Introduction
This paper discusses aspects of designing of «network arch» type tied-arch bridge superstructures with the polygonal top chord and flexible inclined hangers. The P. Tveit and B. Brunn – F. Schanack theories are explained. In addition, the authors present results of their own research on synthesis and classification of foreign practices and recommendations as to design of such systems.
Current trends in the engineering of artificial structures are largely driven by the increasingly stringent requirements for their reliability, carrying capacity, durability, cost effectiveness and aesthetic qualities [3, 5]. Concurrently, designed structures must be sufficiently adaptable to streamlined practical implementation at construction, operation and necessary maintenance. The above listed requirements condition need for the use and development of new innovative solutions both at design and construction of artificial structures. The first arch bridge with multiple crossing hangers was designed by Per Tveit and was built in 1963 in Steinkjer in Norway spanning 80 m, see Figure 1 [10, 11].
Fig.
1. Network arch at Steinkjer (first)
and Bolstadstraumen
(second)
In that same year, another two network arches were constructed. The Bolstadstraumen bridge spanning 84 m was also designed by Per Tveit and built in Norway (Fig. 1). Also the Fehmarnsund bridge in Germany spanning 248 m (Fig. 2). The Fehmarnsund bridge is clearly a class bigger than the two Norwegian bridges, not only in span, but also in load carrying capacity. This bridge accommodates two road lanes and a single railway track.
Fig. 2. Fehmarnsund bridge
Purpose
A network arch bridge is a tied arch bridge with inclined hangers intersecting at least twice [3, 5, 10]. Compared with regular tied arch bridges, i.e. those with vertical hangers, the network arch bridge exhibits low moments in both of the chords, which typically leads to important material savings. In Figure 3 it can be seen that the network arch tends to behave like a simple beam, due to its higher stiffness, leading to small deflections. As Figures show, partial loading on half of the span will lead to deflections on the upper and lower chord in the arch with vertical hangers while the arch with inclined hangers only observes deflections on the lower chord.
Fig.
3. Tied arch with one set of inclined hangers
submitted to
partial loading
As a consequence, in the arch with vertical hangers, bending is a decisive factor when it comes to the choice of the cross-section of the chords. In the network arch, bending will only occur due to local loading, and therefore the arch and the tie are only subjected to axial forces.
At the practical engineering of such type structures, engineer faces the necessity of solution of the following problems:
– Definition of optimum design parameters of the arch, in particular, outline shape, rise height, construction;
– Choice of hanger net construction scheme;
– Determination of reasonable hanger inclination angle;
– Calculation of the most rational number of hangers;
– Analysis of solution cost-effectiveness as compared to superstructures with vertical hangers;
This paper presents selection of information both from foreign practice of this type bridge designing and construction, and from the author’s own researches. It outlines and formulates general principles of designing the superstructures with the inclined hangers and provides general recommendations for the practical designing of such superstructures.
Methodology
An analysis was made for a road bridge with 100 meters span consisting of two circular hollow steel arches with a radius of 82 meters and a maximum height of 17 meters, connected at ends by circular hollow section tie-beams. The arches are inclined inward 15 degrees after the tie-beam axis. Arches are connected at the bottom by means of variable height double T section crossbeams positioned at equal distance of 5 meters and at the top are connected by means of circular hollow sections bracings. A reinforced concrete top slab linked by elastic connectors to the crossbeams completes the composite deck. The virtual prototype of the model is performed by finite element simulator, Midas Civil.
Findings
Vertical Hanger System (Langer System)
Fig.
4. Tied arch bridge with Langer configuration
of hangers
In this configuration the compression forces in the arch increases with the number of hangers as shown in Figure 6. It was observed that with increasing number of hangers, compression increases in the arches, while the hanger’s axial efforts decrease as in Figure 5.
Bending moment decreases with the increasing number of hangers, and this difference is remarkable when the number of hangers is lower and the bending moments in the arch grow rapidly as shown in Figure 7.
The tie beam axial efforts variations do not appear in the system with vertical hangers, but the hanger number variation significantly influences the bending moment in the beam because the hangers play the role of elastic supports for tie beam as in Figure 8.
In this configuration the bending moment dictate the arch sections and the best results for the 100 meters span studied was found for the 20 hanger configuration.
As a consequence, in the arch with vertical hangers, bending is a decisive factor when it comes to the choice of the cross-section of the chords.
Fig. 5. Variation of axial force in hangers in vertical system depending on the hanger number
Fig.
6. Variation of axial force in arch in vertical
system depending
on the hanger number
Fig. 7. Bending moment variation in arch in vertical system depending on the hanger number
Fig. 8. Bending moment variation in tie beam in vertical system depending on the hanger number
Inclined
Hanger System with Constant Slope
(P.
Tveit theory, Nielsen System)
P. Tveit theory, this method follows the same concept as the previous one [9, 10]. In this case, however, the slope of each hanger varies following, for instance, a linear function like Δφ= α.х+β, where х is the number of the hanger, α and β are the parameters that make the hanger arrangement vary along the length of the arch [4, 5, 10]. A general case is illustrated in Figure 9 Assigning a constant slope is a particular case of this configuration, when α is taken as 0.
Fig. 9. Tveit net construction scheme
The construction of reverse hangers is carried in inversed manner. Thus the hanger inclination angle variation is considered positive from the right to the left; in the shown scheme, hanger inclination angle becomes steeper from the left to the right [7].
To simplify the manufacturing process and for a uniform distribution of the moment, and to reduce the buckling length in many cases the hangers are disposed at equal distances along the arc [8]. In this case, the unknowns are the locations of nodes on the tie beam. An alternative is to arrange the hangers at equal distances along the tie beam and the arc node locations are the unknowns.
