Quick assessment of high-speed railway bridges based on a non-dimensional parameter representation

Abstract

The objective of this study is to provide the engineering practice with a tool for simplified dynamic response assessment of high-speed railway bridges in the pre-design phase. To serve this purpose, a non-dimensional representation of the characteristic parameters of the train–bridge interaction problem is described and extended based on a beam bridge model subjected to the static axle loads of the crossing high-speed train. The non-dimensional parameter representation is used to discuss several code-related design issues. It is revealed that in an admitted parameter domain, a code-regulated static assessment of high-speed railway bridges may under-predict the actual dynamic response. Furthermore, the minimum mass of a bridge as a function of the characteristic parameters is presented to comply with the maximum bridge acceleration specified in standards.

Keywords

dynamic response high-speed railway bridge non-dimensional parameter space simplified assessment

Introduction

In general, structural design and assessment of railway bridges crossed by trains with moderate speed are based on outcomes of static analyses amplified by a dynamic load factor. Trains traveling with high speed, however, may excite railway bridges to resonance. Excitation to resonance at critical speeds is the result of repetitive loading of the crossed bridge, transferred through the train wheels to the structure. At a critical speed, the frequency of the repetitive axle loads is close to a natural bridge frequency or a multiple of it (Ju and Lin, 2003). Other sources of resonance are rail irregularities and wheel hunting oscillations (Xia et al., 2006). In a state of resonance, the bridge response can only be predicted by realistically conducting dynamic analysis. The phenomenon of resonance of railway bridges is described in detail, for instance, in the works by Ju and Lin (2003), Yang et al. (2004), and Xia et al. (2006, 2014). In some situations, however, the free vibrations induced by moving loads are canceled. An elaborate discussion on vibration cancelation and resonance disappearance is provided by Xia et al. (2014).

Several standards, design guidelines, and technical notes, such as EN 1991-2 (2003), ÖBB Infrastruktur (2011), and UIC 776-2 (2009), regulate whether the structural railway bridge assessment may be based on a simplified static analysis or a more elaborate dynamic computation must be conducted. A flow chart included in EN 1991-2 (2003) supports the engineer in this decision. The involved decision variables are the maximum line speed $V_{0}$ , the maximum nominal speed $v_{0}$ ( $> V_{0}$ ), the structural system (simply supported or continuous bridge), the span length L, the first natural bending frequency $f_{1}$ , and the first torsional frequency $f_{T}$ of the unloaded bridge.

For many bridges, dynamic computations can be avoided if the maximum line speed is less than or equal to $55.6 m / s$ ( $200 km / h$ ). Figure 1 shows the domain of the fundamental bending bridge frequency as a function of span L (shaded in gray), where EN 1991-2 (2003) admits a bridge assessment based on static analysis. In some cases, additionally, the speed to fundamental frequency limit ratio $(v / f_{1})_{l i m}$ must be checked as tabulated in EN 1991-2 (2003), Annex F.

Figure 1.

Domain (gray shaded) where no dynamic analysis is required as a function of the bridge fundamental frequency $f_{1}$ and span length L.

In the design phase, vertical and horizontal deformation and twist of the bridge deck, internal forces, and in particular the vertical bridge deck acceleration must be assessed. The peak acceleration of the bridge deck is limited to satisfy the traffic safety of the train. For instance, at a certain bridge acceleration, the ballast becomes unstable, and consequently, the track quality is impaired. As outlined in several studies, such as Gulvanessian et al. (2009) and Ülker-Kaustell and Karoumi (2012), the bridge acceleration response is often the limiting and decisive quantity when a dynamic assessment is necessary.

For dynamic analysis of the complex train–bridge interaction system, models of various degrees of sophistication can be employed. Different modeling strategies and simplifications have, however, a significant effect on the accuracy of the response prediction. In a model close to reality, the bridge is discretized by means of three-dimensional finite elements, and a multi-body system composed of rigid masses connected through spring-damper elements is utilized for the train vehicles (see, for example, Gonzalez and Karoumi, 2014; Liu et al., 2009). Several interaction models have been developed to couple the train and the bridge subsystem, as outlined, for instance, in the work by Salcher (2015). Complex numerical modeling is time consuming and it requires special trained and experienced engineers. Furthermore, the mechanical properties of the crossing trains are often not disclosed by the manufacturer and for trains developed in the future naturally not available at present. However, the verification of a preliminary bridge design must be both time-efficient and cost-effective, and thus, elaborate modeling is usually not feasible.

