Asymmetric network arch bridges

Transcription

1 Asymmetric network arch bridges Structures and Architecture Cruz (Ed.) 2010 Taylor & Francis Group, London, ISBN B. Zwingmann & S. Marx Institute of Concrete Structures, Dresden University of Technology, Georg-Bähr-Straße 1, Dresden, Germany F. Schanack Institute of Civil Structures, Universidad Austral de Chile, General Lagos 2086, Valdivia, Chile ABSTRACT: Asymmetric network arch bridges offer new possibilities to design engineers. At the same time, they ensure both economic and static efficiency. In contrast to tied arch bridges with vertical hangers, in network arches without hanger relaxation, the line of thrust lies within the arch for all load cases. Therefore, the arch shape does not have to be adapted to a special load case and may be asymmetric. In order to design an asymmetric network arch, a hanger arrangement that prevents hanger relaxation has to be found. Internal forces and technical feasibility are basically the same as in symmetric network arches. Possible applications are asymmetric bridge locations, skew bridges or bridges with variable width. 1 MOTIVATION An adequate and high-quality visual appearance of bridges is an essential task of bridge engineers, just like the assurance of safety, functionality and economic efficiency. In the recent past, this aspect has been expressed more and more often by the public opinion. For example, in 2005 the German Federal Foundation for Architecture and Culture of Construction was installed, which awards the German Bridge Prize every second year. Also, in 2008, the German Railway Authority published the Guidelines for Railway Bridge Design (Schlaich et al. 2008). In the latter publication, not only is the aesthetical appearance of bridges addressed, but it is also stressed that there is a permanent need for innovation in bridge design. In the present paper, this last topic is treated, and new possibilities for the aesthetical design of arch bridges, which arise from the development of asymmetric arch shapes, are shown. 2 INTRODUCTION Arch structures principally support loads by normal forces. That is why the arch cross section is used in an optimal way, and an outstanding efficiency is obtained. The requirement for this is that the arch geometry corresponds exactly to the line of thrust. This is the case, for example, in a circular arch under constant radial load or in a parabolic arch with constant vertical load. However, in real bridges, these load cases do not exist or do not exist exclusively. There are several reasons for it, for example the non-uniform distribution of the self-weight, the discontinuities of the load models of EC1 (CEN 2005) or the non continuous load introduction into the arch when the deck is suspended from hangers or supported by posts. However, the decisive reason for the deviation of the line of thrust from the arch centre line is the variable distribution of the live load. Therefore, the demands for a load bearing behaviour with exclusive normal force are not met, and it is unavoidable that there are bending moments in arch bridges. However, the actual value of these bending moments is very different from one arch type to another. 1211

2 As each load case and each load position cause a different line of thrust, the arch geometry may only be chosen in such a way that it corresponds to only one line of thrust. Since the beginnings of the construction of arches, according to the line of thrust, the same load case has always been chosen: self-weight and dead load. This load case is in almost every case symmetric with respect to the arch centre. That is why practically all arch bridges constructed so far are symmetric to their centre line. A symmetric and very accentuated structure, like an arch, may be aesthetically unfavourable, for example in a location with an asymmetric landscape. This might be due to different elevations, a city limit or simply an asymmetric valley. In these and similar cases, using an asymmetric arch structure may improve aesthetics (WSA 2007). In conventional arch bridges, such a design would cause large bending moments under self-weight and dead load, which would be unacceptable from an economic efficiency point of view. However, if a network arch bridge is used, then it is completely possible to design asymmetric arch structures without a serious increasing of bending moments, as explained in the following. 3 THE STRUCTURAL CONCEPT First, it is necessary to recall the structural behaviour of conventional network arches. Network arch bridges are tied arch bridges with an arch above the deck. The deck acts as a tie. Consequently, no arch thrust needs to be supported by the bearings. From the statics point of view, it is a simply supported beam with a variable height and a very light web. The normal forces in the arch and in the deck are obtained from the bending moment of the simply supported beam divided by the vertical distance. Tied arch bridges may be classified according to their hanger arrangement into bridges with either vertical or inclined hangers. Network arch bridges, whose hangers have multiple intersections, are found in the second group. While in tied arch bridges with vertical hangers the loads are only suspended by the arch, the network arch bridges have hanger nets which cause further stiffening between the arch and the deck. The load bearing behaviour is similar to a truss. The hanger net aligns the line of thrust almost perfectly to the arch centre line. Therefore, the bending moments are significantly smaller than in similar tied arch bridges with vertical hangers. This is especially evident for asymmetric loads (Schanack & Brunn 2009). The difference in the load bearing behaviour of these two bridge types is shown in Figure 1. Figure 1. Comparison of network arch and tied arch with vertical hangers and partial loading. The tension and compression forces in the diagonals of the hanger net due to asymmetric live load must be taken by the hangers. Usually, the hangers are cables or steel rods that cannot take compression. They can only participate in the structural behaviour if the existent tension force from the permanent load is bigger than the compression caused by live load. If this is not the case, the hanger will buckle and the bending moments in the arch will rise significantly. This hanger relaxation must be avoided, at least under service loads (Geißler et al. 2008). The tendency for hanger relaxation to occur depends primarily on the hanger inclination or, if the radial hanger arrangement is used, on the hanger-arch angle (Brunn & Schanack 2003). In the design of network arches, the parameter hanger-arch angle needs special attention. Usually, 1212

