Lateral Pedestrian Excitation of Footbridges

The phenomenon of synchronized lateral pedestrian excitation of footbridges has been highlighted in recent years by the widely publicized problems encountered with the London Millennium Footbridge. On opening day, during which a large number of pedestrians crossed the bridge, it exhibited alarming and unstable amplitudes of lateral vibration, which resulted in an 18 months closure and a 7 million dollar retrofit. Subsequent investigations revealed that the dynamic characteristics of the bridge did not differ significantly from those of other footbridges of similar span, and that the phenomenon of synchronized lateral excitation had recently been exhibited by several other bridges. An analysis of lateral pedestrian excitation of bridges, based on the governing differential equations of motion, is presented. Analytical models of lateral pedestrian loading are developed which can be used to assess both the subcritical response and the critical number of pedestrians likely to induce synchronized excitation. Theoretical predictions of synchronization are compared, and show close correlation, with the results of full-scale tests on the Millennium Footbridge.     

Combined Experimental-Operational Modal Testing of Footbridges

In combined vibration testing, an artificial, measured force is used in operational conditions. This requires the identification of a system model that takes both the measured and the operational excitation into account. Advantages with respect to the classical operational modal analysis approach are the possibility of obtaining mass-normalized mode shapes and the increase of the excitation level and its frequency content. An advantage with respect to the classical experimental modal analysis approach, where the ambient excitation is not modeled, but considered as disturbing noise, is the possibility of using excitation levels that are of the same amplitude, or even smaller, than the ambient excitation levels. In this paper, combined modal testing of footbridges is explored using two case studies: a steel arch footbridge with spans of 75.2 m and 30.3 m and a concrete stress-ribbon footbridge with spans of 30 m and 28 m. The comparison of the modal parameters (eigenfrequencies, damping ratios, mode shapes, and modal scaling factors) obtained from a combined vibration test with the ones obtained from other modal tests and from a finite-element model, demonstrates the feasibility of using small and practical excitation devices for the modal testing of footbridges.     

Reliability-Based Optimum Design of Glulam Cable-Stayed Footbridges

This work presents a procedure for finding the reliability-based optimum design of cable-stayed bridges. The minimization problem is stated as the minimization of stresses, displacements, reliability, and bridge cost. A finite-element approach is used for structural analysis. It includes a direct analytic sensitivity analysis module, which provides the structural behavior responses to changes in the design variables. An equivalent multicriteria approach is used to solve the nondifferential, nonlinear optimization problem, turning the original problem into sequential minimization of unconstrained convex scalar functions, from which a Pareto optimum is obtained. Examples are given illustrating the procedure.     

Vandal Loads and Induced Vibrations on a Footbridge

The paper describes a study to characterize intentional human dynamic loads (vandal loads) and their induced effects on a footbridge. A numerical model to characterize the dynamic behavior of the footbridge has been developed, is experimentally validated, and is used to numerically simulate the response induced by groups of pedestrians synchronized at critical bridge frequencies. The vandal load associated with a single pedestrian is characterized and compared with definitions in the literature. The response is then calculated considering the measured load and compared with the measured response to the excitation induced by a single pedestrian or a group with varying dimensions. The definition of a general load model for vandal excitation, including a tentative definition for synchronization is discussed.     

Natural Frequency of Railway Girder Bridges under Vehicle Loads

To investigate the natural frequency of a railway girder bridge under vehicle loads, two methods are presented. First, the natural frequency of a railway girder bridge under vehicle loads is obtained by solution of the eigenvalue of the vehicle-bridge interaction equation at each step of the numerical integration. Second, based on the vehicle-bridge interaction equation, an approximate formula is developed. The results show that the natural frequency of a railway girder bridge under vehicle loads varies periodically as the vehicles pass over the bridge. The results obtained with the two methods are then compared, showing that a good agreement is achieved. From parametric studies, the effects of the unsprung mass, the sprung mass, and the stiffness of the vehicle suspension are discussed.     