Consequently, many less elaborated modeling strategies have been proposed and employed in engineering practice. Simplifications apply to both the train and the bridge subsystem. For these models, the number of parameters is assessable, and these parameters are readily available. Additionally, results are obtained in a timely manner. In the simplest approach, the effect of the train on the bridge is described by a series of concentrated forces (representing the static axle loads) crossing the bridge model with constant speed. This approach neglects the beneficial effect of bridge–vehicle interaction on the dynamic bridge response in the critical state of resonance (Museros et al., 2002; Yang et al., 2004). Dynamic analyses according to EN 1991-2 (2003) are based on high-speed load model sets referred to as HSLM-A and HSLM-B, which comprise several single force load models with different spacing and load amplitude, and additionally on single force load models representing real operating trains.

The one-dimensional beam is the simplest and oldest approach to model the entire bridge composed of load bearing system, ballast, and track. It is still extensively used in the research community (see, for instance, Frýba, 2001; Xia et al., 2006; Yang et al., 2004). Salcher (2015) has shown that a beam model captures the global dynamic deflection, and, with some limitations, the overall acceleration response behavior of slender beam-like bridges. These simplified models are, however, also useful for preliminary studies on more complex bridges.

This article provides a description of the Euler–Bernoulli beam bridge model by means of characteristic variables in the non-dimensional parameter space as introduced previously by Yang et al. (2004) and Adam and Salcher (2014). This non-dimensional representation allows the derivation of so-called response spectra, which (when readily available) can be used to assess the dynamic peak response of a high-speed railway bridge in the pre-design phase without costly response history analysis. These spectra reveal global interdependencies between the parameters of the problem. The spectra presented subsequently are based on simply supported beam bridges because this type of structure is in particular prone to vibrations. However, the concept can be used for any type of beam bridge. Additionally, it is discussed how code-based regulations concerning the range of application of static and dynamic methods of analysis can be globally assessed. In a further application, it is shown how the minimum mass of a bridge required for non-exceedance of the maximum admissible bridge deck acceleration can be estimated. The focus of this article is on provisions regulated in EN 1991-2 (2003). For more details, refer to Salcher (2015). First, the underlying equations of the considered beam bridge problem are briefly recapitulated.

Beam bridge model

In this study, the considered simply supported single-span bridges are modeled as linear elastic Euler–Bernoulli beam of constant mass per unit length $ρ A$ and constant bending stiffness EI (Figure 2). The $N_{F}$ axle loads $F_{i}$ , $i = 1, \dots, N_{F}$ , which cross the bridge with constant speed v, induce vibrations expressed through the lateral displacement variable $w (x, t)$ . $w (x, t)$ is governed by the following equation of motion (Adam and Salcher, 2014; Museros and Alarcon, 2005; Yang et al., 2004)

\begin{matrix} ρ A \overset{\cdot\cdot}{w} + E I w_{, x x x x} = \\ \sum_{i = 1}^{N_{F}} F_{i} δ (x - ξ_{i}) [H (t - t_{i}^{0}) - H (t - t_{i}^{E})] \end{matrix}

(1)

Figure 2.

Lateral single force crossing a simply supported beam bridge with constant speed v.

Dirac delta function $δ$ determines the location $ξ_{i}$ of the lateral force $F_{i}$ at time t on the bridge, compare with Figure 2. Unit step functions H indicate arrival and departure of $F_{i}$ on the beam at time instants $t_{i}^{0}$ and $t_{i}^{E}$ , respectively. The actual location $ξ_{i}$ of $F_{i}$ and time instants $t_{i}^{0}$ and $t_{i}^{E}$ depend on speed v and initial location $s_{i}$

ξ_{i} = v t - s_{i}, t_{i}^{0} = \frac{s_{i}}{v}, t_{i}^{E} = \frac{s_{i} + L}{v}

(2)

