Dynamic load allowance of long-span modular steel bridges

Engineering Structures 282 (2023) 115835 Available online 22 February 2023 0141-0296/© 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct Dynamic load allowance of long-span modular steel bridges J. Chordà-Monsonís a,b,∗, E. Moliner b, P. Galvín a,d, M.D. Martínez-Rodrigo b, E. Zacchei c,e, A. Tadeu c,e, I. Ferraz f, A. Romero a aEscuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain bUniversitat Jaume I, Department of Mechanical Engineering and Construction, Avda. Sos Baynat s/n, ES-12071 Castelló, Spain cITeCons, Institute for Research and Technological Development in Construction, Energy, Environment and Sustainability, Rua Pedro Hispano s/n, 3030-289 Coimbra, Portugal dENGREEN, Laboratory of Engineering for Energy and Environmental Sustainability, Universidad de Sevilla, Camino de los Descubrimientos s/n, ES-41092, Spain eUniversidade de Coimbra, CERIS, Department of Civil Engineering, 3030-289 Coimbra, Portugal fMBS, BERD, Edíficio Olympus, Av. D. Alfonso Henriques, 1462-2◦, 4450-013, Matosinhos, Portugal ARTICLE INFO Keywords: Vehicle–bridge coupled model Road roughness Dynamic load allowance indices Impact factor Finite element analysis ABSTRACT Modular steel bridges are structures whose construction is based on regular prefabricated truss units. This presents several advantages, such as rapid and easy deployment, high adaptability to the terrain and reduced construction costs. However, they generally face operational restrictions for span lengths greater than 60 m. Recent technological innovations search to overcome these limitations and develop modular structures with larger spans. Hence, the main objective of this work is to evaluate the dynamic effects on long-span modular steel bridges. The present contribution provides a study on two modular bridge typologies, considering different span lengths from 120 to 140 m. A 3D coupled vehicle–bridge model is used to analyse the vehicle–bridge interaction and the dynamic load allowance of the structures. The vehicle is represented as a multibody truck system and the bridges are modelled with the finite element method. Several types of randomly-generated road irregularities are considered on the bridge deck. The effect of each type of irregularity is evaluated on the dynamic load allowance of the bridges. The results obtained reveal the notable influence of road irregularities that involve abrupt vertical displacements that excite the vehicle mode shapes. In addition, it is observed that dynamic load allowance indices tend to decrease with longer spans and higher speeds, except when a resonance is produced. 1. Introduction Since moving vehicles act as oscillators inducing forces on bridges, the dynamic interaction between both elements is one of the main concerns regarding the design, monitoring, and maintenance of the roadway bridge infrastructure. As experimental and theoretical investigations have shown, the impact of a moving vehicle on a bridge depends on many factors, including the vehicle and bridge characteristics, the speed of the vehicle and the roughness of the pavement [1]. Furthermore, the total response of the vehicle–bridge system is coupled. As a result, the complexity of the phenomenon increases and the study of the entire Vehicle–Bridge Interaction (VBI) is required to analyse the problem [2], which is a key issue in bridge engineering [3]. The dynamic effect of vehicles moving on bridges is generally evaluated with dynamic load allowance (IM) indices in many design codes [4]. Other common denominations for these indices are impact factors and dynamic amplification factors. ∗Corresponding author at: Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain. E-mail addresses: [email protected],[email protected] (J. Chordà-Monsonís). Modular bridges are steel truss structures constructed by means of the repetition of a prefabricated regular unit (or module) along the length of the bridge (see Fig. 1). This provides additional adaptability to the terrain, leads to a faster and easier construction process and reduces costs. Consequently, this type of bridge has been widely used as a temporary rapid erectable structure to recover damaged or destroyed communication routes [5]. Nevertheless, modular bridges are also often used as permanent structures [6]. The main limitation of modular steel bridges is attributed to the deformation caused by the combination of dynamic loads, leading to a maximum span length of 60 m, which is rarely exceeded [7]. In the case of longer spans, operational restrictions related to load capacity and circulation speed are necessary. As the effects of dynamic coupling between vehicle and bridge have become more and more significant due to the increase in operating speeds and axle loads in highway traffic [8], it is crucial to understand the dynamic behaviour of these bridges to improve their capabilities. https://doi.org/10.1016/j.engstruct.2023.115835 Received 9 November 2022; Received in revised form 30 January 2023; Accepted 16 February 2023 Engineering Structures 282 (2023) 