Structural Analysis of Historic Construction – D’Ayala & Fodde (eds)© 2008 Taylor & Francis Group, London, ISBN 978-0-415-46872-5

On the theory of the ellipse of elasticity as a natural discretisationmethod in the design of Paderno d’Adda Bridge (Italy)

R. Ferrari & E. RizziDip.to di Progettazione e Tecnologie, Facoltà di Ingegneria, Università di Bergamo, Dalmine (BG), Italy

ABSTRACT: The Paderno d’Adda Bridge, Lombardia, northern Italy, is one of the very first great iron con-structions designed through the practical application of the theory of the ellipse of elasticity, a graphical-analyticalmethod of structural analysis that was developed in the 19th century. It embeds a natural discretisation of thestructure into a series of elastic elements, treated then with standard tools of geometry of masses. In this work,the application of such theory to the calculation of the parabolic arch of the bridge is inquired, attempting tobreathe, at the same time, the beauty of the architectonic and structural conception directly linked to that; later,results are compared with much modern approaches that also consider now-available numerical discretisationmethods. A further, definite aim of this work is also that of trying to promote interest on the bridge, on its actualstate of conservation and future destinations. Not only it represents a true industrial monument and a living tes-timony of the scientific and technological developments of the time but also a beautiful, effective achievementof architecture and engineering through the methods of Strength of Materials.

1 INTRODUCTION

1.1 The Paderno d’Adda Bridge

The Paderno d’Adda Bridge, also called San Miche-leBridge, is a metallic viaduct that crosses theAdda riverbetween Paderno and Calusco d’Adda to a height ofapproximately 85 m from water, allowing to connectthe two provinces of Lecco and Bergamo, near Milano,in the Lombardia region, northern Italy (SNOS 1889,Nascè et al. 1984). At that location the river flows-down from the exit of Lecco’s branch of Como’s laketo the river Po through an impressive natural scenerythat even seems to have inspired celebrated paintingsby Leonardo (Fig. 1).

The main upper continuous beam, 5 m wide, isformed 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 bankPaderno). A crossing train is visible inside the upper hori-zontal continuous beam. Automotive and pedestrian trafficruns on top.

nine bearings. Four of these supports are providedby a marvellous doubly built-in parabolic metallicarch of about 150 m of span and 37.5 m of rise. Thebridge shares its architectural style with similar archbridges built in Europe at the time (Timoshenko 1953,Benvenuto 1981, Nascè et al. 1984), like e.g. that ofGarabit (1884, France, Eiffel and Boyer) and Maria Pia(1887, Oporto, Eiffel and Seyrig), both doubly hingedat the shoulders, and the Dom Luiz I (1886, Oporto,Seyrig), doubly built-in as that of Paderno.The viaductwas quickly constructed between 1887 and 1889 (thuspractically at the same time of the most celebratedTour Eiffel), to comply with the needs of the rapidlygrowing industrial activities in Lombardia. It was builtby the Società Nazionale delle Officine di Savigliano(SNOS), Cuneo, Italy, under the technical directionof Swiss Engineer Giulio Röthlisberger (1851–1911),the man whom the design of the bridge is normallyattributed to. He was formed at the Polytechnic ofZürich, graduated in 1872 and got later in charge of theTechnical Office of the SNOS since 1885, for 25 years.The bridge is still in service, with alternated one-way automotive traffic, restricted to no heavy-weightvehicles, and trains crossing at slow speed.

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

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This theory was originally conceived by Karl Culmann(1821–1881) and then systematically developed andapplied by his pupil Wilhelm Ritter (1847–1906). Itrepresents a very elegant method for the analysis ofthe flexural elastic response of a structure and is basedon an intrinsic discretisation of a continuous beam ina series of elements, each with a proper elastic weight,directly proportional to its length and inversely pro-portional to its bending stiffness. The theory of theellipse of elasticity is based on the concepts of projec-tive geometry, which lead to a correspondence betweenthe ellipse of elasticity of the structure and the centralellipse of inertia of the distribution of the elastic weightof the structure. This correspondence brings back theproblem of the determination of the flexural elasticdeformation of a beam to a problem of pure geome-try of masses, of more convenient solution and directinterpretation in terms of the design of the structure.

