Stiffness Of Girder Research Articles

Nonlinear free damped vibrations of suspension bridges with uncertain fractional damping

Nonlinear free damped vibrations of a suspension bridge with a bisymmetric stiffening girder are considered under the conditions of the internal resonance one-to-one, i.e., when natural frequencies of two dominating modes - a certain mode of vertical vibrations and a certain mode of torsional vibrations - are approximately equal to each other. Damping features of the system are defined by fractional derivatives with fractional parameters (the orders of the fractional derivatives) changing from zero to one. It is assumed that the amplitudes of vibrations are small but finite values, and the method of multiple scales is used as a method of solution. The influence of uncertainty in choosing the fractional parameters on the character of nonlinear damped vibrations of suspension bridges is investigated.

Approach for analyzing the ultimate strength of concrete filled steel tubular arch bridges with stiffening girder

A convenient approach is proposed for analyzing the ultimate load carrying capacity of concrete filled steel tubular (CFST) arch bridge with stiffening girders. A fiber model beam element is specially used to simulate the stiffening girder and CFST arch rib. The geometric nonlinearity, material nonlinearity, influence of the construction process and the contribution of prestressing reinforcement are all taken into consideration. The accuracy of this method is validated by comparing its results with experimental results. Finally, the ultimate strength of an abnormal CFST arch bridge with stiffening girders is investigated and the effect of construction method is discussed. It is concluded that the construction process has little effect on the ultimate strength of the bridge.

Floating-roof steel tanks under harmonic settlement: FE parametric study and design criterion

Large vertical steel tanks for fluid storage are usually constructed on soft foundations, so it is not surprising that the tank wall will settle unevenly with the settlement of the foundation, thus inducing deformations and stresses in the tank. This work investigates the linear static behavior of floating-roof tanks under harmonic settlement through finite element (FE) analyses. The influences of the radius-to-thickness ratio, the height-to-radius ratio and the wind girder stiffness on the structural behavior are first analyzed. Comparisons between the circumferential stresses in the wind girder and the vertical stresses in the tank bottom are then made. The displacement and the stress along the tank height are also discussed, and the concept of tank division along its height is presented. Finally, a design approximation for the radial displacement at the tank top is developed based on FE results, and a settlement criterion based on the top radial displacement is proposed which can be used in practical design.

Computer simulations and parametric studies of GFRP bridge deck systems

This research studies the structural behavior of a GFRP bridge deck system. The study develops FEM models of the GFRP bridge deck system using ANSYS 7.0 software package. Data collected from field measurements and laboratory testing is used to validate and calibrate the proposed FEM models. Parametric studies through computer simulations are conducted to investigate the characteristics of the system. Four parameters are considered for service load conditions: (a) presence of diaphragms, (b) girder stiffness, (c) girder spacing, and (d) composite action. In addition, geometric and material nonlinearities are considered in the investigation of the capacity of the new bridge system. The analysis results are discussed and conclusions on the GFRP deck system are presented.

