Nonlinear dynamic analysis of Qom Monorail Bridge considering Soil-Pile-Bridge-Train Interaction

Journal Pre-proofs Nonlinear dynamic analysis of Qom Monorail Bridge considering Soil-PileBridge-Train Interaction Mohammad Shamsi, Ali Ghanbari PII: DOI: Reference:

S2214-3912(19)30222-3 https://doi.org/10.1016/j.trgeo.2019.100309 TRGEO 100309

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Transportation Geotechnics

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4 June 2019 11 December 2019 12 December 2019

Please cite this article as: M. Shamsi, A. Ghanbari, Nonlinear dynamic analysis of Qom Monorail Bridge considering Soil-Pile-Bridge-Train Interaction, Transportation Geotechnics (2019), doi: https://doi.org/10.1016/ j.trgeo.2019.100309

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Nonlinear dynamic analysis of Qom Monorail Bridge considering Soil-Pile-Bridge-Train Interaction

Mohammad Shamsi1*, Ali Ghanbari1 1Department

of Civil Engineering, Kharazmi University of Tehran, No. 49 Mofattah Ave. Tehran, Iran

* Corresponding author. Tel: +98-21-88830891; Fax: +98-26-34569555. E-mail addresses: [email protected] (M. Shamsi), [email protected] (A. Ghanbari)

Abstract Railway tracks experience has shown that considering the effects of Soil-Structure Interaction (SSI) especially for soft soil sites is necessary. These effects can also be important for monorail bridges on soft soil beds. The main aim of this study is to investigate the effects of Soil-Pile-Bridge-Train Interaction (SPBTI) on the Qom Monorail Bridge (QMB) responses. In spite of many studies on the effects of monorail train-bridge (fixed-base structure) interaction, very little information is available in the literature on the effects of monorail trains on SPB systems. In this paper, an advanced threedimensional (3D) continuum finite element analysis of QMB subjected to train moving loads was developed. The SPBT models have been validated using three case studies (two bridge-monorail train studies and a soil-pile-structure study) available in the literature. The maximum displacements of the guideway beam at the different train speeds were obtained for various SPBT system conditions. The effects of the stiffness and thickness of the soil, the bridge span length, and the amplitude, length, and geometry of the train loading on the critical speed of the SPBT system are discussed in detail. Finally, the results have been synthesized into simple design charts to select the appropriate straddle-type monorail train and determine its critical speed for a particular soil-bridge condition. In addition, a new simple method for simulating the behavior of the finger-bands (as one of the excitation source of vertical bridge vibrations) in commercial software was presented. The results show that the position of a monorail train bogies plays an important role in determining critical speed so that with uniformity the spacing between the bogies decreases the critical speed in the vertical direction.

Keywords: Monorail transportation system; Soil-Structure Interaction; Finite element modeling; Train moving loads; Finger-band equivalent.

1

1. Introduction With the recent fast development of urban traffic, many developed countries such as China, Japan, India, Iran, Thailand, Malaysia, and Pakistan put the straddle-type monorail transportation system into operation. This new type of urban railway transit traffic is an active technology and a popular solution to the serious traffic jam problem, which has the advantages of comfort, low noise, low costs, short construction period, strong climbing ability, and little influence on the surrounding environment. In contrast with the conventional railway bridges, a limited number of studies have been carried out for straddle monorail transportation systems. In the references [1,2] a coupled dynamic model of the suspension-type monorail vehicle-bridge system was developed based on multi-body dynamics (MBD) and finite element (FE) method, and the coupled dynamic interaction between the train and bridge was systematically investigated based on theoretical and experimental analyses. Whereas the focus of this manuscript is on the SPBTI, but the monorail vehicle and bridge are much simplified. The components of the superstructure of the monorail bridges are usually modeled by beam elements [3–5]. The train is also modeled in different ways, such as the MBD approach [6,7], a dynamical system of the masses, springs, and dashpots (sprung mass model) with different degrees of freedom (DOFs) [4,8–10], and concentrated moving load [11–14]. The numerical results indicate that the moving load model is generally accurate if only the bridge response is desired [15–17]. However, the use of the sprung mass model is necessary whenever the riding comfort of passengers carried by train is of concern [15–18]. Although factors such as the rail irregularity, suspension stiffness, and suspension damping can affect drastically the riding comfort of the train cars, these factors have little influence on the impact response of the bridges [15]. Since the main purpose of this study is to investigate the maximum displacement response of the guideway beam to calculate the critical speed, the moving load can be used instead of the moving sprung mass. Also, similar to some previous studies on the interaction between monorail wheels and guideway beams [2,8,19,20], it is assumed that the running surface of QMB rails is smooth without irregularity. Lee et al. [3] and Wang et al. [8], using numerical modeling, proved that vehicle speed is the main factor that affects the vibration responses of the bridge-vehicle interaction system. Xu et al. [21] studied the vertical coupling dynamics of the monorail vehicle at the finger-band (a bridge connecting different track beams) based 2

on the MBD approach. They simulated the finger-bands as a rigid body (only considering the fingerband geometry and ignoring its deformation) and showed that finger-band is the maximum excitation source of vertical vibration of monorail vehicle. A simple method of describing the rail joint excitation was used by [22–24] in which the defect is represented by sinusoidal profiles that describe the instantaneous bending of the joint in terms of its wavelength and the maximum depth of the defect. Although using a sinusoidal joint profile can be useful for finger-band simulation, the application of this method is very complicated in FE commercial software. Previous studies [25–28] on Soil-Structure Interaction (SSI) showed that considering SSI usually increases structural period and displacement and decreases base shear. Taghavi et al. [29] and Rayhani et al. [30] carried out a series centrifuge test in order to evaluate the effects of SSI in soft clay. They reported that considering SSI especially for low levels of vibration (train loading for example) is necessary. Furthermore, Railway tracks experience has shown that soft ground can increase the rail deflections under a moving train [31–33]. Since the Qom monorail site (the site studied in this paper) is mainly composed of medium to soft soils such as clay or silty clay, the effects of SPBTI can be important. In the continuum numerical models of the train-track-ground (or SPBT) system on the half-space, the finite element region is usually surrounded by infinite elements to simulate unbounded domains and absorb body waves using decay functions [12,34,35]. If the soil is not homogeneous or consists of layers of materials, it is practical to use an equivalent elastic modulus (or reaction modulus) for the infinite boundaries by a matching procedure [12]. This procedure is very time-consuming. In this study, an equivalent depth (Heq) for simulating monorail bridges on the layered or inhomogeneous soils under train loading (at speeds of less than 80 km/h) is estimated. The speed of the monorail train, which creates maximum displacement responses at the center of the guideway beam, is defined as critical speed (CS). Many references have used this definition for the train-bridge systems [15,36] or the train-track-ground [11,31,37,38]. Field-test data [39] and numerical simulations [4] of monorail track beams have shown that the critical speed can be distinguished from the maximum speed allowed for the system operation. It is recognized that the critical speed will always be smaller or equal to the maximum speed of straddle-type monorail trains. This speed is equal to 80 km/h for most trains in the world such as monorail projects of Qom, 3

