Structural Optimization of Network Arch Bridges with Hollow Tubular Arches and Chords

In the framework of bridge engineering, cable structure represent a key argument. The development of innovative bridge solutions are needed in order to keep time and costs at a reasonable level, maintaining at the same time an high level of structural safety and functionality. The network arch bridge solution completely respond to these requirements, and in addition could be designed as a very pleasant and formally elegant structure, because of its slenderness and lightness. In this paper, an introduction on the network arch solution is presented, together with structural analysis and data of relevant structures realized. Furthermore, an optimization of specific types of road and railway bridges is presented in the particular case of hollow sections, considering three alternative of cable disposition.


Introduce the Problem
Bridge engineers should deserve a particular attention for network arches.These structures in fact are highly performant for medium span bridge solutions, and almost ever preferably if compared to traditional arch, both in the road and the railway case.Reference works on the matter includes Tveit (1966) Shanack (2003), Tveit (2010), Tveit and Pipinato (2011), describing the structural performance of net-arches compared to traditional solutions.However similar works exists on the matter (e.g.Teich 2012;Brunn andSchanack 2003, Dimopoulos andGantes 2007), the paper innovates the state of knowledge by the use of a new software able to: (a) compare roadway and railway solutions according to EN loads, and (b) exploring different span lengths.The idea comes from the past, as first drawing of similar solution appeared in 1482, in a manuscript of Leonardo Da Vinci, designing in the "Codex Atlanticus" a truss arch with inclined hangers, probably wooden made.In 1796, American engineer Robert Fulton published a design of a bridge with a type of truss beam that he called "bowstring."His idea is based on an arch where concentrated loads are uniformly distributed by a truss and the lateral abutments are replaced by a tie between the arch ends (Shanak, 2015).Similar archetypes of the network arch could be found among some parental bridges: the Whipple bridges (Griggs, 2002), the lenticular bridge, or Pauli girder (a truss with a curved top and a curved bottom chord); the Lohse bridge (1872, Hermann Lohse lenticular girder with only vertical members between the chords); the Nielsen bridge (1925, Octavius F. Nielsen inclined hangers intersecting once at most).Finally, in 1955, the norwegian engineer and profesor Per Tveit started working on Nielsen bridges using new analysis methods that allowed him to calculate and develop tied arch bridges with multiple intersecting hangers, which he called "network arch bridges" (Tveit, 1966) (Figure 1).More publications on the matter exists, however, a systematic overview on the matter, including structural details, analysis, and optimization is not available.Recent investigation of the author includes a multi-span network arch project case for large span (Pipinato, 2016).As a first definition, network arches could be defined as a simply supported beam, in which the arch is the compression zone, the tie is the tension zone, and the inclined hangers are the web.Most of the shear force is taken by the vertical component of the compressive force in the arch.Some of the variation in the shear force is taken by the hangers.Throughout this structural model, bending is reduced at minimum in all elements.This bridge solution is generally suitable for medium span, as mentioned before, and optimal between 60 and 200 m for road bridges, and up to 150m for railway bridges.The diagram in Figure 2 has been presented to find network arch bridge alternatives, giving an estimate of the materials needed for double track road bridges spanning in the field-span of network arches (Tveit, 2014).Similarly, the second diagram in Figure 3 represents an estimate for two lane railway bridges up to 150m span (Shanack 2003).
Figure 2. Steel needed for various two-lane road bridges between 60m and 200m span lenght (Tveit, 2011).In the first stage of the network arch study and design, some of the hangers where not in tension for particular loading condition (asymmetric).To avoid this and to optimize the model, the geometry is changed, by the introduction of new hangers, and in this way all the hangers were in tension, consequently structural steel and shape was optimal, the arch doesn't have instability problems, and finally the bending stress on the deck is lower.While arches are usually made up of structural steel, the deck could be realized with a post-tensioned deck, or with a concrete-steel mixed section, or with a steel deck, or finally a combination thereof.One of the most performing solution is the post-tensioned deck, even if the choice of the effective structure should be chosen according to the design specifications and requirements.The hangers geometry and the hangers disposition is strictly related to the deck type adopted: if an all-concrete deck solution is adopted, the hangers design could be varied along the deck without any fixed position that is imposed from the geometry of the transverse beam in the steel solution.Some suggestions are given in literature (Tveit 1966;Tveit 2010;Tveit and Pipinato 2011) in order to design the final shape of the structure.In this particular case, transverse are present as a result of the structural optimization performed.Hangers geometry/inclination is given by non-dimensional numerical values related to Live Load vs. Dead Load ratio and Live Load vs. bridge span.In order to understand this, a traditional arch with inclined hangers and subjected to a non-symmetric loading could be considered: hangers are alternatively in tension and compression (Figure 4).To optimize this model, in order to have all hangers in tension and a lower effect of bending on the chord, another series of hangers should be added, as described in the following figure (Figure 5).Adding a new series of hangers, the beneficial aspects includes a minor arch buckling value, and minor bending effects on the chord and on the arch, developing a so called network arch (Figure 6).

