The bridge response under a moving load can be solved using the classical Euler–Bernoulli beam theory, represented by Eq. 1, for a single moving load, P, traversing a beam (Svedholm, 2017). ∑ i = 1 ∞   E I ∂ 4 ϕ i x ∂ x 4 q i t + μ   ∂ 2 q i t ∂ t 2 ϕ i x + c ∂ q i t ∂ t ϕ i x = δ x − v t P , (1)

E I Flexural rigidity of the beam with a constant moment of inertia,

Eq. 1 describes the motion of a beam with a flexural rigidity, EI, and a uniformly distributed mass, μ , along which the force, P , moves at constant velocity, v . The parameters on the right-hand side of Eq. 1 represent the motion of the constant force, which is described by the Dirac function δ x (Yang et al., 2004b). The function of the vertical deflection of the beam, y x , t , can be expressed as a product of two functions, the mode shape function (Eigen-function) ϕ i x , and the function of the generalized coordinates q i t . Eq. 1 forms the basis of the moving load model, which is then extended for a series of moving loads to the form given by Eq. 2 (Frýba, 1999; Yang et al., 2004a). E I   ∂ 4 y x , t ∂ x 4 + μ   ∂ 2 y x , t ∂ t 2 + 2   μ   ω d ∂ y x , t ∂ t = ∑ n = 1 N ε n t   δ   x − x n   F n , (2)

y x , t Vertical deflection of the bridge at position x and time t,

ω d Circular damped frequency of the bridge,

ε n t Describes the Heaviside unit step function for the arrival (turning on) and departure (turning off) of the n ^th axle force, F _n ,

F n Constant magnitude concentrated axle force,

X n Position of the n ^th axle force, F _n , from the first axle,

v Train constant speed,

x n = v t −   X n Position of the n ^th axle force, F _n , from the bridge origin.

Eq. 2 is solved using the fundamental relationships of the Fourier sine integral transformation that presents the problem in the frequency domain. The Laplace–Carson integral transformation method is then applied to present the problem in the complex domain. The inversion of the Laplace–Carson transform presents the problem in real space, and the Fourier transform is then used to reduce the equation to the time domain. This method enables an analytical closed-form solution of Eq. 2. This method has been applied by Frýba (1999) to provide the closed-form solution, enabling the calculation of the vertical deflections of the bridge at any specific location x along the bridge as a function of time as given by Eq. 3. The equation expresses the forced vibration of the bridge due to the moving loads and the free transient damped vibrations after the train has left the bridge. The acceleration response of the bridge can be obtained by the double differentiation of Eq. 3. y x , t = ∑ j = 1 ∞ ∑ n = 1 N y 0 F n j ω ω 1 2 f t − t n H t − t n − − 1 j f t − T n H t − T n sin j π x , L (3)

y 0 Unit load deflection,

L Bridge span,

j j^th modal frequency (j = 1 for first vertical bending mode),

ω Forcing frequency,

ω 1 Circular natural frequency of vibration of the bridge (first vertical bending mode)

The solution is facilitated by introducing an incremental distance, L i n c , which must be a factor of the bridge span and train axle spacing and coupling distance dimensions. This value is based on the rightmost non-zero significant figure of any of the dimensions, and L i n c must also be an integer ≥ 1 . With the first axle position on the bridge at time t = 0 , the complete train set length with the first axle as the origin is given by X n = k , which is the position of the last axle, k , from the first. Dummy axle loads, F d , 1 → F d , n , are introduced between each axle with zero values. These are positioned using the increment distance L i n c to provide a series of equally spaced loads, F 1 → F k , according to Figure 1. With reference to Figure 1, the time when the n^th axle force, F n , enters the bridge is given by Eq. 4, whereas the time when it leaves the bridge is given by Eq. 5. t n = X n v , (4) T n = L + X n v , (5)

For the closed-form solution, Eq. 6 describes the Heaviside unit step function, H t , for the arrival (turning on) and departure (turning off) of the n^th axle force, F n , and their time shifts t − t n and t − T n , respectively. ε n t = H t − t n − H t − T n   H t =   0 , t < 0 1 , t ≥ 0 . (6)

The parameter f in Eq. 3 is a function of the inverse Laplace–Carson transformation, as given by Eq. 7. The first term expresses the response of the bridge due to the moving loads, and the second term is the transient response. The subscript j is the jth mode of vibration, where in this case, as we are only concerned with the bridge’s fundamental vertical bending mode of vibration, j = 1. f t = 1 ω j ′   D   ω j ′ j   ω   sin j ω t + θ + e − ω d   t   sin ω j ′   t + φ   . (7)

