Numerical dynamic characterisation of concrete bridge stays

Following the recent collapse of a cable‐stayed bridge in Genoa, Italy, an interest has arisen to understand if it would have been possible for unknown localised material deterioration and/or decrease in prestress levels to introduce noticeable changes in the dynamic behaviour of RC stays. As such, in this study we start by reviewing past research work, experimental and analytical, on the effects that prestress level may or may not have on the dynamic response of RC beam elements. We then review also available analytical formulations used to determine the natural frequency of vibration of prestressed beams, and, subsequently, complete the preliminary investigation on the effects of prestress levels, and local damage, through the analysis of finite element models of prestressed beams and stays. The comparison of analytical and numerical estimations with results obtained from an in‐situ dynamic characterisation campaign on a RC bridge stay is also undertaken. In the second part of the study, we explicitly consider the Morandi bridge case‐study, for which finite element models of the stays alone, as well as of the full bridge, are developed and analysed. The obtained results are then also compared with the observations made in pre‐collapse in situ dynamic characterisation endeavours. All results obtained and discussed lead to the conclusion that, most regrettably, dynamic characterisation endeavours do not have the capability of providing insight on possible localised material deterioration or partial reduction of the average state of compression in this type of structural elements.


NOVELTY
-Review of past experimental and analytical research work on the effects that prestress level may or not have on the dynamic response of post-compressed RC beam elements.-Verification of an analytical formulation for estimating the natural frequency of vibration of prestressed beams through comparison with results obtained from numerical models.-Validation of a numerical methodology to model RC post-compressed stays through the comparison with in-situ dynamic characterisation of the Carpineto bridge's stays.-Modelling and dynamic (eigenvalue) analysis of the collapsed segment of the Morandi bridge, and comparison with the results from pre-collapse in situ dynamic characterisation endeavours.

INTRODUCTION
The dynamic response of reinforced concrete (RC) bridge stays has so far received only limited attention in the literature (e.g., Gentile and Martinez y Cabrera 1 ), most likely because cable-stayed bridges with concrete stays are not a structural system that has been widely used.However, the recent collapse, on August 2018, of a bridge of such typology (the Polcevera viaduct in Genoa, Italy), involving the failure of one of its RC stays, has given rise to the need for better and further studying this type of structural elements.Indeed, amongst the many issues that have been intensely debated in the aftermath of the bridge collapse, the analysis of the dynamic behaviour of its stays has featured prominently, given its relevance in the attempt to establish if results from in-situ dynamic characterisation efforts carried out prior to the collapse could have in some manner provided a pre-warning for the localised rupture that subsequently occurred.Within the above context, it has become of interest to understand if it would have been possible for unknown localised material deterioration and/or decrease in prestress levels to introduce noticeable changes in the dynamic behaviour of RC stays.As such, in this study, we start by reviewing past research work, experimental and analytical, on the effects that prestress level may or may not have on the dynamic response of RC beam elements; no past studies have been specifically carried out on stays, but the configuration and behaviour of the latter do share similarities to those of beams.We then review also available analytical formulations used to determine the natural frequency of vibration of prestressed beams, and, subsequently, complete the preliminary investigation on the effects of prestress levels, and local damage, through the analysis of finite element models of prestressed beams and stays.Finally, the comparison of analytical and numerical estimations with results obtained from an in situ dynamic characterisation campaign on a RC bridge stay is undertaken.
In the second part of the study, we explicitly consider the Morandi bridge case-study, for which finite element models of the stays alone, as well as of the full bridge, are developed and analysed.The obtained results are then also compared with the observations made in pre-collapse in situ dynamic characterisation endeavours; from such comparison, a number of conclusions and recommendations are derived and discussed.