In this system, the hangers are disposed at equal distances along the arches. Angle with the horizontal plane was set 40 degrees.
As shown in Figure 11, relaxed hangers number is relatively high in this arrangement. As with the horizontal angle is greater, the higher the number of the relaxed hangers. In each case analyzed the hangers at the ends were always relaxed.
In Figure 14 we can see that arch compression tends to decrease with the increasing angle to the horizontal plane. This is explained by the fact that more inclined hangers are less tensioned, due to the small horizontal component of the force. The range most effective for this opening is between 60 to 80 degrees. Bending moments in arches shown in Figure 12. Indicate that the suspensions above 75 degrees inclinations involve large bending moments in arches.
In Figure 15 we can see that tie beam axial force tend to increase with the increasing angle while bending moment shown in Figure 13 is influenced only by the angles over 70 degrees .
As a conclusion to this configuration, the lighter the bridge, more inclined hangers are necessary and more hangers are relaxed. Still, this configuration determines sections that lead to about 40% smaller material consumption than in the vertical arrangement of hangers.
Fig.
10. Variation of axial force in hangers
in NIELSEN system
depending on the angle
Fig.
11. Number of relaxed hangers in NIELSEN
system depending on the
angle
Fig.
12. Bending moment variation in arch
in NIELSEN system depending
on the angle
Fig. 13. Bending moment variation in tie beam in NIELSEN system depending on the angle
Fig.
14. Axial force variation in arch in NIELSEN
system
depending on the angle
Fig. 15. Axial force variation in tie beam in NIELSEN system depending on the angle
Inclined
Hanger System with Variable Slope
(B.
Brunn and F. Schanack theory, Nielsen System)
B. Brunn and F. Schanack theory is based on the arch thrust line and implies the constant value of hanger inclination angle [1, 2, 5]. This model shows that if the forces in each hanger were approximately equal, the «resulting force» would lie on the radii of the arch circle Figure16.
Fig. 16. B. Brunn and F. Schanack net construction scheme
Hence, and in order to simulate a similar structural behavior in the network arch configuration, Brunn and Schanack have defined that the first intersection between hangers below the arch should aim the radii of the arch circle [1, 2]. This way, the only variable involved is the angle between hangers when they cross each other, as illustrated in Figure 17. Here, the hangers are placed with equal space along the upper chord. Throughout this investigation, the angle marked in grey will be the key variable.
Fig. 17. The variable involved
In this system, each set of hangers starting at angle start and then increase or decrease along the bridge. In this study it was considered a first angle of 55 degrees and a variation of 0.5 degrees/hanger.
Fig. 22 shows that maximum axial force in the arch tends to be smaller as the inclination is greater. Bending moment results from the analysis show that the more inclined hangers, the smaller bending moment as in Figure 20.
The hanger angle variation does not appear to significantly influence the tie beam axial force Figure 23. Bending moments along the beam decreases with increasing angle in Figure 21.
Fig. 18. Axial force variation in hangers in inclined hanger system with variable slope depending on the angle variation
Fig.
19. Number of relaxed hangers in inclined hanger system with
variable slope depending on the angle
variation
Unlike hanger system with constant inclination, for this span were obtained unfavorable results, which in turn lead to larger sections, namely higher costs. However, comparing to a system with vertical hangers, in this configuration we get a 30% lighter structure and relaxation of the hangers remains the problem.
Fig. 20. Bending moment variation in arch in inclined hanger system with variable slope depending on the angle variation
Fig.
21. Bending moment variation in tie beam in
inclined
hanger system with variable slope depending on the angle variation
Fig.
22. Axial force variation in arch in inclined hanger system with
variable slope depending on the angle
variation
Fig. 23. Axial force variation in tie beam in inclined hanger system with variable slope depending on the angle variation
Originality and practical value
The research conducted by author shows that both nets could be recommended for the practical implementation, and difference between the values of strength criteria of superstructure elements makes about 5%. It may be said that the Brunn-Schanack net provides somewhat less stress value over the top chord (since it generally provides steeper hanger inclination angles), while the Tveit net allows reduction of maximum stresses in hangers and bottom chord of superstructure.
Conclusions
Hanger inclination angles
Following key parameter at the superstructure design is value of hanger inclination angles with respect to the top chord. Considering fatigue, a crossing angle slightly bigger than 45° will give best results, whereas small maximum internal forces occur for angles of about 55°.Results of the research conducted by author show that the best recommended range of values of hanger inclination angle with respect to the top chord makes 55-70 degrees.
Too small angles (<45–50°) result in increased forces in hangers, in addition, at the extreme shallow inclination angles, net is overly condensed to the middle of superstructure, which in turn impairs its static performance.
When angle values tend to 90°, the system shows increase in strain-stress state indices of superstructure elements (due to the system tendency to work as the configuration with vertical hangers).
Number of hangers
Important
criterion of the inclined ties system performance evaluation is
determination of number of hangers, panel pitch over the top chord.
Figure 25 shows the curve of effective
superstructure element stress coefficient vs. number of hangers.
Fig.
24. Curve of superstructure strain-stress state
indices vs.
number of hangers
Relationship between the number of hangers and superstructure strain-stress state is not linear, but rather hyperbolic with expressed asymptote. Further reduction of number of hangers results in increase in the rate of change of superstructure strain-stress state criteria. The hangers and hanger connections 100 m network arch should be equipped with about 48 hangers per arch plane, which has economical and structural reasons. The extra costs caused by additional hangers and their connections have to be balanced against the material costs that can be saved due to smaller internal forces in arches and tie.
Generally, tied-arch bridge superstructures with the inclined hangers are very efficient systems, and under most circumstances, these can successfully compete with other superstructure configurations even in case of relatively long (400–600 m) span length.