1.2 Main technical features of the bridge

The main technical features of the bridge are reportedin details in Nascè et al. (1984), which is, to our knowl-edge, the most comprehensive publication, and one ofthe very few, concerning the bridge. We rely very muchon this very valuable contribution and on the Techni-cal Report (SNOS 1889) that was originally issuedat the time of the first try-out. Here, the essentialcharacteristics are reported.

The 266 m long upper flyover is made by a continu-ous box girder with nine equally-distributed supports,at 33.25 m distance from each other. Four of the sup-ports are sustained by a big parabolic metallic arch; twoof them bear directly on the same arch’s masonry abut-ments (made with Moltrasio masonry, with Bavenogranite coverings); a seventh, on the Calusco bank,rests on a smaller masonry foundation placed betweenthe arch shoulder and the higher bridge supports; thelast two, in masonry work as well, are the two directbeam bearings at its two ends, on top of the two riverbanks. The four piers resting on the arch are placedsymmetrically, in between keystone, haunches andshoulders of the arch.The inner side of the beam girder,on which the railway is located, runs at about 255.00 mon the sea level (osl); the rails are placed at 255.45 mosl, the upper road at 261.75 m osl. The main verti-cal longitudinal trussed beams of the upper continuousgirder are 6.25 m high and placed at a respective trans-verse distance of 5.00 m, leaving a free passage forthe trains of 4.60 m. They are composed of two mainT-ribs connected by a metallic truss. The upper-levelroad is 5.00 m wide and includes also two additionalcantilever sidewalks, each 1 m long, with iron parapets1.50 m high.

The big arch is composed by two couples of sec-ondary inclined arches. Each couple is formed by twoarches posed at a respective distance of 1 m and lay-ing symmetrically to a mean plane inclined of about

±8.63◦ to the vertical. The parabolic axis of the archhas a span of 150.00 m and rise of 37.50 m in theinclined plane; the transverse arch’s cross section is4.00 m high at the keystone and 8.00 m high at theabutments (i.e. in the same 1:2 ratio between rise andhalf span). The two mean inclined planes of the archesare located at a distance of 5.096 m at the keystone and16.346 m at the shoulders.The wall of each composingarch is also a truss structure with two main T-ribs con-nected by vertical and inclined bars. The two couplesof twin arches are gathered together by two transversebrace systems located at the extrados and at the intra-dos of the arch’s body. In essence, the resulting crosssection of the main parabolic arch supporting the hor-izontal beam is trapezoidal, with variable, increasingcross section from the crown to the shoulders. This,and specifically the inclination of the twin arches, is abeautiful key feature of Röthlisberger’s conception ofthe bridge, in view of counteracting effectively windand transverse horizontal actions in spite of the con-siderable slenderness of the structure. The arch crosssection at the impost is inclined of 45◦ to the hori-zontal, so as the local tangent to the parabolic axisof the arch to the vertical. The vertical bridge piersthat sustain the upper continuous beam are made byeightT-section columns, linked to each other by a bracesystem with horizontal bars and St. Andrew’s crossesand, on top, by transverse beams that directly serve assupports for the bearing devices of the upper beam.For inspection and maintenance purposes a 1 m largeboardwalk is provided into the body of the arch and,inside the bridge piers, a system of ladders along theirheight.The bridge is a riveted wrought iron structure ofabout 2600 t of metals, with near 100000 rivets just inthe arch.

1.3 Aim of this work

In this work, which in its main part largely refers toa study developed in a Laurea Thesis (Ferrari 2006),a detailed analysis of the SNOS Report (1889) is pre-sented. The point of view here is the following: inquirethe application of the theory of the ellipse of elastic-ity to the calculation of the bridge, breathe the beautyof the architectonic and structural conception directlylinked to that, compare results with modern structuralapproaches that also consider now-available numericaldiscretisation methods.

After a careful review of the Report by the SNOS,a full 3D truss Finite Element model of the arch ofthe bridge has been elaborated, based also on directinspections of the bridge and on the screening ofthe marvellous original drawings that are guardedat the Archivio Storico Nazionale di Torino. Dif-ferent loading conditions have been considered andresults compared with those reported in the SNOSReport, showing the remarkable accuracy of theadopted graphical-analytical methods and allowing to

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experience the unrepeated beauty of the original anal-ysis with respect to rather impersonal computer struc-tural analysis. Moreover, the model that has been putin place shows promise for possible further analysesthat could inspect other behaviours of the bridge, as forexample dynamical and inelastic, also connected to thepresent and future state of conservation of the struc-ture. These aspects are left for further developmentsof the present study.

