Theoretical and experimental analysis of the quasi-static and dynamic behaviour of the world's longest suspension footbridge in 2020

Engineering Structures 253 (2022) 113830 Available online 7 January 2022 0141-0296/© 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Theoretical and experimental analysis of the quasi-static and dynamic behaviour of the world’s longest suspension footbridge in 2020 Ant´ onio Tadeu a , c , * , Ant´ onio Romero b , Filipe Bandeira a , Filipe Pedro a , Sara Dias a , Miguel Serra a , Michael Brett a , Pedro Galvín b , d a Itecons – Institute of Research and Technological Development in Construction, Energy, Environment and Sustainability, Rua Pedro Hispano, 3030-289 Coimbra, Portugal b Escuela T´ ecnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, ES-41092 Sevilla, Spain c ADAI-LAETA, University of Coimbra, Department of Civil Engineering, Faculty of Sciences and Technology, P´ olo II, Rua Luís Reis Santos, 3030-788 Coimbra, Portugal d ENGREEN, Laboratory of Engineering for Energy and Environmental Sustainability, Universidad de Sevilla, Camino de los Descubrimientos s/n, ES-41092 Sevilla, Spain ARTICLE INFO Keywords: Longest suspension footbridge in 2020 Theoretical and experimental analysis Dynamic behaviour Nonlinear analysis Modal identification Damping coefficients ABSTRACT This work validates the simplified theoretical, analytical and numerical models used in the preliminary design stage of the 516 Arouca footbridge over the River Paiva (Portugal), the world’s longest suspension footbridge. The models were used to define the configuration of the bridge under static loading and the eigenfrequencies excited under dynamic loading. The three-dimensional finite element model used in the detailed design of the bridge is briefly described. The paper also presents in situ experimental results. Tests were performed to study the static and dynamic behaviour of the footbridge under service loads and to assess the analytical/numerical modelling assumptions. The structure was subjected to loads generated by the wind and by a group of people crossing the bridge. Global Navigation Satellite System (GNSS) antennas were used to record the displacements of the bridge under quasistatic loadings caused by the people crossing the bridge at a slow pace. The data recorded by a set of seismometers allowed us to identify the natural frequencies and modes of vibration. The agreement between all the analytical/numerical solutions and the experimental data was found to be very good. The data recorded also allowed one to evaluate the damping coefficients of the bridge for the different vibration modes, something that is very difficult to predict in the design stage. 1. Introduction Footbridges are being designed with increasingly innovative features and different structural forms. A structural type that attracts the eye is the suspension footbridge, mainly due to its length, height, and location. In addition to serving as the link between two points, this type of bridge usually provides an exciting experience and astonishing views for pedestrians as they use it and often becomes a tourist attraction. The 516 Arouca footbridge (Portugal) is located in the town that bears its name in the area of the Arouca Geopark, recognized as a geological heritage of international relevance, classified as a UNESCO Global Geopark. Currently, the 516 Arouca bridge is the longest suspension footbridge in the world with a span of 516 m. The bridge is in a mountainous area over the Paiva river, at a height of 175 m. In November 2020, a pilot study was conducted to evaluate the static and dynamic behaviour of the structure under pedestrian crossing loads, and some of the findings have been used in this paper. One of the difficulties inherent in the design of pedestrian suspension bridges is the nonlinear behaviour of the cables. Different models have been proposed to take into account the nonlinear behaviour of the main cables and deck hangers [1–7]. In addition to the nonlinear behaviour, there are other difficulties in design related to the dynamic behaviour of the structure. Maraveas et al. [8] reviewed different dynamic responses of footbridges under pedestrian loads with the focus on different load models in the time domain. At the design stage, uncertainty about the predicted dynamic properties of the footbridge is inevitable [9]. Nimmen et al. [9] showed that even with the development of a detailed finite element model, deviations up * Corresponding author at: Itecons – Institute of Research and Technological Development in Construction, Energy, Environment and Sustainability, Rua Pedro Hispano, 3030-289 Coimbra, Portugal. E-mail address: [email protected] (A. Tadeu). Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct https://doi.org/10.1016/j.engstruct.2021.113830 Received 31 August 2021; Received in revised form 16 November 2021; Accepted 29 December 2021 Engineering Structures 253 (2022) 113830 2 to 10% in terms of natural frequencies can be expected. Dynamic effects on pedestrian bridges are increasingly important given the tendency to build lighter and slender structures. In addition to the action of the wind, which can lead to scenarios of resonance with vortex-induced vibrations or wind flutter [10], people crossing is another factor that can induce dynamic effects in these structures [11,12]. Pedestrians crossing the bridge produce a force with a characteristic frequency between 1.5 and 2.5 Hz for walking and jogging, and amplitude-dependent on the density of the pedestrian group [13]. Two types of synchronization mechanisms have been suggested, namely, human-to-human synchronization and human-to-structure synchronization or pedestrian lock-in. The lock-in effect is an undesirable phenomenon produced by synchronizing people’s gait with the movement of the structure, leading to excessive lateral acceleration in low-damped footbridges [13,14]. The influence of several parameters on the critical number of pedestrians that trigger excessive vibrations (like the frequency of pedestrian walk steps, the sensitivity of the crowd to bridge motion, the frequency and damping), and different load conditions have been studied [15]. The walking behaviour of pedestrians during an experimental test on a footbridge is sometimes used to improve the agreement of the measured response and the simulation using a finite element model [16–18]. The vibration of slender suspension footbridges caused by eccentrically distributed dynamic walking loads has been the focus of some studies [19–21] due to the importance of torsional vibration mode shapes, which can induce a large amplitude response when they are excited. The development of theoretical or numerical models for the dynamic analysis of this type of structure requires knowledge of natural frequencies, vibration modes, and damping factors. Although it is possible to develop numerical models to identify natural frequencies and mode shapes, structural damping can be estimated, but it is difficult to obtain reliable information until the structure has been built. This is why many footbridges undergo intervention after construction to implement control devices to mitigate the dynamic response. The Millennium Bridge in London would be the best known case [14,22]. Damping can be seen as the most important and uncertain factor that affects the response of the structure [23], as it depends on the design and materials used and the amplitude of the structural response. Typically, experimental tests are performed to estimate structural damping and validate assumptions of the design stage [23–27]. The experimental tests are based mostly on the analysis of the response produced by ambient excitation. Operational Modal Analysis (OMA) is widely used in the dynamic characterisation of structures [28–31]. Some reports discuss the application of OMA to suspension footbridges. Cunha et al. [32] discussed the importance of output-only dynamic testing to assess dynamic structural behaviour, presenting a few examples of its use in railway, road, and pedestrian bridges. Several studies experimentally identified modal parameters and dynamically characterized suspension bridges based on OMA [33–36]. Ricciardelli et al. [37] presented a study of models to study footbridge behaviour against different walking actions and concluded that design procedures should be revised and validated through full-scale data analysis. In the design phase of the Arouca bridge, a set of simplified structural methods, including both linear and nonlinear approximations of the cables behaviour, were developed to define the initial geometrical configuration of the bridge (static analysis), displacements and the expected mode shapes and associated eigenfrequencies (dynamic analysis). The proposed formulations require less computation time and use less memory than numerical methods such as the finite element method (FEM). The use of these models is significant when defining the geometric configuration of the bridge (which also determines the tensile forces in the main cables and the natural frequencies of the in-plane and out-plane modes), which need to be applied several times to obtain the desired bridge configuration. This work presents the design approach to enable the quasi-static and dynamic behaviour of the 516 Arouca bridge to be studied through theoretical, analytical, and