The Analysis of Rational Component Parameters for Stress Ribbon Through Steel Arch Footbridges

Engineering Structures and Technologies ISSN 2029-882X / eISSN 2029-8838 2018 Volume 10 Issue 2: 72–77 https://doi.org/10.3846/est.2018.6480

THE ANALYSIS OF RATIONAL COMPONENT PARAMETERS FOR STRESS RIBBON THROUGH STEEL ARCH FOOTBRIDGES

Vilius KARIETA*

Vilnius Gediminas Technical University, Saulėtekio al. 11, LT-10233 Vilnius, Lithuania

Received 18 May 2018; accepted 17 October 2018

Abstract. The article discusses the main component parameters and their interdependencies for a composite stress ribbon trough arch structure. The paper presents a methodology for balancing the combined suspension through arch steel bridge structure, suggests analytical methods for putting together the bearing components of the bridge and considers rational component parameters. Keywords: stress ribbon bridge, steel bridge, arch bridge, equilibrium structure, rational component parameters, symmetrical loads, geometrically nonlinear analysis.

Introduction A stress ribbon suspension structure appears as one of the century, to create the new forms and to extend the limits most efficient, cost-effective and graceful types of suspen- of applying stress ribbon bridges, these structures were sion bridge constructions. Due to its properties and geo- started to be combined with other bearing components metric shape, these structures are most commonly used seeking to use them in the structures of combined bridges for building footbridges (Schlaich et al., 2005; Strasky, (Karieta, 2017). The stress ribbon through the steel arch 2005). The main bearing components of the stress ribbon structure is one of such combined structures. This con- bridge include high-strength steel cables, sheets or beams struction is specific because the arrangement of bearing (Juozapaitis, Vainiūnas, & Kaklauskas, 2006; Strasky, 2005; components with opposing internal forces allows creating Schlaich et al., 2005; Sandovič, Juozapaitis, & Kliukas, an equilibrium structure transmitting only vertical bear- 2011). Stress ribbon suspension bridges are very effective ing reactions to the foundations (Strasky, 2005, 2008; Ka- when a strong natural foundation is used for rear sup- rieta, 2010, 2017). However, not much information about ports (Strasky, 2005; Kulbach, 2007). Since the efficiency the behaviour and rational parameters of the composite of stress ribbon suspension bridges increases along with a stress ribbon through steel arch structure is provided. rise in the number of spans, multi-span structures are fre- The paper describes the main components geometric quently employed (Troyano, 2003; Strasky, 2005; Sandovič, parameters and their interdependencies of the combined Juozapaitis, & Gribniak, 2017). Nowadays, in order to stress ribbon through steel arch structure, presents a meth- avoid unwanted vibrations, a variety of mechanical and odology for balancing combined suspension through arch hydraulical dampers and shock absorbers are applied for bridge structure, deals with analytical methods for putting flexible and lightweight stress ribbon bridge structures together the bearing components of the composite bridge (Bleicher, Schauer, Valtin, Raisch, & Schlaich, 2011). To and considers rational component parameters. increase the efficiency of suspension structures, composite materials started to be used for bearing components (Liu, 1. Component parameters for Zwingmann, & Schlaich, 2016). The main disadvantage of the equilibrium structure stress ribbon structures is large horizontal bearing reactions transmitted to the foundations, which often deter- In his book, J. Strasky was one of the first to describe the mines the cost of these structures (Schlaich et al., 2005; idea of arranging the equilibrium stress ribbon supported Kulbach, 2007; Karieta, 2010). At the end of the 20th on arch structure presented in Figure 1. (Strasky, 2005).

*Corresponding author. E-mail: [email protected]

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unre- stricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Engineering Structures and Technologies, 2018, 10(2): 72–77 73