The paper is organized as follows: Section 2 pro-vides a short account on the theory of the ellipseof elasticity; Section 3 reports its application to thestructural analysis of the arch of the bridge; Sec-tion 4 presents an independent validation of the orig-inal design results with present analytical-numericalmethods.

2 ON THE THEORY OF THE ELLIPSE OFELASTICITY

2.1 Fundamentals

The theory of the ellipse of elasticity can be consid-ered as a main icon of the so-called Graphical Statics,the discipline which often characterised the resolvingapproach of practical design problems during the 2ndhalf of the 19th century. It represents a very elegantand practical method for the analysis of the flexuralresponse of an elastic structure. It is based on an intrin-sic discretisation of a continuous elastic problem. Thistheory is basically associated to the two outstandingfigures of Culmann and Ritter, but also of people, likeGiulio Röthlisberger, that were formed at the time atthe Polytechnical Schools in Europe and that becamelater structural engineers and designers and largelycontributed in the practical and effective applicationof the method.

The theory is based on the following main hypothe-ses (we refer here to the Italian text by Belluzzi 1942,which reports results from the technical literatureof the time, basically ascribed to the two names ofCulmann and Ritter): (a) linear elastic behavior ofthe material and the structure, which leads to theproportionality between acting forces and (reversible)displacements provoked by them (property that in turnimplies the validity of the principle of superpositionof effects); (b) existence of the ellipse of elasticity,referred to a section of a structure; (c) correspondencebetween the latter and the central ellipse of inertia ofthe distribution of the so-called elastic weight of thestructure.This correspondence transforms the problemof the determination of the elastic response of a con-tinuous structure to a task of pure geometry of masses.The latter can be feasibly handled by taking advantageof the assumed discrete character of the distribution ofthe elastic weight and is endowed with a visible inter-pretation of the elastic performance of the structure,

Figure 2. Representation of a force R applied along line ofaction r to a beam section A and the centre of rotation C ofA. C′ is the symmetric of C with respect to the centre S of theellipse of elasticity of beam AB referred to section A.

in view of its conception and design. Furthermore, theellipse itself may actually play the role of an hidden,underlying, graphical construct. Indeed, the proper-ties of projective geometry that are attached to thatallow the elastic solution of the structure even withoutthe explicit drawing of the ellipse itself. The methodsare also said graphical-analytical because, in practice,main technical steps that are framed on the graphi-cal constructions may be carried-out analytically, byworking-out formulas that arise from the inspectionof the drawings (Belluzzi 1942).

The concept of the ellipse of elasticity referred to asection of an elastic structure is achieved by inspect-ing the correspondence existing between the line ofaction r of a force R applied to a section A of a gen-eral, curvilinear, elastic beam (with little curvature andcontinuously-varying cross section) and the centre ofrotation C of the same section (Fig. 2).

In particular, refer to a planar beam acted uponby forces laying in the same plane and cross sectionsof the beam that, during the beam’s deformation, areassumed to remain plane and perpendicular to the geo-metric axis, also deforming in its original plane. Thetheory states that there exists an involutory relationshipbetween the line of action r and the centre of rotationC of the section. Moreover, the ellipse of elasticityis the fundamental real conic of the polarity existingbetween the line of action r and the point C′, which isthe symmetric of C with respect to the centre S of theellipse. In other words, the ellipse of elasticity can bedefined as the fundamental conic with respect to whichthe lines of action r and the respective centres of rota-tion C correspond to each other through an antipolarityrelationship.