Chongqing, Kuala Lumpur, Palm Jumeirah, Tokyo, Osaka, Shonan, and Sao Paulo [40–42]. The magnitude of displacement at the guideway beam (induced by the moving loads) and the level of car body vibrations are directly related. The vibration caused by monorail trains can produce a series of problems in bogie systems (i.e. vertical and horizontal air suspensions and rubber pneumatic tires of driving or guiding wheels) failures. For example, at line 3 of the Chongqing straddle monorail transit, there are 20 rubber tires on each wagon, more than 20,000 waste tires accumulate each year [43]. Also, in this line, the average number of failures in air suspensions and bogie systems is more than 1500 times a year [44]. The rapid consumption of rubber tires and bogie systems failures will undoubtedly increase the operation cost and environmental problems. In order to increase the durability of bogie systems and reduce the bridge vibrations, the key parameters of a train such as the geometry of train loading and the number of wagons must be selected appropriately. When the monorail train speed reaches the critical speed of the SPBT system, large vibrations occur, leading to possible damages in bogie systems. To avoid this undesirable scenario, an investigation that aims at determining the influence of various conditions of the SPBT system on the critical speed is essential for geotechnical engineers. In spite of many studies on the effects of monorail train-bridge (fixed-base structure) interaction, very little information is available in the above literature on the critical speed of SPBT systems. In addition, little research [21] has yet been conducted on finger-bands (for monorail bridges) as one of the important components of the bridge and causing large vertical vibrations on the train. Therefore, in this research, an advanced 3D numerical modeling is developed to simulate the dynamic response of QMB under monorail train moving loads, with special reference to the critical speed. This study aimed at the investigation of the SPBTI effects on the critical speed for various conditions. For this purpose, the effects of amplitude, length, and geometry of train moving load on critical speed were investigated. The various conditions of soil and bridge include the soil stiffness, the depth of engineering bedrock, and the bridge span length (Ls). The obtained results are synthesized into simple design charts to select the suitable straddle-type monorail train and determine its critical speed under different conditions. Also, a new simple method for the simulation of the finger-bands behavior in the commercial software was presented. 4

2. Description of the QMB The case chosen for modeling is a monorail bridge project located in the city of Qom, Iran. The first phase of the project which comprises a 6.8 kilometers line with eight stations, linking the northeastern part of the city with the grand mosque in the center of the city. This monorail bridge system typically comprises entirely guideway beams (track beams), decks, piers, caps, and piles. This bridge includes many typical structural modules (Fig. 1). Each module has 5 spans with a length of 20 m and includes a continuous guideway beam resting on two side piers and four middle piers. The pile caps supported by 4 concrete piles and the cross-section of the piles is circular with diameters of 1.2 m. the cross-section of the middle piers is also circular with diameters of 1.6 m. The cross-section of the side piers consists of two discrete columns of semicircles with a diameter of 1.6 m to separate the structural modules. Any of the 100 m-long guideway beams are linked with finger-bands (see Figs. 1 and 2(a)). Each set of finger-band consists of two plates of the same structure with a thickness of about 3 cm. These plates are made of C45E Carbon Steel (elastic modulus of elasticity of E = 200 GPa) that fixed on the base of the concrete with anchor bolts (SadraPol Company, Iran). When the monorail train passes through the finger plate, each finger-band plate group carries the side of the vertical driving wheel. These finger-bands are designed to control the movement (caused by temperature changes) between bridge modules. Based on Qom Urban Railway Organization (QURO), the geometry of the structure in the soil layers (Qom monorail site) is shown in Fig. 2(b). The guideway beam typically consists of a box with a void in the middle in order to reduce the weight of the bridge. The height of the piers and the length of piles were considered typically 9 m and 8 m, respectively. 3. Validation of 3D modeling of the QMB This section presents the validation of the methodology of the modal and dynamic FE analyses. Three case studies have been used to validate the FE models of the current research. These very well documented case studies were considered to develop a robust numerical model. It should be noted that due to the political and military sensitivities of monorail projects in the world, full access to geotechnical reports, geometric details, and measured responses of soil and bridge is almost impossible. Therefore, in order to fully investigate the SPBTI, the validation of the Monorail Bridge5

train Interaction (BTI) model and the Soil-Pile-Structure Interaction (SPSI) model would separately be inevitable. The specifications of the validation problems used in this research are summarized in Table 1. 3.1. Validation problem 1 Lee et al. [45] investigated the dynamic response of the Osaka monorail steel bridge under moving train loads using field-test data and analytical studies. They simulated the fixed-base bridge structure without piles and soil; while in this study, the SPSI effects are also taken into account. The bridge’s span length was 34.8 m (Fig. 3) and the average traveling speed of the Hitachi train (with 144 passengers) was 55 km/h. The bridges were modeled using elastic beam elements with six DOFs at each node and Rayleigh damping was adopted as provided in [45]. The driving wheels of the moving monorail train were modeled as concentrated moving loads using a train dynamic load table available in MIDAS GTS/NX software [46]. The simulation of the guiding wheels was discarded. For a more detailed description of this technique, the reader is referred to [11,46,47]. During the simulation of moving loads, the time step (Δt) was considered in the analysis based on the minimum size of the guideway beam elements (Lmin), the Courant number (Cn), and the monorail train speed (V) following the Courant-Friedrichs-Lewy condition [48] which is represented as Eq. (1).