The diagram in Figure 1 shows that the arch can be used of the bridge under permanent loads; α – the angle of the as an intermediate support. At the initial stage, the stress tilt of the bearing strut. ribbon component can be considered as the structure of Geometric parameters presented in Figure 2 can be two spans supported on the saddle in the middle (Fig- divided into groups to distinguish the main variables: ure 1b). The analysis of the arch is carried out by loading ––Group 1. At the beginning of the arrangement, the it with self-weight, the weight of the saddle components initial parameters for the bridge La, fa, Lsum and h and the pressure transmitted by the tendons of the stress should be always known since they define the obsta- ribbon component (Figure 1c). Structural parameters and cle the bridge must overcome. initial strains in the arch and suspension component can ––Group 2. The initial parameters for the suspension be selected such that the horizontal forces of these struc- part βg, i (%), fv describe suspension component sags tures be identical (Ha = Hv). To uniform bearing reactions, usually limited by the needs of people with disabili- struts can be used for combining the abutments of the ties. Based on these parameters, the sag of the sus- stress ribbon component and the arch thus creating an pension component and its intermediate support, i.e. equilibrium structure under vertical bearing reactions arch saddle, are determined. only (Figure 1d). ––Group 3. The main variable Lb. Since the previously For simulating a composite structure, the applica- discussed geometric parameters should be always tion of digital methods is a complex and time-consuming known at the beginning of arrangement, changing task. This section provides a methodology for balancing the value of parameter Lb may assist in simultane- the structure of the equilibrium suspension through arch ously adjusting the span of the suspension bridge and bridge considering each particular case. The basic com- arch loading. The approximation method may help ponent parameters of the stress ribbon through the arch with balancing the bearing reactions of the main sus- structure are provided in Figure 2: La – arch span; fa – pension components. the rising part of the arch; Lb – the loaded part of the The arrangement of the balanced geometry of the arch (length of the stress ribbon saddle); Lv – the length composite bridge takes place under permanent loads of of stress ribbon component; fv – a sag in the stress ribbon bearing components, and therefore defining materials, component; h – the vertical distance between the sup- the cross-sections of bearing components and calculation ports; S , S , S – the length of the geometric length of v b a methods for estimating horizontal bearing reactions is a stress ribbon component, the saddle and the arch; β – the g crucial point at the beginning of the conducted analysis. tangent angle with a horizontal of the stress ribbon section

L 2. The arrangement of the stress a) sum ribbon bearing section

Hv Hv Ha In the general case, the bearing reactions of stress ribbon L Ha a section should be calculated taking into account the installation of the suspension component, the material of the b) bearing component (cables, steel sheets), cross-sectional Hv Lv Lb Lv structure (using steel profiles or reinforced concrete), etc. rT Subject to the type of stress ribbon structure, calculation Tv Tv c) methods are selected. In order to discuss the rational

Ha Ha component parameters for the composite bridge, the pa- La per presents the methodology for calculating the bearing d) reactions of the flexible single span cable, cross-sectional area and displacements under symmetrical loads. For cal- h DR DR culating suspension structures and for determining the DR a Lb a DR cross-sectional area of the bearing component, it is neces- b La b sary to estimate the nonlinear behaviour of the bearing component. Thus, knowing the displacement of the bear- Figure 1. A component scheme for suspension ing component at each stage of construction and assess- through arch bridge ing technology for construction and installation works are b important points to be considered. Lv g Figure 3 presents schemes for calculating the displace- v i(%) Sv i(%) f ment of the stress ribbon component where L – span; a f

h f – the initial sag defining the initial geometry; ∆f – a 0 gnk S a displacement under permanent characteristic loads; ∆fvk – a Lb a displacement under variable characteristic loads; fgk – sag L b a b under permanent characteristic loads; f – sag under Lsum gvk permanent and variable characteristic loads; β – the tan- Figure 2. The main parameters gent angle of sag with a horizontal under permanent loads. 74 V. Karieta. The analysis of rational component parameters for stress ribbon through steel arch footbridges

a) Sag under strain Sag under permanent loads b

Sag under full loads 0 f

H gk f vk g f vk D f

Sag with no technological strain gnk D f

b) Sag under full loads Sag under permanent loads b

Initial simulated sag 0 f

H gk f vk g f vk gnk D f D f

Figure 3. Displacements of symmetrically loaded cable: a) applying technological strain; b) without technological strain