The determination of the central ellipse of inertia ofthe distribution of the elastic weight of the structure,which coincides with the ellipse of elasticity above,is linked first to the definition of the general conceptof elastic weight and then to the quantification of itsdistribution for the structure under consideration. Theconcept of elastic weight goes as follows. If on a sec-tion A of a beam, a moment M acts in the plane which

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contains the geometric axis, it causes a rotation φ ofA, around the centre S of the ellipse of elasticity. Therotation is proportional to the applied moment M as:

where G represents the so-called elastic weight of thebeam. Thus, G can be defined as the angle of rota-tion φ that is caused by the application of a unitarymoment M = 1; it depends on the beam’s geometricaland physical properties; it gives a global measure ofthe beam’s aptitude to deform. In case of a straightcantilever beam of length l loaded by a moment Mat its free end, composed by a linear elastic materialwith Young’s modulus E and endowed with a constantcross section with moment of inertia J with respect tothe axis perpendicular to the beam’s plane, it turns outthat, referring to the case of flexure of de Saint Venant,the rotation of the free end is:

This relation clearly illustrates the physical meaning ofG as the global parameter that expresses the flexuralelastic deformability of the structure.

Now, the idea arises of thinking at the structure asthe assembly of a series of discrete elastic elements oflength �s, each with a proper elastic weight

such that the total elastic weight of the structure isrepresented by the discrete distribution of these elas-tic weights. It is possible to demonstrate that sucha sought distribution of G is univocally known onlyfor a statically-determined structure, whereas for astatically-undetermined structure the distribution ofelastic weights is not univocally defined. This is notsurprising, due to the redundancy of equilibrium in anhyperstatic system. Despite this, an hyperstatic struc-ture can still be solved, via the Forces Method (withhyperstatic quantities as unknown), through the super-position of effects on underlying isostatic structuresand imposition of the corresponding compatibilityconditions. As the underlying isostatic structure canalso be analysed with a univocally-defined distribu-tion of elastic weights, such distribution can also beused to solve the original hyperstatic structure. Thus,indirectly, its ellipse of elasticity can in essence bedetermined, so the corresponding ellipse of inertia ofthe distribution of elastic weights.

This allows one to write the so-called “theoremsof the theory of the ellipse of elasticity” (Belluzzi1942), as a function of the properties of the distri-bution of elastic weights. For example, the rotation φ

Figure 3. Representation of the quantities reported in Eq. (4)for the calculation of the rotation and displacements of ter-minal section A caused by a force Q applied to the samesection.

and horizontal and vertical displacements dx, dy of aterminal beam section A caused by a force Q appliedto the same section along line of action q (Fig. 3), canbe nominally written as follows:

where G = ∑�G represents the total elastic weight of

the structure; Sq, Jxq, Jyq the static moment of G withrespect to q and the centrifugal moments of inertia ofG with respect to q and axis x, and q and axis y. Theseparameters depend only on the distribution of elasticweights and on the position of the applied load Q, andcan be expressed as a function of the quantities xS , yS ,uS , uX , uY depicted in Fig. 3.

The point S in Fig. 3 represents the centre of grav-ity of the elastic weights of the structure; the pointsX,Y represent the antipoles of the reference systemaxes x, y with respect to the central ellipse of inertiaof the elastic weights. It is then apparent that, oncethe position of points S, X, Y and total elastic weightG are found, the elastic response of the structure isdetermined. The coordinates xS , yS , xY , yX definingthe position of these points can be evaluated by stan-dard calculations of geometry of masses, once giventhe discrete distribution of elastic weights.

2.2 Application to a doubly built-inparabolic arch

In the SNOS Report (1889), the remarkable applica-tion of the theory of the ellipse of elasticity to theanalysis of the arch of the bridge refers to the deter-mination of the elastic response of a parabolic arch,built-in at the two extremities, which is subjected to avertical load P (that can be put equal to 1) and actingin an arbitrary position along the arch, at a horizontaldistance a from left extreme A (Fig. 4).

First, the position of the line of action of the reac-tion A needs to be determined. This can be solved by a

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Figure 4. Calculation of line of action LO of left reactionA; the segments FL and VO that the left reaction A locates onthe vertical lines from A and on load P, below and above thehorizontal line from centre S are determined.