Effect of prestressing or post-compression
One of the first papers describing an investigation of the effects that level of prestressing force may have on the dynamic response of prestressed concrete elements was published by Saiidi et al., 2 in which the authors attempted to determine, through both field and laboratory testing, if natural vibration frequencies of two case-studies (a post-compressed simply supported multicell box girder concrete bridge, and a post-compressed beam) could be used to establish prestress loss.The test data obtained by Saiidi et al. 2 showed that, within practical ranges of prestress force levels, changes in natural frequencies of prestress concrete members are very small, even for the fundamental mode of vibration, leading such authors to conclude that it did not appear feasible to use vibration data to assess potential reductions in prestress force levels.The authors expected to reproduce experimental results with Equation ( 1), but, as reported in the subsequent paper discussion by Dall'Asta and Dezi, 3 Deák 4 and Jain and Goel, 5 that analytical formulation represents only the effect of an external compression on a beam, not the effect of an internal precompression.In the same discussion, Jain and Goel 5 concluded that the dynamic behaviour of a prestressed beam is not affected by the level of prestress applied.Subsequently, Grace et al. 6 investigated the dynamic characteristics of post-compressed concrete girders with web openings subjected to repeated cyclic loading, both from the theoretical and the experimental points of view.The authors considered both straight tendons located at the bottom of the cross-section throughout the girders' length, as well as parabolic tendons that progressively descended from mid-height towards the bottom of the girders' cross-section.In the case of straight bottom prestressing strands, the girders showed a slight decrease in the natural frequencies as the prestress force was increased, whilst an opposite trend was instead observed in the case of a parabolically-shaped tendon, for which slight increases in frequency were observed.Noticeably, however, for both cases the modal shapes remained essentially unchanged, similar to what had been observed in the previous study reported above.
A few years later, Dall'Asta and Leoni 7 studied the free vibration of a series of post-compressed RC beams with both straight and parabolic unbonded tendons, analysing the first two flexural modes (horizontal and vertical) and the first twisting mode.The authors concluded, once again, that the natural frequencies of the flexural modes were not affected by a variation in compression force, and only slightly affected by the cables' path.Miyamoto et al. 8 also studied the dynamic behaviour of prestressed RC beams strengthened with external tendons.They used results of experimental tests to validate an analytical formulation that envisaged two counterbalancing effects; the compression-softening effect due to the axial force acting on the concrete, and the stiffening effect given by the tensioned steel tendons-the natural frequencies on an externally post-compressed beam increase if the stiffening effect related to the tensioned steel tendons prevails on the concrete compression-softening effect.
Finally, Hamed and Frostig 9 presented an analytical study on the effects of prestressing force on the natural frequency of RC beams, with bonded and unbonded tendons.Similar to the previously cited authors, Hamed and Frostig 9 further underlined how the behaviour of a prestressed beam can be described as the combination of two substructures; a compressed concrete beam and a tensioned cable, for which expressions to estimate natural frequencies were readily available in the literature.Indeed, the natural frequency of a simply supported axially compressed beam can be obtained from Equation (1), whilst the natural frequency of a tensioned cable (string) can be obtained from Equation (2), where ω n is the natural frequency for n th mode, m is the distributed mass of the element (respectively m c and m t for concrete and cable (string) elements), L its length, EI its flexural rigidity, C is the level of applied compression axial force and T is the level of applied tensile axial force.
The equations above show that by increasing the magnitude of the axial force, the natural frequency decreases in the case of a compressed beam, due to the compression softening effect, and increases in the case of a tensioned cable.This considered, it is thus not surprising that, in their analyses, Hamed and Frostig 9 showed and concluded that the prestress forces levels do not influence the natural frequencies of prestressed RC beams, both for bonded and unbonded tendons, thus corroborating the observations of the earlier research endeavours, described above.
Materazzi et al. 10 investigated the influence of some parameters, such as amount and position of both reinforcing and prestressing steel, type of prestressing (bonded, unbonded, straight, draped, parabolic), influence of the prestressing forces, on the natural frequencies of post compressed beams.Although the authors did note that the effects of the aforementioned parameters should not necessarily be neglected, they also concluded that for the cases where the hypotheses of linear materials and bonded tendons can be assumed, the effect of prestressing forces on the dynamic response characteristics of RC specimens can be neglected.

Effect of concrete cracking/damage
In prestressed concrete structures, temporary overloading or loss of prestress (due, e.g., to corrosion in the prestressing strands) can induce the opening of cracks in the concrete.Past studies on the possible impact that such concrete damage might have on the dynamic response characteristics of RC members are discussed in what follows.
One of the first studies on this topic was carried out by Kato and Shimada, 11 who undertook experimental testing of on an existing prestressed concrete bridge.They observed that, while the prestressing strands remained in their elastic response range, the decrease in the natural frequencies of vibration of the specimen was very modest.When the strands entered the inelastic range, instead, the significantly larger cracks led to more noticeable reduction in frequency of vibration.
Similar results were obtained by Unger et al., 12 who studied the effect of structural damage (i.e., concrete cracking) in prestressed concrete beams by conducting experimental dynamic tests on beams gradually damaged by the application of external loads.They observed that in prestressed concrete beams, the post-loading recovery of stiffness following the opening of a crack may not be complete, thus causing natural frequencies of vibration to decrease.However, the observed variations, in frequencies and mode shapes, were only minor, unless yielding of the reinforcement would occur.
More recently, Pisani et al. 13 numerically studied a series of prestressed RC beams subjected to damage, such as steel corrosion and concrete spalling.Their results showed that reductions of around 10% in the steel strands strength of prestressed beams would lead to 10-fold lower variations in modal frequency values, which were observed to vary in the range of 0.07-1.5%.These results were obtained by considering a loss of steel cross-section along the entire length of the RC elements; if corrosion was instead not assumed as being uniformly distributed throughout the full length of the beam elements, then the variation in frequency values becomes even lower.Pisani et al. 13 also studied the case of non-prestressed RC beams, observing higher variation percentages (4-5%) for such elements, which are not subjected to an active state of compression.The authors thus concluded that whilst modal frequency does exhibit a noticeable sensitivity to losses of the tension steel area for reinforced concrete members, the phenomenon becomes essentially negligible for the case of prestressed elements.

