Development of Improved Frequency Expressions for Composite Horizontally Curved Bridges with High-Performance Steel Girders

Development of Improved Frequency Expressions for Composite Horizontally Curved Bridges with... Composite curved I-girder bridges are often used in modern highway systems, but the open sections of I-girders mean that these structures suffer from low torsional resistance. The curvature also results in quite complex behaviors due to the coupled bending and torsional responses of curved I-girder bridges. High-performance steel, which adds strength, enhances durability and improves weldability, addresses both the economic and structural problems associated with curved bridges. However, as yet, there are no simplified design methods in the form of practical equations with which to optimize the design parameters of curved bridges and their dynamic behavior remains controversial. This study evaluated the effects of various design parameters on the free vibration responses of curved HPS I-girder bridges. A sensitivity analysis of 278 prototype simple-span and continuous bridges was conducted using CSIbridge software to create a set of simple, practical expressions for the fundamental frequencies of these structures. Keywords Curved bridges · Frequency · Finite element method · High-performance steel 1 Introduction curved and straight bridge superstructures [4–8]. Many stud- ies have sought to define the exact behavior of curved or bent Horizontally curved high-performance steel (HPS) I-girder beams. In one of these earlier studies, Culver [9] developed bridges can offer an economical solution for modern high- several analytical models with which to investigate the inter- ways where roadway alignments need a smooth and curved actions between curved girders and cross-frames. The effect transition, reducing costs and controlling pollution due to of erection sequencing on the induced stress and deflections car emissions. In curved bridges, the centerlines of the girder has also been studied in an attempt to enhance the design webs in the sections rising up from the abutments are not guidelines and improve the stability of the bridges [10–12]. collinear with the cords between the abutments. These eccen- To reduce the complexity of curved bridge design and tricities result in high torsional moments, causing high out improve the capacity of design codes to predict the responses of plane deformations and rotations in the bridge cross sec- of curved bridges, several researchers have tried to model tions [1–3]. High-performance steel (HPS) is often utilized bridge superstructures as an assembly of simple systems, to address these problems as it provides more economic and for example by applying the grillage technique, orthotropic durable bridges, a significant advantage that enables engi- plate methods or other relatively simple forms [13–15]. How- neers to design longer and shallower bridges. To enhance ever, the accuracy of these methods remains problematic the torsional resistance of open-section bridges, cross-frames and the results produced may not be acceptable due to the and diaphragms are applied to interconnect girders in both simplifications involved. Although finite element modeling (FEM) has been found to be a reliable method for evaluating Junsuk Kang the performance of curved bridges, the time and resources windkplus@hotmail.com required mean that it is seldom feasible in a typical bridge design office, especially during the preliminary design stages Department of Landscape Architecture and Rural Systems Engineering, Seoul National University, Seoul 08826, [16,17]. Republic of Korea Several analytical and numerical analyses have been per- School of Architecture and Architecture Engineering, formed to evaluate the dynamic behavior of composite Hanyang University, Ansan, Gyeonggi-do 15588, Republic of I-girder bridges. Christiano and Culver [18] developed a the- Korea 123 Arabian Journal for Science and Engineering vibration or ensuring human comfort for those traveling over Wright and Green [29] Wood and Shepherd [31] them [23,25]. Several studies have shown that the current Billing[30] AASHTO serviceability criteria may be insufficient for con- Caneni [32] Trilly [34] trolling bridge vibration and frequency-based limits are more Dusseau [33] rational than span-based limits [25,26]. This paper presents the results of a simulation that applies CSIbridge V20 [27] to design and analyze the fundamental frequencies of curved HPS I-girder bridges. A parametric study examines the variables that could influence the free vibration characteristics of typical composite HPS I-girder 30 45 60 75 90 bridges. The results from a comparison study indicated that Span Length (m) existing frequency prediction equations are not sufficiently Fig. 1 Fundamental frequencies obtained using various equations accurate for curved composite I-girder bridges. Hence, non- linear regression methods are used to derive two sets of more reliable and practical expressions with which to estimate the oretical method using the differential equations governing fundamental frequencies of curved HPS I-girder bridges that the free vibration of a simple-span curved girder, deriving capture the effects of all the important parameters. two sets of equations to determine the fundamental frequen- cies of such bridges based on Lagrange multiplier concepts. 2 Finite Element Modeling and Verification Maneetes and Linzell [19] studied the effects of cross-frame and lateral bracing on the dynamic responses of a single-span curved I-girder bridge using both experimental and numerical In this section, the general attributes of the proposed finite solutions, while Yoon et al. [20] presented a finite element element modeling (FEM) using CSIbridge [27] are described. formulation based on a free vibration analysis of compos- The results obtained from two field tests and a laboratory ite bridges, taking into account the stiffness as well as the study are used to verify the effectiveness of the proposed mass matrices of the curved beam elements. The findings of modeling technique for curved HPS bridges. extensive studies have indicated that grillage methods are not in fact a reliable way to evaluate the dynamic behavior and 2.1 Bridge Sections free vibration of bridges [3]. Barth and Wu [21] performed a simulation and an experimental study to capture the vibra- A four-node shell element with six degrees of freedom at each tion characteristics (bending frequencies and mode shapes) node that combines membranes and plate-bending behavior of straight HPS girder bridges. was used to model the bridge’s concrete slab and steel gird- Figure 1 shows the fundamental frequencies values ers. Each element has its own local coordinate system for defined based on the equations proposed by various defining material properties and loads, and for interpreting researchers for steel girder bridges. According to the graph, the output. Stresses and internal forces and moments in the Trilly’s equation [22] generates the most conservative val- element local coordinate system were evaluated at the 2-by-2 ues for the fundamental frequency, although all follow the Gauss integration points and extrapolated to the joints of the same trend with increasing span length. Some past reports element. The bridge modeler in CSIbridge [27] was used to on vibration analysis are quite controversial, as they suggest model the prototype bridge. This option takes into account the very high values for the fundamental frequency. Wu [23] full composite action between the reinforced concrete slab summarized these reports, concluding that fundamental fre- and the HPS I-girders at the serviceability limit state, as