Cn 

t  V 1 Lmin

(1)

Table 2 shows the natural frequency of the bridge obtained from MIDAS software (the FE software used in this study) compared with the values given by Lee et al. [45]. It can be clearly seen that good agreement exists between two FE predictions. The field measurement results [45], analytical results [45] and the numerical results from the current study are shown in Fig. 3. The numerical results are in good agreement with field data so that there is a difference of less than 8% and 29% between field and numerical results in the vertical and lateral directions, respectively. 3.2. Validation problem 2 A field-test was carried out by Shi et al. [39] to investigate the effects of BTI on dynamic responses of the concrete track beam (Z206-25, with a span of 21.2 m in the straight path) of Chongqing straddle monorail transit system. In this paper, the monorail beam under the Hitachi train moving loads with 6

different speeds of 10, 20, 30, 40, 50, 60, and 65km/h was modeled. Variation of maximum vertical displacement on the middle span of the tested beam under the passing train (with 230 passengers) versus train speed is shown in Fig. 4. It can be seen from this figure that the developed model predicts the span center displacement with very good accuracy compared to the field data (maximum percentage difference is 6%). 3.3. Validation problem 3 Unlike previous validation, the main objective of this verification is to evaluate the nonlinear behavior of layered soils and the effects of SPSI. In most case studies on bridges under moving loads, the effects of soil and pile are ignored, and sensors measure only the response of different points of the bridge. As mentioned earlier, finding a field study that contains structural properties of the bridge, the soil stratigraphy, and the response of piles and soil under moving loads is very difficult. Therefore, in order to investigate the SPSI effects, in this validation problem, the recorded responses of the structure under earthquake motion were used. Although there is a large difference between moving loads and earthquakes in terms of the dynamic loading location, the upward (for earthquake load) or downward (for trainload) propagating seismic waves have no significant effects on verification of the soil nonlinearity and SPSI, especially in low shaking levels. A 3D continuum model of the centrifuge test (named CSP5-A, B, C, D) conducted by Curras et al. [49] was simulated by Midas software in order to capture the nonlinear dynamic SPSI in soft clay. The soil profile, structural model, and instrumentation for the tests are shown in Fig. 5. Details of the centrifuge model and the experimental data are available in [49,50]. Input motions were scaled acceleration time history from Santa Cruz in the 1989 Loma Prieta earthquake (maximum base accelerations of 0.035g, 0.12g, 0.3g, and 0.6g) and all the components of the structure remained elastic during all earthquake events [49]. The HSsmall constitutive model was used to represent the soil nonlinearity. This constitutive model is an isotropic hardening elastoplastic hysteretic model based on the combination of the well-known Hardening Soil (HS) model with the small-strain overlay model. The HSsmall model requires two additional parameters than the HS model: the initial shear modulus G0 and the threshold shear strain γ0.7. One of the main features of this constitutive model is that the shear modulus (shear wave velocity) of the soil varies in depth and the soil is considered as 7

inhomogeneous materials, as expressed by Eq. (2).

G0  G0ref (

c'.cos  ' '3 . sin  ' m 2 )  Vs ref c'.cos  ' p . sin 

(2)

Where G0ref is the reference initial shear modulus at the reference confining pressure pref (often set as the atmospheric pressure, 100 kPa), m is the dimensionless number that depends on the soil type, σ'3 is the minor principal effective stress, Vs is the shear wave velocity, and ρ is the soil density. At higher strain level, expressions similar to the one given in Eq. (2) are used for the three elasticity moduli, e.g. E50, Eoed, and Eur (the secant stiffness in the drained triaxial test, the tangent stiffness for primary oedometer loading condition, and the unloading-reloading modulus, respectively). With increasing strain range, soil stiffness decreases nonlinearly (backbone curves). To reflect this soil behavior, the HSsmall model uses the modified Hardin and Drnevich [51] hyperbolic relationship as the Eq. (3).

Gs 1  G0 1  0.385  /  0.7

(3)

The model parameters employed in this validation problem are discussed. The coefficient of earth pressure at rest (K0nc) for clay was estimated adopting the well-known equations, Eq. (4), [52].

K0

nc

 0.64  0.001PI ,40  PI  80

(4)

Meehan et al. [53,54] calculated the values of shear strength parameters (c and φ) and E50ref for this San Francisco Bay mud by performing a series of Triaxial tests. In the absence of appropriate laboratory tests, the following moduli ratios were adopted: Eoedref = E50ref, and Eurref = 3E50ref [55,56]. A default value of 0.2 was chosen for the unloading-reloading Poisson’s ratio (υur) [46]. Massarsch [57] and Isenhower [58] reported the value of G0 for the Bay mud clay at effective confining stress of σ'3=200 kPa and plasticity index of PI=45 equal to 62 MPa. Therefore, The G0ref value for this soil was obtained from Eq. 2. For soft clays, the value of the parameter m was assumed to be 0.9, as recommended by [46]. The threshold shear strain (γ0.7) for the clay layer was selected based on Eqs. (5)-(7) [59]. In these equations, the mean principal effective stress (σ'm) was calculated in the middle of the clay layer.

Gs  K ( , PI )( ' m ) m ( , PI )  m0 G0

(5)

8

 0.000102  n( PI ) 0.492  ) ] K ( , PI )  0.51  tanh[ln(    0, PI  0  3.37  10 6 PI 1.404 ,0  PI  15  n( PI )   1.976 7 ,15  PI  70 7.0  10 PI 5 1 . 115   , PI  70  2.7  10 PI

(6)

 0.000556 0.4  m( , PI )  0.2721  tanh[ln( ) ] exp(0.0145 PI 1.3 )   

(7)

The parameters for the dense Sandy layer (relative density of 80%) were adopted from the empirical formulas reported by Brinkgreve [60]. The HSsmall model almost provides no hysteretic material damping at very small strains [61], therefore, Rayleigh damping was considered. The coefficients of Rayleigh damping (α and β) can be expressed as Eq. (8).

