long span is supported by 20 elastomeric bearings divided between the two piers and the abutments.  The fifth case study is a long span bridge with a total length equal to 65 m (Figure 9). The overall width of the roadway is 11.25 m and the long span is realized by a precast concrete lattice girder with five longitudinal I beams, seven transverse beams and a 20 cm thick concrete deck. The long span is supported by twenty elastomeric bearings divided between the two frame piers and the abutments. The piers are characterized by a frame system consisting of four columns with a rectangular section 0.8 × 2 m. The fourth pier presents two independent columns characterized by a triangular section and by a rectangular section 3 × 2.4 m. The fifth pier is composed by a rectangular 6 × 2.4 m column. The piers are made of C32/40 concrete with 22Ø20 longitudinal FeB44K steel rebars confined by Ø12/20 cm stirrups.
The fifth case study is a long span bridge with a total length equal to 65 m ( Figure 9). The overall width of the roadway is 11.25 m and the long span is realized by a precast concrete lattice girder with five longitudinal I beams, seven transverse beams and a 20 cm thick concrete deck. The long span is supported by twenty elastomeric bearings divided between the two frame piers and the abutments. Each pier is composed by a reticular concrete frame where the columns are characterized by a rectangular tapered section that varies from 1.06 × 1.61 m at the base to 1.06 × 1.36 m. Concrete material is C40/50 with 30Ø20 longitudinal FeB44K steel rebars confined by Ø12/30 cm stirrups. Lastly, the sixth case study consists in a multi span bridge characterized by two simply supported 39 m spans ( Figure 10). The overall width of the roadway is about 12 m and each span is realized by a precast concrete lattice girder formed by four longitudinal I beams, four transverse beams and a 20 cm thick deck. Each span of the bridge is supported by 2 × 4 elastomeric bearings placed at the end of each longitudinal beam.
The pier is characterized by a spatial frame where four columns present a C section and are made of C28/35 concrete with 50Ø16 longitudinal AQ50 steel rebars and Ø8/20 cm stirrups. The piers of the analyzed case studies are characterized by quite different structural behaviors. Piers of bridge 1 and 3 present a cantilever boundary configuration. Other bridges' piers are characterized by double-clamped (frame) configuration. Figure 11 shows six representative momentcurvature diagrams, one for each of the analyzed bridges. Each pier is composed by a reticular concrete frame where the columns are characterized by a rectangular tapered section that varies from 1.06 × 1.61 m at the base to 1.06 × 1.36 m. Concrete material is C40/50 with 30Ø20 longitudinal FeB44K steel rebars confined by Ø12/30 cm stirrups.
Lastly, the sixth case study consists in a multi span bridge characterized by two simply supported 39 m spans ( Figure 10). The overall width of the roadway is about 12 m and each span is realized by a precast concrete lattice girder formed by four longitudinal I beams, four transverse beams and a 20 cm thick deck. Each span of the bridge is supported by 2 × 4 elastomeric bearings placed at the end of each longitudinal beam. Each pier is composed by a reticular concrete frame where the columns are characterized by a rectangular tapered section that varies from 1.06 × 1.61 m at the base to 1.06 × 1.36 m. Concrete material is C40/50 with 30Ø20 longitudinal FeB44K steel rebars confined by Ø12/30 cm stirrups. Lastly, the sixth case study consists in a multi span bridge characterized by two simply supported 39 m spans ( Figure 10). The overall width of the roadway is about 12 m and each span is realized by a precast concrete lattice girder formed by four longitudinal I beams, four transverse beams and a 20 cm thick deck. Each span of the bridge is supported by 2 × 4 elastomeric bearings placed at the end of each longitudinal beam.
The pier is characterized by a spatial frame where four columns present a C section and are made of C28/35 concrete with 50Ø16 longitudinal AQ50 steel rebars and Ø8/20 cm stirrups. The piers of the analyzed case studies are characterized by quite different structural behaviors. Piers of bridge 1 and 3 present a cantilever boundary configuration. Other bridges' piers are characterized by double-clamped (frame) configuration. Figure 11 shows six representative momentcurvature diagrams, one for each of the analyzed bridges. The pier is characterized by a spatial frame where four columns present a C section and are made of C28/35 concrete with 50Ø16 longitudinal AQ50 steel rebars and Ø8/20 cm stirrups.
The piers of the analyzed case studies are characterized by quite different structural behaviors. Piers of bridge 1 and 3 present a cantilever boundary configuration. Other bridges' piers are characterized by double-clamped (frame) configuration. Figure 11 shows six representative moment-curvature diagrams, one for each of the analyzed bridges. Each pier is composed by a reticular concrete frame where the columns are characterized by a rectangular tapered section that varies from 1.06 × 1.61 m at the base to 1.06 × 1.36 m. Concrete material is C40/50 with 30Ø20 longitudinal FeB44K steel rebars confined by Ø12/30 cm stirrups. Lastly, the sixth case study consists in a multi span bridge characterized by two simply supported 39 m spans ( Figure 10). The overall width of the roadway is about 12 m and each span is realized by a precast concrete lattice girder formed by four longitudinal I beams, four transverse beams and a 20 cm thick deck. Each span of the bridge is supported by 2 × 4 elastomeric bearings placed at the end of each longitudinal beam.