115835 2 J. Chordà-Monsonís et al. Fig. 1. LMB-A120 bridge rendering. Several aspects of the VBI have been extensively studied in the last decades in a variety of solutions. In this sense, Yang et al. [9] evaluated the VBI by conceiving the bridge as a simply-supported beam and the vehicle as a sprung mass. The modal superposition method was used to obtain solutions for the whole system, admitting a small vehicle to bridge mass ratio. The authors found that accurate solutions could be obtained considering only the contribution of the bridge fundamental mode. Chul et al. [10] derived the equations of motion of a threedimensional VBI system using a variational formulation. The coupled VBI system, comprising an 8 Degree-Of-Freedom (DOF) cargo truck, a steel girder bridge and the roughness of the road was solved using Newmark’s 𝛽method. Liu et al. [11] compared the response of a moving load model and a VBI model on railway bridges. The authors found that the vehicle–bridge mass and fundamental frequency ratios can determine whether the VBI effects should be taken into account. More recently, Yang et al. [12] provided a closed-form solution for calculating the dynamic response of a two-axle asymmetric vehicle moving over a bridge. The authors emphasised the utility of this approach to scan the dynamic properties of bridges. Greco et al. [13] developed a model with the Finite Element Method (FEM) consisting of a moving mesh to evaluate the VBI in presence of moving loads. The methodology was validated by comparing their results with the literature. In the last few years, other aspects related to VBI have been studied, such as the influence of the vehicle and the bridge modal parameters on the vehicle–bridge coupled system, as in [3,14,15]; or the additional damping that the vehicle induces on the dynamic vibration of the vehicle–bridge system [16,17]. In this regard, some authors [18,19] investigated how the standards treat this matter and highlighted the complexity of the phenomenon. As part of the excitation mechanisms of the VBI, the effect and importance of road irregularities has also been widely discussed, concluding that they can strongly affect the performance of the coupled vehicle–bridge system causing a greater dynamic response, lowering the comfort for the users and diminishing the fatigue life of bridges with poorer road conditions [20–22]. Other authors have evaluated the effect of road irregularities focusing on the dynamic load allowance. In this sense, Ma et al. [23] carried out a series of VBI analyses to assess the regularity of the IM on 15 continuous beam bridges. The authors concluded that IM indices increase dramatically when resonance occurs and also, current design codes did not take this effect into account. Deng et al. [4] developed a 3D vehicle–bridge coupled model to investigate the VBI and the IM on multi-girder concrete bridges. The authors provided additional expressions to calculate IM indices under different road conditions since they concluded that AASTHO [24] may underestimate the amplification effects on poor condition roads. Jung et al. [25] studied the IM on a total of 256 bridges and found that 32% of the structures exceeded the design criteria of the IM indices based on the span length provided in several design codes. As a result, the authors suggested a method for calculating IM indices on the basis of the natural frequency of the bridges. Mosheni et al. [26] reached similar conclusions. The authors investigated the dynamic behaviour of multicell box-girder bridges under moving loads and concluded that AASHTO [24] formulas may underestimate the IM, especially on bridges with larger spans. The literature about the evaluation of the IM on modular steel bridges is scarce and represents a clear gap of knowledge. In one of the few publications dedicated to the matter, Montenegro et al. [27] described a stochastic methodology to evaluate IM indices considering road irregularities on two modular steel bridges. The bridges were modelled with the FEM and the vehicle as a Multibody System (MBS). The authors concluded that the technique was adequate and could lead to relevant material savings in the construction of this type of structures compared to the use of predefined IM factors proposed in the design codes. Pinkey et al. [28] used an experimental testing programme to assess the dynamic load allowance of different types of military vehicles at different speeds on a modular steel bridge. The authors found that rough bridge surfaces increase the bridge response and tracked vehicles cause less dynamic effects than wheeled ones. Other authors like Kusimba et al. [6] assessed the dynamic performance of a modular bridge and calculated its IM factor, which was well within the threshold given by AASHTO [24]. In general, for this type of bridge, other aspects are usually studied, such as damage assessment of joints and structural elements as in [29,30], or new structural construction techniques as in [5]. In view of this, the present research aims to (i) evaluate the influence of the VBI on the dynamic behaviour of