Effect of Hanger Flexibility on Dynamic Response of Suspension Bridges

Linearized continuum models of a suspended span with unloaded backstays and of a symmetric three-span suspension bridge are used to study the effects of the flexibility of the hangers on the vertical vibrations of suspension bridges. The models include elastic parabolic cables, flexible distributed hangers with variable length, and a stiffening girder represented by an elastic beam. It is shown that the free vibrations of a suspended span with unloaded backstays are controlled by five dimensionless parameters, while six dimensionless parameters control the response of a symmetric three-span suspension bridge. The results indicate that the flexibility of the hangers has a significant effect on the natural frequencies of the higher modes only when the relative stiffness of the girder is very high. The effects of hanger flexibility on the response of a suspension bridge to localized impulsive loads are also found to be small. These findings confirm the traditional, albeit previously untested, assumption of inextensible hangers. Finally, the threshold amplitudes of free vibrations that would result in the incipient slackening of the hangers are determined.     

Lateral Load Distribution in Curved Steel I-Girder Bridges

The purpose of this paper is to develop new formulas for live load distribution in horizontally curved steel I-girder bridges. The formulas are developed by utilizing computer model results for a number of different horizontally curved steel I-girder bridges. The bridges used in this study are modeled as generalized grillage beam systems composed of horizontally curved beam elements for steel girders and substructure elements for lateral wind bracing and cross frames which consist of truss elements. Warping torsion is taken into consideration in the analysis. The effect of numerous parameters, including radius of curvature, girder spacing, overhang, etc., on the load distribution are studied. Key parameters affecting live load distribution are identified and simplified formulas are developed to predict positive moment, negative moment, and shear distribution for one-lane and multiple-lane loading. Comparisons of the formulas with finite element method and grillage analysis show that the proposed formulas have more accurate results than the various available American Association of State Highway and Transportation Officials specifications. The formulas developed in this study will assist bridge engineers and researchers in predicting the actual live load distribution in horizontally curved steel I-girder bridges.     

Influence of Secondary Elements and Deck Cracking on the Lateral Load Distribution of Steel Girder Bridges

The AASHTO LRFD load distribution factor equation was developed based on elastic finite element analysis considering only primary members, i.e., the effects of secondary elements such as lateral bracing and parapets were not considered. Meanwhile, many bridges have been identified as having significant cracking in the concrete deck. Even though deck cracking is a well-known phenomenon, the significance of pre-existing cracks on the live load distribution has not yet been assessed. The purpose of this research is to investigate the effect of secondary elements and deck cracking on the lateral load distribution of girder bridges. First, secondary elements such as diaphragms and parapets were modeled using the finite element method, and the calculated load distribution factors were compared with the code-specified values. Second, the effects of typical deck cracking and crack types that have a major effect on load distribution were identified through a number of nonlinear finite element analyses. It was established that the presence of secondary elements may produce load distribution factors up to 40% lower than the AASHTO LRFD values. Longitudinal cracking was found to increase the load distribution factor by up to 17% when compared to the LRFD value while the transverse cracking was found to not significantly influence the transverse distribution of moment.     

Equivalent Barge and Flotilla Impact Forces on Bridge Piers

Bridge piers located in navigable inland waterways are designed to resist impact forces from barges and flotillas in addition to other design considerations (e.g., scour dead, live loads, etc.). The primary design tool for estimating these forces is the AASHTO Guide Specification that provides a simple hand calculation method to determine an “equivalent impact force.” The simplicity comes at a cost of excluding the effect of the pier shape, impact duration, and interaction between barges in a flotilla. The objective of this paper is to present a hand calculation method for determining barge or flotilla impact forces on bridge piers. The primary advantage of this approach lies in its incorporation of pier geometry, interaction between barges, and impact duration. The proposed method is derived from the conduct of hundreds of finite-element dynamic simulations of jumbo hopper (JH) barges and flotillas, made up of JH barges, impacting bridge piers. Results are presented and compared with those derived from the AASHTO method and detailed finite-element modeling.     