The pier is characterized by a spatial frame where four columns present a C section and are made of C28/35 concrete with 50Ø16 longitudinal AQ50 steel rebars and Ø8/20 cm stirrups. The piers of the analyzed case studies are characterized by quite different structural behaviors. Piers of bridge 1 and 3 present a cantilever boundary configuration. Other bridges' piers are characterized by double-clamped (frame) configuration. Figure 11 shows six representative momentcurvature diagrams, one for each of the analyzed bridges. The idealized moment-rotation relationships of the corresponding plastic hinges are summarized in Table 1. As previously discussed, the brittle failure is governed by the shear response. The idealized shear-displacement curves of the considered piers are reported in Table 2. The idealized moment-rotation relationships of the corresponding plastic hinges are summarized in Table 1. As previously discussed, the brittle failure is governed by the shear response. The idealized shear-displacement curves of the considered piers are reported in Table 2. Eigenvalue analysis has been performed for each case-study bridge. The most significant vibration modes have been used as modal-horizontal load profiles of the pushover analysis. Natural periods (T j ) and corresponding modal participation mass (m j ) of the vibration mode involving at least 5% of the total mass are listed in Table 3 (longitudinal direction) and Table 4 (transversal direction). These vibration modes are characterized by a prevalent value of participant mass in the longitudinal or transversal direction.   Figure 12 shows one relevant pushover curve for each of the analyzed bridges where only the nonlinear bending response is considered (ductile mechanism). Given a specific seismic input (ADRS spectrum), the calculation of the performance point is carried out with the CSM for each relevant capacity curve as in [21]. Subsequently, corresponding internal forces are combined with the CQC technique and compared to the limit state's maximum capacity.
If the verification is satisfied, the PGA of the selected spectra is lower than PGA C . Therefore, the procedure has to be repeated with an increased spectrum until PGA C is detected. This iterative process leads to the calculation of the risk indexes in terms of PGA or return period T R (RI PGA or RI TR , respectively), i.e., the maximum bearable PGA (or T R ) over the corresponding site design values. Table 5 reports the results of the six bridges for the ductile mechanism.
Infrastructures 2020, 5, 52 10 of 15 Eigenvalue analysis has been performed for each case-study bridge. The most significant vibration modes have been used as modal-horizontal load profiles of the pushover analysis. Natural periods (Tj) and corresponding modal participation mass (mj) of the vibration mode involving at least 5% of the total mass are listed in Table 3 (longitudinal direction) and Table 4 (transversal direction). These vibration modes are characterized by a prevalent value of participant mass in the longitudinal or transversal direction.   -------------- Figure 12 shows one relevant pushover curve for each of the analyzed bridges where only the nonlinear bending response is considered (ductile mechanism).  Given a specific seismic input (ADRS spectrum), the calculation of the performance point is carried out with the CSM for each relevant capacity curve as in [21]. Subsequently, corresponding internal forces are combined with the CQC technique and compared to the limit state's maximum capacity.
If the verification is satisfied, the PGA of the selected spectra is lower than PGAC. Therefore, the procedure has to be repeated with an increased spectrum until PGAC is detected. This iterative process leads to the calculation of the risk indexes in terms of PGA or return period TR (RIPGA or RITR, respectively), i.e., the maximum bearable PGA (or TR) over the corresponding site design values. Table 5 reports the results of the six bridges for the ductile mechanism.   Analogously, pushover analyses of relevant vibration modes are performed for the brittle mechanism. Figure 13 shows one relevant pushover curve for each of the analyzed bridges.
The corresponding risk indexes in terms of PGA and T R (RI PGA or RI TR ) for both the longitudinal and transversal directions are listed in Table 6. Analogously, pushover analyses of relevant vibration modes are performed for the brittle mechanism. Figure 13 shows one relevant pushover curve for each of the analyzed bridges. The corresponding risk indexes in terms of PGA and TR (RIPGA or RITR) for both the longitudinal and transversal directions are listed in Table 6.

Discussion
The risk indexes estimated for the six case-studies reflect the well-known seismic deficiencies of existing RC bridges. Looking at the bending (ductile) capacity, all considered viaducts are compliant with the code-prescribed seismic safety level. The piers, properly designed to resist high vertical loads, have a sufficient amount of longitudinal reinforcement to withstand the bending actions generated by the earthquake shaking. Most of the bridges have PGA C larger than 0.2g. The average value is 0.36 g while the coefficient of variation is 0.39. In general, the higher values of PGA C refer to short viaducts or characterized by wall-piers. Each analyzed viaduct has a longitudinal and transversal risk index larger than one. On the contrary, shear (brittle) capacity results quite limited. The corresponding PGA C has average equal to 0.15g and consistently high scatter (1.19 coefficient of variation). In most cases the PGA C is lower than 0.1g for both longitudinal and transversal directions. Only bridges 4 and 6 present risk indexes larger than one, while the other viaducts are affected by the poor construction details of the piers in terms of transversal reinforcement.

Conclusions
In this paper, an efficient procedure to evaluate the seismic vulnerability of existing RC bridges has been described with reference to six typical bridges of the Italian Highway Network. The procedure, based on the modal pushover analysis approach, guarantees a low computational cost resulting in a balanced solution for the assessment of large portfolios of bridges. Risk indexes, expressed in terms of peak ground acceleration or return period, have been calculated (i) considering bending-ductile/shear-brittle collapse mechanisms (ii) for the two principal directions of the structure [32]. The results of the analyses have shown that these bridges are not affected by bending failure of piers (i.e., risk indexes larger than one) but are quite vulnerable with respect to shear-brittle damage. This result reflects the lack of construction details of these types of bridges that were constructed in the post WWII period. These results are not only useful to define the correct seismic retrofitting interventions to be implemented, but are important decision-making parameters for bridge management, investment prioritization and loss assessment at regional scale.