long-span modular steel bridges and (ii) investigate the dynamic load allowance under different types of randomly-generated road irregularities. This represents an important novelty, because, as stated before, this is not usually the matter of research on this type of truss bridges. Furthermore, a significant part of the research effort dedicated to evaluating the IM on bridges has been focused on short-to-medium span bridges [31]. As a result, studies found in the literature generally cover significantly shorter span lengths than the ones considered in this contribution. In addition, for the sake of completeness, the effect of considering increasing span lengths is taken into account in the present paper and two different modular bridge typologies are studied to understand the dynamic effects on the structural elements of the bridges. Lastly, the separate influence of various types of randomly-generated road irregularities and their total contribution considering a wide velocity range are investigated. To pursue the goals of the work, a 3D vehicle–bridge coupled model is implemented. The vehicle is simulated as a MBS based on the AASHTO HS20-44 truck [24]. Two bridge typologies of span lengths from 120 to 140 mfor a total of six bridges are studied. The numerical models of the bridges are defined using the FEM. The VBI approach considered in this work admits that the vehicle does not significantly alter the dynamic behaviour of the bridges, and thus the modal superposition method can be applied to compute the bridge response [4,31]. The pavement of the bridges is made up of steel frames, and in this regard, the effect of three types of road irregularities is considered: (i) roughness of the pavement and (ii) bowing and (iii) misalignment of the pavement frames. The influence of the quasistatic and dynamic load contributions is analysed. Random irregularity profiles are generated on a stochastic basis from experimentally measured manufacturing imperfections. A sufficient number of simulations are carried out to ensure the representativeness of the obtained results. This paper is organised as follows. In Section 2, the vehicle–bridge formulation is presented in detail. In Section 3, the generation of road irregularities is described. Section 4addresses the numerical models of vehicles and bridges used in this work. In Section 5, the dynamic forces resulting from the VBI and the IM indices of the bridges under different types of irregularities are investigated. The results of the work are discussed in Section 6and the main conclusions are derived in Section 7. Engineering Structures 282 (2023) 115835 3 J. Chordà-Monsonís et al. 2. Vehicle–bridge coupled system This section describes the mathematical formulation that defines the dynamic coupling between the vehicle and the bridge. The adopted procedure is based on that proposed in [32]. The vehicle is represented as a MBS and the bridges are modelled with the FEM. 2.1. Equations of motion of the vehicle The equations of motion of the vehicle can be written as follows: 𝐌𝑣 𝐮𝑣+𝐂𝑣 𝐮𝑣+𝐊𝑣𝐮𝑣=𝐅𝑣+𝐅𝑣𝑏 (1) where 𝐌𝑣,𝐂𝑣and 𝐊𝑣are the mass, damping, and stiffness matrices of the vehicle, respectively, and 𝐮𝑣is its displacement vector. A dot over a variable denotes the time derivative. The right-hand side of the equation represents the force vector caused by the effect of gravity 𝐅𝑣 and the interaction force 𝐅𝑣𝑏 at the contact points between the vehicle and the bridge. 2.2. Equations of motion of the bridge The equilibrium equation of the bridge can be written as follows: 𝐌𝑏 𝐮𝑏+𝐂𝑏 𝐮𝑏+𝐊𝑏𝐮𝑏=𝐅𝑏+𝐅𝑏𝑣 ,(2) where 𝐌𝑏,𝐂𝑏and 𝐊𝑏are the mass, damping, and stiffness matrices, respectively. 𝐮𝑏is the displacement vector of the bridge and 𝐅𝑏represents the forces applied to it. The interaction force 𝐅𝑏𝑣 is defined at the contact points as well. The displacement vector can be expressed in modal coordinates as: 𝐮𝑏(𝐱, 𝑡) = 𝑚 ∑ 𝑗=1 𝜙𝑗(𝐱)𝑌𝑗(𝑡) = 𝜱(𝐱)𝐘(𝑡),(3) where 𝜙𝑗(𝑥)and 𝑌𝑗(𝑡)are the 𝑗-th modal shape and generalised coordinate, respectively. Then, the projection of Eq. (2) in the modal space of dimension 𝑚gives the subsequent modal equations, admitting a viscous modal damping approach: 𝐈 𝐘+𝐙 𝐘+𝜴𝐘=𝜱𝑇(𝐅𝑏𝑣 +𝐅𝑏)(4) 𝐈=⎡⎢⎢⎣ ⋱ 1 ⋱⎤⎥⎥⎦𝑚×𝑚 ,𝐙=⎡⎢⎢⎣ ⋱ 2𝜁𝑗𝜔𝑗 ⋱⎤⎥⎥⎦𝑚×𝑚 , 𝜴=⎡⎢⎢⎣ ⋱ 𝜔2 𝑗 ⋱⎤⎥⎥⎦𝑚×𝑚 . (5) where 𝜔𝑗is the 𝑗-th natural frequency and 𝜁𝑗is the 𝑗-th corresponding modal damping. 2.3. Vehicle–bridge interaction The vehicle–bridge interaction force in the 𝑖th wheel is defined as: 𝐅𝑣𝑏𝑖 =𝑘𝑤𝑖𝛥𝑖+𝑐𝑤𝑖  𝛥𝑖(6) where 𝑘𝑤𝑖 and 𝑐𝑤𝑖 represent the stiffness and damping of the tyres, and 𝛥𝑖and  𝛥𝑖are the relative vertical displacement and velocity between the 𝑖-th wheel and the bridge deck at the contact point: 𝛥𝑖=𝐮𝑣𝑖 − (𝑅𝑖+𝐮𝑏𝑖)(7a)  𝛥𝑖= 𝐮𝑣𝑖 − (  𝑅𝑖+ 𝐮𝑏𝑖)(7b) In these equations, 𝐮𝑣𝑖 is the vertical displacement of the 𝑖-th wheel of the vehicle. 