Experimental Assessment of Dynamic Responses Induced in Concrete Bridges by Permit Vehicles

Results from experimental testing of three permit vehicles are presented in the paper. The selected heavy vehicles, which require permits from state DOTs, included two tractor-trailer systems and a midsize crane. The vehicles were experimentally tested on popular existing speed bumps and on a representative highway bridge. The selected bridge was a reinforced-concrete structure constructed in 1999, located on the U.S. 90 in Northwest Florida. The bridge approach depression, combined with a distinct joint gap between the asphalt pavement and the concrete deck, triggered significant dynamic responses of the vehicle-bridge system. Similar dynamic vibrations were observed and recorded when the permit vehicles were driven over the speed bumps. Time histories of relative displacements, accelerations, and strains for selected locations on the vehicle-bridge system were recorded. The analysis of experimental data allowed for assessment of actual dynamic interactions between the vehicles and the speed bumps as well as dynamic load allowance factors for the selected bridge.     

Combined Damping Effect of Two Dampers on a Stay Cable

The combined effect of two dampers, either on the same end or opposite ends of a stay cable, is analytically studied in this paper. By considering small distances of the dampers from the anchorages, an asymptotic formula for the modal damping ratio of the cable is derived from which the total damping effect is presented in an explicit form. It is shown that when two dampers are installed at opposite ends of the cable, the total damping effect is asymptotically the sum of the contributions from single dampers. On the contrast, if two dampers are at the same end, there is no advantage of increasing the maximum modal damping in the cable over the use of a single damper.     

Analysis and Solution to Human-Induced Lateral Vibrations on a Historic Footbridge

Recently, it has come to light that footbridges may have problems with human induced vibrations if their lateral period of vibration is of the order of 1 s. This has been highlighted by the much-publicized case of the Millennium Footbridge in London. This paper describes a similar problem associated with the historic Cragside Bridge built circa 1875. This bridge is a wonderful historic bridge built in the grounds of Cragside Manor in Northumberland, United Kingdom by the famous Industrialist Lord Armstrong. An investigation of the bridge revealed that it suffered vibration problems resulting from insufficient lateral stiffness. When lightly loaded (approximately six people) the fundamental lateral natural frequency of vibration was 2.6 Hz. This mode was predominantly a lateral mode of vibration. A computer model of the bridge was developed and various loading scenarios were investigated. As the bridge is very lightweight, the computer model showed that the natural frequency could reduce to 1.12 Hz if the bridge was loaded to 4 kPa. Extra stiffening elements were added to the computer and, depending on the superimposed load, the frequency could be increased from 1.12 to 1.57 Hz (for a 4 kPa uniform live load) and increased to 3.67 Hz for an imposed load equivalent to approximately nine people. It is believed that this will help to remedy the problem, but further research is being conducted to reduce the problem further.     

Development of FRP Short-Span Deployable Bridge—Experimental Results

For military and civilian applications, there exists a need for lightweight, inexpensive, short-span bridges that can be easily transported and erected with minimal equipment. Owing to its favorable properties, fiber-reinforced polymer (FRP) has been shown to be feasible for the construction of such bridges. Investigations into the behavior of a short-span bridge structural concept, adapted to the material properties of commercially available glass FRP (GFRP) pultruded products, are presented. A 4.8-m span prototype was built from GFRP sections, bonded throughout to form a tapered box beam, with a width of 1.2?m and a height at midspan of approximately 0.5?m. The box beam represents a single trackway of a double-trackway bridge, whose trackways could be connected by light structural elements. The quasi-static and dynamic behavior of the prototype box beam was investigated in ambient laboratory and field conditions to assess the design and construction techniques used, with a view to designing a full-scale 10-m GFRP bridge. Laboratory testing of the prototype box beam used single and pairs of patch loads to simulate wheel loading. These tests confirmed that the box beam had sufficient stiffness and strength to function effectively as a single trackway of a small span bridge. Field testing of the structure was undertaken using a Bison vehicle (13,000?kg), driven at varying speeds over the structure to establish its response to realistic vehicle loads and the effects of their movement across the span.     