  2 1 2        1   2  1 

(8)

The natural circular frequencies of the first two modes of the system response (ω1 and ω2) were adopted. The constant viscous damping ratio (ξ) was assumed equal to 5% and 2% for structure and soil elements, respectively. The procedure for selecting boundary conditions, element size, and interfaces, was the same as that will be explained in Section 4.2. Fig. 6 compares the 5% damped Acceleration Response Spectra of the measured and computed motions at the ground surface and the superstructure during an earthquake with maximum base acceleration (amax) of 0.3g (CSP5-C). As shown, a very satisfactory match is observed between the two spectral responses in both locations, confirming the capability of the HSsmall constitutive model in capturing nonlinear SPSI. For low and moderate shaking levels (amax = 0.035g, 0.12g and 0.3g) the maximum difference between the numerical and experimental piles' head moments is about 14%. The vibration of the soil and piles under a moving monorail train is similar to the low earthquake motions so that for both seismic and moving traffic loads piles and structure remain elastic. Therefore, the continuum model can provide a realistic simulation of the wave propagation problem for SPSI with high precision. 4. Developing the continuum model of QMB According to Figs. 1 and 2(b), a typical module of QMB in the straight track without horizontal or 9

vertical curve was simulated (Fig. 7). The 3D continuum model contains 15366 nodes 67743 elements was developed. Four noded tetrahedron solid elements were used to model the soil layers, while three noded triangular shell elements were used to model the caps. Piles, piers, decks, and guideway beams were modeled by Beam elements. The piles were rigidly connected to the caps. The same thing was applied to the connection of the piers to the caps and the decks. 4.1. Calibration of soil constitutive model The HSsmall behavior model is suitable for simulating the dynamic behavior of clays, silts, sands and gravel soils [61], so it can be applied to layered soils such as Qom monorail project site. Since the monorail train loading causes low vibration levels in the SPBT system, the use of the small-strain soil stiffness leads to acceptable results. Fig. 8 shows the soil layers profile at the site. Description of the soil layers and their properties are presented by ZAminFAnavaran Consulting Engineers (ZAFA). The groundwater level is located at the depth of about 29 m below the ground surface. The geotechnical characterization of the soil deposit is based on the laboratory tests and geotechnical field investigations, as reported in Table 3. The values of shear strength parameters at depths of 29 m to 36 m are in undrained conditions. The parameter E50ref and the strength parameters properties of soil layers are obtained from triaxial tests, as illustrated in Fig. 9. The reference initial shear stiffness modulus G0ref and the parameter m are obtained by best fitting the shear wave velocity profile (or small-strain shear modulus in Eq. (2)), provided by the down-hole test, as shown in Fig. 8(b). The threshold shear strain (γ0.7) for clay or silty clay layers is selected based on Eqs. (5)-(7), while for the coarse-grained layers the threshold shear strain is selected in order to backbone curves proposed by Vucetic and Dobry [62]. In the backbone curve or Eq. (5), when the shear modulus reaches down to 70% of its initial (Gs/G0 = 0.7) the parameter γ0.7 can be obtained. The parameter K0nc for granular soils was estimated adopting the well-known Jaky's expression, Eq. (9), while for the other clay layers Eq. (10) was used. nc

(9)

nc

(10)

K 0  1  sin ' K 0  0.4  0.007PI,0  PI  40

The first two modes of the full model with damping ratios of 2% (for the soil) and 5% (for the bridge) have been selected to calculate the Rayleigh damping coefficients. The values of natural periods for the first and second modes of the system are 0.72 and 0.65 in the SSI model, while for the fixed-base 10

bridge these values are 0.53 and 0.39. 4.2. Boundary conditions, element size, and interfaces Soil dimensions are assumed to be 180 × 50 × 36 and the 3D viscous artificial boundary (viscous dampers) at the lateral sides of the FE mesh (see Fig. 7) is applied to effectively control the size of the FE mesh, according to the formulation proposed by Lysmer and Kuhlemeyer [63]. The damping coefficient with respect to the P wave (CP) and S wave (CS) are calculated automatically in MIDAS GTS/NX using Eqs. (11)-(14) [46].

C P  A

C S  A



  2G 

(11)

G

(12)



E

(13)

(1  )(1  2 ) E G 2(1  )

(14)

where ρ is the mass density of the soil, A is the area shared by one damper, E is the modulus of elasticity of the soil, and ν is the Poisson’s ratio of the soil. The base of the model is restrained against the movement in all directions and the time step is chosen based on Eq. (1). The wave generated by the train passage should be able to transmit in the soil elements. To ensure this, the maximum dimensions of soil elements (Lmax) are determined based on Eq. (15) [64].

Lmax 

min 5



Vs min 5 f max

(15)

In the equation, λmin is the wavelength of the maximum input frequency to the soil system (fmax) and Vsmin is the minimum shear wave velocity in each soil layer (obtained from Fig. 8(b), HSsmall curve). As the train speed increases, the frequency generated in the model increases [48]. The maximum frequencies generated in the system caused by the train passage occurs at a speed of 80 km/h and in the fixed-base condition. By extracting the frequency content of the acceleration responses (horizontal and vertical) in the different points of the bridge piers (fixed-base) under moving load (speed of 80 km/h), it turned out that the maximum frequency of these acceleration responses (fmax) was 18.9 Hz. Soil-pile interaction and soil-cap interaction were modeled using interface elements. The properties of 11

the interfaces were linked to the properties of the soil using a unit strength reduction factor (R = 0.8 for soil-concrete interfaces). The interfacial behavior for piles and soil layers can be divided into two normal direction behavior (gap displacement) and one tangent direction behavior (slip displacement). The interface is described with the shear stiffness modulus (K't), ultimate shear force (Qu), and normal stiffness modulus (K'n). The interfaces between piles and each soil layer had assigned to a “virtual thickness” and this value was taken as 0.1. At the end node of the pile, the pile tip element is used, as it interacts with an elastic-perfectly plastic spring. This element presents the relative behavior between the ground elements and the pile node that is defined by the pile end bearing capacity (Qp) and spring constant (Ktip). The soil-caps interface (The normal stiffness and shear stiffness) was defined based on the Coulomb Friction criterion that is recommended by [46]. The values of all the interface parameters were obtained using the equations suggested by [46]. 4.3. Monorail train moving loads In order to investigate the effect of amplitude and geometry of monorail train loading on the critical speed, two different trains manufactured by Hitachi and FCF companies were used in the numerical model. The influence of loading length on the critical speed was also investigated by considering trains with 4 and 6 wagons. Fig. 10 shows a schematic diagram of a typical monorail train. The standard axle loads (F) and the distances between the axles for these trains are summarized in Table 4. 4.4. Simulation of finger-band As mentioned earlier, there is still no suitable method for simulation of finger-band in commercial software. Therefore, in this section, this issue will be discussed. In this simulation method, each finger-band was characterized by the two beam elements with no stiffness and two nonlinear springs (Fig. 11). When the monorail train passed by the finger-band, the driving wheels momentarily lose contact with the guideway beam in the dipped zone. Due to dynamic actions, wheels jump and land on the guideway beam and impact forces at the finger-band are generated (similar to the dipped rail joint [23,24]). In this simulation, the difference in the stiffness of the beam elements with no stiffness and guideway beams causes the impact forces on the finger-band to be similar to what happens in reality. In this method, the nonlinear springs were modeled only in vertical and axial directions (KFB(X) and 12

