This paper is devoted to identify the effect of soil-structure interaction on the dynamic response of,a portal frame railway bridge. The study aims to validate the accuracy of numerical models in evaluating the dynamic stiffness and modal properties of the bridge-soil system. To achieve this aim, a controlled vibration test has been performed on a full-scale portal frame bridge to determine the modal properties of the system through measuring Frequency Response Functions. The results of the dynamic test provide reference data for FE model calibration as well as valuable information about the dynamic behavior of this type of bridges. Using the experimental data, an FRF-based model updating procedure was used to calibrate a full 3D solid model involving the entire bridge track-soil system. Both measured and computed responses identify the substantial contribution of the surrounding soil on the global damping of the system and highlight the importance of the soil-structure interaction on the dynamic response of this type of bridges. The identified modal damping ratio corresponding to the fundamental bending mode of the studied bridge was nearly 5 times higher than the recommended design values. A simplified model for the surrounding soil was also proposed in order to attain a less complicated model appropriate for practical design purposes.

Short-span portal frame bridges are predominant in Swedish railway lines. Although it is well known that the dynamic response of these partially-buried rigid structures is governed by the surrounding soil, the effect of the soil is usually neglected in the train-induced vibration analysis due to the expensive computational costs. This paper focuses on studying the effect of the surrounding soil conditions on the dynamic response of portal frame railway bridges. The study aims to validate the accuracy of simplified numerical models in evaluating the dynamic stiffness and modal properties of the bridge-soil system. To achieve this aim, a model updating method was used for FE model calibration of a full-scale portal frame bridge using measured frequency response functions. Both measured and computed responses identify the substantial contribution of the surrounding soil on the global damping of the system and highlight the importance of the soil-structure interaction on the dynamic response of these structures.

I följande rapport redovisas så kallade designdiagram för dynamisk kontroll av järnvägsbroar. Syftet är att dessa ska kunna användas i tidiga skeden som en första bedömning om en given bro har möjlighet att klara kraven på komfort och trafiksäkerhet enligt SS-EN 1990, avsnitt A2.4.4.

Kraven baseras på vertikal acceleration och vertikal nedböjning av brodäcket samt vinkeländring vid upplag. Baserat på spännvidd och brons lägsta egenfrekvens kan erforderlig massa samt styvhet läsas av direkt i diagrammen.

Diagrammen gäller endast för broar med ballastfria spårsystem och största hastighet sth = 320 km/h. Diagrammen är framtagna baserat på en 2D balkmodell med konstant massa och styvhet samt fasta upplag och är giltiga endast under dessa förutsättningar.

In this report, so called design charts for dynamic assessment of railway bridges are presented. The aim is to use these diagrams in early stage design of bridges, to determine if they are likely to comply with the requirements for riding comfort and traffic safety stated in SS-EN 1990, section A2.4.4.

The requirements are based on vertical acceleration and vertical displacement of the bridge deck, as well as end rotation at supports. Based on the span length and the fundamental natural frequency, the required mass and stiffness can be obtained directly from the diagrams.

The diagrams are only valid for bridges with ballastless track systems and an allowable speed sth = 320 km/h. The diagrams are developed based on a 2D beam model with constant mass and stiffness and fixed supports. The diagrams are only valid under these assumptions.

I föreliggande rapport redovisas dynamiska analyser av järnvägsbroar för höghastighetståg. En jämförelse i dynamisk respons mellan 2D- och 3D-modeller har utförts för ett mindre urval av plattbroar, balkbroar och lådbroar. Varje tvärsnitt har först optimerats att precis klara de dynamiska kraven avseende 2D-dynamik, utan beaktande av den statiska dimensioneringen. I många fall skulle tvärsnitten troligen behöva ökas för att klara den statiska bärförmågan.

Plattbroar med spännvidder från 10 – 25 m och 1 – 4 fack har analyserats. I flertalet fall, främst vid kortare spännvidder, är egenfrekvensen för böjning lägre i 3D-modellen jämfört med 2D-modellen. Detta beror på mindre medverkande tvärsnitt i böjning i 3D (shear-lag). Detta resulterar i en lägre resonanshastighet och därmed ofta en större dynamisk respons för samma hastighetsintervall. I övrigt överensstämmer det dynamiska verkningssättet väl mellan 2D och 3D. Inverkan av vridning synes inte vara styrande för de studerade fallen.