The parameters of Eq. 7 are defined as follows: ω j ′ = ω j 2 − ω d 2 , (8) D =   ω j 2 − j 2 ω 2 2 + 4 j 2 ω 2 ω d 2 , (9) θ = ⁡ tan − 1 − 2 j ω ω d ω j 2 − j 2 ω 2 , (10) φ = ⁡ tan − 1 2 ω d ω j ′ ω d 2 − ω ′ j 2 + j 2 ω 2 , (11) ω d = f 1 ϑ . (12)

where ϑ is the logarithmic decrement of damping given by Eq. 13 (Frýba, 1999), and f 1 is the bridge’s fundamental vertical bending mode of vibration. ϑ = 1 0.3 L − 0.0012 L 2 . (13)

Further details on the parameters and derivation of the equations can be found in Frýba (1999). The model equations were implemented and solved within MATLAB.

The mid-span bridge deflections are calculated using classical beam equations and employing the principle of superposition. Using this type of analysis, a DAF can be calculated using relevant codes of practice to account for the dynamic effects on deflections, bending moments, and stresses. This type of assessment is generally referred to as a quasi-static (Q-Static) analysis and cannot account for dynamic effects, such as a resonance response of the bridge, or the change in frequency of the bridge due to additional mass imposed on the bridge due to the train. However, as is shown in this article, the method can be used to obtain a general equation that can be used to predict the change in the frequency due to the train as a function of the unladen bridge resonant frequency and the train–bridge mass ratio. The method is also used to calculate a frequency reduction factor for each train–bridge configuration, which is then implemented in the EBB dynamic model to account for the vertical natural frequency reduction due to the mass of the train.

The quasi-static solution is facilitated by dividing the bridge span into equal increments based on the calculated L i n c value, as discussed previously. Starting from position zero, the total number of bridge increments, N b r , is given using Eq. 14. N b r = 1 + L L i n c . (14)

Similarly, the total number of axles, k , including dummy axle loads, is given by Eq. 15, where X T r a i n is the length of the train from the first to the last axle. k = 1 + X T r a i n L i n c . (15)

The deflection influence curve of the bridge at the mid-span, a = 0.5 L , is calculated using Eq 16, where x is the position of the moving force given by x = 0 : L i n c : L , and P is a unit load, which in this case is equal to 1 ton (9806.65 N). D b r = P a L − x 2 L x − x 2 − a 2 6 E I L   x ≥ a P x L − a L 2 − L − a 2 − x 2 6 E I L   x < a . (16)

For the series of axle loads crossing the bridge, the mid-span displacement is given by Eq. 17 using the principle of superposition. δ Q S t a t i c = ∑ j = 1 k   ∑ j + 1 j + N b r F j D b r . (17)

To check the validity of the EBB dynamic and the quasi-static models, both models were run using two typical trains (S-T1 and EMU-T2) on two case study bridges (Bridges 1 and 2, respectively, see Table 1), which represent typical short- to medium-span railway bridges found on the UK rail network. The speed of the EBB dynamic model was set to 1 km/h to minimize any dynamic effects. The displacement response results are presented in Supplementary Figure S1, which shows that both models produce the same response results.

Because the case study bridges can be reasonably represented as uniform beams for predicting the fundamental vertical bending mode of vibration, a finite element analysis was performed to predict the change in frequency as a result of the additional mass imposed on the beam due to passing trains. Supplementary Figure S3 shows the FE models of each Bridges 1–6, which were developed using NX Siemens, represented as simply supported uniform beams with the flexural stiffness and mass properties as given in Table 1. Nodes were created to represent the axle spacings of train S-T1 with concentrated mass (CM) elements to which mass values were assigned. Train S-T1 represents Steel Train 1 as defined in BS-5400 (1980). The positions of the nodes were determined based on the maximum number of axles that could be positioned on the bridge to give a worst-case loading. Different positions of the train axles over the bridge simulating the train traveling across the bridge were considered, and the worst-case scenario was found to be when placing the train axles symmetrically about the mid-span of each beam.

The mass values represent axle masses that were incrementally changed from zero (unloaded beam) to a maximum value of 50,000 kg to study different M _w /M _b ratios. For each bridge model, eigenvalue analysis was performed to estimate the fundamental mode of vibration for each case. Table 2 shows the frequencies for the unloaded bridge and for an axle mass of 25,000 kg. The results in Table 2 are subsequently used to verify the frequency changes obtained using the generalized equations presented in this work.