𝐮𝑏𝑖 and 𝑅𝑖represent the vertical displacement of the bridge and the total irregularity profile of the road surface under the 𝑖-th wheel, respectively. Fig. 2 shows a scheme of the relative displacement components of the system. The displacement of the bridge at the contact point 𝑖is computed using finite elements as: 𝐮𝑏𝑖 = 𝑝 ∑ 𝑘=1 𝑁𝑘𝐮𝑒 𝑏𝑘 =𝐍𝐮𝑒 𝑏(8) where 𝑝is the number of nodes, 𝑒is the element to which the 𝑖-th contact point belongs, 𝑁𝑘represents the element shape functions and 𝐮𝑒 𝑏𝑘 are the nodal displacements. The force component related to the bridge velocity in Eq. (6) is approximated according to:  𝐮𝑏𝑖 + 𝑅𝑖=𝑑𝐍 𝑑𝑥 𝑑𝑥 𝑑𝑡 𝐮𝑒 𝑏+𝐍 𝑑𝐮𝑒 𝑏 𝑑𝑡 +𝑑𝑅𝑖 𝑑𝑥 𝑑𝑥 𝑑𝑡 =𝐍,𝑥𝑣𝐮𝑒 𝑏+𝐍 𝐮𝑒 𝑏+𝑅𝑖,𝑥𝑣 (9) where 𝑣, here assumed constant, represents the passing velocity of the vehicle. Then, introducing Eqs. (3),(8) and (9) in Eq. (6), the vehicle– bridge interaction force at the contact point 𝑖can be expressed as: 𝐅𝑣𝑏𝑖 =𝑘𝑤𝑖(𝐮𝑣𝑖 − (𝑅𝑖+𝐍𝐮𝑏)) + 𝑐𝑤𝑖( 𝐮𝑣𝑖 − (𝐍,𝑥𝑣𝐮𝑒 𝑏+𝐍 𝐮𝑒 𝑏+𝑅𝑖,𝑥𝑣)) =𝑘𝑤𝑖(𝐮𝑣𝑖 − (𝑅𝑖+𝐍𝜱𝐘)) + 𝑐𝑤𝑖( 𝐮𝑣𝑖 − (𝐍,𝑥𝑣𝜱𝐘+𝐍𝜱 𝐘+𝑅𝑖,𝑥𝑣)) (10) Furthermore, the interaction force in the structure is obtained from the total number of vehicle tyres 𝑤according to the following expression: 𝐅𝑏𝑣 = 𝑤 ∑ 𝑖=1 𝐍𝑇 𝑖𝐅𝑏𝑣𝑖 = − 𝑤 ∑ 𝑖=1 𝐍𝑇 𝑖𝐅𝑣𝑏𝑖 (11) where the transposed shape function matrix 𝐍𝑇 𝑖is evaluated on the 𝑖-th wheel. Once the interaction force is known, it is possible to define an equation of motion to represent the dynamic behaviour of the coupled vehicle–bridge system. The equilibrium equation of the bridge defined in Eq. (4) considering the interaction force of each wheel becomes: 𝐈 𝐘+(𝐙+𝜱𝑇(𝑤 ∑ 𝑖=1 𝐍𝑇 𝑖𝑐𝑤𝑖𝐍𝑖)𝜱) 𝐘 +(𝜴+𝜱𝑇(𝑤 ∑ 𝑖=1 𝐍𝑇 𝑖(𝑘𝑤𝑖𝐍𝑖+𝑐𝑤𝑖𝑣𝐍𝑖,𝑥))𝜱)𝐘= −𝜱𝑇 𝑤 ∑ 𝑖=1 𝐍𝑇 𝑖(𝑘𝑤𝑖(𝐮𝑣𝑖 −𝑅𝑖) + 𝑐𝑤𝑖( 𝐮𝑣𝑖 −𝑅𝑖,𝑥𝑣)) (12) Then, the equation of motion of the vehicle in Eq. (1) and the interaction force in Eq. (10) allow the following: 𝐌𝑣 𝐮𝑣+𝐂𝑣 𝐮𝑣+𝐊𝑣𝐮𝑣=𝐅𝑣+ 𝑤 ∑ 𝑖=1 [𝑘𝑤𝑖(𝐮𝑣𝑖 − (𝑅𝑖+𝐍𝑖𝜱𝐘)) +𝑐𝑤𝑖( 𝐮𝑣𝑖 − (𝐍𝑖,𝑥𝑣𝜱𝐘+𝐍𝑖𝜱 𝐘+𝑅𝑖,𝑥𝑣))](13) Eqs. (12) and (13) are written in a coupled system of equations that permits the calculation of the displacements of the bridge and the vehicle. The system matrices vary each time step according to the vehicle position. Then, an explicit Runge–Kutta time marching scheme [33,34] is adopted to solve the VBI problem. 3. Generation of road irregularities In this study, three types of road irregularities are considered: (i) roughness of the pavement surface, (ii) bow defects of the pavement frames due to manufacturing imperfections and (iii) misalignment of the pavement frames due to faulty installation. In the following, the mathematical representation of these irregularity types is explained. First, the pavement roughness is defined as a stationary stochastic process based on a Power Spectral Density function (PSD) as per ISO- 8608 [35]. The irregularity profile 𝑟(𝑥)is given by a zero-mean normal stationary ergodic random process described by its PSD function 𝑆(𝜅): 𝑆(𝜅𝑛) = 𝑆(𝜅0)(𝜅𝑛 𝜅0)−2 (14) Engineering Structures 282 (2023) 115835 4 J. Chordà-Monsonís et al. Fig. 2. Relative displacement components between the vehicle and the bridge. where 𝜅𝑛is the spatial frequency and 𝑆(𝜅0)is the one-sided PSD for the reference spatial frequency 𝜅0. The PSD reference value depends on the road condition and is defined in ISO-8608 [35] for different road classes. Then, the irregularity profile is calculated as the sum of a series of harmonics, as: 𝑟(𝑥) = 𝑁𝑠 ∑ 𝑛=0 √2𝑆(𝜅𝑛)𝛥𝜅 cos(2𝜋 𝜅𝑛𝑥+𝜑𝑛)(15) where 𝜑𝑛is the random phase angle uniformly distributed in the interval [0,2𝜋],𝑁𝑠is the total number of samples and 𝛥𝜅 is the incremental 𝜅to compute the irregularity profile. Second, the irregularity of the pavement bowing corresponds to imperfections in the pavement frames in the form of bow defects. The bowing displacement 𝑏𝑞in the centre of the 𝑞-th frame is calculated on the basis of the relative deflection in the centre of a simply supported beam subjected to a uniformly distributed load: 𝑏𝑞(𝑥𝑞) = 𝐵𝑞 −384 𝑥𝑞 120 𝐿4 𝑝 (𝐿3 𝑝− 2𝐿𝑝𝑥2 𝑞+𝑥3 𝑞)(16) where the local coordinate 𝑥𝑞=𝑥−𝐿𝑝(𝑞− 1) represents the position of the contact point of the vehicle within the length of the pavement frame 𝐿𝑝. On the other hand, the amplitude of 𝑏𝑞is ranged by the random parameter 𝐵𝑞, which is defined as a normal distribution: 𝐵𝑞∼𝑁(𝜇𝐵, 𝜎𝐵)(17) where 𝜇𝐵and 𝜎𝐵are the mean and standard deviation of the distribution, respectively. In a similar way, the misalignment irregularity is defined as a vertical step defect in the transverse joints of the pavement frames. The amplitude of this defect 𝑀𝑞is defined as: 𝑀𝑞∼𝑁(𝜇𝑀, 𝜎𝑀)(18) where 𝜇𝑀and 𝜎𝑀are the mean and standard deviation of the normal distribution. The total unevenness at the bridge coordinate 𝑥is calculated as the combination of these three types of irregularities: 𝑅(𝑥) = 𝑟(𝑥) + 𝑄 ∑ 𝑞=1 (𝑏𝑞+𝑀𝑞)[𝐻(𝑥−𝐿𝑝(𝑞− 1)) − 𝐻(𝑥−𝐿𝑝𝑞)] (19) where 𝑄is the total number of pavement frames and 𝐻represents the Heaviside step function referred to the 𝑞-th frame. 4. Evaluation of dynamic load allowance This section explains the adopted approach to investigate the dynamic load allowance of modular bridges. Sections 4.1 and 4.2 address the truck and the bridges considered for the study, respectively. Then, the configuration of the road irregularities considered in the analyses is detailed in Section 4.3. 