På motsvarande sätt har balkbroar med spännvidder från 20 – 40 m och 1 – 4 fack analyserats. På samma sätt som för plattbroar ger balkbroar lägre böjfrekvens i 3D jämfört med 2D. För dubbelspårsbroar är skillnaden i respons mellan 2D och 3D liknande som för plattbroar. För enkelspåriga balkbroar visar 3D-modellen i några fall en avsevärt lägre respons utan utpräglade resonanstoppar inom samma hastighets­intervall som 2D-modellen. Orsaken tros vara en kombination av upplagens excentricitet och brons massa, vilket vid vertikal böjning bidrar till en horisontell masströghet. Detta visas i de flesta fall kunna beskrivas med en modifierad 2D-modell.

Lådbroar med spännvidd 40 – 70 m i 1 – 3 fack har analyserats. P.g.a. hög vridstyvhet är egenfrekvensen för vridning mycket högre än första böjmoden och p.g.a. mindre shear-lag är egenfrekvensen för böjning likvärdig i 2D och 3D. Detta ger små skillnader i dynamisk respons mellan 2D och 3D-modellerna.

I de fall dynamiska kontroller utförs med förenklade metoder enligt (Svedholm & Andersson, 2016) föreslås att följande beaktas:

• Första böjfrekvensen n₀ bör beakta inverkan av shear-lag och upplagens excentricitet baserat på en 3D-modell, vilket används som indata i design­diagrammen.

• Då första vridmoden n_T < 1.2n₀ bör en fullständig dynamisk kontroll utföras i 3D.

• I de fall en 3D-modell visar flera närliggande egenmoder för böjning med samma form bör en fullständig dynamisk kontroll utföras i 3D.

This report present result from dynamic analyses of railway bridges for high-speed trains. A comparison of the dynamic response in 2D vs. 3D has been performed for a limited selection of slab bridges, beam bridges and box girder bridges. Each cross-section has been optimized based on the dynamic requirements for dynamics in 2D, without any consideration of the static design. In many cases, the cross-section probably needs to be increased to fulfil the static load capacity.

Slab bridges with a span length from 10 – 25 m and 1 – 4 spans have been analysed. In several cases, mostly for shorter spans, the natural frequency for bending is lower in 3D compared to 2D. The reason is due to a smaller contributing width, owing to shear-lag. This results in a lower resonance speed and therefore often a larger dynamic response within the same speed range. Apart from that, the dynamic response is found to be similar in 3D compared to 2D. The influence of torsional does not appear to be governing the response for the studied cases.

Using the same method, beam bridges with span length from 20 – 40 m and 1 – 4 spans have been analysed. Similar to the slab bridges, the 3D-model of the beam bridges show lower natural frequency in bending compared to the 2D-model, owing to shear-lag. For double-track bridges, the difference in response between 2D and 3D-models are similar to the findings for the slab bridges. For single-track bridges, some cases of the 3D-model shows significantly lower response without pronounced resonance peaks in the same speed interval as the 2D-model. The reason is likely a combination of the support eccentricity and the mass of the bridge, which for vertical bending results in horizontal inertia. It is shown that this can be simulated with a modified 2D-model in most cases.

Box girder bridges with span length from 40 – 70 m in 1 – 3 spans have also been analysed. Due to the larger torsional stiffness, the torsional mode is often much higher than the first bending mode. Also, the shear-lag effect seems to be smaller and the response from the 3D-model agrees well with the corresponding 2D-model.

In the case dynamic assessment is performed using the simplified methods according to (Svedholm & Andersson, 2016), it is suggested that the following is considered:

• Shear-lag and the eccentricity at the supports should be considered when estimating the first natural frequency for bending, n₀, preferably using a 3D-model.

• If the first torsional mode n_T < 1.2n₀, a full dynamic analysis in 3D should be performed.

• In the case a 3D-model shows several closely spaced bending modes with similar shape, a full dynamic analysis in 3D should be performed.