KFB(Z)), because a high level of deflections is expected on the finger-band in these two directions compared with those the other directions [21]. Although the calculation of the real stiffness for these springs is quite complex, based on some simplifying assumptions the estimation of these parameters will be possible. Fig. 12(a) shows a typical finger-band without any loading condition. When the gap between two plates of the finger-band closes under the compressive axial forces (Fig. 12(b)), the axial resistance of the finger-band is mobilized. This behavior of the finger-band can be assumed as beams and gap (Fig. 12(c)) that axial stiffness changes from 0 to EA/L. the finger-bands under the tensile axial forces have no resistance. Hence, the variation of the axial load with displacement for the nonlinear spring (captures the behavior of the finger-band) in the axial direction (X) can be illustrated in Fig. 12(d). Similar to the axial behavior of the finger-band, its performance in the vertical direction was modeled by a nonlinear spring. When a vertical load is applied to the finger-band, first, one of the plats with the stiffness of 3EI/L3 deforms. Then, with increasing vertical load and full interlocking of the two plates (Fig. 12(e)), the stiffness increases to 6EI/L3 (Fig. 12(f)). This behavior is conditioned on the joint performance of the plates (Fig. 12(g)), while this may not occur for technical and executive reasons. Therefore, in a conservative manner, the force-displacement diagram of the vertical nonlinear spring is considered as a Fig. 12(h). This modeling method has two main advantages to the sinusoidal joint profile approach: this technique can be easily implemented in any commercial structural analysis software, and it can be easily applied to engineers with minimum computational time requirements. The damping coefficient of the finger-bands is out of the range considered in this paper because the experimental data would be required to ensure the accuracy of results. As a matter of fact, no experimental results are available to qualitatively confirm or falsify the numerical evidence discussed above with reference to finger-bands behavior. To evaluate the general performance of the proposed method in the context of a comparative study, two fixed-base structural modules of the bridge under train loading are modeled and two guideway beams with 100 m length are linked with finger-bands (Fig. 11). 4.5. Analysis procedure The analysis of the QMB is carried out in three calculation phases. In the first phase, the FE model excludes any structural elements, and the weight of the soil is applied to all solid elements by staged 13

construction analysis. Then the structural components of the bridge are placed, and static nonlinear analyze is performed in the second phase. In the third phase, the train dynamic load was applied to one of the guideway beams. For the soil layers above the groundwater level, the dynamic analyses were carried out under the assumption of drained conditions, while for the saturated soil layers fully undrained conditions were considered. 5. Results and discussion The fundamental mechanism of increasing displacement in some speed levels is that the excitation frequencies produced by the moving load are closer to the natural frequencies and mode vibrations of the SPB system. For example, the following discussion is presented to prove this issue for QMB. When a simply-supported beam (as a bridge) is subjected to a series of moving loads (as a train with multiple wagons) with speed v, Mao and Lu [36] presented the resonance severity indicator, called the Z factor, for the assessment of the critical speed at different potential resonance speeds (vre). The Z factor represents the rate of increase of the bridge response amplitudes and is only dependent on the length ratio between the beam and the wagon (R = Lsb/Lw) [36] as stated in Eqs. (16) and (17).

Z [

2nR ] 2  [1  cos(2nR)] 2 (2nR)  1

v re ( m / s ) 

(16)

f1 L w n

(17)

where n denotes the mode number of the apparent trainload excitation frequencies and f1 is the first natural frequency of the beam in the Z direction. For a more detailed description of these formulas, the reader is referred to [36]. The vertical first-order natural frequency of the fixed-base QMB with a span length of 20 m (characteristic length of Lsb ≈ 19 m) is f1 = 2.91 Hz. Fig. 13 (a,b) shows the Z value at the corresponding potential resonant speeds with respect to the nth mode of trainload excitation frequencies. It should be noted that for n = 1, the vre values for both trains is more than 80 km/h (maximum speed of the straddle-type monorail trains). Since the monorail trains cannot experience these speeds, these values are ignored. From the Fig. 13 (a), for the Hitachi train, the third apparent trainload excitation frequency (n = 3) coincides with the first natural frequency of the QMB in the vertical direction (f1). Therefore, the most severe QMB resonance has occurred at a speed of 51.7

14

km/h. Despite the asymmetric loading on the QMB (only one of the guideway beams is loaded), the critical speed of the fixed-base QMB from the numerical results of this study are in good agreement with the resonance speed values calculated by Eq. (17). The same is true for the FCF train. Finding all the resonant criteria and relationships between critical speed and natural frequencies for various conditions of SPBT systems is very complicated and is out of the range considered in this paper. The main reasons for this are (i) the significant effect of the soil modes on the vertical modes of guideway beams, (ii) the soil inhomogeneity in the HSsmall model, and (iii) the lack of superposition principle. 5.1. Effect of geometry, amplitude, and length of train loading The effect of the geometry of the train loading regime on the critical speed (CS) is important for monorail transport authorities since it can guide to the choice of a suitable train for a particular soilbridge condition. Fig. 14 shows the maximum displacement at the center of the guideway beam (Node A) versus train speed, for both trains with two different lengths (i.e. trains with 4 and 6 wagons). This figure clearly demonstrates that both the maximum displacements and corresponding critical speed for the two trains are different and the critical speed is actually affected by the geometry of train loading. When the distance between the train bogies becomes more uniform (approaching La to Lc), the vertical displacements at Node A are reduced and its lateral displacements become almost constant. The critical speed values for the FCF train are smaller than the Hitachi train. Also, the critical speed in the lateral direction is always more than the vertical one, regardless of the train length and train type. This shows that horizontal air suspensions and guiding wheels are more damaged at higher speed levels because the magnitude of guideway beam displacements and the level of car body vibrations have a direct relationship. The increase in the number of wagons from 4 to 6 causes an increase of 2 to 7 percent in the lateral displacements at the center of the guideway beam, while this has no significant effect on vertical displacements. The influence of amplitude of Hitachi train loading (passenger loading) on the critical speed was investigated using two different standard axle load values e.g. 102 kN and 88 kN. The relationship between the displacement at Node A and train speed for the two loading amplitudes considered is shown in Fig. 15. As expected, the displacements increase with increasing the loading amplitude for all speeds. The critical speed and the trend behavior of the