The horizontal bearing reaction of both the suspension loads (f0 = fgk). In this case, at constant loads, the bearing structure and the cross-sectional area are directly subject component will deform and shift at the value of ∆fgnk. In to sag. The bigger is the sag, the weaker is the horizontal order to move back the bearing component to its original reaction. However, for designing footbridges, the sag of the form, it is necessary to restore the initial state of the cross- current part is usually limited by a longitudinal gradient, section internal force. Therefore, the suspension compo- which may reach 8% for stress ribbon suspension bridges nent can be strained by the force equal to the horizontal (Schlaich et al., 2005). This gradient can be expressed in bearing reaction at the initial stage (Ft = Hgk). Next, the degrees, and then angle b = 4.574°. Based on this condi- structure is subjected by variable loads and shifts at the tion, the permissible sag can be expressed through its tan- value of ∆fvk that depends on the selected cross-sectional gent angle with horizontal fLgk =(tg β⋅ ) / 4. Thus, in this area. case, we can always know the value of the sag of the cur- The displacement under characteristic variable loads rent part at permanent loads under variations in its span. is equal to: Depending on the selected technology for installing a L D=f , (1) suspension structure (applying or omitting the strain of vk k the bearing component), the cross-sectional area of the where k – the coefficient of displacement increment bearing component can be calculated thus selecting the (100, 200, 300, …). initial geometric length of the component so that in each The horizontal bearing reaction under characteristic case sag should not exceed the condition of the allowable permanent loads equals to: gradient of the current part and should meet requirements 2 gLk ⋅ for the safety and appropriateness of the limit state. The H gk = . (2) cross-sectional area of the bearing component is selected 8⋅ fgk according to the displacement of the bearing component The horizontal bearing reaction under characteristic under variable loads. The lower are limits on the displace- full loads equals to: ment of the bearing component under variable loads, the 2 ( g+⋅ vLk ) more rational cross-sectional area can be applied to the H = k . (3) gvk ⋅ bearing component. As for the latest European design 8 f gvk standards, footbridges are not limited, which extends ap- The cross-sectional area of the bearing component is plication limits on modern graceful structures. However, equal to: for designing footbridges, pedestrian comfort and techni- (Hgvk−⋅ HS gk) g cal-performance criteria must be taken into consideration. A = , (4)  8 ⋅−⋅22 The cross-sectional area and displacement of the bearing ( ffEgvk gk ) component at various stages of construction can be calcu- 3⋅ L lated according to the following expressions. where E – Young’s modulus of the steel of the bearing When structures or installation technology of the sus- component. pension component allows strain on the bearing compo- For designing a suspension structure with no strain, it nent (Figure 3a), the initial sag of the suspension compo- is necessary to select the starting sag so that, under per- nent can be selected according to its sag under permanent manent loads, the suspension structure should shift at the Engineering Structures and Technologies, 2018, 10(2): 72–77 75

value of ∆fgnk and form the shape satisfying the condition ture, and therefore bending stiffness EI of the arch has an for the permissible gradient of the current section (Fig- effect on its displacement and internal stress. The hori- ure 3b). To meet the condition of the permissible gradient zontal bearing reaction considering the bending stiffness in this particular case, the displacement of the suspen- of the arch can be defined as follows: sion structure must be relatively small. The cross-sectional M(0) 48⋅ EI ⋅D f area of the structures for such construction technology H ≅−ma, (9) ⋅⋅2 is selected according to expressions (1)–(4), though in fa 5 Lfaa this case coefficient k should be sufficiently high (>325). where: Dfa – vertical displacement of the centre of the By applying higher values of coefficientk , a larger cross- arch, La – arch span, EI – bending stiffness of the arch, (0) sectional area of the bearing component is preferred thus M – a bending moment in the arch examining an simultaneously reducing the elastic displacement of the m equivalent beam, fa – the rising part of the arch. suspension component and behavioural non-linearity as One of the main tasks of arch analysis is to determine well as taking easier control over the displacement of the parameters for the arch cross-section in order to precisely suspension component. ascertain self-weight having a significant impact on the The geometrical length of the suspension component horizontal bearing reaction of the arch. To design the arch under permanent characteristic loads is equal to: considering requirements for strength and stability, it is 2 8⋅ f gk required to know its precise design length in the case of SL= + . (5) gk 3⋅ L the various combinations of arch loading. The approxi- The displacement of the suspension component under mate design length of the arch of the combined bridge permanent characteristic loads equals to: can be defined by taking into account arch displacement described by the diagram of the bending moments of the HS⋅ gk gk arch. The analysis of the arch shows that when the stress D=Sgk . (6) EA⋅ ribbon is placed on the arch, the bending moments of the The initial geometrical length of the suspension com- arch are established. The values of the bending moments ponent is equal to: of the arch under permanent loads vary along with the component parameters of the combined bridge, but the S−D S = SL > . (7) gk gk 0 positions of variations in the signs of the diagram of bend- The initial sag of the suspension component under no ing moments remain similar and correspond to Figure 4. strain equals to: Hence, with reference to Figure 4, the approximate design length of the arch leff is a higher value of geometric length 3⋅(S −⋅ LL) f = 0 . (8) S1 and S2. 0 8