4.1. The HS20-44 truck The vehicle used in this work is based on the HS20-44 truck included in the AASHTO specifications [24]. As can be seen in Fig. 3, the truck is represented as a 3D MBS which considers the mass and inertia Table 1 Parameters of the HS20-44 truck. Note that the abbreviation COM stands for Center Of Mass. Parameter Description Value Unit 𝑚𝑣1Mass of truck body 1 2612 kg 𝐼𝜃 𝑣1Pitching moment of inertia of truck body 1 2022 kg m2 𝐼𝜙 𝑣1Rolling moment of inertia of truck body 1 8544 kg m2 𝑚𝑣2Mass of truck body 2 26113 kg 𝐼𝜃 𝑣2Pitching moment of inertia of truck body 2 33153 kg m2 𝐼𝜙 𝑣1Rolling moment of inertia of truck body 2 181216 kg m2 𝑚𝑎1Mass of the first axle suspension 490 kg 𝐼𝜙 𝑎1Rolling moment of inertia of first axle suspension 350 kg m2 𝑘𝑠𝑓 Upper spring stiffness of the first axle 242604 N/m 𝑐𝑠𝑓 Upper damper coefficient of the first axle 2190 Ns/m 𝑘𝑤𝑓 Lower spring stiffness of the first axle 875082 N/m 𝑐𝑤𝑓 Lower damper coefficient of the first axle 2000 Ns/m 𝑚𝑎2Mass of the second axle suspension 808 kg 𝐼𝜙 𝑎2Rolling moment of inertia of second axle suspension 600 kg m2 𝑘𝑠𝑚 Upper spring stiffness of the second axle 1903172 N/m 𝑐𝑠𝑚 Upper damper coefficient of the second axle 7882 Ns/m 𝑘𝑤𝑚 Lower spring stiffness of the second axle 3503307 N/m 𝑐𝑤𝑚 Lower damper coefficient of the second axle 2000 Ns/m 𝑚𝑎3Mass of the third axle suspension 653 kg 𝐼𝜙 𝑎3Rolling moment of inertia of third axle suspension 600 kg m2 𝑘𝑠𝑟 Upper spring stiffness of the third axle 1969034 N/m 𝑐𝑠𝑟 Upper damper coefficient of the third axle 7182 Ns/m 𝑘𝑤𝑟 Lower spring stiffness of the third axle 3507429 N/m 𝑐𝑤𝑟 Lower damper coefficient of the third axle 2000 Ns/m 𝐿1Length from first axle to COM of truck body 1 1.698 m 𝐿2Length from second axle to COM of truck body 1 2.569 m 𝐿3Length from second axle to COM of truck body 2 1.984 m 𝐿4Length from third axle to COM of truck body 2 2.283 m 𝐿5Length from COM of truck body 1 to pivot joint 2.215 m 𝐿6Length from COM of truck body 2 to pivot joint 2.238 m 𝑏Half width of the truck 1.1 m of the axles, front and rear truck bodies, and the stiffness and damping of the suspensions. Tyres and suspensions are modelled as linear elastic springs and dampers. Subscripts 𝑓,𝑚and 𝑟indicate front, middle and rear, and subscripts 𝑣,𝑠,𝑎and 𝑤stand for vehicle, suspensions, axles, and tyres, respectively. The total load from front to rear axles is 𝑃= [35.2,142.4,142.4] kN. Three degrees of freedom are assigned to both car bodies: 𝑢𝑣indicates vertical displacement, 𝜃𝑣corresponds to rotation about the transverse axis (or pitching) and 𝜙𝑣represents rotation about the longitudinal axis (or rolling). The axles are rigid and have two DOFs assigned: 𝑢𝑎and 𝜙𝑎. Furthermore, the DOF corresponding to the vertical displacement of each wheel is also taken into account. In total, 18 DOFs are considered to represent the vehicle behaviour. The parameters of the complete truck model are adopted from Refs. [4,20] and listed in Table 1. The main mode shapes of the vehicle are represented in Fig. 4. The first one corresponds to the rolling of the rear axle and the rear truck body. The second mode involves the vertical displacement of the three axles and the pitching of the front and rear truck bodies with opposite rotations. Similarly, the third mode consists of the vertical displacement of the axles and the pitching of both truck bodies. The fourth mode corresponds to the rolling of the front and middle axles and the front truck body. The fifth and sixth modes involve higher displacements of the axles and the pitch of front and rear truck bodies. Engineering Structures 282 (2023) 115835 5 J. Chordà-Monsonís et al. Fig. 3. HS20-44 vehicle model [24]. Fig. 4. Mode shapes of the HS20-44 truck. 4.2. Long-span modular bridges under study In this contribution, the dynamic allowance of two Long-span Modular Bridges (LMBs) is investigated. The bridges under study are part of a developing project of the bridge design company BERD [36]. In particular, two different modular typologies with three different span lengths in each case are considered. The first typology, designated as LMB-R, corresponds to a modular truss bridge that includes a number of reinforced modules at both ends of the structure. The reinforced modules present the same spatial distribution as the other modules, but are composed of different structural elements in order to resist higher loads close to the supports. The second bridge typology, designated as LMB-A, is composed of structurally identical modules. In addition, an arch is built on top of the modules along the span of the bridge. Fig. 5 shows three images of the LMB-R bridges. For this typology, three different bridges with different span lengths approaching 120, 130 and 140 m are considered. In the following, these bridges are referred to as LMB-R120, LMB-R130 and LMB-R140. In all cases, the bridges are simply-supported, and six reinforced modules are considered in total: three at each end of the bridge, as can be seen in Fig. 5(a). Each module consists of a three-dimensional truss. Modules are connected through bolted connections in the chords and in the posts. In the inferior part, each truss is connected by means of three floor crossbeams, two horizontal and two longitudinal bracings to provide the necessary stability between