1.1 The Paderno d 'Adda Bridge

The Paderno d'Adda Bridge, also called San Miche-le Bridge, is a metallic viaduct that crosses the Adda river between Paderno and Calusco d'Adda to a height of approximately 85 m from water, allowing to connect the two provinces of Lecco and Bergamo, near Milano, in the Lombardia region, northern Italy (SNOS 1889, Nasce et al. 1984). At that location the river flows-down from the exit of Lecco's branch of Como's lake to the river Po through an impressive natural scenery that even seems to have inspired celebrated paintings by Leonardo (Fig. 1).

The main upper continuous beam, 5 m wide, is formed by a 266 m long metallic truss supported by

Figure 1. Front view of the Paderno d'Adda Bridge, 1887-1889 (from up-stream; left bank Calusco, right bank Paderno). A crossing train is visible inside the upper horizontal continuous beam. Automotive and pedestrian traffic runs on top.

Figure 1. Front view of the Paderno d'Adda Bridge, 1887-1889 (from up-stream; left bank Calusco, right bank Paderno). A crossing train is visible inside the upper horizontal continuous beam. Automotive and pedestrian traffic runs on top.

nine bearings. Four of these supports are provided by a marvellous doubly built-in parabolic metallic arch of about 150 m of span and 37.5 m of rise. The bridge shares its architectural style with similar arch bridges built in Europe at the time (Timoshenko 1953, Benvenuto 1981, Nasce et al. 1984), like e.g. that of Garabit (1884, France, Eiffel and Boyer) and Maria Pia (1887, Oporto, Eiffel and Seyrig), both doubly hinged at the shoulders, and the Dom Luiz I (1886, Oporto, Seyrig), doubly built-in as that of Paderno. The viaduct was quickly constructed between 1887 and 1889 (thus practically at the same time of the most celebrated Tour Eiffel), to comply with the needs of the rapidly growing industrial activities in Lombardia. It was built by the Societa Nazionale delle Officine di Savigliano (SNOS), Cuneo, Italy, under the technical direction of Swiss Engineer Giulio Röthlisberger (1851-1911), the man whom the design of the bridge is normally attributed to. He was formed at the Polytechnic of Zürich, graduated in 1872 and got later in charge of the Technical Office of the SNOS since 1885, for 25 years. The bridge is still in service, with alternated oneway automotive traffic, restricted to no heavy-weight vehicles, and trains crossing at slow speed.

The bridge is designed through the graphical-analytical methods of structural analysis that were booming in the 19th century (Culmann 1880, Timoshenko 1953, Benvenuto 1981). Specifically, it is a remarkable application of the so-called theory of the ellipse of elasticity (Culmann 1880, Belluzzi 1942).

This theory was originally conceived by Karl Culmann (1821-1881) and then systematically developed and applied by his pupil Wilhelm Ritter (1847-1906). It represents a very elegant method for the analysis of the flexural elastic response of a structure and is based on an intrinsic discretisation of a continuous beam in a series of elements, each with a proper elastic weight, directly proportional to its length and inversely proportional to its bending stiffness. The theory of the ellipse of elasticity is based on the concepts of projec-tive geometry, which lead to a correspondence between the ellipse of elasticity of the structure and the central ellipse of inertia of the distribution of the elastic weight of the structure. This correspondence brings back the problem of the determination of the flexural elastic deformation of a beam to a problem of pure geometry of masses, of more convenient solution and direct interpretation in terms of the design of the structure.

1.2 Main technical features of the bridge

The main technical features of the bridge are reported in details in Nasce et al. (1984), which is, to our knowledge, the most comprehensive publication, and one of the very few, concerning the bridge. We rely very much on this very valuable contribution and on the Technical Report (SNOS 1889) that was originally issued at the time of the first try-out. Here, the essential characteristics are reported.

