Analysis of composite bridges with intermediate diaphragms & assessment of design guidelines
Introduction
The importance of warping and distortional phenomena in the analysis of both straight and curved beam formulations has already been discussed since 1960s by Vlasov and Dabrowski [1], [2], [3]. It is worth noting that their evaluation when employing a beam element is far more complex in curved geometries due to additional coupling terms. For this reason, shell or solid Finite Element (FE) models have mostly been employed in practice (e.g. open I-girders and box-shaped cross-sections employed for bridge superstructures). However, these modeling elements usually do not permit the isolation of structural phenomena (e.g. magnitude of warping/distortion due to torsion or bending) and direct interpretation of the results [4], [5]. It is also very common in practice to employ straight-line segments in order to approximate the curved geometry ignoring warping and distortion transmissions between these segments. Thus, the importance of accurate curved beam elements for the implementation of structural analysis has been emphasized many decades ago.
Most of the previous formulations have been based on simplifications regarding either the cross section type or the boundary conditions or the implementation of warping/distortion or the application of loading. Regarding the static analysis of beams including warping and distortional effects, most of the relevant literature has been summarized in [4]. Towards studying further the generalized warping/distortional beam formulations, a beam element of 3 nodes and 11 degrees of freedom (dofs) per node is developed in [6] and provides the reader with a valuable insight into the nature of complex structural phenomena. The main difference with author’s previous formulations (e.g. [4], [5]), as well as the current one, is the consideration of the first derivatives, either of the angle of twist or the primary distortional parameter due to torsion, as dofs of the beam instead of considering additional independent parameters. This results in a less general formulation even though a quite accurate one. Moreover, caution should be taken with the description of the derivatives during the analysis. To the authors’ knowledge, there are no research works that consider the full coupling of bending, torsion, warping and distortion in the analysis of curved beams with arbitrarily shaped composite cross sections (except for the previous work of author in [4]). Commonly, different governing differential equations have been developed for the analysis of beams with open-shaped sections from those with closed ones. Additionally, the study of beam formulations with stiffening systems along their length (lateral or vertical), which are actually the means to control warping and distortion during construction or service phase, is quite limited. In most research efforts, the distortional static analysis has been employed in order to suggest design rules or charts [7], [8], [9], [10], [11] related to the placement of intermediate diaphragms of infinite stiffness (more relevant literature is given in [5]). Many of these studies do not account for the most general boundary constraints or make other assumptions. However, exhibit practical interest. Recently reported research works have developed new design guidelines and formulae for specific cases of practice [12], [13], thus not a generalized theory, but have employed diaphragms of finite stiffness. In very recent studies [14], [15], the influence of thickness and number of diaphragms on the overall performance against distortion has been evaluated through parametric studies. Despite their practical importance, the formulation in these studies is developed only for box-shaped straight beams.
Towards studying further various diaphragmatic arrangements and compare with test results, a three-dimensional (3D) solid-shell FE model has been developed in order to investigate the performance of straight Steel - Concrete Composite Box (SCCB) girders between 30 and 60 m lengths for different material models [16]. In this study, nonlinear inelastic analysis has been performed and the models have been verified against experimental results for a simply supported SCCB girder. A test girder with diaphragms and shear studs designed in [17] has been also employed in [16] for validation. However, in [16] a full connection between the concrete slab and top flanges of the steel box girder is considered (through surface-to-surface connection provided in ABAQUS [18]) exhibiting highly accurate results with respect to ultimate load and deflection at mid-span for most of the material models in [16]. The influence of headed studs has been investigated in several studies through the simulation of push-out specimens [19], parametric studies on composite beams with high-strength shear connectors [20] or vibration based testing methods for various shear connection types [21]. Additionally, elaborate one-dimensional (1D) beam models have either been employed to optimize the positions of diaphragms with specified thickness for straight or curved geometries [22] or estimate the shear connectors in longitudinal direction through the utilization of interface lines [23]. However, even though quite novel, the formulation in [22] is developed for box-shaped quadrilateral homogenous straight or curved beams while in [23], instead of a composite cross section, a plate element has been employed for the simulation of the slab together with additional displacement continuity conditions along the interface lines between the beam and the plate. The introduction of interface slip in the beam formulation of the present work by imposing additional continuity conditions between the various interfaces of the materials, but considering a composite section without the addition of plate for the slab, is within the subsequent research but not within the scope of this paper. As it is illustrated in [16], the diaphragms seem to be sufficient for providing an adequate ultimate carrying capacity and a computationally much cheaper model is developed without the modeling of shear studs.