Analytical formulations for natural frequency estimation of prestressed beams
As discussed above, the dynamic response of a prestressed beam can be considered to be a result of the superposition of the dynamic response of the concrete, subjected to a compressive state, and of the tendons (post-compression cables), subjected to a tensile state.In this Section, the natural frequency analytical expressions introduced above for these two sub-systems are hence further explored and discussed with a view to gain further insight on the effects that prestressing forces may or may not have in a post-compressed concrete sleeve of a bridge stay, an example of which is schematically shown in Figure 1.
If one wishes to explicitly consider that the concrete sleeves of bridge stays are pre-or post-compressed with an axial force level C concrete , then Equation (3) can be used to estimate the natural frequencies of vibration (assuming hinged boundary conditions): where C concrete is the level of applied compression axial force on the concrete, T tendons is the level of applied tensile axial force in the tendons (Figure 1) and m is the distributed total mass of the element (i.e. the addition of the masses of the concrete sleeve (m c ) and of the tendons (m t ), which are bonded and thus displace together).Equation (3) can also fully describe the compression process, starting by considering the natural frequency of a stretched cable (concrete stiffness is neglected), passing by the application of the concrete element on the stretched cable (compression is not yet applied on the concrete element, thus C concrete term is neglected), and finally by computing the natural frequency while the concrete element is being compressed by the stretched cable.
The boundary conditions of a bridge concrete sleeve are not necessarily equivalent to perfect hinges, and one might want thus to consider also the case of full restrain against rotations at the two ends of the stay, in which case Equation (4) can then be adopted to estimate the corresponding natural frequencies of vibration (though, clearly, the actual end-restrains of a concrete stay might not necessarily be of the perfectly fully-fixed type).
F I G U R E 1 Representation of the Morandi Bridge stay (the case study is introduced in Section 4) and its two substructures: post-compressed concrete sleeve and main stay cables.
In the subsequent Section, both Equations ( 3) and (4) will be validated through comparisons against results obtained from a finite element model of a generic prestressed concrete beam.Herein, we note only that if in these two equations one considers the stress state in the concrete sleeve and steel tendons to be in equilibrium (i.e., T = C), then the dynamic behaviour of a stay's concrete sleeve can be described by the classical beam Equation ( 5) alone.In other words, the compression state of a concrete beam does not affect its dynamic behaviour if the internal forces are in equilibrium, as experimentally observed in the past research endeavours (previously discussed) and also as analytically deduced by Jain and Goel (1996). 5 Finally, we note also that, considering that in Equations ( 3) and ( 4), the term related to the flexural stiffness of the beam has an exponent of 4, whilst the terms related to the internal forces have exponents of 2, it results that for higher vibration modes the contribution of the internal forces on the dynamic behaviour of the structures becomes less and less relevant.This is again aligned with the observations in past researcher endeavours, described in the previous Section, which noticed that (minor) variations in frequency values could be perceived only for the fundamental mode.

Numerical modelling of prestressed concrete beams
An 86 m long beam featuring a cross-section with properties similar to the prestressed concrete sleeve of the Morandi bridge stay (see Figure 2B) is herein modelled using the finite element program SeismoStruct Seismosoft, 14 through the employment of 17 beam-column elements.Eigenvalue analyses, which do consider geometric nonlinearity effects (including the interaction between axial load and flexural stiffness), are carried out and the ensuing results compared with  those obtained using the analytical formulations described above.We start by modelling a concrete beam subjected to an external compression force (i.e., an approximate modelling of prestressing) depicted in Figure 2, after which we model prestressing tendons alone, and finally, we combine the two models with a view to numerically reproduce a prestressed beam.

Simply supported beam subjected to an external compression force
The compression is applied externally through the employment of a load time-history (i.e., at the first step of the analysis no compression is present, which is then subsequently introduced), so that dynamic behaviour response can be assessed both before and after the application of the external compression force.
As one may gather from Table 1, due to the external compression applied at the beam, the first mode frequency decreased by 61%, whilst the second and third mode reduced by 11% and 5%, respectively.These results are in line with what is expressed in Equation (1), that is, that the influence of an unbalanced state of internal actions (the compression in the concrete is not internally balanced by tensile forces in tendons) on the dynamic characteristics is predominant on the first mode of vibration, while the higher modes are only slightly affected.The same results can be obtained also by applying Equation (3), considering only the concrete distributed mass (m = m c ), and excluding the term related to the tensile force acting on the tendons (and the corresponding mass, m t ).

3.2.2
Clamped beam subjected to an external compression force The model described above was reanalysed with the consideration of fully fixed conditions at the beam ends; results are reported in Table 2. Also in this case, the numerical results matched almost perfectly the results obtained by considering the formulation shown earlier (Equation ( 4); m = m c ).The differences between the frequency of compressed and noncompressed beam (decreases of 11%, 5% and 3% for the first, second and third modes, respectively) are less marked with respect to those observed in the previous configuration.

Cable subjected to a tensile state
A numerical model of a tensioned cable was also analysed in SeismoStruct (see Figure 3).The cable, representing the equivalent tendon shown in Figure 1, was subjected to 6250 kN of tensile force (which is the level of pre-tension necessary to then introduce a prestress compression of 5 MPa in the concrete beam).The results are shown in Table 3, which includes again also the results obtained with analytical formulations Equation (2).

Simply supported prestressed beam
The explicit modelling of the precompression operation was implemented by the following construction sequence: • Tensioning of the steel tendons from the initial configuration (Figure 4.i) to the stretched one (Figure 4.ii).
• Addition of the concrete elements on the stretched configuration of the cable (Figure 4.iii).
• Gradual release of the force applied on the prestressing cable (Figure 4.iv).At the end of this operation, concrete and steel tendons reach a state of equilibrium.
This modelling aims to reproduce the dynamic behaviour of the post-compressed concrete sleeve alone, depicted in Figure 1.
Table 4 provides a comparison between the numerical and analytical results obtained for each construction stage.These results confirm the discussions of Sections 2 and 3.1; the presence of prestressing in an RC beam does not influence its F I G U R E 4 Representation of the construction sequence modelled in SeismoStruct: Stretching of the cable, from the initial (i) to final (ii) configuration, addition of the concrete elements on the cable's deformed shape (iii), release of the external force and compression of the concrete elements (iv).