The solution of equation (1) is found by modal decomposition of deflection w into the mode shapes $ϕ_{n} (x)$ , $n = 1, \dots, \infty$ , of the actual beam problem, $w (x, t) = \sum_{n = 1}^{\infty} Y_{n} (t) ϕ_{n} (x)$ , yielding an infinite set of ordinary oscillator equations of motion for the modal coordinates $Y_{n}$ (see, for example, Chopra, 2011)

{\overset{··}{Y}}_{n} + 2 ζ_{n} ω_{n} {\overset{\cdot}{Y}}_{n} + ω_{n}^{2} Y_{n} = \frac{1}{m_{n}} \sum_{i = 1}^{N_{F}} P_{n}^{(i)}, n = 1, \dots, \infty

(3)

\begin{matrix} P_{n}^{(i)} (t) & = F_{i} ϕ_{n} (x = v t - s_{i}) [H (t - \frac{s_{i}}{v}) - H (t - \frac{s_{i} + l}{v})] \\ m_{n} & = ρ A \int_{l} {ϕ_{n}}^{2} (x) d x \end{matrix}

(4)

In application, the modal series is approximated by a finite number of modes. $ω_{n}$ denotes the $n th$ natural circular frequency of the beam. In equation (3), the effect of structural damping has been considered by modal adding of viscous damping (Chopra, 2011) equal for all modes, $ζ_{n} = ζ$ . Structural bridge damping is a fundamental bridge parameter, in particular for excitation at resonance. In EN 1991-2 (2003), lower limit values for the viscous damping coefficient are defined, depending on the type of bridge and on span L, see Table 1.

Table 1.

Damping values for railway bridges according to EN 1991-2 (2003).

Bridge type	Lower limit of damping value $ζ$ in %
Bridge type	Span $L < 20 m$	Span $L \geq 20 m$
Steel and composite	$ζ = 0.5 + 0.125 (20 - L)$	$ζ = 0.5$
Prestressed concrete	$ζ = 1.0 + 0.07 (20 - L)$	$ζ = 1.0$
Filler beam and reinforced concrete	$ζ = 1.5 + 0.07 (20 - L)$	$ζ = 1.5$

Vehicle–bridge interaction	$Δ ζ = \frac{0.0187 L - 0.00064 L^{2}}{1 - 0.0441 L - 0.0044 L^{2} + 0.000255 L^{3}}$

Characteristic non-dimensional variables of the train–bridge system

It is desirable to rewrite the dimensional variables and parameters in terms of a set of non-dimensional characteristic parameters, whose number is in general smaller than the number of the original variables. A non-dimensional parameter representation provides a global overview of the behavior of the problem and reveals the interdependencies between certain variables. A well-known scheme for non-dimensionalization is the Buckingham $Π$ theorem (see, for example, Gibbings, 2011).

The considered vehicle–bridge interaction problem simplified as an Euler–Bernoulli beam crossed by the static axle loads with speed v is entirely described by the following non-dimensional quantities (Adam and Salcher, 2014):

An excitation parameter, that is, the speed parameter;

Non-dimensional response parameters for kinematic response quantities and internal forces;

A geometric parameter;

The damping coefficient.

The speed parameter S is defined as the ratio of the fundamental excitation circular frequency $Ω_{1} = π v / L$ of a single moving load to the fundamental natural circular frequency $ω_{1} (= 2 π f_{1})$ of the beam bridge (Yang et al., 2004)

S = \frac{Ω_{1}}{ω_{1}} = \frac{π v}{ω_{1} L} = \frac{v}{2 f_{1} L}

(5)

An appropriate non-dimensional characteristic representation of the deflection w, bending moment $M_{y}$ , and acceleration $\overset{\cdot\cdot}{w}$ reads as (Adam and Salcher, 2014)

\begin{matrix} \bar{w} = \max | w (x, t) | \frac{E I}{F_{\max} L^{3}} \\ {\bar{M}}_{y} = \max | M_{y} (x, t) | \frac{1}{F_{\max} L} \\ \overset{\cdot\cdot}{\bar{w}} = \max | \overset{\cdot\cdot}{w} (x, t) | \frac{ρ A L}{F_{\max}} \end{matrix}

(6)

initially proposed by Hauser and Adam (2007) and Salcher (2010). $F_{\max}$ is the maximum static axle force of the considered train. Furthermore, in the work by Adam and Salcher (2014), the bridge span L has been related to the effective axle spacing $d_{V}$ of the train, leading to the characteristic non-dimensional geometric quantity