rec- quencies are highly dependent on the geometry of bridges ommended by AASHTO LRFD [6]. Cross section properties and that beam theory and the simplified equations proposed were defined for all frame-elements and diaphragm proper- so far are not capable of predicting the correct values. Nas- ties imported (X or V types) and manually defined (built-up sif et al. [24] presented an investigation of vibration control plate sections). The parapets were modeled as a single frame (e.g., acceleration and velocity) in HPS bridges using an element. Note that it is critical to place the connecting nodes experimentally validated three-dimensional dynamic com- of the parapets at the centroid of the barrier section. Addition- puter model to evaluate both the deflection criterion and ally, the parapets were attached to the deck using a series of depth-to-span (D:L) limitation produced by the current rigid links in order to contribute to the stiffness of superstruc- design codes. They concluded that neither set of limita- ture. Figure 2 shows the parapet model assumptions applied. tions could provide effective vibration control for steel girder The discretization of the models was performed using bridges under normal truck traffic conditions and these lim- the auto-meshing option, defining a good expectation ratio its therefore represent a poor method for controlling bridge for all elements. The boundary conditions for simple-span Frequency (Hz) Arabian Journal for Science and Engineering of 90 kN, and then quickly releasing the applied load to induce free vibrations in both the longitudinal and transverse directions. Nearby abandoned railway bridge piers provided adequate anchor points for the pullback testing. The software capabilities of hybrid bridge evaluation system (HBES) were used to analyze records on site shortly after the data were collected. The resulting torsional and vertical modes exhib- ited some coupling, which is typical in this type of analysis when periods are close in value [30]. In order to comprehen- sively verify the results of the FEM, the entire bridge was simulated in the present study using the CSIbridge package Fig. 2 Parapet modeling assumptions software according to the prescribed modeling technique. The first three vertical frequencies obtained by the bridge model (with parapet effects) were 6.12, 7.48 and 8.75 Hz, differing from the field test frequencies of 5.95, 7.14 and 8.35 Hz by about 3.50, 4.54 and 4.72%, respectively. With- out modeling the parapet, the first three vertical frequencies obtained by the FEM method were 6.03, 7.25 and 8.50 Hz, which indicates that including the parapet shifts the results by 1.32, 1.52 and 1.76%, respectively. These results indicate that the FEM technique adequately predicts the vibration responses of composite girder bridges. Since the funda- mental frequency (the first natural frequency) is the most pronounced, the parapets may not noticeably influence the vertical frequency of these bridges. Fig. 3 Selected boundary conditions 2.2.2 Laboratory Test of a Quarter-Scale Model Bridge and continuous curved bridges were defined based on the Additional verification was performed through experimental proposed tangential bearing arrangement recommended by tests of a quarter-scale model of a 4.5 m long simple-span Samaan et al. [28], shown in Fig. 3. In the elastic range of straight I-beam bridge with five scaled girders at a spacing structural behavior, any slab cracking present exerts only a of 0.45 m and width of 1.80 m [31]. These I-beams had the negligible effect on the bridge responses [29]; hence, the slab following dimensions: flange thickness of 4.80 mm, flange was assumed to be uncracked and steel reinforcements were width of 58 mm, web thickness of 3.5 mm and total height of therefore not modeled. 203 mm. Channels with dimensions of 25 × 9.5 × 0.30 mm and a spacing of 15.8 cm were used as shear connectors. 2.2 Verification of Finite Element Models Here, solid diaphragms were applied at the abutments and spaced at one-third intervals along each span, consisting of 2.2.1 Colquitz River Bridge 76 × 50 × 52 mm angles welded to the beam webs about 13 mm below the top of the beam. A comparison of the FEM The results of dynamic field tests on the Colquitz river bridge and experimental results for the mid-span deflections of all in British Columbia, Canada [30] were selected to verify girders under four 20 kN concentrated loads at the mid-span the proposed FEM technique. The bridge has five spans and symmetric about the longitudinal axis of bridge is shown (2 × 14 m + 2 × 17 m + 18 m), each composed of six steel in Fig. 4. girders W 33 × 141, and a bridge width and slab thickness Although the experimental and FEM results compare of 11.9 and 0.10 m, respectively. The diaphragms and cross- fairly well, the FEM results are slightly higher than those braces members are made of U-shaped steel (MC18 × 42.7) obtained in the laboratory, varying by up to 1.5, 4.6 and 3.4% and spaced 3.5 and 4.5 m apart along the end and interme- for the external, intermediate and central girders, respec- diate spans, respectively, of the bridge. For the ambient test, tively. This discrepancy is likely because the experimental six force-balanced accelerometers were used to collect and loads were not localized at an exact position but rather dis- process acceleration data during the day and under varying tributed over the entire steel supporting plate. The same traffic conditions. Pullback tests were carried out by load- phenomenon has also been reported by Wegmuller [32]in ing the bridge bents with a special assemblage using a force a comparison between FEM and experimentally obtained 123 Arabian Journal for Science and Engineering Fig. 5 FE model of a 30 m span length bridge with k = 0.1 Fig. 4 Deflection distribution in the bridge girders at the mid-span cross section ues for the slenderness ratio increasing from 0 to 1.20 at 0.2 intervals. Three types of cross section, with girder spacings results. This suggests that the proposed FEM technique of 2.8, 3.5, and 2.5 m, were utilized, along with the Ameri- accurately predicts the response of I-girder bridges to static can Iron and Steel Institute (AISI) standard parapet design, loading. which has a height of 0.87 m and bottom and top widths of 0.4 and 0.15 m, respectively. The prototype bridges were mod- eled using three different high-performance steel materials: 2.3 Geometric and Material Properties of Bridges HPS 50 W (Fy = 345 Mpa), HPS 70 W (Fy = 485 Mpa) and HPS 100 W (Fy = 620 Mpa). The concrete deck was mod- High-performance steel (HPS) bridges need less steel due eled with a modulus of elasticity of 28,000 Mpa, a Poisson’s to their higher yield stress, but consequently the live-load ratio of 0.2 and a density of 24 kN/m . The concrete deck deflections of these bridges are more likely to exceed thickness was 0.20 m for all the cross sections. AASHTO’s deflection limits [5]. As design optimization must take into account the AASHTO limits on performance and economy, an important first step is to select a set of key parameters and establish a matrix that covers a wide range of steel bridge types. Based on this database, bridges can then 3 Sensitivity Analysis be designed and optimized for various combinations of these parameters to capture a least weight approach using CSIb- A sensitivity analysis provides useful information regard- ridge [27]. The initial designs are performed by neglecting ing the influence of various parameters on the free vibration the AASHTO criteria for deflection in order to design girders responses of bridges. The goal here is to characterize the that meet other AASHTO LRFD strength and serviceabil- key parameters