Damage Assessment of the Z24 Bridge by FE Model Updating

Damage to civil engineering structures can be identified with a finite element (FE) model updating method using experimental modal data. In such a procedure the uncertain properties (e.g. stiffness distribution) in the FE model are adjusted by minimizing the differences between the measured modal parameters and the numerical (FE) predictions. In civil engineering the differences in eigenfrequencies and mode shapes are minimized, mostly identified from ambient vibrations. Since the modal data are nonlinear functions of the uncertain properties, an iterative sensitivity-based minimization method is used to solve this inverse problem. In order to reduce the number of unknowns, damage functions are used. The FE model updating technique is applied to a prestressed concrete bridge with 3 spans whose girder is damaged by lowering one of the intermediate piers. The damage pattern is identified (localized and quantified) by updating the Young’s and the shear modulus. Introduction Accurate condition assessment of civil engineering structures has become increasingly important. FE model updating provides a very efficient, nondestructive, global damage identification technique. The uncertain properties of the FE model are updated by minimizing the discrepancies between the measured modal data and those computed with the numerical FE model [1, 2]. The damage identification procedure is performed in two updating processes. In the first the initial FE model is tuned to a reference state, i.e. the undamaged structure. In the second process the reference FE model is updated to obtain a model which can reproduce the experimental modal data of the damaged state. The damage is identified by comparing the differences between the reference and the damaged FE model. The technique is applied to the Z24 bridge in Switzerland. It is a prestressed concrete bridge with three spans which is damaged by lowering one of the intermediate piers. A nonlinear least squares problem is solved. The residual vector contains the test/analysis differences of the first 4 bending and/or torsion modes. Frequency residuals as well as mode shape residuals are minimized. Eigenfrequencies contain global, accurate information, whereas mode shapes provide important local, but more noisy information. Therefore, both types of residuals are weighted with an appropriate factor in the residual vector. The updating parameters are both the Young’s and the shear modulus of all the girder elements. The least squares problem is solved with a sensitivity-based Gauss-Newton algorithm. In order to improve the condition of the sensitivity matrix the number of unknown parameters is reduced by using a limited set of damage functions [2]. The girder stiffness distribution is found by combining these damage functions, multiplied with the appropriate factors which are the actual variables of the minimization problem. Only linear damage functions are used, but the method can be extended by including higher order functions. With this approach always a realistic smooth result is obtained. A damage pattern is identified which resembles the observed one. The general updating procedure and the application to the Z24 bridge are presented in the paper. General FE model updating procedure Objective function. In FE model updating an optimization problem is set up in which the differences between the experimental and numerical modal data have to be minimized by adjusting the uncertain model properties [1]. The experimental modal data, i.e. the eigenfrequencies and mode shapes , are obtained from measurements. In civil engineering, the measurements are often obtained in operational conditions (ambient vibrations), which means that the exciting forces (coming from wind, traffic,. . . ) are unknown. As a consequence, an absolute scaling of the mode shapes is missing. Furthermore, only the translation degrees of freedom of the mode shapes can be measured. The minimization of the objective function is stated as a nonlinear least squares problem:

Field Evaluation of Decommissioned Noncomposite Steel Girder Bridge

Field tests conducted on a noncomposite steel girder bridge are described. Two separate 36.6 m (120 ft) units, each three-span continuous, were subjected to increasing static loads by means of a trailer and concrete barriers. Results show that the girders acted as partially composite sections in the positive moment region up to the onset of yield. Due to curb participation and the transverse location of the applied load, exterior girders exhibited a higher degree of partially composite action. In the negative moment region, partially composite action was evident only in the exterior girders. As a result of partially composite action and curb participation, the yield load was about 7% higher than predicted. Bearing restraint is shown not to have a significant impact on the behavior of the tested bridges. In addition, the stiffness of the interior girders, as measured under the constant weight of a dump truck, are shown to be virtually unaffected by the heavy trailer loads. More significant changes in girder stiffness were observed between different transverse load positions of the dump truck.

Transverse Cracking in Concrete Bridge Decks

The dominant parameters that lead to premature transverse cracking in bridge decks are determined, and recommendations to reduce cracking tendency in bridge decks are developed. The project is divided into two parts: a field study and a parametric study. The objective of the field study is to determine the correlation between the observed cracking and available design-, material-, and construction-related data. Seventy-two bridges in the Minneapolis/St. Paul metropolitan area are included in the field study. According to the results of the study and correlation with other research, restrained concrete deck shrinkage is the leading cause of cracking. The dominant factors affecting transverse cracking are the longitudinal end restraint, girder stiffness, cross-frame location, splice location, deck thickness, cutoff length of the deck supplemental reinforcing bar, size of the top transverse bar, concrete shrinkage, deck concrete modulus of elasticity, cement content, aggregate type and quantity, air content, and ambient air temperature at deck placement. Recommended practical improvements to bridge deck construction include reducing the shrinkage of the deck concrete through mix design or better curing practices and minimizing deck restraint.

Simplified Analysis Method for Slab-on-Girder Highway Bridge Decks

It has been observed from test and analysis results that the differential deflection between girders in a slab-on-girder bridge deck reduces the negative bending moments in the deck slab. A simplified analysis method considering the effect of girder deflection is developed, based on an analytical solution obtained with the elastic plate theory. With this approach, the effects of the girder stiffness and spacing, and the span length of the bridge on the maximum negative bending moments in a deck slab can be assessed. The simplified analysis method is validated with finite-element models. It has been shown that the maximum negative bending moment in a bridge deck can be accurately evaluated with the proposed method.