TA B L E 4
Comparison between numerical (SeismoStruct) and analytical (Equation 3) results; simply supported prestressed beam.natural frequencies of vibration (the values included in the final row of Table 4 for a prestressed beam are identical to those found in Table 1 for an uncompressed beam), given that the internal axial forces are in equilibrium (the compression in the concrete is internally balanced by the tensile force in tendons).By contrast, these results also show once again that when prestressing is approximately modelled through the application of external compression forces, the dynamic behaviour of the beam is not correctly captured (the natural frequencies given in Table 1 for an externally compressed beam are very different from those shown in Table 4), given that such modelling approach introduces an internally unbalanced axial force state, as discussed before.

Analysis
Similar results, not shown here due to space constraints, were obtained both when considering different prestressing levels and/or diverse support considerations (i.e., clamped rather than simply supported).

Analytical formulations for natural frequency estimation of concrete stays
As a first approximation, a concrete bridge stay, i.e., a tensioned cable with a covering concrete sleeve around it (Figure 1), may be considered to be a tensioned cable (not to be confused with prestressing cables or tendons) with a non-negligible stiffness, given by the concrete sleeve.If the pre-or post-compression of the latter is neglected (an assumption that, as shown above, could be deemed as valid), then it is possible to estimate its natural frequencies of vibration through Equation ( 6), in which EI now stands for the concrete-steel composite flexural rigidity, and T cables is the level of applied force on the main cables and m = m c + m t (see Figure 1): More generally, i.e., considering also the pre-or post-compression of the concrete sleeve, the dynamic behaviour of a concrete stay can be described by Equation ( 7), thanks to which is possible to consider also the effect of external loads (such as an increase of the deck dead loads, moving loads, etc.) on the dynamic response of a stay.
The same approach can be followed for the clamped case.

Numerical modelling of a prestressed concrete bridge stay
The development of the stay's numerical model involved the following steps: • Tensioning of the main cables of the stay (i.e., not the prestressing tendons of the concrete sleeve) from the initial configuration to the stretched one (Figure 5A).A force of 22,200 kN was applied.• Addition of the concrete elements, including their prestressing tendons, on the stretched configuration of the cable (Figure 5B).The concrete sleeve prestressing operation was not modelled, given that, as shown already, it does not influence the eigenvalue results.However, if desired, the proposed modelling allows also to simulate the post-compression.• Connection of the two superstructures by means of rigid links, and application of clamped boundary conditions at both ends of the model.
In Table 5, the results obtained with the finite element model are compared with their analytical counterparts, obtained using Equation (8), in which the only unbalanced force is represented by the tensile force in the main cable.The comparisons are, once again, most reassuring.
F I G U R E 5 Numerical modelling procedure adopted to simulate the complete stay.

TA B L E 5
Comparison between numerical (SeismoStruct) and analytical (Equation 8) results; clamped stay.

Verification against in situ dynamic characterisation results
A final check on the analytical formulations and numerical models discussed above is herein undertaken through comparisons with the results obtained during an in-situ dynamic characterisation exercise carried out by Gentile and Gennari-Santori 15 and Della Sala and Gennari-Santori 16 on the concrete stays of the Carpineto viaduct, depicted in Figure 6.This bridge is characterised by monolithic RC post-compressed stays, very similar to the case-study of this paper that will be presented in the following section.The stays are about 64 m long, characterised by a cross-section of 1.10 × 0.8 m, and a sag of about 0.6 m. 16 The dynamic characterisation focused mainly on the longer internal stays of the viaduct, which featured a first global mode frequency of around 1.136 Hz (this corresponds to the average between the two values of 1.1 and 1.172 Hz reported by Gentile and Gennari-Santori 15 for modes of vibration that mobilize only the mass of the stay).Recently, Della Sala and Gennari-Santori 16 measured frequencies of 0.95 and 1.01 Hz, which were however referred to global bridge vibration modes, not constrained to the stay alone, and hence not necessarily comparable to the numerical results obtained in this section.
A comparison with the frequency estimates given by Equation ( 8) shows a difference of about 15% (Table 6), which is expected, given that this analytical expression was developed for a straight beam configuration, and hence does not account for the sag of the stay, which is approximately 0.6 m.As demonstrated in the work of Petyt and Fleischer, 17 even the smallest amount of sagging will impact the dynamic characteristics of a beam, an observation that was also made by Gentile and Gennari-Santori. 15The development of the numerical model followed the same steps described already in Section 3.4, with the difference that the actual sagged, rather than straight, geometrical configuration of the stay was considered.Concrete elastic modulus equal to 36,000 MPa and an axial force of 16,100 kN, proposed in Gentile and Gennari-Santori, 15 were adopted.The comparison between in situ measurements and numerical results (Table 7) shows a very good agreement, thus confirming the adequacy of the adopted modelling approach and assumptions.