3 it is chosen in order to avoid any hanger relaxation under any load case or load position. When the hanger-arch angle is chosen like this, the line of thrust in network arches is practically the same as the arch centre line for all load cases. Under these conditions, the arch shape, i.e. the designed line of thrust, may be of any shape because it no longer needs to correspond to any special load case. Hence, the arch geometry may be chosen freely. This means that it may also be asymmetric. The only restriction is that according to such a case, a hanger arrangement must be found to avoid hanger relaxation in any load case. After these theoretical considerations, it must now be proven that it is really possible to design such hanger arrangements for asymmetric network arch bridges. 4 NUMERICAL STUDY OF ASYMMETRIC NETWORK ARCH BRIDGES In order to prove the idea derived in Section 3, extensive parametric studies and bridge assessments of asymmetric network arch bridges have been carried out (Zwingmann 2009). In the following section, the general procedure and the main results of this work are presented. An asymmetric network arch bridge has one or more tied arches that are not symmetric to their centre plane. The centre plane is perpendicular to the line connecting both arch ends, at half the distance. The hanger net lies between the arch and the deck. The asymmetric arch shape may be described mathematically. See Figure 2 for an example using a two-centre arch. It consists of two circular arch sections with different radii, which both have a horizontal tangent at their connection point. If the ratio of arch rise and the arch span is given, then this geometry is completely defined by the parameter is the distance between the connection point of the two arch sections and one arch end, divided by the total span (Figure = 0.5, the geometry of a symmetric arch is obtained. Alternatively, for the mathematical description of the asymmetric arch geometry, a quadratic function with an eccentric maximum, a polynomial function of higher order or more than two segments of circular arches may also be used. The curve must be convex and its first derivation must be continuous. Figure 2. Parameters of the two-centre arch. A 100 m long two-lane road bridge with a concrete deck is chosen for the studies. The arch rise is 15% of the bridge length. The bridge is analysed first in a Two-Dimensional Beam Elements Model (Figure 3) and loaded by the vertical loads in Ultimate Limit State, which is defined in the EC 1 (CEN is varied from 0.3 to 0.5. Figure 3. Two-Dimensional FE Model. Linear calculations are performed by analysis of the influence lines of all beam elements. The software package SOFiSTiK is used. In order to find an adequate hanger arrangement, the angle between the arch and the hangers is varied according to various methods until there is no longer hanger relaxation for any cases studied. The methods of angle configuration include constant 1213

