The roughness of the bridge frame was considered as the realization of a homogeneous random process in space. The generalized power spectral density of the unevenness of the road surface was considered to model the random entry into the vehicle.
Boundary Conditions 85
Model Description and Evaluation of Material Properties 109
Natural Frequencies and Damping Ratio 117
Bridge Deck Roughness 139
ANALYSIS OF RESULTS AND SUMMARY OF CONCLUSIONS 214
DETERMINATION OF CROSS–SECTIONAL PROPERTIES OF THIN-
FLOW CHART FOR FINITE ELEMENT ANALYSIS (STATIC) OF CURVED
DYNAMIC ANALYSIS PROCEDURE FOR VEHICLE-BRIDGE
DETERMINATION OF EQUIVALENT LOAD VECTOR
FLOW CHART FOR EVALVATION OF STRESSES FOR CURVED BOX- GIRDER BRIDGE
LIST OF TABLES
106 Fig. 3.1 Geometry (inch) and properties of the box girder model 113
NOMENCLATURE
Nj Number of cycles at constant stress range level Srj from (S-N curve) to cause failure.
INTRODUCTION AND LITERATURE REVIEW
INTRODUCTION
It is therefore necessary to use a mathematical model to be able to describe the dynamic behavior of the supporting structure and vehicles in terms of interaction systems. In particular, the expected service life of the highway bridge exposed to random variable amplitude traffic cycles is highly dependent on the accumulation of damage caused by various fatigue mechanisms.
LITERATURE REVIEW
- Structural Action of Thin-Walled Members
- Finite Element Method of Analysis of Thin-Walled Box-Girder Bridges Different numerical methods of analyses of thin-walled box bridge include
- Free Vibration of Thin-Walled Box-Girder Bridges
- Vehicle-Bridge Interaction
- Fatigue due to Vehicle Induced Loads
Moffat and Dowling (1975) investigated the phenomenon of shear layers in steel box bridges using the finite element analysis. The effect of time-dependent forces on bridges was investigated by Coussy et al. 1988). The authors analyzed the influence of surface irregularities on the dynamic response of bridges by modeling the surface irregularities using PSD function.
SCOPE AND OBJECTIVES OF THE PRESENT INVESTIGATION
This is particularly important in the case of curved thin-walled box girder configurations due to the coupling of bending and torsional stresses accompanied by deformations and distortions of the cross section. Considering the increasing use of thin-wall arched box bridge in modern highways, a study has been undertaken to perform static and dynamic analysis by adopting an efficient computer model. As a limited number of studies have been carried out by past researchers on the box girder thin-wall curved bridge, especially on the dynamic behavior of bridges, this study aims to fulfill the following objectives.
To model horizontally curved thin-wall bridge for static and dynamic analysis. To describe a systematic approach for evaluating the fatigue life of a thin-walled curved box girder bridge from vehicle-induced stress.
ORGANIZATION OF THESIS
FINITE ELEMENT FORMULATION OF THIN-WALLED CURVED BOX-GIRDER BRIDGE
INTRODUCTION
A logical alternative for modeling curved box girder bridges is to combine finite element technology with thin-walled beam theory. In the present study, a one-dimensional beam element developed by Zhang and Lyons (1984) was used to model the thin-walled curved box-girder bridge. However, the commonly used truss or beam type elements are considered too simplified to represent curved box girder bridges.
The one-dimensional beam element, which is being considered for this study, can be considered as a general beam element representing the section of thin-walled beams. To evaluate the finite element formulation representing thin-walled box beams, various structural actions of thin-walled box beams are elaborated in the following sections.
BASIC ASSUMPTIONS FOR THIN-WALLED BOX-GIRDER
However, over the past twenty years, the finite element analysis method has quickly become a very popular technique for solving complex problems in engineering. It is clear that although a continuous bridge structure is actually three-dimensional, an idealized one-dimensional shape has certain simplifying advantages in many circumstances. The bending action of an individual component plate perpendicular to its plane is represented by the bending behavior of an equivalent transverse frame.