The 266 m long upper flyover is made by a continuous box girder with nine equally-distributed supports, at 33.25 m distance from each other. Four of the supports are sustained by a big parabolic metallic arch; two of them bear directly on the same arch's masonry abutments (made with Moltrasio masonry, with Baveno granite coverings); a seventh, on the Calusco bank, rests on a smaller masonry foundation placed between the arch shoulder and the higher bridge supports; the last two, in masonry work as well, are the two direct beam bearings at its two ends, on top of the two river banks. The four piers resting on the arch are placed symmetrically, in between keystone, haunches and shoulders ofthe arch. The inner side ofthe beam girder, on which the railway is located, runs at about 255.00 m on the sea level (osl); the rails are placed at 255.45 m osl, the upper road at 261.75 m osl. The main vertical longitudinal trussed beams of the upper continuous girder are 6.25 m high and placed at a respective transverse distance of 5.00 m, leaving a free passage for the trains of 4.60 m. They are composed of two main T-ribs connected by a metallic truss. The upper-level road is 5.00 m wide and includes also two additional cantilever sidewalks, each 1 m long, with iron parapets 1.50 m high.

The big arch is composed by two couples of secondary inclined arches. Each couple is formed by two arches posed at a respective distance of 1 m and laying symmetrically to a mean plane inclined of about

±8.63° to the vertical. The parabolic axis of the arch has a span of 150.00 m and rise of 37.50 m in the inclined plane; the transverse arch's cross section is 4.00 m high at the keystone and 8.00 m high at the abutments (i.e. in the same 1:2 ratio between rise and half span). The two mean inclined planes of the arches are located at a distance of 5.096 m at the keystone and 16.346 m at the shoulders. The wall of each composing arch is also a truss structure with two main T-ribs connected by vertical and inclined bars. The two couples of twin arches are gathered together by two transverse brace systems located at the extrados and at the intrados of the arch's body. In essence, the resulting cross section of the main parabolic arch supporting the horizontal beam is trapezoidal, with variable, increasing cross section from the crown to the shoulders. This, and specifically the inclination of the twin arches, is a beautiful key feature of Rothlisberger's conception of the bridge, in view of counteracting effectively wind and transverse horizontal actions in spite of the considerable slenderness of the structure. The arch cross section at the impost is inclined of 45° to the horizontal, so as the local tangent to the parabolic axis of the arch to the vertical. The vertical bridge piers that sustain the upper continuous beam are made by eight T-section columns, linked to each other by a brace system with horizontal bars and St. Andrew's crosses and, on top, by transverse beams that directly serve as supports for the bearing devices of the upper beam. For inspection and maintenance purposes a 1 m large boardwalk is provided into the body of the arch and, inside the bridge piers, a system of ladders along their height. The bridge is a riveted wrought iron structure of about 26001 of metals, with near 100000 rivets just in the arch.

1.3 Aim of this work

In this work, which in its main part largely refers to a study developed in a Laurea Thesis (Ferrari 2006), a detailed analysis of the SNOS Report (1889) is presented. The point of view here is the following: inquire the application of the theory of the ellipse of elasticity to the calculation of the bridge, breathe the beauty of the architectonic and structural conception directly linked to that, compare results with modern structural approaches that also consider now-available numerical discretisation methods.

After a careful review of the Report by the SNOS, a full 3D truss Finite Element model of the arch of the bridge has been elaborated, based also on direct inspections of the bridge and on the screening of the marvellous original drawings that are guarded at the Archivio Storico Nazionale di Torino. Different loading conditions have been considered and results compared with those reported in the SNOS Report, showing the remarkable accuracy of the adopted graphical-analytical methods and allowing to experience the unrepeated beauty of the original analysis with respect to rather impersonal computer structural analysis. Moreover, the model that has been put in place shows promise for possible further analyses that could inspect other behaviours of the bridge, as for example dynamical and inelastic, also connected to the present and future state of conservation of the structure. These aspects are left for further developments of the present study.

The paper is organized as follows: Section 2 provides a short account on the theory of the ellipse of elasticity; Section 3 reports its application to the structural analysis of the arch of the bridge; Section 4 presents an independent validation of the original design results with present analytical-numerical methods.

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