numerical models. The assumptions made in these analyses are experimentally validated. The quasi-static and dynamic responses of the structure caused by wind action and people crossing were recorded by a set of GNSS antennas and triaxial accelerometers. The agreement between all the analytical/numerical solutions and the experimental data is found to be very good. The recorded data also allowed us to evaluate the damping coefficients of the bridge for the different vibration modes. The novelties and useful contributions of the work presented herein are: (i) The Arouca bridge is the only bridge of its kind with such a span, and therefore the presented results could be very useful for designing suspension footbridges with very big spans; (ii) A simplified structural methodology is developed to obtain a first approximation of the static and dynamic bridge behaviour; (iii) Simplified results are first compared in the design phase with those provided by a FEM model and later by comparing the results of in situ tests after the completion of the bridge. The numerical study includes a parametric analysis; (iv) GNNS antennas were used to measure the static behaviour of the bridge; (v) Experimental data on the response of this bridge, particularly on the modal damping ratios, are provided. 2. The Arouca 516 suspension footbridge The pedestrian suspension bridge under analysis has a span of l= 516m (see Fig. 1). The suspension system comprises two main cables (catenaries) and a set of secondary cables (pendants) that supports 127 metal deck modules (ndecks). The deck modules are connected to each other at their ends, on the same node as that where the pendants are connected. Each module hangs from the catenaries by 8 hanger cables (shared with the adjacent modules), 4 main hangers which were designed to carry the majority of gravity loads, and 4 secondary hangers, posttensioned, designed to minimize longitudinal deck rotation as well as a portion of lateral displacements (see Fig. 2). The modular deck consists of segments 4.0 m long and 2.1 m wide, composed mainly of UPN profiles and a couple of angles, in S275 steel. The deck slab is formed of lightweight steel grid panels with an approximate self-weight of 25 kg/m 2 . Each module has a total mass of M=603.2 kg. Catenary cables are each composed of seven 40 mm nominal diameter steel wire ropes HD 8 K PPI [38] (steel core, Am=5915.0 mm2, and m=51.0 kg/m). The ends of these cables were attached to the anchoring blocks and the cables were placed resting on a saddle at the top of the pillars, 36.3 m high. Hangers are each composed of a single 12 mm wire rope (Acl =75.8 mm2 and mcl =0.65 kg/m). The wire ropes were previously pre-tensioned during fabrication. The Young modulus of the wire ropes was indicated by the manufacturer and experimentally verified in Itecons, with a value of E=100 ±20 GPa. As default, the value E=100 GPa was used in the computations. In addition, a parametric analysis is performed for Young modulus varying in the range 50 −200 GPa is performed to verify its effect. Steel plates with a hexagonal section were fitted at 4.0 m intervals on each catenary to clamp the cables and work as a connection to the secondary cables (pendants). The configuration of the main cables and the length of the hanger cables were defined to ensure that they all remain in the same plane and lead to a maximum longitudinal inclination of the decks of 15% at the bridge ends. It was also important to guarantee that the tensile force of the main cables was very low compared to their tensile strength. After several iterations, the main cables with a length of L0=530.305 m and a cable sag of about dsag =53.50 m, and secondary cables with length varying between 2.361 m and 35.684 m were defined and are used A. Tadeu et al. Engineering Structures 253 (2022) 113830 3 throughout this study. 3. Theoretical analysis of the quasi-static and dynamic behaviour This section briefly describes the analytical models used to evaluate the static and dynamic behaviour of the bridge. 3.1. Linear analysis This subsection presents two simplified structural methods to define the initial geometrical configuration of the bridge (static analysis) and the expected mode shapes and associated eigenfrequencies (dynamic analysis). A linear behaviour was assumed for the suspended main cables. The tension in the main cables is taken to be constant along the cable, and the axial strain was neglected. 