λ = \frac{L}{d_{V}}

(7)

For articulated and conventional trains, the effective axle spacing $d_{V}$ corresponds to the vehicle length. For instance, the Eurostar, which is an example of an articulated train, exhibits an effective axle spacing of $d_{V} =$ $18.7 m$ . The Railjet train with $d_{V} =$ $26.5 m$ and the Shinkansen SKS-700 with $d_{V} =$ $25.0 m$ represent conventional trains. The Talgo AV with effective axle spacing of $d_{V} =$ $13.4 m$ is a so-called regular train. Defining the modal damping coefficient $ζ$ completes the parameter set.

These characteristic parameters allow a global representation of the train-induced bridge response by means of a so-called response spectrum as subsequently discussed.

As an example problem, three simply supported bridges (differentiated by superscript $j = 1, 2, 3$ ) are considered that vary in mass and fundamental frequency but have the same span ( $L = 32.4 m$ ) and the same modal damping coefficient ( $ζ = 1.0 %$ ). Mass per unit length $ρ A L^{(j)}$ and fundamental frequency $f_{1}^{(j)}$ of these bridges are listed in Table 2.

Table 2.

Euler–Bernoulli beam bridge parameters.

Bridge j	Length L ( $m$ )	Distributed mass $ρ A^{(j)}$ ( $kg / m$ )	Fundamental frequency $f_{1}^{(j)}$ . ( $Hz$ )
1 (blue)	32.4	25,000	4.00
2 (red)	32.4	30,000	2.40
3 (green)	32.4	20,000	2.80

The beam bridges are crossed by the so-called HSLM-A10 load model. The HSLM-A10 is one out of ten load models of the HSLM-A load set specified in EN 1991-2 (2003), referred to as HSLM-A1, HSLM-A2,…, HSLM-A10. These load models differ in effective axle spacing, number of axles, and axle load, and they are supposed to represent current and future high-speed trains. The HSLM-A10 load model depicted in Figure 3 has $N_{F} =$ 36 axle loads $F = F_{i}$ ( $i = 1, \dots, 36$ ) of magnitude $210 kN$ each, and the effective axle spacing $d_{V}$ is $27.0 m$ (EN 1991-2, 2003).

Figure 3.

Load model HSLM-A10 specified in EN 1991-2 (2003).

The bridge acceleration $\overset{\cdot\cdot}{w} (x, t)$ based on a five-mode approximation is computed as described before, and its peak value with respect to time t and location x along the span is recorded as a function of speed v. The analysis is repeated by increasing stepwise speed v in the range between 20 and $120 m / s$ . In Figure 4(a), thin lines represent the resulting spectral peak acceleration response $\overset{\cdot\cdot}{w} (v)$ of the three bridges. Each local maximum of the spectral peak acceleration is related to a resonance speed. The resonance speed leading to the global peak response in the considered speed range is referred to as critical speed. For instance, the critical speed of bridge (1) is $108.09 m / s$ . In this representation with dimensional variables, the critical speed and the corresponding maximum peak acceleration differ for each structure.

Figure 4.

(a) Acceleration response of three different simply supported beam bridges of same span length and damping subjected to the HSLM-A10 load model and (b) corresponding non-dimensional representation.

Translating speed v and peak acceleration $\overset{\cdot\cdot}{w} (v)$ of the three bridges into the corresponding non-dimensional quantities S and $\overset{\cdot\cdot}{\bar{w}}$ according to equations (5) and (6) yields the same spectral response for all bridges, as shown in Figure 4(b) by a thin line. This result proves that the non-dimensional spectral response is identical not only for these structures but for all possible bridges of same span and same damping crossed by train model HSLM-A10.