Arches Optimization
The shape of the arch is chosen after a parametric analysis, finalized to minimize the steel amount needed to carry the live and dead load of the particular solution investigated.Circular arch shape are often found to be the optimal solution for the design of network arches (Pipinato 2016;Tveit and Pipinato 2011).The arch rise is normally 0.15-0.20 times the span, according to the coded loading scheme adopted (Table 1).The box section is usually represented by a squared or a circular commercial section for 100m and more span; for shorter span, universal columns or American wide flange beams, with profiles bent to the right curvature would be a correct solution (Tveit 2010); this alternative enable for attaching hangers and wind bracing connections in a easier way.If a parabola shape is chosen, according to the parametric analysis mentioned before, the arch shape should be near parabola of 2nd degree (Tveit 2010).When the arch is steel made, a constant curvature should be preferred.
The constant curvature -if compared to the parabolic solution -gives in the middle of the arch more even force and wind portals shorter, resulting at the same time easier to be built.Other arch curvatures resulted to be in a moderate level of savings.If near the ends of the arches a smaller radius of curvature is used, the wind portal will be shorter and the force in the middle half of the arch will be smaller.It would also lead to the maximum bending stress in the chords (Brunn & Schanack 2001).With elliptic arches a similar effect could be reproduced (Teich 2012).If in the tie the last members are built shorter, the consequent bending and rotation at the end of the bridge would be reduced and the tension in the shortest hanger would be very similar to the tension in the longer one (Tveit 2008).The absence of transversal beams in the tie, suggests to maintain constant the distance between the nodal points in the arch.At the end of the span the arch members could be 1.5-1.7 times longer than the other member of the arch.The concrete filled steel tube solution (CFST) could be evaluated for longer span, and are found to be economically convenient in the nearest of 200m span.The possible form of buckling in the plane of a network arch is represented in Figure 7.The buckling argument have been threated e.g. in Schanack (2009): with reference to the notation reported in Figure 8, it can be demonstrated that sufficiently accurate results are found if it is assumed that the tie remains straight.

Lower Chords Optimization
Minimizing bending in chords is the mainframe in the choice of chord type: the solution to this problem could be reached considering the four sources of bending in the network arches.Bending due to concentrated load and abrupt changes in load intensity (a), bending due to relaxation of hangers (b), bending due to distance between nodal points (c), and secondary bending (d): a) due to concentrated load, as illustrated in figure 9a, bending arises from internal local shear forces originating from the application of the external actions; similarly could be affirmed for abrupt changes, which implies internal shear variations; if a geometry including three set of hangers is adopted (figure 9b) bending is reduced, as the distance that the shear force have to travel in order to make all trusses deflect equally is shortened (Tveit 2008); further deeps on arches with three or four set of hangers are needed, however basing on first estimates on case studies developed by the author, this solution appear to be more costly and not so technologically affordable; b) relaxation of hangers in net-arch bridges complicates design, as the discontinuous system implies new equations for calculating forces and deflections for each combination of relaxed hangers; the easier solution to avoid relaxation consists in the introduction of smaller angles for hangers in the lower chords, however this implies an increase in bending for the aforementioned concentrated load problem; a straightforward solution consists in the dimensioning of the chords with full loads on the entire span, calculating parametrically the hangers slope optimal solution (inclination, number and geometry) (Tveit and Pipinato 2011); c) bending due to distance between nodal points could be reduced introducing more hangers; however, a crowded arch is then not suitable for an easy and economical construction, so also this structural choice should be calibrated with parametrical analysis; d) the global deflection of the whole span induces a predominant positive bending, which originates secondary moments in the chord; this is controlled by the cross section of the chords, from cambering the tie (approximatively of the 1/1000 of the span), and from the optimization of the hanger geometry; moreover, if the deck is built as a composite section, creep and shrinkage will contribute to counteract this type of secondary bending.To ensure the maximum control of secondary moments, a calculation based on the deflected state will give the exact values of bending moments, which resulted to be always very small.