3.1.1. Static behaviour The 2D structural scheme adopted for each of the main cables of the suspension pedestrian bridge under analysis is shown in Fig. 3. The cable tension force is assumed to be large enough to be considered constant. The axial cable stiffness is large enough for the axial strains to be neglected, and the influence of gravity forces on the cable tension variation is also neglected. So, in this phase, the deflections are assumed to be small, and the nonlinear effects will therefore be neglected when computing the in-plane mode shapes. Defining k=T/d as the stiffness of each segment of the cable, where T is the tensile force applied to the cable (assumed to be constant at this stage) and d is the distance between discrete masses, where mldi = m(xi−xi−1)as the mass of each cable segment between two consecutive masses, and taking Mi the mass resulting from the action of the deck boards at the end of each spring, Ki as the stiffness of each spring given by the secondary cables, and Ui the displacement of each mass placed at Fig. 1. View of the 516 Arouca footbridge (Portugal). Fig. 2. Schematic of the structure module of the bridge and profiles of the construction elements. A. Tadeu et al. Engineering Structures 253 (2022) 113830 4 the end of each spring. In this model, the springs will elongate so that at each point we will have two distinct displacements: that of the mass at the end of the spring (Ui), and another at the base of the spring where the mass is lumped to the cable segment (u i ). The masses are assumed in this preliminary design to be prevented from swinging like pendulums. As the length of each segment of the cable is constant mldi =mld. The static equilibrium equations can be written as follows: − (mld)g+k(− ui−1+2ui−ui+1)− Ki(Ui−ui) = 0 − (Mi+mcli)g+Ki(Ui−ui) = 0(1) with u(0) = u(l) = 0; M1=Mn=M/4; Mi=M (for i∕= 1 and i∕= n);Ki=EA Cli (where Cli is the length of each secondary cable), mcli = mcl Cli is the mass of each secondary and g is the acceleration of gravity. The tensile force T is estimated through the relation T=meqgl2 8dsag . The equivalent mass meq combines the weight of the main cable and the weight of the decks suspended by the secondary cables (not including the secondary cables) per unit length, and l is the span of the bridge. Then, Equation (1) can be written in the following matrix form after applying the boundary conditions: Ku =mg(2) with uT= [u1u2⋯unU1U2⋯Un] mT= [mld mld ⋯mld M1+mcl1M2+mcl2⋯Mn+mcln ] K= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 2k+K1−k| − K1 −k2k+K2−k| − K2 ... ... ... |... −k2k+Kn| − Kn −K1|K1 −K2|K2 ... |... −Kn|Kn ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ where the dimension of K is 2n×2n, with n=ndecks +1 (number of hanger elements). Thus, each matrix element depends on the stiffness of each cable segment (k=T/d) and the stiffness of the secondary cables (Ki=EA Cli ). This model does not account for geometric nonlinearity. The direct solution of Equation (2) gives the position of the main cable. It is important to note that modifying Young modulus of the secondary cable would not change the configuration of the main cable. 3.1.2. Dynamic behaviour After computing the static equilibrium position, a preliminary dynamic analysis can be performed using the model illustrated in Fig. 3. The dynamic displacements are now defined as the oscillations of the mass at the end of the spring (Ui), and of the base of the spring where the mass is lumped to the cable segment (ui) about the static equilibrium position. The dynamic equilibrium equations, in free vibration and neglecting the damping, can be described as follows: mld¨ ui+k(− ui−1+2ui−ui+1)− Ki(Ui−ui) = 0 (Mi+mcli)¨ Ui+Ki(Ui−ui) = 0(3) that can be written in the following matrix form: M¨ u+Ku=0(4) where M is the stiffness matrix, M= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ mld | mld | ... | mld | |M1+mcl1 |M2+mcl2 |... |Mn+mcln ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ and uT= [u1u2⋯unU1U2⋯Un]. Defining the non-dimensional eigenvalue problem K− ω 2M=0, the eigenfrequencies and the associated modes can be determined. This model assumes that there is no increase in tension with the movement of the cable, which is absolutely correct for antisymmetric modes. Note that if the main cable is considered to have an insignificant mass, that the hangers are very rigid, and that the mass of the decks plus the hangers is constant, the results obtained would coincide with the analytical solution given in [39]:fj=1 2 π  4T d2meq √sin( π j 2(ndecks+2)), where n decks =127 is the number of discrete deck modules and d=4 m is the distance between discrete masses. As mentioned above, this model assumes that the cable tension is constant along it, which does not allow for the correct evaluation of inplane modes that generate extra tension/deformation in the cable. 