When speed parameter S slightly exceeds a resonance speed, the peak response decreases, as it is observed in Figure 4. For bridge design, however, only the global peak response in the considered speed range $0 \leq S \leq S_{0}$ is of particular significance. Thus, the spectral response presentation shown in Figure 4(b) by a bold line is more meaningful because the train will not always travel with the maximum admissible speed $v_{0}$ but also with lower speeds, where the corresponding peak response is larger if the critical speed is hit. As observed, the ordinate $\overset{\cdot\cdot}{\bar{w}} (S_{0})$ of such so-called design spectrum matches the peak response $\overset{\cdot\cdot}{\bar{w}} (S)$ as far as $\overset{\cdot\cdot}{\bar{w}} (S)$ increases with increasing S. When a resonance speed is exceeded, the ordinate of the design spectrum remains with increasing S constant until the actual peak response at higher S becomes larger. Then, $\overset{\cdot\cdot}{\bar{w}} (S_{0})$ matches again $\overset{\cdot\cdot}{\bar{w}} (S)$ until S hits another resonance speed.

Expressed by means of characteristic non-dimensional parameters, the maximum peak response of a design spectrum in the range $0 \leq S \leq S_{0}$ is computed as (Adam and Salcher, 2014)

\begin{matrix} \bar{w} (S_{0}) = \max \bar{w} (S \leq S_{0}) \\ {\bar{M}}_{y} (S_{0}) = \max \bar{M_{y}} (S \leq S_{0}) \\ \overset{\cdot\cdot}{\bar{w}} (S_{0}) = \max \overset{\cdot\cdot}{\bar{w}} (S \leq S_{0}) \end{matrix}

(8)

This non-dimensional response representation facilitates various applications as subsequently discussed.

Design chart

Plotting the non-dimensional peak acceleration $\overset{\cdot\cdot}{\bar{w}} (S_{0})$ against the speed parameter $S_{0}$ and the length ratio $λ$ provides an overview of the bridge response on its parameter interdependencies. Exemplarily, in Figure 5, such a three-dimensional spectrum is depicted for 1.0% damped single-span beam bridges subjected to the HSLM-A10 load model. This figure confirms that the bridge acceleration response is largest if span L and effective axle spacing $d_{V}$ are about the same (i.e. $λ \approx 1$ ), and speed parameter $S_{0}$ is higher than 0.5.

Figure 5.

Non-dimensional peak bridge acceleration $\overset{\cdot\cdot}{\bar{w}}$ of 1% damped simply supported beam bridges subjected to the HSLM-A10 load model plotted against the speed parameter $S_{0}$ and bridge length to axle spacing ratio $λ$ .

For application in engineering practice, a two-dimensional representation of the peak response as a function of geometric ratio $λ$ , as shown in Figure 6, is more favorable (Adam and Salcher, 2014) and referred to as design chart. With such a design chart readily available, the non-dimensional peak response of a bridge structure $\overset{\cdot\cdot}{\bar{w}} (S_{0})$ can simply be read without computation, after translation of the actual design speed $v_{0}$ into $S_{0}$ (equation (5)). Subsequently, the identified quantity $\overset{\cdot\cdot}{\bar{w}} (S_{0})$ at the present $λ$ ratio is translated into the actual peak value $\overset{\cdot\cdot}{w} (v_{0})$ using the parameters of the actual bridge, equation (6), as it is described in the work by Adam and Salcher (2014). A step-by-step example is provided by Salcher (2010) and Adam and Salcher (2014).

Figure 6.

Design chart for simply supported steel and composite beam bridges subjected to the HSLM-A10 train model.

Design charts may be generated not only for a single train model but for an entire train set, such as for the HSLM-A load set of EN 1991-2 (2003). However, design charts for a train set cannot be represented completely non-dimensionally anymore. Since axle spacing (i.e. $d_{V}$ ) and axle loads (characterized by $F_{\max}$ ) of the $N_{S}$ train models of a train set are different, too much information is lost when enveloping the individual non-dimensional response spectra for each load model. Consequently, the non-dimensional peak acceleration ${\overset{\cdot\cdot}{\bar{w}}}_{k}$ of each load model $k, k = 1, \dots, N_{S},$ must be multiplied by the corresponding maximum axle load $(F_{\max})_{k}$ . The envelope of the individual response spectra

\begin{array}{c} {(\overset{\cdot\cdot}{\bar{w}} F_{\max})}_{e} = \\ \max [{\overset{\cdot\cdot}{\bar{w}}}_{1} {(F_{\max})}_{1}, \dots, {\overset{\cdot\cdot}{\bar{w}}}_{k} {(F_{\max})}_{k}, \dots, {\overset{\cdot\cdot}{\bar{w}}}_{N_{S}} {(F_{\max})}_{N_{S}}] \end{array}

(9)

is then plotted against the actual span L.