affecting a bridge’s natural frequency so that ity criteria for curved bridges. Any girders that then fail the they can be taken into account in the development of effective AASHTO live-load deflection criteria must be redesigned. expressions for natural frequencies. The investigation vari- Table 1 shows the geometric properties and material param- ables are the span length, the span-to-radius curvature ratio eters for three prototype bridges. Figure 5 shows a typical (k), the number of girders, the girder spacing and the number 30 m span length bridge modeled in CSIbridge [27]. and arrangement of the cross-braces. A detailed discussion Four different span lengths (L), 30, 45, 60, and 90 m, were of each of these parameters is presented in this section, and considered in this investigation, with span-to-radius (k)val- the key parameters are identified. Table 1 Characteristics of three Set L (m) HPS types (W) L/D ratio S (m) No. girder W (m) prototype curved I-girder bridges 1 30, 45, 60, 75, 90 50 20 2.80 5 13.0 2 30, 45, 60, 75, 90 70 25 3.50 4 13.0 3 30, 45, 60, 75, 90 100 30 2.50 4 9.50 123 Arabian Journal for Science and Engineering f1 f2 f3 f4 f5 Davidson’s equaon 23 45 67 89 No. cross braces Fig. 6 Effect of the number of cross-bracing lines on a bridge’s natural frequencies Fig. 7 Effect of cross-brace type on a bridge’s natural frequencies 3.1 Effect of the Number of Cross-braces Curved girder bridges have a high torsional moment due frequencies. For example, enhancing the thickness of the to their curvature. The open nature of these bridges results solid steel plates from 10 to 25 mm changes the first and in low torsional resistance to cracking, so cross-braces are second frequencies by up to 2%. However, although increas- applied to enhance their torsional stiffness. It is thus vital to ing the plate thickness also increases the natural frequencies determine the effect of this variable on the bridge’s free vibra- of the higher modes, it should be avoided due to possible tion responses. The effect of the number of cross-bracing brittle connection failures of the diaphragms under seismic lines on the first natural frequency of a 30 m long bridge with loads. Thus, it is preferable to adjust the thickness of the a curvature ratio of 1.20 is shown in Fig. 6. The minimum steel X-bracing rather than the plate thickness to enhance required number of cross-braces obtained from Davidson’s the torsional resistance of both single-span and continuous equation [33] is also indicated. Although the number of bridges. braces appears to have only an insignificant effect on the first and second frequencies and the vertical mode shapes of the bridge, it does decrease the natural frequency values for 3.3 Effect of the Span Length the higher modes of vibration. Thus, for all the prototype bridges modeled in this study, cross-braces are placed 6 m Many researchers have described the importance of span apart, with a minimum of five bracing lines for each span. length for the dynamic responses of straight bridges. For example, the current North American codes [5,6] offer equa- 3.2 Effect of the Cross-brace Type tions for the dynamic impact factor of bridges as a function of span length. Wright and Walker [34] developed the follow- In order to study the effect of the various types and stiffnesses ing theoretical equation based on beam theory to determine of bracing systems on the free vibration responses of curved the fundamental frequencies. Note that this equation incor- HPS bridges, several different bracing configurations were porates the span length as an important factor: modeled for a 90 m long bridge with a curvature ratio of 0.60. The following bracing systems were considered: X- type, V-type, inverted V-type, single beam and solid plates; π E · I · g b b f = (1) the results are shown in Fig. 7. Once again, although the sb 2L w type of bracing systems had an insignificant impact on the first natural frequency of the bridge, it did have an important impact on the higher frequencies. For instance, the second where E · I and w are the flexural rigidity and the weight b b natural frequency of a bridge with X-type bracing was up to per unit length of the composite steel girder, respectively. 12, 48 and 54% higher than bridges with V-, inverted V- and Figure 8 shows the effect of span length on the natural fre- single beam bracing systems, respectively. The effects of the quencies of bridges with a slenderness ratio (L/D)of25and stiffness of the cross-bracing and the end diaphragms on the a curvature ratio of 0.20. Increasing the span length from 30 free vibration of the bridge are also shown in Fig. 7. to 90 m decreases the natural frequencies by up to 65%. From These results suggest that the stiffness of secondary mem- a practical standpoint, the span length is thus an important bers has a significant influence on the bridge’s fundamental parameter affecting the dynamic response of a curved bridge. Frequency (Hz) Arabian Journal for Science and Engineering 10 4 f1 k=0.2, L/D=25 f2 3.5 L=30m, L/D=25, 50W f3 L=30m, L/D=25, 70W f4 L=30m, L/D=30, 70W 2.5 L=60m, L/D=20, 70W L=60m, L/D=20, 50W 1.5 0.5 30 45 60 75 90 0.2 0.4 0.6 0.8 1 1.2 Span Length (m) k=L/R Fig. 8 Effect of span length on the fundamental frequencies of curved Fig. 10 Effects of the bridge curvature ratio on the fundamental fre- bridges (k = 0.20, L/D = 25) quency f1 L=60m, k=0.60 f2 derness ratios (L/D). The graph indicates that the curvature 7 f3 ratio has an important effect on the free vibration of curved f4 f5 superstructures, with an increase in the curvature ratio sig- nificantly reducing the natural frequencies. For instance, for the 60 and 90 m bridges, the natural frequencies decrease by up to 56 and 39%, respectively, when the span-to radius ratio increases from 0 to 1.20. The curvature ratio also has a significant impact on the mode shapes. Figure 11 shows the first three mode shapes for both Number of Girders straight and curved bridges with 30 m long spans. These reveal that any curvature in a bridge exerts a higher torsional Fig. 9 Effect of the number of girders on a bridge’s natural frequencies impact on mode shapes. For example, the first mode shape for a straight bridge is purely flexural, but in a curved bridge it consists of a combination of flexural and symmetric torsional 3.4 Effect of the Number of Girders modes. Thus, the effect of the curvature ratio on both the mode shapes and frequencies will be significant and should Analyzing the performance of a 60 m long curved bridge with thus be included in the parametric study. a curvature ratio of 0.6 to investigate the effect of the number of girders on the bridge’s natural frequencies revealed that 3.6 Effect of Slenderness (Span-to-Depth) Ratio although increasing the number of girders results produced higher torsional stiffness in the structure, the natural fre- AASHTO (5) specifications limit the value of the slenderness quencies of the bridge’s superstructure were enhanced only ratio to a maximum of 25 in order to control the maximum slightly. The results, presented in Fig. 9, show that the first live-load deflection through controlling the bridge’s bending and second frequencies increase by approximately 3% as the stiffness. However, this approach became somewhat con- number of girders increases from three to seven, for example. troversial with the introduction of high-performance steel In practice, this does not influence the free vibration (HPS), which has approximately 40% higher yield stress than sufficiently to justify its inclusion as an important model conventional steel and hence allows engineers to design shal- parameter. These relatively small changes are due to the fact lower and longer