Application of Fractional Calculus for Analysis of Nonlinear Damped Vibrations of Suspension Bridges

Free damped vibrations of a suspension bridge with a bisymmetric stiffening girder are considered under the conditions of the internal resonance one-to-one: when natural frequencies of two dominating modes--a certain mode of vertical vibrations and a certain mode of torsional vibrations--are approximately equal to each other. Damping features of the system are defined by a fractional derivative with a fractional parameter (the order of the fractional derivative) changing from zero to one. It is assumed that the amplitudes of vibrations are small but finite values, and the method of multiple scales is used as a method of solution. It is shown that in this case the amplitudes of vertical and torsional vibrations attenuate by an exponential law with the common damping ratio, which is an exponential function of the natural frequency. Analytical solitonlike solutions have been found. A numerical comparison between the theoretical results obtained and the experimental data is presented. It is shown that the theoretical and experimental investigation agree well with each other at the appropriate choice of the parameters of the exponential function determining the damping coefficient.

Stability and load-carrying capacity of three-dimensional long-span steel arch bridges

The behavior of several models of three-dimensional long-span steel arch bridges is investigated for evaluating the effects of various design parameters on both the strength and stability of these special structures. The major concerns in the design of a long-span steel arch bridge, from the structural safety point of view, are the yield and buckling failures. Different design parameters may affect the failure load for either type of failure in various ways. This study investigates how changes in certain design parameters would affect the behavior of steel arch bridges, which could lead to an optimum design of this type of bridge structures. The effects of the plate girder stiffness and arch bracing stiffness as well as the rise-to-span ratio and inclination of the arches towards each other are examined in this study. Both critical buckling load and the load-carrying capacity of each design alternative are investigated using the finite element method. All design alternatives are based on the latest AASHTO code for highway bridge design. It is concluded from this study that the inclined arch bridge using the maximum practical rise-to-span ratio (which is about 0.25) is the most favorable design. In addition, the increase in the stiffness of the plate girder does not reduce the bending moments in the arch ribs. However, providing a lateral bracing system with sufficient stiffness greatly reduces the out-of-plane bending moments and increases the load-carrying capacity and the critical buckling load of a long-span arch bridge.

Analysis of nonlinear free vibrations of suspension bridges

Nonlinear free vibrations of suspension bridges with a bisymmetrical stiffening girder are investigated by using the method of multiple scales in two cases: when a two-to-one or one-to-one internal resonance takes place. The influence of the initial conditions on the vibrational processes’ character is shown. Three types of the energy-exchange mechanism between vertical and torsional modes are determined: two-sided energy exchange (a periodic motion); one-sided energy interchange (an aperiodic motion); and, energy exchange does not occur (stationary vibrations). The solutions describing the one-sided energy exchange are demonstrated to correspond with soliton-like solutions in the form of a single kink. On the hydrodynamic analogy, the qualitative method of analysis of suspension bridge non-linear free vibrations is proposed. This procedure allows one to determine the types of vibrational process, to investigate the stability of each vibrational regime, to define the character of amplitude and phase dependences from the initial conditions etc. Using five examples, the non-linear free vibrations of the Golden Gate Bridge in San Francisco are considered for the cases of two-to-one and one-to-one internal resonances. All quoted examples verify the qualitative hydrodynamic analysis.

Rating and Preliminary Analysis of Suspension Bridges

Formulas for different loading cases of suspension bridges are presented. Although theoretically not correct, the principle of superposition can be used to get practical answers that are sufficiently accurate for determining stresses, deflections, and rotation in a stiffening truss or girder. The horizontal cable tension H is computed using the more accurate method by Steinman or the approximate method by Tsien. With H known, the differential equation for the various loadings can be resolved, and the results of such computations are shown in this paper. A numerical example is worked out to clarify the mathematical procedure. The data are from the Angostura suspension bridge over the Orinoco River in Venezuela.

Wind tunnel experiments using 3D models and response prediction for a long-span suspension bridge

A series of wind tunnel experiments on a long-span suspension bridge has been conducted. A proposed design for Akashi Straits Bridge having a box stiffening girder was selected as the prototype for full aeroelastic models and taut strip models to a scale of 1:350. The wind-induced response (flutter) under smooth flow and the prediction calculated from the results of sectional model tests using the strip theory are presented. It is found that agreement between experiment and prediction is very good. Response under turbulent flow and its prediction are also presented.