For this study, the standard set of trains defined in BS-5400 (1980), shown in Supplementary Figure S4 and Supplementary Table S1, are included within the model, enabling the selection of any one type of train for analysis. Supplementary Table S1 also includes an additional hypothetical train with equally spaced axles, ESA-10. The train dimensions are as given in Supplementary Figure S4, and those that are used for the calculation of the wagon pass frequencies are in accordance with the train axle spacings as illustrated in Supplementary Figure S5 and Supplementary Table S2.

The assessment of bridge frequency considering train mass is performed following the steps outlined in the flow chart shown in Figure 2. The first step uses a quasi-static analysis to determine the mid-span displacement response for each train–bridge combination. The displacements are then used to calculate the mean bridge frequency, which represents the first vertical bending mode of the loaded bridge. For a simply supported beam, where the loading is mainly distributed uniformly, and the bridge is subjected to bending only, the fundament vertical bending mode natural frequency can be estimated using Eq. 12 (NR/GN/CIV/025, 2006) and EN1992-2, 2003. f b = 17.75   δ o   δ o = m i d − s p a n   d e f l e c t i o n   d u e   t o   p e r m a n e n t   a c t i o n s , m m .   (18)

In this quasi-static assessment, the time history of mid-span deflection due to the moving train loads is accounted for by considering an additional permanent action leading to a total mid-span deflection of δ o + y t . This enables the bridge’s fundamental vertical resonant frequency change to be obtained as a function of time. The effective bridge vertical bending mode resonant frequency is then taken as the mean frequency. Eq. 18 is therefore rewritten as follows: f b , t = 17.75   δ o + y t     y t = m i d − s p a n   d e f l e c t i o n   a s   a   f u n c t i o n   o f   t i m e , m m . (19)

It should be noted that using the lowest frequency would lead to the most undesirable scenario, resulting in the highest deflection. However, it is deemed that the mean frequency value is more reasonable to use to avoid overly conservative scenarios/cases. In a similar study, Mao and Lu (2013) also used a mean frequency value and specified this as being the effective natural frequency.

To account for the additional mass of the train on the bridge, the equivalent mass, M w , t , is calculated as a function of mid-span deflection of y t , Eq. 20. This is then used to calculate the mass ratio, M _w /M _b , for the given bridge–train configurations. M w , t = 48 E I y t g L 3   g = G r a v i t a t i o n a l   a c c e l e r a t i o n   m s 2 y t = m i d − s p a n   d e f l e c t i o n   a t   t i m e   t   m . (20)

The frequency variations for each train type are shown in Figure 3 and Figure 4. For train MF-T9, which is a mixed freight train, only wagons that lead to the longest periodic signal are considered. The results are also shown for a hypothetical train, Train ESA-10, which has equally spaced axles. The results for the frequency reduction due to the mass of the trains are given in Supplementary Table S3.

By plotting the mean bridge frequency for each train as a function of the train–bridge mass ratio (M _W /M _b ), it becomes apparent that a power law of the form Y = A X c (Roman, 2022) can describe the non-linear frequency–mass ratio relationship, as shown in Figure 5. To express the power-law relationship in the general form, as given by Eq. 21, the constant A and exponent c are determined by considering a range of bridges with different spans and resonant frequencies. Figure 5 shows that the first term in the equation represents the resonant frequency of the bridge. Therefore, the first expression in Eq. 21 is the fundamental bending frequency of vibration, f _n , of the bridge. f b , w = f n − A M w M b c . (21)

The bridge frequency reduction factor, R _f , can also be obtained from Eq. 21. This is of particular use for the EBB model as it allows for the unloaded bridge fundamental resonant frequency to be adjusted for a particular bridge–train configuration without the need for any additional model amendments. The frequency reduction factor can be represented in the general form according to Eq. 22. R f = 1 − A f n M w M b c . (22)

The reduced frequency can be implemented in the EBB model, but this would require significantly more model code adjustment. This has been calculated separately, which is what is presented in the article, and the expressions have been simply included in the EBB model that calculates this reduction. As the EBB model already contains the databases for the bridges and the different BS-5400 trains, it is believed that this provides a simpler and more efficient means by which bridge frequency can be adjusted for the train mass.