Regarding the free vibration and dynamic response of beams, the importance of cross sectional deformations has been reported decades ago when it had been observed that thin-walled pipes’ performance can be highly affected by the cross section’s in-plane modes even though in-plane and rotary inertias had been neglected in those formulations with shell elements [24]. Immediately after, a simple cubic linear beam element having 4 degrees of freedom per node has been developed to consider the influence of shear deformation together with rotatory inertia on curved beams’ frequencies for various slenderness ratios and curvatures [25]. Additionally, some years ago, the dynamic analysis of thin-walled straight or curved beams has been studied including warping effects (out-of-plane) due to torsion [26], [27], [28]. In the same concept, the influence of cross-sectional characteristics (I-shape or box-shaped) has also been studied in [29]. It was noticed that the shear effect influenced the frequencies associated both with higher and lower modes. Some years later, various approaches have been proposed in order to determine the cross sectional deformations which are either restricted to quadrilateral cross sections [30], [31] or to the evaluation of two distortional modes that are roughly approximated by cubic polynomials [32]. The distortion of thin-walled box-shaped beams resulted in the significant lowering of the natural frequencies of the torsion-dominant vibration modes. One common aspect of the aforementioned studies is that sections of symmetric shape have only been investigated. Recently, towards establishing more general one-dimensional models, beam formulations which can be exploited for the analyses of compact thin-walled structures [33], [34], but for a limited range of deformation types, have been developed. More refined straight beam models with open- or closed-shaped cross sections have been proposed for the vibration problem towards studying arbitrarily shaped sections [35], [36]. Additionally, the generalized beam theory (GBT) is employed in order to develop a model based on piece-wise beam description of thin-walled sections, which is the linear combination of deformation modes [36]. However, the cross sectional analyses in the previously mentioned models are based on frame and/or plate models (assuming mid-surfaces) for the discretization of the cross section. Thus, irrespectively to the longitudinal geometry of the beam, these approaches depend on the cross sectional type, the nodal positions as well as the intermediate nodes employed. These make the procedure quite complicated while initialization of the deformation modes becomes important for the analysis. One-dimensional higher-order beam theories (HOBT) based on generalized displacement variables lead to less complex models in order to perform vibration analysis of compact and thin-walled structures with or without the inclusion of warping [37]. However, most of the formulations in these studies have not been developed for curved geometries and general cross sections. Towards studying the dynamic response of circular arches with variable cross section under seismic ground motions, several studies have been reported during the last years [38], [39] showing promising results, but still without the consideration of warping and/or distortion. Lately [40], the dynamic characteristics of curved nanobeams subjected to free and forced vibrations have been studied using nonlocal (Hookean solid) higher-order curved beam theories. However, the formulation is based on the assumption of rectangular cross section. In order to study the vibrational problem of curved beams including both warping and distortional effects, a curved beam element has been developed in [41] to study the out-of- and in-plane vibration problems of horizontally curved thin-walled beams based on the displacement fields proposed in [30]. However, this approach is implemented only for a rectangular hollow section.