4 hanger-arch angles in different arch sections and a linear change of the angle along the arch. It is found that despite different the same hanger arrangement are suitable. In Figure 4, a hanger net geometry is shown, which has no hanger relaxation in the Ultimate Limit State ger arrangement suggested in (Zwingmann 2009) for the design of asymmetric network arch bridges. The hanger-arch angles are either constant or change linearly in different sections along the arch. The values for the beginning and the end of each arch segment are given. By means of the number of hangers in each segment, the angle increment from one hanger to another may be calculated. These requirements are less strict if the parameter is close to 0.5, i.e. the arch is almost symmetric. It might be changed gradually to a radial h, hanger relaxation does not occur in the Serviceability Limit State smaller than 0.3 are not recommended because the arch curvature of the larger section is reduced significantly. There, the arch acts more like a curved beam, and bending moments increase considerably. Figure 4. Example of a hanger arrangement of an asymmetric arch. In the section of a high arch curvature, the arch has a higher stiffness for loads in its plane. Hangers connected in such a section have a stiffer support than other hangers and participate to a relatively larger extent in the load bearing behaviour. Consequently, larger maximum hanger forces are obtained. Therefore, it is recommended to decrease the hanger distances in this area. Then, the maximum tension is similar in all hangers, which allows a more efficient design if hanger diameters are constant. The hanger distances should be increased and decreased gradually, in order to obtain a harmonic appearance of the hanger arrangement. Based on static analysis of the structural behaviour of asymmetric network arch bridges, it may be stated that the general properties of network arches described in Section 3 are also found in asymmetric network arches. First of all, there are almost exclusive normal forces in all structural parts and only small local bending moments. However, the design of a suitable hanger arrangement, without any hanger relaxation, is more complicated. The asymmetric network arch, with a good net geometry, has very small structural disadvantages compared to the symmetric network arch. In Table 1, the biggest internal forces obtained from the analysis of the influence lines are shown. Four different designs are compared: a network arch bridge and a tied arch bridge with vertical hanger, both in symmetric and asymmetric alternatives. It may be seen that the difference between the symmetric and the asymmetric alternative is much smaller in the case of a network arch than in a case of a tied arch with vertical hangers. This is especially true for the bending moments in the arch and in the deck. Table 1. Comparison of internal forces. Internal force Network arch bridge Tied arch bridge with vertical hangers symmetric asymmetric symmetric asymmetric N arch * N hanger* M arch * M deck* *Maximum values after analysis of influence lines. On the basis of these results, it may be concluded that the arch shape of a network arch bridge does not need to be symmetric. The design engineering is not obliged to use symmetric circular or parabolic arches. This creates certain design freedom to choose the morphology of the arch shape. New possibilities arise that are presented in the following section. 1214

5 5 DESIGN AND APPLICATION POSSIBILITIES Asymmetric network arch bridges may at least be used in all the cases where symmetric network arch bridges or tied arch bridges with vertical hangers are usually suited. From the aesthetics point of view, they are suited, for example, in valleys with asymmetric profiles (Figure 5). Furthermore, they might be appealing with an inclined gradient (Figure 6) or as final spans of a sequence of several arches. In Figure 7, some possible arrangements of symmetric and asymmetric arch sequences are shown. In an asymmetric valley the bridge fits well into the landscape without a forced axis of symmetry. In the case of an inclined gradient, the bridge seems to fight against the hillside. Figure 5. Asymmetric valley with a network arch bridge. Figure 6. Network arch bridges with inclined gradient. Figure 7. Sequence of several network arches. In Figure 8, one of the three designs carried out in Zwingmann (2009) is shown. It is a = 0.4) with two arches, asymmetric in the same direction. It is 100 m long, has a width of 14.1 m and an arch rise of 15 m. The deck is a concrete slab with longitudinal pre-stressing and a variable thickness of 30 cm to 50 cm. The arch cross section is a welded and inverted U profile with dimensions of 600 mm 600 mm. The hangers are steel rods with a diameter of 60 mm. 1215