Discussion
The set of analyses included in this Section show that prestressing force levels do not influence the natural frequency of vibration of concrete beams or stays, thus confirming the experimental observations from past researchers (Section 2) and their conclusion that in-situ dynamic characterisation endeavours cannot provide insight on prestress levels of RC beams or stays.In addition, these analytical and numerical results also highlighted how the employment of an approximative modelling of prestressing through the application of external compression forces (Sections 3.2.1 and 3.2.2),rather than its explicit numerical modelling (Section 3.2.4),leads to an erroneous estimation of the dynamic characteristics of prestressed elements, given that the equilibrium/balancing between the concrete compression and tendons tension is missing from the model.In other words, this simplified numerical modelling will only consider the compression softening effect (Equation 1), disregarding the stiffening provided by the tensioned steel cables (Equation 2).
Instead, both the comparison of the results obtained in Section 3.2.4 for a prestressed beam and in Section 3.2.1 for an uncompressed beam, as well as the verification against in situ measurements given in Section 3.5, show that, if explicit reproduction of the prestressing operation cannot be for some reason introduced in a given numerical model, it is then better not to include any external compression forces at all, in order to obtain a correct numerical estimation of the natural frequencies of vibration of a prestressed beam or stay.

Structural system
The Morandi bridge, depicted in Figure 7, represented an important infrastructure for the Italian road network, part of the motorway connecting several North-West urban centres to the French border.While the entire structure, as well as its foundation scheme, is comprehensively described in, e.g., Morandi,19 in what follows special attention will be given to the collapsed cable-stayed 'balanced system 9' (see, e.g., Calvi et al., 20 Malomo et al. 21and Scattarreggia et al. 22 ), and in particular to its post-compressed RC concrete stays.
F I G U R E 7 (A) Aerial picture of the bridge (left) and of a balanced system (right), (B) schematic of the bridge and (C) main sub-components constituting the balanced system and employed nomenclature. 21.
With reference to the nomenclature reported in Figure 7C, the considered cable-stayed balanced system essentially comprised the following main elements: • A pier with eight inclined struts (with cross-section varying between 4.5 × 1.2 to 2.0 × 1.2 m) that props the deck over a distance of about 42 m.• An antenna with two 90 m height A-shaped structures (element cross-section varying between 4.5 × 0.9 and 2.0 × 3.0 m) that converge about 45 m above the deck level.• A main deck with a five-sector box of depth variable between 4.5 and 1.8 m, an upper and lower slab 16 cm thick, and six deep webs with thickness varying between 18 and 30 cm.In its final configuration, the deck of the balanced system 9 was 172 m long and supported at four points: from below by the piers at the aforementioned spacing of 42 m and from above by the cable stays at a distance of 152 m.Two 10 m cantilevers were thus completing the deck length.Four transverse link girders connected stays and pier trusses to the deck.• Two simply supported Gerber beam spans connecting the cable-stayed balanced system to the adjacent part of the bridge.Each span was 36 m long and comprised six precast prestressed beams, with a variable depth equal to 2.20 m at mid-span, sitting on Gerber saddles protruding from the main deck.• Four cable stays, hanging from the antenna's top and intersecting the deck at an angle of about 30 • .The first 38.60 m, starting from the top of the Antenna, are characterised by a monolithic section, whilst the final 46 m feature a bifurcated section.
Further details on the bridge design and construction sequence are available in Orgnoni et al.  ('secondary cables'), with dimensions equal to 0.98 × 1.22 m and 0.98 × 0.61 m, respectively, for the monolithic and bifurcated part.The construction of the stays consisted of the following phases: • Installation of the primary cables, made of 352 ½ inch high-resistance steel strands, to sustain the weight of the whole deck and of the simply supported Gerber beams.• Installation of the secondary cables, made of 112 ½ inch high-resistance steel strands, necessary to compress the concrete elements that make up the stay's concrete sleeve.• Casting of the concrete sleeve in 5.9 m long segments, separated by a space of 10 cm between each other.Thanks to this operation, it was possible to reach the deformed catenary shape without inducing stresses in the concrete.The segments weight was carried by the primary cables, whilst the secondary ones were still slack; sliding of the concrete elements on the cables was allowed (see photographs included in Orgnoni et al.) 23 ; the 10 cm gaps between the elements were kept open through the use of spacers.• Filling of the empty spaces between the concrete sleeve segments.
• Post-compression of the concrete sleeve segments through the secondary cables starting from the lower end of the stays.
A stress of about 686 MPa (the same acting in the primary cables) was applied to the tendons, generating a compression of around 7.2 MPa in the concrete.• 'Homogenization' of the concrete sleeve, through mortar injection/filling of the secondary cable ducts.It is noted, however, that there is evidence showing that this operation was not fully completed, with portions of the ducts remaining unfilled, as noted, e.g., in Mortellaro et al. 24 and Rosati et al. 25