Figure 5. Calculation of line of action HK of left reaction A;segments a, b, c to be drawn at points S,X,Y are determinedanalytically. Points H,K are then located, so the direction of A.

graphical-analytical procedure. Figs. 4 and 5 representtwo ways to solve this problem (SNOS 1889, Ferrari2006). Underlying to these constructions lay the com-patibility conditions φ = 0, dx = 0, dy = 0 for the leftbuilt-in constraint at terminal section A. These can beworked-out from Eq. (4), leading to:

where A is now taking the role that force Q had in(4) and the superposition of effects is considered withload P, so that u′

S = a, u′X = b, u′

Y = c (a, b, c denotesegments used below in Fig. 5) are quantities similar tothose entering Eq. (4), but related to load P at position afrom A. They can be calculated analytically as follows,given the distribution of elastic weights:

Once scalars u′S = a, u′

X = b, u′Y = c are evaluated, the

construction in Fig. 5 determines the position of A.

Figure 6. Equilibrium requires that the left and right reac-tions A and B form with load P a closed polygon of forces.Segments V = OQ and V ′ = QP represent the vertical com-ponents of A and B, with V + V ′ = P; segment H = TQ theirhorizontal component.

Alternatively, one may also proceed as sketched inFig. 4, by calculating parameters µ, ν and segmentsFL, VO, with same results. The last relations in (5)also give the magnitude of reaction A, given P.

Once the position of the lines of action of left built-in reaction A, and, by equilibrium, of right reactionB are known, the value of their vertical componentsV , V ′ and (common) horizontal component H can befound as follows:

where quantities a, a′, f, f′ are represented in Fig. 6.Following similar arguments, it is possible to derive

the in-plane arch’s deflections dy at any position x ofthe arch, for the different locations a of the load Pacting on the arch (SNOS 1989). The equation thatgives the mean parabolic line at the arch is taken as:

where f and l represent now the rise and span of thearch. The profile of the arch is symmetric with respectto the vertical axis x = l/2 at half span. The formulasfor dy obtained by the SNOS with this graphical-analytical procedure correspond indeed to those thatmay be obtained by the application of theVirtualWorksPrinciple (VWP):

where H , V , M represent the “components” of thereaction force A (built-in moment M is positive clock-wise) and C = EJdx/ds = cost is a quantity related to

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Figure 7. Discretisation of the half arch in 14 elementswith �x = cost, with indication of the geometrical quanti-ties useful to determine the corresponding elastic weights(quotes in m).

the bending stiffness and local inclination of the arch.C is assumed by the designer constant along the arch,since while J decreases from the shoulder to the key-stone, the ratio dx/ds between horizontal projection dxof infinitesimal element length ds of the arch and dsitself increases (SNOS 1889).

3 STRUCTURAL ANALYSIS OFTHE BRIDGE

3.1 Explicit application of the theory ofthe ellipse of elasticity to the analysisof the arch of the bridge

Going to the explicit definition of the distribution ofelastic weights made by the SNOS (1989), all the cal-culations reported there were developed consideringan arch which is the projection on their mean inclinedplane of one of the two couples of inclined parabolicarches that are placed symmetrically to the verticallongitudinal median plane of the bridge. Such arch isplaced on a plane inclined of about α = 8.63◦ to thevertical (such that sin α = 0.15) and consists of a trussbeam with parabolic axis of 150 m of span and 37.5 mof rise in such plane, having extrados and intrados linesboth described by parabolic functions so as to deter-mine a cross-high of the arch of 4 m at the keystoneand of 8 m at the abutment. On the basis of this model,the elastic weights of the structure have been calcu-lated according to a symmetric structural discretisationwith 28 elements of different �s extensions as reportedin Fig. 7.

In the Report, the procedure adopted to calculatethe elastic weights �G = �s/EJ is not really appar-ent. An attempt of careful analysis is provided inFerrari (2006). Also, the elastic weights are actually

determined without the constant proportionality fac-tor E = 17000000 t/m2, Young’s modulus of the iron,i.e. �G ′ = �s/J . The coordinates xS , yS , xY , yX andtotal elastic weight G ′ = EG are finally found as:

The constant C in Eq. (9) is also evaluated in55539000 t · m2.

Anyway, tables are presented in which the left reac-tion and the deflections of the arch are determined for aunitary load located at the various elastic elements. Inpractice, these influence coefficients, which are laterused for the design of the truss members, are deter-mined by a true application of the theory of the ellipseof elasticity as applied to the arch.