15

displacement responses are not significantly affected by the magnitude of train loading. It should be noted that all results obtained in Section 5.1 (effect of geometry, amplitude, and length of train loading on displacement responses and critical speed) for both situations of with and without SSI are approximately the same. 5.2. Comparison between fixed-base and SSI models Fig. 16 shows the maximum displacement at Node A versus train speed for both fixed-base and SSI (bridge located on the Qom monorail site) models under both trains moving loads with 4 wagons. As expected, SSI responses in vertical and lateral directions are always more than fixed-base ones. The figure also has shown that soil modeling for the Qom monorail bridge under train load is necessary so that the SSI model responses in vertical and lateral directions are about 25% and 60% more than the responses of the fixed-base model. Considering SSI usually can change the critical speed. As the propagation wave velocity of any soil medium is highly dependent on its stiffness and depth of the engineering bedrock, a series of numerical analyses were performed to study the effects of the soil stiffness and soil thickness (Hs) on the critical speed of SPBT system. To investigate the impacts of the soil stiffness, a uniform soil layer with five different values of reference shear wave velocity (V's = 170, 250, 365, 465, and 510 m/s) was modeled. These values cover the range of soft to stiff soils. The properties of all soil materials that were extracted from the literature [65–68] are summarized in Table 5. Similarly, the effects of the depth of the engineering bedrock were investigated by considering a uniform soil layer with four different thicknesses (Hs = 15, 25, 35, and 45 m). The impacts of soil stiffness and soil thickness are presented in Fig. 17 in terms of the evolution of the maximum displacements of Node A with train speed. As shown in Fig. 17(a,b), with increasing soil stiffness, the response of the Node A decreases and the change in the critical speed is not clear. As expected, the impacts of SSI on increasing bridge responses are higher in soft soils. Fig. 17(c,d) illustrated the envelope of axial forces in the pile group (1) for FCF train in 30 km/h (critical speed in the vertical direction) and 40 km/h. as expected, the maximum axial force in the piles for the critical speed of the train is about 5% more than 40 km/h one. It can be seen from Fig. 17(e,f) that the maximum displacement (vertical and horizontal) at the center of the guideway beam generally increases with the increase of soil thickness, so that in the thicknesses of about 25 meters, the 16

responses are insensitive to the increase of Hs. The critical speed is almost the same regardless of the soil thickness. These trends are also valid for other situations (a uniform soil layer with different shear wave velocities). Therefore, for monorail bridges on the layered or inhomogeneous soils under train loading (at speeds of less than 80 km/h), the soil thickness can be assumed to be about Heq = 25 m and ignored the lower layers. In other words, the responses obtained from this condition are in good agreement with layered or inhomogeneous half-space responses. 5.3. Effect of bridge span length The longest span of the guideway beams in the Qom monorail transportation system is 25 m. In order to evaluate the impacts of bridge span length (Ls) on the critical speed of the SPBT system, four different values of Ls (17.5, 20, 22.5, and 25 m; e.g. structural module lengths of 87.5, 100, 112.5 and 125 m) were considered in the analysis. In all models, the soil was assumed to have a width of 50 meters and a length of 1.8 times the module lengths. The results of changes in the length of the bridge span are shown in Fig. 18. It can be seen in this figure that as span length increases, displacement responses of Node A increases, but the critical speed variations are not regular. 5.4. The response of finger-band under train moving load Fig. 19(a) shows the vertical acceleration response of the finger-band (on the traveling path) under FCF train loads with a speed of 40 km/h. Impact forces on the nonlinear spring (finger-band) caused by each of the monorail train wheels generate impulse excitations with high frequency (predominant frequency of 556 Hz) and high amplitude. This result is consistent with the numerical predictions of the dipped rail joint reported by [23,24]. As shown in Fig. 19(b), the maximum axial displacement of the finger-band is 1.9 mm that due to the 2.5 cm spacing between the plates, the nonlinear axial springs are practically non-effective (the gap between plats remains open). The maximum vertical displacements of the finger-band for the different train speeds are listed in Table 6. These values are slightly conservative because the damping coefficient of the finger-bands has been neglected. In contrast with Node A, in this node, with the increase of speed displacement responses are increased (due to increasing the effects of impact forces). The critical speed for both trains at the finger-band (in the vertical direction) is 80 km/h. therefore, when the train passes on the finger-band at high speed, the vertical air suspensions, driving wheels, and the finger-band are more damaged. 17

5.5. Development of design charts for calculation of critical speed As mentioned earlier, the critical speed is highly dependent on bridge span length and the soil stiffness. Therefore a parametric study has been carried out to determine the critical speed for any train with 4 wagons, axle load of 88 kN, and particular loading geometry. Design charts (Fig. 20) are presented using the results obtained from previous Sections (5.1 to 5.3). Using these charts the critical speed of three monorail trains for different soil stiffnesses and span lengths can be directly obtained. The critical speed can be useful in designing the air suspensions of the trains. It is worth noting that the number of wagons and the axle load of the train does not affect the critical speed values, as mentioned in Section 5.1. Hence, these charts apply to any type of straddle monorail train with any amplitude of loading and number of optional wagons. The critical speed can be estimated from Eq. (18) and Table 7 as:

G 0ref H s Ls (18) ,V ' s   V 's Where λ is the project condition parameter. This equation is applicable to a wide range of soil-bridgeCS  a.(  ) b ,  

train conditions; e.g. 17.5m
18

suspensions for the monorail trains in the horizontal direction is more than the vertical direction, this solution can be useful in reducing the environmental problems of solid waste. 6. Conclusions In spite of many studies on the effect of monorail train-bridge (fixed-base) interaction, very limited studies exist on the effect of SPBTI. In this paper, a 3D numerical approach was used to investigate the dynamic response of QMB under train moving loads, with special reference to the critical speed of the SPBT system. A parametric study was then carried out to investigate the effect of the train geometry, the soil stiffness, and the bridge span length on the maximum displacements of the guideway beam induced by the moving train at different speeds. Also, a new simple model of the finger-band was presented. The following conclusions can be drawn from this numerical study: 1. The critical speed of the SPBT system (corresponding to the maximum displacements of the guideway beam in the vertical or lateral direction) is independent of the train loading length and amplitude; however, it is significantly influenced by the monorail train loading geometry. 2. The bogies position for a particular train plays an important role in determining critical speed so that in the vertical direction the critical speed decreases by approaching La to Lc (i.e. the more uniform spacing between the train bogies). In the lateral direction, when the parameter λ is smaller than 2.82, the critical speed decreases by approaching La to Lc, while for values higher than 2.82 contradicting results were found. 3. The simulation of soil in numerical models of monorail bridges under train load is necessary, especially for soft soil sites. Considering SSI usually can change the critical speed. The maximum displacement (vertical and horizontal) at the center of the guideway beam generally increases with the increase of Hs, so that in the thicknesses of about Heq = 25 meters, the responses are not affected by the increase of Hs. 4. The soil stiffness and the bridge span length were found to have significant influences on the critical speed of the SPBT system. Changes in critical speed are irregular in relation to these two factors. The effect of soil thickness on the critical speed is almost negligible. 5. The proposed method for the simulation of the finger-bands can be easily implemented in

19

commercial software with minimum computational time requirements. Although this model captures the behavior of the finger-bands, the obtained responses are slightly conservative because the damping coefficient of the finger-bands has been neglected. This coefficient can easily be obtained using some experiments. 6. The design charts and a simple relationship for estimation of critical speed of SPBT systems are applicable to any straddle-type monorail trains (with different bogies positions, any amplitude of loading and number of optional wagons) and a wide range of soils with different stiffness and thickness.

Acknowledgment The authors are grateful to Qom Urban Railway Organization (QURO), SadraPol and FCF companies, ZAFA Consulting Engineers, and the Keyson-Mapna consortium of contractors for providing general specifications and practical information of the train, bridge, and soil.

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List of Tables Table 1. Specifications of validation problems used in this research Table 2. Results of natural frequency Table 3. HSsmall model parameters for the soil layers at the Qom monorail site Table 4. Geometry and axle loads of monorail trains used in the FE simulation Table 5. Properties used to investigate the impact of soil stiffness on the critical speed Table 6. Maximum vertical displacement of the finger-band (Ls = 20 m) Table 7. The constants for calculation of critical speed

24

Table 1 Specifications of validation problems used in this research Validation Validation problem target

1

2

3

BTI

BTI

SPSI

Description

Limitation

Verification of the time displacement relationship, and the trend of the vertical and horizontal displacements at the center of the guideway beam under monorail train loading Verification of the trend of the maximum vertical displacements at the center of the guideway beam under train loading with different speeds Verification of the soil (soft clay) nonlinearity, piles’ head moment, and superstructure response for different amplitudes of input motion in the horizontal direction using centrifuge tests

Lack of evaluation of the train with different speeds and passenger loading

25

Validation results

The developed model is able to predict the displacement responses at the center of the guideway beam for all of the considered BTI conditions.

Disregarding soil effects

Earthquake loading instead of train loading

The HSsmall constitutive model can provide a realistic simulation of the wave propagation problem and the developed model is able to predict the nonlinear dynamic responses for all of the considered SPSI conditions.

Table 2 Results of natural frequency Natural frequency (Hz) 1 2 3 4 5 Lee et al. (2006) 2.806 2.982 5.096 5.267 8.091 MIDAS software 2.859 3.302 5.412 5.821 8.513

26

Table 3 HSsmall model parameters for the soil layers at the Qom monorail site

Parameters γ (kN/m3) G0ref (MPa) γ0.7 (%) E50ref (MPa) Eoedref (MPa) Eurref (MPa) υur m K0nc c (kPa) φ (°) ψ (°)

Silty clay 0-4 m 19.87 350 0.03 3.7 3.7 11.1 0.2 0.9 0.44 13 27.6 0

Clay 4-8 m 20.1 530 0.07 7.4 7.4 22.2 0.2 0.9 0.55 57 28.6 0

Sandy gravel 8-13 m 20 440 0.01 15 15 45 0.3 0.5 0.43 1 33 3

Clay 13-24 m

Clay 24-29 m

19.8 640 0.09 5 5 15 0.2 0.2 0.55 10 20.1 0

20.1 480 0.11 12 12 36 0.2 0.5 0.46 15 24 0

27

Sandy gravel occasionally with silt 29-33 m 20.3 500 0.01 17 17 51 0.3 0.5 0.47 19 35.3 5.3

Clay 33-36 m 20.6 530 0.12 12.8 12.8 38.4 0.2 0.5 0.49 17 32.2 0

Table 4 Geometry and axle loads of monorail trains used in the FE simulation Company Hitachi, Japan FCF, Italy city Chongqing Qom La (m) 8.1 5.5 Lb (m) 1.5 1.5 Lc (m) 3.5 5 axle load in full passenger condition (kN) 102 88 number of wagons 4 or 6 4 or 6 loading length (m) 54.9 or 84.1 49 or 76

28

Table 5 Properties used to investigate the impact of soil stiffness on the critical speed Parameters 𝑉′𝑠 = 𝐺𝑟𝑒𝑓 0 /𝜌(m/s) E50ref (MPa) c (kPa) φ (°)

Taha et al. (2015) 170 4.91 5 25

Rampello et al. (2019) 250 5.52 28 27

29

Akdag (2016) 365 30 10 18

Mathew & Lehane (2014) 465 22.8 20 35

This study 510 7.4 57 28.6

Table 6 Maximum vertical displacement of the finger-band (Ls = 20 m) Monorail Train Speed (km/h) 10 20 30 40 50 60 70 80 Hitachi Train 9.49 9.97 10.08 10.74 10.99 11.66 12.03 12.31 FCF Train 9.31 9.59 9.95 10.11 10.73 11.49 11.98 12.23

30

Table 7 The constants for calculation of critical speed a Vertical direction b a Lateral direction b

La/Lc= 1.1 La/Lc= 1.7 La/Lc= 2.3 33.85 37.24 38.43 0.19 0.17 0.16 60.27 62.78 64.69 0.087 0.052 0.019