3.2. Nonlinear analysis This subsection uses a nonlinear analysis to evaluate the final geometrical configuration of the bridge for different static loads. It also computes the in-plane and out-of-plane eigenmodes. As the whole theory for this problem was derived by Irvine for suspended cables [40], only the main equations are presented. 3.2.1. Static behaviour The following static model assumes that the main cable, with length L0(unstrained length), has an undetermined shape that is the length of the cable without strain in its free form. The cable length L0 is longer than the distance between the supports D. This means that the tension of the cable varies with its position, depending on the L0−D difference. The greater this difference, the greater the decline of the cable. Consider that the cable is suspended between two fixed points horizontally distanced by l and a relative vertical offset of h (points with Cartesian coordinates A (0,0)and B (l,h)). The cable in its free form L0 is greater than D= l2+h2 √. A point on the cable situated at a distance s from the origin, in a free state without traction, when subjected to the Fig. 3. Structural scheme for one of the main cables of the pedestrian suspension bridge adopted in the preliminary design stage. A. Tadeu et al. Engineering Structures 253 (2022) 113830 5 own weight of the cable W=mgL0 moves and occupies a new position in its strained form, described by the Cartesian coordinates x and y, and the Lagrangian coordinate p. In the presence of suspended N discrete masses (load), the shape of the cable will be a discontinuous catenary with changes in the inclination in each point of mass. A segment of the cable (see Fig. 4) will have to obey the static equilibrium equations, along the horizontal and vertical directions. In this figure, Fi is a set of N vertical loads, T is the tension in the cable, V is the vertical reaction of the support, H is the horizontal component of the traction on the cable and s is the curvilinear abscissa. Adopting the dimensionless relationships, ξ=x L0, η =y L0, σ =s L0, σ k=sk L0 (k=1,2, ...., N), γ=l L0, τ =T W, χ =H W, ϕ=V W, ψ n=Fk W (k=1, 2, ...., N)and β=W EA0 (A0 is the transverse area of the not traction cable), defining σ 0=0, ψ −1=0 and ψ 0=0, using ϑn=∑N j=− 1 ψ j with ϑ−1=0 and ϑ0=0, and imposing the boundary conditions in A and B (x=0, y=0,p=0 in s=0; x=l,y=h,p=L in s=L0, with L being the length of the cable in the strained form), the following equations can be derived [40]: τ ( σ ) =  χ 2+(ϕ−ϑn− σ )2 √(5) valid for the interval σ k< σ < σ k+1 for k=0,1,2, ....,N. ξ( σ ) = χ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ β σ +sinh−1(ϕ χ )−sinh−1(ϕ−ϑn− σ χ ) +∑ n i=0(sinh−1(ϕ−ϑi− σ i χ )−sinh−1(ϕ−ϑi−1− σ i χ ))⎤ ⎥ ⎥ ⎥ ⎥ ⎦(6) η ( σ ) = β σ (ϕ− σ 2)+ χ 2+ϕ2 √− χ 2+(ϕ−ϑn− σ )2 √ +∑n i=0[β ψ i( σ i− σ )+  χ 2+(ϕ−ϑi− σ i)2 √− χ 2+(ϕ−ϑi−1− σ i)2 √](7) also valid for the interval σ k< σ < σ k+1 for k=0,1,2, ....,N. The values χ and ϕ can be obtained from the following equations, γ χ −β=sinh−1(ϕ χ )−sinh−1(ϕ−ϑN−1 χ ) +∑ N i=0(sinh−1(ϕ−ϑi− σ i χ )−sinh−1(ϕ−ϑi−1− σ i χ )) (8) δ=β(ϕ−1 2)+ χ 2+ϕ2 √− χ 2+(ϕ−ϑN−1)2 √ +∑N i=0[β ψ i( σ i−1)+  χ 2+(ϕ−ϑi− σ i)2 √− χ 2+(ϕ−ϑi−1− σ i)2 √](9) The solution of these equations makes it possible to obtain V and H. These equations have been solved using the modified bi-dimensional Newton-Raphson method. 3.2.2. Dynamic behaviour Disregarding the horizontal component of the motion because it is small and neglecting the second-order terms, the equilibrium equations of a suspended cable supported at the same level can be expressed by the following equations [40]: TH ∂ 2u(x,t) ∂ x2+Th ∂ 2y ∂ x2−m ∂ 2u(x,t) ∂ t2=0(10) TH ∂ 2v(x,t) ∂ x2−m ∂ 2v(x,t) ∂ t2=0(11) where T H is the horizontal component of the cable tension at the midspan, and T h is the additional horizontal component of the cable tension. These equations correspond to the in-plane vertical u(x,t)and out-of-plane motionv(x,t), respectively. Th can be computed by the following integral [40]: Th=EAmeqg THLe∫l 0 u(x,t)(12) where Le=∫l o(ds dx)3dx ≈∫l ol(1+8(d1 l)2)is the strained length of the cable. In-plane motion The in-plane motion includes antisymmetric and symmetric modes. In the presence of antisymmetric in-plane modes the additional horizontal component of the cable tension Th=0 because ∫l 0u(x,t) = 0. In this specific case, Equation (11) can be written in the frequency domain after the substitution of u(x,t) = u ⌢(x)e ιω t. Knowing that u(0) =u(l/2) = 0, the following eigenfrequencies and associated eigenmodes can be computed [41]: ω j=2j π l TH meq √ u ⌢ j(x) = Ajsin(2j π x l)(13) where Aj is the amplitude, and j=1,2, ...... In the presence of symmetric in-plane modes, additional cable tension is induced, Th=T ⌢ he ιω t, that