As an example, Figure 7 shows the corresponding design chart $(\overset{\cdot\cdot}{\bar{w}} (S_{0}) F_{\max})_{e}$ for the HSLM-A train set. It should be noted that for each load model of this set, the axle loads are the same, that is, $(F_{\max})_{k} = F_{k}$ . Thus, in this figure, subscript “ $\max$ ” could be omitted. However, it is kept to avoid a change of notation. Design charts for train sets are also derived for other response quantities such as the bridge deflection and the bending moment.

Figure 7.

Design chart for simply supported steel and composite beam bridges subjected to the HSLM-A train set.

Providing that such a design chart is readily available, the maximum peak response of the bridge crossed by the considered train set can be identified without response history analysis. However, the information of the location of the peak response and of the particular train, which induces the peak response, is not available.

Evaluation of code-regulated static analysis approach

The non-dimensional representation of response and excitation parameters allows the comparison of the outcomes of different design proposals in the full parameter space.

According to EN 1991-2 (2003), for many simply supported bridges crossed by trains with operating speed $v_{0} \leq 55.6 m / s$ , dynamic assessment is not required as discussed in section “Introduction.” In such a case, static analysis of the bridge model is conducted to predict deflection and internal forces based on the static load model LM71 (EN 1991-2, 2003), shown in Figure 8, amplified by a dynamic factor.

Figure 8.

Load model LM71 for a static assessment according to EN 1991-2 (2003).

As an example, the yellow shaded plane depicted in Figure 9 represents the maximum static deflection of simply supported beam bridges, subjected to the LM71 load model amplified by the dynamic factor $Φ_{2}$ , as a function of span L. Factor $Φ_{2}$ is specified in EN 1991-2 (2003) and used to reflect the worst case scenario. The corresponding dynamic peak bridge response induced by the HSLM-A10 load model, plotted against span L and speed parameter $S_{0}$ , is shown by a blue surface. Damping for steel and composite bridges as specified in Table 1 is assumed. The blue solid line represents the intersection of the plane and the surface, which splits for the HSLM-A10 load model the parameter domains of under-predicting and over-predicting the bridge deflection by means of static analysis.

Figure 9.

Peak deflection parameter $\bar{w} (S_{0}) F_{\max}$ of simply supported steel and composite bridges subjected to the HSLM-A10 train. Static analysis versus dynamic analysis according to EN 1991-2 (2003).

Dynamic analyses have been conducted for the 10 load models of the HSLM-A train set. Projecting for each load model, the lines of intersection in the $L - S_{0}$ plane delivers for the entire train set the parameter domain ( $L - S_{0}$ ), where the deflection based on dynamic analyses exceeds the outcomes of the static approach. In Figure 10, for the 10 load models, these lines of intersection are depicted. Additionally, the red shaded area shows the parameter domain, where EN 1991-2 (2003) admits for common high-speed railway bridges a static analysis. In this representation, the upper and lower frequency boundaries of the static analysis domain specified in Figure 1 have been converted into speed parameter $S_{0}$ boundaries, substituting the limit speed $v_{0} = 55.6 m / s$ into equation (5).

Figure 10.

Domains where the dynamic peak deflection of simply supported steel and composite beam bridges exceeds the corresponding static response prediction. Load models A1–A10 of the HSLM-A train set (EN 1991-2, 2003). Red shaded area: static analysis admitted according to EN 1991-2 (2003).

It is observed that the red shaded area and the areas limited by the intersection lines overlap in certain domains. That is, in these overlapping domains, a static (EN 1991-2, 2003) analysis under-predicts the actual dynamic response and thus is unsafe although admitted according to EN 1991-2 (2003). Only for spans $L \geq 55.0 m$ , the static approach yields for all load models conservative predictions of the deflection.

From the results of this study, it can be concluded that the lower frequency limit confining the domain according to EN 1991-2 (2003) (which corresponds to the upper limit of the boundaries expressed in $S_{0}$ ), where a static bridge assessment is admitted, should be increased. Spengler (2010) draws similar conclusions, however, using an alternative approach of response representation.