spans. Since HPS bridges are more likely that the prototype bridges have almost the same flexural stiff- to suffer from large deflections, the slenderness ratio plays ness, irrespective of the number of girders. Previous research a significant role in the dynamic behavior of these systems. has also indicated that the number of girders has no impact The effects of different slenderness ratios on the natural fre- on the natural frequencies of straight bridges [35]. quencies of a 30 m long bridge are presented in Fig. 12.The natural frequencies decrease almost linearly as the L/D ratio 3.5 Effect of the Span-to-Radius Curvature Ratio increases, irrespective of the curvature ratio. For example, in a bridge with a curvature ratio of 0.80, the first natural Figure 10 shows the relationship between the first natural frequency decreases by approximately 28% when the L/D frequencies (f1) and curvature ratios (k = L/R)offive ratio increases from 20 to 30. From a practical standpoint, bridges with lengths of 30, 60 and 90 m and various slen- the impact of the slenderness ratio is considerable and should Frequency (Hz) Frequency (Hz) First Frequency (Hz) Arabian Journal for Science and Engineering Fig. 11 First three modes of vibration for straight and curved bridges. a Straight bridges and b curved bridges 1.8 proposed equations adequately predict the fundamental fre- k=0.20 simple supported bridge, L=30m 1.6 quency of curved composite girder bridges constructed using k=0.40 k=0.80 high-performance steel. While Wood and Shepherd’s equa- 1.4 k=1.20 tion [36] estimates highly conservative values, the models 1.2 proposed by Dusseau [37] and Wright and Walker [38] both significantly underestimate the fundamental frequencies of 0.8 such bridges. To address this problem, the data for 180 sim- 0.6 ple supported and 85 two- and three-span continuous bridges were collected and analyzed to develop a set of proposed 0.4 20 25 30 expressions to describe their fundamental frequencies. The Slenderness rao (L/D) proposed expressions take the following form: Fig. 12 Effect of slenderness ratio (L/D) on a bridge’s natural fre- quency f = ψ · f (2) c·b sb thus be taken into account in any parametric study of curved HPS I-girder bridges. where f is the fundamental frequency of a straight bridge sb from Eq. (1). A modification factor, ψ, is applied to take into account the effect of the parameters from the previous 4 Empirical Expressions for Fundamental section that were found to affect the free vibration of curved Frequency HPS I-girder bridges. A regression analysis using a statisti- cal computer package for best fit based on the least squares As the comparison of the various previous studies that have method for nonlinear data was used to deduce appropriate investigated this issue presented in Fig. 1 shows, none of the modification factors as follows: Frequency (Hz) Arabian Journal for Science and Engineering Fig. 13 Comparison of the 4 4 results obtained by the proposed expression and the FEM analysis. a Simple-span and b R² = 0.9739 3 3 continuous span R² = 0.9772 2 2 0 0 012 34 Frequency FEM Analysis (Hz) Frequency FEM Analysis (Hz) (a) (b) Table 2 Comparison between No. spans L/DL (m) L/Rf (Hz) ψ Frequency (Hz) sb the FEM results and those f = ψ · f FE c·b sb obtained using the proposed expressions 120 30 1.0 0.5966 2.240 1.337 1.375 125 30 0.80 0.6865 1.930 1.328 1.281 130 30 1.0 0.5910 1.980 1.172 1.102 230 30 1.20 0.8435 2.290 1.936 2.020 230 60 0.20 1.2063 1.050 1.275 1.250 � Simple-span bridges I-girder bridges. This numerical approach includes an exten- sive sensitivity study of simple supported and continuous −0.82k 0.018S bridges to determine the effect of a number of different vari- ψ = 1.160 × e × e × 0.80 + 0.008 (3) ables on the natural frequencies and mode shapes of the bridge structures. Empirical expressions for the fundamental � Continuous bridges frequencies of such bridges were derived that are suitable for use in design codes and engineering offices. The proposed −0.37k 0.005 ψ = 1.137 × e × e (4) expressions are a function of the natural frequency obtained from flexural beam theory and are hence applicable to bridges It should be recognized that the effect of the slenderness with various slenderness ratios. Based on the results obtained ratio in the proposed expressions for the modification from the sensitivity study, the following conclusion can be factor is negligible because the impact of the span length derived: and depth has already been taken into account in Eq. (1). The results of the verification analyses for the proposed � The slenderness ratio significantly affects the natural expressions are shown in Fig. 13 and Table 2. frequency of such bridges: the fundamental frequency decreases with increasing span-to-depth ratio. Examining the first natural frequency for both simple-span � The magnitude of the fundamental frequency decreases and continuous curved bridges reveals that the difference with increasing span-to-radius curvature ratio for hor- between the FEM analyses results and those obtained using izontally curved I-girder bridges. The curvature ratio the expressions proposed here are within 1%. The high coef- also has a significant effect on the bridge’s vibration ficient of determination, R , obtained in both cases indicates modes due to the high contribution of the torsional effects the minimal variation in the results calculated by the pro- induced in curved superstructures. posed expressions and the FEM analysis. � Cross-braces between the abutments increase the tor- sional stiffness of open-section bridges. It is recom- mended that the maximum spacing for bracing lines be 5 Conclusions limited to 6 m in order to exert a meaningful effect on bridge responses. This study has presented a detailed numerical analysis of the free vibration characteristics of horizontally curved HPS Frequency Eq.(3) Frequency Eq.(4) Arabian Journal for Science and Engineering Acknowledgements The work reported herein was supported by a 15. Linzell, D.; Chen, A.; Sharafbayani, M.; Seo, J.; Nevling, D.; Grant (18CTAP-C132633-02) funded by the Ministry of Land, Infras- Ard, T.J.; Ashour, O.: Guidelines for Analyzing Curved and tructure and Transport (MOLIT) of the Korean Agency for Infras- Skewed Bridges and Designing Them for Construction. Project No. tructure Technology Advancement (KAIA), the National Research PSU-009. Commonwealth of Pennsylvania Department of Trans- Foundation of Korea (NRF) Grant (NRF-2016R1C1B1015711) funded portation. Final Report, US (2010) by the Korea government (Ministry of Science, ICT & Planning) and the 16. Samaan, M.; Sennah, K.; Kennedy, J.B.: Distribution factors 2017 Seoul National University Invitation Program for Distinguished for curved continuous composite box-girder bridges. J. Bridge Scholars. This financial support is gratefully acknowledged. 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Kim, S.K.; Laman, J.A.; Linzell, D.: Live load radial moment dis- 34. Wright, R.N.; Walker, W.H.: Criteria for the deflection of steel tribution for horizontally curved bridges. J. Bridge Eng. 12(6), 727 bridges. Bulletin for the America Iron and Steel Institute, No. 19 (2007). https://doi.org/10.1061/(ASCE)1084-0702(2007) (1971) 14. Hajjar, J.F.; Krzmarzick, D.; Pallares, L.: Measured behavior of a curved composite I-girder bridge. J. Constr. Steel Res. (2010). 35. Mohseni, I.; Kalim, A.R.A.; Kang, J.: A simplified method to https://doi.org/10.1016/j.jcsr.2009.10.001 estimate the fundamental frequency of skew continuous multicell 123 Arabian Journal for Science and Engineering box-girder bridges. Lat. Am. J. Solids Struct. (2014). https://doi. 37. Dusseau, R.A.: Natural Frequencies of Highway Bridges in the org/10.1590/S1679-78252014000400006 New Madrid Region. Wayne State University, Detroit, Final Report, 36. Wood, J.H.; Shepherd, R.: Vehicle induced vibrations. Ministry of Civil Engineering Department (1996) Works and Development and the University of Auckland, Bulletin 38. Wright, D. T.; Green, R.: Highway Bridge Vibration. Part II: Report 44 (1977) No. 5 Ontario Test Programme. Ontario Department of Highways and Queen’s University. Kingston, Ontario (1964) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Arabian Journal for Science and Engineering Springer Journals