STRUCTURAL ACTIONS IN BOX-GIRDERS
- Shear Lag Effect
- Torsion of Thin-walled Girder
- Distortion of Thin-walled Girder
- Definition of Element Geometry
- Displacement Field and Degrees of Freedom
- Strain Components and Stress Resultants
- Stress - Strain Relationship
- Shape Functions to Define the Displacement Field
- Formulation of Displacements and Strains
- The Element Stiffness, Mass Matrix and the Equivalent Nodal Force Vector
- Boundary Conditions
For an open section, the shear strain is assumed to vanish at the center line of the section. The deformation of the diameter of a thin-walled bridge with a box girder is the main source of torsional stress and can therefore make a significant contribution to the bending stresses. To represent the resultant of the torsional stresses, a torsional bi-moment is defined as.
For loads applied at the cantilever portion of the cross section, the distorting moment is given by (Zhang 1983). The local x-axis is tangential to the element axis in the direction of node 1 towards node 3. The three displacement components can be related to the twist angle and the distortion angle of the cross section as.
The common displacements in the local coordinate system are. where { }δe are the nodal values of the global displacements and.
NUMERICAL EXAMPLES
- A Straight Box-Girder Subjected to an Eccentric Load
- A Simply-Supported Curved Box-Girder Bridge
Thin-walled box beam elements were used with an 8 element mesh to analyze the beam. The bridge was discretized using thirty thin-walled box girder elements to evaluate different response parameters. The longitudinal normal stresses were calculated for both nine degrees of freedom thin-walled box beam as well as conventional six degrees of freedom beam element.
To compare the evaluated values of different response parameters, the thin-walled box-girder bridge was analyzed using shell elements in the general finite element software ANSYS. A continuous two-cell, straight box girder model with two spans was analyzed, subjected to two eccentric point loads.
CLOSURE
FREE VIBRATION CHARACTERISTICS
INTRODUCTION
EIGEN VALUE PROBLEM FOR UN-DAMPED SYSTEM
VIBRATION TESTING
- Model Description and Evaluation of Material Properties
- Fabrication of Support
- Sensor Locations and Fittings
- Instrument Setup
- Test Procedure
- Modal Data Extraction
- Experimental Results and Discussion
Frequencies, damping ratio and state form of the test item are shown in the following sections. The Half Power Band Width Method (Inman 2001) has been used to evaluate the damping ratio (ζ) of the model bridge. Eq.(3.20) only gives the measurement of the size of an element of [u ui i sr]T . The phase plot of H( ) ωi is used to determine the sign of element.
The peaks of the FRF correspond to the natural frequencies of the bridge under test. The first mode shape of the bridge model, as observed from ANSYS, is shown in Fig.
59.1726 60.13 Second mode (lateral
PARAMETRIC STUDY
Mainly the effect of curvature on the free vibration characteristics of box girder bridges was studied. The modes shapes show the presence of coupled modes even in the early modes of the curved box girder bridge. Furthermore, in order to demonstrate the effect of curvature on the free-vibration, analyzes were carried out with varying radii of curvature (40 m – 200 m) of the trestle bridge.
The table shows that the frequencies change slightly with the increase in the radius of curvature of a simply supported box girder bridge. In general, the effect of curvature of practical arched box girder bridges on frequency is not significant.
44.8571 Table 3.4 (Contd.)
A computationally less expensive and realistic three-node thin-walled box girder element was used for modeling the bridge which is curved in plan. The applicability of such an element for the dynamic analysis was verified by evaluating the modal parameters of a curved box beam model theoretically as well as experimentally. The credibility of the adopted finite element for dynamic analysis was further improved by performing analysis in ANSYS (version 6.0) general purpose finite element software for the estimation of modal parameters.