It should be emphasized that the cross-sectional area of h S1 S2 S1 the bearing components of single-lane suspension struc- a Lb a tures is selected according to the condition for the appro- La priateness of the limit state. Nevertheless, due to the geo- Figure 4. A diagram of the bending moments of the arch metrically nonlinear behaviour of suspension structures, under permanent loads this cross-sectional area will always meet the limit state of safety. Calculations according to expressions (1)–(8) show that the results are accurate and basically coincide with the 4. Rational component parameters for the stress results of the numerical geometrically nonlinear analysis. ribbon through steel arch equilibrium bridge With reference to the expressions given, it is possible to determine the exact horizontal bearing reactions of the Calculations referring to formulas (1)–(9) and the scheme suspension component taking control over the displace- presented in Figure 2 show that each individual case pro- ment of the suspension component and evaluating the vides a possibility of arranging the equilibrium structure actual weight of the suspension component of the bridge of the combined bridge (when the bearing reactions of in each case. suspension and arched sections of the bridge are equal) (Figure 5). In order to determine the parameters and their inter- 3. The arrangement of the arched section dependencies of the combined steel equilibrium bridge, When the internal forces of the suspension section are behavioural analysis due to permanent loads was per- known, it is possible to prepare a calculation scheme for formed using the following initial data: arch span – the component of the arch and to estimate the horizon- La = 65.0 m; loads of the stress ribbon section of the tal bearing reaction under permanent loads (Figure 1c). bridge: g = 7.5 kN/m, v = 7.5 kN/m, γ = 1.0; stressed rib- The internal force and behaviour of the arch can be de- bon supports are at the same level (Figure 3). The small termined on the basis of the behaviour of the equivalent difference in horizontal bearing reactions is due to the beam having the same geometric length and stiffness. In vertical displacement of the arch, which should be taken this case, the arch is the once-statically unresolved struc- into account in the detailed analysis of the bridge. 76 V. Karieta. The analysis of rational component parameters for stress ribbon through steel arch footbridges

Figure 5. The bearing reactions of the nonlinear FEM analysis of the equilibrium composite structure: stress ribbon section 1770.6 kN; arched section 1752.1 kN

The analysis (using expressions (1)–(9)) of rational pa- The arch is a curve which has a different radius than a rameters for the combined bridge (when the horizontal re- curvature of the smooth part of the pedestrian path (with action of the stress ribbon component is equal to the hori- allowable gradients). The arched part of the saddle must zontal reaction of the arch) resulted in the diagram shown properly support the stress ribbon section of the bridge. in Figure 6. The diagram illustrates variations in the length Increasing the cross-section of the arch leads to the ad- of the intermediate support (saddle) of the stress ribbon ditional amount of steel in the edges of the saddle. For under changes in the rising part and weight of the arch. this reason, the right geometric shape of the arc should be Arch flexibility: chosen in each case (Figure 7). λ=li/ . (10) With reference to Figure 6, the range of the rising eff parts of the rational steel arches of the bridge can be de- The radius of the inertia of the arch cross-section: termined. The presented diagram clearly shows the ef- i= IA/ , (11) ficiency of extremely flat arches. While applying arches with low rising parts (f = (1/16…1/15)L ), the area of where leff – the calculated length of the arch; I – the mo- a a ment of inertia of the arch cross-section; A – the cross- the arched saddle is very small. When the rising part of sectional area of the arch. the arch makes (1/12)La, the length of the saddle is equal to 42–29% of the arch span, which is the approximate 45 fa = La/12 limit from which the geometric shapes of the arched and 40 fa = La/13 suspension sections start to differ significantly and the f = L /14 35 a a length of the part of the suspension section of the bridge fa = La/15 decreases sharply. Thus, it can be stated that the effective 30 f = L /16 a a range of the rising parts of the steel arches for combined 25 [%] a

L footbridge is (1/16 ... 1/12) of the arch span. /

b 20 L 15 Conclusions 10

5 The proper selection of component parameters for the bearing components of stress ribbon through arch bridg- 0 350 375 400 425 450 475 500 525 550 575 600 es shows this structure can be balanced in such a way Self weight of the arch [kg/m] it should transfer only vertical bearing reactions to the Figure 6. The dependence of the length of the loaded part of the foundations affected by loads (equilibrium structure is arch on the self-weight and the rising part of the arch when the obtained). horizontal bearing reaction of the stress ribbon structure is equal The cross-sectional area of stress ribbon bearing com- to the thrust of the arch (H = H ). L – arch span, f – rising part v a a a ponent is selected according to the condition for the ap- of the arch, Lb – the length of the intermediate support (saddle) of stress ribbon component propriateness of the limit state. The cross-sectional area Engineering Structures and Technologies, 2018, 10(2): 72–77 77