Development of Improved Frequency Expressions for Composite Horizontally Curved Bridges with High-Performance Steel Girders

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Engineering; Engineering, general; Science, Humanities and Social Sciences, multidisciplinary
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Abstract

Composite curved I-girder bridges are often used in modern highway systems, but the open sections of I-girders mean that these structures suffer from low torsional resistance. The curvature also results in quite complex behaviors due to the coupled bending and torsional responses of curved I-girder bridges. High-performance steel, which adds strength, enhances durability and improves weldability, addresses both the economic and structural problems associated with curved bridges. However, as yet, there are no simplified design methods in the form of practical equations with which to optimize the design parameters of curved bridges and their dynamic behavior remains controversial. This study evaluated the effects of various design parameters on the free vibration responses of curved HPS I-girder bridges. A sensitivity analysis of 278 prototype simple-span and continuous bridges was conducted using CSIbridge software to create a set of simple, practical expressions for the fundamental frequencies of these structures. Keywords Curved bridges · Frequency · Finite element method · High-performance steel 1 Introduction curved and straight bridge superstructures [4–8]. Many stud- ies have sought to define the exact behavior of curved or bent Horizontally curved high-performance steel (HPS) I-girder beams. In one of these earlier studies, Culver [9] developed bridges can offer an economical solution for modern high- several analytical models with which to investigate the inter- ways where roadway alignments need a smooth and curved actions between curved girders and cross-frames. The effect transition, reducing costs and controlling pollution due to of erection sequencing on the induced stress and deflections car emissions. In curved bridges, the centerlines of the girder has also been studied in an attempt to enhance the design webs in the sections rising up from the abutments are not guidelines and improve the stability of the bridges [10–12]. collinear with the cords between the abutments. These eccen- To reduce the complexity of curved bridge design and tricities result in high torsional moments, causing high out improve the capacity of design codes to predict the responses of plane deformations and rotations in the bridge cross sec- of curved bridges, several researchers have tried to model tions [1–3]. High-performance steel (HPS) is often utilized bridge superstructures as an assembly of simple systems, to address these problems as it provides more economic and for example by applying the grillage technique, orthotropic durable bridges, a significant advantage that enables engi- plate methods or other relatively simple forms [13–15]. How- neers to design longer and shallower bridges. To enhance ever, the accuracy of these methods remains problematic the torsional resistance of open-section bridges, cross-frames and the results produced may not be acceptable due to the and diaphragms are applied to interconnect girders in both simplifications involved. Although finite element modeling (FEM) has been found to be a reliable method for evaluating Junsuk Kang the performance of curved bridges, the time and resources windkplus@hotmail.com required mean that it is seldom feasible in a typical bridge design office, especially during the preliminary design stages Department of Landscape Architecture and Rural Systems Engineering, Seoul National University, Seoul 08826, [16,17]. Republic of Korea Several analytical and numerical analyses have been per- School of Architecture and Architecture Engineering, formed to evaluate the dynamic behavior of composite Hanyang University, Ansan, Gyeonggi-do 15588, Republic of I-girder bridges. Christiano and Culver [18] developed a the- Korea 123 Arabian Journal for Science and Engineering vibration or ensuring human comfort for those traveling over Wright and Green [29] Wood and Shepherd [31] them [23,25]. Several studies have shown that the current Billing[30] AASHTO serviceability criteria may be insufficient for con- Caneni [32] Trilly [34] trolling bridge vibration and frequency-based limits are more Dusseau [33] rational than span-based limits [25,26]. This paper presents the results of a simulation that applies CSIbridge V20 [27] to design and analyze the fundamental frequencies of curved HPS I-girder bridges. A parametric study examines the variables that could influence the free vibration characteristics of typical composite HPS I-girder 30 45 60 75 90 bridges. The results from a comparison study indicated that Span Length (m) existing frequency prediction equations are not sufficiently Fig. 1 Fundamental frequencies obtained using various equations accurate for curved composite I-girder bridges. Hence, non- linear regression methods are used to derive two sets of more reliable and practical expressions with which to estimate the oretical method using the differential equations governing fundamental frequencies of curved HPS I-girder bridges that the free vibration of a simple-span curved girder, deriving capture the effects of all the important parameters. two sets of equations to determine the fundamental frequen- cies of such bridges based on Lagrange multiplier concepts. 2 Finite Element Modeling and Verification Maneetes and Linzell [19] studied the effects of cross-frame and lateral bracing on the dynamic responses of a single-span curved I-girder bridge using both experimental and numerical In this section, the general attributes of the proposed finite solutions, while Yoon et al. [20] presented a finite element element modeling (FEM) using CSIbridge [27] are described. formulation based on a free vibration analysis of compos- The results obtained from two field tests and a laboratory ite bridges, taking into account the stiffness as well as the study are used to verify the effectiveness of the proposed mass matrices of the curved beam elements. The findings of modeling technique for curved HPS bridges. extensive studies have indicated that grillage methods are not in fact a reliable way to evaluate the dynamic behavior and 2.1 Bridge Sections free vibration of bridges [3]. Barth and Wu [21] performed a simulation and an experimental study to capture the vibra- A four-node shell element with six degrees of freedom at each tion characteristics (bending frequencies and mode shapes) node that combines membranes and plate-bending behavior of straight HPS girder bridges. was used to model the bridge’s concrete slab and steel gird- Figure 1 shows the fundamental frequencies values ers. Each element has its own local coordinate system for defined based on the equations proposed by various defining material properties and loads, and for interpreting researchers for steel girder bridges. According to the graph, the output. Stresses and internal forces and moments in the Trilly’s equation [22] generates the most conservative val- element local coordinate system were evaluated at the 2-by-2 ues for the fundamental frequency, although all follow the Gauss integration points and extrapolated to the joints of the same trend with increasing span length. Some past reports element. The bridge modeler in CSIbridge [27] was used to on vibration analysis are quite controversial, as they suggest model the prototype bridge. This option takes into account the very high values for the fundamental frequency. Wu [23] full composite action between the reinforced concrete slab summarized these reports, concluding that fundamental fre- and the HPS I-girders at the serviceability limit state, as rec- quencies are highly dependent on the geometry of bridges ommended by AASHTO LRFD [6]. Cross section properties and that beam theory and the simplified equations proposed were defined for all frame-elements and diaphragm proper- so far are not capable of predicting the correct values. Nas- ties imported (X or V types) and manually defined (built-up sif et al. [24] presented an investigation of vibration control plate sections). The parapets were modeled as a single frame (e.g., acceleration and velocity) in HPS bridges using an element. Note that it is critical to place the connecting nodes experimentally validated three-dimensional dynamic com- of the parapets at the centroid of the barrier section. Addition- puter model to evaluate both the deflection criterion and ally, the parapets were attached to the deck using a series of depth-to-span (D:L) limitation produced by the current rigid links in order to contribute to the stiffness of superstruc- design codes. They concluded that neither set of limita- ture. Figure 2 shows the parapet model assumptions applied. tions could provide effective vibration control for steel girder The discretization of the models was performed using bridges under normal truck traffic conditions and these lim- the auto-meshing option, defining a good expectation ratio its therefore represent a poor method for controlling bridge for all elements. The boundary conditions for simple-span Frequency (Hz) Arabian Journal for Science and Engineering of 90 kN, and then quickly releasing the applied load to induce free vibrations in both the longitudinal and transverse directions. Nearby abandoned railway bridge piers provided adequate anchor points for the pullback testing. The software capabilities of hybrid bridge evaluation system (HBES) were used to analyze records on site shortly after the data were collected. The resulting torsional and vertical modes exhib- ited some coupling, which is typical in this type of analysis when periods are close in value [30]. In order to comprehen- sively verify the results of the FEM, the entire bridge was simulated in the present study using the CSIbridge package Fig. 2 Parapet modeling assumptions software according to the prescribed modeling technique. The first three vertical frequencies obtained by the bridge model (with parapet effects) were 6.12, 7.48 and 8.75 Hz, differing from the field test frequencies of 5.95, 7.14 and 8.35 Hz by about 3.50, 4.54 and 4.72%, respectively. With- out modeling the parapet, the first three vertical frequencies obtained by the FEM method were 6.03, 7.25 and 8.50 Hz, which indicates that including the parapet shifts the results by 1.32, 1.52 and 1.76%, respectively. These results indicate that the FEM technique adequately predicts the vibration responses of composite girder bridges. Since the funda- mental frequency (the first natural frequency) is the most pronounced, the parapets may not noticeably influence the vertical frequency of these bridges. Fig. 3 Selected boundary conditions 2.2.2 Laboratory Test of a Quarter-Scale Model Bridge and continuous curved bridges were defined based on the Additional verification was performed through experimental proposed tangential bearing arrangement recommended by tests of a quarter-scale model of a 4.5 m long simple-span Samaan et al. [28], shown in Fig. 3. In the elastic range of straight I-beam bridge with five scaled girders at a spacing structural behavior, any slab cracking present exerts only a of 0.45 m and width of 1.80 m [31]. These I-beams had the negligible effect on the bridge responses [29]; hence, the slab following dimensions: flange thickness of 4.80 mm, flange was assumed to be uncracked and steel reinforcements were width of 58 mm, web thickness of 3.5 mm and total height of therefore not modeled. 203 mm. Channels with dimensions of 25 × 9.5 × 0.30 mm and a spacing of 15.8 cm were used as shear connectors. 2.2 Verification of Finite Element Models Here, solid diaphragms were applied at the abutments and spaced at one-third intervals along each span, consisting of 2.2.1 Colquitz River Bridge 76 × 50 × 52 mm angles welded to the beam webs about 13 mm below the top of the beam. A comparison of the FEM The results of dynamic field tests on the Colquitz river bridge and experimental results for the mid-span deflections of all in British Columbia, Canada [30] were selected to verify girders under four 20 kN concentrated loads at the mid-span the proposed FEM technique. The bridge has five spans and symmetric about the longitudinal axis of bridge is shown (2 × 14 m + 2 × 17 m + 18 m), each composed of six steel in Fig. 4. girders W 33 × 141, and a bridge width and slab thickness Although the experimental and FEM results compare of 11.9 and 0.10 m, respectively. The diaphragms and cross- fairly well, the FEM results are slightly higher than those braces members are made of U-shaped steel (MC18 × 42.7) obtained in the laboratory, varying by up to 1.5, 4.6 and 3.4% and spaced 3.5 and 4.5 m apart along the end and interme- for the external, intermediate and central girders, respec- diate spans, respectively, of the bridge. For the ambient test, tively. This discrepancy is likely because the experimental six force-balanced accelerometers were used to collect and loads were not localized at an exact position but rather dis- process acceleration data during the day and under varying tributed over the entire steel supporting plate. The same traffic conditions. Pullback tests were carried out by load- phenomenon has also been reported by Wegmuller [32]in ing the bridge bents with a special assemblage using a force a comparison between FEM and experimentally obtained 123 Arabian Journal for Science and Engineering Fig. 5 FE model of a 30 m span length bridge with k = 0.1 Fig. 4 Deflection distribution in the bridge girders at the mid-span cross section ues for the slenderness ratio increasing from 0 to 1.20 at 0.2 intervals. Three types of cross section, with girder spacings results. This suggests that the proposed FEM technique of 2.8, 3.5, and 2.5 m, were utilized, along with the Ameri- accurately predicts the response of I-girder bridges to static can Iron and Steel Institute (AISI) standard parapet design, loading. which has a height of 0.87 m and bottom and top widths of 0.4 and 0.15 m, respectively. The prototype bridges were mod- eled using three different high-performance steel materials: 2.3 Geometric and Material Properties of Bridges HPS 50 W (Fy = 345 Mpa), HPS 70 W (Fy = 485 Mpa) and HPS 100 W (Fy = 620 Mpa). The concrete deck was mod- High-performance steel (HPS) bridges need less steel due eled with a modulus of elasticity of 28,000 Mpa, a Poisson’s to their higher yield stress, but consequently the live-load ratio of 0.2 and a density of 24 kN/m . The concrete deck deflections of these bridges are more likely to exceed thickness was 0.20 m for all the cross sections. AASHTO’s deflection limits [5]. As design optimization must take into account the AASHTO limits on performance