Minimum mass needed for passing code acceleration limits

In codes and standards, the vertical bridge acceleration is limited to guarantee the traffic safety. The ballast of ballasted tracks becomes unstable if the bridge deck acceleration exceeds a certain value, and thus, the safety of the train passage is not ensured anymore. EN 1991-2 (2003) specifies for bridges with ballasted tracks an acceleration threshold of $γ_{b t} = 0.35 g$ . Bridges with unballasted tracks exhibit a larger threshold of $γ_{c t} = 0.50 g$ .

Subsequently, a quick pre-design check is implemented to verify whether the mass of the bridge is sufficiently large to satisfy the bridge acceleration limits specified in codes and standards.

Substituting, for instance, the lower acceleration threshold of $γ_{b t} = 0.35 g$ ( $\approx$ 3.5 m/s²) into equation (6) yields for the non-dimensional acceleration response

\overset{\cdot\cdot}{\bar{w}} = \max | \overset{\cdot\cdot}{w} (x, t) | \frac{ρ A L}{F_{\max}} \leq 3.5 \frac{ρ A L}{F_{\max}}

(10)

This expression is solved for the mass per unit length, leading to the minimum distributed mass $ρ A_{\min}$ for non-exceedance of this acceleration threshold as a function of $\overset{\cdot\cdot}{\bar{w}}$

ρ A_{\min} = \frac{F_{\max} \overset{\cdot\cdot}{\bar{w}}}{3.5 L}

(11)

The derivation of response quantity $F_{\max} \overset{\cdot\cdot}{\bar{w}}$ representing the numerator of equation (11) has been discussed in the previous sections, see, for instance, Figure 7. When dividing $F_{\max} \overset{\cdot\cdot}{\bar{w}}$ by $3.5 L$ , the minimum mass $ρ A_{\min}$ for non-exceedance of $\overset{\cdot\cdot}{\bar{w}} = 3.5 m / s^{2}$ is obtained as a function of span L and speed parameter $S_{0}$ .

As an example, Figure 11 shows for steel and composite beam bridges subjected to the HSLM-A load set the minimum mass $ρ A_{\min}$ as contour plot in the parameter domain $S_{0} - L$ . This figure has been derived from the results shown in Figure 7 divided by $3.5 L$ . It gives an overview of the required bridge mass as a function of geometry and maximum speed, a useful information particularly in the pre-design phase of a bridge.

Figure 11.

Minimum mass per unit length of simply supported steel and composite beam bridges subjected to the HSLM-A train set for non-exceedance of the acceleration limit of $3.5 {m / s}^{2}$ specified in EN 1991-2 (2003).

Note that in speed parameter $S_{0}$ , the mass is included through the fundamental bridge frequency $f_{1}$ , compare with equation (5). Thus, changing the mass of the bridge also changes $S_{0}$ , and thus, an additional check of required mass is necessary.

Additionally, the red shaded area in Figure 11 represents the domain, where a static assessment according to EN 1991-2 (2003) is admitted, compare also with Figure 10.

As an application example, a simply supported steel bridge with span $L = 60 m$ and speed parameter $S_{0} = 0.5$ is considered, indicated by a yellow circle in this Figure. For this bridge, a minimum distributed mass $ρ A_{\min}$ of $1 . 5^{*} 10^{4} kg / m$ can be identified. Thus, if the actual distributed mass of this bridge is equal or larger than this amount, the bridge acceleration complies with the threshold of $γ_{b t} = 0.35 g$ .

Summary and conclusion

It has been shown that a non-dimensional representation of excitation, response, and geometric variables of the high-speed train–bridge interaction problem allows the derivation of design spectra and design charts in the full parameter space, which can be used for a quick assessment of the peak response in the pre-design phase of a bridge.

These spectra have been used to evaluate a regulation of EN 1991-2 (2003) concerning the assessment of bridges by means of static and dynamic methods of analysis. From this evaluation, it can be concluded that this regulation should be modified because for certain admitted parameter combinations, the outcome of a static analysis under-predicts the actual dynamic response.

In an additional application, it has been shown how the required minimum mass of a bridge can be estimated to avoid an exceedance of code-regulated bridge acceleration thresholds in the pre-design phase.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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