Truck Loading and Fatigue Damage Analysis for Girder Bridges Based on Weigh-in-Motion Data

Based on data collected by weigh-in-motion (WIM) measurements, truck traffic is synthesized by type and loading condition. Three-dimensional nonlinear models for the trucks with significant counts are developed from the measured data. Six simply supported multigirder steel bridges with spans ranging from 10.67 m (35 ft) to 42.67 m (140 ft) are analyzed using the proposed method. Road surface roughness is generated as transversely correlated random processes using the autoregressive and moving average model. The dynamic impact factor is taken as the average of 20 simulations of good road roughness. Live-load spectra are obtained by combining static responses with the calculated impact factors. A case study of the normal traffic from a specific site on the interstate highway I-75 is illustrated. Static loading of the heaviest in each truck type is compared with that of the American Association of State Highway and Transportation Officials standard design truck HS20-44. Several important trucks causing fatigue damage are found.     

Dynamic Response of Suspension Bridge to High Wind and Running Train

A framework is presented for predicting the dynamic response of long suspension bridges to high winds and running trains. A three-dimensional finite-element model is used to represent a suspension bridge. Wind forces acting on the bridge, including both buffeting and self-excited forces, are generated in the time domain using a fast spectral representation method and measured aerodynamic coefficients and flutter derivatives. Each 4-axle vehicle in a train is modeled by a 27-degrees-of-freedom dynamic system. The dynamic interaction between the bridge and train is realized through the contact forces between the wheels and track. By applying a mode superposition technique to the bridge only and taking the measured track irregularities as known quantities, the number of degrees of freedom of the bridge-train system is significantly reduced and the coupled equations of motion are efficiently solved. The proposed formulation is then applied to a real wind-excited long suspension bridge carrying a railway inside the bridge deck of a closed cross section. The results show that the formulation presented in this paper can predict the dynamic response of the coupled bridge-train systems under fluctuating winds. The extent of interaction between the bridge and train depends on wind speed and train speed.     

Experimental and Analytical Studies of a Horizontally Curved Steel I-Girder Bridge during Erection

A series of studies on an experimental, full-scale curved steel bridge structure during erection are discussed. The work was part of the Federal Highway Administration’s curved steel bridge research project (CSBRP). The CSBRP is intended to improve the understanding of curved bridge behavior and to develop more rational design guidelines. The main purpose of the studies reported herein was to assess the capability of analytical tools for predicting response during erection. Nine erection studies, examining six different framing plans, are presented. The framing plans are not necessarily representative of curved bridge subassemblies as they would be erected in the field; however, they represent a variety of conditions that would test the robustness of analysis tools and assess the importance of erection sequence on initial stresses in a curved girder bridge. The simply supported, three I-girder system used for the tests is described and methods for reducing and examining the data are discussed. Comparisons between experimental and analytical results demonstrate that analysis tools can predict loads and deformations during construction. Comparison to the V-load method indicates that it predicts stresses in exterior girders well, but can underpredict them for interior girders.     

Equivalent Wheel Load Approach for Slender Cable-Stayed Bridge Fatigue Assessment under Traffic and Wind: Feasibility Study

In the current AASHTO LRFD specifications, the fatigue design considers only one design truck per bridge with 15% dynamic allowance. While this empirical approach may be practical for regular short and medium span bridges, it may not be rational for long-span bridges (e.g., span length >152.4?m or 500?ft) that may carry many heavy trucks simultaneously. Some existent studies suggested that fatigue may not control the design for many small and medium bridges. However, little research on the fatigue performance of long-span bridges subjected to both wind and traffic has been reported and if fatigue could become a dominant issue for such a long-span bridge design is still not clear. Regardless if the current fatigue design specifications are sufficient or not, a real understanding of the traffic effects on bridge performance including fatigue is desirable since the one truck per bridge for fatigue design does not represent the actual traffic condition. As the first step toward the study of fatigue performance of long-span cable-stayed bridges under both busy traffic and wind, the equivalent dynamic wheel load approach is proposed in the current study to simplify the analysis procedure. Based on full interaction analyses of a single-vehicle–bridge–wind system, the dynamic wheel load of the vehicle acting on the bridge can be obtained for a given vehicle type, wind, and driving condition. As a result, the dimension of the coupled equations is independent of the number of vehicles, through which the analyses can be significantly simplified. Such simplification is the key step toward the future fatigue analysis of long-span bridges under a combined action of wind and actual traffic conditions.     