Configuration and Optimization
The The optimization of network arches up to the span length of 200m has been performed and is presented herein for the particular case of hollow tubular sections.The choice of the hollow section has been based on past studies (Tveit and Pipinato 2011) ensuring the best optimal use of the arches and chords, a more pleasant aesthetic, and reduced maintenance.The calculation of a ten thousands of models of net arch bridges, has enhanced the optimization model, providing recommendations for practical application.For this purpose a stepwise procedure allows the designer to have precise information on the optimal hanger geometry considering different parameters, thus the exact solution should be achieved on the base of the specific design situation.The optimization procedure has taken into consideration three network arch models considered for the optimization (Figure 11): a) CSM, constant slope model: the constant slope of hangers solution is the most diffused construction solution, e.g. in Nielsen-Lohse bridges; built bridges of this types implies angles varying between 45°-75°; b) V-CSM, variant-constant slope model: the Tveit model ( 2004), is a particular subset of the CSM bridge alternative, with a constant hangers start angle of 30° at the abutment and angle changes between 0.5° and 3°; c) CARSM, constant angle to radius slope model: the model (Brunn and Schanack 2003), with inclination of hangers between 30° < 50.Lower values of  especially for 0°<  <30° are not recommended for unexpected bending concentration in the arch.
The parametrical analysis aiming to identify the correct number and disposition of hangers, is based on the optimization of the member capacity towards the loading conditions superimposed, minimizing the bending components on the whole principal members.In the following, the result of the optimization has been presented both for roadway and railway bridges; the main description of the analysed arches are provided herein.-Presence of foot paths: yes on both directions, out of arches.

Optimization of the Bridge Network Arches
The following step have been performed for every type of models presented before (CSM, V-CSM, CARSM), with reference to Figure 13   According to the previous steps performed, a synthesis of the results is reported in the figures below: in Figure 14 an example of the railway arch compression optimization is reported for the CSM alternative, 120m span, varying the angle of hangers; the other alternatives are quite similar to these results; a further comparison of the whole alternatives (CSM, V-CSM, CARSM), is reported in figure 15; analogous figures (16-17) are reported for the roadway case; -in Figure 18 an example of the railway arch bending optimization is reported for the CSM alternative, 120m span, varying the angle of hangers; the other alternatives are quite similar to these results; the figure 19 illustrates the roadway case; -in Figure 20 an example of the arch stress optimization is reported for a 120m span railway bridge, varying the angle of hangers, and considering a hollow arch and wind-bracings cross section (arch: B = 740 mm, h = 740 mm, t = 100 mm; wind-bracings: B = 560 mm, h = 560 mm, t = 80 mm) including the comparison of the whole alternatives (CSM, V-CSM, CARSM); the roadway case is reported in figure 21.
The further implementation of the different alternatives, has been implemented with Midas Civil (2016), supported by a computer routine in C++ providing the different parameter variation in the FEM input/output.According to the analysis performed, resume tables has been prepared for the hanger net optimal geometry: Tables 2-3 represent the optimal geometry alternative starting from the choice of the hanger number and the bridge span for the road and the railway alternative.The determination of the optimal inclination parameters for the selected alternative should be made according to Figure 22-23, while for CSM alternative, indications are given in Table 4.

