A few studies can be found in the literature concerning the structural behavior of the collapsed self-standing system, all of them involving very complex finite element or discrete element calculations. These analyses were able to provide, starting from the assumption of a weak section, the kinematics of the collapse with great details. In contrast, they were quite ineffective in describing the possible mechanism that was at the origin of the failure mechanism. Alternatively, a simplified analytical model can be exploited to gain more insight about the damage accumulation. The Polcevera bridge was composed of several statically determinate Gerber beams of 36 m span each and three main self-standing structural systems. Each balanced system designed by Morandi was a quite complicated structure itself and it is worth, as a first step, limiting the analysis to the bridge deck. The deck of the self-sustained system was a multicell box girder 4.5 m deep and 18 m wide (Fig. 4), which was supported at four points by inclined piers and two couples of stay cables hung at the top of the A-shaped antenna. The link of the deck with the piers and the stay cables is guaranteed by four transverse beams. In other words, each self-standing structural system is a three-span continuous beam with two terminal cantilever beams, which sustain the two lateral simply supported beams by means of Gerber saddles (Fig. 5). Therefore, in order to simplify the analysis and better understand the failure mechanism, our attention has been focused exclusively on the self-standing structural system, which collapsed independently of the remaining parts of the viaduct. Furthermore, for each self-standing structural system, two main substructures can be considered: the first is the deck of Polcevera bridge; the second is made up by the two couples of stays. Therefore, in the following subsection, a detailed analysis of the two main substructures is proposed in order to assess the internal reactions and the corresponding stresses in the deck and, in particular, in the stays.Static Analysis of the Bridge DeckAccording to Morandi’s design concept (Morandi 1967), the 352 inner tendons were supposed to support the permanent loads. Thus, the stress in the inner tendons was set, such as the vertical displacement of the deck at the connection point with the stay cables would vanish. As a consequence, the static scheme shown in Fig. 6(a) allows for an easy determination of the axial force in the two couples of stay cables due to the permanent loads (Invernizzi et al. 2019, 2020b). In fact, the axial force in the cable-stays at the connection with the deck can be estimated with no special efforts, given that the cross-section of the main deck is assumed constant, and considering that the 352 steel strands, which were put in place, were continuously retensioned during the removal of provisional tendons (Morandi 1967), in order to vanish the vertical displacement of the hanging section [point A in Fig. 6(a)]. At the same time, the internal actions in the deck can be assessed due to itself self-weight [Fig. 6(b)].Stress Field in the Post-Tensioned Concrete StayIn order to evaluate the fatigue damage accumulation in the stay, it is crucial to our purpose to obtain the cable-stay geometry and distribution of stresses with sufficient detail and high-precision (Invernizzi et al. 2020a). This can be achieved exploiting the analytical solution for the elastic catenary (Irwine 1981), with respect to the different construction phases of the cable-stay. In addition, a simple but effective model is introduced in the following to allow for the determination of the state of stress in the cross-section, even in the case of corrosion of the steel strands.The elastic catenary scheme is represented in Fig. 7, whereas the parametric solution of the deformed elastic line and of the axial force, as a function of the curvilinear coordinate s, are given by the following equations:(1) x(s)=HEA0s+HL0W[sinh−1⁡(VH)−sinh−1⁡(V−Ws/L0H)](2) y(s)=WEA0s(VW−s2L0)+HL0W[1+(VH)2−1+(V−Ws/L0H)2](3) According to the construction phases, at the end of phase 1, the 352 inner steel strands (Fig. 8) hung at the top of the antenna were set up to constrain the vertical displacement of the deck in correspondence of the transverse girder beam [point A in Fig. 6(a)].Therefore, the kinematic boundary conditions for the elastic catenary equation are known at both extremities of the cable, whereas the vertical component V–W of the axial force is known at the lower end of the cable, because it must be equal to the vertical reaction at point A [Fig. 6(a)] obtained from the previous analysis of the deck. It is, thus, possible to assume the initial length of the cable L0 and the horizontal component H of the reaction of axial force T as unknown discrete parameters, to be obtained by solving the corresponding nonlinear algebraic system (Irwine 1981). Those values can be finally substituted into the general solutions [Eqs. (1)–(3)] that provide both the axial force diagram, T(1)(s), and the elastic line as functions of the curvilinear coordinate, s, at the end of phase 1 (Fig. 9, thick continuous line). Therefore, the normal stress in the cross-section of the 352 inner strands, Ainn_st, at the end of phase 1 is calculated as follows:(4) σinn_st(1)=T(1)(s)Ainn_stAt the second stage of the cable-stay construction (phase 2), the concrete covering was cast in several distinct segments in order to allow for the sensible change of the elastic line without cracking of the concrete. In this phase, the concrete segments act as an external