Stays' geometric characterization
The actual structural geometry of the viaduct was taken from a 3D in situ survey of the bridge, 25 which, in addition to providing the necessary information to model the collapsed portion of the cable-stayed viaduct (balanced system 9), also rendered evident that the latter was markedly different from its balanced system 10 counterpart (this fact is particularly relevant for some of the analyses and discussions carried out in subsequent sections of this manuscript).The balanced systems 9 and 10 (which, is recalled, are identified in Figure 7) were not symmetrical along the North-South axis.The top transverse beam of the antenna, particularly for balanced system 10, was not perpendicular to the longitudinal axis of the roadway (inclination of about 1 • ).The stays were consequently not identical and not symmetrical, featuring different total lengths between each other; regarding the southern stays, both for balanced system 9 and 10, the maximum length difference is between the South-Eastern stay of balanced system 9 (84.14 m) and the South-Eastern stay of balanced system 10 (84.73 m), contrary to what was envisaged in the original design (Design table 327) 26 where equal lengths of 84.75 m were prescribed.Regarding the northern stays, not explicitly analysed in this paper, the maximum length difference is between the North-Western stay of balanced system 9 (84.77m) and the North-Eastern stay of balanced system 10 (83.67 m).The most important difference between System 9 and System 10 stays is perhaps the length of the bifurcated part; as shown in Figure 9, the average length of the bifurcated part of System 9 is about 42.55 m, while for System 10 it is around 46.44 m, thus with a difference of about 3.89 m.The same can be stated for the northern stays, with a difference of 3.55 m (BS9 42.55 m, BS10 46.10 m).It is noted that the length envisaged in the original design was about 46.15 m.
Other discrepancies between the built structure and the original project regard the stays cross-section dimensions: the designed cross-section height was 1.22 and 0.61 m, respectively, for the monolithic and bifurcated parts, whilst the actual height, identified in the aforementioned 3D in situ survey of the bridge, varies from 1.21 to 1.45 m and from 0.58 to 0.77 m, respectively, for the monolithic and bifurcated parts; in Figure 9, it is possible to appreciate the cross-section height variation (expressed as a percentage variation with respect to the design values) along the entire length of the stays.

Material properties characterization
Concrete compressive resistance in the stays of the balanced system 9, 1990−1991 [27][28][29] In the period of 1990−1991, the state of maintenance of the stays' concrete was evaluated.Pull-out tests, as well as the combined sclerometer and ultrasound method, were carried out in order to assess the concrete resistance at the top, middle Actual geometry of the modelled stays and percentage variation of cross-section height with respect to design.
and bifurcated sections of the Southwest, Northwest and Northeast stays.The mechanical properties of the concrete were deemed to be reasonably homogeneous along the stays, with a compressive strength generally higher than 45 MPa.

Concrete compressive resistance in the bridge deck, 2010 30
This 2010 material resistance survey for the bridge's deck, preceded by a visual examination, was carried out on a spotcheck basis and consisted of sclerometer and pull-out tests, aimed at assessing the state of maintenance and homogeneity of the concrete in different sections of the bridge deck.Table 8 summarizes the most significant results of the survey, noting that the Elastic modulus values were estimated trough the equation reported in table 3.1 of Eurocode 2. 31 Concrete compressive resistance in the stays of balanced systems 9 and 10, 2015 32 In 2015, a new survey of the mechanical characteristics of the concrete, as well as of the steel (both standard reinforcement and tendons), was carried out in the stays of balanced systems 9 and 10.These investigations included the execution of non-destructive in situ tests, in tandem with the removal of cores that were then subjected to laboratory testing.The tests on balanced system 9 involved the bifurcated part of the Southwest and Southeast stays, about 6−7 m above the deck level whilst the tests on balanced system 10 involved the monolithic part close to the bifurcation of the Southwest and the bifurcated part of the Northeast and Northwest; the mechanical characteristics obtained are summarized in Table 9.The Elastic modulus was again estimated trough the equation reported in table 3.1 of Eurocode 2. 31

Deck loads/mass characterization
In addition to its own self-weight, numerous permanent non-structural loads were applied to the deck during the life of the structure, such as the first layer of road pavement in 1967 (design dead loads are described in the original project 26 ), the introduction of the New Jersey barriers and the widening of the carriageway in the 1990s, and the application of additional layers of asphalt over the years (Orgnoni et al.). 23The distribution of these loads was not uniform, especially due to the various asphalt layers.In fact, from surveys carried out on the collapsed portions of the viaduct, 25 it was possible to define the thickness of the asphalt as listed below: • Simply supported span between balanced systems 8 and 9: average asphalt thickness equal to 14 cm.
• Balanced system 9: average asphalt thickness equal to 17 cm.
• Simply supported span between balanced systems 9 and 10: average asphalt thickness equal to 22 cm.
The variations in deck loads/mass introduced by the uneven asphalt thickness described above need not only to be explicitly accounted for in modal analyses, but also constitute yet another source of asymmetry of the considered system.

4.2
Experimental dynamic characterisation campaigns

Balanced system 11
The first experimental dynamic analyses were carried out by Camomilla et al. 33 on the stays of balanced system 11.The tests, carried out for experimental purposes, were performed to assess the tensile stress acting on the steel cables, by measuring the acceleration on lower branch of the bifurcated part of one of the Northern stays of such balanced system.The layout of the instrumentation used was relatively simple, consisting in the application of two accelerometers positioned on two different sides of the same cross section.Frequencies of vibration were obtained using both natural excitations with and without traffic loads.To try to assess possible effects of material degradation on the natural frequencies of vibration, Camomilla and his team developed numerical models in which some parameters were investigated, such as corrosion level, presence of cracking and tensile stress in steel cables.Their conclusion was, once again, that the natural frequencies were essentially insensitive to corrosion in the steel tendons.