modules. At the top of the module, the truss connection is carried out by lateral bracings. The lower chords are connected to the upper chords by means of posts and diagonals. The length, width and height of each module are represented as 𝐿𝑚, 𝑊𝑚, and 𝐻𝑚, respectively. The length of the end truss is denoted as 𝐿𝑒𝑛𝑑 . Details of the LMB-A typology are shown in Fig. 6. In the same way as in the previous case, three simply-supported bridges are defined for this typology, approaching 120, 130 and 140 m of span length. These bridges are designated as LMB-A120, LMB-A130 and LMB-A140, respectively. In this case, the modules form a more complex threedimensional structure that includes an arch of height 𝐻𝑎that extends throughout the length of the bridge. Each side of the module is formed Engineering Structures 282 (2023) 115835 6 J. Chordà-Monsonís et al. Fig. 5. LMB-R bridge typology. (a): schematic front and side view, (b): isometric view of two modules, and (c): module cross-section. Table 2 General properties of the LMB-R and LMB-A bridges. Bridges 𝑀 𝐿𝑚 [m] 𝐿𝑒𝑛𝑑 [m] 𝐿𝑏 [m] 𝑊𝑚 [m] 𝐻𝑚 [m] 𝑚𝑏 [kg] 𝑓𝑏1 [Hz] LMB-R120 19 6 3 120 9 10 5.843 × 1051.48 LMB-R130 21 6 3 132 9 10 6.580 × 1051.23 LMB-R140 23 6 3 144 9 10 7.345 × 1051.04 LMB-A120 20 5.6 2.8 117.6 11.5 8.3 9.992 × 1051.39 LMB-A130 22 5.6 2.8 128.8 11.5 8.3 1.100 × 1061.17 LMB-A140 24 5.6 2.8 140 11.5 8.3 1.211 × 1060.99 by a double truss that comprises lower, upper chords and diagonals. These elements are linked by the lower posts. The bottom connection between modules is provided by three floor cross-beams and two horizontal bracings. At the top of the module, four lateral bracings and a strut attach both trusses. The arch is connected to the modules by means of the upper posts and two sway bracings that join at the centre of the strut. The bridge deck is composed of light pavement steel frames of length 𝐿𝑝= 2.8 m. The equivalent density and thickness of the pavement frames is set as 𝜌𝑝= 753 kg∕m3and ℎ𝑝= 0.072 m on the basis of experimental measurements. The main properties of the bridges are listed in Table 2, where 𝑀 indicates the number of modules, 𝐿𝑏is the length of the bridge, 𝑚𝑏is the total mass and 𝑓𝑏1 is the fundamental frequency, which corresponds to the first longitudinal bending mode of the bridges. In the case of the LMB-A typology, 𝐻𝑚refers to the height of the module without considering the arch. The maximum height of the arch 𝐻𝑎is reached at mid-span and is equal to 7 m. Fig. 7 shows the FEM of the LMB-R120 and LMB-A120 bridges. The bridge is modelled using beam elements based on the Timoshenko formulation and shell elements to represent the pavement frames. The normalised steel profiles of the main structural elements of the bridges (see Figs. 5(b) and 6(b)) are listed in Table 3. For the LMB-R bridges, standard and reinforced modules are differentiated. The first three longitudinal bending modes of the LMB-R120 and LMB-A 120 bridges are shown in Fig. 8. These mode shapes are representative of the other bridge lengths considered in each typology. Table 3 Normalised steel profiles for the main structural elements of the LMB-R and LMB-A bridges. Element LMB-R (Standard modules) LMB-R (Reinforced modules) LMB-A Arch – – 2 HEA 300 Upper posts – – IPE 300 Upper chords 450 ×35-450 ×15 450 ×25-450 ×12 HEA 300 Posts IPE 450 IPE 450 IPE 500 Diagonals 2 UPN 220 2 UPN 260 SHS 180 ×5 Lower chords 450 ×30-450 ×15 450 ×20-450 ×12 HEA 400 4.3. Road irregularities The road unevenness considered in the dynamic analyses includes five different irregularity cases. First, the quasistatic excitation mechanism is considered. In this case, the dynamic problem is solved by only taking into account the self-weight of the vehicle, and no irregularities are considered. Then, three irregularity cases are defined on the basis of the unevenness types detailed in Section 3: the pavement roughness, bowing and misalignment. The pavement roughness is defined according to ISO-8608 [35], where the degree of unevenness is determined on the basis of the road surface condition, varying from road class A (very good) to road class E (very poor). However, this classification is provided for asphalt roads, which are more prone to exhibit important defects. In this study, the bridge pavement consists of a 1mm thick coating that acts as a protective layer of the steel frames, and the maximum roughness amplitude of this layer is expected to be equal to its thickness in the worst-case scenario. Thus, a value of 𝑆(𝜅0) = 0.35 × 10−6 m3∕rad is adopted with a reference spatial frequency of 𝜅0= 0.1m−1. The 𝛥𝜅 increment and the number of spatial frequency samples are set to 𝛥𝜅 = 4.5 × 10−5 m−1 and 𝑁𝑠= 2000 . Table 4 summarises the irregularity parameters. The first column indicates if the effect of gravity is considered in the dynamic analyses, while 𝑆𝑛is the number of simulations performed in each case per vehicle speed. To define the bowing and misalignment irregularities, a Engineering Structures 282 (2023) 115835 7 J. Chordà-Monsonís et al. Fig. 6. LMB-A bridge typology. (a): schematic front and side view, (b): isometric