Lb = 23.78 m l = 97, m = 4.522 kN/m Parabolic arch a) La/12

l Lb = 18.76 m = 96, m = 4.522 kN/m Parabolic arch b) La/13

Lb = 13.95 m l = 106, m = 4.522 kN/m Hyperbolic arch r = 0.7 c) La/14

Lb = 9.40 m l = 115, m = 4.522 kN/m Hyperbolic arch r = 0.8 d) La/15

Ha = Hv b = 10 m La = 65 m b = 10 m 85 m

Figure 7. Parameters for the combined steel footbridge has been found will always meet requirements for the Karieta, V. (2017). Vienajuosčių kabamųjų paremtų arka pėsčiųjų limit state of safety. tiltų pusiausvirosios sistemos komponavimas. Science – Fu- According to expressions (1)–(8), the calculation re- ture of Lithuania, 9(5), 495-499. sults of the suspension bearing component is sufficiently https://doi.org/10.3846/mla.2017.1084 Karieta, V. (2010). Analysis of the single-lane suspension supported accurate and basically coincide with the numerically ob- on arch pedestrian multispan bridge behaviour and research ra- tained results of the geometrically nonlinear analysis. tional parameters (Master’s thesis). Vilnius Gediminas Tech- Based on the provided expressions, it is possible to deter- nical University, Lithuania. mine the horizontal bearing reactions of the suspension Kulbach, V. (2007). Cable structures. Design and analysis. Tallin: bearing component taking control over the displacement Estonian Academy Publisher. 224 p. of this component in each individual case. Liu, Y., Zwingmann, B., & Schlaich, M. (2016). Advantages of In order to create an equilibrium structure of the com- using CFRP cables in orthogonaly loaded cable structures. posite bridge, the main rational component parameters for AIMS Materials Science, 3(3), 862-880. the bridge arch have been determined. The rational range https://doi.org/10.3934/matersci.2016.3.862 Sandovič, G., Juozapaitis, A., & Gribniak, V. (2017). Experimen- of the rising parts of the arched section varies from 1/16 tal and analytical investigation of deformations and stress to 1/12 of the arch span. distribution in steel bands of a two-span stress-ribbon pe- Stress ribbon structures of bridges are rational in terms destrian bridge. Mathematical Problems in Engineering, 2017. of the cost of materials, and therefore it is important to Article ID 9324520. https://doi.org/10.1155/2017/9324520 consider the advantages of the suspension section in the Sandovič, G., Juozapaitis, A., & Kliukas, R. (2011). Simplified structures of combined bridges. However, a rise in the engineering method of suspension two-span pedestrian steel length of the suspension section simultaneously increases bridges with flexible and rigid cables under action of asym- axial forces in both bearing components. Hence, for cre- metrical loads. The Baltic Journal of Road and Bridge Engi- ating the analyzed structure, it is important to exclusively neering, 6(4), 267-273. https://doi.org/10.3846/bjrbe.2011.34 Schlaich, M., Brownlie, K., Conzett, J., Sobrino, J., Strasky, J., & focus on selecting the geometric shape of the arch and, Takenouchi, K. (2005). Guidelines for the design of footbridges. certainly, on its stability analysis. Stuttgart: Sprint-Digital-Druck. 154 p. Strasky, J. (2005). Stress ribbon and cable-supported pedestrian References bridges. London: Thomas Telford Ltd. 232 p. https://doi.org/10.1680/sracspb.32828 Bleicher, A., Schauer, T., Valtin, M., Raisch, M., & Schlaich, M. Strasky, J. (2008, July 10-14). Stress – ribbon pedestrian bridges (2011, August). Active vibration control of a light and flex- supported or suspended on arches. In Chinese-Croatian Joint ible stress ribbon footbridge using pneumatic muscles. In Pre- Colloquium “Long arch bridges”, (pp. 135-148). Brijuni Islands. prints of the 18th IFAC World Congress, (pp. 911-916). Milano, Troyano, L. F. (2003). Bridge engineering: a global perspective. Italy. https://doi.org/10.3182/20110828-6-IT-1002.02781 London: Tomas Telford Ltd. 775 p. Juozapaitis, A., Vainiūnas, P., & Kaklauskas, G. (2006). A new https://doi.org/10.1680/beagp.32156 steel structural system of a suspension pedestrian bridge. Journal of Constructional Steel Research, 62(12), 1257-1263. https://doi.org/10.1016/j.jcsr.2006.04.023