3.2 Evaluation of the loads

The SNOS Report analyses independently, one byone, the various loadings on the arch, for subsequentsuperposition of effects:

– permanent weight of the arch;– permanent weight of the upper girder beam, of the

bridge piers and vertical actions induced by thewind acting on the girder beam;

– accidental vertical load on the upper girder beam;– temperature effects and compression on the arch

due to the horizontal thrust H ;– direct horizontal wind action on the arch.

For each listed item, the SNOS reports the calculationof the stresses in the various arch’s elements, as wellas the final value that arises by the superposition ofeffects.

3.3 Dimensioning of the structural elements

The Report by the SNOS provides the calculationof the stresses in the various bar elements of thearch and at the stone abutments as compared to thetarget admissible values that are summarised below.This is done for the final geometries of the struc-tural members. On the other end, the Report does notprovide specific information about the design proce-dure that has lead, through pre-dimensioning, to suchfinal structural dimensions.As the overall architecturaland structural conception of the entire viaduct, thesephases seem to be linked to the engineering practiceand experience of the designer with the standards ofmetallic carpentry in use at the time.

The material employed in the structural membersof the bridge is a wrought iron, with very low car-bon percentage of about 0.01% (Nascè et al. 1984).The admissible stresses are taken differently for eachstructural component of the bridge: for the main upperand lower arch’s ribs 6.0 kg/mm2; for the vertical and

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Figure 8. Scheme with four test loading configurations,with indication of the four piers resting symmetrically onthe arch (view from down-stream; Paderno left side, Caluscoright side). Pier III is at half length of the upper continuousbeam.

diagonal bars linking the inferior and superior ribs ofthe arch 6.0 kg/mm2; for the elements that composethe transverse bridge’s brace system counteractingwind 4.2 kg/mm2. For each arch’s stone abutment,the allowable compression stress is assumed in nearly31 kg/cm2. Stresses are all found safely below thesetarget values.

The executive drawings, including the various crosssections of the structural elements, the details of theriveted joints and part of the graphical/analytical calcu-lations are reported in 147 marvellous drawing tables.They are still in an excellent state of conservation andshow the meticulous design of the complex structureof the bridge. Often, the calculus is integrated in thedrawing itself, which testimonies the beautiful, inti-mate link between conception, structural analysis andexecutive design.

3.4 Viaduct’s tests

The first viaduct’s tests took place from 12th to 19thMay 1889. The measured deflections for loading con-ditions conforming to those considered at design stagewere compared to the corresponding theoretical val-ues. To this purpose, the tests were carried out in twomoments: first, the different road loads were obtainedby deposition of gravel on the upper deck; second, witha uniformly distributed gravel load of 3.9 t/m all overthe road, 6 locomotives with tender, each of 83 t ofweight, corresponding to a distributed load of 5.1 t/m,were displaced on the railway track following fourdifferent loading configurations (Fig. 8). Tables 1–2below list the calculated and measured vertical deflec-tions. The good overall agreement doubtlessly showsthe validity of the structural approach used by theSNOS at design stage, in species the effectiveness ofthe theory of the ellipse of elasticity as applied to theexplicit calculation of the arch of the bridge.

A final test was also run with 3 locomotives and30 wagons, loaded by gravel, for a total weight of theconvoy of 600 t. The train was let running 3 times,

Table 1. Arch’s vertical deflections calculated at designstage for four different loading tests (SNOS 1889, p. 71).Negative values indicate downward displacements.

Pier I Pier II Vertex Pier III Pier IVDeflections (m) (m) (m) (m) (m)

Test I +0.0033 −0.0010 −0.0045 −0.0108 −0.0066Test II −0.0016 −0.0080 −0.0097 −0.0080 −0.0016Test III −0.0040 −0.0101 −0.0094 −0.0022 +0.0027Test IV −0.0068 −0.0064 −0.0010 +0.0035 +0.0031

Table 2. Arch’s vertical deflections measured in situ at thetry-out for the corresponding loading tests (SNOS 1889,p. 71).