view of two modules, and (c): module cross-section. Fig. 7. View of the FEMs of the bridges. (a): LMB-R120 bridge, and (b): LMB-A120 bridge. Table 4 Configuration and main parameters that define the five irregularity cases. Irregularity case Gravity 𝑆𝑛Parameters Quasistatic Yes 1 – Pavement roughness No 256 𝑆(𝑘0)=0.35 × 10−6 m3∕rad Pavement bowing No 256 𝜇𝐵= 2.3296 mm, 𝜎𝐵= 1.0738 mm, 𝜎2 𝐵= 1.1530 mm2 Pavement misalignment No 256 𝜇𝑀= −0.1250 mm, 𝜎𝑀= 1.4857 mm, 𝜎2 𝑀= 2.2072 mm2 Total Yes 256 All irregularities combined large sample of already fabricated pavement frames were analysed following the procedure detailed in Refs. [27,37]. A Kolmogorov–Smirnov non-parametric test was used to determine that these two types of unevenness can be statistically characterised as a normal distribution. Fig. 9 shows a randomly generated unevenness profile for each type of irregularity. 5. Results and analysis The main results of the research are presented in this section. In order to evaluate the influence of each mode on the dynamic response of the bridges, the cumulative participation of the first 100 modes on the bridge displacement and velocity at mid-span is calculated. On Engineering Structures 282 (2023) 115835 8 J. Chordà-Monsonís et al. Fig. 8. Longitudinal bending mode shapes of: (a, c, e) the LMB-R120, and (b, d, f) the LMB-A120 bridges. Table 5 Frequency ranges used to calculate the bridge response using the modal superposition method of the LMB-R and LMB-A bridges. Bridge Frequency range 𝑓𝑏[Hz] LMB-R120 𝑓1–𝑓22 [1.47, 13.47] LMB-R130 𝑓1–𝑓18 [1.23, 12.06] LMB-R140 𝑓1–𝑓18 [1.04, 10.85] LMB-A120 𝑓1–𝑓14 [1.39, 7.29] LMB-A130 𝑓1–𝑓13 [1.17, 6.46] LMB-A140 𝑓1–𝑓12 [0.99, 5.76] this basis, the frequency range to compute the bridge responses is defined between the fundamental frequency and the fifth longitudinal bending modal frequency of each bridge. Fig. 10 shows the modal Cumulative Contribution in terms of Displacement (DCC) and Velocity (VCC) at mid-span of the LMB-A120 bridge considering a vehicle speed of 90 km∕h. The vertical dashed line represents the 14th mode shape, which corresponds to the fifth in-plane bending mode shape. At that point, the modal contribution in the frequency range reaches values greater than 90% for both displacement and velocity. Then, it can be admitted that the frequency range considered is adequate. In the remaining bridges, the same trend is observed. Table 5 lists the selected frequency ranges 𝑓𝑏of all bridges. The nominal speed considered for the dynamic analyses corresponds to 90 km/h, as it is the maximum design speed allowed in the modular bridges under study. A modal damping ratio 𝜁= 1 % is set for all the modes according to the AASHTO [24] recommendation for steel structures. As each bridge has two lanes, vehicle passages are simulated with the truck circulating centred in one lane. 5.1. Dynamic forces In this section, the contact forces produced on the vehicle as a result of the VBI and the bridge acceleration response are studied. Fig. 11 shows the frequency content of the contact force of the front, middle and rear axles of the vehicle considering the quasistatic excitation and the pavement roughness, bowing and misalignment irregularities. The contact force is higher on the middle and rear axles of the truck, which is consistent, as they are located below the rear truck body, which is heavier than the front truck body. In Fig. 11(a), the highest amplitude in the quasistatic contribution is mainly found at low frequencies. The pavement roughness contribution (Fig. 11(b)) is found in the frequency range from 2 to 5 Hz, coinciding with three vertical bouncing modes of the vehicle (𝑓𝑣2 = 2.15 Hz,𝑓𝑣3 = 3.07 Hz,𝑓𝑣5 = 4.72 Hz). Regarding the bowing imperfections in Fig. 11(c), the forces are higher than in the previous cases. A narrow peak is found near 8.93 Hz, which is related to the loading frequency induced by the vehicle 𝑓=𝑣∕𝐿𝑝, where 𝐿𝑝is the length of the pavement frames and 𝑣= 90 km∕h. In addition, a lower peak appears near 2.23 Hz, approaching the second frequency mode of the vehicle 𝑓𝑣2 = 2.15 Hz. The misalignment irregularity produces the highest contact forces. A notable peak occurs close to 2.08 Hz. Again, this maximum is caused due to the proximity of the second frequency mode of the vehicle, which involves the vertical displacement of the axles and the pitch of the traction and body cars. Fig. 12 shows the frequency content and the time-history of the total contact force in the front, middle and rear axles of the vehicle considering the combination of the quasistatic and dynamic contributions detailed above. In both cases, the contact force is clearly higher in the middle and rear axles of the truck. Regarding the frequency content, due to the influence of the pavement misalignment, a notable amplification occurs in the proximity of the second frequency mode of the vehicle 𝑓𝑣2 = 2.15 Hz. In the same way, the peak at 8.93 Hz is produced by the effect of bowing imperfections. It can be concluded that these are the two types of irregularities that affect the most the total response. On the basis of