In this research effort, shear lag and distortion due to both flexure and torsion are considered in the analysis of straight or horizontally curved composite beams with intermediate diaphragms for arbitrary cross section, loading and boundary conditions. The 1D beam element developed for the analysis of straight or curved bridge decks has an inclined plane of curvature (with respect to the horizontal one), which is closer to reality, leading to a curvature equal to . Additionally, the diaphragmatic stiffness has been considered in the formulation and has a finite value (instead of employing rigid diaphragms). These are the main two differences from the previous work of the authors [4]. Moreover, this work aims to study further the dynamic behavior of curved beams of arbitrary composite cross section with intermediate diaphragms. By employing a distributed mass model system accounting for longitudinal, transverse, rotatory, torsional, warping and distortional inertia, dynamic characteristics can be evaluated. The numerical procedures and schemes developed in [4] for the cross sectional (Boundary Element Method-BEM) and longitudinal analysis (Non Uniform Rational B-Splines in the Analogue Equation Method) of the one-dimensional curved beam model are also adopted in this study. Additionally to the formulation in [4], a topology optimization procedure is employed in order to define the optimum positions of diaphragms. The problem can be solved numerically using the method of moving asymptotes (MMA) [42]. Additionally, the bridge design specifications [43], [44], [45], discussed also in [5], have been employed in order to define the positions of the inner diaphragms and study the dynamic characteristics of the composite beam models considered. However, these design specifications do not consider the boundary conditions, the cross section shape or any dynamic factors. It is worth noting that the design of a bridge deck has mostly been based on dynamic amplification factors determined without considering the effects of the transverse inertial forces due to distortion. Hamed and Frosting [46] revealed that the assumption of the vertical flexural eigenmode as the fundamental one in the conventional dynamic design of bridges differs from the reality endangering the safety of the structure. Finally, 3D shell models employing the FEM for composite open and box-shaped cross sections are developed in this effort and analyzed for the static problem with geometry and material nonlinearities as well as surface-to-surface contact interaction, which is available in ABAQUS [18], between the concrete and steel parts. The dynamic problem is also studied.
The essential features and novel aspects of the present formulation comparing with previous ones are summarized as follows.
- i.
The developed beam formulation is capable of the warping and distortional analysis of spatial curved beams of arbitrary composite thin-walled cross section with inclined curvature plane and intermediate diaphragms.
- ii.
New 3D models of shell or solid FEs with composite open or closed cross sections, contact interaction between materials and diaphragms are proposed and employed for linear and nonlinear analysis.
- iii.
The importance of the diaphragms’ plate thickness, their optimum positions and their influence on the nonlinear behavior of bridge decks are investigated through the proposed formulations.
- iv.
The assessment of the design guidelines regarding the maximum spacing of intermediate diaphragms is achieved through the analyses of the developed models and parametric studies in order to consider further aspects on the placement of diaphragms.
Section snippets
General theory
Let us consider a curved prismatic element (Fig. 1) of length with a thick- or thin-walled arbitrarily shaped composite cross section of homogenous, isotropic and linearly elastic materials with modulus of elasticity , shear modulus and and Poisson ratio , occupying the region of the plane (Fig. 2). Let also the boundaries of the regions be denoted by . This boundary curve is piecewise smooth, i.e. it may have a finite number of corners. is the principal bending
Composite beam model
The evaluation of the warping and distortional functions is mainly accomplished as it has already been described in [4], namely employing the BEM within the context of the method of subdomains and the BEM for the Navier operator. These functions are employed in order to obtain the cross sectional operation factors given in Eq. (3). These are used as input values to solve the beam’s longitudinal problem. When intermediate diaphragms are employed, their stiffness values are added as described in
Examples
The computer programs formulated for the proposed 1D beam model are employed and compared to shell or solid FEs. Both the static and dynamic problems are examined. The design guidelines [43], [44], [45] for specifying the maximum spacing of intermediate diaphragms have been employed in order to assess the suggested number of diaphragms and its efficiency in the design. Three different examples of composite open or closed thin-walled cross sections with shapes and materials usually employed in
Conclusions
The proposed beam model is employed for the static and dynamic analysis of composite thin-walled straight or curved beams together with linear and nonlinear analyses of solid/shell FE models in order to assess the design guidelines regarding the placement of intermediate diaphragms to prevent distortion and compare the results. The influence of plate thickness and optimum positions of diaphragms are investigated through the proposed beam model. Additionally, in the solid/shell FE models the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work is part of a research project funded by Alexander von Humboldt Foundation.
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