load on the 352 inner steel tendons, without providing effective bending stiffness to the cable-stay. The new elastic line can thus be obtained again by solving the catenary equation, simply accounting for the increased dead weight per unit length of the cable due to both the self-weight of the inner steel strands and of the concrete segments. The elastic line of phase 2 is shown in Fig. 9 with a dotted line. Fig. 10 shows the normal stress in the steel strands depending on the considered construction phase , which are given by the following equation:(5) σinn_st(2)=T(2)(s)Ainn_stwhere T(2)(s) is the axial force at the end of phase 2. This mechanical quantity is assessed according to (3), once that the horizontal force H and L0 have been calculated by the Eqs. (1) and (2) by considering the self-weight of the concrete covering and the inner tendons.After curing of the concrete segments and filling of the joints, the 112 outer tendons were added and prestressed to provide a negative stress state to the concrete covering and ensuring protection of the steel elements against aggressive environment (phase 3).During post-tensioning, the outer strands are looped over the top of the antenna, and equal pretensioning is provided in correspondence to the transverse girder beam at the bridge deck. The strands are disposed symmetrically with respect to the cable-stay cross-section. The post-tensioning of the concrete covering does not affect the elastic line configuration because the contraction of the concrete can take place independently of the inner tendons because the relative axial displacement are not yet constrained, so that the normal stress in the inner strands is not affected by the concrete pretensioning:(6) In Table 1 the principal mechanical properties of the construction materials used to make the stay-cables, such as the elastic modulus, the characteristic ultimate tensile strength, and the characteristic compression strength of steel tendons and concrete, are reported.Table 1. Principal mechanical properties of the construction materials used to make the stay-cablesTable 1. Principal mechanical properties of the construction materials used to make the stay-cablesElasticUltimateCompressivemodulustensile strengthstrengthMaterial(MPa)(MPa)(MPa)Prestressing steel195,0001,700—Concrete35,000—35On the other hand, the stress state of the outer tendons is not uniform nor constant with time. The stress losses due to relaxation and friction along the curved axis of the cable-stay can be assessed according to the European Standards (CEN 2011). The initial pulling stress of 1,200 MPa at the bridge deck connection (Morandi 1967), σp_0_max, is thus reduced to 900 MPa due to long-term phenomena. In addition, the prestress losses due to friction can be assessed, being known as the curvature of the elastic line in phase 2. Therefore, the tensile stress in the prestressing outer tendons can be evaluated as follows:(7) σout_st(3)=σp0(s)−Δσp,c+s+r(s)The term σp_0(s) is the prestressing stress along the curvilinear coordinate by considering the friction loss, whereas Δσp,c+s+r(s) takes into account the stress losses due to the long-term phenomena, that is, creep and shrinkage of concrete and the relaxation of steel tendons. The negative stress state induced in the concrete covering can be calculated, for each section of the cable state, thanks to the compatibility equation for deformation, namely:(8) σc(3)=σout_st(3)(s)Aout_stAcwhere Aout_st is the cross-section area of the outer tendons, whereas Ac is the reacting cross-section area of the concrete covering.After post-tensioning of the outer strands, the ducts containing the tendons were injected, and the 112 tendons were linked to the transverse beams of the deck. From this stage on (phase 3) the cable-stay performed as a unique solid prestressed concrete element.Subsequently, the Gerber beams were put in place,as well as New Jersey barriers, and the flexible road pavement was made (phase 4). At the end of this stage, the normal stress in the three resisting elements is given by the following expressions:(9a) σinn_st(4)=σinn_st(3)+nΔTdlAequi_c(9b) σout_st(4)=σout_st(3)+nΔTdlAequi_c(9c) σc(4)=σc(3)+ΔTdlAequi_cwhere ΔTdl is the increment in the axial force of the stay cable due to the dead load, the term Aequi_c is the equivalent area of the section in terms of concrete, whereas n is the modular ratio.The effects of variable loads on the bridge deck is considered in phase 5 according to European Standards (CEN 2003), which are evaluated similarly to what has been done above, that is,(10a) σinn_st(5)=σinn_st(4)+nΔTvlAequi_c(10b) σout_st(5)=σout_st(4)+nΔTvlAequi_c(10c) σc(5)=σc(4)+ΔTvlAequi_cwhere ΔTvl is the increment in the axial force of the stay cable due to the variable load.It is worth noting that a comparison with the variable loads adopted by Morandi at the design time provided no significant difference as far as the amplitude of the loads is concerned.Figs. 10–12 show the normal stress in the inner strands, in the outer strands, and in the concrete for the five different construction phases, respectively. It appears evident how the stress losses diminish both the stress level in steel and in the concrete. On the other hand, nonstructural permanent loads increase the level of stress in steel and decrease the absolute value of stress in concrete.It is worth noting that the structural system conceived by Morandi would have been effective, although not everywhere at the same extent, if only the degradation could be carefully avoided.

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