In this paper, an analytical solution for evaluating the dynamic behaviour of a non-proportionally damped Bernoulli–Euler beam under a moving load is derived. The novelty of this paper, when compared with other publications along this line of work is that general boundary conditions are assumed throughout the derivation. Proper orthogonality conditions are then derived and a closed form solution for the dynamical response for a given eigenmode is developed. Based on this, the dynamical response of the system to any load can be determined by mode superposition. The proposed method is particularly useful for studying various types of damping mechanisms in bridges, such as soil–structure interaction, external dampers, and material damping. Several numerical examples are presented to validate the proposed method and provide insight into the problem of non-proportionally damped systems. The numerical examples also allow for some interesting observations concerning the behaviour of modal damping for closely spaced modes (with respect to undamped natural frequencies).

Resonance vibration of railway bridges has become an important design issue during the last decade due to the development of new high-speed lines. At resonance it is well known that the damping of the structure has a great influence on the final response. The current paper presents modal parameters, such as modal damping ratios and natural frequencies, derived based on a simply supported Bernoulli-Euler beam on an elastic half-space. In the current chapters of Eurocode dealing with dynamical analysis of railway bridges, the expression for the modal damping ratio is essentially empirical. The novelty of this paper is therefore two-fold: (1) provide modal damping ratios for soil-structure interaction, and (2) show that the damping should not be considered as constant for all modes but it varies depending on the particular mode shape and interaction with the soil.

The Swedish government is considering upgrading the train speed along three railway lines in the Southern part of Sweden from 200 km/h to 250 km/h. According to the current design code, this requires that the bridges be examined with dynamic simulations to avoid excessive vibrations. This paper employs a method that can be used at an early stage to estimate the expected cost of upgrading a bridge network. The results revealed that 70% of the plate/beam bridges, 64% of the closed slab-frame bridges, and 41% of the open slab-frame bridges are expected to not fulfill the requirement on the maximum bridge deck acceleration for ballasted tracks.

In fatigue assessments of existing bridges, the influence of dynamics is conventionally considered by a dynamic amplification factor on the static response. This quasi-static approach gives a multiplicative enlargement of the stress ranges but doesn't necessarily represent the actual dynamic response. Moving loads on a bridge usually gives a more complicated time response than the corresponding static load. Redistribution of the stress range spectrum is expected dependent on the train speed and the dynamic properties of the bridge. Typical bridge beams found in open deck steel bridges are studied using two analytical methods for analysis. The methods are based on the Euler-Bernoulli beam theory and the Timoshenko beam theory, respectively. Using both methods enable a study on the influence of shear deformation and rotational inertia. The study also comprises the discrepancies in dynamic response of simply supported beams and continuous beams. The numerical results support the use of the quasi static approach for one span beams. For continuous beams on the other hand, the quasi static approach does not correctly reflect the dynamic behavior.

In this paper, a closed-form solution for evaluating the dynamical behavior of a general multi-span Bernoulli-Euler beam is derived. The natural frequencies of vibration and corresponding mode shapes are obtained by applying the boundary conditions to the characteristic function of a beam. A Laplace transformation is applied to the governing differential equation which is then solved for each normal mode in the frequency domain. The main contribution of this paper is to provide a closed-form solution for the vibration of continuous stepped beams under constant moving loads. Several numerical examples are included.

In the last few years the usage of 3D finite element analyses has increased substantially in the bridge design community. Such analyses provide the possibility for a more accurate study of the structure than what is possible by using more traditional design tools. However, in order to use the full strength of the finite element method in daily design practice a number of critical issues have to be addressed. These issues are related either to the FE-modelling itself (geometry, support conditions, mesh density, etc.) or to the post processing of the obtained results (stress concentrations, choice of critical sections, distribution widths and so on).  In the latter category, one problem of special significance refers to concrete structures subjected to restrained forces caused by temperature loading or shrinkage.

In this context, the present report addresses the problem of crack width control in transversal direction for concrete slab frame bridges subjected to restrained thermal or shrinkage loading. The recommendations given herein are based not only on the existing literature but also on the authors own investigations using non-linear finite element analyses.

The authors want to express their gratitude to the members of the reference group. Their comments and suggestions have been invaluable in shaping the report.

This report is a part of a larger research project entitled “Recommendations for finite element analysis of structures whose coordinators are Costin Pacoste and Mario Plos. The work has been financially supported by Trafikverket and also ELU Konsult. This support is gratefully acknowledged.