The power-law constants and exponents for each bridge are shown in Table 3. The power-law parameters are plotted against the fundamental bridge vertical bending frequency in Figure 6. Both curves can be approximated to a linear curve from which the following expressions for the constant A and exponent c, Eqs 23, 24, respectively, can be derived. For simplicity, the exponent c can be approximated as equal to the intercept as the curve slope is effectively zero. A = 0.3775 f n + 0.021 . (23) c = − 0.0006 f n + 0.6006 ≈ 0.6 . (24)

Substituting Eqs 23, 24 into Eqs 21, 22 gives a general equation for the laden bridge resonant frequency, f b , w , Eq. 25, and the frequency reduction factor, R f , Eq. 26, for any given bridge where the fundamental bending frequency and train masses are known. f b , w = f n − 0.3775 f n + 0.021 M w M b 0.6 . (25) R f = 1 − 0.3775 + 0.021 f n M w M b 0.6 . (26)

Using Eqs 25, 26, the bridge frequency reduction and frequency reduction factors versus the train–bridge mass ratio are calculated for Bridges 1–6. Figure 7 shows that for low mass ratios (typically <0.3), the average reduction factor, R f , is approximately 0.8. This means that a frequency reduction of 20% can be expected for bridges whose fundamental frequencies fall within the band 3–14 Hz.

Figure 7B shows the frequency reduction factor does not change significantly between bridges. As can be seen in Figure 7B, the change in the reduction factor between the highest and lowest bridge frequencies is <1%. It can also be seen that the ratio of 0.021 f n does not significantly contribute to the frequency reduction for the considered bridge group. For the lowest bridge frequency of 5.3 Hz (Bridge 2), the ratio is 0.00396, and for the highest bridge frequency of 14 Hz (Bridge 3), it is 0.0015.

As the bridge span is known, the additional mass on the bridge would need to be calculated for any given train. Therefore, it would be convenient to establish a train mass factor, T _mf , for the BS5400 trains, which would further simplify Eqs 25, 26. For the BS-5400 trains considered, the equivalent train mass is calculated using Eq. 14. The variation of the equivalent train mass for each train as a function of bridge span is shown in Figure 8. The curves allow the train mass factors, T _mf , to be calculated for each train type as a function of bridge span. The results are given in Table 4.

The train–bridge mass ratio, M w / M b , can be written using the train mass factors of Table 4, considering any of the BS-5400 trains, using Eq. 27. The term μ in this equation represents the uniformly distributed mass of the bridge. M w M b = T m f L μ L = T m f μ . (27)

The method and the frequency equations obtained can now be generalized in a more convenient form, offering the practicing engineer an efficient approach to assess bridge frequency under standard BS-5400 trains without the need for any complex dynamic modal numerical models. The method presented can also be extended to include other train types.

Using the general equation for calculating the fundamental bending frequency of a beam with a uniformly distributed mass, the frequency and the train–bridge mass terms in Eqs 25, 26 can now be replaced. Using Frýba (1999)’s definition of circular frequency at the j^th mode of vibration for a simply supported beam, as shown in Eq. 28, and substituting both Eqs 27, 28 into Eqs 25, 26, the following general equations for the BS-5400 trains can be derived to calculate the bridge resonance frequency and the frequency reduction factor (Eqs 29, 30, respectively). The equations are easily adaptable for any train type by establishing other train mass factors, T _mf , for non-BS-5400 trains. f n = ω n 2 π = π 2 E I μ L 4 . (28) f n , l a d e n = π 2 E I μ L 4 − 151 π 800 E I μ L 4 + 0.021 T , m f μ 0.6 . (29) R f = 1 − 0.3775 + 0.021 f n T , m f μ 0.6 . (30)

In Section 4.2, it was shown that the ratio of 0.021 f n does not significantly contribute to the frequency reduction for the considered bridge group. Therefore, for simplicity, this ratio can be ignored as it can be taken as zero. In addition, the remaining coefficient of 0.3775 can be rounded to 0.4, leading to a simplified equation for the frequency reduction factor, R f , as given by Eq. 31. R f = 1 − 0.4 T , m f μ 0.6 . (31)

The Siemens NX FE models created in Section 2.1 for each Bridge 1–6 are used to calculate the frequency change for different train–bridge mass ratios, as summarized in Table 2. These results are now compared with those obtained using the methodology presented in the previous section. Comparisons are made for the following types of assessment against the FE model results:

- EBB dynamic model

The laden bridge frequencies for each bridge are shown in Figure 9 for Bridges 1, 3, and 5 and Figure 10 for Bridges 2, 4, and 6. The results for Bridges 1, 3, and 5, which are considered short-span bridges (<10 m span), show a good correlation between each of the assessment methods; however, the results using the general equation start to diverge for the higher mass ratios. It is noteworthy that the higher mass ratios do not represent real train scenarios.