Pier I Pier II Vertex Pier III Pier IVDeflections (m) (m) (m) (m) (m)

Test I +0.0038 +0.0001 −0.0044 −0.0106 −0.0056Test II 0 −0.0079 −0.0080 −0.0102 −0.0012Test III −0.0016 −0.0102 −0.0073 −0.0014 +0.0025Test IV −0.0063 −0.0058 −0.0007 +0.0042 +0.0026

with speed up to 45 km/h. Transverse oscillations atthe keystone were recorded in less than 3.142 mm andvertical deflections in less than 10 mm. According toNascè et al. (1984) a 2nd, final, try-out took place inJune 1892, with different modalities and applied loadsbut with similar results.

4 VALIDATIONS OF THE REPORTS’ RESULTSBY ANALYTICAL-NUMERICAL METHODS

4.1 Analysis of the elastic arch by the VWP

In order to compare the results presented by the SNOS,the hyperstatic scheme of the doubly built-in arch wassolved by the Virtual Works Principle. Such analy-sis allowed to notice some little inconsistencies in theSNOS results. First, the reactions A and B due to P = 1at distance a from left support A were evaluated andcompared to the Report’s results. The match was notperfect, particularly for the bending moments at theextremes of the arch. The differences did not seem tobe due to transcription errors, since the values reportedby the SNOS look coherent with formulas and tablespresented within the text. Second, further validationconcerning the arch’s deflections was attempted, sincethey also did not seem to be calculated through thereactions that are listed in the Report (Ferrari 2006).However, the final values reported by the SNOS cor-respond, to a good degree of accuracy, to the presentresults obtained by the VWP (Tables 3–4). The valuesreported inTables 3–4 refer to a precise load configura-tion (1st distribution, see Section 4.2.2 andTable 5) and

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Table 3. Bending deflections listed in SNOS (1889, p. 59).

Load Pier I Pier II Pier III Pier IVPoint (m) (m) (m) (m)

Pier I −0.0089 −0.0029 +0.0071 +0.0042Pier II −0.0011 −0.0042 +0.0026 +0.0030Pier III −0.0004 −0.0003 +0.0005 +0.0001Pier IV +0.0001 +0.0001 −0.0000 −0.0001

Total −0.0103 −0.0073 +0.0102 +0.0071

Table 4. Bending deflections calculated by the VWP.

Load Pier I Pier II Pier III Pier IVPoint (m) (m) (m) (m)

Pier I −0.00893 −0.00298 +0.00703 +0.00422Pier II −0.00126 −0.00425 +0.00255 +0.00297Pier III −0.00038 −0.00033 +0.00055 +0.00016Pier IV +0.00006 +0.00010 −0.00004 −0.00013

Total −0.01051 −0.00746 +0.01009 +0.00722

Table 5. Total vertical deflections (1st distribution) reportedby SNOS (1889, p. 60) and present results by the FE model.

Vertical loads SNOS deflections FE deflectionsPoint (t) (m) (m)

Pier I +340.6 −0.0120 −0.015903Pier II +144.0 −0.0113 −0.013847Pier III −18.5 +0.0062 +0.010322Pier IV +4.9 +0.0054 +0.007222

regard the vertical deflections (due to pure bending) atthe four piers due to forces acting at their locations.

The reason of the discrepancies above may be dueto the fact that the SNOS might have used in the finalReport also data concerning a preliminary project.Indeed, the bridge’s layout was slightly modified inthe executive project, due to new requirements on therailway trace that were posed by the Strade FerrateMeridionali, after checks on the Adda’s banks (Nascèet al. 1984). It is therefore possible that some specificdata were referring to a previous project, while impor-tant global quantities, such as the deflections causedby external actions, were indeed corresponding to thefinal one. As a matter of fact, even if this Report hasbeen probably conceived to present to a general audi-ence the main steps of the calculations, including avery valuable account on the theory of the ellipse ofelasticity and on its explicit application to the analysisof the arch, it is doubtlessly very concise and it obvi-ously presents concepts and practical considerationsthat may not be directly apparent to the contemporaryreader.

4.2 Structural analysis of the arch by the FEM

Currently, an attempt is made of building a full FEmodel of the bridge. So far, a truss mesh of the arch ofthe bridge has been developed and appropriate loadingconditions has been considered for validation of theprevious results. The FE analysis has been run withthe commercial code ABAQUS®.