Balanced systems 9 and 10 -global vibration modes of the structure
During the life of the structure, only two experimental campaigns were carried out on Systems 9 and 10′s stays, respectively, in 2015 by CESI 34 and in 2017 by Gentile. 35Both on-site tests were characterized by an instrumentation layout spread over the entire structure, but slightly different from each other; whilst CESI 34 used triaxial accelerometers, Gentile 35 employed instead a higher number of biaxial accelerometers, though not monitoring the antenna.In both campaigns, the bifurcated portion of the stays were instrumented with the sensors installed only at the upper branch, a detail that will be further discussed in Section 4.2.3.Both CESI 34 and Gentile 35 reported a substantial homogeneity between the global vibration modes (i.e., which mobilized participating masses throughout the entire structure) identified in the two balanced systems.Some deformed shapes, identified on System 9 and 10, are depicted in Figure 10.
Although the experimental results obtained by CESI 34 revealed the presence of vibrating modes also out-of-plane (thanks to the use of triaxial accelerometers), while the results obtained by Gentile only detected modes of vibrating in the plane of the viaduct, due to the use of biaxial instruments, it was still possible to make a comparison between the two experimental campaigns.In particular, for what concerns the dynamic behaviour of System 9, it was possible to compare four vibrational modes (Table 10), while for System 10 only three (Table 11).As can be observed, the difference between the frequencies obtained in the two experimental campaigns is negligible.It is noted that, notwithstanding the aforementioned similarity with the results obtained by CESI 34 two years before, as well as the previous findings, documented in Section 2 and corroborated in Section 3, on the impossibility of dynamic characterisation being able to identify partial reduction of the average state of compression of prestressed RC stays, Gentile 35 stated that the lack of longitudinal (North vs.South sides) and transversal (Genoa vs. Savona ends) symmetry in the stays' deformed shapes for the global modes in both systems could be ascribed to differences in mechanical characteristics and tension forces in stays.However, this lack of perfect symmetry in the global vibration of the stays was, instead, certainly a result of the geometrically and mechanically different, as well as asymmetrical, configurations of balanced systems 9 and 10, described in Sections 4.1.3,4.1.4and 4.1.5.

4.2.3
Balanced systems 9 and 10 -local vibration modes of the stays As for the local vibration modes (i.e., modes in which only the mass of the stays is mobilised), these were reported by Gentile 35 alone, who identified four local modes in System 9 (only on the southern stays, two modes for each stay were found), and a total of 11 local modes on System 10 (for both southern and northern stays).Gentile 35 considered anomalous . the fact that two of the local vibration modes identified for System 9 featured frequencies of vibrations 15−20% higher than their counterparts in System 10, as well as modal shapes (Figure 11) that he deemed as being 'not entirely in accordance with expectations', as the experimental shapes appeared in a different way from those expected, that is, a sequence of half-waves of sinusoids.However, the former was a natural consequence of the previously discussed different geometric characteristics of the two systems, whilst the latter was actually a consequence of the fact that the sensors had been positioned only in the upper branch of the bifurcated part of stays, as will be shown in Section 4.3.This asymmetric disposition of the instruments had been previously adopted by Gentile and Martinez y Cabrera 1 for the dynamic characterisation of balanced System 11, but the 1993 retrofitting of the latter involved the linking of the bifurcated branches of the stays, this being the reason why the type of local vibration modes captured in System 9 had not been observed in System 11.

Numerical dynamic characterisation using local models of the stays
Four local models of the southern stays of Systems 9 and 10 were developed using the modelling approach discussed in Section 3.4 (Figure 12), and adopting as value of concrete elastic modulus the average of the experimentally derived values reported in Section 4.1.4(hence 36 GPa), as well as an axial force of about 25,400 kN (as derived in Orgnoni et al.). 23he numerical model yielded results very much in-line with the experimental results obtained by Gentile. 35The numerical frequencies are slightly higher than the experimental ones (see Tables 12 and 13), but an average difference of around just 5% can be deemed as satisfactory. 1The model managed to reproduce the experimentally observed differences in the frequency values for the second local modes of vibration of Systems 9 and 10, given that it duly considered their differences in cross-section, lengths of bifurcated part and rotation of the antennas around the vertical axis (which affects the effective length of each stay).
The numerical modal shapes also managed to capture very well the experimentally collected data points, as shown in Figures 13 and 14; as previously discussed, the fact that only the upper branches of the bifurcated part of the stays were  instrumented inevitably implied that the experimental shapes of some of their modes of vibration resulted incomplete.
(note: to better understand the comparison between numerical and experimental data, the reader should compare the experimental black dots in the bifurcated portion with the numerical solid red line of the Upper branch alone, given that only this one was instrumented.)Also, the fact that a numerical model that does not consider loss of post-compression force levels is capable of reproducing experimental characterisation results for these concrete stays confirms once again that dynamic characterisation F I G U R E 1 4 Comparison between numerical and experimental results, System 10 southern stays.Dots represent the experimental data, solid lines the numerical mode shape, dashed lines the system undeformed configuration.
endeavours cannot be used to gain insight on the state of this type of structural elements (unless the compression stresses are completely annulled and the stay becomes subjected to a generalised tensile stress state).

4.4
Numerical dynamic characterisation using full models of the bridge

Models developed
Three global models of the case-study bridge were developed in Midas Civil. 36The first numerical model, herein termed 'Simplified', adopted the material properties reported by Morandi, 19 as indicated in Figure 15A.The second numerical model, labelled 'Detailed', featured a distribution of the mechanical properties of the materials consistent with the in situ measurements reported in Section 4.1.4(see Figure 15B).In the third model, named 'Damaged', the modulus of elasticity at the top 5 m of the South-East stay was reduced to 25% of its original value (Figure 15C), with a view to simulate the damage (cracking and cavities) that was noticed when inspecting that portion of the collapsed structure. 25

Comparison between numerical and experimental results
The numerical frequencies of vibration are compared with their experimental counterparts (Gentile) 35 in Table 14, which shows differences in general below 10%, notwithstanding the fact that the numerical models do not consider the reduction of post-compression force in the concrete stays that would have been induced by the significant corrosion in the tendons observed in the aftermath of the bridge collapse. 37An excellent result is also obtained by comparing the modal shapes, ad depicted in Figure 16.In addition, it can also be observed that the simulation of local damage at the top of the   stay introduced an almost negligible variation in the results (see comparison between 'Simplified' and 'Damaged' models results), again highlighting how dynamic characterisation efforts cannot provide an alert on this type of phenomena.