31

List of Figures Fig. 1. A typical module of the monorail bridge system Fig. 2. (a) Photograph of QMB and finger-band on a side pier; (b) The cross-section of the QMB Fig. 3. FE model of validation problem 1, and displacement of the span center of the track girder Fig. 4. FE model versus field-test; maximum vertical displacement of track beam in various train speeds Fig. 5. Specifications of centrifuge models (prototype scale) Fig. 6. Comparison between the results of experimental and simulated SPSI models Fig. 7. The developed finite element continuum model of the QMB Fig. 8. Local soil profile at the Qom monorail site: (a) soil stratigraphy; (b) shear wave velocity profile from the down-hole test Fig. 9. Stress-strain curves of the triaxial tests for soil layers under confining pressures of 100, 200, and 400 kPa Fig. 10. A typical monorail train specifications (half of the train is shown due to the symmetry) Fig. 11. Two fixed-base structural modules of the QMB and finger-band Fig. 12. A simple representation of the axial and the vertical behavior of the finger-band: (a) Axial, before loading; (b) Axial, after gapping displacement; (c) Axial, as a beam; (d) Axial, nonlinear spring as the finger-band; (e) Vertical, after gapping displacement; (f) Vertical, as a beam; (g) Vertical, nonlinear spring as the finger-band; (h) Vertical, nonlinear spring in a conservative manner Fig. 13. Comparison between the critical speeds of the fixed-base QMB: (a) Hitachi, obtained from Eqs. (16)-(17); (b) FCF, obtained from Eqs. (16)-(17); (c) Hitachi, derived from numerical modeling; (d) FCF, derived from numerical modeling Fig. 14. Evolution of maximum displacement at the center of the guideway beam (Ls=20 m) with train speed for trains of different geometry: (a) vertical displacement; (b) lateral displacement Fig. 15. Evolution of maximum displacement at the center of the guideway beam (Ls=20 m) with train speed for two loading amplitudes

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Fig. 16. Evolution of maximum displacements at the center of the guideway beam with train speed for both trains in situations with SSI (Qom monorail site) and fixed-base bridge (Ls=20 m): (a) vertical displacement; (b) lateral displacement Fig. 17. Evolution of maximum displacements at the center of the guideway beam (Ls=20 m) with train speed for FCF train (4 wagons, F = 88 kN) in a situation with a soil layer: (a) vertical, different Vs; (b) lateral, different Vs; (c) axial force envelope, 20 km/h; (d) axial force envelope, 30 km/h; (e) vertical, different Hs; (f) lateral, different Hs Fig. 18. Evolution of maximum vertical displacements at the center of the guideway beam with train speed for different bridge span lengths and trains (4 wagons, F = 88 kN) in a situation with a soil layer: (a) FCF train; (b) Hitachi train Fig. 19. Responses of the finger-band under FCF train loads (F = 88 kN) with speed of 40 km/h: (a) vertical acceleration; (b) vertical and axial displacement Fig. 20. Design charts to calculate the critical speed of any type of monorail train for different ground and bridge span conditions: (a) vertical train vibrations; (b) lateral train vibrations

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Fig. 1. A typical module of the monorail bridge system

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Fig. 2. (a) Photograph of QMB and finger-band on a side pier; (b) The cross-section of the QMB

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Fig. 3. FE model of validation problem 1, and displacement of the span center of the track girder

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Fig. 4. FE model versus field-test; maximum vertical displacement of track beam in various train speeds

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Fig. 5. Specifications of centrifuge models (prototype scale)

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Fig. 6. Comparison between the results of experimental and simulated SPSI models

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Fig. 7. The developed finite element continuum model of the QMB

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Fig. 8. Local soil profile at the Qom monorail site: (a) soil stratigraphy; (b) shear wave velocity profile from the down-hole test

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Fig. 9. Stress-strain curves of the triaxial tests for soil layers under confining pressures of 100, 200, and 400 kPa

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Fig. 10. A typical monorail train specifications (half of the train is shown due to the symmetry)

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Fig. 11. Two fixed-base structural modules of the QMB and finger-band

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Fig. 12. A simple representation of the axial and the vertical behavior of the finger-band: (a) Axial, before loading; (b) Axial, after gapping displacement; (c) Axial, as a beam; (d) Axial, nonlinear spring as the finger-band; (e) Vertical, after gapping displacement; (f) Vertical, as a beam; (g) Vertical, nonlinear spring as the finger-band; (h) Vertical, nonlinear spring in a conservative manner

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Fig. 13. Comparison between the critical speeds of the fixed-base QMB: (a) Hitachi, obtained from Eqs. (16)-(17); (b) FCF, obtained from Eqs. (16)-(17); (c) Hitachi, derived from numerical modeling; (d) FCF, derived from numerical modeling

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Fig. 14. Evolution of maximum displacement at the center of the guideway beam (Ls=20 m) with train speed for trains of different geometry: (a) vertical displacement; (b) lateral displacement

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Fig. 15. Evolution of maximum displacement at the center of the guideway beam (Ls=20 m) with train speed for two loading amplitudes

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Fig. 16. Evolution of maximum displacements at the center of the guideway beam with train speed for both trains in situations with SSI (Qom monorail site) and fixed-base bridge (Ls=20 m): (a) vertical displacement; (b) lateral displacement

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Fig. 17. Evolution of maximum displacements at the center of the guideway beam (Ls=20 m) with train speed for FCF train (4 wagons, F = 88 kN) in a situation with a soil layer: (a) vertical, different Vs; (b) lateral, different Vs; (c) axial force envelope, 20 km/h; (d) axial force envelope, 30 km/h; (e) vertical, different Hs; (f) lateral, different Hs

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Fig. 18. Evolution of maximum vertical displacements at the center of the guideway beam with train speed for different bridge span lengths and trains (4 wagons, F = 88 kN) in a situation with a soil layer: (a) FCF train; (b) Hitachi train

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Fig. 19. Responses of the finger-band under FCF train loads (F = 88 kN) with speed of 40 km/h: (a) vertical acceleration; (b) vertical and axial displacement

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Fig. 20. Design charts to calculate the critical speed of any type of monorail train for different ground and bridge span conditions: (a) vertical train vibrations; (b) lateral train vibrations

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Author contribution statement 

Mohammad Shamsi: Software, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing - Original Draft, Writing - Review & Editing, Visualization



Ali Ghanbari: Conceptualization, Methodology, Supervision, Project administration

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