is taken to be constant along the span [42]. After the mathematical manipulation of Equation (10) and imposing the zero boundary conditions, the following equation is obtained: THu ⌢(x) meqgl2= T ⌢ h/TH ω 2(1.0−tan( ω 2)sin( ω x l)−cos( ω x l)) (14) ω = ω l  TH meq √ Using Equation (12) to eliminate T ⌢ h/TH the following transcendental equation is obtained [40]: Fig. 4. Forces to which a cable segment is subjected in its equilibrium position, when submitted to N vertical hanging masses and its own weight W s. A. Tadeu et al. Engineering Structures 253 (2022) 113830 6 tan( ω 2)= ω 2−4.0 λ2( ω 2)3(15) λ2=(meqgl TH)2 l/(THLe EA ) The solution of Equation (15) gives the natural frequencies of the symmetric in-plane modes. Out-of-plane motion Assuming v(x,t) = v ⌢(x)e ιω t in Equation (11), the following natural frequencies and associated modes can be computed after imposing the boundary conditions v(0) = v(l) = 0: ω j=j π l TH meq √;v⌢ j(x) = Bjsin(j π x l)(16) where Bj is the amplitude, and j=1,2, .... 3.2.3. Cable configuration The cable configuration is estimated from the static analysis described in the previous sections. A preliminary analysis was performed using the linear model presented in Section 3.1. The tensile force is estimated as T =780.2 kN (assuming dsag =53.00 m) and the equivalent mass is meq =126.4 kg/m. The system of Equation (1) has 256 unknown degrees of freedom (DOF). Its solution gives the initial position of the main cable, dsag =53.50 m (see Fig. 5a). In addition, a nonlinear analysis allows identification of the main cable position and variation of the tensile force according to Equations (7–9). Fig. 5a and 5b show the position and the tensile force of the main cable obtained. The sag of the main cable obtained in the nonlinear analysis dsag =53.56 m, is similar to that given by the linear model (dsag =53.50 m), and the value measured in the bridge after construction (dsag =53.72 m). The tensile force in the cable changes from 848.1 kN at the cable ends to 781.9 kN at the midspan, higher than estimated in the linear analysis (T =780.2 kN). The result of a parametric analysis for the Young modulus varying in the range 50 −200 GPa shows: i) the sag changes from 54.83 m to 52.88 m; and ii) the tensile forces are in the ranges 763.1–830.9 kN (E= 50 GPa) and 792.1–857.6 kN (E=200 GPa). 4. Numerical analysis using a finite element model A numerical finite element model was developed in Autodesk® Robot Structural Analysis Professional software [43], using its capabilities to model cable structures subjected to large displacements, to define initial stress conditions, and to perform a modal analysis considering a stress state at the end of static analysis. The elements forming the deck were modelled with 12DOF Euler- Bernoulli beam elements, while the cables were modelled with cable elements (tension only, 6DOF) (see Fig. 6). For load application, three nodes are added at the ends of each deck module defined by L, C, and R. These correspond to the theoretical centre of mass of human bodies for concentric and eccentric loads coupled with the centre of mass of the deck (see Fig. 6b). All the loads are applied to the above-mentioned points. Loads arising from self-weight and wind action are modelled as a concentrated load at point C. Loads arising from human activity are modelled at any of the 3 points, depending on the load case under study. The masses used in the modal analysis are equally modelled at these 3 points, with the same assumptions. The bridge model was analysed in two steps taking into account that tension stiffening phenomena make a large contribution to its overall stiffness: 1) a nonlinear analysis to determine the shape and stress state Fig. 5. Cable design obtained from the linear and nonlinear analysis: a) position; b) tensile force. A. Tadeu et al. Engineering Structures 253 (2022) 113830 7 of the structure; 2) a modal analysis using the stress state at the end of stage 1. After defining an undeformed structure, the final lengths of the catenaries upon self-weight loading and pretension levels of the hanger cables, a large displacement geometrically nonlinear analysis was carried out using a full Newton-Raphson solver. In addition to determining the final shape of the bridge under self-weight and imposed conditions, this analysis allows us to save the total stiffness and mass matrices at the end of the iterative process. The resulting stiffness matrix, also called a secant stiffness matrix, is then used as an input for the modal analysis and accounts for the extra geometric stiffness arising from tensioned elements. 