Experimental Dynamic Response of a Short-Span Composite Bridge to Military Vehicles

A continued desire for increased mobility in the aftermath of natural disasters or on the battlefield has lead to the need for improved lightweight bridging solutions. Currently, within the U.S. military, there is a need for a lightweight bridging system for crossing short-span gaps up to 4 m (13.1 ft) in length. This paper describes the field testing of a newly developed lightweight fiber-reinforced polymer bridging system to meet the U.S. militaries needs. The study investigates dynamic impact loads of track and wheel vehicles at different crossing speeds to increase understanding of appropriate impact factors used in design. It was found that the impact loads for the bridge treadways were most sensitive to vehicle crossing speed and vehicle type (wheel versus track and axle spacing) with observed impact factors as high as 1.71.     

Dynamic Modeling of Large-Scale Magnetorheological Damper Systems for Civil Engineering Applications

Magnetorheological (MR) dampers are one of the most promising new devices for structural vibration mitigation. Because of their mechanical simplicity, high dynamic range, low power requirements, large force capacity, and robustness, these devices have been shown to mesh well with earthquake and wind engineering application demands and constraints. Quasistatic models of MR dampers have been investigated by researchers. Although useful for damper design, these models are not sufficient to describe the MR damper behavior under dynamic loading. This paper presents a new dynamic model of the overall MR damper system which is comprised of two parts: (1) a dynamic model of the power supply and (2) a dynamic model of the MR damper. Because previous studies have demonstrated that a current-driven power supply can substantially reduce the MR damper response time, this study employs a current driver to power the MR damper. The operating principles of the current driver, and an appropriate dynamic model are provided. Subsequently, MR damper force response analysis is performed, and a phenomenological model based on the Bouc–Wen model is proposed to estimate the MR damper behavior under dynamic loading. This model accommodates the MR fluid stiction phenomenon, as well as fluid inertial and shear thinning effects. Compared with other types of models based on the Bouc–Wen model, the proposed model has been shown to be more effective, especially in describing the force rolloff in the low velocity region, force overshoots when velocity changes in sign, and two clockwise hysteresis loops at the velocity extremes.     

Dynamic Response of Three Fiber Reinforced Polymer Composite Bridges

Fiber reinforced polymer (FRP) composite bridge decks are gaining the attention of bridge owners because of their light self-weight, corrosion resistance, and ease of installation. Constructed Facilities Center at West Virginia University working with the Federal Highway Administration and West Virginia Department of Transportation has developed three different FRP decking systems and installed several FRP deck bridges in West Virginia. These FRP bridge decks are lighter in weight than comparable concrete systems and therefore their dynamic performance is equally as important as their static performance. In the current study dynamic tests were performed on three FRP deck bridges, namely, Katy Truss Bridge, Market Street Bridge, and Laurel Lick Bridge, in the state of West Virginia. The dynamic response parameters evaluated for the three bridges include dynamic load allowance (DLA) factors, natural frequencies, damping ratios, and deck accelerations caused by moving test trucks. It was found that the DLA factors for Katy Truss and Market Street bridges are within the AASHTO 1998 LRFD specifications, but the deck accelerations were found to be high for both these bridges. DLA factors for Laurel Lick bridge were found to be as high as 93% against the typical design value of 33%; however absolute deck stress induced by vehicle loads is less than 10% of the deck ultimate stress.