Conclusion
In this study, network arch bridges optimization with hollow structural member has been presented.The network arch main positive issues have been explained and presented also with reference with real case bridges.The idea comes from the past, as first drawing of similar solution appeared in 1482, in a manuscript of Leonardo Da Vinci, designing in the "Codex Atlanticus" a truss arch with inclined hangers, probably wooden made.In 1796, American engineer Robert Fulton published a design of a bridge with a type of truss beam that he called "bowstring."A wide amount of studies have been published also in the recent years, however, the diffusion of this type of bridge solution has not reached the clear and hopeful evidence that should really deserve.
The shape of the case study arches considered in this study is chosen after a parametric analysis, finalized to minimize: -the steel amount needed to carry the live and -the dead load of the solution investigated.
In the second part, the paper illustrates an optimization study performed according to three main models: -the CSM, constant slope model, -the V-CSM, variant-constant slope model, -and the CARSM, constant angle to radius slope solution.
For each alternative a roadway and a railway application has been dimensioned for different span length, illustrating the possible configuration of the cables in each arch.According to the optimization performed, the designer could focus directly to the alternative that is considered more adherent to the specific design, avoiding for successive optimizations of the cables disposition, and going directly to a FEM model and to structural design and calculation of details.The further implementation of the different alternatives, has been implemented with Midas Civil (2016), supported by a computer routine in C++ providing the different parameter variation in the FEM input/output.According to the analysis performed, resume tables has been prepared for the hanger net optimal geometry: Tables 2-3 represent the optimal geometry alternative starting from the choice of the hanger number and the bridge span for the road and the railway alternative.The determination of the optimal inclination parameters for the selected alternative should be made according to Figure 22-23, while for CSM alternative, indications are given in Table 4.
The study will be in the near future enlarged, including: -the alternative with four lane, which is more interesting especially for increasing span length and highway solution; -the analysis of theoretical span-limit of CFST network arches, which appear to be the frontier of medium span bridges, for their easy construction and robust structural solution; -the convenience of multi-span net-arch bridges compared to traditional long-span bridge types (e.g., suspension or cable stay).

Figure 1 .
Figure 1.Hanger layout in tied arch bridges a) Steel weight in two-track railway arches with concrete deck bridges between 60m and 150m span lenght (Shanack, 2003); b) Steel weight of railway bridges with ballast

Figure 4 .
Figure 4. Traditional arch bridge with inclined strands with an asymmetric loading condition.

Figure 5 .
Figure 5. Arch bridge with a double net of inclined strands.

Figure 7 .
Figure 7. Possible form of buckling in the plane of a network arch.

Figure 8 .
Figure 8. Strait compressive member with elastic support.

Figure 9 .
Figure 9. Deflection and shear forces due to concentrated load in network arches with two(a) and three sets of hangers (b)

Figure 11 .
Figure 11.Network arch models considered for the optimization of network arches: CSM, constant slope model, the angle is constant (a); V-CSM, variant-constant slope model, the angle varies with a costant the (b); CARSM, constant angle to radius slope model arch portion: -minimization of the compression N in the arch, analysing both the lower part of the arch between the portal transverse and the road (A), and the centre of the arch between the two portal transverse (B); -minimization of the bending moments M in the arch, analysing both the lower part of the arch between the portal transverse and the road (A), and the centre of the arch between the two portal transverse (B); -minimization of the stress level in the arches.

Figure 13 .
Figure 13.Identification of the arch portions: A, up to the portal transverse; B, central part.

Figure 14 .
Figure 14.Maximum N-compression force in the arch as a function of the hanger angle inclination (CSM configuration), railway bridge alternative -Full bridge loaded Arch-B -Full bridge loaded Arch-A -Half bridge loaded Arch-B -Half bridge loaded

Figure 16 .
Figure 16.Maximum N-compression force in the arch as a function of the hanger angle inclination (CSM configuration), roadway bridge alternative

Figure 21 .
Figure 21.Mean values of the stress the arch (all configuration), roadway bridge alternative

Table 1 .
European and American built network arches (in order of L/f ratio).