4.2.1 Structural modelThe FE model consists of a 3D truss frame, reproduc-ing as much as possible the actual arch geometry. Itconsists of two planar parabolic trusses referring tothe in-plane geometry of the arch (see Fig. 7), placedin two inclined planes (of ±8.63◦ to the vertical). Theinclined planes are placed at a distance of 5.096 m fromeach other at the axis of the arches at the keystone.The truss nodes are linked to each other through areticular system that corresponds to the actual brac-ing of the arch. To each bar of the model, a crosssection with equivalent geometrical characteristics isattributed (area, principal moments of inertia, torsionalstiffness). Some approximations were made as regardsto the section’s attribution to the superior and inferiorarch ribs, which are made with a variable number oflongitudinal plates. Also, at the node junctions, thereare additional reinforcing plates, for local stiffening.In spite of this, the model has been simplified withbars of constant average cross section. The model iscomprised of 752 beam elements and 266 nodes. Built-in constraints are imposed at the nodes of the archshoulders.

4.2.2 Obtained resultsFirst trial loading cases considered a unitary load (1 t)applied at the piers and at the keystone, in view ofverifying the order-of-magnitude agreement with thedeflections that were listed in the Report and calculatedhere by the VWP. After these preliminary checks, thedeflections were evaluated for five given load distribu-tions with vertical loads acting at the four piers (SNOS1889, p. 58–62).

A sample of these outcomes is given in Figure 9for the 1st load distribution already considered inTables 3–4, with main results summarized in Table 5.Such distribution considers spans 2–3 charged by auniformly distributed load of 9 t/m, leads to a max-imum pressure on Pier I and is somehow similar toTest IV considered in the try-outs (Fig. 8). The maxi-mum compressive axial force at the intrados of the leftshoulder is found in 377.0 t, in good agreement withthe value 751.6/2 = 375.8 t calculated by the SNOS(1889, p. 62). Agreement is also found for the hor-izontal thrust H and vertical reactions V , V ′ at theshoulders, with H = 185.8 t, V = 417.2 t, V ′ = 53.8 t.The FE outcomes confirm, to a quite good degree ofaccuracy, the SNOS results, showing the true potential

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Figure 9. FE deformed configuration of the arch for the1st load distribution reported in Table 5 (amplificationfactor = 500).

of the theory of the ellipse of elasticity as applied tothe elastic analysis of the arch of the bridge.

5 CLOSING REMARKS

An attempt to scrutinise in details the SNOSReport (1889) has been made in view of breathing theoriginal conception of the design of the bridge and itsspecific calculation through the elegant method of thetheory of the ellipse of elasticity. The intrinsic discreti-sation in elastic elements is remarkable and makes anatural connection with discretisations that can nowbe provided by FE codes. The beauty of the executiveproject can be totally appreciated only by the comple-mentary screening of the 147 drawing tables of the

bridge. All this shows the ingenuous, effective andbeautiful approach to the design of the bridge. Onemight think at this once contemplating the giant stillstanding there silently, serving since almost 120 yearsof duty, with an actual state of conservation that actu-ally poses serious questions about its future survival.We should take care of it.

REFERENCES

Belluzzi, O. 1942. Scienza delle Costruzioni. Bologna:Zanichelli.

Benvenuto, E. 1981. La Scienza delle Costruzioni e il suoSviluppo Storico. Firenze: Sansoni.

Culmann, K. 1880. Traitè de Statique Graphique. Paris:Dunod.

Ferrari, R. 2006. Sulla concezione strutturale ottocentescadel ponte in ferro di Paderno d’Adda secondo la teoriadell’ellisse d’elasticità. Laurea Thesis in Building Engi-neering. Advisor E. Rizzi, Università di Bergamo, 228pages.

Nascè, V., Zorgno, A.M., Bertolini, C., Carbone, V. I.,Pistone, G., Roccati, R. 1984. Il ponte di Paderno: sto-ria e struttura – Conservazione dell’architettura in ferro.Restauro, Anno XIII, n. 73–74, 215 pages.

Società Nazionale delle Officine di Savigliano 1889.Viadottodi Paderno sull’Adda (Ferrovia Ponte S. Pietro-Seregno).Torino: Tip. e Lit. Camilla e Bertolero.

Timoshenko, S.P. 1953. History of Strength of Materials. NewYork: McGraw-Hill.

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