Pontoon Torsional Strength Analysis Related to Ships with Large Deck Openings

The torsional problem of a pontoon, consisting of channel middle part and rectangular tube peaks, isconsidered within the higher-order beam theory. The cross section and the contour compatibility conditions for assembling of the pontoon parts are investigated. The acceptability of the introduced assumptions is checked by three-dimensional finite-element model analysis. Some deficiencies of the classical beam theory regarding the girder stiffness are noticed. The finite-element formulation to be used for the torsional analysis of the ship's hull with large hatch openings is given.

On torsional and warping stiffness of thin-walled girders

The torsional and warping stiffness of the beam idealization of thin-walled girders is defined from equivalence of their natural frequencies for torsional vibration of sinusoidal modes. The membrane and plate strip theory are employed for the girder structural members. The stiffness parameters are determined analytically for a few girders of simple cross-section, while for those of complex cross-section a numerical procedure is formulated.

Wind tunnel experiment of long-span suspension bridge using 3D models

A series of wind tunnel experiments on a long-span suspension bridge was conducted. A proposed design for the Akashi Straits Bridge having a box stiffening girder was selected as the prototype for full aeroelastic models and taut strip models to scale 1: 350.In this paper, an outline of the experiment, the wind-induced response (flutter) under smooth flow, and its prediction computed from the results of sectional model tests using “strip theory” are presented. It is found that agreement between experiment and prediction is good.

Wind tunnel study of long-span suspension bridge under smooth and turbulent flow

Wind tunnel experiments of long-span suspension bridge are described. The proposed design of the Akashi Straits Bridge having box stiffening girder (Ref. 1) is selected as the prototype for the aeroelastic full models with scale 1:350. Tests are conducted under smooth and turbulent flow. Preliminary results are presented. Wind-induced oscillations of the towers on completion are also mentioned.

Investigation of Effective Bending and Shear Stiffness of Thin-Walled Girders Related to Ship Hull Vibration Analysis

Application of beam theory in flexural vibration analysis of thin-walled girders is extended for the high-frequency domain by introducing the concept of effective values of beam parameters, that is, cross-sectional moment of inertia, shear area, mass, and mass moment of inertia. Formulation of these parameters is based on equivalence of deformation energy and inertia work, respectively, for a considered structure and its beam model, resulting in the same values of their natural frequencies. For illustration, the natural vertical vibration of a simply supported pontoon has been considered, where it was possible to obtain the analytical solution due to sinusoidal mode shapes. The effective values of cross-sectional moment of inertia and shear area show significant variation in frequency domain. Transfer of effective values of beam parameters, determined for simply supported structure, in the case of other boundary conditions is suggested, based on equal mode wavelengths, and checked for the free pontoon. The results show very low discrepancies compared with a three-dimensional finite-element model solution, so this procedure may be applied generally, as well as to the problem of ship hull vibration. In conclusion, the possibility of calculating the values of effective parameters for multicell ship cross sections, utilizing the theory of folded structure and the finite-element method, is pointed out.

A SIMPLIFIED METHOD OF ESTIMATING NATURAL FREQUENCIES OF CONTINUOUS TYPE CABLE-STAYED BRIDGES

The cable-stayed bridges having the new structural system such as the Annacis Bridge, stiffened by the plate girder and with precast concrete slabs, have been interested from a viewpoint of economy. It will be more necessary for the Annacis type bridges as well as the ordinary type bridges to estimate the natural frequencies in the initial stage of design in order to check the aerodynamic stability.In this paper, the effect of cable stiffness on the natural frequencies of the cablestayed bridges is investigated with respect to the relation between a cable-stayed bridge and a beam on the elastic support for their dynamic behaviour. And the simplified method of estimating the natural frequencies of the continuous type cable-stayed bridges with the various girder and cable stiffness is newly proposed for the aerodynamic resistant design.

A PARAMETRIC STUDY ON CABLE-STAYED BRIDGES BY THE LIMIT STATES DESIGN

Cable-stayed bridges with multi-stays have been recently planned and constructed in many countries probably because of aesthetic and economical superiority. It would be generally possible that the girder could become much slender by properly controlled cable prestresses. The system has been designed in this paper by mainly BS 5400, one of the few codes which have introduced the limit states design. The consistently optimised designs have been achieved by Linear Programming in order to compare the various bridge configuration. A simple method has been proposed to estimate the buckling capacity of the girder. The girder stiffness obtained is substantially lower than that of existing bridges. The bridge with multi-cables could have much slender girders than that with few-cables.