It was observed that the frequencies as obtained from ANSYS are very close to the finite element solution obtained using Zhang and Lyons (1984) element. However, it has been observed that the effect of curvature of most of the practical bridges is insignificant.
BRIDGE-VEHICLE INTERACTION
INTRODUCTION
SYSTEM MODELS
- Vehicle Model
- Bridge Deck Roughness
The vehicle model chosen for this dynamic interaction investigation is a seven degrees of freedom, climb and roll model (Yadav and Upadhyay 1993) as shown in Figure 4.1. The model consists of a single rigid beam representing the vehicle body, the mass of which is concentrated at the center of the rigid beam and is known as the "sprung mass" of the vehicle. The total seven independent degrees of freedom of the vehicle model can be written as
The random road surface roughness hr(x) of the bridge can be described by the zero-mean, stationary ergodic Gaussian process in space. The value of N depends on the speed of the vehicle (ie the total time needed to cross the bridge) and the size of the time increment chosen for the dynamic response analysis (N=Total time/ Δt).
- Bridge Model
- COUPLED BRIDGE-VEHICLE SYSTEM EQUATIONS The equation of heave motion of the sprung mass can be written as
- CONSTRUCTION OF DAMPING MATRIX
- SOLUTION TECHNIQUE
- Explicit Predictor-Corrector Algorithm
- Begin predictor phase by setting
- Performing factorization, forward reduction and back substitution as required to solve K * Δd [ ]0 = Ψ [ ]0
- NUMERICAL SIMULATION
The governing differential equation of motion of the box girder bridge can be expressed as. 4.13). The Newmark-β (mean acceleration) scheme with the predictive-corrective algorithm ( Owen and Hinton 1960 ) was used to evaluate the dynamic response of the bridge due to vehicle-induced vibrations. The span of the simply supported bridge is 30 m with a radius of curvature of 150 m and the bridge has diaphragms at the abutments.
The histories are obtained at the mid-span of the curved thin-wall box bridge, which corresponds to good road surface roughness (ref. The abscissa in these time histories is the distance measured from the left side of the bridge to the front axle of the bridge). vehicle.
Parameter Study
However, the variation of the impact factor of torsion, distortion and their corresponding bi-moments are relatively significant, while the variations of the vertical deflection, bending moment and shear force are almost unaffected for the span considered in this study. The influence factor corresponding to different response parameters for different categories of road surface is shown in table 4.8. It can be observed that surface roughness has more significant effect on the influence factor of the vertical deflection, bending moment and shear force than any other parameters considered in the study.
Table 4.9 shows that the impact factors of most reactions decrease as the spring mass increases. It also indicates that with the reduction of vehicle mass, as the fundamental natural frequency of the vehicle increases, the impact factor also increases.
GENERAL DESIGN GUIDELINE USING IMPACT FACTOR .1 Existing Codal Provisions
The fr value for the actual vehicle parameter considered in the study was calculated as 4.15, and significant DAF values were observed even at a frequency ratio of 4.5. Bridge codes usually provide dynamic effects for straight bridges, although some attention has recently been given to the analysis of curved bridges. One is the influence factor to span length ratio and the other is the DLA-natural frequency ratio.
In 1927, a joint committee of the American Railway Engineering Association (AREA) and the American Association of State and Highway Transportation Officials (AASHTO) recommended the dynamic effect through an impact factor as the standard specification for highway bridges.
Suggested Guidelines for Curved Box Girder Bridge
The present study attempts to establish general guidelines for the designer to take into account the dynamic effect of live load by using a simplified formula which incorporates the effect of dominant parameters affecting the dynamic response of bridges. The most important parameters affecting curved box girder bridge design are length, radius of curvature and vehicle speed. A multiple regression analysis has been performed to fit a linear relationship for the influence factor.
The set of simultaneous equations are constructed and solved to form the regression matrix (X). The regression equation for the bending moment about the transverse axis( )Mz, shear force( )Qy, twisting moment( )Mx, twisting bimoment( )BI, distortion moment( )Md.