and economy, an important first step is to select a set of key parameters and establish a matrix that covers a wide range of steel bridge types. Based on this database, bridges can then 3 Sensitivity Analysis be designed and optimized for various combinations of these parameters to capture a least weight approach using CSIb- A sensitivity analysis provides useful information regard- ridge [27]. The initial designs are performed by neglecting ing the influence of various parameters on the free vibration the AASHTO criteria for deflection in order to design girders responses of bridges. The goal here is to characterize the that meet other AASHTO LRFD strength and serviceabil- key parameters affecting a bridge’s natural frequency so that ity criteria for curved bridges. Any girders that then fail the they can be taken into account in the development of effective AASHTO live-load deflection criteria must be redesigned. expressions for natural frequencies. The investigation vari- Table 1 shows the geometric properties and material param- ables are the span length, the span-to-radius curvature ratio eters for three prototype bridges. Figure 5 shows a typical (k), the number of girders, the girder spacing and the number 30 m span length bridge modeled in CSIbridge [27]. and arrangement of the cross-braces. A detailed discussion Four different span lengths (L), 30, 45, 60, and 90 m, were of each of these parameters is presented in this section, and considered in this investigation, with span-to-radius (k)val- the key parameters are identified. Table 1 Characteristics of three Set L (m) HPS types (W) L/D ratio S (m) No. girder W (m) prototype curved I-girder bridges 1 30, 45, 60, 75, 90 50 20 2.80 5 13.0 2 30, 45, 60, 75, 90 70 25 3.50 4 13.0 3 30, 45, 60, 75, 90 100 30 2.50 4 9.50 123 Arabian Journal for Science and Engineering f1 f2 f3 f4 f5 Davidson’s equaon 23 45 67 89 No. cross braces Fig. 6 Effect of the number of cross-bracing lines on a bridge’s natural frequencies Fig. 7 Effect of cross-brace type on a bridge’s natural frequencies 3.1 Effect of the Number of Cross-braces Curved girder bridges have a high torsional moment due frequencies. For example, enhancing the thickness of the to their curvature. The open nature of these bridges results solid steel plates from 10 to 25 mm changes the first and in low torsional resistance to cracking, so cross-braces are second frequencies by up to 2%. However, although increas- applied to enhance their torsional stiffness. It is thus vital to ing the plate thickness also increases the natural frequencies determine the effect of this variable on the bridge’s free vibra- of the higher modes, it should be avoided due to possible tion responses. The effect of the number of cross-bracing brittle connection failures of the diaphragms under seismic lines on the first natural frequency of a 30 m long bridge with loads. Thus, it is preferable to adjust the thickness of the a curvature ratio of 1.20 is shown in Fig. 6. The minimum steel X-bracing rather than the plate thickness to enhance required number of cross-braces obtained from Davidson’s the torsional resistance of both single-span and continuous equation [33] is also indicated. Although the number of bridges. braces appears to have only an insignificant effect on the first and second frequencies and the vertical mode shapes of the bridge, it does decrease the natural frequency values for 3.3 Effect of the Span Length the higher modes of vibration. Thus, for all the prototype bridges modeled in this study, cross-braces are placed 6 m Many researchers have described the importance of span apart, with a minimum of five bracing lines for each span. length for the dynamic responses of straight bridges. For example, the current North American codes [5,6] offer equa- 3.2 Effect of the Cross-brace Type tions for the dynamic impact factor of bridges as a function of span length. Wright and Walker [34] developed the follow- In order to study the effect of the various types and stiffnesses ing theoretical equation based on beam theory to determine of bracing systems on the free vibration responses of curved the fundamental frequencies. Note that this equation incor- HPS bridges, several different bracing configurations were porates the span length as an important factor: modeled for a 90 m long bridge with a curvature ratio of 0.60. The following bracing systems were considered: X- type, V-type, inverted V-type, single beam and solid plates; π E · I · g b b f = (1) the results are shown in Fig. 7. Once again, although the sb 2L w type of bracing systems had an insignificant impact on the first natural frequency of the bridge, it did have an important impact on the higher frequencies. For instance, the second where E · I and w are the flexural rigidity and the weight b b natural frequency of a bridge with X-type bracing was up to per unit length of the composite steel girder, respectively. 12, 48 and 54% higher than bridges with V-, inverted V- and Figure 8 shows the effect of span length on the natural fre- single beam bracing systems, respectively. The effects of the quencies of bridges with a slenderness ratio (L/D)of25and stiffness of the cross-bracing and the end diaphragms on the a curvature ratio of 0.20. Increasing the span length from 30 free vibration of the bridge are also shown in Fig. 7. to 90 m decreases the natural frequencies by up to 65%. From These results suggest that the stiffness of secondary mem- a practical standpoint, the span length is thus an important bers has a significant influence on the bridge’s fundamental parameter affecting the dynamic response of a curved bridge. Frequency (Hz) Arabian Journal for Science and Engineering 10 4 f1 k=0.2, L/D=25 f2 3.5 L=30m, L/D=25, 50W f3 L=30m, L/D=25, 70W f4 L=30m, L/D=30, 70W 2.5 L=60m, L/D=20, 70W L=60m, L/D=20, 50W 1.5 0.5 30 45 60 75 90 0.2 0.4 0.6 0.8 1 1.2 Span Length (m) k=L/R Fig. 8 Effect of span length on the fundamental frequencies of curved Fig. 10 Effects of the bridge curvature ratio on the fundamental fre- bridges (k = 0.20, L/D = 25) quency f1 L=60m, k=0.60 f2 derness ratios (L/D). The graph indicates that the curvature 7 f3 ratio has an important effect on the free vibration of curved f4 f5 superstructures, with an increase in the curvature ratio sig- nificantly reducing the natural frequencies. For instance, for the 60 and 90 m bridges, the natural frequencies decrease by up to 56 and 39%, respectively, when the span-to radius ratio increases from 0 to 1.20. The curvature ratio also has a significant impact on the mode shapes. Figure 11 shows the first three mode shapes for both Number of Girders straight and curved bridges with 30 m long spans. These reveal that any curvature in a bridge exerts a higher torsional Fig. 9 Effect of the number of girders on a bridge’s natural frequencies impact on mode shapes. For example, the first mode shape for a straight bridge is purely flexural, but in a curved bridge it consists of a combination of flexural and symmetric torsional 3.4 Effect of the Number of Girders modes. Thus, the effect of the curvature ratio on both the mode shapes and frequencies will be significant and should Analyzing the performance of a 60 m long curved bridge with thus be included in the parametric study. a curvature ratio of 0.6 to investigate the effect of the number of girders on the bridge’s natural frequencies revealed that 3.6 Effect of Slenderness (Span-to-Depth) Ratio although increasing the number of girders results produced higher torsional stiffness in the structure, the natural fre- AASHTO (5) specifications limit the value of the slenderness quencies of the bridge’s superstructure were enhanced only ratio to a maximum of 25 in order to