The same is true for Bridges 2, 4, and 6, which are considered medium-span bridges (10–25 m span), as shown in Figure 10. One reason for this is that the general equations have been derived based on the BS-5400 trains and do not extend to the higher mass ratios as in the case of the FE model assessment, which are hypothetical as they do not represent real trains.

The general Eq. 27 is also used to calculate the bridge frequency reduction factor for a given M w / M b ratio. The frequency reduction curves for Bridges 1, 3, and 5 are shown in Figure 11A, and those for Bridges 2, 4, and 6 are shown in Figure 11B. These provide a convenient graphical means by which a bridge’s fundamental vertical bending mode resonant frequency can be approximated for a given mass ratio based on the two bridge groups considered in this assessment, short span (<10 m) and medium span (10–25 m).

In this section, the results of the simplified model are compared with those of similar independent studies. Hora et al. (2023), using a multiple-moving mass and load model representing a train, show that for mass ratios < 1.0, both models produced similar bridge displacement responses. The result between the two started to diverge for mass ratios greater than 1.0, with the moving mass model producing higher bridge displacements (on the order of 12%) when the mass ratio reached 1.5. The authors show that the train’s critical speeds shift to the left for the moving mass analysis. The investigations conducted in this study show that the change in vertical resonant frequency is not greatly affected when the train–bridge mass ratios are <1.0, as can be seen in Figure 9 and Figure 10. The same divergence is generally seen when the train–bridge mass ratios are >1.0. As demonstrated in the Campbell diagrams in Figure 14 and Figure 15, showing the dynamic amplification factor against train speed, there is also a shift in DAF peaks between the unladen and laden cases. The DAF peaks represent a train’s critical speed, where the wagon pass frequency, or its multiples, coincides with the bridge’s vertical natural frequency of vibration.

In a recent publication, Kohl et al. (2023) analyzed a comprehensive set of trains and bridges using multi-body vehicle models. The main aim of the work is to establish the additional damping that could be incorporated into a moving load model to give the same maximum acceleration at the bridge mid-span obtained from a multi-body model. The results of the multi-body interaction model also showed a horizontal left shift and a reduction of the resonant speed of the ICE 2 and Eurostar trains due to the unsprung mass of the trains. To account for this reduction in the MLM model, the authors used an iterative method to determine the additional mass that could be added to the model to give the same effect. It is also worth noting that the suspension system frequencies are generally lower than the vertical bending mode of the bridge, which is typically >5 Hz, as is the case for the bridges considered in this work. This dynamically isolates the train body masses from the bridge, but as the authors note, this depends on the particular train–bridge configuration. Although this work is not directly comparable to the investigations in the current study, the general idea that an additional mass can be added to the MLM to account for the reduction in train critical speed is, in principle, the same.

Mao and Lu (2013) use the classical beam equation to represent a bridge with a moving vehicle to investigate the resonance phenomenon in the railway bridge response. In their study, the influence of the moving mass explicitly incorporates the coupled system dynamic properties. The study introduces a Z-factor that allows the prediction of the resonance effect but only if effective natural frequency is used in the calculation of resonance speeds. In their study, the effective natural frequency lies between the lowest natural frequency, when considering train mass, and the bridge’s fundamental vertical natural frequency of vibration. A similar methodology is incorporated in this study, where a mean train–bridge mass ratio and frequencies are used in the formulation of a set of generalized equations for predicting bridge resonance effects and train critical speeds. The main difference in this study is that actual assessment trains from BS-5400 are used as opposed to a hypothetical lumped mass model used by Mao and Lu (2013). Both studies present an approach by which bridge resonance effects and train critical speeds can be approximated. Both provide a means by which train critical speed bounds, accounting for train mass, can be approximated. The present study, which directly incorporates the BS-5400 trains, is considered to be of more practical relevance in real train–bridge assessments, particularly when considering fatigue.

Although more accurate and complex train–bridge multi-body system models are widely presented in the literature, these studies tend to look specifically at different areas of the train–bridge interaction, such as noise, wear, and passenger comfort. In such cases, modeling the train suspension systems, track irregularity, effects of sleepers, and even the ballast are necessary. However, whether this level of complexity is necessary when considering bridge fatigue effects is not a subject that has been adequately presented in the literature. Furthermore, many of the complex models presented in literature would not be easily understood or utilized by a practicing engineer, who, in most cases, is interested in a first approximation to see whether a potential problem exists for a particular train–bridge configuration. What this work aims to provide is a more efficient and practical means by which a practicing engineer can make a relatively simple assessment without resorting to complex models and assessment techniques.