CONCLUSION
The dynamic response of post-compressed reinforced concrete (RC) bridge stays has been analysed deeply in this work.
After the review of the existing past research work, experimental and analytical, on the effects that prestress level may or not have on the dynamic response of post-compressed RC beam elements, an analytic formulation to estimate the natural frequency of vibration of prestressed beams was presented.The implementation of two different numerical models, in which the post-compression was applied to a beam both in a simplified way and explicitly, demonstrated the validity of the proposed formulation and confirmed what was found by Hamed and Frostig 9 and others: the beam's compression level does not influence its dynamic behaviour, if the forces (compression in concrete, tension in tendons) are internally equilibrated and a generalised tensile stress state is not reached.
A numerical methodology to model RC post-compressed stay was then proposed in Section 3.4: this methodology was validated through the comparison between numerical and experimental results, obtained by Gentile and Gennari-Santori. 15on the Carpineto bridge's stays.The Morandi bridge case-study was subsequently introduced.Reliable numerical models of both stays alone and the complete structure, with different degrees of accuracy, were carried out using the SeismoStruct 14 and Midas Civil 36 computer programs.Results obtained from the accurate local modelling of the stays alone allowed to reproduce experimental data obtained by Gentile, 35 both from frequency and modal shape points of view; local models, in which both the actual geometry and the sag of the stays is correctly implemented, showed a better agreement with experimental results than the global models, in which the sag effect is not considered, specially from a mode shape point of view.
It was demonstrated that in order to match and replicate accurately experimental results, RC post-compressed stays numerical models should reproduce correctly the actual geometry, the sag, and the tensile force in the main cables, as well as the mechanical properties.The behaviour of post-compressed structures should thus be simulated explicitly, rather than through the application of an equivalent external force.
Finally, and in addition to confirming that the variation of the state of stress in post-compressed RC stays does not induce noticeable changes in their dynamic response, neither from a theoretical or numerical viewpoint, it was also shown how the introduction of a state of local damage in the top part of the stay does not lead to perceptible variations in the stays' vibration properties, hence precluding the possibility of identifying the presence of such type of unknown defects through in-situ dynamic characterisation efforts.Only in a scenario of a generalised tensile stress state being reached throughout the stay, will recognisable changes in the dynamic behaviour of these structural elements be developed.On the contrary, a partial reduction of the average state of compression in a stay cannot be detected by in situ dynamic characterisation endeavours.
In closing, it is underlined that the Polcevera viaduct case-study modelling work herein described was facilitated by the extensive details and evidence that has now become available through the ongoing post-collapse court proceedings (e.g., Rosati et al. 25 ).

A C K N O W L E D G E M E N T S
The authors would like to acknowledge two anonymous reviewers, whose constructive feedback led to a non-negligible improvement of the original version of the manuscript.

D ATA AVA I L A B I L I T Y S TAT E M E N T
Data sharing is not applicable to this article as no new data were created or analysed in this study.

F I G U R E 2
Representation of the external compression numerical model, implemented in SeismoStruct.TA B L E 1 Comparison between numerical (SeismoStruct) and analytical (Equation (3)) results: Simply supported beam subjected to an external compression force.

F I G U R E 6
Carpineto I Viaduct aerial view and schematic.TA B L E 7 Comparison between experimental and numerical results, Carpineto bridge stay.

4. 1 . 2
Stays' construction sequence Each one of the RC prestressed stays, depicted in Figure 8, comprised 352 ½ inch high resistance steel strands ('primary cables') enclosed by a series of concrete sleeve segments post-compressed together by 112 strands of the same type F I G U R E 8 Schematic of the RC prestressed stays.

F I G U R E 1 0 1
Shapes of the first, second and fifth vibrational mode, identified on Systems 9 and 10.35.TA B L E 1 0 Comparison between frequencies obtained by CESI34 and Gentile35 -System 9. Comparison between frequencies obtained by CESI34 and Gentile35 -System 10.

3
Comparison between numerical and experimental results, System 9 southern stays.Dots represent the experimental data, solid lines the numerical mode shape, dashed lines the system undeformed configuration.

F I G U R E 1 5 TA B L E 1 4
Numerical models implemented in Midas Civil.Experimental and numerical frequencies for the global modes of vibration.

F I G U R E 1 6
Experimental and numerical shapes for main global modes of vibration (note: experimental modal shapes are represented in black colour).

state Cable Tensile force (kN) Concrete Compression force (kN) Mode SeismoStruct f (Hz) Equation (3) f (Hz) Δ [%]
a The employed eigenvalue solver cannot capture the first mode of vibration of a non-tensioned tendon.
Concrete mechanical properties of system 9 deck.Concrete mechanical properties of systems 9 and 10 stays.
TA B L E 8

2
Example of numerical model implemented in SeismoStruct.Comparison between results, System 9 stays.Comparison between results, System 10 stays.