6 Figure 8. Visualization of an asymmetric network arch bridge. This bridge, and the other two, are analysed in a Three-Dimensional Finite Element Model (Figure 9). Calculation concentrates on the analysis of the influence lines of the beam elements. Subsequently the main structural parts are assessed and drawings are elaborated (Figures 10, 11). In the Figures 10, 11, the extraordinary slenderness that is possible with network arch bridges may be observed. This aesthetic advantage is obtained for symmetric as well as asymmetric arches. Figure 9. Three-Dimensional Finite Element Model. Figure 10. Lateral view of an example design. Figure 11. Plan view and longitudinal section of an example design. As mentioned in Section 3, the normal force in the arch depends on the moment distribution of an equivalent simply supported beam. Consequently, it suggests itself to look for applications of asymmetric arches, where this property could be a special advantage. This is the case in bridges, which have larger permanent loads on one side of the bridge than on the other. Two examples for such a case are shown in Figure 12. The first is a skew bridge with two arches, asymmetric in different directions. The second is a bridge with increasing width and two arches, 1216

7 asymmetric in the same direction. In skew bridges, the load on the arch in the acute corner is reduced due the smaller load influence area. When the bridge width is variable, obviously the load is smaller at the narrow end than on the broad one. The asymmetric network arch bridge with increasing width is shown in Figure 13 as a photorealistic visualisation. Figure 12. Skew bridge and bridge with increasing width. Figure 13. Visualization of an asymmetric network arch bridge with increasing width. However, the analysis of the internal force in these cases shows that there are only very small structural advantages of the asymmetric arch compared to a symmetric one. These advantages are primarily a more even normal force distribution, or smaller deformations. 6 SUMMARY AND EVALUATION All internal forces of asymmetric network arch bridges are similar to those of symmetric network arch bridges. The main difference is found in the arch compression forces that are a bit larger in the less curved arch section, but a bit smaller in the other one. Consequently, asymmetric network arch bridges are feasible solutions from the technical and economic point of view. Decisive structural advantages or disadvantages as compared to symmetric network arch bridges cannot be found in the cases studied. The design and connection details remain almost unchanged. The additional costs from the fabrication of the arches are negligible, because arch profiles are fabricated in sections. A fabrication of different sections with different radii is possible without any difficulty. Some hangers are longer than in a symmetric hanger arrangement. Costs for such a design should be similar to that of other network arch bridges. Asymmetric network arches offer new design possibilities in modern bridge construction. The design engineer obtains more creative freedom. They are interesting due to their unusual appearance. This is especially obvious in the Figures 14, 15. These figures show a skew network arch bridge with two arches, asymmetric in different directions. Asymmetric network arches are suitable to demonstrate the advantageous structural behaviour of the hanger net, because the arch has only small local bending moments in spite of the asymmetry. Just like symmetric 1217

8 network arch bridges, they stand for an efficient and slender structure, where normal forces are predominant. Figure 14. Visualization of a skew and asymmetric network arch bridge. Figure 15. Visualization of a skew and asymmetric network arch bridge. 7 ACKNOWLEDMENTS This interesting research was inspired by Prof. Dr. Ing. Javier Torres Ruiz from the University of Cantabria, Santander, Spain. It was his unconventional view of structures that allowed him to predict the described potential of the network arch. 8 REFERENCES Brunn, B. & Schanack, F Calculation of a Double Track Railway Network Arch Bridge applying the European Standards. Diploma thesis. Dresden: TU-Dresden. Comité Europeén de Normalisation (CEN) Eurocode 1 Actions on structures Part 2: Traffic loads on bridges. Geißler, K. & Steimann, U. & Graße, W Netzwerkbogenbrücken Entwurf, Bemessung, Ausführung. Stahlbau 77: Berlin: Ernst & Sohn. Schanack, F. & Brunn, B Analysis of the structural performance of network arches. The Indian Concrete Journal 38(1). Schlaich, J. & Fackler, T. & Weißbach, M. & Schmitt, V. & Ommert, C. & Marx, S. & Krontal, L Leitfaden Gestalten von Eisenbahnbrücken. Peine: Fischer Druck Gmbh. Wasser- und Schifffahrtsamt Rheine (WSA) Neubau von vier Straßenbrücken Bereich DEK Stadtstrecke Münster Ergebnis der Variantenuntersuchung. Anlage 2 zur Vorlage V/0554/2007. Zwingmann, B Unkonventionelle Gestaltungsmöglichkeiten mit Netzwerkbögen. Student Research Project, Dresden: TU-Dresden. 1218