5. Experimental test and results This section presents the experimental test performed to evaluate the footbridge behaviour under service loading and identify the dynamic properties. The bridge was subjected to both a quasi-static load induced by people walking at a slow pace and the dynamic load due to the wind action. Moreover, the experimental results allow assessment of the analytical and numerical modelling assumptions. The equipment used to evaluate the quasi-static and dynamic behaviour of Ponte 516 Arouca included ten triaxial accelerometers (model SYSCOM MR3000C), precision timing system clock <1 ppm, three GNSS antennas (Leica model GS15) and an anemometer (Gill windsonic 2D) (see Fig. 7). GNNS antennas were found to be the best system for measuring the static behaviour of the bridge. Alternative Fig. 6. Partial view of the numerical model: a) longitudinal view; b) cross-sectional view. A. Tadeu et al. Engineering Structures 253 (2022) 113830 8 systems, such as a displacement transducer or digital image correlation (DIC), are not available due to the characteristics of the valley. Triaxial accelerometers synchronized by GPS and wireless connectivity allowed measuring the dynamic response of the bridge at the required low frequency range. Other equipment was used, but it is not listed because it is not relevant to the content of the present paper. The triaxial accelerometers, with a nominal sensitivity of 1.25 V/g and a frequency range from DC to 600 Hz, were distributed along the span of the bridge at the locations indicated in Fig. 8, on the upstream side of the platform. In the middle, another triaxial accelerometer was placed on the downstream side of the platform. These triaxial accelerometers provided communication with a computer via the 4G network. The equipment was operated wirelessly to reduce any electrical interference or unwanted movement produced by the operator. This communication option meant that a GPS antenna was connected to each triaxial accelerometer, thus ensuring that the internal clock of each item of equipment was synchronized. The A/D conversion was carried out at a sampling frequency, fs =200 Hz, which avoided aliasing effects. The structural responses were filtered by applying a 50 Hz Bessel low pass filter of the 8th order to limit the analysis in the frequency range of interest 0–20 Hz. The GNSS antennas were placed at the longitudinal coordinates l/2, 2 l /3 and 3 l /4. In addition to the GNSS antennas used on the bridge, a fourth antenna, designated as a reference antenna, was also placed nearby in a stable and open area (900 m away), which allowed the GNSS antennas to be used in relative mode, thus increasing their precision. This reference antenna used data from the closest reference station (Bai˜ ao, 24 km away) in the National Network of Permanent Stations, ReNEP, to obtain high precision coordinates, using the PT-TM06/ ETRS89 system. This reference data was then used to get the rest of the coordinates of the GNSS antennas at an acquisition rate of 20 Hz and an accuracy of 3 mm (Root Mean Square (RMS)). The anemometer was placed at the bridge midspan. It was placed close to the middle of the deck module and above the deck platform (see Fig. 7c). It should also be noted that the porosity of the bridge modules is very high, which allows the wind speed to be measured accurately. To perform the quasi-static test, a group of volunteers was invited to cross the bridge. The group consisted of 34 people, which is approximately a total mass of 2778 kg on the footbridge. The group crossed the bridge from the Arouca side to the Alvarenga side at a slow pace, approximately 0.5 m/s (1.8 km/h), moving along the longitudinal axis. The pace was controlled by means of a digital metronome. The people were distributed throughout the various platform modules so that no more than three people were on any module (see Fig. 9). Fig. 7. Sensors used to monitor the bridge: a) Triaxial accelerometers; b) GNSS Antenna; c) anemometer. Fig. 8. Location of the sensors along the span of the bridge. a) Seismometer placed on the upstream side; b) Seismometer placed on the downstream side; c) GNSS antenna. A. Tadeu et al. Engineering Structures 253 (2022) 113830 9 5.1. Bridge response due to a quasi-static load The loads generated by the group of volunteers were considered as discrete masses suspended on the main cables. After the mathematical manipulation of Equations (8) and (9) the displacements were computed. This nonlinear model was used to calculate the displacements Fig. 9. The group of volunteers crossing the bridge. Point 5 Point 7 Point 8 Fig. 10. Experimental displacement recorded with the GNSS antennas and analytical results at points 5, 7, and 8 due to pedestrians walking at 0.5 m/s. A. Tadeu et al.