the results from Figs. 11 and 12, it can be deduced that a resonance amplification could occur when the effects of bowing and misalignment are combined. First, the pavement misalignment notably excites the second vertical bouncing frequency of the vehicle (𝑓𝑣2 = 2.15 Hz), as shown in Fig. 11(d). Then, since the length of the pavement frames is 𝐿𝑝= 2.8 m, a resonance speed takes place at 𝑣= 21 km∕h (i.e., 5.83 m/s) because the loading frequency generated at this speed by the bowing irregularities (𝑓=𝑣∕𝐿𝑝= 5.83∕2.8=2.08 Hz) excites the same frequency mode of the vehicle (𝑓𝑣2). Therefore, a notable amplification in the bridge and vehicle responses should be expected when both conditions are met. Next, the dynamic effects at the resonance and the nominal speeds of 21 and 90 km∕h are compared. Fig. 13 presents a comparison of the contact force and acceleration at mid-span for a vehicle travelling at a speed of 90 and 21 km∕h. A combination of pavement bowing and misalignment is considered, as they have the highest impact on the total response of the bridge. As can be seen, the amplitude of the contact force at 21 km∕h is higher than at 90 km∕h, which implies that the contribution of the bowing and misalignment irregularities is also higher at that speed. In Fig. 13(a), the force peaks at 2.15 Hz when 𝑣= 21 km∕h reach maximum values that are much higher than the force at 𝑣= 90 km∕h. Regarding the frequency content of the acceleration in Fig. 13(b), higher accelerations are found Engineering Structures 282 (2023) 115835 9 J. Chordà-Monsonís et al. Fig. 9. Randomly generated unevenness profiles of the pavement roughness, bowing, misalignment and total irregularity cases. Fig. 10. Cumulative modal contribution for the LMB-A120 bridge in terms of (a) displacement and (b) velocity, at mid-span for a vehicle velocity of 90 km/h and considering all irregularities. Engineering Structures 282 (2023) 115835 16 J. Chordà-Monsonís et al. Fig. 21. Comparison of the IM indices calculated from the dynamic analyses and on the basis of several design codes for the (a) LMB-A and (b) LMB-R bridges. AASHTO [24] provides suitable IM values for the LMBs, but at resonance, IM indices increase and the values given by AASHTO [24] are exceeded. In summary, the main reasons for these divergences are (i) the lack of adaptability to the type of bridge of the formulas given in the design codes (in this case, to the long-span modular bridges) and (ii) the fact that resonance and road deterioration are not considered in those expressions. This investigation explores the VBI phenomenon and the dynamic load allowance of long-span modular steel bridges, which is a key aspect in order to develop this technology and to build larger, better and safer structures. In this work, conclusions are drawn from the study of two bridge typologies that represent two different approaches based on the concept of modular construction. However, future investigations are still required to fully understand the dynamic effects of these structures to overcome their limitations and enhance their potential on the global transportation infrastructure. CRediT authorship contribution statement J. Chordà-Monsonís: Conceptualization, Methodology, Software, Validation, Formal analysis, Writing – original draft, Writing – review & editing. E. Moliner: Formal analysis, Writing – original draft, Writing – review & editing. P. Galvín: Conceptualization, Methodology, Formal analysis, Writing – original draft, Writing – review & editing, Funding acquisition. M.D. Martínez-Rodrigo: Formal analysis, Writing – original draft, Writing – review & editing, Funding acquisition. E. Zacchei: Formal analysis, Writing – original draft, Writing – review & editing. A. Tadeu: Conceptualization, Funding acquisition. I. Ferraz: Conceptualization. A. Romero: Conceptualization, Methodology, Software, Validation, Formal analysis, Writing – original draft, Writing – review & editing, Supervision, Funding acquisition. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability Data will be made available on request. Acknowledgements The authors would like to acknowledge the financial support provided by the Spanish Ministry of Science and Innovation under the research project PID2019-109622RB; Generalitat Valenciana, Spain under the research project AICO/2021/200; Junta de Andalucía, Spain and the European Social Fund through the contract USE-22311-R with the Universidad de Sevilla and the Andalusian Scientific Computing Centre (CICA), Spain. This work was also carried out under POCI-01-0247-FEDER-039742 (SELF-Bridges-Long-Span Modular Bridges: Smarter, Extensible, Lighter and Fast Assembly). Project funded by Portugal through COMPETE 2020. References [1] Kim CW, Kawatani M, Kim KP. Three-dimensional dynamic analysis for bridge–vehicle interaction with roadway roughness. Comput Struct 2005;83:1627–45. [2] Li H, Wang T, Wu G. A Bayesian deep learning approach for random vibration analysis of bridges subjected to vehicle dynamic interaction. Mech Syst Signal Process 2022;170:108799. 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