control the maximum slightly. The results, presented in Fig. 9, show that the first live-load deflection through controlling the bridge’s bending and second frequencies increase by approximately 3% as the stiffness. However, this approach became somewhat con- number of girders increases from three to seven, for example. troversial with the introduction of high-performance steel In practice, this does not influence the free vibration (HPS), which has approximately 40% higher yield stress than sufficiently to justify its inclusion as an important model conventional steel and hence allows engineers to design shal- parameter. These relatively small changes are due to the fact lower and longer spans. Since HPS bridges are more likely that the prototype bridges have almost the same flexural stiff- to suffer from large deflections, the slenderness ratio plays ness, irrespective of the number of girders. Previous research a significant role in the dynamic behavior of these systems. has also indicated that the number of girders has no impact The effects of different slenderness ratios on the natural fre- on the natural frequencies of straight bridges [35]. quencies of a 30 m long bridge are presented in Fig. 12.The natural frequencies decrease almost linearly as the L/D ratio 3.5 Effect of the Span-to-Radius Curvature Ratio increases, irrespective of the curvature ratio. For example, in a bridge with a curvature ratio of 0.80, the first natural Figure 10 shows the relationship between the first natural frequency decreases by approximately 28% when the L/D frequencies (f1) and curvature ratios (k = L/R)offive ratio increases from 20 to 30. From a practical standpoint, bridges with lengths of 30, 60 and 90 m and various slen- the impact of the slenderness ratio is considerable and should Frequency (Hz) Frequency (Hz) First Frequency (Hz) Arabian Journal for Science and Engineering Fig. 11 First three modes of vibration for straight and curved bridges. a Straight bridges and b curved bridges 1.8 proposed equations adequately predict the fundamental fre- k=0.20 simple supported bridge, L=30m 1.6 quency of curved composite girder bridges constructed using k=0.40 k=0.80 high-performance steel. While Wood and Shepherd’s equa- 1.4 k=1.20 tion [36] estimates highly conservative values, the models 1.2 proposed by Dusseau [37] and Wright and Walker [38] both significantly underestimate the fundamental frequencies of 0.8 such bridges. To address this problem, the data for 180 sim- 0.6 ple supported and 85 two- and three-span continuous bridges were collected and analyzed to develop a set of proposed 0.4 20 25 30 expressions to describe their fundamental frequencies. The Slenderness rao (L/D) proposed expressions take the following form: Fig. 12 Effect of slenderness ratio (L/D) on a bridge’s natural fre- quency f = ψ · f (2) c·b sb thus be taken into account in any parametric study of curved HPS I-girder bridges. where f is the fundamental frequency of a straight bridge sb from Eq. (1). A modification factor, ψ, is applied to take into account the effect of the parameters from the previous 4 Empirical Expressions for Fundamental section that were found to affect the free vibration of curved Frequency HPS I-girder bridges. A regression analysis using a statisti- cal computer package for best fit based on the least squares As the comparison of the various previous studies that have method for nonlinear data was used to deduce appropriate investigated this issue presented in Fig. 1 shows, none of the modification factors as follows: Frequency (Hz) Arabian Journal for Science and Engineering Fig. 13 Comparison of the 4 4 results obtained by the proposed expression and the FEM analysis. a Simple-span and b R² = 0.9739 3 3 continuous span R² = 0.9772 2 2 0 0 012 34 Frequency FEM Analysis (Hz) Frequency FEM Analysis (Hz) (a) (b) Table 2 Comparison between No. spans L/DL (m) L/Rf (Hz) ψ Frequency (Hz) sb the FEM results and those f = ψ · f FE c·b sb obtained using the proposed expressions 120 30 1.0 0.5966 2.240 1.337 1.375 125 30 0.80 0.6865 1.930 1.328 1.281 130 30 1.0 0.5910 1.980 1.172 1.102 230 30 1.20 0.8435 2.290 1.936 2.020 230 60 0.20 1.2063 1.050 1.275 1.250 � Simple-span bridges I-girder bridges. This numerical approach includes an exten- sive sensitivity study of simple supported and continuous −0.82k 0.018S bridges to determine the effect of a number of different vari- ψ = 1.160 × e × e × 0.80 + 0.008 (3) ables on the natural frequencies and mode shapes of the bridge structures. Empirical expressions for the fundamental � Continuous bridges frequencies of such bridges were derived that are suitable for use in design codes and engineering offices. The proposed −0.37k 0.005 ψ = 1.137 × e × e (4) expressions are a function of the natural frequency obtained from flexural beam theory and are hence applicable to bridges It should be recognized that the effect of the slenderness with various slenderness ratios. Based on the results obtained ratio in the proposed expressions for the modification from the sensitivity study, the following conclusion can be factor is negligible because the impact of the span length derived: and depth has already been taken into account in Eq. (1). The results of the verification analyses for the proposed � The slenderness ratio significantly affects the natural expressions are shown in Fig. 13 and Table 2. frequency of such bridges: the fundamental frequency decreases with increasing span-to-depth ratio. Examining the first natural frequency for both simple-span � The magnitude of the fundamental frequency decreases and continuous curved bridges reveals that the difference with increasing span-to-radius curvature ratio for hor- between the FEM analyses results and those obtained using izontally curved I-girder bridges. The curvature ratio the expressions proposed here are within 1%. The high coef- also has a significant effect on the bridge’s vibration ficient of determination, R , obtained in both cases indicates modes due to the high contribution of the torsional effects the minimal variation in the results calculated by the pro- induced in curved superstructures. posed expressions and the FEM analysis. � Cross-braces between the abutments increase the tor- sional stiffness of open-section bridges. It is recom- mended that the maximum spacing for bracing lines be 5 Conclusions limited to 6 m in order to exert a meaningful effect on bridge responses. This study has presented a detailed numerical analysis of the free vibration characteristics of horizontally curved HPS Frequency Eq.(3) Frequency Eq.(4) Arabian Journal for Science and Engineering Acknowledgements The work reported herein was supported by a 15. Linzell, D.; Chen, A.; Sharafbayani, M.; Seo, J.; Nevling, D.; Grant (18CTAP-C132633-02) funded by the Ministry of Land, Infras- Ard, T.J.; Ashour, O.: Guidelines for Analyzing Curved and tructure and Transport (MOLIT) of the Korean Agency for Infras- Skewed Bridges and Designing Them for Construction. Project No. tructure Technology Advancement (KAIA), the National Research PSU-009. Commonwealth of Pennsylvania Department of Trans- Foundation of Korea (NRF) Grant (NRF-2016R1C1B1015711) funded portation. Final Report, US (2010) by the Korea government (Ministry of Science, ICT & Planning) and the 16. Samaan, M.; Sennah, K.; Kennedy, J.B.: Distribution factors 2017 Seoul National University Invitation Program for Distinguished for curved continuous composite box-girder bridges. J. Bridge Scholars. This financial support is gratefully acknowledged. 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Arabian Journal for Science and EngineeringSpringer Journals

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